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# Polarized currents in Coulomb blockade and Kondo regimes without magnetic
fields
Anh T. Ngo Department of Physics and Astronomy, and Nanoscale and Quantum
Phenomena Institute, Ohio University, Athens, Ohio 45701-2979 Edson Vernek
Department of Physics and Astronomy, and Nanoscale and Quantum Phenomena
Institute, Ohio University, Athens, Ohio 45701-2979 Instituto de Física -
Universidade Federal de Uberlândia, Uberlândia, MG 38400-902, Brazil Sergio
E. Ulloa Department of Physics and Astronomy, and Nanoscale and Quantum
Phenomena Institute, Ohio University, Athens, Ohio 45701-2979
###### Abstract
We present studies of the Coulomb blockade and Kondo regimes of transport
through a quantum dot connected to current leads through spin-polarizing
quantum point contacts (QPCs). This structure, arising from the effect of
lateral spin-orbit fields defining the QPCs, results in spin-polarized
currents even in the absence of external magnetic fields and greatly affects
the correlations in the dot. Using equation-of-motion and numerical
renormalization group calculations we obtain the conductance and spin
polarization for this system under different parameter regimes. We find that
the system exhibits spin-polarized conductance in both the Coulomb blockade
and Kondo regimes, all in the absence of applied magnetic fields. We analyze
the role that the spin-dependent tunneling amplitudes of the QPC play in
determining the charge and net magnetic moment in the dot. These effects,
controllable by lateral gate voltages, may provide an alternative approach for
exploring Kondo correlations, as well as possible spin devices.
###### pacs:
72.15.Qm, 72.25.-b, 72.10.-d, 73.23.Hk
## I Introduction
Electronic transport in semiconducting nanostructures is studied both as it
has great potential applications in spintronics, 5 and because of its
exquisite control of parameters, which allow insightful probes into
fundamental physical phenomena. Electrons in such systems experience
externally controlled confining environments that result in strong Coulomb
interactions with other electrons. As such, quantum dot (QD) structures
provide well-characterized and defined systems for studying quantum many-body
physics. QDs may also allow the realization of solid state quantum computation
devices as well as spintronic semiconductor devices with unprecedented
functionalities. QDref
Manipulation of spin-polarized current sources is crucial in spintronics. This
typically requires efficient spin injection into conventional semiconductors.
The difficulties with spin injection from ferromagnetic metal leads has
stimulated extensive efforts to produce spin polarized currents out of
unpolarized sources. Schmidt In this context, the Rashba spin-orbit (RSO)
coupling mechanism provides a basis for possible device applications. Rashba
This coupling, arising from interfacial structure asymmetries, depends on the
materials used as well as on the confinement geometry of the structures. 6 ; 9
Most interestingly, the Rashba effect allows external tunability, which has
been studied experimentally in QDs 7 and quantum point contacts. Bird ; 8
In this paper we study the electronic transport through a quantum dot
connected to polarizing quantum point contacts (QPCs) in both the Coulomb
blockade (CB) and Kondo regimes. Due to strong spin-orbit interactions, 8 ; 9
QPCs can exhibit spin-dependent hybridization of the QD states with the leads,
without applied magnetic fields, opening the possibility for generating spin-
polarized transport in an all-electrical setup. These effects are controllable
by lateral gate voltages applied on QPCs, resulting in spatially asymmetric
structures, as in recent experiments.8 Using the equation-of-motion technique
and numerical renormalization group (NRG) calculations we obtain the
electronic Green’s function, conductance and spin polarization in different
system regimes. Our results show that both the CB and Kondo regimes exhibit
non-zero spin-polarized conductance in this system. We analyze how the spin-
dependent hybridization of the QPC modifies the charge accumulation in the
dot, as well as the density of states (spectral functions) of the system.
Interestingly, we find that the polarizing QPCs produce spin polarization and
split DOS in the Kondo regime akin to that reported for current injection from
ferromagnetic leads, Martinek although here it occurs for unpolarized
reservoirs. Our theoretical studies suggest that these effects could be
accessible in experiments and result in future spintronic devices.
The paper is organized as follows. First, we present the main features of the
current polarization in quantum point contact systems with lateral spin-orbit,
obtained by scattering matrix methods in section II. The next section
describes the model of a quantum dot connected to current leads via polarizing
QPCs. In section III.1 we use the equation-of-motion (EOM) technique to obtain
the dot Green’s function and discuss numerical results in both the CB and
Kondo regimes. In section III.2 we revisit the problem using the numerical
renormalization group approach, which provides the most reliable information
in the Kondo regime for realistic system parameters. The paper concludes with
final remarks and summary in section IV.
## II Polarizing QPC
The effective electric field in the $z$ direction that creates a two-
dimensional electron gas (2DEG) confined to the $x$-$y$ plane results in the
well-known Rashba spin orbit interaction, Rashba ; 6
$\displaystyle
H_{SO}^{R}=\frac{\alpha}{\hbar}\left(\sigma_{x}P_{y}-\sigma_{y}P_{x}\right).$
(1)
Here, $\sigma_{x}$ and $\sigma_{y}$ are Pauli matrices, and $\alpha$ is Rashba
spin orbit coupling constant, which is proportional to the field and is
therefore material and structure dependent. Electrons on this 2DEG entering a
quantum dot, pass through QPCs defined via a confining potential $V(x,y)$
which can be thought as made of two parts: $U(y)+V_{b}(x,y)$, where $U(y)$
defines a hard-wall potential of width $W$ (related in the experiment to the
side-wall etching defining the structure) outlining the overall channel
structure, while $V_{b}(x,y)$ is the potential generating the QPC barrier, and
effectively modulated by the side gate potentials of the structure. 8 We
model such barrier by 12
$\displaystyle V_{b}(x,y)=\frac{1}{2}V_{g}\left(1+\cos\frac{\pi
x}{L_{x}}\right)+\frac{1}{2}m\omega^{2}\bar{y}^{2}\Theta(\bar{y})$ (2)
with $\bar{y}=y-y_{s}$, and
$\displaystyle y_{s}=W_{1}\left(1-\cos\frac{\pi x}{L_{x}}\right)\,,$ (3)
where $\Theta(x)$ is the step function, $m$ is the effective mass of the
electrons, $L_{x}$ is the unit length of the structure in the $x$ direction
(along the current direction) and $\omega$ is the confinement potential
frequency. Notice that this potential form is asymmetric in the $y$ direction,
to reflect an essential ingredient in the experiments: the QPC potential must
lack $y$-reflection symmetry in order to generate the polarizing effect along
the $z$-direction. 12 ; 12b ; 12c In fact, the fields forming $V_{b}$
generate a spin-orbit coupling given by 6
$\displaystyle V_{SO}^{\beta}=-\frac{\beta}{\hbar}\nabla
V_{b}\cdot(\mathbf{\hat{\sigma}}\times\mathbf{\hat{P}}),$ (4)
where $\beta$ is material-specific. Notice that $\nabla V_{b}$ lies in the
$x$-$y$ plane, so that the barrier fields induce a lateral spin-orbit
coupling. The total Hamiltonian of the QPC will then be given by,
$\displaystyle
H=\frac{P_{x}^{2}+P_{y}^{2}}{2m}+H_{SO}^{R}+V(x,y)+V_{SO}^{\beta}.$ (5)
Figure 1: (color online) Total and spin-dependent conductances for an
asymmetric QPC as function of Rashba spin-orbit coupling $\alpha$, obtained
from a scattering matrix approach.9 The interplay of vertical and lateral SO
effects may result in large asymmetry for the up and down spin components for
realistic structure parameters, even in the tunneling regime shown here
($G_{\sigma}\ll 1$).
Using a scattering-matrix formalism to study the spin-dependent electron
transport in this QPC, 11 ; 9 we find that a net spin-polarized conductance
is produced only for $y$-asymmetric potentials. 9 We can calculate the
conductance of the structure, assuming that the SO coupling $\alpha$ and
$V(x,y)$ are zero at the source and drain 2DEG reservoirs, while both of these
terms in the Hamiltonian are turned on in the QPC region. Typical structure
parameters in experiments can be cast in terms of characteristic length and
energy scales, $L_{0}=32.5$ nm, and $E_{0}=3.12$ meV, with
$\alpha_{0}=E_{0}L_{0}=1.0\times 10^{-12}$ eVm, a typical value of spin-orbit
coupling. Using $W=L_{x}=2L_{0}$ and $W_{1}=0.6L_{0}$ as width/length of the
confining potential, with $\omega=6\times 10^{13}$s-1, and $\beta=0.97\times
10^{-16}$m2, gives results as shown in Fig. 1. This figure shows spin-
dependent conductances as functions of Rashba coupling $\alpha$, for a
potential barrier which is near its conduction onset (or “pinch-off”, as
controlled by the value of $V_{g}$); arrows indicate the results of spin up
and down conductances. We see that conductances $G_{\uparrow}$ and
$G_{\downarrow}$ can be very different from each other, even in the tunneling
regime (where each $G_{\sigma}\ll 1$) in which the QPC would operate to create
a quantum dot in the 2DEG. We stress that for these realistic values of
structure parameters, one obtains non-zero spin polarization even when there
is no external magnetic field and the injection is unpolarized. This
interesting result can be understood from anticrossing features in the subband
energy structure in the channel region defining the QPC. 8 ; 9 The spin
mixings and avoided crossings generate spin rotation as electrons pass through
the narrow constriction of the QPC, and can generate large values of the ratio
$G_{\uparrow}/G_{\downarrow}$, even in the tunneling regime. Two of these QPCs
can then be used to define the QD and result in interesting charging and
conductance regimes, as we will see below.
## III Quantum dot with polarizing QPCs
In order to address the transport through a quantum dot formed with polarizing
QPCs, we consider the single impurity Anderson model given by the following
Hamiltonian:
$\displaystyle H=\sum_{\ell k\sigma}\varepsilon_{\ell k}c^{\dagger}_{\ell
k\sigma}c_{\ell
k\sigma}+\sum_{\sigma}\varepsilon_{d}c^{\dagger}_{d\sigma}c_{d\sigma}+Un_{d\uparrow}n_{d\downarrow}$
$\displaystyle+\sum_{\ell k\sigma}t_{\sigma}(c^{\dagger}_{d\sigma}c_{\ell
k\sigma}+c^{\dagger}_{\ell k\sigma}c_{d\sigma}),$ (6)
where $c^{\dagger}_{d\sigma}(c_{d\sigma})$ is the creation (annihilation)
operator of an electron of spin $\sigma$ in the dot. The quantities
$\varepsilon_{\ell k}$, $\varepsilon_{d}$ are the energies of the electrons in
the $\ell^{\it th}$ conduction band channels ($\ell=L,R$) and the single local
energy level in the dot, respectively. $U$ is the Coulomb repulsion between
electrons occupying the QD with
$n_{d\sigma}=c^{\dagger}_{d\sigma}c_{d\sigma}$, while $t_{\sigma}$ represents
the lead-QD hybridization occurring via tunneling through the QPC, and which
is assumed to be $k$-independent. The density of states for conduction
electrons in each lead is taken to be constant,
$\rho_{L}(\varepsilon)=\rho_{R}(\varepsilon)\equiv\rho=(1/2D)\Theta(D-|\varepsilon|)$,
where $D$ is the conduction band halfwidth (hereafter taken as our energy
unity).
The theoretical description of such quantum dot system, especially in the
strong correlations regime, has been greatly developed over the years. Mahan
Techniques of note include quantum Monte-Carlo, 2 equations of motion for the
Green’s functions,3 and the numerical renormalization group approach.costi
In what follows, we explore the role that SO interactions play on the Coulomb
blockade and Kondo regimes of transport of the QD, utilizing equations of
motion and numerical renormalization group formalisms.
### III.1 Equation of motion approach and numerical results
To calculate the charge and conductance of the system we calculate Green’s
functions (GFs), which allow us to take into account the correlations induced
by the Coulomb interaction in the QD. The retarded double-time Green’s
functions are defined as ($\hbar=1$)13
$i\langle\langle
A;B\rangle\rangle=\int_{-\infty}^{\infty}\langle[A(\tau),B(0)]_{+}\rangle\Theta(\tau)e^{-i\omega\tau}d\tau,$
(7)
where $A$ and $B$ are generic fermionic operators, $[A,B]_{+}$ indicates their
anticommutator and $\langle\cdots\rangle$ indicates the thermodynamic average
for $T>0$, or the ground state expectation value for $T=0$. The GF can be
obtained using equation of motion (EOM) techniques, so that
$\displaystyle\omega\langle\langle
A;B\rangle\rangle=\langle[A,B]_{+}\rangle+\langle\langle[A,H];B\rangle\rangle,$
(8)
where $[A,B]$ represents a commutator. Iteration of this formula generates a
hierarchy of expressions, starting with the local one-particle GF as
$\left(\omega-\varepsilon_{d}-\sum_{k}\frac{\tilde{t}_{\sigma}^{2}}{\omega-\varepsilon_{k}}\right)\langle\langle
c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle=1+U\langle\langle
c_{d\sigma}n_{d\bar{\sigma}};c^{\dagger}_{d\sigma}\rangle\rangle,$ (9)
where $\tilde{t}_{\sigma}=\sqrt{2}t_{\sigma}$ and $\bar{\sigma}=-\sigma$. The
new (higher order) GF on the right hand side of Eq. (9) can also be determined
from (8), giving
$\displaystyle(\omega-\varepsilon_{d}-U)\langle\langle
c_{d\sigma}n_{d\bar{\sigma}};c^{\dagger}_{d\sigma}\rangle\rangle=\langle
n_{d\bar{\sigma}}\rangle+\tilde{t}_{\sigma}\sum_{k}\left(\langle\langle
c_{k\sigma}n_{d\bar{\sigma}};c^{\dagger}_{d\sigma}\rangle\rangle-\langle\langle
c_{k\bar{\sigma}}c^{\dagger}_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle+\langle\langle
c^{\dagger}_{k\bar{\sigma}}c_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle\right).$
(10)
#### III.1.1 Coulomb blockade regime
Although the EOM in Eq. (10) is exact, a solution of the impurity GF requires
a procedure to truncate and/or decouple the higher order terms appearing on
the right hand side of (10). A solution that captures the Coulomb blockade
physics is given by the Hubbard-I approximation:14
$\displaystyle\langle\langle
c_{k\sigma}n_{d\bar{\sigma}};c^{\dagger}_{d\sigma}\rangle\rangle$
$\displaystyle\simeq$ $\displaystyle\langle
n_{d\bar{\sigma}}\rangle\langle\langle
c_{k\sigma};c^{\dagger}_{d\sigma}\rangle\rangle$ $\displaystyle\langle\langle
c_{k\bar{\sigma}}c^{\dagger}_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle$
$\displaystyle\simeq$ $\displaystyle\langle
c_{k\bar{\sigma}}c^{\dagger}_{d\bar{\sigma}}\rangle\langle\langle
c_{k\sigma};c^{\dagger}_{d\sigma}\rangle\rangle$ $\displaystyle\langle\langle
c^{\dagger}_{k\bar{\sigma}}c_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle$
$\displaystyle\simeq$ $\displaystyle\langle
c^{\dagger}_{k\bar{\sigma}}c_{d\bar{\sigma}}\rangle\langle\langle
c_{k\sigma};c^{\dagger}_{d\sigma}\rangle\rangle,$ (11)
which allows one to write
$\displaystyle G_{d\sigma}(\omega)\equiv\langle\langle
c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle_{\omega}=\frac{G^{0}_{d\sigma}(\omega)}{1-G^{0}_{d\sigma}(\omega)\tilde{t}^{2}_{\sigma}\tilde{g}(\omega)},$
(12)
where $G^{0}_{d\sigma}(\omega)=\frac{1-\langle
n_{d\bar{\sigma}}\rangle}{\omega-\varepsilon_{d}}+\frac{\langle
n_{d\bar{\sigma}}\rangle}{\omega-\varepsilon_{d}-U}$ is the local GF in the
“atomic” approximation (the exact result for $t_{\sigma}=0$), and
$\tilde{g}(\omega)=\sum_{k}(\omega-\epsilon_{k})^{-1}$ is the non-interacting
GF of the leads. The DOS of the system (proportional to the imaginary part of
$G_{d\sigma}$) contains two Hubbard peaks of width proportional to
$\Gamma_{\sigma}=\pi t^{2}_{\sigma}/D$, resulting in the broadening of the
poles of $G^{0}_{d\sigma}$. The spectral weights of these peaks are controlled
by the dot level occupancy with opposite spin, and caused by the Coulomb
interaction in the dot. Notice that the SO-induced polarization of the QPC
results in different peak widths for the different spins. The Hubbard-I
approximation (11) is known to be valid for a large $U/\Gamma$ ratio, when the
Hubbard subbands are well separated in energy scale. Mahan It is the simplest
scheme which describes correlated electrons, although, since it ignores the
Kondo effect, it is a reasonable description only at temperatures higher than
the Kondo scale ($T\gg T_{K}$–see next section).
Figure 2: (color online) Occupancies for spin-up $\langle
n_{d\uparrow}\rangle$, spin-down $\langle n_{d\downarrow}\rangle$ and total
spin $\langle n_{d\uparrow}\rangle+\langle n_{d\downarrow}\rangle$ vs.
$\varepsilon_{d}$ at zero temperature. Parameters used are
$\Gamma_{\uparrow}=0.06$, $\Gamma_{\downarrow}=0.03$, $U=0.3$, with $D=1$.
Notice asymmetry in $\langle n_{d\sigma}\rangle$ on each side of the plateau.
Figure 3: (color online) Spin-dependent conductance and polarization as
function of $\varepsilon_{d}$ at zero temperature. The other parameters are as
in Fig. 2.
The occupancies of spin $\uparrow$ and $\downarrow$ are calculated self-
consistently from the equation
$\displaystyle\langle n_{d\sigma}\rangle=\int
f(\omega)\left(-\frac{1}{\pi}\text{Im}\left[G_{d\sigma}(\omega)\right]\right)d\omega\,,$
(13)
where $f(\omega)$ is the Fermi function. It is clear that when the QPCs are
not polarizing, $t_{\uparrow}=t_{\downarrow}$, the occupancy curves for spin-
up and down coincide and a plateau of width $\sim U$ appears when the QD level
moves below the Fermi level ($\varepsilon_{d}+U\gtrsim
E_{F}\gtrsim\varepsilon_{d}$). This situation changes when the QPCs are
polarized (see Fig. 2), as the different $t_{\sigma}$ result in
$\Gamma_{\uparrow}\neq\Gamma_{\downarrow}$, which in turn produce different
$\langle n_{d\uparrow}\rangle$ and $\langle n_{d\downarrow}\rangle$,
especially on both sides of the plateau.
The zero-bias conductance is calculated using a Landauer formula generalized
for interacting systems 15
$\displaystyle G_{\sigma}=\frac{e^{2}}{h}\Gamma_{\sigma}\int
d\epsilon\frac{\partial
f(\epsilon)}{\partial\epsilon}\,\text{Im}[G_{d\sigma}(\epsilon)],$ (14)
for symmetric coupling of the leads for each spin. One can also calculate the
polarization factor
$\displaystyle\eta=\frac{G_{\uparrow}-G_{\downarrow}}{G_{\uparrow}+G_{\downarrow}},$
(15)
which gives a measure of current polarization in the system. Figure 3 shows
the spin-dependent conductances and polarization for the system in the
Hubbard-I approximation. As $\langle n_{d\uparrow}\rangle\neq\langle
n_{d\downarrow}\rangle$ in general (except at the particle-hole symmetry
point, $\varepsilon_{d}=-U/2$), the conductance per spin $G_{\sigma}$ are also
different. Notice that the spin-dependent conductance peaks are very
asymmetric and non-Lorentzian, due to the peculiar behavior of the occupancies
and their different up and down-spin couplings. As we consider here the case
$\Gamma_{\uparrow}>\Gamma_{\downarrow}$, one clearly sees that generally
$G_{\uparrow}>G_{\downarrow}$ over the entire range of $\varepsilon_{d}$
values. As a consequence, there is a net up-spin polarization ($\simeq 60$%)
and conductance around the resonant peaks, with the latter reaching $\simeq
0.3(e^{2}/h)$.
#### III.1.2 Kondo regime
To study the low-temperature behavior of the system within the EOM we need to
consider higher order GFs in Eq. (10). Using Lacroix’s approach, 3 one can
obtain relations for the three GFs on the right side of (10) as:
$\displaystyle(\omega-\varepsilon_{k\sigma})\langle\langle
c_{k\sigma}n_{d\bar{\sigma}};c^{\dagger}_{d\sigma}\rangle\rangle$
$\displaystyle=$
$\displaystyle\langle[c_{k\sigma}n_{d\bar{\sigma}};c^{\dagger}_{d\sigma}]_{+}\rangle+\langle\langle[c_{k\sigma}n_{d\bar{\sigma}};H];c^{\dagger}_{d\sigma}\rangle\rangle$
(16) $\displaystyle=$ $\displaystyle\tilde{t}_{\sigma}\langle\langle
n_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle+\tilde{t}_{\bar{\sigma}}\sum_{k^{\prime}}[\langle\langle
c_{k\sigma}c^{\dagger}_{d\bar{\sigma}}c_{k^{\prime}\bar{\sigma}};c^{\dagger}_{d\sigma}\rangle\rangle-\langle\langle
c^{\dagger}_{k^{\prime}\bar{\sigma}}c_{d\bar{\sigma}}c_{k\sigma};c^{\dagger}_{d\sigma}\rangle\rangle],$
$\displaystyle(\omega-\varepsilon_{d\sigma}+\varepsilon_{d\bar{\sigma}}-\varepsilon_{k\bar{\sigma}})\langle\langle
c_{k\bar{\sigma}}c^{\dagger}_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle$
$\displaystyle=\langle[c^{\dagger}_{d\bar{\sigma}}c_{k\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}]_{+}\rangle+\langle\langle[c^{\dagger}_{d\bar{\sigma}}c_{k\bar{\sigma}}c_{d\sigma};H];c^{\dagger}_{d\sigma}\rangle\rangle$
(17) $\displaystyle=\langle
c^{\dagger}_{d\bar{\sigma}}c_{k\bar{\sigma}}\rangle+\tilde{t}_{\bar{\sigma}}\langle\langle
n_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle+\sum_{k^{\prime}}[-\tilde{t}_{\bar{\sigma}}\langle\langle
c^{\dagger}_{k^{\prime}\bar{\sigma}}c_{k\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle$
$\displaystyle+\tilde{t}_{\sigma}\langle\langle
c^{\dagger}_{d\bar{\sigma}}c_{k\bar{\sigma}}c_{k^{\prime}\sigma};c^{\dagger}_{d\sigma}\rangle\rangle],$
and
$\displaystyle(\omega-\varepsilon_{d\sigma}-\varepsilon_{d\bar{\sigma}}+\varepsilon_{k\bar{\sigma}}-U)\langle\langle
c^{\dagger}_{k\bar{\sigma}}c_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle$
$\displaystyle=\langle[c^{\dagger}_{k\bar{\sigma}}c_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}]_{+}\rangle+\langle\langle[c^{\dagger}_{k\bar{\sigma}}c_{d\bar{\sigma}}c_{d\sigma};H];c^{\dagger}_{d\sigma}\rangle\rangle$
(18) $\displaystyle=\langle
c^{\dagger}_{k\bar{\sigma}}c_{d\bar{\sigma}}\rangle-\tilde{t}_{\bar{\sigma}}\langle\langle
n_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle+\sum_{k^{\prime}}[\tilde{t}_{\sigma}\langle\langle
c^{\dagger}_{k^{\prime}\bar{\sigma}}c_{d\bar{\sigma}}c_{k^{\prime}\sigma};c^{\dagger}_{d\sigma}\rangle\rangle$
$\displaystyle-\tilde{t}_{\bar{\sigma}}\langle\langle
c^{\dagger}_{k\bar{\sigma}}c_{d\sigma}c_{k^{\prime}\bar{\sigma}};c^{\dagger}_{d\sigma}\rangle\rangle].$
Following the decoupling procedure in Ref. 3, , each GF of the type
$\langle\langle A^{*}BC,D^{*}\rangle\rangle$ is replaced by
$\displaystyle\approx\langle A^{*}B\rangle\langle\langle
C,D^{*}\rangle\rangle-\langle A^{*}C\rangle\langle\langle
B,D^{*}\rangle\rangle,$ (19)
resulting in an equation for the dot GF given by
$\displaystyle G_{d\sigma}(\omega)$ $\displaystyle=$
$\displaystyle\left[U(\omega)-\langle n_{d\bar{\sigma}}\rangle-
B_{\bar{\sigma}}(\omega)-B_{\bar{\sigma}}(\omega_{1})\right]$ (20)
$\displaystyle\times\left\\{U(\omega)[\omega-\varepsilon_{d\sigma}-\Sigma_{\sigma}(\omega)]+[B_{\bar{\sigma}}(\omega)+B_{\bar{\sigma}}(\omega_{1})\right]\Sigma_{\sigma}(\omega)-A_{\bar{\sigma}}(\omega)+A_{\bar{\sigma}}(\omega_{1})\\}^{-1}\,,$
where
$\Sigma_{\sigma}(x)=\sum_{k}|\tilde{t}_{\sigma}|^{2}/(x-\varepsilon_{k\sigma})$,
$U(\omega)=[U-\omega+\varepsilon_{d\sigma}-\Sigma_{\sigma}(\omega)+\Sigma_{\bar{\sigma}}(\omega)-\Sigma_{\bar{\sigma}}(\omega_{1})]/U$
and $\omega_{1}=-\omega+\varepsilon_{d\sigma}+U$. The functions
$A_{\sigma}(\omega)$ and $B_{\sigma}(\omega)$ are given by
$\displaystyle B_{\sigma}(\omega)$ $\displaystyle=$
$\displaystyle\frac{i}{2\pi}\int
d\omega^{\prime}f(\omega^{\prime})\left[G_{d\sigma}(\omega^{\prime})\frac{\Sigma_{\sigma}(\omega^{\prime})-\Sigma_{\sigma}(\omega)}{\omega-\omega^{\prime}-i\delta}-G^{*}_{d\sigma}(\omega^{\prime})\frac{\Sigma^{*}_{\sigma}(\omega^{\prime})-\Sigma_{\sigma}(\omega)}{\omega-\omega^{\prime}+i\delta}\right]$
$\displaystyle A_{\sigma}(\omega)$ $\displaystyle=$
$\displaystyle\frac{i}{2\pi}\int
d\omega^{\prime}f(\omega^{\prime})\left[\left(1+G_{d\sigma}(\omega^{\prime})\Sigma_{\sigma}(\omega^{\prime})\right)\frac{\Sigma_{\sigma}(\omega^{\prime})-\Sigma_{\sigma}(\omega)}{\omega-\omega^{\prime}-i\delta}-\left(1+G^{*}_{d\sigma}(\omega^{\prime})\Sigma^{*}_{\sigma}(\omega^{\prime})\right)\frac{\Sigma^{*}_{\sigma}(\omega^{\prime})-\Sigma_{\sigma}(\omega)}{\omega-\omega^{\prime}+i\delta}\right],$
(21)
and have to be calculated self-consistently. In the limit
$U\rightarrow\infty$, the GF (20) acquires a simpler form,
$\displaystyle G_{d\sigma}(\omega)=\frac{1-\langle n_{d\bar{\sigma}}\rangle-
B_{\bar{\sigma}}(\omega)}{\omega-\varepsilon_{d\sigma}-(1-B_{\bar{\sigma}}(\omega))\Sigma_{\sigma}(\omega)-A_{\bar{\sigma}}(\omega)}.$
(22)
Figure 4: (color online) Density of states in QD as a function of $\omega$ in
the Kondo regime. Parameters used are $T=10^{-7}$, $\varepsilon_{d}=-0.136$,
$\Gamma_{\uparrow}=0.06$ and $\Gamma_{\downarrow}=0.03$ ($D=1$). Sharp feature
near the Fermi level ($\omega=0$) is the signature of the Kondo screening,
although asymmetric here for the different spin species.
In the following, we solve numerically for the spectral function (DOS) in the
Kondo regime by the self-consistent iteration of Eqs. (13) and (22). Figure 4
shows the DOS vs. $\omega$ at low temperature ($T=10^{-7}$) for both spin
orientations (and total), when the QD electron level $\varepsilon_{d}$ is
taken to be at $-0.136$. The three curves exhibit Kondo resonance peaks near
the Fermi level ($\omega\simeq 0$), in addition to a much broader peak at
$\omega\simeq\varepsilon_{d}$. 3 Since the hybridization between leads and QD
are spin dependent in this case, the DOS clearly splits into two different
components for spin up and down. The DOS for spin down is shifted upwards in
energy with respect to the spin up component, resulting in a lower occupancy
for the down spin. Figure 5 shows indeed the occupancy curves for $\langle
n_{d\uparrow}\rangle$, $\langle n_{d\downarrow}\rangle$ and total $\langle
n\rangle$ vs. $\varepsilon_{d}$ at a given temperature ($T=10^{-7}$).
Qualitatively similar to the results in the Coulomb blockade regime, the
occupancy curves show that $\langle n_{d\uparrow}\rangle>\langle
n_{d\downarrow}\rangle$ for $\omega\gtrsim-0.05$, while the relation is
reversed for smaller $\omega$ values, reflecting the asymmetry introduced by
the polarizing QPCs, through $\Gamma_{\uparrow}$ and $\Gamma_{\downarrow}$.
Notice also that the plateaus at 0.5 are not as well defined here, due to the
enhanced spin and charge fluctuations in the Kondo regime. The spin-orbit
effect can be understood qualitatively as arising from a shift in the dot
level (as well as depending on the occupancy factors): since the effective
level position of the electron with spin $\sigma$ is given by
$\varepsilon_{d\sigma}\approx\varepsilon_{d}+\text{Re}\Sigma^{{}^{\prime}}_{\sigma}(\omega)$,
where $\Sigma^{{}^{\prime}}_{\sigma}(\omega)\propto t_{\sigma}^{2}$ is the
self-energy, and thus naturally causes the spin-dependent occupancy seen in
the figure.
Figure 5: (color online) Occupation in the QD as function of $\epsilon_{d}$.
Parameters used as in Fig. 4. Figure 6: (color online) (a) Conductance, and
(b) polarization and difference between spin up and down conductance, as
functions of $\epsilon_{d}$. Parameters as in Fig. 4.
Figure 6(a) shows the spin-dependent conductance curves $G_{\sigma}$ vs.
$\varepsilon_{d}$. Several features are noteworthy. As $\varepsilon_{d}$
changes, the spin-dependent conductances exhibit the anticipated peaks at low
temperature, with spin up conductance dominating (naturally, as
$\Gamma_{\uparrow}>\Gamma_{\downarrow}$). Figure 6(b) shows the difference
between spin up and down conductances as well as the spin polarization vs.
$\varepsilon_{d}$. In this regime, the difference between the conductance for
the two spin orientations reaches $\simeq 0.4(e^{2}/h)$. Correspondingly, the
net spin polarization reaches $\eta\simeq 60$%.
Let us now analyze in more detail the effect of temperature on the conductance
of the system and especially its drop for $\varepsilon_{d}\ll 0$. The
conductance curves in Fig. 6(a) are highly asymmetric about the Fermi energy
and vanish rapidly away from it. This vanishing for very negative values of
$\varepsilon_{d}$ is due to the the Kondo temperature, $T_{K}$, becoming
smaller than the temperature of the system. Figure 7 presents the total zero
bias conductance as function of $\varepsilon_{d}$ for several values of
temperature. As $T$ is lowered, the total conductance increases for a given
$\varepsilon_{d}$, and the width of the conductance peak increases in
$\varepsilon_{d}$. This explicitly reflects the existence of the Kondo
resonant peak in the spectral function, and how the system will reach the
unitary limit of conductance for $T=0$ for $\varepsilon_{d}$ well below the
Fermi level.
Figure 7: (color online) Total conductance as function of $\varepsilon_{d}$
for several temperatures $T$. Notice that $T_{K}$ is strongly suppressed for
more negative $\varepsilon_{d}$ values, which lowers the conductance at a
given $T$. Parameters as in Fig. 4.
Our discussions above are applied to the case of infinite $U$, where the EOM
method gives qualitatively accurate results in the Kondo regime. Extension of
this approach for finite $U$ is known to be problematic, including its failure
to exhibit a Kondo resonance at the particle-hole symmetry point
($\varepsilon_{d}=-U/2$). 18 To carry out our study in the finite $U$ case,
we use instead the essentially exact numerical renormalization group approach,
as we discuss in the following section.
### III.2 NRG results for finite U
For the finite-$U$ case we study the spin polarized conductance using the
standard numerical renormalization group approach.Wilson75 ; costi In this
case, unlike the previous infinite-$U$ case, the processes involving double
occupied states are naturally present in the dynamics of the system, and allow
for a reliable description of the low-energy behavior. We set $U=0.5$ and
$T=0$. Figure 8 depicts the local density of states calculated with NRG for
the same system parameters as before, $\Gamma_{\uparrow}=0.06$,
$\Gamma_{\downarrow}=0.03$, and with $\epsilon_{d}=-0.2$. This value of
$\epsilon_{d}$ corresponds to a situation where the system is away from the
particle-hole symmetric point ($=-U/2$), more suitable to compare to the
previous infinite-$U$ calculation (where the system never reaches the p-h
symmetry point). Notice that there is a strong spin asymmetry in the DOS; on
the negative side of the $\omega$-axis, the DOS for spin up (solid curve)
presents a peak near $-0.2$, corresponding to the energy of the local bare
orbital $\varepsilon_{d}$, slightly shifted by the real part of the proper
self-energy. On the positive side of the $\omega$-axis, the peak near
$\epsilon_{d}+U$ would result in the succeeding CB peak, which appears only
for the spin-down DOS (dashed curve). In this regime as well, the fact that
$\Gamma_{\uparrow}>\Gamma_{\downarrow}$ favors the spin-up occupancy to the
detriment of the spin-down occupancy. One also notices the important peaks
close to the Fermi level, signature of the Kondo effect, which are slightly
split away and suppressed by a seemingly effective magnetic field induced by
the spin-asymmetric coupling to the leads. This phenomenon is akin to the
suppression discussed by Martinek et al.,Martinek in the context of a QD
coupled to ferromagnetic leads. Notice, however, that no external
magnetization is present in our system and that the polarization is only
arising from the QPCs and due to the lateral SO interaction.
Figure 8: (color online) Density of states in the QD as function of energy
obtained from NRG calculations. Polarizing QPCs generate effective splitting
of the DOS for different spins. System parameters used are $U=0.5$,
$\Gamma_{\uparrow}=0.06$, $\Gamma_{\downarrow}=0.03$, $D=1$, and $T=0$.
Analogously to the infinite-$U$ case, the spin-asymmetry discussed above
induces spin polarized transport in the system. In Fig. 9 we show the
conductance as a function of $\varepsilon_{d}$ for the same parameters as Fig.
8. Notice that away from the p-h symmetric point ($\varepsilon_{d}=-0.25$) the
conductance for spin up is much larger than for spin down, resulting in a
sizeable polarization ($\eta\simeq 70$%), as shown by the (green) dotted
curve. At the p-h symmetric point the conductance for both spins reaches the
unitary limit and $\eta\rightarrow 0$; this is consistent with the restoration
of the Kondo state of the system at the p-h point for ferromagnetic leads.
Sindel
Figure 9: (color online) Total conductance as function of $\varepsilon_{d}$
obtained from NRG calculations. Same system parameters as in Fig. 8.
## IV Summary
In summary, we have investigated the spin-dependent transport properties of
quantum dot structures with polarizing quantum point contacts. We have shown
that as QPCs can generate finite spin-polarized currents, due to the
combination of lateral and perpendicular spin-orbit interactions, they also
induce current polarization in quantum dots made with these QPCs. Using
equation-of-motion techniques and numerical renormalization group
calculations, we obtained the electronic Green’s function, conductance and
spin polarization in different parameter regimes. Our results demonstrate that
both in the Coulomb blockade and Kondo regimes, the quantum dot exhibits non-
zero spin-polarized conductance, even when the injection is unpolarized and
there are no applied magnetic fields. The spin-dependent coupling is shown to
give rise to nontrivial effects in the density of states of the single QD,
resulting in strong modification of the charge distribution in the system.
Most importantly, these effects are controllable by lateral gate voltages
applied to the QPCs, and together with the ability to create quantum dots,
they provide a new approach for exploring spintronic devices, spin polarized
sources and spin filters.
## V Acknowledgements
We thank helpful discussions with P. Debray and N. Sandler, as well as
financial support from CNPq, CAPES, and FAPEMIG in Brazil, and NSF-PIRE, and
NSF-MWN/CIAM in the US.
## References
* (1) S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnár, M. L. Roukes, A. Y. Chtchelkanova, D. M. Treger, Science 294, 1488 (2001).
* (2) R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. Vandersypen, Rev. Mod. Phys. 79, 1217 (2007).
* (3) G. Schmidt, D. Ferrand, L. W. Molenkamp, A. T. Filip and B. J. van Wees, Phys. Rev. B 62, R4790 (2000).
* (4) Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984).
* (5) R. Winkler, Spin-Orbit Coupling Effects in Two Dimensional Electron and Hole Systems (Springer, Berlin, 2003).
* (6) A. T. Ngo, P. Debray and S. E. Ulloa (arXiv:0908.1080) Phys. Rev. B 81, 115328 (2010).
* (7) J. A. Folk, S. R. Patel, K. M. Birnbaum, C. M. Marcus C. I. Duroz and J. S. Harris, Jr. Phys. Rev. Lett. 86, 2101 (2001).
* (8) J. P. Bird and Y. Ochiai, Science 303, 1621 (2004).
* (9) P. Debray, S. M. S. Rahman, J. Wan, R. S. Newrock, M. Cahay, A. T. Ngo, S. E. Ulloa, S. T. Herbert, M. Muhammad, and M. Johnson, Nature Nanotech. 4, 759 (2009).
* (10) J. Martinek, M. Sindel, L. Borda, J. Barnaś, J. König, G. Schön, and J. von Delft, Phys. Rev. Lett. 91, 247202 (2003).
* (11) M. Eto, T. Hayashi and Y. Kurotani, J. Phys. Soc. Japan. 74, 1934 (2005).
* (12) A. Reynoso, Gonzalo Usaj, C. A. Balseiro, Phys. Rev, B 75, 085321 (2007).
* (13) E. N. Bulgakov and A. F. Sadreev, Phys, Rev. B 66, 075331 (2002).
* (14) H.Q. Xu, Phys. Rev. B 72, 045347 (2005); H.Q. Xu, Phys. Rev. B 52,5803 (1995).
* (15) G. D. Mahan, Many Particle Physics (Plenum, New York, 1981).
* (16) J. E. Hirsch and R.M. Fye, Phys. Rev. Lett. 56, 2521 (1986).
* (17) C. Lacroix, J. Phys. F: Met. Phys. 11, 2389 (1981).
* (18) R. Bulla, T. A. Costi, and T. Pruschke, Rev. Mod. Phys. 80, 395 (2008).
* (19) D. N. Zubarev, [Usp. Fiz. Nauk 71, 71 (1960)], Sov. Phys. Usp. 3, 320 (1960).
* (20) J. Hubbard, Proc. R. Soc. London, Ser. A 276, 238 (1963).
* (21) Y. Meir, N. S. Wingreen and P. A. Lee, Phys. Rev. Lett. 66, 3048 (1991).
* (22) V. Kashcheyevs, A. Aharony and O. Entin-Wohlman, Phys, Rev. B 73, 125338 (2005).
* (23) K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975); H. R. Krishna-murty, J. W. Wilkins and K. G. Wilson Phys. Rev. B 21, 1003 (1980), and 21, 1044 (1980).
* (24) M. Sindel, L. Borda, J. Martinek, R. Bulla, J. König, G. Schön, S. Maekawa, and J. von Delft, Phys. Rev. B 76, 045321 (2007).
|
arxiv-papers
| 2010-12-19T15:36:31 |
2024-09-04T02:49:15.827718
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Anh T. Ngo, Edson Vernek, and Sergio E. Ulloa",
"submitter": "Anh Ngo",
"url": "https://arxiv.org/abs/1012.4175"
}
|
1012.4290
|
# Bit recycling for scaling random number generators
Andrea C. G. Mennucci Scuola Normale Superiore, Pisa, Italy
(a.mennucci@sns.it)
###### Abstract
Many Random Number Generators (RNG) are available nowadays; they are divided
in two categories, _hardware RNG_ , that provide “true” random numbers, and
_algorithmic RNG_ , that generate pseudo random numbers (PRNG). Both types
usually generate random numbers $(X_{n})_{n}$ as independent uniform samples
in a range $0,\ldots 2^{b}-1$, with $b=8,16,32$ or $b=64$. In applications, it
is instead sometimes desirable to draw random numbers as independent uniform
samples $(Y_{n})_{n}$ in a range $1,\ldots M$, where moreover $M$ may change
between drawings. Transforming the sequence $(X_{n})_{n}$ to $(Y_{n})_{n}$ is
sometimes known as _scaling_. We discuss different methods for scaling the
RNG, both in term of mathematical efficiency and of computational speed.
###### Contents
1. 1 Introduction
2. 2 Process splitting
1. 2.1 Mathematical formulation
3. 3 Recycling in uniform random number generation
4. 4 Mathematical analysis of the efficiency
5. 5 Speed, simple _vs_ complex algorithms
6. 6 Numerical tests
1. 6.1 Architectures
2. 6.2 Back-end PRNGs
3. 6.3 _Ad hoc_ functions
4. 6.4 Uniform RNGs
5. 6.5 Timing
6. 6.6 Conclusions
7. A Test results
1. A.1 Speed of back-end RNGs and _Ad hoc_ functions
2. A.2 Graphs
1. A.2.1 Uniform random generators
2. A.2.2 Integer arithmetic
8. B Code
1. B.1 Back-end RNGs
2. B.2 _ad hoc_ functions
3. B.3 Uniform random generators
## 1 Introduction
We consider the following problem. We want to generate a sequence of random
numbers $(Y_{n})_{n}$ with a specified probability distribution, using as
input a sequence of random numbers $(X_{n})_{n}$ uniformly distributed in a
given range. There are various methods available; these methods involve
transforming the input in some way; for this reason, these methods work
equally well in transforming both pseudo-random and true random numbers. One
such method, called the acceptance-rejection method, involves designing a
specific algorithm, that pulls random numbers, transforms them using a
specific function, tests whether the result satisfies a condition: if it is,
the value is accepted; otherwise, the value is rejected and the algorithm
tries again.
This kind of method has a defect, though: if not carefully implemented, it
throws away many inputs. Let’s see a concrete example. We suppose that we are
given a RNG that produces random bits, evenly distributed, and
independent111For example, repeatedly tossing a coin with the faces labeled
$0,1$.. We want to produce a random number $R$ in the range $\\{1,2,3\\}$,
uniformly distributed and independent. Consider the following method.
###### Example 1
We draw two random bits; if the sequence is $11$, we throw it away and draw
two bits again; otherwise we return the sequence as $R$, mapping $00,01,10$ to
$1,2,3$.
This is rather wasteful! The entropy in the returned random $R$ is
$\log_{2}(3)=1.585$bits; there is a $1/4$ probability that we throw away the
input, so the expected number of (pair of tosses) is $4/3$ and then expected
number of input bits is $8/3$; all together we are effectively using only
missing$$$\frac{\log_{2}(3)}{8/3}=59\%$$oftheinput.\par
ThewastemaybeconsiderasanunnecessaryslowdownoftheRNG:iftheRNGcangenerate1bitin$1μs$,then,aftertheexampleprocedure,theratehasdecreasedto$1.6
μs$perbit.SincealotofeffortwasputindesigningfastRNGinthenearpast,thenslowingdowntherateby$+60%$issimplyunacceptable.\par
Anotherproblemintheexamplemethodaboveisthat,althoughitisquiteunlikely,wecanhaveaverylongrunof"00"bits.Thismeansthatwecannotguaranteethattheaboveprocedurewillgeneratethenextnumberinapredeterminedamountoftime.\par
Thereareofcoursebettersolutions,asthis\emph{ad hoc
method}.\begin{Example}[\cite{DJ}]Wedraweightrandombits,andconsiderthemasanumber$x$intherange$0\ldots
255$;ifthenumberismorethan$3^{5}-1=242$,wethrowitawayanddraweightbitsagain;otherwisewewrite$x$asfivedigitsinbase3andreturnthesedigitsas5randomsamples.\end{Example}Thisismuchmoreefficient!Theentropyinthereturnedfivesamplesis$5
log_2(3)=7.92$bits;thereisa$13/256$probabilitythatwethrowawaytheinput,sotheexpectednumberof8-tuplesofinputsis$256/253$andthenexpectednumberofinputbitsis$2048/253$;alltogetherwearenowusingmissing$
$\frac{5\log_{2}(3)}{2048/253}=97\%$
of the input.
In this paper we will provide a mathematical proof (section 2), and discuss
some method (section 3), to optimize the scaling of a RNG. Unfortunately after
writing this paper it turned out that one of the ideas we are presenting in
section 3, was already described in [1]. This paper contains the mathematical
proof of the method, the discussion of how best to choose parameters,
discussion of its efficiency, and numerical speed tests.
###### Remark 3
A different approach may be to use a decompressing algorithm. Indeed, e.g.,
the _arithmetic encoder_ decoding algorithm, can be rewritten to decode a
stream of bits to an output of symbols with prescribed probability
distributions 222If interested, I have the code somewhere in the closet.
Unfortunately, it is quite difficult to mathematically prove that such an
approach really does transform a stream of independent equidistributed bits
into an output of independent random variables. Also, the _arithmetic encoder_
is complex, and this complexity would slow down the RNG, defeating one of the
goals. (Moreover, the _arithmetic encoder_ was originally heavily patented.)
A note on notations. In all of the paper, ns is a _nanosecond_ , that is
$10^{-9}$seconds. When $x$ is a real number, $\lfloor
x\rfloor=\mathtt{floor}(x)$ is the largest integer that is less or equal than
$x$.
## 2 Process splitting
Let ${\mathrm{l\hskip-1.49994ptN}}=\\{0,1,2,3,4,5\ldots\\}$ be the set of
natural numbers.
Let $(\Omega,{\mathcal{A}},{\mathbb{P}})$ a probability space, let
$(E,{\mathcal{E}})$ be a measurable space, and ${\overline{X}}$ a process of
i.i.d. random variables $(X_{n})_{n\in{\mathrm{l\hskip-1.04996ptN}}}$ defined
on $(\Omega,{\mathcal{A}},{\mathbb{P}})$ and each taking values in
$(E,{\mathcal{E}})$. We fix an event $S\in{\mathcal{E}}$ such that
${\mathbb{P}}\\{X_{i}\in S\\}\neq 0,1$; we define
$p_{S}\stackrel{{\scriptstyle\mbox{\tiny{def}}}}{{=}}{\mathbb{P}}\\{X_{i}\in
S\\}$.
We define a formal method of process splitting/unsplitting.
The splitting of ${\overline{X}}$ is the operation that generates three
processes ${\overline{B}},{\overline{Y}},{\overline{Z}}$, where
${\overline{B}}=(B_{n})_{n\in{\mathrm{l\hskip-1.04996ptN}}}$ is an i.i.d.
Bernoulli process with parameter $p_{S}$, and
${\overline{Y}}=(Y_{n})_{n\in{\mathrm{l\hskip-1.04996ptN}}}$ and
${\overline{Z}}=(Z_{n})_{n\in{\mathrm{l\hskip-1.04996ptN}}}$ are processes
taking values respectively in $S$ and $E\setminus S$. The unsplitting is the
opposite operation. These operations can be algorithmically and intuitively
described by the following pseudocode (where processes are thought of as
_queues of random variables_).
procedure
Splitting(${\overline{X}}\mapsto({\overline{B}},{\overline{Y}},{\overline{Z}})$)
initialize the three empty queues
${\overline{B}},{\overline{Y}},{\overline{Z}}$ repeat pop X from
${\overline{X}}$ if $X\in S$ then push 1 onto ${\overline{B}}$ push X onto
${\overline{Y}}$ else push 0 onto ${\overline{B}}$ push X onto
${\overline{Z}}$ end if until forever end procedure procedure
Unsplitting($({\overline{B}},{\overline{Y}},{\overline{Z}})\mapsto{\overline{X}}$)
initialize the empty queue ${\overline{X}}$ repeat pop B from ${\overline{B}}$
if $B=1$ then pop Y from ${\overline{Y}}$ push Y onto ${\overline{X}}$ else
pop Z from ${\overline{Z}}$ push Z onto ${\overline{X}}$ end if until forever
end procedure
The fact that the _splitting_ operation is invertible implies that no entropy
is lost when splitting. We will next show a very important property, namely,
that the splitting operation preserve probabilistic independence.
### 2.1 Mathematical formulation
We now rewrite the above idea in a purely mathematical formulation.
We define the Bernoulli process $(B_{n})_{n\in{\mathrm{l\hskip-1.04996ptN}}}$
by
$B_{n}\stackrel{{\scriptstyle\mbox{\tiny{def}}}}{{=}}\begin{cases}1&X_{n}\in
S\\\ 0&X_{n}\not\in S\end{cases}$ (1)
and the times of return to success as
$\displaystyle U_{0}$ $\displaystyle=$ $\displaystyle\inf\\{k:k\geq
0,B_{k}=1\\}$ (2) $\displaystyle U_{n}$ $\displaystyle=$
$\displaystyle\inf\\{k:k\geq 1+U_{n-1},B_{k}=1\\},\leavevmode\nobreak\
\leavevmode\nobreak\ n\geq 1$ (3)
whereas the times of return to unsuccess are
$\displaystyle V_{0}$ $\displaystyle=$ $\displaystyle\inf\\{k:k\geq
0,B_{k}=0\\}$ (4) $\displaystyle V_{n}$ $\displaystyle=$
$\displaystyle\inf\\{k:k\geq 1+U_{n-1},B_{k}=0\\},\leavevmode\nobreak\
\leavevmode\nobreak\ n\geq 1$ (5)
it is well known that $(U_{n}),(V_{n})$ are (almost certainly) well defined
and finite.
We eventually define the processes
$(Y_{n})_{n\in{\mathrm{l\hskip-1.04996ptN}}}$ and
$(Z_{n})_{n\in{\mathrm{l\hskip-1.04996ptN}}}$ by
$Y_{n}=X_{U_{n}}\quad Z_{n}=X_{V_{n}}$ (6)
###### Theorem 4
Assume that ${\overline{X}}$ is a process of i.i.d. random variables. Let
$\mu$ be the law of $X_{1}$. Then
* •
the random variables $B_{n},Y_{n},Z_{n}$ are independent; and
* •
the variables of the same type are identically distributed: the variables
$B_{n}$ have parameter ${\mathbb{P}}\\{B_{n}=1\\}=p_{S}$; the variables
$Y_{n}$ have law $\mu(\cdot\leavevmode\nobreak\ |\leavevmode\nobreak\ S)$; the
variables $Z_{n}$ have law $\mu(\cdot\leavevmode\nobreak\
|\leavevmode\nobreak\ E\setminus S)$.
* Proof.
It is obvious that ${\overline{B}}$ is a Bernoulli process of independent
variables with parameter ${\mathbb{P}}\\{B_{n}=1\\}=p_{S}$.
Let $K,M\geq 1$ integers. Let $u_{0}<u_{1}<\ldots u_{K}$ and
$v_{0}<v_{1}<\ldots v_{M}$ be integers, and consider the event
$A\stackrel{{\scriptstyle\mbox{\tiny{def}}}}{{=}}\\{U_{0}=u_{0},\ldots
U_{K}=u_{K},V_{0}=v_{0},\ldots V_{M}=v_{M}\\}$ (7)
If $A\neq\emptyset$ then
$A=\\{B_{0}=b_{0},\ldots,B_{N}=b_{N}\\}$ (8)
where $N=\max\\{u_{K},v_{M}\\}$ and $b_{j}\in\\{0,1\\}$ are suitably chosen.
Indeed, supposing that $N=u_{K}>v_{M}$, then we use the success times, and set
that $b_{j}=1$ iff $j=u_{k}$ for a $k\leq K$; whereas supposing that
$N=v_{M}>u_{K}$, then we use the unsuccess times, and set that $b_{j}=0$ iff
$j=v_{m}$ for a $m\leq M$.
Let ${\mathcal{F}}_{K,M}$ be the family of all above events $A$ defined as per
equation (7), for different choices of $(u_{i}),(v_{j})$; let
missing$$${\mathcal{F}}=\bigcup_{K,M\geq
1}{\mathcal{F}}_{K,M}\leavevmode\nobreak\ \leavevmode\nobreak\
;$$let$A^¯B⊂A$bethesigmaalgebrageneratedbytheprocess$¯B$.\par
Theaboveequality\eqref{eq:A_as_B}provesthat$F$isa\emph{base}for$A^¯B$:itisstablebyfiniteintersection,anditgeneratesthesigmaalgebra$A^¯B$.\par
Consideragain$K,M≥1$integers,andevents$F_i,G_j∈E$for$i=0,…K,j=0,…M$,andtheeventmissing$
$C=\\{Y_{0}\in F_{0},\ldots Y_{K}\in F_{K},Z_{0}\in G_{0},\ldots Z_{M}\in
G_{M}\\}\in{\mathcal{A}}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ ;$
let $A\in{\mathcal{F}}_{K,M}$ non empty; we want to show that
missing$$${\mathbb{P}}(C\leavevmode\nobreak\ |\leavevmode\nobreak\
A)={\mathbb{P}}(C)$$thiswillprovethat$(¯Y,¯Z)$areindependentof$¯B$,byarbitrinessof$(F_i),(G_j),K,M$andsince$F$isabasefor$A^¯B$.\par
Wefix$(u_i),(v_j)$anddefine$A$asinequation\eqref{eq:A};welet$N=max{u_K,v_M}$anddefine$(b_n)$asexplainedafterequation\eqref{eq:A_as_B}.Bydefiningmissing$
$S^{1}\stackrel{{\scriptstyle\mbox{\tiny{def}}}}{{=}}S,S^{0}\stackrel{{\scriptstyle\mbox{\tiny{def}}}}{{=}}E\setminus
S$
for notation convenience, we can write equation (8) as missing$$$A=\\{X_{0}\in
S^{b_{0}},\ldots X_{N}\in S^{b_{N}}\\}\leavevmode\nobreak\
\leavevmode\nobreak\ .$$Let$E_0…E_N∈E$bedefinedbymissing$
$E_{n}\stackrel{{\scriptstyle\mbox{\tiny{def}}}}{{=}}\begin{cases}F_{k}&\text{if
}n=u_{k}\text{ for a }k\leq K\\\ G_{m}&\text{if }n=v_{m}\text{ for a }m\leq
M\\\ E&\text{else}\end{cases}$
then we compute
$\displaystyle{\mathbb{P}}(C\leavevmode\nobreak\ |\leavevmode\nobreak\ A)$
$\displaystyle=$ $\displaystyle{\mathbb{P}}(\\{X_{u_{0}}\in F_{0},\ldots
X_{u_{K}}\in F_{K},X_{v_{0}}\in G_{0},\ldots X_{v_{M}}\in
G_{M}\\}\leavevmode\nobreak\ |\leavevmode\nobreak\ \\{X_{0}\in
S^{b_{0}},\ldots X_{N}\in S^{b_{N}}\\})=$ $\displaystyle=$
$\displaystyle\frac{{\mathbb{P}}\\{X_{u_{0}}\in F_{0},\ldots X_{u_{K}}\in
F_{K},X_{v_{0}}\in G_{0},\ldots X_{v_{M}}\in G_{M}\leavevmode\nobreak\
,\leavevmode\nobreak\ X_{0}\in S^{b_{0}},\ldots X_{N}\in
S^{b_{N}}\\}}{{\mathbb{P}}\\{X_{0}\in S^{b_{0}},\ldots X_{N}\in
S^{b_{N}}\\}}=$ $\displaystyle=$
$\displaystyle\frac{\prod_{n=0}^{N}{\mathbb{P}}\\{X_{n}\in S^{b_{n}}\cap
E_{n}\\}}{\prod_{n=0}^{N}{\mathbb{P}}\\{X_{n}\in
S^{b_{n}}\\}}=\prod_{n=0}^{N}{\mathbb{P}}(X_{n}\in E_{n}|X_{n}\in S^{b_{n}})=$
$\displaystyle=$ $\displaystyle\prod_{k=0}^{K}\mu(F_{k}\leavevmode\nobreak\
|\leavevmode\nobreak\ S^{1})\prod_{m=0}^{M}\mu(G_{m}\leavevmode\nobreak\
|\leavevmode\nobreak\ S^{0})\leavevmode\nobreak\ \leavevmode\nobreak\ ;$
the last equality is due to the fact that: when $n=u_{k}$ then $b_{n}=1$, when
$n=v_{m}$ then $b_{n}=0$, and for all other $n$ we have $E_{n}=E$. Since the
last term does not depend on $A$, that is, on $(u_{i}),(v_{j})$, we obtain
that $({\overline{Y}},{\overline{Z}})$ are independent of ${\overline{B}}$.
The above equality then also shows that missing$$${\mathbb{P}}\\{Y_{0}\in
F_{0},\ldots Y_{K}\in F_{K},Z_{0}\in G_{0},\ldots Z_{M}\in
G_{M}\\}=\prod_{k=0}^{K}\mu(F_{k}\leavevmode\nobreak\ |\leavevmode\nobreak\
S)\prod_{m=0}^{M}\mu(G_{m}\leavevmode\nobreak\ |\leavevmode\nobreak\
S^{c})$$andthisimpliesthat$¯Y,¯Z$areprocessesofindependentvariables,distributedasinthethesis.Byassociativityoftheindependence,weconcludethattherandomvariables$B_n,Y_n,Z_n$areindependent.\quad\hbox{\leavevmode\hbox
to7.77786pt{\hfil\vrule\vbox to6.75003pt{\hrule
width=6.00006pt\vfil\hrule}\vrule\hfil}}\par\endtrivlist$
## 3 Recycling in uniform random number generation
We now restrict our attention to the generation of uniformly distributed
integer valued random variables. We will say that _$R$ is a random variable of
modulus $M$_ when $R$ is uniformly distributed in the range $0,\ldots(M-1)$.
We present an algorithm, that we had thought of, and then found (different
implementation, almost identical idea) in [1]. We present the latter
implementation.
The following algorithm Uniform random by bit recycling in figure 1, given
$n$, will return a random variable of modulus $n$; note that $n$ can change
between different calls to the algorithm.
1:initialize the static integer variables $m=1$ and $r=0$
2:procedure Uniform random by bit recycling(n)
3: repeat
4: while $m<N$ do$\triangleright$ fill in the state
5: r : = 2*r + NextBit();
6: m : = 2*m; $\triangleright$ r is a random variable of modulus m
7: end while
8: q := $\lfloor m/n\rfloor$; $\triangleright$ integer division, rounded down
9: if $r<n*q$ then
10: d : = $r\mathop{\operator@font mod}\nolimits n$ $\triangleright$
remainder, is a random variable of modulus n
11: r : = $\lfloor r/n\rfloor$ $\triangleright$ quotient, is random variable
of modulus q
12: m : = q
13: return d
14: else
15: r : = r - n*q $\triangleright$ r is still a random variable of modulus m
16: m : = m - n*q $\triangleright$ the procedure loops back to line 3
17: end if
18: until forever
19:end procedure
Figure 1: Algorithm Uniform random by bit recycling
It uses two internal integer variables, m and r, which are not reset at the
beginning of the algorithm (in C, you would declare them as "static").
Initially, $m=1$ and $r=0$.
The algorithm has an internal constant parameter $N$, which is a large integer
such that $2N$ can still be represented exactly in the computer. We must have
$n<N$. 333We will show in next section that it is best to have $n<<N$. The
algorithm draws randomness from a function NextBit() that returns a random
bit.
Here is an informal discussion of the algorithm, in the words of the original
author [1]. _At line 10, as r is between 0 and $(n*q-1)$, we can consider r as
a random variable of modulus $n*q$. As this is divisible by n, then
$d:=(r\mathop{\operator@font mod}\nolimits n)$ will be uniformly distributed,
and the quotient $\lfloor r/n\rfloor$ will be uniformly distributed between 0
and $q-1$. _
Note that the theoretical running time is unbounded; we will though show in
the next section that an accurate choice of parameters practically cancels
this problem.
## 4 Mathematical analysis of the efficiency
We recall this simple idea.
###### Lemma 5
Suppose $R$ is a random variable of modulus $MN$; we perform the integer
division $R=QN+D$ where $Q\in\\{0,\ldots(M-1)\\}$ is the quotient and
$D\in\\{0,\ldots(N-1)\\}$ is the remainder; then $Q$ is a random variable of
modulus $M$ and $D$ is a random variable of modulus $N$; and $Q,D$ are
independent.
###### Proposition 6
Let us assume that repeated calls of NextBit() return a sequence of
independent equidistributed bits. Then the above algorithm Uniform random by
bit recycling in figure 1 will return a sequence of independent and uniformly
distributed numbers.
* Proof.
We sketch the proof. We use the lemma above 5 and the theorem 4. Consider the
notations in the second section. The bits returned by the call NextBit()
builds up the process ${\overline{X}}$. When reaching the if (line 9 in the
pseudocode at page 1), the choice $r<n*q$ is the choice of the value of
$B_{n}$ in equation (1). This (virtually) builds the process ${\overline{B}}$.
At line 10 $r$ is a variable in the process ${\overline{Y}}$; since it is of
modulus $nq$, we return (using the lemma) the remainder as $d$, that is a
random variable of modulus $n$, and push back the quotient into the state. At
line 16 we would be defining a variable in the process ${\overline{Z}}$, that
we push back into the state.
The “pushing back” of most of the entropy back into the state recycles the
bits, and improves greatly the efficiency.
The only wasted bits are related to the fact that the algorithm is throwing
away the mathematical stream ${\overline{B}}$. Theoretically, if this stream
would be feeded back into the state (for example, by employing Shannon-Fano-
Elias coding), then efficiency would be exactly $100\%$.444But this would
render difficult to prove that the output numbers are independent…
Practically, the numbers $N$ and $n$ can be designed so that this is totally
unneeded.
###### Remark 7
Indeed, consider the implementation (see the code in the next sections) where
the internal state is stored as 64bit unsigned integers, whereas $n$ is
restricted to be 32bit unsigned integer; so the internal constant is
$N=2^{62}$ while $n\in\\{2\ldots 2^{32}-1\\}$, When reaching the if at line 9,
$m$ is in the range $2^{62}\leq m<2^{64}$, and $r$ is uniform of modulus $m$;
but $m-n*\lfloor m/n\rfloor$ is less than $n$, that is, less than $2^{32}$; so
the probability that $r\geq n*q$ at the if is less than $1/2^{30}$.
In particular, this means that each $B_{n}$ in the mathematical stream
${\overline{B}}$ contains $\sim 10^{-8}$bits of entropy, so there is no need
to recycle them.
Indeed, in the numerical experiments we found out that the following algorithm
wastes $\sim 30$ input bits on a total of $\sim 10^{9}$ input bits (!) this is
comparable to the entropy of the internal state (and may also be due to
numerical error in adding up $\log_{2}()$ values).
Also, this choice of parameters ensures that the algorithm will never
practically loop twice before returning. When the condition in the if at line
9 is false, we will count it as a failure. In $\sim 10^{10}$ calls to the
algorithm, we only experienced 3 failures. 555For this reason, the else block
may be omitted with no big impact on the quality of the output – we implement
this idea in the algorithm uniform_random_by_bit_recycling_cheating.
## 5 Speed, simple _vs_ complex algorithms
We now consider the algorithm Uniform random simple in 2.
1:procedure Uniform random simple(n)
2: repeat
3: r : = GetRandomBits(b); $\triangleright$ fill the state with b bits
4: q = $\lfloor N/n\rfloor$; $\triangleright$ integer division, rounded down
5: if $r<n*q$ then
6: return $r\mathop{\operator@font mod}\nolimits n$ $\triangleright$
remainder, is random variable of modulus n
7: end if$\triangleright$ otherwise, start all over again
8: until forever
9:end procedure
Figure 2: Uniform random simple ; in our tests $N=2^{b}$ or $N=2^{b}-1$,
whereas $b=32,40,48,64$
Again, when the condition in the if at line 5 is false, we will count it as a
failure.
This algorithm will always call the original RNG to obtain $b$ bits,
regardless of the value of $n$. When the algorithm fails, it starts again and
again draws $b$ bits. This is inefficient in terms of entropy: for small
values of $n$ it will produce far less entropy than it consume. But, will it
be slower or faster than our previous algorithm? It turns out that the answer
pretty much depends on the speed of the back-end RNG (and this is
unsurprising); but also on how much time it takes to compute the basic
operations “integer multiplications” $q*n$ and “integer division” $\lfloor
N/n\rfloor$: we will see that, in some cases, these operations are so slow
that they defeat the efficiency of the algorithm Uniform random by bit
recycling.
## 6 Numerical tests
### 6.1 Architectures
The tests were performed in six different architectures,
(HW1)
_Intel® Core ™ 2 Duo CPU E7500 2.93GHz_ , in i686 mode,
(HW2)
_Intel® Core ™ 2 Duo CPU P7350 2.00GHz_ , in i686 mode,
(HW3)
_AMD Athlon™ 64 X2 Dual Core Processor 4200+_ , in x86_64 mode,
(HW4)
_AMD Athlon ™ 64 X2 Dual Core Processor 4800+_ , in x86_64 mode,
(HW5)
_Intel® Core ™ 2 Duo CPU P7350 2.00GHz_ , in x86_64 mode,
(HW6)
_Intel ® Xeon ® CPU 5160 3.00GHz_ , in x86_64 mode.
In the first five cases, the host was running a _Debian GNU/Linux_ or _Ubuntu_
O.S. , and the code was compiled using _gcc 4.4_ , with the optimization flags
` -march=native -O3 -finline-functions -fno-strict-aliasing -fomit-frame-
pointer -DNDEBUG ` .
In the last case, the O.S. was _Gentoo_ and the code was compiled using _gcc
4.0_ with flags
` -march=nocona -O3 -finline-functions -fno-strict-aliasing -fomit-frame-
pointer -DNDEBUG `.
### 6.2 Back-end PRNGs
To test the speed of the following algorithms, we used four different back-end
PRNGs.
(sfmt_sse)
The _SIMD oriented Fast Mersenne Twister(SFMT)_ ver. 1.3.3 by Mutsuo Saito and
Makoto Matsumoto [2] (compiled with SSE support)
(xorshift)
The _xorshift_ generator by G. Marsaglia [4]
(sfmt_sse_md5)
as sfmt_sse above, but moreover the output is cryptographically protected
using the MD5 algorithm
(bbs260)
The Blum-Blum-Shub algorithm [5], with two primes of size $\sim 130$bit.
The last two were home-made, as examples of slower but (possibly)
cryptographically strong RNG 666The author makes no guarantees, though, that
the implemented versions are really good and cryptographically strong RNGs —
we are interested only in their speeds.. All of the above were uniformized to
implement two functions, my_gen_rand32() and my_gen_rand64(), that return
(respectively) a 32bit or a 64bit unsigned integer, uniformly distributed. The
C code for all the above is in the appendix B.1. The speeds of the different
RNGs are listed in the tables in sec. A.1.
We also prepared a simple _counter_ “RNG” algorithm, that returns numbers that
are in arithmetic progression; since it is very simple, it is useful to assess
the overhead complexity in the testing code itself; this overhead is on the
order of 2 to to 4 ns, depending on the CPUs.
### 6.3 _Ad hoc_ functions
We implemented some _ad hoc_ functions, that are then used by the uniform RNGs
(that are described in the next section).
NextBit
returns a bit
Next2Bit
returns two bits
NextByte
returns 8 bits
NextWord
returns 16 bits
For any of the above, we prepared many variants, that internally call either
the my_gen_rand32() or my_gen_rand64() calls (see the C code in sec. B.2) and
then we benchmarked them in all architecture, to choose the faster one (that
is then used by the uniform RNGs). 777We had to make an exception for when the
back-end RNG is based on SFMT, since SFMT cannot mix 64bit and 32bit random
number generations: in that case, we forcibly used the 32bit versions (that in
most of our benchmarks are anyway slightly faster).
We also prepared a specific method (that is not used for the uniform RNGs):
NextCard
returns a number uniformly distributed in the range $0\ldots 51$ (it may be
thought of as a card randomly drawn from a deck of cards).
The detailed timings are in the tables in sec. A.1.
### 6.4 Uniform RNGs
We implemented nine different versions of uniform random generators. The C
code is in B.3; we here briefly describe the ideas. Four versions are based on
the “simple” generator in fig. 2:
uniform_random_simple32
uses 32bit variables internally, $N=2^{32}-1$, and consumes a 32bit random
number, (a call to my_gen_rand32()) each time
uniform_random_simple40
uses 64bit variables internally, $N=2^{40}$, and calls my_gen_rand32() and
NextByte() each time
uniform_random_simple48
uses 64bit variables internally, $N=2^{48}$, and calls my_gen_rand32() and
NextWord() each time
uniform_random_simple64
uses 64bit variables internally, $N=2^{64}-1$, and calls my_gen_rand64() each
time.
Then there are three versions based on the “bit recycling” generator in fig. 1
(all use 64bit variables internally):
uniform_random_by_bit_recycling
is the code in fig. 1 (but it refills the state by popping two bits at a time)
uniform_random_by_bit_recycling_faster
it refills the state by popping words, bytes and pairs of bits, for improved
efficiency
uniform_random_by_bit_recycling_cheating
as the “faster” one, but the _if/else_ block is not implemented, and the
modulus $r\mathop{\operator@font mod}\nolimits n$ is always returned; this is
not mathematically exact, but the probability that it is inexact is $\sim
2^{-30}$.
Moreover there are “mixed” methods
uniform_random_simple_recycler
uses 32bit variables internally, keeps an internal state that is sometimes
initialized but not refilled each time (so, it is useful only for small $n$),
uniform_random_by_bit_recycling_32
when $n<2^{29}$, it implements the “bit recycling” code using 32bit variables;
when $2^{29}\leq n<2^{32}$, it implements a “simple”–like method, using only
bit shifting.
We tested them in all of the architectures, for different values of the
modulus $n$, and graphed the results (see appendix A.2.1).
### 6.5 Timing
To benchmark the algorithms, we computed the process time using both the Posix
call clock() (that returns an approximation of processor time used by the
program) and the CPU TSC (that counts the number of CPU ticks). When
benchmarking one of the above back-end RNGs or the _ad hoc_ functions, we
called it in repeated loops of $2^{24}$ iterations each, repeating them for at
least 1 second of processor time; and then compared the data provided by TSC
and clock(). We also prepared a statistics of the values
missing$$$\mathtt{cycles\\_per\\_clock}:=\frac{\Delta\texttt{TSC}}{\Delta\texttt{clock()}}$$sothatwecouldconvertCPUcyclestonanoseconds;weverifiedthatthestandarddeviationofthelogarithmoftheabovequantitywasusuallylessthan$1%$.\par
Toavoidover-
optimizationofthecompiler,theresultsofanybenchmarkedfunctionwas\emph{xor}-edina\emph{bucket}variable,thatwasthenprintedonscreen.\par
Duringbenchmarking,wedisabledtheCPUpower-
savingfeatures,forcingtheCPUtobeatmaximumperformance(usingthe\texttt{cpufreq-
set}command)andalsowetiedtheprocesstoonecore(usingthe\texttt{taskset}command).\par
Unfortunatelythe\texttt{clock()}call,inGNU/Linuxsystems,hasatimeresolutionof$0.01sec$,soitwastoocoarsetobeusedforthegraphsinsection\ref{sec:graph_speeds}:forthosegraphs,onlythe\texttt{TSC}wasused(andthencycleswereconvertedtonanoseconds,usingtheaveragevalueof\texttt{cycles\\_per\\_clock}).\par$
### 6.6 Conclusions
While efficiency is exactly mathematically assessed, computational speed is a
more complex topic, and sometimes quite surprising. We report some
considerations.
1. 1.
In our Intel™ CPUs running in 32bit mode, integers divisions and remainder
computation using 64bit variables are quite slow: in HW1, each such operations
cost $\sim 15$ns.
2. 2.
Any bit recycling method that we could think of needs at least four arithmetic
operations for each result it produces; moreover there is some code to refill
the internal state.
3. 3.
In the same CPUs, the cost of _bit shifting_ or _xor_ operations are on the
order of 3 ns, even on 64bit variables; moreover the back-ends sfmt_sse and
xorshift can produce a 32bit random number in $\sim 5$ ns.
4. 4.
So, unsurprisingly, when the back end is sfmt_sse and xorshift, and the Intel™
CPU runs in 32bit mode, the fastest methods are the “simple32” and
“simple_recycler” methods, that run in $\sim 10$ns; and the bit recycling
methods are at least 5 times slower than those.
5. 5.
When the back-end is sfmt_sse and xorshift, but the the Intel™ CPU runs in
64bit mode, the fastest method are still the “simple32” and “simple_recycler”
methods; the bit recycling methods are twice slower.
6. 6.
When the back-ends RNGs are sfmt_sse or xorshift, in the AMD™ CPUs, the
“simple32” and “simple_recycler” take $\sim 40$ns; this is related to the fact
that 32bit division and remainder computation need $\sim 20$ns (as is shown in
sec. A.2.2). So these methods are much slower than the back-ends RNGs, that
return a 32bit random number in $\sim 6$ns.
One consequence is that, since the uniform_random_by_bit_recycling_32 for
$n>2^{29}$ uses a “simple”–like method with only bit shifting, then it is much
faster than the “simple32” and “simple_recycler”.
What we cannot explain is that, in the same architectures, the NextCard32
function, that implements the same type of operations, runs in $\sim 8$ns (!)
(We also tried to test the above with different optimizations. Using the
xorshift back-end, setting optimization flags to be just -O0, NextCard32
function takes $\sim 21$ns; setting it to -O, it takes $\sim 12$ns.)
7. 7.
The back-end RNGs sfmt_sse_md5 or bbs260, are instead much slower, that is,
sfmt_sse_md5 needs $\sim 300$ ns to produce a 32bit number, and bbs260 needs
$\sim 500$ ns when the CPU runs in 64bit mode and more than a microsecond (!)
in 32bit mode.
In this case, the bit recycling methods are usually faster. Their speed is
dominated by how many times the back-end RNGs is called, so it can be
estimated in terms of _entropy bitrate_ , and indeed the graphs are (almost)
linear (since the abscissa is in logarithmic scale).
8. 8.
One of the biggest surprises comes from the NextCard functions: there are four
implementations,
* –
using 64bit or 32bit variables;
* –
computing a result for each call, or precomputing them and storing in an array
(the “prefilled” versions).
The speed benchmarks give discordant results. When using the faster back-ends
sfmt_sse and xorshift, the “prefilled” versions are slower. When using the
slower back-ends sfmt_sse_md5 or bbs260, the 64bit “prefilled” version is the
fastest in Intel™ CPUs; but it is instead much slower than the “non prefilled”
version in AMD™ CPUs. It is possible that the cache misses are playing a rôle
in this, but we cannot provide a good explanation.
9. 9.
Curiously, in some Intel™ CPUs, the time needed for an integer arithmetic
operation depends also on the _values_ of the operands (and not only on the
bit sizes of the variable)! See the graph in sec. A.2.2. So, the speed of the
functions depend on the value of the modulus $n$. This is the reason why some
the graphs are all oscillating in nature.
In particular, when we looked at the graphs for _Core 2_ architectures in
32bit mode, by looking at the graphs of the functions simple_40, simple_48
(where $N=2^{40},2^{48}$ constant) we noted that the operations
$q:=N/n,qn:=n*q$ are $\sim 10$ ns slower when $n<N2^{-32}$ than when
$n>N2^{-32}$. This is similar to what is seen in the graphs in sec. A.2.2.
Instead the speed graphs in AMD™ CPUs are almost linear, and this is well
explained by the average number of needed operations.
Summarizing, the speeds are quite difficult to predict; if a uniform random
generator is to be used for $n$ in a certain range, and the back-end RNG takes
approximatively as much time as 4 integer operations in 64bits, then the only
sure way to decide which algorithm is the fastest one is by benchmarking. If a
a uniform random generator is to be used for a constant and specific $n$ (such
as in the case of the NextCard function), there may be different strategies to
implement it, and again the only sure way to decide which algorithm is the
fastest one is by benchmarking.
## Appendix A Test results
### A.1 Speed of back-end RNGs and _Ad hoc_ functions
These tables list the average time (in nanoseconds) of the back-end RNGs and
the _ad hoc_ functions (see the C code in sec. B.2); these same data are
plotted as red crosses in the plots of the next section. For each family, the
fastest function is marked blue; competitors that differ less than $10\%$ are
italic and blue; competitors that are slower more than $50\%$ are red.
| sfmt_sse | xorshift
---|---|---
| HW1 | HW2 | HW3 | HW4 | HW5 | HW6 | HW1 | HW2 | HW3 | HW4 | HW5 | HW6
Next2Bit32 | 2.6 | 3.9 | 4.9 | _4.8_ | _3.9_ | _2.6_ | 2.5 | 3.7 | _4.7_ | _4.2_ | 3.7 | 2.8
Next2Bit64 | 4.2 | 4.7 | _5.0_ | 4.4 | 3.7 | 2.5 | 3.3 | 6.3 | 4.5 | 4.0 | 3.7 | 2.4
NextBit32 | 2.5 | 3.7 | 4.6 | 4.2 | _3.7_ | _2.8_ | 2.5 | 3.6 | _4.4_ | _3.9_ | 3.6 | 2.4
NextBit32_by_mask | 2.5 | 3.7 | _4.9_ | _4.4_ | _3.7_ | _2.8_ | 2.5 | 3.6 | _4.4_ | _3.9_ | 3.6 | 2.7
NextBit64 | 3.2 | 4.6 | _4.7_ | 4.2 | 3.6 | 2.8 | 3.2 | 4.7 | 4.3 | 3.8 | 3.6 | 2.4
NextByte32 | 3.5 | 5.1 | 6.1 | 5.4 | 5.1 | 3.8 | _3.0_ | _4.4_ | 5.1 | 4.5 | _4.5_ | 3.3
NextByte64 | _3.3_ | 5.6 | _6.5_ | _5.8_ | 4.4 | 3.1 | _3.0_ | _4.4_ | 6.2 | 5.2 | 4.3 | 2.8
NextByte64_prefilled | 3.1 | 4.6 | _6.3_ | _5.6_ | _4.5_ | 3.4 | 2.9 | 4.3 | 5.8 | 6.0 | _4.6_ | _3.0_
NextCard32 | 3.5 | 5.1 | 8.0 | 7.1 | 5.5 | 7.0 | 3.4 | 4.9 | 5.6 | 5.0 | 5.2 | 6.9
NextCard32_prefilled | 6.1 | 9.0 | 24.4 | 21.7 | 11.4 | 7.1 | 6.0 | 15.6 | 22.2 | 19.7 | 11.1 | 7.1
NextCard64 | 22.0 | 32.2 | 6.5 | 5.8 | 7.0 | 4.9 | 22.4 | 32.8 | 6.6 | 5.8 | 7.0 | 4.9
NextCard64_prefilled | 22.0 | 32.2 | 37.9 | 33.5 | 20.5 | _5.4_ | 24.0 | 37.0 | 37.4 | 33.1 | 20.4 | _5.3_
NextWord32 | 4.2 | 6.2 | 7.5 | 6.7 | 6.5 | 4.5 | 3.4 | 5.0 | 5.7 | 5.0 | _5.0_ | 3.3
NextWord64 | 4.2 | _6.5_ | 6.5 | 5.8 | 5.2 | 3.7 | 4.2 | 5.9 | 5.7 | 5.0 | 4.9 | _3.4_
my_gen_rand32 | 3.7 | 5.4 | 6.5 | 5.7 | 5.4 | 3.9 | 3.5 | 5.1 | 5.0 | 4.4 | 5.4 | 3.5
my_gen_rand64 | 4.2 | 6.3 | 8.4 | 7.4 | 6.3 | 5.2 | 4.5 | 6.9 | 6.8 | 6.0 | 7.4 | 4.8
| sfmt_sse_md5 | bbs260
---|---|---
| HW1 | HW2 | HW3 | HW4 | HW5 | HW6 | HW1 | HW2 | HW3 | HW4 | HW5 | HW6
Next2Bit32 | 18.7 | 27.5 | 30.2 | 26.7 | 32.0 | 23.1 | 55.2 | 81.0 | 32.9 | 31.4 | 36.3 | 28.6
Next2Bit64 | 11.6 | 17.0 | 17.6 | 15.6 | 18.1 | 12.9 | 35.6 | 52.2 | 18.9 | 17.5 | 20.0 | 15.2
NextBit32 | 10.5 | 15.5 | 17.2 | 15.2 | 17.8 | 12.6 | 28.7 | _42.2_ | 19.4 | 16.4 | 30.4 | 15.3
NextBit32_by_mask | 10.3 | 15.2 | 17.6 | 15.6 | 18.2 | 12.9 | 28.7 | _42.4_ | 18.9 | 16.8 | 25.0 | 15.5
NextBit64 | 7.5 | 10.9 | 11.1 | 9.8 | 11.1 | 7.8 | 19.7 | 41.5 | 11.7 | 10.4 | 11.8 | 9.8
NextByte32 | 67.6 | 99.5 | 108.1 | 95.7 | 117.9 | 84.8 | 212.5 | 315.5 | 118.2 | 113.7 | 136.3 | 102.7
NextByte64 | _36.2_ | _53.5_ | 57.1 | 50.5 | 60.5 | 43.8 | 130.6 | 192.9 | 62.4 | _59.7_ | 69.2 | _56.2_
NextByte64_prefilled | 35.7 | 52.5 | 56.9 | 50.3 | 60.6 | 43.7 | 131.0 | 192.7 | 62.6 | 56.4 | 69.7 | 54.1
NextCard32 | 56.2 | 83.3 | 88.1 | 77.9 | 96.5 | 71.7 | 173.9 | 255.6 | 97.9 | 85.9 | 110.4 | 86.4
NextCard32_prefilled | 60.7 | 89.7 | 104.3 | 92.2 | 102.9 | _37.6_ | 176.3 | 259.6 | 115.6 | 106.3 | 117.5 | 45.3
NextCard64 | 46.4 | 68.1 | 43.5 | 38.4 | 49.1 | 34.8 | 119.0 | 193.4 | 48.5 | 43.1 | 55.3 | 40.7
NextCard64_prefilled | 46.0 | 67.5 | 74.7 | 66.0 | 62.0 | _35.2_ | 118.7 | 173.6 | 79.2 | 72.9 | 68.1 | _43.8_
NextWord32 | 132.7 | 195.7 | 209.2 | 185.4 | 228.3 | 166.7 | 429.4 | 633.0 | 236.0 | 204.2 | 265.3 | 202.9
NextWord64 | 69.3 | 102.8 | 108.1 | 95.6 | 117.8 | 84.5 | 264.2 | 387.4 | 117.9 | 105.6 | 136.2 | 108.7
my_gen_rand32 | 262.5 | 391.5 | 413.3 | 366.7 | 457.3 | 328.7 | 859.8 | 1263.3 | _459.2_ | 403.1 | 525.4 | 399.5
my_gen_rand64 | 263.0 | 390.4 | 413.1 | 368.4 | 457.7 | 328.4 | 1018.1 | 1513.1 | 454.3 | 402.9 | 523.9 | 398.5
| counter | | | | | |
---|---|---|---|---|---|---|---
| HW1 | HW2 | HW3 | HW4 | HW5 | HW6 | | | | | |
Next2Bit32 | 2.5 | 3.6 | _4.6_ | _4.1_ | _3.6_ | _2.4_ | | | | | |
Next2Bit64 | 3.4 | 4.9 | 4.4 | 3.9 | 3.6 | 2.4 | | | | | |
NextBit32 | 2.4 | 3.6 | _4.3_ | _3.8_ | 3.6 | 2.4 | | | | | |
NextBit32_by_mask | 2.4 | 3.6 | _4.4_ | _3.9_ | 3.6 | 2.4 | | | | | |
NextBit64 | 3.1 | 4.6 | 4.2 | 3.7 | 3.6 | 2.7 | | | | | |
NextByte32 | _2.9_ | 4.3 | 4.8 | 4.2 | 4.3 | 2.8 | | | | | |
NextByte64 | _2.9_ | 4.3 | 5.3 | 4.7 | _3.8_ | _2.9_ | | | | | |
NextByte64_prefilled | 2.8 | 3.8 | _5.2_ | _4.6_ | 3.6 | _3.0_ | | | | | |
NextCard32 | 2.8 | 4.2 | 5.1 | 4.5 | 4.5 | 6.4 | | | | | |
NextCard32_prefilled | 5.2 | 7.7 | 21.9 | 19.4 | 8.9 | 6.9 | | | | | |
NextCard64 | 22.0 | 32.7 | 6.0 | 5.4 | 6.0 | 4.2 | | | | | |
NextCard64_prefilled | 21.7 | 31.9 | 36.9 | 32.7 | 19.7 | 4.9 | | | | | |
NextWord32 | 2.9 | 4.3 | _5.0_ | _4.4_ | 4.8 | 3.2 | | | | | |
NextWord64 | 3.3 | 4.9 | 4.6 | 4.1 | 4.0 | 2.7 | | | | | |
my_gen_rand32 | 2.0 | 3.0 | 3.6 | 3.2 | 3.5 | 2.0 | | | | | |
my_gen_rand64 | 2.4 | 3.5 | 3.6 | 3.2 | 3.5 | 2.3 | | | | | |
### A.2 Graphs
In all of the following graphs, the abscissa is $n$, (that is the modulus of
the uniform RNGs); the abscissa is in log-scale (precisely, it contains all
$n$ from 2 to 32, and then $n$ is incremented by $\lfloor n/32\rfloor$ up to
$2^{32}$, for a total of 733 samples). The number in parentheses near the
graph labels are the average time for call (in nanoseconds; averaged in the
aforementioned log scale).
#### A.2.1 Uniform random generators
To reduce the size of the labels, we abbreviated
uniform_random_by_bit_recycling as bbr, and uniform_random_simple as simple.
#### SFMT
#### xorshift
#### SFMT + MD5
#### bbs260
#### A.2.2 Integer arithmetic
typedef uint64_t st;
st div32(uint32_t n) {
return my_gen_rand32() / n ;
}
st div32_24(uint32_t n) {
return (my_gen_rand32() & 0xFF000000) / n ;
}
st div48(uint32_t n) {
uint64_t r = my_gen_rand32(), m = r << 16;
return m / n ;
}
st div64(uint32_t n) {
uint64_t m = my_gen_rand64() / n ;
return m;
}
st mod32(uint32_t n) {
return my_gen_rand32() % n ;
}
st mod32_24(uint32_t n) {
return (my_gen_rand32() & 0xFF000000) % n ;
}
st mod48(uint32_t n) {
uint64_t r = my_gen_rand32(), m = r << 16;
return m % n ;
}
st mod64(uint32_t n) {
uint64_t m =my_gen_rand64() % n ;
return m;
}
st prod32(uint32_t n) {
return my_gen_rand32() * n ;
}
st prod32_24(uint32_t n) {
return (my_gen_rand32() & 0xFF) * n ;
}
st prod48(uint32_t n) {
uint64_t r = my_gen_rand32(), m = r << 16;
return m * n ;
}
st prod64(uint32_t n) {
uint64_t m = my_gen_rand64() * n ;
return m;
}
## Appendix B Code
The complete C code is available on request. The code that we wrote is
licensed according to the _Gnu Public License v2.0_. (The _SFMT_ code is
licensed according to a modified BSD license — they are considered to be
compatible).
In the following code, the macro COUNTBITS was used in testing efficiency; and
was disabled while testing speeds.
### B.1 Back-end RNGs
//
//to compile this code, define RNG, by using ’gcc .... -DRNG=n’, where n is
// 1 -> SFMT , by M. Saito and M. Matsumoto
// 2 -> SFMT + md5
// 3 -> xorshift , by Marsaglia
// 4 -> Blum Blum Shub with ~128bit (product of two ~31bit primes) (only on amd64, using gcc 128 int types)
// 5 -> Blum Blum Shub with ~260bit modulus (product of two ~130bit primes)
//disclaimer: methods 2,4,5 are not guaranteed to generate high quality pseudonumbers; they were used
// only to test the code speed
/*************** SFMT ***/
#if 1==RNG
#ifdef HAVE_SSE2
char *RNGNAME="SFMT (sse)";
char *RNGNICK="sfmt_sse";
#else
char *RNGNAME="SFMT";
char *RNGNICK="sfmt";
#endif
#include "SFMT.h"
uint32_t my_gen_rand32()
{
uint32_t r=gen_rand32();
COUNTBITS(32);
return r;
}
uint64_t my_gen_rand64()
{
uint64_t r=gen_rand64();
COUNTBITS(64);
return r;
}
void my_init_gen_rand(uint32_t seed)
{
init_gen_rand(seed);
}
/*************** SFMT + md5 ***/
#elif 2==RNG
#ifdef HAVE_SSE2
char *RNGNAME="SFMT (sse) + md5";
char *RNGNICK="sfmt_sse_md5";
#else
char *RNGNAME="SFMT + md5";
char *RNGNICK="sfmt_md5";
#endif
#include "SFMT.h"
#include "md5.h"
uint32_t my_gen_rand32()
{
uint32_t I[8];
for(int i=0;i<8;i++) I[i]=gen_rand32();
unsigned char *UCp, output[16];
UCp=(unsigned char*)I;
md5(UCp, 32, output);
uint32_t *U32p=(uint32_t *)output, r=*U32p;
COUNTBITS(32);
return r;
}
uint64_t my_gen_rand64()
{
uint32_t I[8];
for(int i=0;i<8;i++) I[i]=gen_rand32();
unsigned char *UCp, output[16];
UCp=(unsigned char*)I;
md5(UCp, 32, output);
uint64_t *U64p=(uint64_t *)output, r=*U64p;
COUNTBITS(64);
return r;
}
void my_init_gen_rand(uint32_t seed)
{
if(md5_self_test(0)) exit(4);
init_gen_rand(seed);
}
/*************** xorshift , by Marsaglia***/
#elif 3==RNG
char *RNGNAME="xorshift";
char *RNGNICK="xorshift";
static uint32_t __xorshift__x = 123456789;
static uint32_t __xorshift__y = 362436069;
static uint32_t __xorshift__z = 521288629;
static uint32_t __xorshift__w = 88675123;
uint32_t my_gen_rand32(void) {
uint32_t t;
t = __xorshift__x ^ (__xorshift__x << 11);
__xorshift__x = __xorshift__y; __xorshift__y = __xorshift__z; __xorshift__z = __xorshift__w;
COUNTBITS(32);
return __xorshift__w = __xorshift__w ^ (__xorshift__w >> 19) ^ (t ^ (t >> 8));
}
uint64_t my_gen_rand64()
{
uint64_t a=my_gen_rand32(), r = (a << 32) | my_gen_rand32();
return r;
}
void my_init_gen_rand(uint32_t seed)
{
__xorshift__x = 123456789 ^ seed;
__xorshift__y = 362436069;
__xorshift__z = 521288629;
__xorshift__w = 88675123;
}
/******************** Blum Blum Shub (needs amd64 arch) *******************/
#elif 4==RNG
char *RNGNAME="Blum Blum Shub (64bit)";
char *RNGNICK="bbs64";
uint64_t my_seed=0x987fed5;
typedef __uint128_t uint128_t;
uint32_t my_gen_rand32(void) {
const uint64_t p = 4222234259UL; //~ 2^31.9
const uint64_t q = 4222231271UL;
const uint64_t M = p * q ;
const uint64_t mask = 0xFFFFFFFFFFFFFFFFUL;
uint128_t a = my_seed, b = a * a , my_seed = b % M;
my_seed = c & mask ;
COUNTBITS(32);
return my_seed & 0xFFFFFFFFF;
}
uint64_t my_gen_rand64()
{
uint64_t a=my_gen_rand32(), r = (a << 32) | my_gen_rand32();
return r;
}
void my_init_gen_rand(uint32_t seed)
{
my_seed=seed ^ 0x987fed5 ;
}
/***********************************************************/
/******************** Blum Blum Shub *******************/
#elif RNG==5
char *RNGNAME="Blum Blum Shub (260bit)";
char *RNGNICK="bbs260";
#include "gmp.h"
mpz_t bbs__pq__ , bbs__n__, bbs__64__, bbs ;
int initialized = 0;
void my_init_gen_rand(uint32_t seed)
{
if(!initialized) {
//http://primes.utm.edu/lists/small/small.html
mpz_t p; mpz_init_set_str (p, "3615415881585117908550243505309785526231", 10);
assert(mpz_probab_prime_p(p,12));
mpz_t q; mpz_init_set_str (q, "5570373270183181665098052481109678989411", 10);
assert(mpz_probab_prime_p(q,12));
mpz_mul(bbs__pq__ ,p ,q);
mpz_clear(p); mpz_clear(q);
mpz_init(bbs__n__);
initialized=1;
}
unsigned long s=seed+0x100000000;
mpz_set_ui(bbs__n__, s);
}
void bbs_step()
{
mpz_t sqr; mpz_init(sqr);
mpz_mul(sqr, bbs__n__, bbs__n__);
mpz_tdiv_r(bbs__n__, sqr, bbs__pq__);
mpz_clear(sqr);
}
uint32_t my_gen_rand32()
{
bbs_step();
unsigned long int r=mpz_get_ui(bbs__n__);
COUNTBITS(32);
if( sizeof(unsigned long int) == 4) {
return r;
} else {
uint32_t rr=r & 0xFFFFFFFF;
return rr;
}
}
uint64_t my_gen_rand64()
{
bbs_step();
unsigned long int r=mpz_get_ui(bbs__n__);
COUNTBITS(64);
if ( sizeof(unsigned long int) == 8 ) {
return r;
} else {
mpz_t q; mpz_init(q); mpz_tdiv_q_2exp (q, bbs__n__, 32);
uint64_t r2=mpz_get_ui(q);
mpz_clear(q);
uint64_t rr=r | (r2 << 32);
return rr;
}
}
/*************** a counter , to test speeds ***/
#elif 11==RNG
char *RNGNAME="counter";
char *RNGNICK="counter";
static uint32_t my_seed32=0;
static uint64_t my_seed64=0;
uint32_t my_gen_rand32(void) {
COUNTBITS(32);
return my_seed32 += 0x4c1;
}
uint64_t my_gen_rand64()
{
COUNTBITS(64);
return my_seed64 += 0x4c7000004c1;
}
void my_init_gen_rand(uint32_t seed)
{
my_seed32=seed;
my_seed64=seed;
}
#else
#error "please define RNG"
#endif
//
### B.2 _ad hoc_ functions
//
unsigned int NextByte32() {
static int l=0;
static uint32_t R=0;
if(unlikely(l<=0)) {
R=my_gen_rand32();
l=4;
}
unsigned int byte=R&255;
l--;
if(l) R>>=8;
return byte;
}
unsigned int NextByte64() {
static int l=0;
static uint64_t R=0;
if(unlikely(l<=0)) {
R=my_gen_rand64();
l=8;
}
unsigned int byte=R & 255;
l--;
if(l) R>>=8;
return byte;
}
unsigned int __saved_bytes[8];
unsigned int NextByte64_prefilled() {
static int l=0;
if(unlikely(l<=0)) {
uint64_t R=my_gen_rand64();
__saved_bytes[0] = R & 255;
R >>= 8;
__saved_bytes[1] = R & 255;
R >>= 8;
__saved_bytes[2] = R & 255;
R >>= 8;
__saved_bytes[3] = R & 255;
R >>= 8;
__saved_bytes[4] = R & 255;
R >>= 8;
__saved_bytes[5] = R & 255;
R >>= 8;
__saved_bytes[6] = R & 255;
R >>= 8;
__saved_bytes[7] = R & 255;
l=8;
}
l--;
return __saved_bytes[l];
}
unsigned int NextWord32() {
static int l=0;
static uint32_t R=0;
if(l<=0) {
R=my_gen_rand32();
l=2;
}
unsigned int bytes=R & 0xFFFF;
R>>=16;
l--;
return bytes;
}
unsigned int NextWord64() {
static int l=0;
static uint64_t R=0;
if(unlikely(l<=0)) {
R=my_gen_rand64();
l=4;
}
unsigned int bytes=R & 0xFFFF;
R>>=16;
l--;
return bytes;
}
unsigned int Next2Bit32() {
static int l=0;
static uint32_t R=0;
if(unlikely(l<=0)) {
R=my_gen_rand32();
l=16;
}
unsigned int bit=R&3;
R>>=2;
l--;
return bit;
}
unsigned int Next2Bit64() {
static int l=0;
static uint64_t R=0;
if(unlikely(l<=0)) {
R=my_gen_rand64();
l=32;
}
unsigned int bit=R&3;
R>>=2;
l--;
return bit;
}
const static uint32_t bitmasks[32] = {
0x1, 0x2, 0x4, 0x8 , 0x10, 0x20, 0x40, 0x80,
0x100, 0x200, 0x400, 0x800, 0x1000, 0x2000, 0x4000, 0x8000,
0x10000, 0x20000, 0x40000, 0x80000, 0x100000, 0x200000, 0x400000, 0x800000,
0x1000000, 0x2000000, 0x4000000, 0x8000000, 0x10000000, 0x20000000, 0x40000000, 0x80000000
};
unsigned int NextBit32_by_mask() {
static int l=0;
static uint32_t R=0;
if(unlikely(l<=0)) {
R=my_gen_rand32();
l=32;
}
l--;
return (R & bitmasks[l]) ? 1 : 0;
}
unsigned int NextBit32() {
static int l=0;
static uint32_t R=0;
if(unlikely(l<=0)) {
R=my_gen_rand32();
l=32;
}
unsigned int bit=R&1;
R>>=1;
l--;
return bit;
}
unsigned int NextBit64() {
static int l=0;
static uint64_t R=0;
if(unlikely(l<=0)) {
R=my_gen_rand64();
l=64;
}
unsigned int bit=R&1;
R>>=1;
l--;
return bit;
}
//draws one card from a deck of 52 cards, using 32bit variables
unsigned int NextCard32() {
static int l=0;
static uint32_t R=0;
if(unlikely(l<=0)) {
R=my_gen_rand32();
l=5;
}
l--;
unsigned int c = R % 52;
R /= 52;
return c;
}
//draws one card from a deck of 52 cards, using 64 bit variables
unsigned int NextCard64()
{
static int l=0;
static uint64_t R=0;
if(unlikely(l<=0)) {
R=my_gen_rand64();
l=11;
}
l--;
unsigned int c = R % 52;
R /= 52;
return c;
}
//draws one card from a deck of 52 cards, using 32 bit variables
//and prefilling an array in memory
unsigned char __32saved_cards[5];
unsigned int NextCard32_prefilled() {
static int l=0;
if(unlikely(l<=0)) {
uint32_t R=my_gen_rand32(); //52**5 < 2**32
__32saved_cards[0] = R % 52;
R /= 52;
__32saved_cards[1] = R % 52;
R /= 52;
__32saved_cards[2] = R % 52;
R /= 52;
__32saved_cards[3] = R % 52;
R /= 52;
__32saved_cards[4] = R % 52;
l=5;
}
l--;
return __32saved_cards[l];
}
//draws one card from a deck of 52 cards, using 64 bit variables
//and prefilling an array in memory
unsigned char __64saved_cards[11];
unsigned int NextCard64_prefilled() {
static int l=0;
if(unlikely(l<=0)) {
uint64_t R=my_gen_rand64(); //52**11 < 2**64
__64saved_cards[0] = R % 52;
R /= 52;
__64saved_cards[1] = R % 52;
R /= 52;
__64saved_cards[2] = R % 52;
R /= 52;
__64saved_cards[3] = R % 52;
R /= 52;
__64saved_cards[4] = R % 52;
R /= 52;
__64saved_cards[5] = R % 52;
R /= 52;
__64saved_cards[6] = R % 52;
R /= 52;
__64saved_cards[7] = R % 52;
R /= 52;
__64saved_cards[8] = R % 52;
R /= 52;
__64saved_cards[9] = R % 52;
R /= 52;
__64saved_cards[10] = R % 52;
l=11;
}
l--;
return __64saved_cards[l];
} //
### B.3 Uniform random generators
In the following code, the macro COUNTFAILURES was used in testing efficiency;
and was disabled while testing speeds.
//
/********** uniform_random_by_bit_recycling
this implements the pseudocode in page 5
(but pops 2 bits at a time)
*******************/
uint32_t uniform_random_by_bit_recycling(uint32_t n)
{
static uint64_t m = 1, r = 0;
const uint64_t N62=((uint64_t)1)<<62;
while(1) {
while(m<N62) { //fill the state
r = (r<<2) | Next2Bit();
m = m<<2;
}
const uint64_t q=m/n, nq = n * q;
if( likely(r < nq) ) {
uint32_t d = r % n; //remainder, is a random variable of modulus n
r = r/n; //quotient, is random variable of modulus q
m = q;
return d;
} else {
COUNTFAILURES();
m = m - nq;
r = r - nq; // r is still a random variable of modulus m
}
}
}
/********** uniform_random_by_bit_recycling
this implements the pseudocode in page 5 ,
but pops words,bytes,and pairs of bits
*******************/
uint32_t uniform_random_by_bit_recycling_faster(uint32_t n)
{
static uint64_t m = 1, r = 0;
const uint64_t N62=((uint64_t)1)<<62, N56=((uint64_t)1)<<56, N48=((uint64_t)1)<<48;
while(1) {
//fill the state
if(m<N48) {
r = (r<<16) | NextWord();
m = m<<16;
}
while(m<N56) {
r = (r<<8) | NextByte();
m = m<<8;
}
while(m<N62) {
r = (r<<2) | Next2Bit();
m = m<<2;
}
const uint64_t q=m/n, nq=n*q;
if( likely(r < nq) ) {
uint32_t d = r % n; //remainder, is a random variable of modulus n
r = r/n; //quotient, is random variable of modulus q
m = q;
return d;
} else {
COUNTFAILURES();
m = m - nq;
r = r - nq; // r is still a random variable of modulus m
}
}
}
/********** uniform_random_by_bit_recycling_cheating
this implements the pseudocode in page 6 ,
but does not implement the "else" block, so it is not
mathematically perfect; at the same time, since
the probability of the "else" block would be less than 1/2^24
the random numbers generated by this function are good enough
for most purposes
*******************/
uint32_t uniform_random_by_bit_recycling_cheating(uint32_t n)
{
static uint64_t m = 1, r = 0;
const uint64_t N48=((uint64_t)1)<<48, N56=((uint64_t)1)<<56;
if(m<N48) { //fill the state
r = (r<<16) | NextWord();
m = m<<16;
}
while(m<N56) {
r = (r<<8) | NextByte();
m = m<<8;
}
uint32_t d = r % n;
r = r/n;
m = m/n;
return d;
}
/********** uniform_random_by_bit_recycling32
this implements the pseudocode in page5
but only using 32bit variables, so it has some special
methods when n>=2^29
*******************/
uint32_t uniform_random_by_bit_recycling32(uint32_t n)
{
const uint32_t N29=((uint32_t)1)<<29, N30=((uint32_t)1)<<30, N31=((uint32_t)1)<<31, N24=((uint32_t)1)<<24;
//special methods
if(n>=N31) {
while(1) {
uint32_t r = my_gen_rand32();
if( likely(r < n) ) {
return r;
} else {
COUNTFAILURES();
}
}}
if(n>=N30) {
while(1) {
uint32_t r = my_gen_rand32() >> 1;
if( likely(r < n) ) {
return r;
} else {
COUNTFAILURES();
}
}}
if(n>=N29) {
while(1) {
uint32_t r = my_gen_rand32() >> 2;
if( likely(r < n) ) {
return r;
} else {
COUNTFAILURES();
}
}}
//usual bit recycling
static uint32_t m = 1, r = 0;
while(1) {
while(m<N24) { //fill the state
r = (r<<8) | NextByte();
m = m<<8;
}
while(m<N30) { //fill the state
r = (r<<2) | Next2Bit();
m = m<<2;
}
uint32_t q=m/n, nq=q*n;
if(likely( r < nq) ) {
uint32_t d = r % n; //remainder, is a random variable of modulus n
r = r/n; //quotient, is random variable of modulus q
m = q;
return d;
} else {
COUNTFAILURES();
m = m - nq;
r = r - nq; // r is still a random variable of modulus m
}
}
}
/********** uniform_random_simple_64
this is a simple implementation, found in many random number libraries
this version uses 64 bit variables
*******************/
uint32_t uniform_random_simple_64(uint32_t n)
{
const uint64_t N=0xFFFFFFFFFFFFFFFFU; //2^64-1;
uint64_t q = N/n, nq=((uint64_t)n) * q;
while(1) {
uint64_t r = my_gen_rand64();
if( likely(r < nq) ) {
uint32_t d = r % n; //remainder, is a random variable of modulus n
return d;
} else {
COUNTFAILURES();
}
}
}
/********** uniform_random_simple
this is a simple implementation, found in many random number libraries
this version uses 32 bit variables
*******************/
uint32_t uniform_random_simple_32(uint32_t n)
{
const uint32_t N=0xFFFFFFFFU; // 2^32-1;
uint32_t q = N/n;
while(1) {
uint32_t r = my_gen_rand32();
if( likely(r < n * q)) {
uint32_t d = r % n; //remainder, is a random variable of modulus n
return d;
} else {
COUNTFAILURES();
}
}
}
/********** uniform_random_simple_recycler
this is a simple implementation, with recycling for small n
this version uses 32 bit variables
*******************/
uint32_t uniform_random_simple_recycler(uint32_t n)
{
static uint32_t _r=0, _m=0;
const uint32_t N=0xFFFFFFFFU; //2^32-1
if (_m>n) {
uint32_t d = _r % n;
_m/=n;
_r/=n;
return d;
}
const uint32_t q = N/n, nq=n*q;
while(1) {
uint32_t newr = my_gen_rand32();
if(newr < nq) {
uint32_t d = newr % n; //newr is a random variable of modulus n
if(_m<q) { //there is more entropy in newr than in _r
_m=q;
_r=newr/n;
}
return d;
} else {
COUNTFAILURES();
}
}
}
/********** uniform_random_simple
some alternative versions, using 64 bit variables
*******************/
uint32_t uniform_random_simple_40(uint32_t n)
{
const uint64_t N=((uint64_t)1)<<40;
uint64_t q = N/n, nq=((uint64_t)n) * q;
while(1) {
uint64_t r = my_gen_rand32();
r=(r<<8) | NextByte(); //create 40 bits random numbers
if( likely(r < nq) ) {
uint32_t d = r % n; //remainder, is a random variable of modulus n
return d;
} else {
COUNTFAILURES();
}
}
}
uint32_t uniform_random_simple_48(uint32_t n)
{
const uint64_t N=((uint64_t)1)<<48;
uint64_t q = N/n, nq=((uint64_t)n) * q;
while(1) {
uint64_t r = my_gen_rand32();
r=(r<<16) | NextWord(); //create 48 bits random numbers
if( likely(r < nq) ) {
uint32_t d = r % n; //remainder, is a random variable of modulus n
return d;
} else {
COUNTFAILURES();
}
}
}
//
## Acknowledgments
The author thanks Professors S. Marmi and A. Profeti for allowing access to
hardware.
## Bibliography
## References
* [1] “Doctor Jacques” at the Math forum http://mathforum.org/library/drmath/view/65653.html (2004)
* [2] Mutsuo Saito and Makoto Matsumoto, _SIMD oriented Fast Mersenne Twister(SFMT): a 128-bit Pseudorandom Number Generator_ Monte Carlo and Quasi-Monte Carlo Methods 2006, Springer, 2008, pp. 607 – 622. DOI 10.1007/978-3-540-74496-2_36. Code (ver. 1.3.3 ) available from http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/SFMT/index.html (See Mutsuo Saito’s Master’s Thesis for more detailed information).
* [3] Mutsuo Saito and Makoto Matsumoto, _"A PRNG Specialized in Double Precision Floating Point Number Using an Affine Transition"_ , Monte Carlo and Quasi-Monte Carlo Methods 2008, Springer, 2009, pp. 589 – 602. DOI 10.1007/978-3-642-04107-5_38
* [4] George Marsaglia _Xorshift RNGs_ Journal of Statistical Software, 8, 1-9 (2003). http://www.jstatsoft.org/v08/i14/.
* [5] L. Blum, M. Blum, and M. Shub, _A Simple Unpredictable Pseudo-Random Number Generator_ SIAM J. Comput. 15, 364 (1986), DOI 10.1137/0215025
|
arxiv-papers
| 2010-12-20T11:27:57 |
2024-09-04T02:49:15.837560
|
{
"license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/",
"authors": "Andrea C. G. Mennucci",
"submitter": "Andrea Carlo Giuseppe Mennucci",
"url": "https://arxiv.org/abs/1012.4290"
}
|
1012.4292
|
# Infall and outflow detections in a massive core JCMT 18354-0649S
Tie Liu11affiliation: Department of Astronomy, Peking University, 100871,
Beijing China; liutiepku@gmail.com, ywu@pku.edu.cn , Yuefang Wu11affiliation:
Department of Astronomy, Peking University, 100871, Beijing China;
liutiepku@gmail.com, ywu@pku.edu.cn , Qizhou Zhang22affiliation: Harvard-
Smithsonian Center for Astronomy, 60 Garden St., Cambridge, MA 02138, USA ,
Zhiyuan Ren11affiliation: Department of Astronomy, Peking University, 100871,
Beijing China; liutiepku@gmail.com, ywu@pku.edu.cn , Xin Guan11affiliation:
Department of Astronomy, Peking University, 100871, Beijing China;
liutiepku@gmail.com, ywu@pku.edu.cn 33affiliation: Current affiliation: I.
Physikal. Institut, Universität zu Köln, Zülpicher St.77,D-50937 Köln, Germany
and Ming Zhu44affiliation: National Astronomical Observatories, Chinese
Academy of Sciences, Beijing, 100012
###### Abstract
We present a high-resolution study of a massive dense core JCMT 18354-0649S
with the Submillimeter Array. The core is mapped with continuum emission at
1.3 mm, and molecular lines including CH3OH ($5_{23}$–$4_{13}$) and HCN (3–2).
The dust core detected in the compact configuration has a mass of
$47~{}M_{\odot}$ and a diameter of $2\arcsec$ (0.06 pc), which is further
resolved into three condensations with a total mass of $42~{}M_{\odot}$ under
higher spatial resolution. The HCN (3–2) line exhibits asymmetric profile
consistent with infall signature. The infall rate is estimated to be
$2.0\times 10^{-3}~{}M_{\odot}\cdot$yr-1. The high velocity HCN (3-2) line
wings present an outflow with three lobes. Their total mass is
$12~{}M_{\odot}$ and total momentum is $121~{}M_{\odot}\cdot$km s-1,
respectively. Analysis shows that the N-bearing molecules especially HCN can
trace both inflow and outflow.
Massive core:pre-main sequence-ISM: molecular-ISM: kinematics and dynamics-
ISM: jets and outflows-stars: formation
††slugcomment: Accepted by APJ
## 1 Introduction
Studies of high-mass star formation have received much attention during recent
years. One of the main questions is whether massive stars form through an
accretion-disk-outflow process, similar to low-mass counterparts (Shu, Adams,
& Lizano 1987), or via collision-coalescence (Wolfire & Cassinelli 1987;
Bonnell, Bate, & Zinnecker. 1998). Studying the characteristics of massive
cores at the early stages is critical for understanding their formation
process. High-Mass Protostellar Objects (HMPOs) are precursors of UC Hii
regions, and represent an essential phase in high-mass star formation
(Churchwell 2002). HMPOs often have strong dust emission and high bolometric
luminosity. But their radio emission is weak or non detectable at a level of
approximately 1 mJy (Molinari et al. 1996, 2000; Sridharan et al. 2002;
Beuther, Schilke, & Menten. 2002; Wu et al. 2006).. Their natal clouds have
not been affected significantly by the star forming process. Thus, they
present the information about the early kinematic processes of high mass star
formation.
The dense core JCMT 18354-0649S was first detected in an ammonia survey of
high-mass star forming regions with Max-Planck-Institut für Radioastronomie
(MPIfR) 100 m telescope at Effelsberg (Wu et al. 2006), and was later
confirmed by the observation with the Submillimeter Common-User Bolometric
Array (SCUBA) of James Clerk Maxwell telescope (JCMT) (Wu et al. 2005).
Another SCUBA core which harbors a UC H ii region G35.4NW is located about
$1\arcmin$ north of JCMT 18354-0649S. The kinetic distance of the two SCUBA
cores is 5.7 or 9.6 kpc (Wu et al. 2005), and 5.7 kpc was adopted in this
paper. Core JCMT 18354-0649S has no counterpart in radio continuum. Multiple
lines towards this core including HCN (3–2), H13CO+ (3–2), and C17O (2–1)
reveal typical ”blue profile” (Wu et al. 2005), indicating that the core is
undergoing gravitational collapse (Keto, Ho & Haschick. 1988; Zhou et al.
1993; Zhang, Ho, & Ohashi. 1998; Wu & Evans. 2003; Wu et al. 2005; Fuller,
Williams, & Sridharan. 2005; Wyrowski 2006; Wu et al. 2007; Birkmann et al.
2007; Klaassen & Wilson 2007; Sun & Gao 2008; Velusamy et al. 2008). The core
is also associated with a near-infrared point source, corresponding to a star
of 6-11 $M_{\odot}$ (Zhu et al.2010, submitted). Carolan et al. (2009)
observed sixteen different molecular line transitions including CO, HCN, HCO+
and their isotopes in this region, and modeled the source with a chemically
depleted rotating envelope collapsing onto a central protostellar source which
has evolved sufficiently to generate a molecular outflow. All the evidence
suggests that JCMT 18354-0649S is forming high mass protostellar
object(s)(HMPO). However, single-dish observations with a resolution of
$15\arcsec-40\arcsec$ can not reveal detailed kinematics in the core at a
distance as large as 5.7 kpc. In this paper we report the results of a high
resolution study with the Submillimeter Array (SMA111Submillimeter Array is a
joint project between the Smithsonian Astrophysical Observatory and the
Academia Sinica Institute of Astronomy and Astrophysics and is funded by the
Smithsonian Institution and the Academia Sinica.) in order to probe the
details of the core and its kinematic features. The observation and initial
results are presented in sections 2 and 3. Properties of infall and outflow
motions are discussed further in section 4, and a brief summary is given in
section 5.
## 2 Observations
The observations of JCMT 18354-0649S was carried out with the SMA in July 2005
with seven antennas in its compact configuration and in September 2005 with
six antennas in its extended configuration. The 345 GHz receivers were tuned
to 265 GHz for the lower sideband (LSB) and 275 GHz for the upper sideband
(USB). The frequency spacing across the spectral band is 0.8125 MHz or $\sim$1
km s-1 for both configurations. The phase reference center of both
observations was R.A.(J2000) = 18h38m08.10s and DEC.(J2000) = -$6\arcdeg
46\arcmin 52.17\arcsec$.
In the observations with the compact configuration, Jupiter, Uranus and QSO
3c454.3 were observed for antenna-based bandpass correction. An amplitude
offset was found on some baselines and the baseline-based errors in bandpass
were further corrected using the point source QSO 3c454.3. QSOs 1741-038 and
1908-201 were employed for antenna-based gain correction. Uranus was observed
for flux-density calibration. The synthesized beam size is $3.76\arcsec\times
2.72\arcsec$ (PA=-54$\arcdeg$).
For the extended configuration, QSO 3c454.3 was used as bandpass calibrator,
QSOs 1741-038 and 1908-201 as gain calibrators, and Uranus as a flux
calibrator, respectively. The synthesized beam size is about $1\arcsec$.
Miriad222http://carma.astro.umd.edu/miriad was employed for calibration and
imaging. The 1.3 mm continuum data were acquired by averaging all the line-
free channels over both the 2 GHz of upper and lower spectral bands. MIRIAD
task ”selfcal” was employed to perform self-calibration on the continuum data.
Since the dust emission is weak, self-calibration with phase only was
performed. The gain solutions from the self-calibration were applied to the
line data.
The continuum data combined from both configurations yield a synthesized beam
of $1.63\arcsec\times 1.28\arcsec$ (PA=-81.4$\arcdeg$), and 1 $\sigma$ rms of
2.5 mJy in the naturally weighted maps. HCN (3-2) and CH3OH
($5_{23}$–$4_{13}$) were detected in the compact configuration.
The shortest baseline in compact configuration observations is 16.5 m,
corresponding to a spatial scale of $20\arcsec$. Spatial structures more
extended than this limit, such as HCN maps close to the cloud velocity, would
be filtered out. HCN (3–2) data in JCMT archive (Wu et al. 2005; Carolan et
al. 2009) were used to recover the missing flux. The JCMT archive data were
reduced using the KAPPA and GAIA packages in the STARLINK suite. The JCMT beam
size for HCN (3–2) was $18.3\arcsec$, and the main beam efficiency was 0.69.
The combination of the SMA compact and JCMT HCN (3–2) data was done using the
task ”immerge” in MIRIAD.
## 3 Results
### 3.1 Dust core
The 1.3 mm continuum images are shown in Fig.1. The left panel is obtained in
the compact array and the right panel from the combined data of the compact
and extended configurations. An elongated core is revealed with the 1.3 mm
continuum emission observed with the compact array, and is further resolved
into three condensations by the continuum emission using the combined data
from both configurations. The three condensations are named as MM1, MM2 and
MM3. The peak position of MM1 is R.A.(J2000)=$18^{\rm h}38^{\rm m}08.1^{\rm
s}$, DEC.(J2000)=$-6\arcdeg 46\arcmin 52.98\arcsec$. MM2 peaks at
R.A.(J2000)=$18^{\rm h}38^{\rm m}07.9^{\rm s}$, DEC.(J2000)=$-6\arcdeg
46\arcmin 51.36\arcsec$, and MM3 peaks at R.A.(J2000)=$18^{\rm h}38^{\rm
m}08.05^{\rm s}$, DEC.(J2000)=$-6\arcdeg 46\arcmin 51.36\arcsec$.
From the fit of an elliptical Gaussian, the core revealed by the compact array
is found to be elongated from south-east to north-west. It has an average FWHM
diameter of 0.06 pc ($\sim 2\arcsec$) at a distance of 5.7 kpc, smaller than
the beam size of the compact configuration. The total integrated flux is 0.47
Jy. The total dust and gas mass can be obtained with the formula
$M=S_{\nu}D^{2}/\kappa_{\nu}B_{\nu}(T_{d})$, where $S_{\nu}$ is the flux at
1.3 mm, D is the distance, and $B_{\nu}(T_{d})$ is the Planck function. We
adopt a dust opacity $k_{1330}$=$1.4\times 10^{-2}$ cm2g-1 at 1.3 mm
calculated from Ossenkopf & Henning (1994) with a dust opacity index
$\beta=2$. Here the ratio of gas to dust is taken as 100. Molecular line CH3OH
($5_{23}$–$4_{13}$) is detected, and its emission peak coincides with the dust
core very well (see Sec.3.2). The upper energy level of the CH3OH
($5_{23}$–$4_{13}$) line is 57 K above the ground, indicating a relatively
warm conditions. Assuming $T_{d}$=57 K, a total dust and gas mass of
$47~{}M_{\odot}$ is derived. A beam-average gas/dust density amounts to
$2.0\times 10^{6}$ cm-3, which is larger than $1.1\times 10^{6}$ cm-3 obtained
from single-dish telescope (Wu et al. 2005).
The three condensations (MM1, MM2 and MM3) have total integrated flux of 0.42
Jy, leading to a total mass of $42~{}M_{\odot}$ (assuming Td = 57 K as above).
MM1 has a diameter of 0.02 pc, and a mass of $30~{}M_{\odot}$. MM1 is
centrally concentrated and compact, while MM2 and MM3 are much more diffuse
and extended.
With UKIRT (United Kingdom Infrared Telescope) Zhu et al. (2010, in
preparation) detected three near-infrared sources IRS1a, IRS1b, IRS1c in H, K
and L bands. Their positions are marked with crosses in Fig.1. IRS1a ($18^{\rm
h}38^{\rm m}08.135^{\rm s}$, $-6\arcdeg 46\arcmin 51.57\arcsec$) lies about
$1.6\arcsec$ north-east of MM1. IRS1b ($18^{\rm h}38^{\rm m}08.026^{\rm s}$,
$-6\arcdeg 46\arcmin 56.24\arcsec$) and IRS1c ($18^{\rm h}38^{\rm
m}07.929^{\rm s}$, $-6\arcdeg 46\arcmin 55.34\arcsec$) are about $3\arcsec$
southwest from MM1 and are much fainter.
### 3.2 Gas core
The molecular line CH3OH ($5_{23}$–$4_{13}$) is detected in the compact
configuration. Fig.2 presents its spectrum at three positions and the
integrated emission overlaid on the 1.3 mm continuum image. The central
velocity of the CH3OH ($5_{23}$–$4_{13}$) spectra is 96.7 km s-1, which is
taken as the systemic velocity of the core. The central velocity of CH3OH
($5_{23}$–$4_{13}$) does not shift at different positions (see Fig.2), which
should exclude rotation at the core. The P-V diagram of CH3OH
($5_{23}$–$4_{13}$) is shown in Fig.3, indicating a compact gas core without
rotation. The emission center of CH3OH ($5_{23}$–$4_{13}$)
(R.A.(J2000)=$18^{\rm h}38^{\rm m}08.092^{\rm s}$, DEC.(J2000)=$-6\arcdeg
46\arcmin 52.318\arcsec$) coincides with MM1 very well. While there are no
CH3OH components corresponding with MM2 and MM3. The deconvolved size of the
gas core revealed by CH3OH ($5_{23}$–$4_{13}$) is $3.78\arcsec\times
2.76\arcsec$ (PA=-29$\arcdeg$), comparable to the synthesized beam size of the
compact array.
The HCN (3–2) (265.886GHz) spectra obtained from the SMA compact configuration
and from the data combined from both compact and extended configurations are
presented in the left panel of Fig.4. Both of the two spectra are averaged
over a region of $5\arcsec\times 5\arcsec$, which show a redshifted absorption
dip and broad wings. The line profiles observed with the SMA and JCMT, as well
as a combination of the two are presented in the right panel of Fig.4. All the
spectra in the right panel of Fig.4 are convolved with the JCMT beam
($18.3\arcsec$) for comparing. One can see that the SMA compact array
observations recover less than 10$\%$ of JCMT flux around the systematic
velocity, but recover more than 30$\%$ flux at the wings. The combination of
the SMA and JCMT data recovers more than 70$\%$ of the JCMT flux at all the
velocity channels.
### 3.3 Kinematic signatures of lines
#### 3.3.1 Infall motion
The left panel of Fig.4 shows the most prominent feature (”blue profile”) of
the HCN (3–2) line at the core. The absorption gap is more than 8 km s-1 wide,
ranging from 93 to 101 km s-1. Fig.5 presents the channel maps of the HCN
(3–2) emission from 80 km s-1 to 109 km s-1 constructed from the combined
data, which is convolved with the beam of the SMA compact configuration. The
absorption is obvious in the velocity range (95,99) km s-1. The absorption dip
is also clearly seen in the P-V diagrams (see Fig.6), which is much deeper
than that revealed by the single-dish observation (Wu et al. 2005).
From the left panel of Fig.4, it is clearly to see that the centeral velocity
of the absorption dip (98 km s-1) is redshifted from the systematic velocity
(96.7 km s-1) by 1.3 km s-1. Such a blue asymmetric line profile where the
blue emission peak is at a higher intensity than the red one is a collapse
signature of molecular cores (Zhou et al. 1993). The spectra constructed from
JCMT data and the combined data (the right panel of Fig.4) also show
significant ”blue profile”, confirming the existence of infall motions.
#### 3.3.2 Molecular outflow
Besides the absorption dip, the HCN (3–2) line exhibits remarkable broad wings
extending more than 40 km s-1. High-velocity gas also can be easily identified
in P-V diagrams of the HCN (3–2) emission along the direction of
P.A.=15$\arcdeg$ and P.A.=90$\arcdeg$ as shown in Fig.6. The HCN (3–2)
emission obtained from SMA compact configuration is integrated from 80 to 87
km s-1 for the blue lobe and from 103 to 109 km s-1 for the red lobe,
respectively. The contour map of the integrated flux are shown in Fig.7. As in
the channel maps (Fig.5), we can see several clumps in each lobe in the
integrated map. The integrated HCN (3–2) emission seems to comprise an
S-shaped structure from north-east to south. Another jet-like structure
extended more than $10\arcsec$ is also seen at the west of the continuum
emission center. IRS1a seems to be the driving source of the outflow.
The southern redshifted lobe (S-lobe) comprises two clumps named ”Clump1” and
”Clump2”. In the north-east blueshifted lobe (NE-lobe), two clumps are also
found and named ”Clump3” and ”Clump4”. These clumps are distributed along the
direction of the outflow and likely to be outward gas knots. They are probably
not physically related with other stellar sources except the driving source
though Clump2 is close to IRS1b and IRS1c.
## 4 Discussion
### 4.1 Infall motion
Although the HCN emission is extended over a region larger than the compact
configuration beam, the infall region is still difficult to confine due to the
contamination of the outflow. Since the size of the gas core traced by CH3OH
($5_{23}$–$4_{13}$) is comparable to the compact configuration beam size, the
beam size of the compact configuration was taken as the radius ($R_{in}$) of
the infall region (Wu et al. 2009). The kinematic mass infall rate can be
calculated using dM/dt=$4{\pi}n\mu_{G}m_{H_{2}}R_{in}^{2}V_{in}$, where
$V_{in}$, $\mu_{G}=1.36$, $m_{H_{2}}$ and n=$2.0\times 10^{6}$ cm-3 are the
infall velocity, the mean molecular weight, the H2 mass, and the beam-average
gas/dust density, respectively. The infall velocity $V_{in}$ is 1.3 km s-1 by
comparing the systemic velocity (96.7 km s-1) and the velocity of the
redshifted absorbing dip (98 km s-1) in the HCN (3-2) spectrum (Welch et al.
1987), leading to a kinematic mass infall rate of 2.0$\times
10^{-3}~{}M_{\odot}\cdot$yr-1. In core G10.6-0.4, the redshifted NH3 indicates
large infall velocity $5.0\pm 1.7$ km s-1 at about 0.05 pc, and a mass infall
rate as high as 5$\times 10^{-3}~{}M_{\odot}\cdot$yr-1 (Keto, Ho & Haschick.
1987). Also with NH3 inverse lines, Zhang & Ho (1997) obtained a high infall
velocity $\sim 3.5$ km s-1 within a region smaller than 0.02 pc towards core
W51e2. Large infall velocities ($>1.5$ km s-1) and mass infall rates
($>1\times 10^{-3}~{}M_{\odot}\cdot$yr-1) were also detected towards G10.47
and G34.26 with HCO+ (4–3) line (Klaassen & Wilson 2007). In core G19.61+0.23,
an infall velocity of 2.5 km s-1 and a mass infall rate as high as $6.1\times
10^{-3}~{}M_{\odot}\cdot$yr-1 were derived (Wu et al. 2009). It seems high
mass infall rate is required by high-mass star formation. The results of the
core JCMT 18354-0649S are comparable with those of the above sources. For
comparison, the $V_{in}$ from pure free-infall assumption is also derived with
the formula $V_{in}^{2}~{}=~{}2GM/R_{in}$. The pure free-infall velocity
inferred is 2.9 km s-1, larger than the infall velocity obtained from the
spectrum.
Wu et al. (2005) obtained a small infall velocity ($\sim 0.3~{}km~{}s^{-1}$)
at a radius of 4$\arcsec$. The absorption dip of HCN (3–2) line seen by the
SMA is much deeper and broader than that observed by JCMT. However, the
kinematic mass infall rate $\dot{M}_{in}$ ($2.0\times
10^{-3}~{}M_{\odot}\cdot$ yr-1) obtained here is well coincident with that
obtained with JCMT (Wu et al. 2005), $3.4\times 10^{-3}~{}M_{\odot}\cdot$
yr-1.
### 4.2 Properties of HCN (3-2) outflow
The column density of HCN at each velocity channel in each outflow lobe can be
obtained through (Garden et al. 1991):
$N_{HCN}(v)=\frac{3k}{8\pi^{3}B\mu^{2}}\frac{exp[hBJ(J+1)/kT_{ex}]}{(J+1)}\frac{(T_{ex}+hB/3k)}{1-exp(-h\nu/kT_{ex})}\int\tau_{v}dv$
(1)
Where $v$ is the central velocity of the channel relative to the systemic
velocity, the rotational constant B=44.315976 GHz and permanent dipole moment
$\mu=3$ debye for HCN, the velocity channel width is smoothed to be
$1~{}km~{}s^{-1}$. Assuming HCN emission in the line wings to be optically
thin and excitation temperature of Tex=30 K (Wu et al. 2004), the optical
depth $\tau_{v}$ can be derived with the equation:
$\tau_{v}=\frac{kT_{r}(v)}{h\nu}(\frac{1}{exp(h\nu/kT_{ex})-1}-\frac{1}{exp(h\nu/kT_{bg})-1})^{-1}$
(2)
where $T_{r}(v)$ is the excess brightness temperature of HCN(3–2) emission at
$v$. Adopting $X_{HCN}=[HCN]/[H_{2}]=1\times 10^{-10}$ (Carolan et al. 2009),
the mass of each lobe at $v$ can be calculated with:
$M(v)=X_{HCN}^{-1}\mu_{G}m_{H_{2}}D^{2}{\int}N_{HCN}(v)d{\Omega}$ (3)
where D, $\Omega$ are the cloud distance and the solid angle. Thus the total
mass of each lobe is given by $M=\sum$$M(v)$, the total momentum by
$P=\sum$$M(v)v$, and the energy by $E={\frac{1}{2}}\sum$$M(v)v^{2}$. The
dynamical timescale $t_{dyn}$ is estimated as $R/V_{max}$, where R is the
outflow extent, and $V_{max}$ is the maximum velocity of the outflow lobe. The
mechanical luminosity L, and the mass-loss rate $\dot{M}$ are calculated as
L=E/t, $\dot{M}=P/(tV_{w}$), where the wind velocity $V_{w}$ is assumed to be
500 km s-1 (Lamers et al. 1995). The derived parameters are listed in Table.1.
The total mass, momentum, energy of the three lobes are 12 $M_{\odot}$, 121
$M_{\odot}\cdot$ km s-1 and $1.3\times 10^{46}$ erg, respectively. The average
dynamical timescale is about $1.6\times 10^{4}$ yr, and the total mass-loss
rate $1.6\times 10^{-5}~{}M_{\odot}\cdot$ yr-1. The outflow is massive with
parameters similar to that of IRAS 05274+3345E and the other outflows detected
towards five massive star formation regions (Zhang et al. 2007b; Klaassen &
Wilson 2008).
The Position-Velocity diagram at the left panel of Fig.6 shows that at low
velocities, the NE-lobe and S-lobe both have compact morphology near the core
center. At higher velocities the southern lobe becomes further away from the
center. As shown in Fig.7, the outflow axis traced by Clump1 and Clump2
differs from that traced by Clump3 and Clump4. The different outflow
orientations in the large and small scales may be attributed to the precession
of the outflow axis (Su et al. 2007). From the right panel of the P-V diagram,
a high-velocity component (V $<$ 85 km s-1) with velocities decreasing with
distance from the protostar, and a second component tracing the low-velocity
material (V $>$ 85 km s-1) extending about $15\arcsec$ along the axis of the
W-lobe are clearly seen. Such convex spur PV structure was also revealed in a
simulation of a pulsed jet driven outflow (Lee et al. 2001).
### 4.3 HCN — tracer of both inflow and outflow motions
HCN is among the most abundant molecular species with a high critical density
larger than $10^{6}$ cm-3 (for HCN (1–0)) (Carolan et al. 2009), and is
believed to trace dense molecular cores. HCN is detected in both low mass
class 0 and I sources (Park, Kim, & Ming. 1999; Yun et al. 1999), and high-
mass hot cores (Boonman et al. 2001).
HCN is thought as a good tracer of inflow motions (Wu & Evans. 2003). The
infall asymmetry in the HCN spectra is found to be more prevalent, and more
prominent than in any other previously used infall tracers such as CS (2–1),
DCO+ (2–1), and N2H+(1–0) during a survey toward 85 starless cores (Sohn et
al. 2007). Among the small group of pre- and protostellar objects in L1251B,
infall signature was also detected in the HCN emission (Lee et al. 2007). HCN
also traces inflow motions very well in massive star-formation regions. Wu and
Evans. (2003) found 12 sources showing ”blue profile” in the HCN lines during
a spectroscopic survey of 28 massive cores with water maser. Besides HCN,
other nitrogen bearing molecules such as N2H+ are also tracers of inflow
motions (Tsamis et al. 2008; Schnee et al. 2007; Crapsi et al. 2005). Recently
inverse P Cygni profile of CN line in hot cores was found (Zapata et al. 2008;
Wu et al. 2009). These results suggest nitrogen bearing molecular species be
good tracers of inflowing motions in star-formation regions.
Outflows traced by HCN are often detected not only in low-mass star-formation
regions but also in massive star-formation regions (Bachiller, Gutiérrez, &
Pérez. 1997; Choi 2001; Su et al. 2007; Zhang et al. 2007a). HCN outflow of
the core JCMT 18454-0649S is another good sample. Additionally, Zhu et al.
(2010 in preparation) and Cyganowski (2008) found excess emission at 4.5
$\micron$ at the position of source IRS1a, which is close to the center of the
NE-lobe and S-lobe. Such excess emission at the 4.5 $\micron$ band could be
shock-excited. In IRAS 20126+4104, HCN emission is also found to be closely
related to the shock-excited near-IR H2 knots and was identified to be
associated with shock wings (Su et al. 2007). The inner clumps (Clump1 in the
S-lobe and Clump3 in the NE-lobe) of the core JCMT 18354-0649S should also be
coincident with shocks. In fact, models have already demonstrated a dramatic
increase of HCN molecules, during the intense interaction between outflow and
ambient gas, or slow shock front (Mitchell 1984; Nejad, Williams, & Charnley.
1990). In this process sulfur and nitrogen react with hydrocarbons to produce
various compounds, wherein HCN abundance gets higher than the rest of the
products (Nejad, Williams, & Charnley. 1990). Thus HCN may trace the outflow
even better than sulfur containing molecules.
## 5 Summary
Both dust continuum at 1.3 mm and CH3OH emission detected with SMA reveal a
compact core in JCMT 18354-0649S. The core observed with the compact
configuration has a mass of $47~{}M_{\odot}$ and an average density of
$2.0\times 10^{6}$ cm-3. With the combination of the compact and extended
configurations, the core is resolved to three condensations with a total mass
of $42~{}M_{\odot}$.
HCN (3-2) spectra exhibit an infall signature in this region. The red shifted
absorption seen in the SMA observation is deeper and broader than that in the
JCMT observation. The infall rate is $2.0\times 10^{-3}~{}M_{\odot}\cdot$
yr-1. High velocity gas is detected in HCN (3-2) emission. The outflow has
three lobes and their total mass is 12 $M_{\odot}$ and momentum of 121
$M_{\odot}\cdot$ km s-1. The average dynamical timescale and the total mass-
loss rate are about $1.6\times 10^{4}$ yr and $1.6\times
10^{-5}~{}M_{\odot}\cdot$ yr-1, respectively. All the findings indicate a
high-mass protostar is forming via rapid accretion. Our results suggest that
nitrogen bearing molecules especially HCN are good for probing both infall and
outflows.
## Acknowledgment
We are grateful to the SMA staff. We also thank Dr. Shengli Qin. for his help
with data reduction and discussion. This work was funded by Grants of NSFC No
10733030 and 10873019.
## References
* Arce et al. (2007) Arce, H. G., Shepherd, D., Gueth, F., Lee, C.-F., Bachiller, R., Rosen, A., Beuther, H., 2007, arXiv:astro-ph/0603071
* Bachiller, Gutiérrez, & Pérez. (1997) Bachiller, R., Gutiérrez, M., & Pérez. 1997, ApJ, 487, L93
* Beuther, Schilke, & Menten. (2002) Beuther, H., Schilke, P., & Menten, K. M. 2002, ApJ, 566, 945
* Birkmann et al. (2007) Birkmann, S. M., Krause, O., Hennemann, M., Henning, Th., Steinacker, J., Lemke, D. 2007, A&A, 474, 883
* Bonnell, Bate, & Zinnecker. (1998) Bonnell, I. A., Bate, M. R., & Zinnecker, H. 1998, MNRAS, 298, 93
* Boonman et al. (2001) Boonman, A. M. S., Stark, R., Van der tak, F. F. S., Van dishoeck, E. F., Van der wal, P. B., Scháfer F., De lange, G., & Laauwen, W. M. 2001, ApJ, 553, L63
* Carolan et al. (2009) Carolan, P. B., Khanzadyan, T., Redman, M. P., Thompson, M. A., Jones, P. A., Cunningham, M. R., Loughnane, R. M., Bains, I., Keto, E. 2009, MNRAS, 400, 78
* Choi (2001) Choi, Minho. 2001, ApJ, 553, 219
* Churchwell (2002) Churchwell, E. 2002, ARA&A, 40, 27
* Crapsi et al. (2005) Crapsi, A., Caselli, P., Walmsley, C., M., Myers, P., C., Tafalla, M., Lee, C., W., Bourke, T., L. 2005, ApJ, 619,379
* Cyganowski (2008) Cyganowski, C. J. et al. 2008, AJ, 136, 2391
* Fuller, Williams, & Sridharan. (2005) Fuller, G. A., Williams, S. J., & Sridharan, T. K. 2005, A&A, 442, 949
* Garden et al. (1991) Garden, R. P., Hayashi, M., Gatley, I., Hasegawa, T., & Kaifu, N. 1991, ApJ, 374, 540
* Keto, Ho & Haschick. (1987) Keto, E. R., Ho, P. T. P., & Haschick, A. D. 1987, apj, 318, 712
* Keto, Ho & Haschick. (1988) Keto, E. R., Ho, P. T. P., & Haschick, A. D. 1988, ApJ. 324, 920
* Klaassen & Wilson (2007) Klaassen, P. D. & Wilson, C. D. 2007, ApJ, 663, 1092
* Klaassen & Wilson (2008) Klaassen, P. D., & Wilson, C. D. 2008, ApJ, 684, 1273
* Lamers et al. (1995) Lamers, H. J. G. L. M., Snow, T. P., Lindholm, D. M. 1995, ApJ, 455, 269
* Lee et al. (2001) Lee, C.-F., Stone, J. M., Ostriker, E. C., & Mundy, L. G. 2001, ApJ, 557, 429
* Lee et al. (2007) Lee, J.-E., Di Francesco, J., Bourke, T. L., Evans, N. J. II & Wu, J. 2007, ApJ, 671, 1748
* Lester et al. (1985) Lester, D. F., Dinerstein, H. L., Werner, M. W., Harver, P. M., Evans, N. J. II, Brown, R. L. 1985, ApJ, 296, 565L
* Mitchell (1984) Mitchell, G. F. 1984, ApJ, 287, 665
* Molinari et al. (1996) Molinari, S., Brand, J., Cesaroni, R., Palla, F. 1996, A&A, 308, 573
* Molinari et al. (2000) Molinari, S., Brand, J., Cesaroni, R., & Palla, F. 2000, A&A, 355, 617
* Motoyama & Yoshida (2003) Motoyama, K., & Yoshida T. 2003, MNRAS, 344, 461
* Myers et al. (1996) Myers, P. C., Mardones, D., Tafalla, M., Williams, J. P. & Wilner, D. J. 1996, ApJ, 465, L133
* Nejad, Williams, & Charnley. (1990) Nejad, L. A. M., Williams, D. A., & Charnley, S. B. 1990, MNRAS, 246, 183
* Ossenkopf & Henning (1994) Ossenkopf, V., Henning, Th. 1994, A&A, 291, 943
* Park, Kim, & Ming. (1999) Park, Y. S., Kim, J., & Ming Y. C. 1999, ApJ, 520, 223
* Schnee et al. (2007) Schnee, S., Caselli, P., Goodman, A., Arce, H. G., Ballesteros-Paredes, J., Kuchibhotla, K. 2007, ApJ, 671, 1839
* Shu (1977) Shu, F. H. 1977, ApJ, 214, 488
* Shu, Adams, & Lizano (1987) Shu, F. H., Adams, F. C., & Lizano, S. 1987, ARA&A, 25, 23
* Sohn et al. (2007) Sohn, J. J., Lee, C. W., Park, Y.-S., Lee, H. M., Myers, P. C., & Lee Y. 2007, ApJ, 664,928
* Sridharan et al. (2002) Sridharan, T. K., Beuther, H., Schilke, P., Menten, K. M., Wyrowski, F. 2002, ApJ, 566, 931
* Su et al. (2007) Su, Y.-N., Liu, S.-Y., Chen, H.-R., Zhang, Q., Cesaroni, R. 2007, ApJ, 671,571
* Sun & Gao (2008) Sun, Y., & Gao, Y. 2008, MNRAS, 392, 170
* Tsamis et al. (2008) Tsamis, Y. G., Rawlings, J. M. C., Yates, J. A., Viti, S. 2008, MNRAS, 388, 898
* Velusamy et al. (2008) Velusamy, T., Peng, R., Li, D., Goldsmith, P. F., Langer, William D. 2008, ApJ, 688, L87
* Welch et al. (1987) Welch, Wm. J., Dreher, J. W., Jackson, J. M., Terebey, S., Vogel, S. N. 1987, Science, 238, 1550
* Wolfire & Cassinelli (1987) Wolfire, M. G. & Cassinelli, J. P. 1987, ApJ, 319, 850
* Wu & Evans. (2003) Wu, J., Evans N. J. II. 2003, ApJ, 592, L79
* Wu et al. (2004) Wu, Y., Wei, Y., Zhao, M., Shi, Y., Yu, W., Qin, S., Huang, M. 2004, A&A, 426, 503
* Wu et al. (2005) Wu, Y., Zhu, M., Wei, Y., Xu, D., Zhang, Q., & Fiege, J. D. 2005, ApJ, 628, L57
* Wu et al. (2006) Wu, Y., Zhang, Q., Yu, W., Miller, M., Mao, R., Sun, K., & Wang, Y. 2006, A&A, 450, 607
* Wu et al. (2007) Wu, Y., Henkel, C., Xue, R., Guan, X., Miller, M. 2007, ApJ, 669, L37
* Wu et al. (2009) Wu, Y., Qin, S.-L., Guan, X., Xue, R., Ren, Z., Liu, T., Huang, M., Chen, S. 2009, ApJ, 697, L116
* Wyrowski (2006) Wyrowski, F., Heyminck, S., G$\ddot{u}$sten, R., Menten, K. M. 2006, A&A, 454, L95
* Yun et al. (1999) Yun, J. L., Moreira, M. C., Afosso, J. M. & Clemens, D. P. 1999, ApJ, 118, 990
* Zapata et al. (2008) Zapata, L. A., Palau, A., Ho, P. T. P., Schilke1, P., Garrod, R. T., Rodr guez, L. F., & Menten, K. 2008, A&A, 479, L25
* Zhang & Ho (1997) Zhang, Q., & Ho, P. T. P. 1997, ApJ, 488,241
* Zhang, Ho, & Ohashi. (1998) Zhang, Q., Ho, P. T. P., & Ohashi, N. 1998, ApJ, 494, 636
* Zhang et al. (2007a) Zhang, Q., Sridharan, T. K., Hunter, T. R., Chen, Y.; Beuther, H., Wyrowski, F. 2007a, A&A, 470, 269
* Zhang et al. (2007b) Zhang, Q., Sridharan, T. K., Hunter, T. R., Chen, T., Beuther, H., & Wyrowski, F. 2007b, ApJ, 658, 1152
* Zhou et al. (1993) Zhou, S., Evans, N. J. II., Koempe, C., Walmsley, C. M. 1993, ApJ, 404, 232
Table 1: Outflow parameters of each lobe. outflow | Vmax | $t_{dyn}$ | Mass | momentum | Energy | $L$ | $\dot{M}_{out}$
---|---|---|---|---|---|---|---
| km s-1 | (10${}^{4}~{}yr$) | ($M_{\odot}$) | ($M_{\odot}$ km s-1) | (10${}^{45}~{}erg$) | ($L_{\odot}$) | ($10^{-6}M_{\odot}$ yr-1)
southern lobe | 11.7 | 1.4 | 3.3 | 29 | 3.0 | 1.8 | 4.0
north-eastern lobe | 15.3 | 1.0 | 4.2 | 42 | 4.5 | 3.6 | 8.0
western lobe | 15.3 | 2.3 | 4.6 | 50 | 5.5 | 2.1 | 4.0
Figure 1: The 1.3 mm continuum emission towards JCMT 18354-0649S. The left one
is obtained with the compact configuration. The rms level is 3 mJy beam-1 (1
$\sigma$). The contours are at -6, 3, 6, 12, 21, 33, 48, 66, 87$\sigma$. The
right panel gives the contours of the continuum emission combined from both
configurations. The rms level is 2.5 mJy beam-1 (1 $\sigma$) and the contours
are at -6, 3, 6, 12, 21, 33, 48$\sigma$. The three near-infrared sources are
marked with crosses. Figure 2: The lower-left panel is the contours of the
CH3OH integrated intensity overlaid on the 1.3 mm continuum image (grey
scale). The contours start from 30$\%$ in steps of 10$\%$ of the peak emission
(15 Jy beam${}^{-1}\cdot$km s-1). The three near-infrared sources are marked
with crosses. The beam-averaged spectrum of CH3OH at three positions are
presented in the other panels. The gaussian fit towards each spectrum is shown
with solid lines. Figure 3: Position-velocity diagrams of CH3OH along a P.A.
of 0$\arcdeg$. The contour levels are from 15% to 90% in steps of 15% of the
peak intensity in both panels. The intensity at the peak is 3.73 Jy beam-1.
Figure 4: The HCN (3-2) spectra. Left: the solid black line exhibits the
spectrum constructed from SMA compact array and the dashed line shows the
spectrum obtained from combining the compact and extended data together. Both
of the two spectra are integrated over a region of $5\arcsec\times 5\arcsec$.
Right: the solid black line shows the spectrum constructed from the combined
SMA and JCMT data, which is convolved with the JCMT beam ($18.3\arcsec$); the
dashed line shows the spectrum from JCMT only; the dash-dotted gray line shows
the spectrum obtained with the SMA compact array and convolved with the JCMT
beam. The vertical dashed lines in both panels mark the position of the
systematic velocity (96.7 km s-1). Figure 5: Combined JCMT and SMA HCN (3-2)
channel maps from 80 km s-1 to 109 km s-1, which is convolved with the beam of
the SMA compact configuration ($3.76\arcsec\times 2.72\arcsec$,
PA=-54$\arcdeg$). The contours are in steps of 0.5 Jy beam-1 (3$~{}\sigma$)
from 0.5 Jy beam-1 (3$~{}\sigma$). The velocity of each channel is plotted at
the upper-left of each panel, and the beam size at the lower-right. The
positions of MM1, MM2 and MM3 are marked with crosses.
Figure 6: Position-velocity diagrams of HCN (3-2) outflow observed by the SMA
along a P.A. of 15$\arcdeg$ (left panel) and 90$\arcdeg$ (right panel). The
image is smoothed to 2 km s-1 velocity resolution. The contour levels are from
15% to 90% in steps of 15% of the peak intensity in both panels. The peak is
1.84 Jy beam-1 in the left panel and 1.92 Jy beam-1 in the right panel. The
four clumps are labeled by solid lines with arrows. The clumps are distinguish
by the thick dashed lines. Figure 7: The high-velocity HCN (3-2) intensity
contours overlaid on the 1.3 mm continuum image, integrated from 80 to 87 km
s-1 for the blueshifted lobes (solid contours) and from 103 to 109 km s-1 for
the redshifted lobe (dashed contours), with contours from 30$\%$ in steps of
10$\%$ of the peak emission. The peak is 8.55 Jy beam-1$\cdot$km s-1 for the
blueshifted lobes and 7.16 Jy beam-1$\cdot$km s-1 for the redshifted lobe. The
empty and solid ellipses in the lower-right corner represent the synthesized
beams of HCN (3-2) emission and 1.3 mm continuum emission combined from both
configurations, respectively.
|
arxiv-papers
| 2010-12-20T11:43:11 |
2024-09-04T02:49:15.846574
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tie Liu, Yuefang Wu, Qizhou Zhang, Zhiyuan Ren, Xin Guan and Ming Zhu",
"submitter": "Tie Liu",
"url": "https://arxiv.org/abs/1012.4292"
}
|
1012.4308
|
11institutetext: 1 Euratom-VR Association, Department of Radio and Space
Science, Chalmers University of Technology, SE-412 96 Göteborg, Sweden.
2 Max-Planck-Institut für Plasmaphysik EURATOM-IPP, D-85748 Garching, Germany.
# Impurity transport in ITG and TE mode dominated turbulence
A. Skyman1 H. Nordman1 P. Strand1 F. Jenko2 F. Merz2
## 1 Introduction
Figure 1: Illustration of $PF_{0}$ and the linearity of
$\Gamma_{Z}\left(\nabla n_{Z}\right)$; ITG dominated quasilinear GENE result
for $Ne$ with $k\rho=0.3$
The transport properties of impurities is of high relevance for the
performance and optimisation of magnetic fusion devices. For instance, if
impurities from the plasma-facing surfaces accumulate in the core, wall-
impurities of relatively low density suffice to dilute the plasma and lead to
unacceptable energy losses in the form of radiation.
In the present study, turbulent impurity transport in Deuterium tokamak
plasmas, driven by Ion Temperature Gradient (ITG) and Trapped Electron (TE)
modes, has been investigated using fluid and gyrokinetic models. The impurity
diffusivity ($D_{Z}$) and convective velocity ($V_{Z}$) are calculated, and
from these the zero-flux peaking factor ($PF_{0}$) is derived. This quantity
expresses the impurity density gradient at which the convective and diffusive
transport of impurities are exactly balanced. The sign of $PF_{0}$ is of
special interest, as it determines whether the impurities are subject to an
inward ($PF_{0}>0$) or outward ($PF_{0}<0$) pinch.
Quasilinear results obtained from the GENE code [1, 2] are compared with two-
fluid results [3] for both ITG and TE mode dominated turbulence. Scalings of
$PF_{0}$ with impurity charge ($Z$) and various plasma parameters, such as
magnetic shear ($\hat{s}$), are studied. Of particular interest are conditions
favouring an outward convective impurity flux.
## 2 Theoretical background
The transport of a trace impurity species can locally be described by a
_diffusive_ and a _convective_ part. The former is characterized by the
diffusion coefficient $D_{Z}$, the latter by a convective velocity or “pinch”
$V_{Z}$, see equation (1) [4]. From these, the _zero flux peaking factor_ is
defined as $PF_{0}=\frac{-R\,V_{Z}}{D_{Z}}|_{\Gamma=0}$, see figure 1.
$PF_{0}$ is important in reactor design, as it quantifies the balance of
convective and diffusive transport. This can be seen from equation (1), where
$\Gamma_{Z}$ is the impurity flux, $n_{Z}$ the density of the impurity species
and $R$ the major radius of the tokamak. For the domain studied – a narrow
flux tube – the gradient of the impurity density is constant: $\nabla
n_{Z}/n_{Z}=1/L_{n_{Z}}$. Setting $\Gamma_{Z}=0$ in equation (1) yields the
interpretation of $PF_{0}$ as the gradient of zero impurity flux.
$\Gamma_{Z}=-D_{Z}\nabla
n_{Z}+n_{Z}V_{Z}\Leftrightarrow\frac{R\Gamma}{n_{Z}}=-D_{Z}\frac{R}{L_{n_{Z}}}+RV_{Z}$
(1)
## 3 Fluid model
Though the main results presented in this study have been obtained using
quasilinear gyrokinetic simulations, their physical meaning is interpreted by
comparing with the Weiland multi-fluid model [3]. The fluid equations for each
included species ($j=i,\,te,\,Z$, representing Deuterium ions, trapped
electrons, and trace impurities) are:
$\displaystyle\frac{\partial n_{j}}{\partial
t}+\nabla\cdot\left(n_{j}\boldsymbol{v}_{j}\right)=0$ (2) $\displaystyle
m_{i,Z}n_{i,Z}\frac{\partial v_{||i,Z}}{\partial
t}+\nabla_{||}\left(n_{i,Z}T_{i,Z}\right)+n_{i,Z}e\nabla_{||}\varphi=0$ (3)
$\displaystyle\frac{3}{2}n_{j}\frac{\mathrm{d}T_{j}}{\mathrm{d}t}+n_{j}T_{j}\nabla\cdot\boldsymbol{v}_{j}+\nabla\cdot\boldsymbol{q}_{j}=0$
(4)
Here $\boldsymbol{q}_{j}$ is the diamagnetic heat flux, and
$\boldsymbol{v_{j}}$ is the sum of the $\boldsymbol{E}\times\boldsymbol{B}$,
diamagnetic drift, polarization drift, and stress-tensor drift velocities. To
solve the equations, it is assumed that $\boldsymbol{q}_{j}$ is the only heat
flux for all species, that passing electrons are adiabatic, and that
quasineutraility (equation (5)) holds. Going to the trace limit for the
impurities, i.e. letting $Zf_{Z}\rightarrow 0$ in equation (5), an eigenvalue
equation for ITG and TE modes is obtained. The impurity particle flux in
equation (1) is then obtained from $\Gamma_{nj}=\langle\delta
n_{j}\boldsymbol{v}_{\boldsymbol{E}\times\boldsymbol{B}}\rangle$, where the
averaging is performed over all unstable modes for a fixed length scale
$k\rho$ of the turbulence.
$\frac{\delta n_{e}}{n_{e}}=\left(1-Zf_{Z}\right)\frac{\delta
n_{i}}{n_{i}}+Zf_{Z}\frac{\delta n_{Z}}{n_{Z}},\enspace
f_{Z}=\frac{n_{Z}}{n_{e}}$ (5)
## 4 Quasilinear gyrokinetic simulations
GENE is a parallel gyrokinetic code employing a fixed grid in five dimensional
phase space and a flux-tube geometry [1]. The simulations were performed on
the _HPC-FF_ cluster111HPC-FF (_High Performance Computing For Fusion_) is an
EFDA funded computer situated at Forschungszentrum Jülich. Germany, dedicated
to fusion research with GENE running in eigenvalue mode. Growth rates and
impurity fluxes were thus computed for ITG and TE mode dominated cases, for
which a number of parameters were varied and trends observed. The main
parameters used are presented in table 1.
Table 1: Parameters used in all simulations | ITG: | TEM:
---|---|---
$T_{D}/T_{e}$: | $1.0$ | $1.0$
$\hat{s}$: | $0.8$ | $0.8$
$q_{0}$: | $1.4$ | $1.4$
$\varepsilon$: | $0.14$ | $0.14$
$R/L_{T_{D}},R/L_{T_{Z}}$: | $7.0$ | $3.0$
$R/L_{T_{e}}$: | $3.0$ | $7.0$
$N_{x}\times N_{ky}\times N_{z}$: | $5\times 1\times 24$ | $4\times 1\times 24$
$N_{v_{||}}\times N_{\mu}$: | $64\times 12$ | $64\times 12$
## 5 Results
### Impurity charge $Z$:
The main results obtained are the scalings of the peaking factor with the
charge of the impurity species. These are presented in figures 2(a) and 2(b),
showing ITG and TE mode dominated turbulence respectively.
(a) ITG mode dominated case
(b) TEM dominated case
Figure 2: Scalings of $PF_{0}$ with impurity charge $Z$; quasilinear GENE and
fluid results
The difference between figure 2(a) and 2(b) can be understood from the
properties of the convective velocity in (1). $V_{Z}$ contains a
thermodiffusive term $V_{T_{Z}}\sim\frac{1}{Z}\frac{R}{L_{T_{Z}}}$ and a
parallel impurity compression term
$V_{p_{Z}}\sim\frac{Z}{A_{Z}}k_{||}^{2}\sim\frac{Z}{A_{Z}q^{2}}$. The former
is generally outward ($V_{T_{Z}}>0$) for ITG and inward ($V_{T_{Z}}<0$) for TE
mode dominated transport, whereas for the latter the opposite is generally the
case.
### Magnetic shear $\hat{s}$:
The effect of magnetic shear on the peaking factor is shown in figures 3(a)
and 3(b). It is worth noting that a flux reversal, i.e. a change of sign in
$PF_{0}$, owing to a change in sign of $V_{Z}$, occurs for negative $\hat{s}$
for $Z\gtrsim 6$ in the TE mode dominated case, indicating a net outward
transport of the heavier elements. Similar trends are not seen in fluid
simulations, and this warrants further investigation.
(a) ITG mode dominated case
(b) TEM dominated case
Figure 3: Scalings of $PF_{0}$ with magnetic shearing $\hat{s}$; quasilinear
GENE results with $k\rho=0.3$
### Other parameters:
Scans of the dependence $PF_{0}$ on other parameters, such as $k\rho$ and
$L_{T}$, have also been carried out. The results are similar to those reported
in [5], [6] and [7] respectively. In most cases, only a weak dependence of
$PF_{0}$ is observed.
## 6 Conclusions and outlook
Quasilinear GENE simulations and fluid results show that peaking factor
increases with impurity charge $Z$ for ITG mode dominated transport, whereas
the opposite holds for TE mode dominated transport. In both cases $PF_{0}$
saturates for high $Z$.
For magnetic shear, a flux reversal is observed for negative magnetic shear in
the TEM dominated case. This is not seen in fluid simulations, and will be a
focus of future studies.
For other parameters investigated, weak scalings for $PF_{0}$ are observed, in
agreement with previous work.
## References
* [1] F. Jenko, W. Dorland, M. Kotschenreuther, and B. N. Rogers. Electron temperature gradient driven turbulence. Physics of Plasmas, 7(5):1904–10, May 2000.
* [2] F. Merz. Gyrokinetic Simulation of Multimode Plasma Turbulence. Monography, Westfälischen Wilhelms-Universität Münster, 2008.
* [3] J. Weiland. Collective Modes in Inhomogeneous Plasma. Institute of Physics Publishing, London, UK, 2000.
* [4] C. Angioni and A. G. Peeters. Direction of impurity pinch and auxiliary heating in tokamak plasmas. PRL, 96:095003–1–4, 2006.
* [5] T. Dannert. Gyrokinetische Simulation von Plasmaturbulenz mit gefangenen Teilchen und elektromagnetischen Effekten. Monography, Technischen Universität München, January 2005.
* [6] H. Nordman, R. Singh, and T. Fülöp et al. Influence of the radio frequency ponderomotive force on anomalous impurity transport in tokamaks. PoP, 15:042316–1–5, 2007.
* [7] T. Fülöp and H. Nordman. Turbulent and neoclassical impurity transport in tokamak plasmas. PoP, 16:032306–1–8, 2009.
This work benefited from an allocation on the EFDA HPC-FF computer.
|
arxiv-papers
| 2010-12-20T12:55:46 |
2024-09-04T02:49:15.853512
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Andreas Skyman, Hans Nordman, P\\\"ar Strand, Frank Jenko, Florian Merz",
"submitter": "Andreas Skyman",
"url": "https://arxiv.org/abs/1012.4308"
}
|
1012.4309
|
Core transport studies in fusion devices Pär Strand, Andreas Skyman and Hans
Nordman Department of radio and space science, Chalmers university of
technology, EURATOM-VR association
SE-412 96 Göteborg, Sweden & EFDA Task Force on Integrated Tokamak Modelling
elfps@chalmers.se http://www.chalmers.se/rss/EN/research/research-
groups/transport-theory/
## 1 Introduction
Comprehensive first principles modelling of fusion plasmas is a numerically
challenging: the complicated magnetic geometry and long range electromagnetic
interactions between multiple species introduce complex collective behaviour
in the plasma. In addition, steep density and temperature gradients combined
with an inhomogeneous magnetic field drives instabilities, resulting in non-
linear dynamics and turbulence.
The turbulence in magnetically confined fusion plasmas has important and non-
trivial effects on the quality of the energy confinement. These effects are
hard to make a quantitative assessment of analytically. The problem
investigated in this article is the transport of energy and particles, in
particular impurities, in a Tokamak plasma. Impurities from the walls of the
plasma vessel cause energy losses if they reach the plasma core. It is
therefore important to understand the transport mechanisms to prevent impurity
accumulation and minimize losses. This is an area of research where turbulence
plays a major role and is intimately associated with the performance of future
fusion reactors, such as ITER.
With the rapid growth and increased accessibility of high performance
computing (HPC) over the last few decades, plasma modelling has matured
towards an increased predictive capability. Particular emphasis has been put
on simulation of drift wave physics, widely accepted as the main source of
transport in the plasma core. Theory, reduced physics as well as first
principles modelling, and software are developed in a coordinated European
effort to produce a virtual Tokamak, a tool that will become indispensable,
both when it comes to developing and running ITER, and in the planning of
future reactors aimed at energy production ParFusEngDes.
## 2 Physical background
To arrive at a set of equations that are both meaningful and solvable, some
approximation is necessary. The advances in high performance computing have
allowed fusion modellers to move from fluid descriptions of the plasma to
kinetic descriptions as the basis for turbulence modelling. In _kinetic
theory_ the plasma is described through distribution functions of velocity and
position for the plasma species. Hence, kinetic equations are inherently six-
dimensional, however, magnetically confined particles are constrained to tight
orbits along field lines. This motivates averaging over the gyration, reducing
the problem to five-dimensional _gyrokinetic_ equations
## References
* [1]
* [2] This is a considerable gain, and the foundation of most current plasma codes. In this project, GENE, a European code developed by IPP-Garching
* [3] , has been used. GENE employs a second order accurate explicit finite difference scheme, and has demonstrated excellent parallel performance using in excess of $10000$ cores
* [4] .
* [5]
* [6]
## 3 Modelling plasmas
As mentioned above, gradients drive turbulence. Here, plasma core turbulence
induced by the so called _ion temperature gradient_ (ITG) mode
* [7] , has been studied. Parameters were taken from discharge #67730 of the _Joint European Torus_ (JET). A slice of the simulation domain, illustrating the turbulence, is shown in figure 1(a).
* [8] The GENE code employs a fixed grid in five dimensional phase space and a flux-tube geometry. For a typical simulation for main ions and one trace species, with electrons considered adiabatic, a resolution of $n_{x}\times n_{ky}\times n_{z}=48\times 48\times 32$ grid points in real space and of $n_{v}\times n_{\mu}=64\times 12$ in velocity space is necessary. A normal run with these parameters uses a minimum of $8000$ CPU hours and $384$ cores. Incorporating kinetic electrons increases the demand for high resolution in all of phase space, but most notably in velocity space, and also requires a shorter time step. Typically, such simulations require $40000$ CPU hours, occupying $1024$ cores.
* [9] The computations produce tens of gigabyte of data to be analysed. The non-linear data presented in figure 1(b) is the result of approximately twenty runs on the HPC cluster _Akka_ , and from the discussion above, it is readily understood that HPC is vital for this kind of study.
* [10]
* [11]
## 4 Transport in ITER-like plasmas
In general, transport of a species with atomic number $Z$ can locally be
described by a diffusive and a convective contribution. The former is
characterized by the diffusion coefficient $D_{Z}$, the latter by a convective
velocity or “pinch” $V_{Z}$. The _zero flux peaking factor_ , defined as
$PF_{0}=-R\,V_{Z}/D_{Z}$, is important in reactor design because it quantifies
the balance of convective and diffusive transport. This can be understood from
equation (1), where $\Gamma_{Z}$ is the impurity flux, $n_{Z}$ the density of
the impurity species and $R$ the major radius of the tokamak HansArtikel. For
the regime studied $\nabla n_{Z}$ is regarded as a constant, such that
$-\nabla n_{Z}/n_{Z}=1/L_{n_{Z}}$. Setting $\Gamma_{Z}=0$ in equation (1)
yields the interpretation of $PF_{0}$ as the gradient at which the impurity
flux vanishes.
* [12] $\Gamma_{Z}=-D_{Z}\nabla n_{Z}+n_{Z}V_{Z}\Leftrightarrow\frac{R\Gamma}{n_{Z}}=D_{Z}\frac{R}{L_{n_{Z}}}+RV_{Z}$ (1)
* [13] A positive sign of $PF_{0}$ indicates a net inward transport. This might lead to an accumulation of wall impurities in the plasma core, which can seriously hamper the efficiency of the fusion device. Understanding under what circumstances a negative peaking factor can be achieved is therefore an important issue for ITER and future fusion reactors.
* [14] Time series were generated by GENE on the _Akka_ HPC cluster for multiple values of $L_{n_{Z}}$, and from these $\Gamma_{Z}$ was extracted. The parameters $D_{Z}$ and $RV_{Z}$ were estimated and the peaking factor calculated as their quotient. This was repeated for several different nuclear charges $Z$. Results have been reported in HansArtikel, HansEPS and SkymanEPS.
* [15]
(a) crossection of the Deuterium density profile showing turbulent features
(b) zero-flux peaking factor for different $Z$ according to three different
models with adiabatic and one with kinetic electrons
Figure 1: Results from the non-linear simulations on Akka
* [16] Investigating where different models are in agreement is one of the aims of this kind of study: it is the first step towards an understanding of the physics that underlie their differences. In figure 1(b) the non-linear results from _Akka_ are compared with quasi-linear kinetic results and results from a fluid model developed at Chalmers
* [17] . As can be seen, the three models are in good qualitative agreement with one another for $Z>4$ (Be). This owes to the domination of the single strong ITG mode for these particular parameters, and it cannot be guaranteed to hold for other cases. Also in figure 1(b), the quasi-linear result with kinetic electrons is shown for comparison. From that one expects kinetic effects to be most pronounced for low $Z$. At the time of writing, the fully kinetic non-linear case is still being simulated.
* [18]
* [19] 00 ParFusEngDes P. Strand et al.: Simulation & high performance computing – building a predictive capability for Fusion, Fusion Engineering and Design (accepted), 2010 HansArtikel H. Nordman, A. Skyman, P Strand et al: Fluid and gyrokinetic simulations of impurity transport in JET, (manuscript), 2010 HansEPS H. Nordman, A. Skyman, P. Strand et al.: Modelling of impurity transport experiments at the Joint European Torus, Proceedings of EPS 2010 (accepted), 2010 SkymanEPS A. Skyman, H. Nordman, P. Strand et al.: Impurity transport in ITG and TE mode dominated turbulence, Proceedings of EPS 2010 (accepted), 2010
* [20] 00 Merz2008 F. Merz: Gyrokinetic Simulation of Multimode Plasma Turbulence, PhD thesis, Westfälischen Wilhelms-Universität Münster, 2008 Jenko2000 F. Jenko et al.: Electron temperature gradient driven turbulence, Physics of Plasmas, 7, pp. 1904–10, 2000 Weiland2000 J. Weiland: Collective Modes in Inhomogeneous Plasma, Institute of Physics Publishing, London (UK), 2000 [‡]GENE http://www.ipp.mpg.de/~fsj/gene/
* [21]
|
arxiv-papers
| 2010-12-20T12:56:13 |
2024-09-04T02:49:15.858012
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "P\\\"ar Strand, Andreas Skyman, Hans Nordman",
"submitter": "Andreas Skyman",
"url": "https://arxiv.org/abs/1012.4309"
}
|
1012.4507
|
020008 2010 A. Vindigni A. A. Fedorenko, CNRS-Lab. de Physique, ENS de Lyon,
France. 020008
We numerically study the geometry of a driven elastic string at its sample-
dependent depinning threshold in random-periodic media. We find that the
anisotropic finite-size scaling of the average square width $\overline{w^{2}}$
and of its associated probability distribution are both controlled by the
ratio $k=M/L^{\zeta_{\mathrm{dep}}}$, where $\zeta_{\mathrm{dep}}$ is the
random-manifold depinning roughness exponent, $L$ is the longitudinal size of
the string and $M$ the transverse periodicity of the random medium. The
rescaled average square width $\overline{w^{2}}/L^{2\zeta_{\mathrm{dep}}}$
displays a non-trivial single minimum for a finite value of $k$. We show that
the initial decrease for small $k$ reflects the crossover at $k\sim 1$ from
the random-periodic to the random-manifold roughness. The increase for very
large $k$ implies that the increasingly rare critical configurations,
accompanying the crossover to Gumbel critical-force statistics, display
anomalous roughness properties: a transverse-periodicity scaling in spite that
$\overline{w^{2}}\ll M$, and subleading corrections to the standard random-
manifold longitudinal-size scaling. Our results are relevant to understanding
the dimensional crossover from interface to particle depinning.
# Anisotropic finite-size scaling of an elastic string at the depinning
threshold in a random-periodic medium
S. Bustingorry [inst1] A. B. Kolton[inst1] E-mail: sbusting@cab.cnea.gov.arE-
mail: koltona@cab.cnea.gov.ar
(20 October 2010; 1 December 2010)
††volume: 2
99 inst1 CONICET, Centro Atómico Bariloche, 8400 San Carlos de Bariloche, Río
Negro, Argentina.
## 1 Introduction
The study of the static and dynamic properties of $d$-dimensional elastic
interfaces in $d+1$-dimensional random media is of interest in a wide range of
physical systems. Some concrete experimental examples are magnetic [1, 2, 3,
4] or ferroelectric [5, 6] domain walls, contact lines of liquids [7], fluid
invasion in porous media [8, 9], and fractures [10, 11]. In all these systems,
the basic physics is controlled by the competition between quenched disorder
(induced by the presence of impurities in the host materials) which promotes
the wandering of the elastic object, against the elastic forces which tend to
make the elastic object flat. One of the most dramatic and worth understanding
manifestations of this competition is the response of these systems to an
external drive.
The mean square width or roughness of the interface is one of the most basic
quantities in the study of pinned interfaces. In the absence of an external
drive, the ground state of the system is disordered but well characterized by
a self-affine rough geometry with a diverging typical width $w\sim
L^{\zeta_{\mathrm{eq}}}$, where $L$ is the linear size of the elastic object
and $\zeta_{\mathrm{eq}}$ is the equilibrium roughness exponent. When the
external force is increased from zero, the ground state becomes unstable and
the interface is locked in metastable states. To overcome the barriers
separating them and reach a finite steady-state velocity $v$ it is necessary
to exceed a finite critical force, above which barriers disappear and no
metastable states exist. For directed $d$-dimensional elastic interfaces with
convex elastic energies in a $D=d+1$ dimensional space with disorder, the
critical point is unique, characterized by the critical force $F=F_{c}$ and
its associated critical configuration [12]. This critical configuration is
also rough and self-affine such that $w\sim L^{\zeta_{\mathrm{dep}}}$ with
$\zeta_{\mathrm{dep}}$ the depinning roughness exponent. When approaching the
threshold from above, the steady-state average velocity vanishes like
$v\sim(F-F_{c})^{\beta}$ and the correlation length characterizing the
cooperative avalanche-like motion diverges as $\xi\sim(F-F_{c})^{-\nu}$ for
$F>F_{c}$, where $\beta$ is the velocity exponent and $\nu$ is the depinning
correlation length exponent [13, 14, 15, 16]. At finite temperature and for
$F\ll F_{c}$, the system presents an ultra-slow steady-state creep motion with
universal features [17, 18] directly correlated with its multi-affine geometry
[19, 20]. At very small temperatures the absence of a divergent correlation
length below $F_{c}$ shows that depinning must be regarded as a non-standard
phase transition [20, 21] while exactly at $F=F_{c}$, the transition is
smeared-out with the velocity vanishing as $v\sim T^{\psi}$, with $\psi$, the
so-called thermal rounding exponent [22, 23, 24, 25, 26, 27].
During the last years, numerical simulations have played an important role to
understand the physics behind the depinning transition thanks to the
development of powerful exact algorithms. In particular, the development of an
exact algorithm able to target efficiently the critical configuration and
critical force for a given sample [28, 29] has allowed to study, precisely,
the self-affine rough geometry at depinning [7, 28, 29, 30, 31], the sample-
to-sample critical force distribution [32], the critical exponents of the
depinning transition [26, 27, 33], the renormalized disorder correlator [34],
and the avalanche-size distribution in quasistatic motion [35]. Moreover, the
same algorithm has allowed to study, precisely, the transient universal
dynamics at depinning [36, 37], and an extension of it has allowed to study
low-temperature creep dynamics [20, 21].
In practice, the algorithm for targeting the critical configuration [28, 29]
has been numerically applied to directed interfaces of linear size $L$
displacing in a disordered potential of transverse dimension $M$, applying
periodic boundary conditions in both directions in order to avoid border
effects. This is thus equivalent to an elastic string displacing in a
disordered cylinder. The aspect ratio between longitudinal $L$ and transverse
$M$ periodicities must be carefully chosen, in order to have the desired
thermodynamic limit corresponding to a given experimental realization. In Ref.
[32] it was indeed shown that the critical force distribution $P(F_{c})$
displays three regimes associated with $M$: (i) At very small $M$ compared
with the typical width $L^{\zeta_{\mathrm{dep}}}$ of the interface, the
interface wraps the computational box several times in the transverse
direction, as shown schematically in Fig. 1(b), and therefore the periodicity
of the random medium is relevant and $P(F_{c})$ is Gaussian; (ii) At very
large $M$ compared with $L^{\zeta_{\mathrm{dep}}}$, as shown schematically in
Fig. 1(c), periodicity effects are absent but then the critical force, being
the maximum among many independent sub-critical forces, obeys extreme value
statistics and $P(F_{c})$ becomes a Gumbel distribution; (iii) In the
intermediate regime, where $M\approx L^{\zeta_{\mathrm{dep}}}$ and periodicity
effects are still irrelevant, as shown schematically in Fig. 1(a), the
distribution function is in between the Gaussian and the Gumbel distribution.
It has been argued that only the last case, where $M\approx
L^{\zeta_{\mathrm{dep}}}$, corresponds to the random-manifold depinning
universality class (periodicity effects absent) with a finite critical force
in the thermodynamic limit $L,M\to\infty$. This criterion does not give,
however, the optimal value of the proportionality factor between $M$ and
$L^{\zeta_{\mathrm{dep}}}$, and must be modified at finite velocity since the
crossover to the random-periodic universality class at large length-scales
depends also on the velocity [38]. To avoid this problem, it has been
therefore proposed to define the critical scaling in the fixed center of mass
ensemble [39]. The crossover from the random-manifold to the random-periodic
universality class is, however, physically interesting, as it can occur in
periodic elastic systems such as elastic chains. Remarkably, although the
mapping from a periodic elastic system (with given lattice parameter) in a
random potential to a non-periodic elastic system (such as an interface) in a
random potential with periodic boundary conditions is not exact, it was
recently shown that the lattice parameter does play the role of $M$ for
elastic interfaces with regard to the geometrical or roughness properties
[38]. Since the periodicity can often be experimentally tuned in such periodic
systems it is thus worth studying in detail the geometry of critical
interfaces of size $L$ as a function of $M$ with periodic boundary conditions,
and thus complement the study of the critical force in such systems [32].
In this paper, we study in detail, using numerical simulations, the
geometrical properties of the one-dimensional interface or elastic string
critical configuration in a random-periodic pinning potential as a function of
the aspect ratio parameter $k$, conveniently defined as
$k=M/L^{\zeta_{\mathrm{dep}}}$. We show that $k$ is indeed the only parameter
controlling the finite-size scaling (i.e. the dependence of observables with
the dimensions $L$ and $M$) of the average square width and its sample-to-
sample probability distribution. The scaled average square width
$\overline{w^{2}}L^{-2{\zeta_{\mathrm{dep}}}}$ is described by a universal
function of $k$ displaying a non-trivial single minimum at a finite value of
$k$. We show that while for small $k$ this reflects the crossover at $k\sim 1$
from the random-periodic to the random-manifold depinning universality class,
for large $k$ it implies that in the regime where the depinning threshold is
controlled by extreme value (Gumbel) statistics, critical configurations also
become rougher, and display an anomalous roughness scaling.
Figure 1: (a) Elastic string driven by a force $F$ in a random-periodic
medium with periodic boundary conditions. It is described by a displacement
field $u(z)$ and has a mean width $w$. The anisotropic finite-size scaling of
width fluctuations are controlled by the aspect-ratio parameter
$k=M/L^{\zeta_{\mathrm{dep}}}$, with ${\zeta_{\mathrm{dep}}}$ the random-
manifold roughness exponent at depinning. In the case $k\ll 1$ (b) periodicity
effects are important, while when $k\gg 1$ (c) they are not important but the
roughness scaling of the critical configuration is anomalous.
## 2 Method
The model we consider here is an elastic string in $(1+1)$ dimensions
described by a single valued function $u(z,t)$, which gives the transverse
displacement $u$ as a function of the longitudinal direction $z$ and the time
$t$ [see Fig. 1(a)]. The zero-temperature dynamics of the model is given by
$\gamma\,\partial_{t}u(z,t)=c\,\partial^{2}_{z}u(z,t)+F_{p}(u,z)+F,$ (1)
where $\gamma$ is the friction coefficient and $c$ the elastic constant. The
first term in the right hand side derives from an harmonic elastic energy. The
effects of a random-bond type disorder is given by the pinning force
$F_{p}(u,z)=-\partial_{u}U(u,z)$. The disorder potential $U(u,z)$ has zero
average and sample-to-sample fluctuations given by
$\overline{\left[U(u,z)-U(u^{\prime},z^{\prime})\right]^{2}}=\delta(z-z^{\prime})\,R^{2}(u-u^{\prime}),$
(2)
where the overline indicates average over disorder realizations and $R(u)$
stands for a correlator of finite range $r_{f}$ [18]. Finally, $F$ represents
the uniform external drive acting on the string. Physically, this model can
phenomenologically describe, for instance, a magnetic domain wall in a thin
film ferromagnetic material with weak and randomly located imperfections [1],
being $F$ proportional to an applied external magnetic field pushing the wall
in the energetically favorable direction.
In order to numerically solve Eq. (1), the system is discretized in the
$z$-direction in $L$ segments of size $\delta z=1$, i.e. $z\to j=0,...,L-1$,
while keeping $u_{j}(t)$ as a continuous variable. To model the continuous
random potential, a cubic spline is used, which passes through $M$ regularly
spaced uncorrelated Gaussian number points [30]. For the numerical simulations
performed here we have used, without loss of generality, $\gamma=1$, $c=1$ and
$r_{f}=1$ and a disorder intensity $R(0)=1$. In both spatial dimensions we
have used periodic boundary conditions, thus defining a $L\times M$ system.
The critical configuration $u_{c}(z)$ and force $F_{c}$ are defined from the
pinned (zero-velocity) configuration with the largest driving force $F$ in the
long time limit dynamics. They are thus the real solutions of
$c\,\partial^{2}_{z}u(z)+F_{p}(u,z)+F=0,$ (3)
such that for $F>F_{c}$ there are no further real solutions (pinned
configurations). Middleton theorems [12] assure that for Eqs. (3) the solution
exists and it is unique for both $u_{c}(z)$ and $F_{c}$, and that above
$F_{c}$ the string trajectory in an $L$ dimensional phase-space is trapped
into a periodic attractor (for a system with periodic boundary conditions as
the one we consider). In other words, the critical configuration is the
marginal fixed point solution or critical state of the dynamics, being $F_{c}$
the critical point control parameter of a Hopf bifurcation. Solving the
$L$-dimensional system of Eqs. (3) for large $L$ directly is a formidable
task, due to the non-linearity of the pinning force $F_{p}$. On the other
hand, solving the long-time dynamics at different driving forces $F$ to
localize $F_{c}$ and $u_{c}$ is very inefficient due to the critical slowing
down. Fortunately, Middleton theorems, and in particular the “non-passing
rule”, can be used again to devise a precise and very efficient algorithm
which allows to obtain the critical force $F_{c}$ and the critical
configuration $u^{c}_{j}$ for each independent disorder realization
iteratively without solving the actual dynamics nor directly inverting the
system of Eqs. (3) [30]. Once the critical force and the critical
configuration are determined with this algorithm, we can compute the different
observables. In particular, the square width or roughness of the string at the
critical point for a given disorder realization is defined as
$w^{2}=\frac{1}{L}\sum_{j=0}^{L-1}\left[u^{c}_{j}-\frac{1}{L}\sum_{k=0}^{L-1}u^{c}_{k}\right]^{2}.$
(4)
Computing $w^{2}$ for different disorder realizations allows us to compute its
disorder average $\overline{w^{2}}$ and the sample-to-sample probability
distribution $P(w^{2})$. In addition, the average structure factor associated
to the critical configuration is
$S_{q}=\frac{1}{L}\overline{\left|\sum_{j=0}^{L-1}u^{c}_{j}\,e^{-iqj}\right|^{2}},$
(5)
where $q=2\pi n/L$, with $n=1,...,L-1$. One can show, using a simple
dimensional analysis, that given a roughness exponent $\zeta$, such that
$\overline{w^{2}}\sim L^{2\zeta}$, the structure factor behaves as $S(q)\sim
q^{-(1+2\zeta)}$ for small $q$, thus yielding an estimate to $\zeta$ without
changing $L$. To compute averages over disorder and sample-to-sample
fluctuations, we consider between $10^{3}$ and $10^{4}$ independent disorder
realizations depending on the size of the system.
## 3 Results
### 3.1 Roughness at the critical point
Figure 2: The scaling of $\overline{w^{2}}$ for the critical configuration at
different $M$ values as indicated. The curves for $M=64$ and $16384$ are
shifted upwards for clarity. The dashed and dotted lines are guides to the eye
showing the expected slopes corresponding to the different roughness
exponents.
Figure 2 shows the scaling of the square width of the critical configuration
$\overline{w^{2}}$ with the longitudinal size of the system $L$ for
$L=32,64,128,256,512$ and different values of $M$. When $M$ is small, $M=8$,
for all the $L$ values shown we observe $\overline{w^{2}}\sim
L^{2\zeta_{\mathrm{L}}}$ with $\zeta_{\mathrm{L}}=1.5$, corresponding to the
Larkin exponent in $(1+1)$ dimensions. This value is different from the value
$\zeta_{\mathrm{dep}}=1.25$ [33, 40] expected for the random-manifold
universality class, and is thus indicating that the periodicity effects are
important for this joint values of $M$ and $L$. This situation is
schematically represented in Fig. 1(b). This result is a numerical
confirmation of the two-loop functional renormalization group result of Ref.
[16] which shows that the $\zeta=0$ fixed point, leading to a universal
logarithmic growth of displacements at equilibrium is unstable. The
fluctuations are governed, instead, by a coarse-grained generated random-force
as in the Larkin model, yielding a roughness exponent
$\zeta_{\mathrm{L}}=(4-d)/2$ in $d$ dimensions [16], which agrees with our
result for $d=1$. We can thus say that for small enough $M$ (compared to $L$)
the system belongs to the same random-periodic depinning universality class as
charge density wave systems [14, 41], which strictly correspond to $M=1$.
When $M$ is large, on the other hand, $M=16384$ in Fig. 2, for all the $L$
values considered the exponent is consistent with $\zeta_{\mathrm{dep}}$, of
the random-manifold universality class. This situation is schematically
represented in Fig. 1(c), and we will show later that, for this elongated
samples, the effects of extreme value statistics are already visible.
For intermediate values of $M$, such as $M=64$ in Fig. 2, we can observe the
crossover in the scale-dependent roughness exponent
$\zeta(L)\sim\frac{1}{2}\frac{d\log w^{2}}{d\log L}$ changing from
$\zeta_{\mathrm{dep}}$ to $\zeta_{\mathrm{L}}$ as $L$ increases, as indicated
by the dashed and dotted lines. This crossover, from the random-manifold to
the random-periodic depinning geometry, occurs at a characteristic distance
$l^{*}\sim M^{1/\zeta_{\mathrm{dep}}}$, when the width in the random-manifold
regime reaches the transverse dimension or periodicity $M$. At finite
velocity, this crossover length remains constant up to a non-trivial
characteristic velocity and then decreases with increasing velocity [38].
Figure 3: Structure factor of the critical configuration for $L=256$ and
different $M$ values, as indicated. The curves for $M=64$ and $16384$ are
shifted upwards for clarity. The dashed and dotted lines are guides to the eye
showing the expected slopes corresponding to the different roughness
exponents.
The above mentioned geometrical crossover can be studied in more details
through the analysis of the structure factor $S(q)$, for a line of fixed size
$L$. In Fig. 3 we show $S(q)$ for $L=256$ and $M=8,64,16384$. For the
intermediate value $M=64$ a crossover between the two regimes is visible, and
can be described by
$S_{q}\sim\left\\{\begin{array}[]{ll}q^{-(1+2\zeta_{\mathrm{L}})}&q\ll
q^{*},\\\ q^{-(1+2\zeta_{\mathrm{dep}})}&q\gg q^{*}.\\\ \end{array}\right.$
(6)
with $q^{*}$ expected to scale as $q^{*}\sim l^{*-1}\sim
M^{-1/\zeta_{\mathrm{dep}}}$. Therefore, the structure factor should scale as
$S_{q}M^{-(2+1/\zeta_{\mathrm{dep}})}=H(x)$, where the scaled variable is
$x=q\,M^{1/\zeta_{\mathrm{dep}}}\sim q/q^{*}$ and the scaling function behaves
as
$H(x)\sim\left\\{\begin{array}[]{ll}x^{-(1+2\zeta_{\mathrm{L}})}&x\ll 1,\\\
x^{-(1+2\zeta_{\mathrm{dep}})}&x\gg 1.\\\ \end{array}\right.$ (7)
The collapse of Fig. 4 for $L=256$ and different values of $M=2^{p}$ with
$p=3,4,...,14$ shows that this scaling form is a very good approximation.
However, as we show below, small corrections can be expected fully in the
random-manifold regime in the large $ML^{-\zeta_{\mathrm{dep}}}$ limit of very
elongated samples.
Figure 4: Scaling of the structure factor of the critical configuration for
$L=256$ and different values of the transverse size $M=2^{p}$ with
$p=3,4,...,14$ $M$. Although the values of the two exponents are very close,
the change in the slope of the scaling function against the scaling variable
$x=q\,M^{1/\zeta_{\mathrm{dep}}}$ is clearly observed. Figure 5: (a) Squared
width of the critical configuration as a function of $M$ for different system
sizes $L$ as indicated. (b) Scaling of the width in (a), showing that the
relevant control parameter is $M/L^{\zeta_{\mathrm{dep}}}$. The dashed line in
(a) and (b) corresponds to $\overline{w^{2}}=M^{2}$, which is always to the
left of the minimum of $\overline{w^{2}}$ occurring at
$k^{*}=m^{*}L^{-\zeta_{\mathrm{dep}}}$. The solid line indicates
$k^{2(1-\zeta_{\mathrm{L}}/\zeta_{\mathrm{dep}})}$ which is the behavior
expected purely from the random-periodic to random-manifold crossover at the
characteristic distance $l^{*}\sim M^{1/\zeta_{\mathrm{dep}}}$.
In Fig. 5(a), we show $\overline{w^{2}}$ as a function of the transverse
periodicity $M$ for different values of the longitudinal periodicity $L$.
Remarkably, $\overline{w^{2}}$ is a non-monotonic function of $M$. For small
$M$ it decreases towards an $L$ dependent minimum $m^{*}$, and then increases
with increasing $M$, in the regime where the extreme value statistics starts
to affect the distribution of the critical force [32]. Since the only typical
transverse scale in Fig. 5(a) is set by the minimum $m^{*}$, we can expect
$\overline{w^{2}}\sim m^{*2}G(M/m^{*})$ with $G(x)$ some universal function.
On the other hand, since the only relevant characteristic length-scale of the
problem is set by the crossover between the random-periodic regime and the
random-manifold regime, we can simply write $m^{*}\sim
L^{\zeta_{\mathrm{dep}}}$ and therefore
$\overline{w^{2}}\,L^{-2\zeta_{\mathrm{dep}}}\sim
G(M\,L^{-\zeta_{\mathrm{dep}}}).$ (8)
This scaling form is confirmed in Fig. 5(b) and shows that the aspect-ratio
parameter $k=ML^{-\zeta_{\mathrm{dep}}}$ fully controls the anisotropic
finite-size scaling of the problem. It is worth, however, noting some
interesting consequences of the result of Fig. 5(b), as we describe below.
Since at very small $k$ the interface is in the random-periodic regime, Eq.
(8) should led to $\overline{w^{2}}\sim L^{2\zeta_{\mathrm{L}}}$ and therefore
one deduces that,
$G(k)\sim k^{2(1-{\zeta_{\mathrm{L}}}/{\zeta_{\mathrm{dep}}})},\;\;\;k\ll
k^{*},$ (9)
where $k^{*}=m^{*}L^{-\zeta_{\mathrm{dep}}}$. The fact that the random-
periodic roughness exponent ${\zeta_{\mathrm{L}}}=3/2$ is larger than the
random-manifold one ${\zeta_{\mathrm{dep}}}\approx 5/4$ consequently implies
an initial decrease of $G(k)$ as $G(k)\sim k^{-2/5}$, as shown in Fig. 5(b) by
the solid line. Periodicity effects, or the crossover from random-periodic to
random-manifold, thus explain the initial decrease of $G(k)$ seen in Fig.
5(b), or the initial decrease of $\overline{w^{2}}$ against $M$ for fixed $L$,
seen in Fig. 5(a). At this respect, it is then worth noting that the line
$\overline{w^{2}}=M^{2}$, shown by a dashed line, lies completely in the
regime $k<k^{*}$ implying that the naive criterion $\overline{w^{2}}<M^{2}$ is
not enough to avoid periodicity effects, and to have the system fully in the
random-manifold regime. As we show later, this is related with the shape of
the probability distribution of $P(w^{2})$ which displays sample-to-sample
fluctuations of the order of the average $\overline{w^{2}}$.
The presence of a minimum at $k^{*}$ in the function $G(k)$ and in particular
its slower than power-law increase for $k>k^{*}$ is non-trivial and
constitutes one of the main results of the present work. This result shows
that corrections to the standard scaling $\overline{w^{2}}\sim
L^{\zeta_{\mathrm{dep}}}$ may arise from the aspect-ratio dependence of the
prefactor $G(k)$. On the one hand, $\overline{w^{2}}$ now grows with $M$ for
$L$ fixed, in spite that $\overline{w^{2}}\ll M^{2}$, i.e. transverse-
size/periodicity scaling is present. On the other hand, the scaling of
$\overline{w^{2}}$ with $L$ is slower in this regime, due to subleading
scaling corrections coming from $G(k)$. The precise origin of these
interesting leading and subleading corrections in the finite-size anisotropic
scaling are highly non-trivial. Since the critical configurations in this
regime have the constant roughness exponent $\zeta_{\mathrm{dep}}$ of the
random-manifold universality class, the slow increase of $G(k)$ cannot be
attributed to a geometrical crossover effect, as for the case $k<k^{*}$.
However, we might relate this effect to the crossover in the critical force
statistics, from Gaussian to Gumbel, in the $k\gg k^{*}$ limit [32]. In the
Gumbel regime, the average critical force is expected to increase as
$F_{c}\sim\log(M/L^{\zeta}_{\mathrm{dep}})\equiv\log k$ [39], since the sample
critical force can be roughly regarded as the maximum among
$M/L^{\zeta}_{\mathrm{dep}}$ independent sub-critical forces and
configurations [32]. The increase in the critical force might be therefore
correlated with the slow increase of roughness. The physical connection
between the two is subtle though, since a large critical force in a very
elongated sample could be achieved both by profiting very rare correlated
pinning forces such as accidental columnar defects, or by profiting very rare
non-correlated strong pinning forces. Since in the first case the critical
configuration would be more correlated and in general less rough than for less
elongated samples (smaller $k$), contrary to our numerical data of Fig. 5(b),
we think that the second cause is more plausible. We can thus think that in
the $k\gg k^{*}$ limit of extreme value statistics of $F_{c}$, the effective
disorder strength on the critical configuration increases with $k$. This might
be translated into the universal function $G(k)$, such that
$\overline{w^{2}}\approx L^{2\zeta_{\mathrm{dep}}}G(k)$ can increase for
increasing values of $k$ at fixed $L$ in such regime. A quantitative
description of these scaling corrections remains an open challenge.
### 3.2 Distribution function
Figure 6: Scaling function $\Phi(x)$ for $L=256$ and different values of
$M=8,128,2048,16384$, which shows the change with the transverse size $M$.
We now analyze sample-to-sample fluctuations of the square width $w^{2}$ by
computing its probability distribution $P(w^{2})$. This property is relevant
as $w^{2}$ fluctuates even in the thermodynamic limit for critical interfaces
with a positive roughness exponent [42]. It has been computed for models with
dynamical disorder such as random-walk [43] or Edwards–Wilkinson interfaces
[44, 45], the Mullins Herrings model [46] and for non-Markovian Gaussian
signals in general [47, 48]. It has also been calculated for non-linear models
such as the one-dimensional Kardar–Parisi–Zhang model [49, 50] and for the
quenched Edwards–Wilkinson model at equilibrium [51].
In particular, the probability distribution $P(w^{2})$ of critical interfaces
at the depinning transition was studied analytically [52], numerically [31]
and also experimentally for contact lines in partial wetting [7]. Remarkably,
non-Gaussian effects in depinning models are found to be smaller than $0.1\%$
[31, 52], thus showing that $P(w^{2})$ is strongly determined by the self-
affine (critical) geometry itself, rather than by the particular mechanism
producing it. As in all the above mentioned systems the width distribution
$P(w^{2})$ at different universality classes of the depinning transition was
found to scale as
$\overline{w^{2}}P(w^{2})\approx\Phi_{\zeta}\left(\frac{w^{2}}{\overline{w^{2}}}\right).$
(10)
with $\Phi_{\zeta}$ an universal function, which only depends on the roughness
exponent $\zeta$ and on boundary conditions when the global width is
considered [47, 48]. In this way, $\overline{w^{2}}$ is the only
characteristic length-scale of the system, absorbing the system longitudinal
size $L$, and all the non-universal parameters of the model such as the
elastic constant of the interface, the strength of the disorder and/or the
temperature. Since $\Phi_{\zeta}$ can be easily generated using non-Markovian
Gaussian signals [53], the quantity $\overline{w^{2}}P(w^{2})$ is a good
observable to extract the roughness exponent of a critical interface from
experimental data.
Figure 7: Scaling function $\Phi(x)$ for different values of
$L=32,64,128,256$ while keeping (a) $k=M/L^{\zeta_{\mathrm{dep}}}\approx 1$
and (b) $k=M/L^{\zeta_{\mathrm{dep}}}\approx 0.025$. The dotted line
corresponds to the scaling function of the non-disordered Edwards–Wilkinson
equation [43], while the continuous and dashed lines correspond to the scaling
functions of Gaussian signals with $\zeta=1.25$ and $\zeta=1.5$, respectively
[31, 53].
In Fig. 6, we show how the scaled distribution function
$\Phi(x)\equiv\overline{w^{2}}\,P(x\;\overline{w^{2}})$ looks like for the
depinning transition in a random-periodic medium for a fixed value $L=256$ and
different values of $M$. We see that $\Phi(x)$ depends on $M$ for small $M$
but converges to a fixed shape for large $M$. We also note that for all $M$
$\Phi(x)$ extends appreciably beyond $x=1$ explaining why the criterion
$\overline{w^{2}}\lesssim M^{2}$ is not enough to be fully in the random-
manifold regime, as noted in Fig. 5.
In Fig. 7, we show the scaling function $\Phi(x)$ for different values of $L$
and $M$ but fixing the aspect-ratio parameter $k=M/L^{\zeta_{\mathrm{dep}}}$,
$k\approx 1>k^{*}$ in Fig. 7(a) and $k\approx 0.025\ll k^{*}$ in Fig. 7(b),
with $k^{*}$ the minimum of $\overline{w^{2}}$. Since data for the same $k$
practically collapses into the same curve, we can write for our case:
$\overline{w^{2}}P(w^{2})=\Phi\left(\frac{w^{2}}{\overline{w^{2}}},k\right).$
(11)
Therefore, the anisotropic scaling of the probability distribution is fully
controlled by $k$, as it was found for $\overline{w^{2}}$.
In Figs. 7(a) and (b), we also show the universal functions $\Phi_{\zeta_{L}}$
and $\Phi_{\zeta_{\mathrm{dep}}}$ generated using non-Markovian Gaussian
signals [31, 53], and for comparison we also show $\Phi_{1/2}$ corresponding
to the Markovian periodic Gaussian signal or the Edwards–Wilkinson equation
[43]. Comparing this with the collapsed data for depinning, we see that the
function $\Phi\left(\frac{w^{2}}{\overline{w^{2}}},k\right)$ respects the
limits
$\displaystyle\Phi\left(x,k\to 0\right)$ $\displaystyle=$
$\displaystyle\Phi_{\zeta_{\mathrm{L}}}(x),$
$\displaystyle\Phi\left(x,k\gtrsim k^{*}\right)$ $\displaystyle\approx$
$\displaystyle\Phi_{\zeta_{\mathrm{dep}}}(x),$ (12)
as expected from the existence of the geometric crossover between the
roughness exponents ${\zeta_{\mathrm{L}}}$ for $k\to 0$ and
${\zeta_{\mathrm{dep}}}$ for $k>k^{*}$. For intermediate values $k<k^{*}$,
however, $\Phi\left(\frac{w^{2}}{\overline{w^{2}}},k\right)$ does not
necessarily coincide with the one of a Gaussian signal function $\Phi_{\zeta}$
for a given $\zeta$, since the critical configuration includes a crossover
length $l^{*}\lesssim L$. Whether multi-affine or effective exponent self-
affine non-Markovian Gaussian signals can be used to describe satisfactorily
these intermediate cases is an interesting open issue.
## 4 Conclusions
We have numerically studied the anisotropic finite-size scaling of the
roughness of a driven elastic string at its sample-dependent depinning
threshold in a random medium with periodic boundary conditions in both the
longitudinal and transverse directions. The average square width
$\overline{w^{2}}$ and its probability distribution are both controlled by the
parameter $k=M/L^{\zeta_{\mathrm{dep}}}$. A non-trivial single minimum for a
finite value of $k$ was found in $\overline{w^{2}}/L^{2\zeta_{\mathrm{dep}}}$.
For small $k$, the initial decrease of $\overline{w^{2}}$ reflects the
crossover from the random-periodic to the random-manifold roughness. For very
large $k$, the growth with $k$ implies that the crossover to Gumbel statistics
in the critical forces induces corrections to $G(k)$, that grow with $k$, to
the string roughness scaling $\overline{w^{2}}\approx
G(k)L^{2\zeta_{\mathrm{dep}}}$. These increasingly rare critical
configurations thus have an anomalous roughness scaling: they have a
transverse-size/periodicity scaling in spite that its width is
$\overline{w^{2}}\ll M^{2}$, and subleading (negative) corrections to the
standard random-manifold longitudinal-size scaling.
Our results could be useful for understanding roughness fluctuations and
scaling in finite experimental systems. The crossover from random-periodic to
random-manifold roughness could be studied in periodic elastic systems with
variable periodicity, such as confined vortex rows [54] and single-files of
macroscopically charged particles [55] or colloids [56], with additional
quenched disorder. The rare-event dominated scaling corrections to the
interface roughness scaling could be studied in systems with a large
transverse dimension, such as domain walls in ferromagnetic nanowires [57].
For the later case, it would be interesting to have a quantitative theory,
making the connection between the extreme value statistics of the depinning
threshold and the anomalous scaling corrections to the roughness of such rare
critical configurations. This would allow to understand the dimensional
crossover, from interface to particle depinning.
###### Acknowledgements.
This work was supported by CNEA, CONICET under Grant No. PIP11220090100051͒,
and ANPCYT under Grant No. PICT2007886. A. B. K. acknowledges Universidad de
Barcelona, Ministerio de Ciencia e Innovación (Spain) and Generalitat de
Catalunya for partial funding through I3 program.
## References
* [1] S Lemerle, J Ferré, C Chappert, V Mathet, T Giamarchi, P Le Doussal, Domain wall creep in an Ising ultrathin magnetic film, Phys. Rev. Lett. 80, 849 (1998).
* [2] M Bauer, A Mougin, J P Jamet, V Repain, J Ferré, S L Stamps, H Bernas, C Chappert, Deroughening of domain wall pairs by dipolar repulsion, Phys. Rev. Lett. 94, 207211 (2005).
* [3] M Yamanouchi, D Chiba, F Matsukura, T Dietl, H Ohno, Velocity of domain-wall motion induced by electrical current in the ferromagnetic semiconductor (Ga,Mn)As, Phys. Rev. Lett. 96, 096601 (2006).
* [4] P J Metaxas, J P Jamet, A Mougin, M Cormier, J Ferré, V Baltz, B Rodmacq, B Dieny, R L Stamps, Creep and flow regimes of magnetic domain-wall motion in ultrathin Pt/Co/Pt films with perpendicular anisotropy, Phys. Rev. Lett. 99, 217208 (2007).
* [5] P Paruch, T Giamarchi, J M Triscone, Domain wall roughness in epitaxial ferroelectric PbZr0.2Ti0.8O3 thin films, Phys. Rev. Lett. 94, 197601 (2005).
* [6] P Paruch, J M Triscone, High-temperature ferroelectric domain stability in epitaxial PbZr0.2Ti0.8O3 thin films, Appl. Phys. Lett. 88, 162907 (2006).
* [7] S Moulinet, A Rosso, W Krauth, E Rolley, Width distribution of contact lines on a disordered substrate, Phys. Rev. E 69, 035103(R) (2004).
* [8] N Martys, M Cieplak, M O Robbins, Critical phenomena in fluid invasion of porous media, Phys. Rev. Lett. 66, 1058 (1991).
* [9] I Hecht, H Taitelbaum, Roughness and growth in a continuous fluid invasion model, Phys. Rev. E 70, 046307 (2004).
* [10] E Bouchaud, J P Bouchaud, D S Fisher, S Ramanathan, J R Rice, Can crack front waves explain the roughness of cracks?, J. Mech. Phys. Solids 50, 1703 (2002).
* [11] M Alava, P K V V Nukalaz, S Zapperi, Statistical models of fracture, Adv. Phys. 55, 349 (2006).
* [12] A A Middleton, Asymptotic uniqueness of the sliding state for charge-density waves, Phys. Rev. Lett. 68, 670 (1992).
* [13] D S Fisher, Sliding charge-density waves as a dynamic critical phenomenon, Phys. Rev. B 31, 1396 (1985).
* [14] O Narayan, D S Fisher, Critical behavior of sliding charge-density waves in $4-\varepsilon$ dimensions, Phys. Rev. B 46, 11520 (1992).
* [15] T Nattermann, S Stepanow, L H Tang, H Leschhorn, Dynamics of interface depinning in a disordered medium, J. Phys. II 2, 1483 (1992).
* [16] P Le Doussal, K J Wiese, P Chauve, Two-loop functional renormalization group theory of the depinning transition, Phys. Rev. B 66, 174201 (2002).
* [17] L B Ioffe, V M Vinokur, Dynamics of interfaces and dislocations in disordered media, J. Phys. C: Solid State Phys. 20, 6149 (1987).
* [18] P Chauve, T Giamarchi, P Le Doussal, Creep and depinning in disordered media, Phys. Rev. B 62, 6241 (2000).
* [19] A B Kolton, A Rosso, T Giamarchi, Creep motion of an elastic string in a random potential, Phys. Rev. Lett. 94, 047002 (2005).
* [20] A B Kolton, A Rosso, T Giamarchi, W Krauth, Creep dynamics of elastic manifolds via exact transition pathways, Phys. Rev. B 79, 184207 (2009).
* [21] A B Kolton, A Rosso, T Giamarchi, W Krauth, Dynamics below the depinning threshold in disordered elastic systems, Phys. Rev. Lett. 97, 057001 (2006).
* [22] L W Chen, M C Marchetti, Interface motion in random media at finite temperature, Phys. Rev. B 51, 6296 (1995).
* [23] D Vandembroucq, R Skoe, S Roux, Universal depinning force fluctuations of an elastic line: Application to finite temperature behavior, Phys. Rev. E 70, 051101 (2004).
* [24] U Nowak, K D Usadel, Influence of temperature on the depinning transition of driven interfaces, Europhys. Lett. 44, 634 (1998).
* [25] L Roters, A Hucht, S Lübeck, U Nowak, K D Usadel, Depinning transition and thermal fluctuations in the random-field Ising model, Phys. Rev. E 60, 5202 (1999).
* [26] S Bustingorry, A B Kolton, T Giamarchi, Thermal rounding of the depinning transition, Europhys. Lett. 81, 26005 (2008).
* [27] S Bustingorry, A B Kolton, T Giamarchi, (unpublished).
* [28] A Rosso, W Krauth, Monte Carlo dynamics of driven elastic strings in disordered media, Phys. Rev. B 65, 012202 (2001).
* [29] A Rosso, W Krauth, Origin of the roughness exponent in elastic strings at the depinning threshold, Phys. Rev. Lett. 87, 187002 (2001).
* [30] A Rosso, W Krauth, Roughness at the depinning threshold for a long-range elastic string, Phys. Rev. E 65, 025101(R) (2002).
* [31] A Rosso, W Krauth, P Le Doussal, J Vannimenus, K J Wiese, Universal interface width distributions at the depinning threshold, Phys. Rev. E 68, 036128 (2003).
* [32] C Bolech, A Rosso, Universal statistics of the critical depinning force of elastic systems in random media, Phys. Rev. Lett. 93, 125701 (2004).
* [33] O Duemmer, W Krauth, Critical exponents of the driven elastic string in a disordered medium, Phys. Rev. E 71, 061601 (2005).
* [34] A Rosso, P L Doussal, K J Wiese, Numerical calculation of the functional renormalization group fixed-point functions at the depinning transition, Phys. Rev. B 75, 220201 (2007).
* [35] A Rosso, P Le Doussal, K J Wiese, Avalanche-size distribution at the depinning transition: A numerical test of the theory, Phys. Rev. B 80, 144204 (2009).
* [36] A B Kolton, A Rosso, E V Albano, T Giamarchi, Short-time relaxation of a driven elastic string in a random medium, Phys. Rev. B 74, 140201 (2006).
* [37] A B Kolton, G Schehr, P Le Doussal, Universal nonstationary dynamics at the depinning transition, Phys. Rev. Lett. 103, 160602 (2009).
* [38] S Bustingorry, A B Kolton, T Giamarchi, Random-manifold to random-periodic depinning of an elastic interface, Phys. Rev. B 82, 094202 (2010).
* [39] A A Fedorenko, P Le Doussal, K J Wiese, Universal distribution of threshold forces at the depinning transition, Phys. Rev. E 74, 041110 (2006).
* [40] A Rosso, A K Hartmann, W Krauth, Depinning of elastic manifolds, Phys. Rev. E 67, 021602 (2003).
* [41] O Narayan, D Fisher, Threshold critical dynamics of driven interfaces in random media, Phys. Rev. B 48, 7030 (1993).
* [42] Z Rácz, Scaling functions for nonequilibrium fluctuations: A picture gallery, SPIE Proc. 5112, 248 (2003).
* [43] G Foltin, K Oerding, Z Rácz, R L Workman, R K P Zia, Width distribution for random-walk interfaces, Phys. Rev. E 50, R639 (1994).
* [44] T Antal, Z Rácz, Dynamic scaling of the width distribution in Edwards–Wilkinson type models of interface dynamics, Phys. Rev. E 54, 2256 (1996).
* [45] S Bustingorry, L F Cugliandolo, J L Iguain, Out-of-equilibrium relaxation of the Edwards–Wilkinson elastic line, J. Stat. Mech.: Theor. Exp. P09008 (2007).
* [46] M Plischke, Z Rácz, R K P Zia, Width distribution of curvature-driven interfaces: A study of universality, Phys. Rev. E 50, 3589 (1994).
* [47] A Rosso, R Santachiara, W Krauth, Geometry of Gaussian signals, J. Stat. Mech.: Theor. Exp. L08001 (2005).
* [48] R Santachiara, A Rosso, W Krauth, Universal width distributions in non-Markovian Gaussian processes, J. Stat. Mech.: Theor. Exp. P02009 (2007).
* [49] E Marinari, A Pagnani, G Parisi, Z Rácz, Width distributions and the upper critical dimension of Kardar–Parisi–Zhang interfaces, Phys. Rev. E 65, 026136 (2002).
* [50] S Bustingorry, Aging dynamics of non-linear elastic interfaces: The Kardar–Parisi–Zhang equation, J. Stat. Mech.: Theor. Exp. P10002 (2007).
* [51] S Bustingorry, J L Iguain, S Chamon, L F Cugliandolo, D Domínguez, Dynamic fluctuations of elastic lines in random environments, Europhys. Lett. 76, 856 (2006).
* [52] P Le Doussal, K J Wiese, Higher correlations, universal distributions, and finite size scaling in the field theory of depinning, Phys. Rev. E 68, 046118 (2003).
* [53] W Krauth, Statistical mechanics: Algorithms and computations, Oxford University Press, New York (2006).
* [54] N Kokubo, R Besseling, P Kes, Dynamic ordering and frustration of confined vortex rows studied by mode-locking experiments, Phys. Rev. B 69, 064504 (2004).
* [55] C Coste, J B Delfau, C Even, M S Jean, Single-file diffusion of macroscopic charged particles, Phys. Rev. E 81, 051201 (2010).
* [56] S Herrera-Velarde, A Zamudio-Ojeda, R Castañeda-Priego, Ordering and single-file diffusion in colloidal systems, J. Chem. Phys. 133, 114902 (2010).
* [57] K J Kim, J C Lee, S M Ahn, K S Lee, C W Lee, Y J Cho, S Seo, K H Shin, S B Choe, H W Lee, Interdimensional universality of dynamic interfaces, Nature (London) 458, 740 (2009).
|
arxiv-papers
| 2010-12-20T23:42:44 |
2024-09-04T02:49:15.869618
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sebastian Bustingorry, Alejandro B. Kolton",
"submitter": "Luis Ariel Pugnaloni",
"url": "https://arxiv.org/abs/1012.4507"
}
|
1012.4542
|
# Impact of Mistiming on the Achievable Information Rate of Rake Receivers in
DS-UWB Systems
Chunhua Geng, Yukui Pei, Jiaqi Zhang and Ning Ge This work is supported by
National Nature Science Foundation of China No. 60928001 and 60972019,
National Basic Research Program of China under grant No. 2007CB310608, and the
National Science & Technology Major Project under grant No. 2009ZX03006-007-02
and 2009ZX03006-009. State Key Laboratory on Microwave and Digital
Communications
Tsinghua National Laboratory for Information Science and Technology
Department of Electronic Engineering, Tsinghua University, Beijing 100084,
China
Email: {gengch07,zhangjq06}@mails.tsinghua.edu.cn,
{peiyk,gening}@tsinghua.edu.cn
###### Abstract
In this paper, we investigate the impact of mistiming on the performance of
Rake receivers in direct-sequence ultra-wideband (DS-UWB) systems from the
perspective of the achievable information rate. A generalized expression for
the performance degradation due to mistiming is derived. Monte Carlo
simulations based on this expression are then conducted, which demonstrate
that the performance loss has little relationship with the target achievable
information rate, but varies significantly with the system bandwidth and the
multipath diversity order, which reflects design trade-offs among the system
timing requirement, the bandwidth and the implementation complexity. In
addition, the performance degradations of Rake receivers with different
multipath component selection schemes and combining techniques are compared.
Among these receivers, the widely used maximal ratio combining (MRC)
selective-Rake (S-Rake) suffers the largest performance loss in the presence
of mistiming.
## I Introduction
Ultra-wideband (UWB) is promising for wireless high rate and short range
communications [1]. Direct-sequence UWB (DS-UWB) [2] has received considerable
interest due to its fine properties of coherent processing of the occupied
bandwidth and the widest contiguous bandwidth [3].
To exploit the ample multipath diversity, the Rake reception is widely
employed in DS-UWB systems [4]. Various types of Rake receivers, like
selective Rake (S-Rake) and partial Rake (P-Rake), are proposed recently [5].
However, Rake receivers have stringent requirements for timing accuracy [6].
In practical DS-UWB systems, mistiming due to acquisition and tracking errors
is inevitable, thus its effects on the performance degradation is worthy of
investigation. Several studies have explored this issue in UWB systems
[7]-[9]. In [7], it is shown that the system throughput degrades significantly
with relatively modest increase in timing errors over additive white Gaussian
noise (AWGN) channels. In [8] and [9], the authors analyze the bit error rate
(BER) degradation induced by mistiming for both fixed and random channels in
UWB systems based on Rake reception. Compared with throughput and BER, the
achievable information rate, which identifies the maximum mutual information
between the input and output of one communication system, is a more
fundamental measurement for system performance, and is also a subject of
continuing research in UWB systems [10]. To the best of the authors’
knowledge, the effect of mistiming on the achievable information rate of Rake
receivers in DS-UWB systems has not been investigated yet to date.
In this paper, a systematic approach is presented to evaluate the impact of
imperfect timing on Rake receivers in DS-UWB systems from the perspective of
the achievable information rate. The influence of key system parameters on the
performance of various types of Rake receivers is also investigated. In our
analysis, a two-step procedure is adopted. First, a generalized expression of
the system performance degradation due to timing mismatch is derived. Then
based on this expression, the numerical results are obtained by averaging over
a sufficiently large number of channel realizations. The major contributions
of this paper lie in the following: (1) As for the widely used maximal ratio
combining (MRC) S-Rake receiver, we observe that the performance degradation
has little relationship with the target information rate, but varies
significantly with the occupied bandwidth and the diversity order, which
reflects design trade-offs among the system timing requirement, the bandwidth
and the implementation complexity. (2) The performance degradation of various
Rake receivers, including MRC S-Rake, MRC P-Rake and equal gain combining
(EGC) P-Rake, are compared. Such comparisons shed light on the robustness of
various multipath component selection schemes and combining techniques to the
variation of system parameters in the presence of mistiming.
This paper is organized as follows: Section II describes the DS-UWB system
model. In Section III, from the perspective of the achievable information
rate, we derive a generalized expression for the system performance
degradation induced by timing mismatch. In Section IV, Monte Carlo simulations
based on the analytic derivation are conducted to investigate the influence of
some key parameters on the system performance and compare the performance
degradation of various Rake receivers under mistiming. Section V draws
conclusions.
## II System Models
Motivated by current DS-UWB system implementations, we confine our discussions
to binary phase-shift keying (BPSK) modulation. The equivalent complex-valued
system baseband model considered throughout this paper is shown in Fig.1.
Figure 1: Block diagrams for the Rake receiver with pulse shaping filters in
DS-UWB systems
### II-A Transmitter Model
In DS-UWB systems, the random source symbol is spreaded and then modulated
with chip pulse $p_{T}(t)$. For each symbol, the transmitted waveform is
defined as
$S(t)=\sum\limits_{n=0}^{N-1}c[n]p_{T}(t-nT_{c})$ (1)
where $c[n]$ denotes the $n$-th chip of the spreading code of length $N$, and
$T_{c}$ is the chip duration.
### II-B UWB Channel Model
In this model, the IEEE802.15.3a UWB indoor channel for wireless personal area
networks (WPAN) is considered [11]. It states that the magnitude of channel
amplitude better agrees with the lognormal distribution, corresponding to the
shadowing phenomenon which arises from a more serious fluctuation than
ordinary fading in the impulse response [12]. In addition, multipath arrivals
are grouped into two categories: cluster arrivals, and ray arrivals within
each cluster. The channel impulse response is defined as:
$H(t)=X\sum\limits_{l=0}^{L-1}\sum\limits_{k=0}^{K-1}\alpha_{k,l}\delta(t-T_{l}-\tau_{k,l})$
(2)
where $\delta(t)$ represents the impulse function, $X$ stands for the log-
normal shadowing, $\alpha_{k,l}$ denotes the multipath gain coefficient,
$T_{l}$ is the delay of $l$-th cluster and $\tau_{k,l}$ is the delay of the
$k$-th multipath component relative to the $l$-th cluster arrival time
($T_{l}$). By definition, we have $\tau_{0,l}=0$ for $l\in\\{0,1,...,L-1\\}$.
### II-C Reception Model
At the receiver, $p_{R}(t)$ is matched to the impulse response of the transmit
filter $p_{T}(t)$. In current DS-UWB systems, the raised cosine filter is
commonly employed as the pulse shaping filter, which is always achieved by
implementing root raised cosine filters as the transmit and receive filters
[13]. Therefore, throughout the rest of this paper, we will consider that the
overall impulse response $p(t)=p_{T}(t)*p_{R}(t)$ corresponds to a raised
cosine filter, which means that $p(t)$ can be written as
$p(t)=p_{T}(t)*p_{R}(t)=\frac{\sin(\pi t/T)}{\pi t/T}\frac{\cos(\alpha\pi
t/T)}{1-4\alpha^{2}t^{2}/T^{2}}$ (3)
where $\alpha$ is the roll-off factor, which represents the excess bandwidth
of the filter and is a real number ranging from 0 to 1.
In this reception model, we assume that the perfect channel information of the
UWB channel is available at the Rake receiver. The impulse response of the
Rake receiver can be written as
$k(t)=\sum\limits_{j=1}^{J}w_{j}\sigma(t-t_{j})$ (4)
where $J$ stands for the Rake diversity orders i.e. finger numbers, $w_{j}$ is
the path weights, and $t_{j}$ denotes the path delay satisfying
$t_{j}<t_{j+1}$. In P-Rake receiver, the first arrival $J$ multipath
components are combined, while the S-Rake receiver selects out the most $J$
strongest multipath components and then combines them together. The Rake
weights $w_{j}$ are selected according to different linear combining
techniques. For MRC, $w_{j}=a^{*}_{j}$, while for EGC,
$w_{j}=sign(\alpha_{j})$, where $a_{j}$ denotes the actual path amplitude,
$(.)^{*}$ represents complex conjugation, and $sign(.)$ is the signum
function.
Finally, de-spreading is performed to get the symbol-level estimation of
transmitted data $y(n)$. The whole DS-UWB receiver, including the matched
filter $p_{R}(t)$, the Rake receiver $k(t)$ and the de-spreading operation,
can be expressed as
$\displaystyle R(t)=$
$\displaystyle\sum\limits_{j=1}^{J}\sum\limits_{n=0}^{N-1}c[n]w_{j}p_{T}(t-nT_{c}-t_{j})$
(5) $\displaystyle=$ $\displaystyle\sum\limits_{j=1}^{J}w_{j}S(t-t_{j})$
Finally, let the impulse response given by $S(t)*H(t)*R(t)$ be denoted by
$g(t)$, and its symbol-sampled version be $g(n)$. Then we can write $y(n)$ as
$y(n)=\sum\limits_{k}x(n-k)g(k)+w(n)$ (6)
where $w(n)$ represents the noise component at the Rake output. In (6), $w(n)$
is the symbol-sampled version of $w(t)$, and
$w(t)=Z(t)*R(t)$ (7)
where $Z(t)$ represents the channel noise which is modeled as AWGN.
## III Performance Analysis under Mistiming
In this section, the performance degradation induced by timing mismatch for
Rake receivers in DS-UWB systems is derived in terms of the achievable
information rate.
In the DS-UWB system model, when the length of spreading code $N$ is
sufficiently large, the autocorrelation property of spreding code is ideal.
Hence the equivalent channel response between the source symbols $x(n)$ and
the symbol-level received data $y(n)$ can be simplified to
$h(t)=p_{T}(t)*H(t)*p_{R}(t)*k(t)$ (8)
Its symbol-sampled version is denoted as $h(k)$.
When mistiming is caused by acquisition or tracking errors in the DS-UWB
receiver, the branch delays in the Rake receiver get inaccurate. Denote this
timing mismatch in all branches as
$\Delta t:=t^{\prime}_{j}-t_{j}(\forall j\in\\{1,2...J\\})$ (9)
where $t_{j}$ is the actual path delay for path $j$, and $t^{\prime}_{j}$ is
the estimated path delay. In this case, the impulse response of the Rake
receiver is given by
$\displaystyle k^{\prime}(t)=$
$\displaystyle\sum\limits_{j=1}^{J}w_{j}\sigma(t-t^{\prime}_{j})$ (10)
$\displaystyle=$ $\displaystyle\sum\limits_{j=1}^{J}w_{j}\sigma(t-t_{j}-\Delta
t)$ $\displaystyle=$ $\displaystyle k(t-\Delta t)$
The corresponding equivalent channel with timing errors is then expressed as
$\displaystyle f(t)=$ $\displaystyle p_{T}(t)*H(t)*p_{R}(t)*k(t-\Delta t)$
(11) $\displaystyle=$ $\displaystyle h(t-\Delta t)$
Its symbol-sampled version is written as $f(k)$. This timing mismatch will
result in performance loss in DS-UWB systems.
It is also worthwhile noting that, when the spreading code attains ideal
autocorrelation property and the amplitude modulation schemes, especially
those with bipolar modulation, e.g. BPSK, are employed, the inter-chip
interference (ICI) can be reduced to a negligible order [14]. Furthermore, to
simplify the analysis, we also assume the excess multipath delay is smaller
than several symobl periods, therefore the effect of inter-symbol interference
(ISI) on the DS-UWB system is also limited. In this case, where the ICI and
ISI are ignorable, the noise component at the Rake output $w(n)$ can be
regarded as AWGN [14].
In the system model described in section II, the source data are constrained
to be independent and identically distributed (i.i.d). The system achievable
information rate, which corresponds to the maximum mutual information between
the input $x(n)$ and the output $y(n)$, is given by [15] 111When the white
Gaussian noise at the Rake output is not valid, i.e. in the case of colored
Gaussian noise, the calculation of achievable information rate can be
performed by using the method of water pouring [16].
$C=\frac{1}{4\pi}\int\limits_{-\pi}^{\pi}\log_{2}[1+2\frac{E_{s}}{N_{0}}|H(e^{j\theta})|^{2}]d\theta$
(12)
where $E_{s}$ is the symbol energy, and $H(e^{j\theta})$ is the Fourier
transform of the equivalent channel impulse response. In the scenario of
perfect synchronization, $H(e^{j\theta})$ stands for the Fourier transform of
$h(k)$; in the scenario with timing errors, $H(e^{j\theta})$ represents the
Fourier transform of $f(k)$.
From (12), it is obvious that the achievable information rate $C$ is derived
from $E_{s}/N_{0}$ considerations, and a certain $E_{s}/N_{0}$ is required to
achieve a specified $C$. Let $R$ be the target achievable information rate. In
the perfect synchronization scenario, we have
$C|_{H(e^{j\theta})=F[h(k)]}=R$ (13)
where $F\\{\\}$ represents the Fourier transform operator. Assume the needed
$E_{s}/N_{0}$ in dBs at this point is $SNR_{h}$.
AS for the scenario with timing errors, let $SNR_{f}$ be the $E_{s}/N_{0}$ in
dBs at which
$C|_{H(e^{j\theta})=F[f(k)]}=R$ (14)
Finally, the performance degradation $L$ induced by timing mismatch is
obtained as
$L=SNR_{f}-SNR_{h}$ (15)
## IV Numerical Results and Discussions
In this section, Monte Carlo simulations based on the generalized expressions
(12) and (15) are conducted to evaluate the effect of various system
configurations on the performance degradation $L$ induced by timing mismatch
in DS-UWB systems. In the following simulations, in order to keep the
simulation complexity on a reasonable level, the timing mismatch $\Delta t$ on
Rake diversity branches is set as $(0,0.1,0.2,...0.9)\times T_{s}$, where
$T_{s}$ denotes the duration of one symbol. In addition, the numerical results
are averaged over the best 900 out of 1000 IEEE 802.15.3a CM1 channel
realizations, following the recommended instructions in [11] that the worst
10% channels are ignored in the simulation. The rest of this section consists
of two parts: In the first part, the influence of timing mismatch on the
performance of the widely used MRC S-Rake receiver is investigated in details
under different system parameters; In the second part, we compare the
performance loss of various Rake receivers, including MRC S-Rake, MRC P-Rake
and EGC P-Rake, in the presence of mistiming.
The Influence of Timing Mismatch on MRC S-Rake Receivers: The needed
$E_{s}/N_{0}$ curves for MRC S-Rake receivers in DS-UWB systems with different
roll-off factors $\alpha$ are plotted in Fig.2. It is observed that if no
timing mismatch exists, as $\alpha$ increases, the needed $E_{s}/N_{0}$ to
achieve the target information Rate $R$ gets smaller. However, the receiver
with larger $\alpha$ is more sensitive to the timing mismatch $\Delta t$. As
seen in this figure, when mistiming is small, the receiver with larger
$\alpha$ needs less $E_{s}/N_{0}$ to obtain the target $R$; however, they
requires more $E_{s}/N_{0}$ as mistiming aggravates.
Figure 2: The needed $E_{s}/N_{0}$ (dB) for MRC S-Rake receivers in DS-UWB
systems with different roll-off factors $\alpha$ when $J=8$ and $R=0.3$
Fig.3 depicts the $E_{s}/N_{0}$ needed for MRC S-Rake receivers with different
diversity orders to achieve $R=0.3$ when $\alpha$ equals to 1.0. As one can
expect, when no timing errors exhibit, the MRC S-Rake receiver with more
diversity branches needs less $E_{s}/N_{0}$ to achieve the target $R$. It is
further seen that the sensitivity to timing mismatch increases with increasing
$J$ in the MRC S-Rake receiver. Hence there also exists a design trade-off
between the robustness to mistiming and the implementation complexity.
Figure 3: The needed $E_{s}/N_{0}$ (dB) for MRC S-Rake receivers with
different diversity orders $J$ when $\alpha=1.0$ and $R=0.3$
The needed $E_{s}/N_{0}$ for MRC S-Rake receivers to achieve various target
information rates $R$ is illustrated in Fig.4 when $J$ is 8 and $\alpha$
equals to 0.3. It shows that as $R$ increases, the needed $E_{s}/N_{0}$
increases obviously. However, the performance loss induced by timing mismatch
rarely varies with the increase of the target achievable information rate in
MRC S-Rake receivers.
Figure 4: The needed $E_{s}/N_{0}$ (dB) for MRC S-Rake receivers with
different target achievable information rates $R$ when $\alpha=0.3$ and $J=8$
The Comparison of Various Rake Receivers under Mistiming: In this part, the
mistiming is assumed to follow uniform distribution [17]. We consider two
cases: one is the worst case with the maximum performance degradation; the
other is the average case, which represents the average degradation over the
set of timing mismatch $\Delta t$.
Fig.5 \- Fig.7 demonstrate the performance degradation of three types of Rake
receivers, including MRC S-Rake, MRC P-Rake and EGC P-Rake, with respect to
the variation of the excess bandwidth, the diversity order and the target
achievable information rate under timing mismatch respectively. Fig.5 shows
that with roll-off factor increasing, all types of Rake receivers obtain worse
performance due to timing errors. Among all the receivers, MRC S-Rake is most
sensitive to the increase of excess bandwidth and the EGC P-Rake is least
sensitive. From Fig.6, it is observed that as the diversity order increases,
the performance degradation of all the three kinds of Rake receivers gets
larger. Fig.6 also shows that when MRC is employed, P-Rake is more sensitive
to the change of Rake finger numbers than S-Rake, and in P-Rake receiver the
EGC technique is more robust compared with MRC. Fig.7 demonstrates that the
target information rate rarely impacts the performance loss of all the Rake
receivers under timing mismatch, and the MRC S-Rake suffers the largest
performance loss compared with other Rake receivers.
## V Conclusion
The effect of imperfect timing has been evaluated for the Rake reception in
DS-UWB systems from the perspective of the achievable information rate. A
generalized expression for the system performance degradation is derived, then
corresponding simulations are conducted to investigate the effect of timing
mismatch on the widely used MRC S-Rake receiver with respect to the excess
bandwidth induced by the roll-off factor in RRC filters, the multipath
diversity order and the target information rate. Simulation results illustrate
that the performance loss has little relationship with the target information
rate, but varies significantly with the system bandwidth and the diversity
order, which further demonstrates that there exist fundamental trade-offs
among the system timing requirement, the occupied bandwidth and the
implementation complexity of DS-UWB systems. In addition, the performance
degradation of various types of Rake receivers, including MRC S-Rake, MRC
P-Rake and EGC P-Rake, is compared, and the sensitivity of different multipath
component selection schemes and diversity combining techniques to the
variation of system parameters are obtained. The numerical results also show
that among the three types of Rake receivers, the MRC S-Rake suffers the most
performance degradation in the presence of mistiming.
Figure 5: Performance degradation $L$ (dB) as a function of the roll-off
factor $\alpha$ for various Rake receivers when $J=8$ and $R=0.3$ Figure 6:
Performance degradation $L$ (dB) as a function of the diversity order $J$ in
various Rake receivers when $\alpha=1.0$ and $R=0.3$ Figure 7: Performance
degradation $L$ (dB) as a function of the target achievable information rate
$R$ for various Rake receivers when $\alpha=0.3$ and $J=8$
## References
* [1] S. Roy, J. R. Foerster, V. S. Somayazulu, and D. G. Leeper, ”Ultrawideband radio design: the promise of high-speed, short-range wireless connectivity,” Proceedings of the IEEE, vol.92, no.2, pp. 295-311, Feb. 2004.
* [2] R. Fisher, R. Kohno, M. McLaughlin, et al. DS-UWB physical layer submission to IEEE 802.15 task group 3a (Doc. Number P802.15-04/0137r4), IEEE P802.15, 2005.
* [3] P. Runkle, J. McCorkle, T. Miller, and M. Welborn, ”DS-CDMA: the modulation technology of choice for UWB communications”, IEEE Conference on Ultra Wideband System and Technologies (UWBST), pp.364-368, 2003.
* [4] J. D. Choi, and W. E. Stark, ”Performance of ultra-wideband communications with suboptimal receivers in multipath channels,” IEEE Journal on Selected Areas in Communications, vol.20, no.9, pp. 1754-1766, Dec. 2002.
* [5] D. Cassioli, M. Z. Win, F. Vatalaro, and A. F. Molisch, ”Low Complexity Rake Receivers in Ultra-Wideband Channels,” IEEE Transactions on Wireless Communications, vol.6, no.4, pp.1265-1275, Apr. 2007.
* [6] Y. Yin, J. P. Fonseka, and I. Korn, ”Sensitivity to timing errors in EGC and MRC techniques”, IEEE Transactions on Communications, vol.51, no.4, pp.530-534, Apr. 2003.
* [7] W. M. Lovelace, and J. K. Townsend, ”The effects of timing jitter and tracking on the performance of impluse radio”, IEEE Journal on Selected Areas in Communications, vol.20, no.12, pp.1646-1651, Dec. 2002.
* [8] Z. Tian, and G. B. Giannakis, ”BER sensitivity to mistiming in ultra-wideband impluse radios - part I: nonrandom channels”, IEEE Transactions on Signal Processing, vol.53, no.4, pp.1550-1560, Apr. 2005.
* [9] Z. Tian, and G. B. Giannakis, ”BER sensitivity to mistiming in ultra-wideband impluse radios - part II: fading channels”, IEEE Transactions on Signal Processing, vol.53, no.5, pp.1897-1907, May 2005\.
* [10] N. Guney, H. Deli, and F. Alagoz, ”Achievable information rates of PPM impulse radio for UWB channels and rake reception” , IEEE Transactions on Communications, vol.58, no.5, pp.1524-1535, May 2010\.
* [11] Channel-Modeling-Subcommittee-Report-Final, IEEE P802.15. Dec. 2002.
* [12] R. C. French, ”The effect of fading and shadowing on channel reuse in mobile radio”, IEEE Transactions on Vehicular Technology, vol.28, no.3, pp.171-181, Aug. 1979.
* [13] A. Parihar, L. Lampe, R. Schober, and C. Leung, ”Equalization for DS-UWB Systems Part I: BPSK Modulation,” IEEE Transactions on Communications, vol.55, no.6, pp.1164-1173, Jun. 2007.
* [14] J. Zhang, R. A. Kennedy, and T. D. Abhayapala, ”Performance of Rake reception for ultra-wideband signals in a lognormal-fading channel”, Proc. IWUWBS, pp. 5-9, 2003.
* [15] W. Hirt, and J. L. Massey, ”Capacity of the discrete-time Gaussian channel with intersymbol interference,” IEEE Transactions on Information Theory, vol.34, no.3, pp.380-388, May 1988.
* [16] W. Xiang, and S. S. Pietrobon. ”On the capactity abd normalization of ISI channels”, IEEE Transactions on Information Theory, vol.49, no.9, pp.2263-2268, Sep. 2003.
* [17] M. O. Sunay, and P. J. Mclane, ”Diversity combining for DS CDMA systems with synchronization errors”, IEEE International Conference on Communications (ICC), pp.83-89, 1996.
|
arxiv-papers
| 2010-12-21T04:28:18 |
2024-09-04T02:49:15.877654
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chunhua Geng, Yukui Pei, Jiaqi Zhang and Ning Ge",
"submitter": "Chunhua Geng",
"url": "https://arxiv.org/abs/1012.4542"
}
|
1012.4556
|
# Selective Multipath Interference Canceller with Linear Equalization for DS-
UWB Systems with Low Spreading Factor
Chunhua Geng, Yukui Pei, and Ning Ge This paper was presented in part at the
IEEE International Conference on Ultra-Wideband (ICUWB), Vancouver, Canada,
2009. The authors are with the State Key Laboratory on Microwave and Digital
Communications, Tsinghua National Laboratory for Information Science and
Technology, Tsinghua University. (email: gengch07@mails.tsinghua.edu.cn,
peiyk@tsinghua.edu.cn, gening@tsinghua.edu.cn)
###### Abstract
In high rate DS-UWB systems with low spreading factor, the selective multipath
interference canceller with linear equalization (SMPIC-LE) is developed to
alleviate severe multipath interferences induced by the poor orthogonality of
spreading codes. The SMPIC iteratively mitigates the strongest inter-path
interference, inter-chip interference and inter-symbol interference, while the
former two are unresolvable in conventional RAKE-decision feedback equalizer
(DFE) receivers. The numerical results and complexity analysis demonstrate
that SMPIC-LE with proper parameters provides an attractive overall advantage
in performance and computational complexity compared with RAKE-DFE. In
addition, it approaches the matched filter bound well as the RAKE finger in
SMPIC increases.
###### Index Terms:
_Equalization, iterative receiver, multipath interference, RAKE, ultra-
wideband_
## I Introduction
Ultra-wideband (UWB) is a promising technology for wireless high rate and
short range communications [2]. Direct-sequence spreading based UWB (DS-UWB)
and multiband orthogonal frequency-division multiplexing UWB (MB-OFDM UWB) are
two main physical layer standards for high data rate wireless personal area
networks (WPAN) [3], [4]. Because of the fine properties of coherent
processing of the occupied bandwidth and the widest contiguous bandwidth, DS-
UWB has received considerable attention from both academia and industry [5],
[6].
For high data rate DS-UWB systems supporting transmission rate ranging from
several Megabit per second to more than one Gigabit per second, most recent
research on the receiver design focuses on the RAKE reception with symbol
level decision feedback equalizer (DFE) [7]-[9]. In practical high rate DS-UWB
systems, limited by state-of-the-art ADC technology, the spreading factor (SF)
cannot be large enough to maintain the ideal orthogonality between spreading
codes [10]. Therefore, the conventional RAKE-DFE receiver would suffer
significant performance loss due to severe inter-path interference (IPI),
inter-chip interference (ICI) and inter-symbol interference (ISI) [11]-[14].
The former two kinds of interference can not be mitigated by the RAKE-DFE
receiver effectively. Furthermore, in order to combat the severe ISI induced
by long channel delay spread, the DFE tap number has to be quite large. The
demanding computational complexity of DFE always exceeds that of the RAKE
receiver significantly by far and becomes a heavy burden for system design.
To resolve the above problems, the selective multipath interference canceller
(SMPIC) with symbol level linear equalization (LE) is proposed in this paper
for practical high rate DS-UWB systems with low SF. The SMPIC is capable of
mitigating the IPI, ICI and ISI by reconstructing and subtracting the selected
strongest multipath interferences from the received signal in an iterative
way. Then the symbol level LE is concatenated to alleviate the remaining ISI.
In addition, to validate the effectiveness of the SMPIC-LE receiver, we derive
the matched filter bound (MFB), which takes into account such practical
constrains as the sampling rate and the RAKE diversity order, i.e. the finger
number. Simulation results and complexity analysis show that the proposed
SMPIC-LE can achieve similar or even better performance with much lower
computational complexity compared with the conventional RAKE-DFE receiver in
various realistic UWB channels. Moreover, as the RAKE diversity order
increases, the performance of SMPIC-LE receivers can approach the derived MFB
well.
The remainder of this paper is organized as follows. The DS-UWB system model
with the proposed SMPIC-LE receiver is introduced in Section II. In Section
III, the computational complexity and performance of the SMPIC-LE receiver are
analyzed, and the MFB is also derived. In Section IV, the corroborating
simulation results are presented. Section V summarizes the whole paper.
## II System Model
In this paper, the IEEE802.15.3a UWB indoor channel model is employed [15].
The equivalent complex-valued baseband model with the proposed SMPIC-LE
receiver is shown in Fig.1.
Figure 1: Diagram of the DS-UWB System model with the SMPIC-LE receiver
### II-A Transmitter
In this paper, we only focus on binary phase-shift keying (BPSK) modulation,
which is the mandatory transmission mode for DS-UWB systems. At the
transmitter, the random source symbol is spread and modulated with chip pulse
$g_{T}(t)$. For each symbol, the pulse shape is defined as
$g(t)=\sum\limits_{n=0}^{N-1}c[n]g_{T}(t-nT_{c})$ (1)
where $c[n]$ denotes the $n$-th chip of the spreading code of length $N$, and
$T_{c}$ is the chip duration. Assume $M$ symbols are contained in each frame,
and each transmitted frame can be written as
$s(t)=\sum\limits_{m=0}^{M-1}b[m]g(t-mT_{b})$ (2)
where $b[m]\in\\{-1,+1\\}$ represents the $m$-th symbol of each frame, and
$T_{b}=NT_{c}$ is the symbol interval.
### II-B UWB channels
In order to compare standardization proposals for high data rate WPANs,
IEEE802.15.3a task group developed a standard channel model for UWB systems
[15]. This model is based on the Saleh-Valenzuela model [16] with some
modification to account for the properties of realistic UWB channels. In this
model, multipath arrivals are grouped into two categories: cluster arrivals
and ray arrivals within each cluster. The channel impulse response is defined
as:
$h(t)=X\sum\limits_{l=0}^{L-1}\sum\limits_{k=0}^{K-1}\alpha_{k,l}\delta(t-T_{l}-\tau_{k,l})$
(3)
where $X$ represents the log-normal shadowing, $\alpha_{k,l}$ is the multipath
gain coefficient, $T_{l}$ is the delay of $l$-th cluster and $\tau_{k,l}$ is
the delay of the $k$-th multipath component relative to the $l$-th cluster
arrival time ($T_{l}$). By definition, we have $\tau_{0,l}=0$ for
$l\in\\{0,1,...,L-1\\}$.
### II-C SMPIC-LE Receiver
In the DS-UWB system, the SMPIC-LE receiver is developed to alleviate the
severe multipath interferences induced by the poor orthogonality of spreading
codes. In the first stage, the SMPIC is employed to specifically mitigate the
strongest multipath interference components, including IPI, ICI and ISI. Then,
a conventional symbol level LE with small tap number is concatenated to combat
the residual ISI. In the sequel of this section, we mainly focus on the
proposed SMPIC scheme.
The structure of SMPIC is presented in Fig.2. The SMPIC works in an iterative
manner. Its purpose is to eliminate the interference induced by multipath
delay at each RAKE finger. Similar with selective-RAKE (SRAKE) [17], the SMPIC
selects the instantaneously strongest $J$ multipath components and combines
them together at first. Then the interference is estimated and subtracted from
the received data at each RAKE finger to get more precise input signals for
the RAKE reception in the next iteration. In order to reduce the complexity,
the interference is reconstituted by using the hard decision of the SRAKE
output. Through this iterative process, the precision of the correlation
result in each RAKE finger is improved, so is the output of the receiver.
In DS-UWB systems, the received signal of each frame is given by
$\displaystyle r(t)=$
$\displaystyle\sum\limits_{l=0}^{L-1}\sum\limits_{k=0}^{K-1}\sum\limits_{m=0}^{M-1}\sum\limits_{n=0}^{N-1}a_{k,j}b[m]c[n]$
(4) $\displaystyle g_{T}(t-T_{l}-\tau_{k,j}-nT_{c}-mT_{b})+z(t)$
where $z(t)$ is additive white Gaussian noise (AWGN) with mean being zero and
power spectral density being $N_{0}/2$ W/Hz.
We assume the receiver can get the perfect channel knowledge. Received data
$r(t)$ is first fed into maximal ratio combining (MRC) SRAKE in SMPIC. After
conventional RAKE processing, the output is sent to hard-decision module. The
estimation of $m$-th bit of each frame at the output of hard-decision module
is denoted as $\widetilde{b}^{(0)}[m]$. This estimated sequence is then
spread, modulated and processed by a very simple multipath interference
regenerator (MIR). In this sub-module, the modulated sequence is multiplied by
the amplitude of selected $J$ paths and delayed by corresponding time. So the
reconstituted $j$-th path signal can be expressed as
$\displaystyle\widetilde{r}_{j}^{(0)}(t)=$
$\displaystyle\sum\limits_{m=0}^{M-1}\sum\limits_{n=0}^{N-1}a_{k_{j},l_{j}}\widetilde{b}^{(0)}[m]c[n]$
(5) $\displaystyle g_{T}(t-T_{l_{j}}-\tau_{k_{j},l_{j}}-nT_{c}-mT_{b})$
where $j\in\\{1,2,...,J\\}$, and $\alpha_{k_{j},l_{j}}$ is the multipath gain
coefficient corresponding to the $j$-th RAKE finger. $T_{l_{j}}$ and
$\tau_{k_{j},l_{j}}$ denote the delays.
In the next iteration, the input signal to the $j$-th finger is represented as
$\displaystyle r_{j}^{(1)}(t)=r(t)-w\sum\limits_{j^{\prime}=1,j^{\prime}\neq
j}^{J}\widetilde{r}_{j^{\prime}}^{(0)}(t)$ (6)
where $w\in[0,1]$ is a constant named as interference rejection weight, which
allows to reduce the impact of possible errors presented in the estimated
multipath interference replicas. Then $r_{j}^{(1)}(t)$ is delivered to the MRC
SRAKE receiver in the next iteration. The SRAKE output can be used as either
the input of MIR for the following iterations, or the output of the SMPIC
receiver if the pre-defined iteration time $p$ is achieved. Finally, the
output of the SMPIC is sent to the LE to reduce the remaining ISI.
Figure 2: Block diagram of SMPIC (the iteration times $p$ are 2 in this
diagram)
## III Performance and Computational Complexity Analysis
### III-A Performance Analysis and the Matched Filter Bound
In this subsection, the effect of multipath components (MPCs) on conventional
SRAKE receivers and the validity of SMPIC are analyzed first.
The energy of $g_{T}(t)$ is defined as $E_{g}$,
$E_{g}=\int_{-\infty}^{+\infty}g_{T}^{2}(t)dt$ (7)
The normalized autocorrelation function of $g_{T}(t)$ expresses as
$R_{g}(\triangle t)=\frac{1}{E_{g}}\int_{-\infty}^{+\infty}g(t)g(t+\triangle
t)dt$ (8)
The channel is assumed perfectly known at the receiver. The local template of
the $\widetilde{m}$-th bit in the $j$-th finger of SRAKE is given by
$v_{j}(t)=\sum\limits_{\widetilde{n}=0}^{N-1}\alpha_{k_{j},l_{j}}^{*}c[\widetilde{n}]g_{T}(t-T_{l_{j}}-\tau_{k_{j},l_{j}}-\widetilde{m}T_{b}-\widetilde{n}T_{c})$
(9)
where $(.)^{*}$ denotes complex conjugation. The correlation output of the
$j$-th finger is
$R_{j}(t)=\int\limits_{\widetilde{m}T_{b}}^{(\widetilde{m}+1)T_{b}}r(t)v_{j}(t)dt=b[\widetilde{m}](S+I_{1}+I_{2})+I_{3}+Z$
(10)
where $Z$ represents the effect of noise, and
$S=NE_{g}|\alpha_{k_{j},l_{j}}|^{2}$ (11)
is the signal component. $I_{1}$, $I_{2}$ and $I_{3}$ are the IPI, ICI and ISI
respectively,
$I_{1}=NE_{g}\sum\limits_{l=0}^{L-1}\sum\limits_{k=0}^{K-1}a_{k,j}a_{k_{j},l_{j}}^{*}R_{g}(T_{l}-T_{l_{j}}+\tau_{k,j}-\tau_{k_{j},l_{j}})$
(12)
where $l\neq l_{j}$ or $k\neq k_{j}$,
$\displaystyle I_{2}=$ $\displaystyle
E_{g}\sum\limits_{l=0}^{L-1}\sum\limits_{k=0}^{K-1}\sum\limits_{n=0}^{N-1}\sum\limits_{\widetilde{n}=0}^{N-1}a_{k,j}a_{k_{j},l_{j}}^{*}c[n]c[\widetilde{n}]$
(13) $\displaystyle
R_{g}(T_{l}-T_{l_{j}}+\tau_{k,j}-\tau_{k_{j},l_{j}}+(n-\widetilde{n})T_{c})$
where $n\neq\widetilde{n}$,
$\displaystyle I_{3}=$ $\displaystyle
E_{g}\sum\limits_{m=0}^{M-1}\sum\limits_{l=0}^{L-1}\sum\limits_{k=0}^{K-1}\sum\limits_{n=0}^{N-1}\sum\limits_{\widetilde{n}=0}^{N-1}a_{k,j}a_{k_{j},l_{j}}^{*}b[m]c[n]c[\widetilde{n}]$
(14) $\displaystyle
R_{g}(T_{l}-T_{l_{j}}+\tau_{k,j}-\tau_{k_{j},l_{j}}+(n-\widetilde{n})T_{c}$
$\displaystyle+(m-\widetilde{m})T_{b})$
where $m\neq\widetilde{m}$.
The accuracy of the RAKE output is closely related to the statistical
properties of IPI, ICI, and ISI, which follow an impulsive distribution [11].
The conventional symbol level equalizer can only combat long ISI at the cost
of high computational complexity, but fails to mitigate IPI and ICI
effectively. When the SF is small, which means that the autocorrelation
property of the spreading code is poor, the multipath interferences degrade
the performance of the RAKE-DFE receiver dramatically. From (5) and (6), we
can see that the proposed SMPIC can subtract the $J$-1 strongest interference
components in every finger before the correlation and combining at each
iteration, hence the strongest interferences, including $I_{1}$, $I_{2}$, and
$I_{3}$, in (10) can be mitigated effectively.
To validate the effectiveness of the SMPIC-LE receiver, in the following
simulation part, the performance of SMPIC-LE is compared with the MFB of DS-
UWB systems, which yields the absolute performance limit for equalization
schemes. In order to obtain expressions for the bit error rate (BER) of the
MFB, we define the signal-to-noise ratio (SNR) as
$\gamma_{r}=\frac{E_{b}(r)}{N_{0}}$ (15)
where $E_{b}(r)$ is the received energy per bit for the $r$th UWB channel
realization. The corresponding BER($\gamma_{r}$) for BPSK in one particular
channel realization can be written as
$BER(\gamma_{r})=Q(\sqrt{2\gamma_{r}})$ (16)
where $Q(*)$ stands for $Q$ function. The average BER is obtained semi-
analytically by averaging over $R$ channel realizations
$BER_{MFB}=\frac{1}{R}\sum_{r=1}^{R}BER(\gamma_{r})$ (17)
As for the DS-UWB systems employing $J$-finger RAKE receiver, where $J$ is
much smaller than the total number of resolvable multipath components for
complexity reasons, we obtain
$E_{b}(r)=\sum_{j=1}^{J}|h(t_{j})|^{2}$ (18)
where $t_{j}$ ($j\in{1,2,...J}$) denotes the positions of the strongest $J$
multipath components in one channel realization. The resolution of $t_{j}$
equals to the sampling rate at the receiver. In this paper, the derived MFB
takes into account the sampling rate at the receiver and the effect of
selective RAKE combining with a limited number of RAKE fingers. Therefore, it
demonstrates an accurate performance bound for practical receivers in DS-UWB
systems.
### III-B Computational Complexity Analysis
In this paper, the computational complexity of SMPIC-LE and SRAKE-DFE receiver
is calculated in terms of multiplications and divisions per output symbol
(MADPOS) [18].
The proposed SMPIC is comprised of two kinds of basic sub-modules: one is the
MRC SRAKE, and the other is the MIR. When the SF is small, correlators, the
main part of SRAKE, are quite simple. The MIR can be seen as the inverse
procedure of RAKE processing from Section II. Therefore, the computational
complexity of MIR is comparable with that of RAKE combining. Moreover, the
computational complexity of SMPIC is independent of the magnitude of path
delays, hence it can be kept under a relatively low level in various
transmission scenarios. The computational complexity of SRAKE and SMPIC is
given by
$\displaystyle C_{SRAKE}=2\times J$ (19) $\displaystyle
C_{SMPIC}=2\times(p+1)\times J+p\times 3J$ (20)
where $J$ stands for the RAKE finger number and $p$ denotes the iteration time
in SMPIC.
For equalization, the widely used adaptive Kalman recursive least-square
(K-RLS) algorithm is employed for adjusting tap coefficients to ensure fast
convergence and lower stead-state mean square error (MSE), and hence a
favorable detection performance in UWB system [19]. The adaptive equalizer
works in two stages: in training stage, a training sequence is employed to
initially adjust the tap weights; in decision directed stage, the decisions at
the output of the equalizer are used to continue the coefficients adaption
process. The computational complexity of equalizers based on K-RLS is
approximately given by [18]
$\displaystyle C_{K-RLS}=2.5\times N^{2}+4.5\times N$ (21)
where $N$ represents the total tap number in equalizers.
## IV Numerical Results and Discussion
### IV-A System Parameters
Monte Carlo simulations have been run to access the performance of the
proposed receiver in high rate DS-UWB systems with low SF. The spreading code
is set as [-1 +1] with SF being an extreme of 2. The sampling rate is
$T_{c}/4$. At the receiver, the value of interference rejection weight $w$ in
SMPIC is chosen as 0.9 by investigating the effect of different weights on the
system BER performance. The iteration time $p$ is set as 2, which can
guarantee the performance convergence in most cases through our simulations.
The forgetting factor in K-RLS algorithm is 0.99999. As for equalization,
without notable instructions, the lengths of LE tap $L$, DFE feedforward tap
$FF$ and feedback tap $FB$ are set as 15, 25 and 20, respectively. The IEEE
802.15.3a CM1 line-of-sight (LOS) and CM4 extreme non-LOS (NLOS) UWB indoor
channel models are considered here. According to the recommended instructions
in [15], the numerical results are averaged over the best 90 out of 100
channel realizations.
### IV-B Bit Error Rate Performance
As a function of $E_{b}/N_{0}$ at the input of receivers, the BER performance
of SRAKE-DFE and SMPIC-LE is evaluated and compared with MFB.
Figure 3: BER performance of SRAKE-DFE and SMPIC-LE receivers in CM1 channel
Figure 4: BER performance of SRAKE-DFE and SMPIC-LE receivers in CM4 channel
First, we present BER curves of SMPIC-LE and SRAKE-DFE receivers with
different RAKE finger numbers and transmission data rates. Fig.3 shows the
system performance in the CM1 channel model. As seen in this figure, the
SMPIC-LE outperforms the conventional SRAKE-DFE receiver. When the
transmission data rate equals to 250Mbps, the $J$=32 SMPIC-LE gets a
performance gain about 0.2dB over $J=32$ SRAKE-DFE at a BER of $10^{-4}$, and
the loss in power efficiency compared with the derived MFB is within 1dB. As
the data rate increases to 1.5Gbps, the advantage of SMPIC-LE over SRAKE-DFE
gets more significant. It is shown when $J$ equals to 32, the performance gain
is up to about 1dB, and the performance of SMPIC-LE approaches the MFB well.
From Fig.4, it is observed that in the case of CM4 channels, when data rate is
250Mbps, the proposed SMPIC-LE receiver only suffers negligible performance
loss compare with SRAKE-DFE with the same RAKE fingers. As the data rate
increases to 1.5Gbps, the SMPIC-LE receiver can lower the error floor. From
the above two figures, we can conclude that as the data rate increases, the
performance gain of SMPIC-LE over SRAKE-DFE improves. This can be attributed
to the fact that with the data rate increasing, more resolvable strong
interferences, which degrade the system performance dramatically, occur at the
receiver, and the proposed SMPIC-LE receiver can alleviate these interferences
in a more effective iterative way compared with the conventional SRAKE-DFE
receiver.
Figure 5: BER performance of SMPIC-LE receivers with different equalizer tap
lengths in CM4 channel when the transmission data rate is 250Mbps
Then the effect of the LE tap number on the BER performance of the SMPIC-LE
receiver is also investigated. From Fig.5, it shows that as LE tap number $L$
increases, the system performance improves as well. In addition, as the RAKE
finger $J$ increases, the SMPIC-LE receiver yields a close-to-optimum
performance and the performance gain by increasing LE taps become unobvious.
For instance, when the finger number $J$ is 16, with $L$ increasing from 15 to
25, the SMPIC-LE receiver can obtain a performance gain of more than 1dB. When
$J$ increases to 32, the performance improvement is only about 0.4dB. This is
due to the fact that as the RAKE finger number gets larger, the strong ISIs
are mitigated by SMPIC effectively, hence increasing LE taps cannot get
additional significant performance gain. This fact provides a useful pointer
for system designers when specifying system parameters. Our findings also
suggest that the SMPIC-LE receiver with more RAKE fingers outperforms the
receiver with less RAKE fingers but more equalizer taps, which demonstrates
the key role of the proposed SMPIC scheme to mitigate severe multipath
interferences in UWB channels.
### IV-C Computational Complexity Comparison
Finally, the computational complexity of SMPIC-LE and SRAKE-SFE receivers
adopted in the simulations are compared. The MADPOS of both SMPIC-LE and
SRAKE-DFE is shown in Table I. As seen in this table, when $J$ equals to 16,
the MADPOS in the SMPIC-LE receiver with $L$ = 15 is 822, which is only 15.5%
of that in the SRAKE-DFE with $FF$ = 25 and $FB$ = 20. When $J$ increases to
32, the SMPIC-LE can still save 81% MADPOS than SRAKE-DFE. These results
demonstrate that the computational complexity of SMPIC-LE scheme is much less
than that of conventional SRAKE-DFE receivers.
TABLE I: Computational Complexity Comparison (MADPOS) | | | SRAKE-DFE
---
($FF$=25, $FB$=20)
| SMPIC-LE
---
($L$=15, $p$=2)
Saving
| $J$ = 16 | 5297 | 822 | 84.5%
| $J$ = 32 | 5329 | 1014 | 81.0%
## V Conclusions
The scheme presented in this paper offers a low computational complexity
alternative to the conventional SRAKE-DFE receiver, which provides a more
efficient way for UWB signal detection by mitigating significant multipath
interference components specifically. In this proposed SMPIC-LE scheme, the
receiver can alleviate the strongest IPI, ICI and ISI, while the former two
interferences are unresolvable in conventional RAKE-DFE receivers. The MFB,
which takes into account the effects of sampling rate and the number of RAKE
fingers at the receiver, is also derived. Numerical results and complexity
analysis show that compared with SRAKE-DFE, the SMPIC-LE receiver, with much
lower computational complexity, can achieve similar or even better performance
in high rate DS-UWB systems with low SF for various UWB propagation scenarios.
In addition, as the RAKE finger number increases, the low-complexity SMPIC-LE
receiver approaches the derived MFB limit well.
## Acknowledgment
This work is supported by National Nature Science Foundation of China No.
60928001 and 60972019, National Basic Research Program of China under grant
No. 2007CB310608, and the National Science & Technology Major Project under
grant No. 2009ZX03006-007-02 and 2009ZX03006-009.
## References
* [1]
* [2] L. Q. Yang, and G. B. Giannakis, “Ultra-wideband communications: an idea whose time has come,” IEEE Signal Process. Mag., vol.21, no.6, pp.26-54, Nov. 2004.
* [3] R. Fisher, R. Kohno, M. McLaughlin, and M. Welbourn, ”DS-UWB physical layer submission to 802.15 task group 3a,” IEEE P802.15-04/0137r4, Jan. 2005.
* [4] IEEE P802.15. Multi-band OFDM physical layer proposal for IEEE 802.15 task group 3a (Doc. Number P802.15-03/268r3). 2004.
* [5] P. Runkle, J. McCorkle, T. Miller, and M. Welborn, “DS-CDMA: the modulation technology of choice for UWB communications,” IEEE Conf. Ultra-Wideband System and Technologies (UWBST), pp.364-368, 2003\.
* [6] N. Boubaker, and K. B. Letaief, “Ultra wideband DSSS for muiltiple access communications using antipodal siganling,” Proc. IEEE Int. Conf. Communications (ICC), pp.2197-2201, 2003.
* [7] A. Parihar, L. Lampe, R. Schober, and C. Leung, “Equalization for DS-UWB Systems - Part I: BPSK Modulation,” IEEE Trans. Commun., vol.55, no.6, pp.1164-1173, Jun. 2007.
* [8] A. Parihar, L. Lampe, R. Schober, and C. Leung, “Equalization for DS-UWB Systems - Part II: 4BOK Modulation,” IEEE Trans. Commun., vol.55, no.8, pp.1525-1535, Aug. 2007.
* [9] M. Eslami, and X. Don, “Rake-MMSE-equalizer performance for UWB,” IEEE Commun. Letters, vol.9, no.6, pp. 502- 504, Jun 2005.
* [10] J. Singh, S. Ponnuru, and U. Madhow, “Multi-Gigabit communication: the ADC bottleneck,” Proc. IEEE Int. Conf. Ultra-Wideband (ICUWB), pp.22-27, 2009.
* [11] H. Shao, and N. C. Beaulieu, “An analytical method for calculating the bit error rate performance of RAKE reception in UWB multipath fading channels,” Proc. IEEE Int. Conf. Communications (ICC), pp.4855-4860, 2008.
* [12] Y. Chen, and N. C. Beaulieu, “Interference analysis iof UWB systems for IEEE channel model using first- and second-order moments,” IEEE Trans. Commun., vol.57, no.3, pp.622-625, Mar. 2009.
* [13] B. Zhao, Y. Chen, and R. J. Green, “Average SINR analysis of DS-BPSK UWB systems with IPI, ICI, ISI, and MAI and its application,” IEEE Trans. Veh. Technol, vol.58, no.8, pp.4690-4696, Oct. 2009.
* [14] C. Unger, and G. P. Fettweis, “Analysis of the Rake receiver performance in low spreading gain DS/SS systems,” Proc. IEEE Global Telecommun. Conf. (GLOBECOM), pp. 830-834, 2002.
* [15] Channel Modeling Sub-Committee Final Report, IEEE P802.15-02/368r5-SG3a, IEEE P802.15, Dec. 2002.
* [16] A. Saleh, and R. Valenzuela, “A statistical model for indoor multipath propagation,” IEEE J. Sel. Areas Commun., vol.5, no.2, pp.128-137, Feb. 1987.
* [17] D. Cossioli, M. Z. Win, F. Vatalaro, and A. F. Molish,“Low complexity Rake receivers in ultra-wideband channels,” IEEE Trans. Wireless Commun., vol.6, no.4, pp.1265-1275, Apr. 2007.
* [18] J. G. Proakis, ed., Digital Communications. New York, NY, McGraw-Hill, Inc., 4th Ed., 2001.
* [19] C. C. Hu, and J. F. Chang, “DS-UWB forward link adaptive chip-equalizer using subband decomposition technique,” Proc. IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), pp.2802-2806, 2009.
|
arxiv-papers
| 2010-12-21T07:11:26 |
2024-09-04T02:49:15.884109
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"authors": "Chunhua Geng, Yukui Pei, and Ning Ge",
"submitter": "Chunhua Geng",
"url": "https://arxiv.org/abs/1012.4556"
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1012.4643
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¡html¿ ¡head¿ ¡title¿CERN-2010-002¡/title¿ ¡/head¿ ¡body¿ ¡h1¿¡a
href=”http://physicschool.web.cern.ch/PhysicSchool/2009/Info/Bautzen_Info.html”¿2009
European School of High-Energy Physics¡/a¿¡/h1¿ ¡h2¿Bautzen, Germany, 14 - 27
June 2009¡/h2¿ ¡h2¿Proceedings - CERN Yellow Report ¡a
href=”http://cdsweb.cern.ch/record/1119304?ln=en”¿CERN-2010-002¡/a¿¡/h2¿
¡h3¿editors: C. Grojean and M. Spiropulu ¡/h3¿
The European School of High-Energy Physics is intended to give young
physicists an introduction to the theoretical aspects of recent advances in
elementary particle physics. These proceedings contain lecture notes on
quantum field theory, quantum chromodynamics, physics beyond the Standard
Model, flavour physics, effective field theory, cosmology, as well as
statistical data analysis.
¡h2¿Lectures¡/h2¿
¡!– Quantum field theory and the standard model –¿ LIST:1012.3883 ¡br¿
¡!– Elements of QCD for hadron colliders –¿ LIST:1011.5131 ¡br¿
¡!– Beyond the Standard Model –¿ LIST:arXiv:1005.1676 ¡br¿
¡!– Introduction to flavor physics –¿ LIST:arXiv:1006.3534 ¡br¿
Title: ¡a href=”http://cdsweb.cern.ch/record/1281952”¿Effective field theory -
concepts and applications¡/a¿ ¡br¿ Author: M. Beneke ¡br¿ Journal-ref: CERN
Yellow Report CERN-2010-002, pp. 145-148 ¡br¿
¡!– The violent universe: the Big Bang –¿ LIST:arXiv:1005.3955 ¡br¿
¡!– Topics in statistical data analysis for high-energy physics –¿
LIST:arXiv:1012.3589 ¡br¿
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arxiv-papers
| 2010-12-21T13:09:41 |
2024-09-04T02:49:15.891319
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"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C. Grojean and M. Spiropulu",
"submitter": "Scientific Information Service Cern",
"url": "https://arxiv.org/abs/1012.4643"
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|
1012.4726
|
# Pathways of Distinction Analysis:
A new technique for multi-SNP analysis of GWAS data
Rosemary Braun and Kenneth Buetow
National Cancer Institute, NIH, Bethesda, MD.
(March 4, 2011)
###### Abstract
Genome-wide association studies have become increasingly common due to
advances in technology and have permitted the identification of differences in
single nucleotide polymorphism (SNP) alleles that are associated with
diseases. However, while typical GWAS analysis techniques treat markers
individually, complex diseases (cancers, diabetes, and Alzheimers, amongst
others) are unlikely to have a single causative gene. There is thus a pressing
need for multi-SNP analysis methods that can reveal system-level differences
in cases and controls.
Here, we present a novel multi-SNP GWAS analysis method called Pathways of
Distinction Analysis (PoDA). The method uses GWAS data and known pathway-gene
and gene-SNP associations to identify pathways that permit, ideally, the
distinction of cases from controls. The technique is based upon the hypothesis
that if a pathway is related to disease risk, cases will appear more similar
to other cases than to controls (or vice versa) for the SNPs associated with
that pathway. By systematically applying the method to all pathways of
potential interest, we can identify those for which the hypothesis holds true,
i.e., pathways containing SNPs for which the samples exhibit greater within-
class similarity than across classes. Importantly, PoDA improves on existing
single-SNP and SNP-set enrichment analyses in that it does not require the
SNPs in a pathway to exhibit independent main effects. This permits PoDA to
reveal pathways in which epistatic interactions drive risk.
In this paper, we detail the PoDA method and apply it to two GWA studies: one
of breast cancer, and the other of liver cancer. The results obtained strongly
suggest that there exist pathway-wide genomic differences that contribute to
disease susceptibility. PoDA thus provides an analytical tool that is
complementary to existing techniques and has the power to enrich our
understanding of disease genomics at the systems-level.
## Author Summary
We present a novel method for multi-SNP analysis of genome-wide association
studies. The method is motivated by the intuition that if a set of SNPs is
associated with disease, cases and controls will exhibit more within-group
similarity than across-group similarity for the SNPs in the set of interest.
Our method, Pathways of Distinction Analysis (PoDA), uses GWAS data and known
pathway-gene and gene-SNP associations to identify pathways that permit the
distinction of cases from controls. By systematically applying the method to
all pathways of potential interest, we can identify pathways containing SNPs
for which the cases and controls are distinguished and infer those pathway’s
role in disease. We detail the PoDA method and describe its results in breast
and liver cancer GWAS data, demonstrating its utility as a method for systems-
level analysis of GWAS data.
## Introduction
Genome-wide association studies (GWAS) have become a powerful and increasingly
affordable tool to study the genetic variants associated with disease. Modern
GWAS yield information on millions of single nucleotide polymorphism (SNPs)
loci distributed across the human genome, and have already yielded insights
into the genetic basis of complex diseases [1, 2], including diabetes,
inflammatory bowel disease, and several cancers [3, 4, 5, 6, 7]; a complete
list of published GWAS can be found at the National Cancer Institute–National
Human Genome Research Institute (NCI-NHGRI) catalog of published genome-wide
association studies [8].
Typically, the data produced in GWAS are analyzed by considering each SNP
independently, testing the alleles at each locus for association with case
status; significant association is indicative of a nearby genetic variation
which may play a role in disease susceptibility. Genomic regions of interest
may also be subject to haplotype analysis, in which a handful of alleles
transmitted together on the same chromosome are tested for association with
disease; in this case, the loci which are jointly considered are located
within a small genomic region, often confined to the neighborhood of a single
gene.
Recently, however, there has been increasing interest in multilocus, systems-
based analyses. This interest is motivated by a variety of factors. First, few
loci identified in GWAS have large effect sizes (the problem of “missing
heritability”) and it is likely that the common–disease, common–variant
hypothesis [9, 10] does not hold in the case of complex diseases. Second,
single marker associations identified in GWAS often fail to replicate. This
phenomenon has been attributed to underlying epistasis [11], and a similar
problem in gene expression profiling has been mitigated through the use of
gene-set statistics. Most importantly, it is now well understood that because
biological systems are driven by complex biomolecular interactions, multi-gene
effects will play an important role in mapping genotypes to phenotypes; recent
reviews by Moore and coworkers describe this issue well [10, 12].
Additionally, the finding that epistasis and pleiotropy appear to be inherent
properties of biomolecular networks [13] rather than isolated occurences
motivates the need for systems-level understanding of human genetics.
The impact that biological interaction networks have on our ability to
identify genomic causes of complex disease is readily apparent. Consider a
biologically crucial mechanism with several potential points of failure, such
that an alteration to any will confer disease risk. Because no single
alteration is predominant amongst cases, none attain a significant
association; indeed, it has long been observed that even in histologically
identical tumors, only a fraction may share the same set of mutations (see
references in [14] for examples). Additionally, in a robust system, multiple
alterations may be necessary to alter the activity of an interaction network;
here, healthy individuals may share a subset of the deleterious alleles found
in cases, and again these loci will not be detected. This complexity, noted by
[10, 12, 13, 14] and others, has generated considerable interest in multi-
locus analysis techniques that take advantage of known interaction
information.
Several multi-SNP GWAS analysis approaches have been described in the
literature. Thorough reviews are provided in [15, 16], and we briefly describe
several here. Building on the well-established Gene Set Enrichment Analysis
[17] method initially developed for gene expression data, two articles have
proposed an extension of GSEA for SNP data [18, 19]. In these techniques, each
SNP is assigned a statistic based on a $\chi^{2}$ test of association with the
phenotype; a running sum is then used to assess whether large statistics occur
more frequently amongst a SNP set of interest than could be expected by
chance. While GSEA-type approaches have proven quite useful, their reliance on
single-marker statistics means that systematic yet subtle changes in a gene
set will be missed if the individual genes do not have a strong marginal
association. In the case of a purely epistatic interaction between two SNPs in
a set, the set may fail to reach significance altogether.
To address this issue, Yang and colleagues proposed SNPHarvester [20],
designed to detect multi-SNP associations even when the marginal effects are
weak. To reduce the search space of possible multi-SNP effects, SNPHarvester
[20] first removes any SNPs with univarite significance. Using a novel
searching algorithm, they identify groups of $l$ SNPs that show association
with status in a $\chi^{2}$ test with $3^{l}-1$ degrees of freedom. While this
approach can reveal epistatic effects that would be missed by the GSEA-type
schemes [18, 19], it has other drawbacks. First, the combinatorial explosion
of SNP groups puts a limit on the number of SNPs $l$ that may simultaneously
be examined. Second, the the arbitrary groupings of SNPs and the exclusion of
SNPs with marginal effects can make the biological interpretation of the
analysis results difficult.
The notion that cases will more closely resemble other cases than they will
controls has motivated a number of interesting distance-based approaches for
detecting epistasis. Multi-dimensionality reduction (MDR) has been proposed
and applied to SNP data [21, 22, 23]. In this technique, sets of $l$ SNPs are
exhaustively searched for combinations that will best partition the samples by
examining the $3^{l}$ cells in that space (corresponding to homozygous minor,
heterozygous, or homozygous major alleles for each locus) for
overrepresentation of cases. While this method both finds epistatic
interactions without requiring marginal effects and can be structured to
incorporate expert knowledge, it is limited by the fact the the total number
of loci to be combinatorially explored must be restricted to limit
computational cost. To address this, an “interleaving” approach in which
models are constructed hierarchically has been suggested [22] to reduce the
combinatorial search space. A recent and powerful MDR implementation [24]
taking advantange of the CUDA parallel computing architecture for graphics
processors has made feasible a genome-wide analysis of pairwise SNP
interactions. Still, MDR remains computationally challenging, such that
expanding the search to other SNP set sizes (rather than restricting to
pairwise interactions) can be impeded by combinatorial complexity if an
exhaustive search is to be performed.
In order to narrow down the combinatorial complexity of discovering SNP sets
using techniques such as MDR, feature selection may be employed. Of particular
importance here is the distance-based approach of the Relief family of
algorithms [25, 26, 27, 28]. These are designed to identify features of
interest by weighting each feature through a nearest-neighbor approach. The
weights are constructed in the following way: for each attribute, one selects
samples at random and asks whether the nearest neighbor (across all
attributes) from the same class and the nearest neighbor from the other class
have the same or different values from the randomly chosen sample. Attributes
for which in-class nearest neighbors tend to have the same value are weighted
more strongly. Because the distances are computed across all attributes,
Relief-type algorithms can identify SNPs that form part of an epistatic group
and they provide a means of filtration that does not have the drawbacks of
other significance filters.
While these methods have so far been applied to finding small groups of
interacting SNPs, one may instead be interested in whether cases and controls
exhibit differential distance when considering a large number of genes. A
multi-SNP statistic has been proposed in the literature [29, 30, 31] for
determining whether an individual of interest is on average (across a large
number of SNPs) “closer” to one population sample than to another. The method,
originally proposed by Homer [29], is motivated by the idea that a subtle but
systematic variation across a large number of SNPs can produce a discernible
difference in the closeness of an individual to one population sample relative
to another. While this statistic was originally designed to identify the
proband as a member of one of the population samples, it was shown in [30]
that out-of-pool cases from a case-control breast cancer study were in general
closer (as defined by the statistic presented in [29]) to in-pool cases than
they were to in-pool controls, suggesting that the combination of multiple
alleles has the potential to distinguish cases from controls.
Building on these ideas, we present a new technique that finds pathway-based
SNP-sets that differentiate cases from controls; we call this method Pathways
of Distinction Analysis (PoDA). In PoDA, SNP sets are defined based on known
relationships (e.g., SNPs in genes sharing a common pathway), and thus
incorporate expert knowledge to reduce the search space and provide biological
interpretability. Motivated by the differential “closeness” of cases and
controls as discussed about and as observed in [30], we hypothesize that if
the SNPs come from a pathway that plays a role in disease, there will be
greater in-class similarity than across-class similarity in the genotypes for
those SNPs; i.e., a case will show greater genetic similarity to other cases
than to controls for the SNPs on a disease-related pathway, but will be
equidistant for the SNPs on a non-disease-related pathway. Based on this
notion, PoDA seeks to identify pathways for which differential heterogeneity
is exhibited in cases and controls. In each pathway, PoDA returns a statistic
$S$ for each sample that quantifies that sample’s distance to the remaining
cases relative to its distance to the remaining controls for a given pathway’s
SNPs. PoDA then examines whether the distributions of $S$ for the controls
differ from those of the cases by computing and testing for significance a
Pathway Distinction Score $DS$ that quantifies the differences in case and
control $S$ distributions. In this manuscript, we detail the PoDA method and
report the results of its application to two data sets.
As we will describe, PoDA improves and complements existing approaches in a
number of respects. First, it permits the investigation of arbitrarily large
pathways, circumventing the dimensionality issues that are encountered with
MDR and SNP-Harvester. Second, it is able to detect pathways that contain an
over-abundance of highly-significant markers as well as pathways whose markers
have a small but consistent association that would be missed by GSEA-type
approaches. Finally, it uses a leave-one-out technique to return for each
sample an unsupervised relative distance statistic that can then be used to
model disease risk via logistic regression. In addition to providing an effect
size for the pathway, this allows the odds of disease for new samples to be
obtained by computing its relative distance statistic with respect to the
known samples and applying the model.
## Methods
Following our conjecture that SNPs associated with the genes in a pathway
involved in disease will exhibit more within-group similarity than across-
group similarity, we propose Pathways of Distinction Analysis (PoDA), a method
designed to address the following questions:
* •
Given some set of SNPs, do we find that, on average, cases are “closer” to
other cases than to controls (or that controls are “closer” to other controls
than to cases)?
* •
If we look for these distinctions systematically over all SNP-sets of
potential interest, can we use it to single out SNP-sets which may be
associated with disease?
In PoDA, a set of SNPs are selected, and for each sample we compute whether it
is closer to the pool of remaining cases or controls across that SNP set,
using the relative distance statistic described below. Once this is done for
every sample, the distribution of the relative distance statistic is compared
in the cases and controls using a nonparametric statistic, addressing the
first question above. This may be carried out amongst all SNP sets of
interest, adjusting the $p$-value for the multiple hypotheses, to find SNP
sets for which cases more strongly resemble the population of remaining cases
while controls more strongly resemble the population of remaining controls.
We begin with a discussion of how we measure the relative distance of an
individual to the other cases vs. other controls. A simple but computationally
intensive approach is to represent each sample by a vector in an
$l$-dimensional space, where $l$ is the number of SNPs in the group of
interest. One can then compute, between each sample pair, their distance in
this $l$-dimensional space using a Euclidean, Manhattan, or Hamming metric.
For each sample, we would define its relative distance statistic as the mean
of its distance to other controls minus the mean of its distance to other
cases; a sample that is more similar to cases will exhibit a positive
statistic, whereas one that is more similar to other controls will exhibit a
negative statistic. For the given SNP set, we would then have for each sample
a value quantifying its relative distance that was computed without knowledge
of that sample’s class (i.e., using a leave-one-out scheme) and could then be
used in further tests. By doing this for all pathways of interest, one derives
a relative distance value for each sample in each pathway.
This brute-force approach, while conceptually clear, has two significant
drawbacks. The first is that the distance computation is $\mathcal{O}(l\cdot
n^{2})$ where $n$ is the total number of samples in the study—a considerable
undertaking, particularly if many SNP sets are to be analyzed, and one that
becomes exceedingly troublesome in the context of permutation tests. The
second drawback is that because we are taking the mean of the distances, a
sample that is situated squarely within a cluster of cases may have a large
case-distance value due to the dispersion of cases around it. Both of these
issues are circumvented by instead considering the relative distance to the
centroids of the cases and controls in the $l$-dimensional space, a
computation that can be performed in $\mathcal{O}(l\cdot n)$ for all $n$
samples. It is this approach that PoDA employs, as follows:
In [30, 29], the authors consider a measure of individual $Y$’s distance to
two population samples $F$ and $G$ at locus $i$,
$D_{Y,i}=\left|{y_{i}-f_{i}}\right|-\left|{y_{i}-g_{i}}\right|\,.$ (1)
where $f_{i}$ and $g_{i}$ are the minor allele frequencies (MAFs) of SNP $i$
in samples $F$ and $G$, and $y_{i}\in\\{0,0.5,1\\}$ is $Y$’s genotype at $i$
corresponding to homozygous major, heterozygous, and homozygous minor alleles,
respectively (i.e., the frequency of minor allele in that individual. The
first term quantifies how different $Y$’s MAF is from $F$’s for a given locus
$i$; the second term quantifies how different $Y$’s MAF is from $G$’s at locus
$i$; and so $D_{Y,i}$ gives the distance of $Y$ relative to $F$ and $G$ at
locus $i$. Since the minor allele frequencies $f_{i}$ and $g_{i}$ are computed
by averaging the genotypes (again, written as $\\{0,0.5,1\\}$) in samples $F$
and $G$ respectively, it is clear that $\left|{y_{i}-f_{i}}\right|$ is the
distance from $Y$ to the centroid of $F$ along the coordinate $i$ (and
likewise for the $g_{i}$ term). It can be seen from Eq. 1 that positive
$D_{Y,i}$ implies that $y_{i}$ is closer to $g_{i}$ than to $f_{i}$, and that
negative $D_{Y,i}$ implies that $y_{i}$ is closer to $f_{i}$ than to $g_{i}$.
By computing $D_{Y,i}$ at each locus $i$ and taking the standardized mean
across the $l$ loci, [29] obtain a distance score $S$ which quantifies how
close $Y$ is relative to $F$ and $G$ across all $l$ loci under consideration,
$S_{Y}=\frac{\mathsf{E}({D_{Y,i}})}{\sqrt{\mathsf{Var}({D_{Y,i}})/l}}\,,$ (2)
where $\mathsf{E}({D_{Y,i}})$ denotes the mean of $D_{Y,i}$ across all loci
$i$. That is, $S$ provides a means to quantify whether $Y$’s MAFs are closer
to $G$’s MAFs or $F$’s MAFs on average for the loci under consideration. It is
instructive to consider the geometrical interpretation of Eq. 2. Is clear upon
inspection that the numerator in Eq. 2 is equal, up to a factor of $l$, to the
difference in Manhattan distances between $Y$ and the (nonstandardized) $G$
centroid and $Y$ and the (nonstandardized) $F$ centroid; in this sense, Eq. 2
resembles a nearest-centroid classifier. However, the denominator scales the
relative distances by their variance across the $l$ SNPs; that is, a sample
$Y$ who is consistently closer to $G$ than to $F$ for each of the $l$ SNPs
will obtain a higher $S$ than an individual who is variously closer to either
across the $l$ SNPs under consideration.
By assigning the (non-$Y$) controls as $F$ and the (non-$Y$) cases as $G$, we
can compute a statistic $S_{Y}$ quantifying $Y$’s distance to other cases
relative to $Y$’s distance to other controls. If we then apply this
systematically to all individuals in the study population (removing that
individual, computing the MAF’s $f_{i}$ and $g_{i}$ for the remaining
individuals who comprise $F$ and $G$, and then computing $S_{Y}$ in a leave-
one-out manner), we can obtain distributions of $S_{Y}$ statistics in cases
and controls that may be compared. Here, the null hypothesis is that case and
control $S_{Y}$ distributions do not differ, with the alternative hypothesis
that the cases have higher $S$ values than do controls, i.e., that they are
closer (via Eqs. 1-2) to other cases than are controls.
We can use $S$ in the following manner to answer the questions posed above by
applying it in a leave-one-out manner in each pathway:
1. 1.
For a given pathway $P$, select the $l_{P}$ SNPs associated with that pathway;
2. 2.
For every sample $Y$, remove $Y$ from the case or control group as needed, and
compute $S_{Y,P}$ with respect to the remaining cases and controls using the
SNPs chosen in step 1.
3. 3.
Quantify the differences in distribution of $S_{Y,P}$’s for the case samples
versus that of the controls and test for significance.
By systematically carrying out the above steps on all pathways of interest, we
can identify pathways for which there appears to be differential homogeneity
in cases and controls, indicating that the pathway may play a disease-related
role. The details of the algorithm are explained below, and summarized in
Table 1.
In [30], we examined Eqs. 1-2 and found that the magnitude of $S$ is
influenced both by the MAF differences $f_{i}-g_{i}$ (that is, how distant the
centroids of $F$ and $G$ are) and by correlations between the SNPs under
consideration (due to the penalization for variance in $D_{i}$ provided by the
denominator of Eq. 2. These properties are extremely well-suited to the
application we propose: pathways with few highly-significant SNPs will yield
large $S$ differences (due to the influence of $f_{i}-g_{i}$), as will
pathways with SNPs that are highly correlated yet have subtle individual MAF
differences, corresponding to the concerted action of multiple SNPs.
At the same time, however, we wish to ensure that the pathways we select as
having differential $S$ are not being influenced large LD blocks covered by
the SNPs in the genes on the pathway. That is, we wish to ensure that the SNP
correlations which drive $S$ are reflective of epistatic effects between
different genes rather than recombination events within a gene. To this end,
we select a single SNP to represent each gene, based on the desire to detect
multi-gene rather than multi-SNP effects.
In practice, SNPs are selected as follows: for each pathway represented in the
Pathway Interaction Database [32] (PID, http://www.pid.nci.gov, containing
annotations from BioCarta, Reactome, and the NCI/Nature database [32]) and
KEGG [33], we select the associated genes. Using dbSNP [34], we retrieve the
SNPs associated with the pathway genes that are present in the data, excluding
those with $>20\%$ missing data or with minor allele frequency $<0.05$ in
either case of control group. We necessarily exclude pathways for which only
one gene is probed by the remaining SNPs. Because we are interested in $S$
values that are driven by correlations across genes (and not in individual
genes covered by many SNPs with high LD), we select for each gene its most
significant SNP in a univariate test of association (Fisher’s exact test). It
should be noted here that while the SNP chosen for each gene is the most
significant of that gene’s SNPs, it is not necessarily significantly
associated with disease. Our goal here is not to filter based on SNP
significance, but rather to select, for each gene, a single marker that is as
informative as possible.
Having selected the SNPs of interest, we compute $D_{Y,i}$ at each locus for
every sample by selectively removing it and comparing it to the remaining
cases and controls, as described above. For each pathway $P$, we compute
$S_{Y,P}$ for $l_{P}$ the SNPs $i$ that comprise it, yielding a distribution
of $S_{Y,P}$ for cases and another distribution for controls. The difference
in the location of the case and control $S_{Y,P}$ distributions is then
quantified nonparametrically by computing the Wilcoxon rank sum statistic,
defined as
$W_{P}=\sum_{Y\in\mathrm{case}}R_{Y,P}-\frac{n_{\mathrm{case}}(n_{\mathrm{case}}+1)}{2}\,$
(3)
where $R_{Y,P}$ is the rank of $S_{Y,P}$ amongst all samples $Y$ for a given
pathway $P$. Eq. 3 thus quantifies, non-parametrically, the degree to which
cases are “closer” to other cases and controls “closer” to other controls
across a set of SNPs for all individuals in the GWAS.
To illustrate the above, we consider a simulated GWAS of $250$ cases and $250$
controls and $50$ SNPs, shown in Figure 1, and ask whether we are able to
detect a 12-SNP pathway in which a subset of SNPs appear to have an epistatic
interaction. Alleles were simulated as binomial samples from a source
population with MAFs ranging from $0.1$ to $0.4$ across the $50$ SNPs, and
case labels were assigned such that a combintion of homozygous minor alleles
at SNPs 1 and 2 or 3 (i.e., $(y_{1}=1)\land((y_{2}=1)\lor(y_{2}=1))$)
conferred a three-fold relative risk, mimicking an epistatic interaction
between SNPs 1 and 2 and SNPs 1 and 3 (Figure 1(a)). Alone, none of the $50$
SNPs showed any association with case status, nor was any SNP significantly
out of HWE in either cases or controls. However, the “shared pattern” in SNPs
1–3 is such that a 12 SNP pathway comprising SNPs 1–12 yields greater $S$ in
cases than in controls as can been seen in Figure 1(b), while a random 12 SNP
pathway selected from the 50 SNPs (that includes SNP 3, but neither SNP 1 or
2) shows no difference in $S$ values as seen in Figure 1(c).
While the Wilcoxon statistic $W$ is normal in the large-sample limit and can
be directly compared to a Gaussian, to truly evaluate the significance of
$W_{P}$ for a given pathway $P$, we must address two sources of bias: the
number of SNPs per gene, and the size of the pathway. To address these issues,
we introduce a normalized Pathway Distinction Score $DS_{P}$ that we test for
significance using a resampling procedure.
First, we expect that because we have selected for each gene the single most
informative SNP, we are pre-disposed to seeing higher $W_{P}$ for pathways
that contain large genes. Because large genes will be more likely to contain
highly-significant SNPs by chance, the concern has been raised that [35, 18]
selecting the single most significant SNP as a proxy for the gene (as is done
here) will lead to a bias toward pathways that contain an abundance of large
genes. To account for this, we follow the approach in [18] and normalize the
score via a permutation-based procedure. First, we permute the phenotype
labels and in each permutation recalculate $W_{P}$ as described above, but
using the permuted case and control labels. The permuted labels are used both
to select the most informative SNP per gene and to compute $f_{i}$, $g_{i}$,
and $W_{P}$ in Eqns. 1–3). This yields a distribution of $W^{*}_{P}$ under the
null hypothesis that the magnitude of $W$ is independent of the true
case/control classifications. We then normalize the true $W_{P}$ by the
distribution from the permutation procedure, yielding a Distinction Score
$DS_{P}$ for pathway $P$ that effectively adjusts for different sizes of genes
and preserves correlations of SNPs in the same gene:
$DS_{P}=\frac{W_{P}-\mathsf{E}({W^{*}_{P}})}{\mathsf{SD}({W^{*}_{P}})}\,,$ (4)
where $W^{*}_{P}$ are the set of $W_{P}$ obtained for pathway $P$ across the
permutations. (In practice, 100 permutations are used.) Because the permuted
labels are used in the SNP selection, this normalization adjusts for the bias
introduced by the fact that large genes have more opportunity to contain
significant SNPs by chance. The Pathway Distinction Score $DS_{P}$ thus
provides a model-free, gene-size adjusted metric for quantifying the degree to
which cases are “closer” to other cases (higher $S_{P}$) than controls.
To test whether $DS_{P}$ is significant, we note that larger pathways may
yield high $DS_{P}$ values simply due to the fact that they sample the case
anc control differences more thoroughly. Indeed, the question of significance
that we wish to address is not simply whether a pathway permits the
distinction of cases and controls, but whether it does so better than a random
collection of as many SNPs, wherein the SNPs are still selected to be the most
informative by gene. To account for the fact that the pathways are of
differing sizes, significance of the Distinction Score for a given pathway is
assessed through resampling by choosing, at random, the same number of SNPs
that are present in that pathway ($l_{P}$) from the total set of most-
informative-SNP-per-gene and recomputing $DS_{P}$ for the random pathway. The
$p$ value is obtained by counting the fraction of random $l_{P}$-SNP sets
which give a larger $DS_{P}$ than the true pathway SNPs in $10^{4}$
resamplings. In this way, we are able to detect pathways that yield large
differences of case and control $S$ distributions due to their particular
SNPs, rather than simply being the result of choosing many SNPs. The $p$ value
obtained addresses the question of whether the pathway under consideration
permits greater separation of cases and controls than would a random
collection of most-informative-SNP-per-gene, i.e., whether there exists a more
extreme aggregated effect in that pathway than expected by chance.
## Results
We applied PoDA to 2287 genotypes obtained from the Cancer Genomic Markers of
Susceptibility (CGEMS) breast cancer study. The samples were sourced as
described in [4]. Briefly, the samples comprised 1145 breast cancer cases and
a comparable number (1142) of matched controls from the participants of the
Nurses Health Study. All the participants were American women of European
descent. The samples were genotyped against the Illumina 550K arrays, which
assays over 550,000 SNPs across the genome.
We also applied it to a smaller liver cancer GWAS [36] comprising 386
hepatocellular carcinoma (HCC) patients and 587 healthy controls from a Korean
population. Samples were genotyped against Affymetrix SNP6.0 arrays, which
provides SNP information at approximately one million loci.
### Breast cancer GWAS results
We begin by applying PoDA to the CGEMS breast cancer GWAS data. Having
observed (Figure 1) that PoDA performs as expected for the simulated data, we
first turn our attention to a simple test in which we select a SNP set
comprising the four SNPs in intron 2 of $\mathit{FGFR2}$ that were reported to
show significant association with case status in [4] (rs11200014, rs2981579,
rs1219648, rs2420946). We expect to see a strong difference in the test case
and test control distributions, and indeed we do: the cases more frequently
have positive $S$ than do controls in Fig. 2. (The discrete peaks in the
distribution are a result of the fact that with four SNPs there exist fewer
available values of $S$.) Using a nonparametric Wilcoxon rank sum test with
the alternative hypothesis that cases have greater $S$ than controls,
$p=1.016\cdot 10^{-6}$ is obtained, confirming our intuition.
We next applied PoDA systematically to the pathways represented in PID [32]
using CGEMS data. Associations between genes and SNPs were made using dbSNP
build 129 [34]. 1081 pathways were non-trivially covered in the data set;
69453 SNPs in the data could be associated with at least one of the pathways.
Because these 69453 SNPs were associated with 4446 unique genes, 4446 were
kept in the analysis (the most significant SNP for each gene of interest). The
1081 pathways ranged from 2 to 229 genes, with a mean of 19. $S_{Y,P}$ was
computed in each pathway $P$ for each of the 2287 samples $Y$ via Eq. 2, and
the distinction score $DS_{P}$ (Eq. 4) quantifying differential $S$
distributions in cases and controls was computed for each pathway.
Significance was assessed as described above, by resampling “dummy” pathways
of the same length and computing the fraction of greater $DS_{P}$ scores.
Because PoDA provides for each sample a measure $S$ (Eq. 2 of that sample’s
relative distance from the remaining ones that is obtained without regard to
that sample’s true class membership, we can use the $S$ value as a metric by
which to predict the odds of disease. Here, we construct a logistic regression
model of case status as a function of $S$ to obtain the odds ratio. $p$-values
were adjusted for the multiplicity of pathways using FDR adjustment [37, 38].
Pathways with significant $DS_{P}$ and odds ratios are reported in Table 2 and
plots of $S$ for four of them are illustrated in Figure 3. Although the cases
and controls are not crisply separable, a unit increase in $S$ over its range
from approximately -3 to 3 yields between a 1.5 and 2.0-fold increase in odds.
Importantly, given known minor allele frequencies for cases and controls for
this set of SNPs, we can model the increase in odds for an unknown individual
based on her “closeness” to other cases.
In order to ensure that the pathways listed were not interrogating the same
set of genes, we carried out two checks. First, we computed the SNP overlap
between all pairs of significant pathways, sequentially removing pathways that
shared in excess of 60% of their genes with another pathway. Because this is
done using a greedy algorithm that depends on the order of the pathways input,
the culling algorithm was run with different starting orders, and the most
frequent output was kept. No pathway remaining in Table 2 shares more than 60%
of its SNPs with another pathway. (An un-culled list may be found in
Supplementary Table S-1.) Second, we computed the correlation of $S$ values
between each pair of pathways to assess whether any pathway’s $S$ statistic
was reflecting the same genetic variation as another (i.e., whether samples
that had high $S$ values for one pathway consistently did so in another). The
maximum correlation of $S$ values observed between any two pathways in Table 2
was 0.58, suggesting that a different subset of samples is affected in each
pathway.
Many of the pathways listed in Table 2 fulfill biological functions that are
well known to be cancer-associated, playing a strong role in cell
proliferation and migration, processes which are perturbed in malignancies.
Purine metabolism—the most significantly associated pathway—has been observed
to be altered in cancer cells [39, 40], and the majority of the other
significant pathways are directly related to cell migration (e.g., ErbB
signaling and gap junction pathways) and cellular signalling (e.g., calcium
signaling, PKC-catalyzed phosphorylation of myosin phosphatase, attenuation of
GPCR signaling, and activation of PKC through GPCRs) processes that have been
implicated in a variety of cancers. In addition, eicosanoids and unsaturated
fatty acid metabolism have been associated with breast cancer specifically
[41]. In general, the findings in Table 2 suggest that there exist germline
genetic differences in these mechanisms that confer a predisposition to
disease.
Interestingly, the GnRH (gonadotropin releasing hormone) signaling pathway
appears to be significant. GnRH has been linked with HR-positive breast cancer
and the use of GnRH analogues in breast cancer treatment has already been
proposed [42, 43]. However, a recent large sequencing study found no
association of GnRH1 or GnRH receptor gene polymorphisms with breast cancer
risk [44], contrary to the author’s hypothesis that common, functional
polymorphisms of GnRH1 and GnRHR could influence breast cancer risk by
modifying hormone production. In contrast to their null findings, our result
suggests that there are system-wide variations in GnRH signalling that
contribute to risk that are not evident when considering the GnRH1 and GnRHR
SNPs independently.
Of the 1081 pathways considered, four—FGF signaling, MAPK signaling,
regulation of actin cytoskeleton, and prostate cancer—contained
$\mathit{FGFR2}$, the gene found to be significantly associated in the initial
CGEMS analysis [4]. However, only one—prostate cancer—was significant in
comparison to randomly generated pathways of the same length. It may
reasonably be asked, then, whether the high significance of the prostate
cancer pathway in Table 2 is a result of $\mathit{FGFR2}$. To address this, we
eliminated the $\mathit{FGFR2}$ SNP from the prostate cancer pathway; the
resampling-based test remained significant ($p(DS_{P})=0.044,OR=0.3,q(OR)=$
8.2e-09), suggesting that the association of the prostate cancer pathway is
not driven solely by differences in $\mathit{FGFR2}$.
### Liver cancer GWAS results
We carried out the same procedure in using data from the liver cancer GWAS
described above. Here, 1049 pathways were non-trivially covered in the data
set; 53079 SNPs in the data could be associated with at least one of the
pathways. Because these 53079 SNPs were associated with 3718 unique genes,
3718 were kept in the analysis (the most significant SNP for each gene of
interest). The 1081 pathways ranged from 2 to 193 genes, with a mean of 16. As
above, $DS_{P}$ scores for differential $S$ distributions in cases and
controls were computed for each pathway, resampled $p$ values obtained for
each pathway size, odds ratios for $S$ were obtained, and the multiple
hypotheses were corrected using FDR adjustment [37, 38]. Significant pathways
are listed in Table 3, and plots of the top three pathways are given in
Figures 4a-d. As in the breast cancer data above, we removed pathways which
had over 60% their SNPs covered by another pathway (a complete list, with
overlapping pathways, is give in Supplementary Table S-2) and examined the
correlation in $S$ for all remaining pathways (maximum $\rho=0.42$).
The results here are interesting. First, we observe that a couple pathways are
significant in both the CGEMS breast and liver GWAS with similar effect sizes,
namely ErbB signaling and biosynthesis of unsaturated fatty acids. ErbB has a
well–established association with cancer; unsaturated fatty acid biosynthesis
may link diet to cancer risk, and its appearance may suggest a gene-
environment interaction. The commonality of these known cancer-associated
pathways across the two studies suggest that there may exist genetic patterns
that confer carcinogenesis risk irrespective of the disease site. Along with
those shared in the breast cancer data, many of the other significant pathways
in the liver cancer data well known to be tumorassociated, including cell
adhesion molecules, Wnt signaling, c-Kit receptor, and angiogenesis pathways,
further supporting the notion that germline genetic differences in these
mechanisms contribute to cancer risk. The appearance of many neuronal pathways
is also supported by our understanding of carcinogenesis: thes contain well-
known signal transduction molecules including Ras and PKA that may both be
driving their conferring increased cancer risk and driving the significance of
the pathway [45].
Additionally, six of the 25 significant liver cancer pathways are immune– and
inflammation–related, namely, antigen processing and presentation (two, with
$<$60% overlap), classical complement pathway, corticosteroids, IL12 signaling
mediated by STAT4, and NO2-dependent IL-12 pathway in NK cells. This is a
particularly interesting finding in light of the fact that the original
analysis of the liver data [36] suggested that altered T-cell activation plays
a direct role in the onset of liver cancer. The involvement of the immune
system in liver cancer development has been established in clinical studies
and research involving model organisms. Increased activity of helper T-cells,
which promote inflammation, is associated with hepatocellular carcinogenesis
[46] while activation and proliferation of cytotoxic T-lymphocytes is
suppressed in liver cancers [47, 48]. The inflammatory immune response,
mediated by interleukins, has also been closely connected to liver cancers in
mice [49] and humans [50, 51, 52]. These findings, coupled with the
observation of several significant immune-related pathways in our data, are
suggestive of germline polymorphisms in immune response that lead to
hepatocellular carcinoma risk.
### Combining pathways
In both the breast and liver cancer results, we see observe that even though
significant pathways yield between a 1.5 and 2.0-fold increase in odds for
each unit increase in $S$ (over its typical range of approximately $-3$ to
$3$), the cases and controls are not crisply separable based on $S$ values.
These findings suggest that it may be possible to combine pathways to yield a
model that is more predictive than a single pathway alone. However, the $S$
values must not simply be put into the regression model because the overlap in
pathways will result in some SNPs being double-counted. Rather, we combine
pathways by taking the union of their SNPs, and recomputing the statistics.
Doing this sequentially for the top pathways in the order as listed in Tables
2 and 3 yields the values given in Tables 4 and 5, respectively. Considerably
higher ORs are obtained when combining the significant pathways. An
illustration of the case and control distributions when using a “superpathway”
comprised of the top three pathways in the breast and liver data,
respectively, is given in Figure 5. These findings support the notion that the
genomic variation contributing to risk is spread over several mechanisms,
rather than being concentrated in a single gene.
## Discussion
We have introduced the Pathways of Distinction analysis method (PoDA) for
identifying pathways which can be used to distinguish between phenotype
groups. PoDA identifies sets of SNPs in GWAS studies for which cases and
controls exhibit differential “closeness” to other cases and controls; that
is, it permits one to infer whether cases are more similar to other cases than
are controls across a given set of SNPs. Because PoDA is designed to detect
the joint effects of multiple SNPs, it presents an approach to GWAS analysis
that augments single-SNP or single-gene tests.
We applied PoDA to two GWAS data sets, with highly promising results. In the
breast cancer data, we found a number of pathways which are known to play a
role in cancers generally and breast cancer specifically, suggesting that
differences in these mechanisms which confer disease risk may exist at the
germline DNA level. In the liver cancer data, we found an extreme abundance of
immune-related pathways, further corroborating the known link between
inflammation and hepatocellular carcinoma, and bolstering the observation in
[36] that germ-line differences in immune function may play a role in liver
carcinogenesis.
PoDA may be used as a complement to other multi-SNP analysis techniques [18,
19, 20, 21]. Unlike gene-set enrichment type approaches [17, 18, 19], which
search for an overabundance of significant markers in a gene set of interest,
PoDA finds both sets containing highly significant markers as well as sets
that have a subtle but consistent pattern across all the markers in the set.
This permits the detection of pathways in which the joint action of several
alterations produce a phenotype and those for which any of several possible
alterations, none of them the dominant one, confer predisposition to disease.
Indeed, many of the pathways indicated in our analysis of the breast cancer
data (Table 2) were not detected using SNP-set enrichment [17, 18, 19] (data
not shown), including the highly significant purine metabolism and GnRH
signaling pathways, both of which are biologically relevant (purine metabolism
has been implicated in cancers generally due to its role in DNA and RNA
synthesis [40], and GnRH has been shown to be clinically important in breast
and gynecological cancers [53]). These pathways, along with others that were
indicated using PoDA but not enrichment analysis (data not shown), have a
statistically significant difference in case and control $S$ distributions and
remain significant in comparison with randomly-generated pathways of the same
length.
Because PoDA effectively measures the closeness of each individual to
remaining cases and controls, it bears a conceptual relationship to nearest-
neighbor and nearest-centroid classifiers [54, 55], as well as to the
distance-based feature selection algorithms like Relief-F and its derivatives
[25, 26, 27, 28]. However, it must be remembered that the goal of PoDA is to
indicate mechanisms that may be deleteriously hit at the genomic level even
when those hits are heterogeneous, whereas the goal of nearest-centroid
classifiers and Relief-F–type feature selection is to derive a minimal set of
markers that best classify cases and controls (and thus are the most
homogeneously hit). These approaches are complementary, and one can easily
envision an application in which (e.g.) Relief-F is run within pathways that
are highly significant in the PoDA analysis in order to single out the SNPs
driving the effect. In fact, this approach may improve ReliefF’s ability to
find those genes, since the nearest neighbors from which the Relief SNP
weights are calculated would be the nearest-neighbors for that specific
pathway, thus discounting heterogeneity introduced by mechanistically
unrelated genes. For instance, in the provided example (Fig 1), ReliefF fails
to identify the significance of SNPs 1–3 when run using the complete 50-SNP
data, but places at least two of SNPs 1, 2 or 3 in the top third of selected
features when restricted to SNPs 1–12.
While PoDA has many benefits, it should be noted that when epistasis drives a
phenotype with no differences in the minor allele frequencies for the
epistatically-interacting genes (as opposed to a slight yet consistent one
shown in the example), PoDA as computed via Eqs. 1,2 will miss the pathway.
Geometrically, such a situation would mean that the case and control groups
have the same centroids while having a different distribution of samples about
those centroids. A famous example of this is provided through the non-linearly
separable XOR (exclusive or): consider two epistatic loci $(X,Y)$ such that
all controls have genotypes in the set $\\{(0,0),(1,1)\\}$ and all cases have
genotypes in the set $\\{(0,1),(1,0)\\}$ (i.e., that a genotype of 1 at either
locus can be compensated by a genotype of 1 at the other, but having just one
alone—1 at exclusively $X$ or $Y$—is deleterious). If the loci $X$ and $Y$
each have the same MAF in cases and controls, it is plain to see that the
centroids will be in the same location for both groups, and Eq. 1 will yield
zero for both cases and controls. If instead of using Eq. 1, we compute
pairwise sample-sample distances, we can circumvent this limitation and find
such epistatic situations (it is this pairwise approach that permits Relief-F
to also uncover nonlinearly interacting SNPs). While we provide the facility
for this in the PoDA package, the cost of carrying out the pairwise
computation is a considerable increase in computational complexity.
A number of potential avenues exist to extend the application of PoDA further.
One possible application is in improving the reproducibility of GWAS results.
We note that several of the pathways identified in the breast cancer GWAS data
were also implicated in the liver cancer data, which suggests that there may
be common features which distinguish individuals to cancer generally. Because
different GWA studies—even those of the same phenotypes—often yield different
results at the SNP level, it may be possible to find common alterations at the
pathway level across disparate GWAS using PoDA.
Extending PoDA further, the $DS_{P}$ scores obtained for each pathway may be
examined for over-representation of extreme values in pathways that comprise a
particular biological subsystem—one may think of this as a “pathway-set”
enrichment analysis (which would be conducted using the a running-sum
statistic analogous GSEA [17]), and could use it to answer whether (e.g.)
immune-related pathways are hit in liver cancer more often than expected by
chance. Alternatively, boosting [56, 57] could be used to find sets of
pathways which are more predictive of case status than individual pathways.
Either of these approaches would yield a richer, systems-wide view of the
connection between genotype and phenotype. Finally, because PID contains
topological information regarding pathway connectivity, one may consider sub-
networks of pathways, permitting one to find potential chemopreventive and
therapeutic targets. Alternatively, Relief-F can be used, as mentioned above,
in a pathway–specific manner to yield the subset of SNPs that drive the
distinction of cases and controls in high-$DS_{P}$ pathways.
PoDA provides an advantage over existing GWAS analysis methods. Because it
does not rely on the significance of individual markers, it has the power to
aid in identifying the genomic causes of complex diseases that would not be
detected in single-gene tests or enrichment analyses. The size of the SNP set
is not limited in PoDA, and since PoDA leverages known biological
relationships to find multi-SNP effects, the results are readily
interpretable. PoDA may thus be used to augment existing analysis techniques
and provide a richer, systems-level understanding of genomics.
## Availability
An R package to carry out PoDA is available upon request from the authors (to
be deposited in the Bioconductor in the near future).
## Acknowledgments
This research was supported by the Intramural Research Program of the National
Cancer Institute, National Institutes of Health, Bethesda, MD. RB was
supported by the Cancer Prevention Fellowship Program, National Cancer
Institute, National Institutes of Health, Bethesda, MD. The authors would like
to that Dr Carl Schaefer (NCI) for helpful discussion and assistance with PID.
## References
* [1] Hirschhorn JN, Daly MJ (2005) Genome-wide association studies for common diseases and complex traits. Nat Rev Genet 6: 95-108.
* [2] McCarthy MI, Abecasis GR, Cardon LR, Goldstein DB, Little J, et al. (2008) Genome-wide association studies for complex traits: consensus, uncertainty and challenges. Nat Rev Genet 9: 356-69.
* [3] Easton DF, Eeles RA (2008) Genome-wide association studies in cancer. Hum Mol Genet 17: R109-15.
* [4] Hunter DJ, Kraft P, Jacobs KB, Cox DG, Yeager M, et al. (2007) A genome-wide association study identifies alleles in FGFR2 associated with risk of sporadic postmenopausal breast cancer. Nature Genetics 39: 870-874.
* [5] Lou H, Yeager M, Li H, Bosquet JG, Hayes RB, et al. (2009) Fine mapping and functional analysis of a common variant in MSMB on chromosome 10q11.2 associated with prostate cancer susceptibility. Proc Natl Acad Sci U S A 106: 7933-8.
* [6] Lou H, Yeager M, Li H, Bosquet JG, Hayes RB, et al. (2009) Fine mapping and functional analysis of a common variant in MSMB on chromosome 10q11.2 associated with prostate cancer susceptibility. Proc Natl Acad Sci U S A 106: 7933-8.
* [7] Thomas G, Jacobs KB, Kraft P, Yeager M, Wacholder S, et al. (2009) A multistage genome-wide association study in breast cancer identifies two new risk alleles at 1p11.2 and 14q24.1 (RAD51L1). Nat Genet 41: 579-84.
* [8] Hindorff LA, Sethupathy P, Junkins HA, Ramos EM, Mehta JP, et al. (2009) Potential etiologic and functional implications of genome-wide association loci for human diseases and traits. Proc Natl Acad Sci U S A 106: 9362-7.
* [9] Schork N, Murray S, Frazer K, Topol E (2009) Common vs. rare allele hypotheses for complex diseases. Current opinion in genetics & development 19: 212–219.
* [10] Moore J, Asselbergs F, Williams S (2010) Bioinformatics challenges for genome-wide association studies. Bioinformatics 26: 445.
* [11] Greene C, Penrod N, Williams S, Moore J (2009) Failure to replicate a genetic association may provide important clues about genetic architecture. PLoS One 4: e5639.
* [12] Moore J (2003) The ubiquitous nature of epistasis in determining susceptibility to common human diseases. Human Heredity 56: 73–82.
* [13] Tyler A, Asselbergs F, Williams S, Moore J (2009) Shadows of complexity: what biological networks reveal about epistasis and pleiotropy. BioEssays 31: 220–227.
* [14] Hanahan D, Weinberg RA (2000) The hallmarks of cancer. Cell 100: 57-70.
* [15] Holmans P (2010) Statistical methods for pathway analysis of genome-wide data for association with complex genetic traits. Advances in genetics 72: 141.
* [16] Wang K, Li M, Hakonarson H (2010) Analysing biological pathways in genome-wide association studies. Nature Reviews Genetics 11: 843–854.
* [17] Subramanian A, Tamayo P, Mootha VK, Mukherjee S, Ebert BL, et al. (2005) Gene set enrichment analysis: a knowledge-based approach for interpreting genome-wide expression profiles. Proc Natl Acad Sci USA 102: 15545-50.
* [18] Wang K, Li M, Bucan M (2007) Pathway-based approaches for analysis of genomewide association studies. Am J Hum Genet 81: 1278.
* [19] Holden M, Deng S, Wojnowski L, Kulle B (2008) GSEA-SNP: applying gene set enrichment analysis to SNP data from genome-wide association studies. Bioinformatics 24: 2784-5.
* [20] Yang C, He Z, Wan X, Yang Q, Xue H, et al. (2009) SNPHarvester: a filtering-based approach for detecting epistatic interactions in genome-wide association studies. Bioinformatics 25: 504-11.
* [21] Motsinger A, Ritchie M (2006) Multifactor dimensionality reduction: an analysis strategy for modelling and detecting gene–gene interactions in human genetics and pharmacogenomics studies. Human Genomics 2: 318–328.
* [22] Moore J, Gilbert J, Tsai C, Chiang F, Holden T, et al. (2006) A flexible computational framework for detecting, characterizing, and interpreting statistical patterns of epistasis in genetic studies of human disease susceptibility. Journal of theoretical biology 241: 252–261.
* [23] Cordell H (2009) Detecting gene–gene interactions that underlie human diseases. Nature Reviews Genetics 10: 392–404.
* [24] Greene C, Sinnott-Armstrong N, Himmelstein D, Park P, Moore J, et al. (2010) Multifactor dimensionality reduction for graphics processing units enables genome-wide testing of epistasis in sporadic als. Bioinformatics 26: 694.
* [25] Kira K, Rendell L (1992) A practical approach to feature selection. Proceedings of the ninth international workshop on Machine learning : 249–256.
* [26] Robnik-Šikonja M, Kononenko I (1997) An adaptation of relief for attribute estimation in regression. Proc Int Conf on Machine Learning ICML-97 : 296–304.
* [27] Moore J (2007) Genome-wide analysis of epistasis using multifactor dimensionality reduction: feature selection and construction in the domain of human genetics. Knowledge Discovery and Data Mining: Challenges and Realities with Real World Data : 17–30.
* [28] Greene C, Penrod N, Kiralis J, Moore J (2009) Spatially Uniform ReliefF (SURF) for computationally-efficient filtering of gene-gene interactions. BioData mining 2: 5.
* [29] Homer N, Szelinger S, Redman M, Duggan D, Tembe W, et al. (2008) Resolving individuals contributing trace amounts of DNA to highly complex mixtures using high-density SNP genotyping microarrays. PLoS Genetics 4: e1000167.
* [30] Braun R, Rowe W, Schaefer C, Zhang J, Buetow K (2009) Needles in the haystack: Identifying individuals present in pooled genomic data. PLoS Genetics 5(10): e1000668.
* [31] Visscher PM, Hill WG (2009) The limits of individual identification from sample allele frequencies: theory and statistical analysis. PLoS Genet 5: e1000628.
* [32] Schaefer CF, Anthony K, Krupa S, Buchoff J, Day M, et al. (2009) PID: the Pathway Interaction Database. Nucleic Acids Res 37: D674–679.
* [33] Kanehisa M, Araki M, Goto S, Hattori M, Hirakawa M, et al. (2008) KEGG for linking genomes to life and the environment. Nucleic Acids Res 36: D480-4.
* [34] Sherry ST, Ward MH, Kholodov M, Baker J, Phan L, et al. (2001) dbSNP: the NCBI database of genetic variation. Nucleic Acids Res 29: 308–311.
* [35] Kraft P, Raychaudhuri S (2009) Complex diseases, complex genes: keeping pathways on the right track. Epidemiology (Cambridge, Mass) 20: 508.
* [36] Clifford RJ, Zhang J, Meerzaman DM, Lyu MS, Hu Y, et al. (2009) Genetic variations at loci involved in the immune response are risk factors for hepatocellular carcinoma. Submitted .
* [37] Benjamini Y, Hochberg Y (1995) Controlling the false discovery rate: a practical and powerful approach to multiple testing. Journal of the Royal Statistical Society 57: 289–300.
* [38] Benjamini Y, Yekutieli D (2001) The control of the false discovery rate in multiple testing under dependency. Annals of Statistics : 1165–1188.
* [39] Weber G (1977) Enzymology of cancer cells. New England Journal of Medicine 296: 541–551.
* [40] Weber G (1983) Enzymes of purine metabolism in cancer. Clinical Biochemistry 16: 57–63.
* [41] Rose D, Connolly J (1990) Effects of fatty acids and inhibitors of eicosanoid synthesis on the growth of a human breast cancer cell line in culture. Cancer research 50: 7139.
* [42] Eidne K, Flanagan C, Harris N, Millar R (1987) Gonadotropin-releasing hormone (GnRH)-binding sites in human breast cancer cell lines and inhibitory effects of GnRH antagonists. Journal of Clinical Endocrinology & Metabolism 64: 425.
* [43] Manni A, Santen R, Harvey H, Lipton A, Max D (1986) Treatment of breast cancer with gonadotropin-releasing hormone. Endocrine reviews 7: 89.
* [44] Canzian F, Kaaks R, Cox D, Henderson K, Henderson B, et al. (2009) Genetic polymorphisms of the GnRH1 and GNRHR genes and risk of breast cancer in the national cancer institute breast and prostate cancer cohort consortium. BMC cancer 9: 257.
* [45] Nakagawara A (2001) Trk receptor tyrosine kinases: a bridge between cancer and neural development. Cancer letters 169: 107–114.
* [46] Pentcheva-Hoang T, Corse E, Allison JP (2009) Negative regulators of T-cell activation: potential targets for therapeutic intervention in cancer, autoimmune disease, and persistent infections. Immunol Rev 229: 67-87.
* [47] Ormandy LA, Hillemann T, Wedemeyer H, Manns MP, Greten TF, et al. (2005) Increased populations of regulatory T cells in peripheral blood of patients with hepatocellular carcinoma. Cancer Res 65: 2457-64.
* [48] Unitt E, Rushbrook SM, Marshall A, Davies S, Gibbs P, et al. (2005) Compromised lymphocytes infiltrate hepatocellular carcinoma: the role of T-regulatory cells. Hepatology 41: 722-30.
* [49] Naugler WE, Sakurai T, Kim S, Maeda S, Kim K, et al. (2007) Gender disparity in liver cancer due to sex differences in MyD88-dependent IL-6 production. Science 317: 121-4.
* [50] Budhu AS, Zipser B, Forgues M, Ye QH, Sun Z, et al. (2005) The molecular signature of metastases of human hepatocellular carcinoma. Oncology 69 Suppl 1: 23-7.
* [51] Budhu A, Wang XW (2006) The role of cytokines in hepatocellular carcinoma. J Leukoc Biol 80: 1197-213.
* [52] Budhu A, Forgues M, Ye QH, Jia HL, He P, et al. (2006) Prediction of venous metastases, recurrence, and prognosis in hepatocellular carcinoma based on a unique immune response signature of the liver microenvironment. Cancer Cell 10: 99-111.
* [53] Emons G, Grundker C, Gunthert A, Westphalen S, Kavanagh J, et al. (2003) GnRH antagonists in the treatment of gynecological and breast cancers. Endocrine-related cancer 10: 291.
* [54] Cover T, Hart P (2002) Nearest neighbor pattern classification. IEEE Transactions on Information Theory 13: 21–27.
* [55] Tibshirani R, Hastie T, Narasimhan B, Chu G (2002) Diagnosis of multiple cancer types by shrunken centroids of gene expressionx. Proc Natl Acad Sci USA 99: 6567-72.
* [56] Buhlmann P, Hothorn T (2007) Boosting algorithms: regularization, prediction and model fitting. Statistical Science 22: 477–505.
* [57] Meir R, Ratsch G (2003) An introduction to boosting and leveraging. Lecture Notes in Computer Science 2600: 118–183.
Figure 1: PoDA applied to simulated data. Alleles at 50 loci for 250 cases
and 250 controls were simulated such that each SNP was in HWE and not
associated with case status, but homozygous minor (red) at both loci 1 and 2
or 1 and 3 yielded a three-fold relative risk (a). A 12-SNP pathway comprising
SNPs 1–12 shows differential $S$ distributions (b); a random 12-SNP pathway
does not (c). Boxplots are overlayed on the scatterplots of $S$ for clarity.
Figure 2: PoDA applied to four highly-significant SNPs. Shown is the
distribution of $S$ values in CGEMS cases (red) and controls (black) for a
SNP-set comprised of four highly-significant SNPs located in the
$\mathit{FGFR2}$ gene [4]. As expected, there is a substantial difference in
case and control $S$ values, with the cases having higher $S$ (i.e., closer to
other cases) than controls. The discreteness of the distributions are due to
the fact that with four SNPs, a finite number of $S$ values are possible.
Figure 3: Four significant pathways in breast cancer data. Scatter plots of
$S_{Y,P}$ for each pathway are overlayed with boxplots are given in the left
panel; higher values of $S$ indicate that the sample is closer to other cases
than it is to other controls. Distributions of $S$ for cases (red) and
controls (black) are given to the right. A significant shift toward higher $S$
values is seen in the cases. Odds ratios and FDR-adjusted OR $p$ values are
given.
Figure 4: Four significant pathways in liver cancer data. Scatter plots of
$S_{Y,P}$ for each pathway are overlayed with boxplots are given in the left
panel; higher values of $S$ indicate that the sample is closer to other cases
than it is to other controls. Distributions of $S$ for cases (red) and
controls (black) are given to the right. A significant shift toward higher $S$
values is seen in the cases. Odds ratios and FDR-adjusted OR $p$ values are
given.
Figure 5: Union of top three pathways. SNPs from the top three pathways are
combined to compute $S$ for the breast cancer data (a) and the liver cancer
data (b). Distributions of $S$ for cases (red) and controls (black) are given
to the right. A significant shift toward higher $S$ values is seen in the
cases.
| Procedure for Pathways of Distinction Analysis
---|---
1. | For a each pathway $P$, select all associated genes from pathway database such as PID [32];
2. | For each gene on the pathway, select associated SNPs (e.g., using dbSNP) and choose the one
| with the strongest association with case status, determined using Fisher’s
exact test;
3. | For each sample $Y$ in the GWAS, select the controls $F$ and cases $G$ which do not include it,
| compute MAFs $f_{i}$ and $g_{i}$ for the SNPs $i$ selected in step 2, and
compute $S_{Y,P}$ for each sample $Y$;
4. | Compare the distribution of $S_{Y,P}$ obtained in step 2 for cases to that of controls by computing
| the Wilcoxon statistic $W_{P}$ based on the $S_{Y,P}$ for that pathway;
5. | Repeat steps 2–5 using permuted case/control labels, and normalize $W_{P}$ by the distribution
| of $W^{*}_{P}$ obtained with permuted labels, yielding the distinction score
$DS_{P}$;
6. | Compare the distinction score $DS_{P}$ obtained in step 5 to that obtained for random sets of $l_{P}$ genes,
| where $l_{P}$ is the number of genes in the pathway of interest.
Table 1: Procedure for Pathways of Distinction Analysis
Pathway | Source | Length | $DS_{P}$ | $p(DS_{P})$ | O.R. | $q$(O.R.)
---|---|---|---|---|---|---
Purine metabolism | Kegg | 136 | 1.86 | 6.36e-03 | 1.59 | 4.15e-21
Calcium signaling pathway | Kegg | 100 | 1.38 | 1.82e-03 | 1.55 | 6.99e-20
Melanogenesis | Kegg | 84 | 2.36 | 4.55e-03 | 1.53 | 1.47e-18
Gap junction | Kegg | 80 | 1.54 | 5.45e-03 | 1.49 | 1.49e-16
ErbB signaling pathway | Kegg | 81 | 1.36 | 1.45e-02 | 1.46 | 4.68e-15
Long-term potentiation | Kegg | 60 | 1.71 | 9.09e-04 | 1.45 | 4.34e-15
GnRH signaling pathway | Kegg | 79 | 1.36 | 1.18e-02 | 1.44 | 1.32e-14
TCR signaling in naive CD4+ T cells | NCI-Nature | 60 | 2.11 | 5.45e-03 | 1.42 | 7.80e-13
Prostate cancer | Kegg | 75 | 1.45 | 4.09e-02 | 1.38 | 4.37e-11
PKC-catalyzed phosphorylation …myosin phosphatase | BioCarta | 20 | 1.97 | $<$1e-04 | 1.30 | 5.82e-09
CCR3 signaling in eosinophils | BioCarta | 21 | 1.59 | 1.09e-02 | 1.29 | 8.86e-08
Biosynthesis of unsaturated fatty acids | Kegg | 18 | 1.69 | 2.45e-02 | 1.26 | 1.38e-06
Attenuation of GPCR signaling | BioCarta | 11 | 1.75 | 1.09e-02 | 1.25 | 2.41e-06
Stathmin and breast cancer resistance to antimicrotubule agents | BioCarta | 18 | 1.84 | 4.82e-02 | 1.24 | 4.96e-06
Visual signal transduction: Cones | NCI-Nature | 20 | 1.56 | 4.73e-02 | 1.24 | 2.24e-06
Dentatorubropallidoluysian atrophy (DRPLA) | Kegg | 11 | 1.84 | 2.73e-03 | 1.24 | 2.24e-06
Intrinsic prothrombin activation pathway | BioCarta | 22 | 1.35 | 3.18e-02 | 1.23 | 4.61e-06
Eicosanoid metabolism | BioCarta | 19 | 1.69 | 1.91e-02 | 1.23 | 3.44e-06
Effects of botulinum toxin | NCI-Nature | 7 | 1.44 | 2.27e-02 | 1.20 | 3.50e-05
Activation of PKC through G-protein coupled receptors | BioCarta | 10 | 1.50 | 9.09e-03 | 1.20 | 8.42e-06
Streptomycin biosynthesis | Kegg | 9 | 1.36 | 3.55e-02 | 1.17 | 1.89e-04
PECAM1 interactions | Reactome | 6 | 2.70 | 5.45e-03 | 1.17 | 7.28e-05
HDL-mediated lipid transport | Reactome | 8 | 1.47 | 2.00e-02 | 1.14 | 1.59e-03
Granzyme A mediated apoptosis pathway | BioCarta | 8 | 1.97 | 1.73e-02 | 1.12 | 6.60e-04
Table 2: PID pathways with significant $DS_{P}$ in the CGEMS breast cancer
GWAS. (Pathways with over 60% SNPs covered by another pathway have been
removed; for the complete list, see Supplemental Table S-1). Pathway-length
based resampled $p$-values, denoted $p(DS_{P})$, are given for significant
pathways, along with the odds ratios and associated FDRs for a logistic
regression model.
Pathway | Source | Length | $DS_{P}$ | $p(DS_{P})$ | O.R. | $q$(O.R.)
---|---|---|---|---|---|---
Cell adhesion molecules (CAMs) | Kegg | 86 | 1.57 | 9.09e-03 | 1.66 | 3.56e-13
ErbB signaling pathway | Kegg | 76 | 1.45 | 3.45e-02 | 1.61 | 2.59e-10
Signaling events mediated by Stem cell factor receptor (c-Kit) | NCI-Nature | 40 | 2.35 | 5.45e-03 | 1.58 | 7.31e-10
Neurotrophic factor-mediated Trk receptor signaling | NCI-Nature | 50 | 1.60 | 2.36e-02 | 1.55 | 2.49e-08
Lissencephaly gene (LIS1) in neuronal migration and development | NCI-Nature | 21 | 2.02 | 7.27e-03 | 1.52 | 1.44e-07
Angiopoietin receptor Tie2-mediated signaling | NCI-Nature | 40 | 2.36 | 1.36e-02 | 1.51 | 5.77e-08
Reelin signaling pathway | NCI-Nature | 28 | 1.62 | 5.45e-03 | 1.46 | 7.35e-08
Syndecan-4-mediated signaling events | NCI-Nature | 27 | 1.74 | 1.64e-02 | 1.46 | 1.19e-06
Galactose metabolism | Kegg | 19 | 1.65 | 2.27e-02 | 1.44 | 5.01e-06
Vibrio cholerae infection | Kegg | 35 | 1.84 | 2.64e-02 | 1.43 | 6.67e-07
Paxillin-independent events mediated by a4b1 and a4b7 | NCI-Nature | 19 | 2.14 | 1.00e-02 | 1.40 | 6.67e-07
Antigen processing and presentation | Kegg | 34 | 3.26 | 1.36e-02 | 1.40 | 3.71e-08
Corticosteroids and Cardioprotection | BioCarta | 21 | 1.98 | 3.55e-02 | 1.39 | 1.24e-05
Lissencephaly gene (Lis1) in neuronal migration and development | BioCarta | 15 | 1.60 | 1.36e-02 | 1.37 | 2.52e-05
IL12 signaling mediated by STAT4 | NCI-Nature | 25 | 1.93 | 4.55e-02 | 1.37 | 1.58e-05
Biosynthesis of unsaturated fatty acids | Kegg | 13 | 1.76 | 1.64e-02 | 1.36 | 6.44e-05
Growth hormone signaling pathway | BioCarta | 18 | 1.75 | 3.18e-02 | 1.36 | 7.46e-05
Canonical Wnt signaling pathway | NCI-Nature | 28 | 1.92 | 4.73e-02 | 1.35 | 9.36e-06
NO2-dependent IL-12 pathway in NK cells | BioCarta | 8 | 1.82 | 2.73e-03 | 1.32 | 5.83e-05
Signaling events mediated by HDAC Class III | NCI-Nature | 19 | 2.12 | 3.91e-02 | 1.32 | 4.19e-05
Removal of aminoterminal propeptides from $\gamma$-carboxylated proteins | Reactome | 7 | 3.12 | 5.45e-03 | 1.29 | 8.46e-05
Aminophosphonate metabolism | Kegg | 13 | 1.91 | 3.36e-02 | 1.26 | 8.17e-04
Antigen processing and presentation | BioCarta | 6 | 2.61 | 1.82e-03 | 1.22 | 3.36e-05
Classical complement pathway | BioCarta | 12 | 2.27 | 1.55e-02 | 1.19 | 1.67e-04
Chylomicron-mediated lipid transport | Reactome | 7 | 1.94 | 3.27e-02 | 1.16 | 1.49e-02
Table 3: PID pathways with significant $DS_{P}$ in the liver cancer GWAS.
(Pathways with over 60% SNPs covered by another pathway have been removed; for
the complete list, see Supplemental Table S-2). Pathway-length based resampled
$p$-values, denoted $p(DS_{P})$, are given for significant pathways, along
with the odds ratios and associated FDRs for a logistic regression model.
Pathway | Length | $p(DS_{P})$ | O.R. | $q$(O.R.)
---|---|---|---|---
Top-2 | 318 | $<$1e-04 | 2.02 | 1.63e-46
Top-3 | 397 | 1.00e-04 | 2.19 | 2.07e-54
Top-4 | 474 | $<$1e-04 | 2.33 | 3.65e-62
Top-5 | 522 | $<$1e-04 | 2.45 | 6.83e-66
Top-6 | 544 | $<$1e-04 | 2.44 | 8.51e-66
Top-7 | 558 | 2.00e-04 | 2.47 | 1.22e-67
Top-8 | 626 | $<$1e-04 | 2.59 | 1.01e-73
Top-9 | 658 | $<$1e-04 | 2.64 | 9.84e-75
Top-10 | 700 | $<$1e-04 | 2.77 | 9.72e-79
Top-11 | 710 | $<$1e-04 | 2.80 | 1.42e-79
Top-12 | 723 | $<$1e-04 | 2.82 | 2.06e-80
Top-13 | 739 | $<$1e-04 | 2.89 | 3.31e-82
Top-14 | 744 | $<$1e-04 | 2.93 | 2.86e-83
Top-15 | 770 | $<$1e-04 | 2.96 | 6.41e-85
Top-16 | 774 | $<$1e-04 | 2.97 | 5.10e-85
Top-17 | 791 | $<$1e-04 | 2.95 | 2.43e-85
Top-18 | 800 | $<$1e-04 | 3.06 | 1.15e-87
Top-19 | 814 | $<$1e-04 | 3.14 | 1.19e-89
Top-20 | 832 | $<$1e-04 | 3.26 | 4.51e-92
Top-21 | 837 | $<$1e-04 | 3.28 | 2.92e-92
Top-22 | 839 | $<$1e-04 | 3.29 | 2.41e-92
Top-23 | 845 | $<$1e-04 | 3.34 | 1.45e-93
Top-24 | 854 | $<$1e-04 | 3.38 | 4.62e-95
Table 4: PoDA results for sucessive unions of significant pathways in the
CGEMS breast cancer data. Pathway-length based resampled $p$ values,
denoted$p(DS_{P})$, are given along with the odds ratios and associated FDRs
for a logistic regression model.
Pathway | Length | $p(DS_{P})$ | O.R. | $q$(O.R.)
---|---|---|---|---
Top-2 | 321 | 5.38e-02 | 2.37 | 1.20e-27
Top-3 | 402 | 2.80e-03 | 2.63 | 1.40e-34
Top-4 | 474 | 1.10e-03 | 2.86 | 6.50e-38
Top-5 | 539 | 9.00e-04 | 3.22 | 4.03e-42
Top-6 | 560 | 1.00e-04 | 3.39 | 1.19e-43
Top-7 | 580 | $<$1e-04 | 3.50 | 1.39e-44
Top-8 | 589 | 6.00e-04 | 3.50 | 1.35e-44
Top-9 | 603 | 4.00e-04 | 3.52 | 1.23e-44
Top-10 | 624 | $<$1e-04 | 3.60 | 1.33e-45
Top-11 | 640 | $<$1e-04 | 3.73 | 3.69e-47
Top-12 | 646 | $<$1e-04 | 3.78 | 1.68e-47
Top-13 | 667 | $<$1e-04 | 3.81 | 9.29e-48
Top-14 | 709 | 3.00e-04 | 3.88 | 1.90e-48
Top-15 | 751 | $<$1e-04 | 4.09 | 2.11e-49
Top-16 | 761 | $<$1e-04 | 4.09 | 1.76e-49
Top-17 | 797 | $<$1e-04 | 4.45 | 1.29e-50
Top-18 | 805 | $<$1e-04 | 4.46 | 5.24e-51
Top-19 | 823 | $<$1e-04 | 4.56 | 2.20e-51
Top-20 | 838 | $<$1e-04 | 4.56 | 1.73e-51
Table 5: PoDA results for sucessive unions of significant pathways in the
liver cancer data. Pathway-length based resampled $p$ values, denoted
$p(DS_{P})$, are given along with the odds ratios and associated FDRs for a
logistic regression model.
Pathway | Source | Length | $DS_{P}$ | $p(DS_{P})$ | O.R. | $q$(O.R.)
---|---|---|---|---|---|---
Purine metabolism | Kegg | 136 | 1.86 | 6.36e-03 | 1.59 | 4.15e-21
Calcium signaling pathway | Kegg | 100 | 1.38 | 1.82e-03 | 1.55 | 6.99e-20
Melanogenesis | Kegg | 84 | 2.36 | 4.55e-03 | 1.53 | 1.47e-18
Gap junction | Kegg | 80 | 1.54 | 5.45e-03 | 1.49 | 1.49e-16
ErbB signaling pathway | Kegg | 81 | 1.36 | 1.45e-02 | 1.46 | 4.68e-15
Long-term potentiation | Kegg | 60 | 1.71 | 9.09e-04 | 1.45 | 4.34e-15
GnRH signaling pathway | Kegg | 79 | 1.36 | 1.18e-02 | 1.44 | 1.32e-14
TCR signaling in naive CD4+ T cells | NCI-Nature | 60 | 2.11 | 5.45e-03 | 1.42 | 7.80e-13
TCR signaling in naive CD8+ T cells | NCI-Nature | 48 | 2.03 | 7.27e-03 | 1.38 | 1.11e-11
Prostate cancer | Kegg | 75 | 1.45 | 4.09e-02 | 1.38 | 4.37e-11
PKC-catalyzed phosphorylation …myosin phosphatase | BioCarta | 20 | 1.97 | $<$1e-04 | 1.30 | 5.82e-09
CCR3 signaling in eosinophils | BioCarta | 21 | 1.59 | 1.09e-02 | 1.29 | 8.86e-08
Biosynthesis of unsaturated fatty acids | Kegg | 18 | 1.69 | 2.45e-02 | 1.26 | 1.38e-06
Attenuation of GPCR signaling | BioCarta | 11 | 1.75 | 1.09e-02 | 1.25 | 2.41e-06
Stathmin and breast cancer resistance to antimicrotubule agents | BioCarta | 18 | 1.84 | 4.82e-02 | 1.24 | 4.96e-06
Visual signal transduction: Cones | NCI-Nature | 20 | 1.56 | 4.73e-02 | 1.24 | 2.24e-06
Dentatorubropallidoluysian atrophy (DRPLA) | Kegg | 11 | 1.84 | 2.73e-03 | 1.24 | 2.24e-06
Intrinsic prothrombin activation pathway | BioCarta | 22 | 1.35 | 3.18e-02 | 1.23 | 4.61e-06
Eicosanoid metabolism | BioCarta | 19 | 1.69 | 1.91e-02 | 1.23 | 3.44e-06
Effects of botulinum toxin | NCI-Nature | 7 | 1.44 | 2.27e-02 | 1.20 | 3.50e-05
Activation of PKC through G-protein coupled receptors | BioCarta | 10 | 1.50 | 9.09e-03 | 1.20 | 8.42e-06
Ca-calmodulin-dependent protein kinase activation | BioCarta | 8 | 1.70 | 1.00e-02 | 1.19 | 5.67e-05
Streptomycin biosynthesis | Kegg | 9 | 1.36 | 3.55e-02 | 1.17 | 1.89e-04
PECAM1 interactions | Reactome | 6 | 2.70 | 5.45e-03 | 1.17 | 7.28e-05
HDL-mediated lipid transport | Reactome | 8 | 1.47 | 2.00e-02 | 1.14 | 1.59e-03
Granzyme A mediated apoptosis pathway | BioCarta | 8 | 1.97 | 1.73e-02 | 1.12 | 6.60e-04
Table S-1: Full list PID pathways with significant $DS_{P}$ in the breast
cancer GWAS, including highly “overlapping” pathways. Pathway-length based
resampled $p$-values, denoted $p(DS_{P})$, are given for significant pathways,
along with the odds ratios and associated FDRs for a logistic regression
model.
Pathway | Source | Length | $DS_{P}$ | $p(DS_{P})$ | O.R. | $q$(O.R.)
---|---|---|---|---|---|---
Cell adhesion molecules (CAMs) | Kegg | 86 | 1.57 | 9.09e-03 | 1.66 | 3.56e-13
ErbB signaling pathway | Kegg | 76 | 1.45 | 3.45e-02 | 1.61 | 2.59e-10
Signaling events mediated by Stem cell factor receptor (c-Kit) | NCI-Nature | 40 | 2.35 | 5.45e-03 | 1.58 | 7.31e-10
Neurotrophic factor-mediated Trk receptor signaling | NCI-Nature | 50 | 1.60 | 2.36e-02 | 1.55 | 2.49e-08
Lissencephaly gene (LIS1) in neuronal migration and development | NCI-Nature | 21 | 2.02 | 7.27e-03 | 1.52 | 1.44e-07
Angiopoietin receptor Tie2-mediated signaling | NCI-Nature | 40 | 2.36 | 1.36e-02 | 1.51 | 5.77e-08
Reelin signaling pathway | NCI-Nature | 28 | 1.62 | 5.45e-03 | 1.46 | 7.35e-08
Syndecan-4-mediated signaling events | NCI-Nature | 27 | 1.74 | 1.64e-02 | 1.46 | 1.19e-06
Galactose metabolism | Kegg | 19 | 1.65 | 2.27e-02 | 1.44 | 5.01e-06
TPO signaling pathway | BioCarta | 17 | 2.61 | 6.36e-03 | 1.44 | 3.80e-06
Vibrio cholerae infection | Kegg | 35 | 1.84 | 2.64e-02 | 1.43 | 6.67e-07
Paxillin-independent events mediated by a4b1 and a4b7 | NCI-Nature | 19 | 2.14 | 1.00e-02 | 1.40 | 6.67e-07
Antigen processing and presentation | Kegg | 34 | 3.26 | 1.36e-02 | 1.40 | 3.71e-08
Corticosteroids and cardioprotection | BioCarta | 21 | 1.98 | 3.55e-02 | 1.39 | 1.24e-05
Lissencephaly gene (Lis1) in neuronal migration and development | BioCarta | 15 | 1.60 | 1.36e-02 | 1.37 | 2.52e-05
IL12 signaling mediated by STAT4 | NCI-Nature | 25 | 1.93 | 4.55e-02 | 1.37 | 1.58e-05
Biosynthesis of unsaturated fatty acids | Kegg | 13 | 1.76 | 1.64e-02 | 1.36 | 6.44e-05
Growth hormone signaling pathway | BioCarta | 18 | 1.75 | 3.18e-02 | 1.36 | 7.46e-05
Canonical Wnt signaling pathway | NCI-Nature | 28 | 1.92 | 4.73e-02 | 1.35 | 9.36e-06
NO2-dependent IL-12 pathway in nk cells | BioCarta | 8 | 1.82 | 2.73e-03 | 1.32 | 5.83e-05
Signaling events mediated by HDAC Class III | NCI-Nature | 19 | 2.12 | 3.91e-02 | 1.32 | 4.19e-05
Removal of aminoterminal propeptides from gamma-carboxylated proteins | Reactome | 7 | 3.12 | 5.45e-03 | 1.29 | 8.46e-05
Gamma-carboxylation, transport, and amino-terminal cleavage of proteins | Reactome | 6 | 3.25 | 1.82e-03 | 1.28 | 6.64e-05
Transport of $\gamma$-carboxylated protein precursors … | Reactome | 6 | 3.25 | 1.82e-03 | 1.28 | 6.64e-05
Paxillin-dependent events mediated by a4b1 | NCI-Nature | 17 | 1.84 | 2.00e-02 | 1.28 | 3.41e-05
Gamma-carboxylation of protein precursors | Reactome | 7 | 2.86 | 3.64e-03 | 1.28 | 1.38e-04
Aminophosphonate metabolism | Kegg | 13 | 1.91 | 3.36e-02 | 1.26 | 8.17e-04
Antigen processing and presentation | BioCarta | 6 | 2.61 | 1.82e-03 | 1.22 | 3.36e-05
Lectin induced complement pathway | BioCarta | 11 | 1.91 | 2.18e-02 | 1.20 | 1.55e-04
Classical complement pathway | BioCarta | 12 | 2.27 | 1.55e-02 | 1.19 | 1.67e-04
Chylomicron-mediated lipid transport | Reactome | 7 | 1.94 | 3.27e-02 | 1.16 | 1.49e-02
Table S-2: Full list PID pathways with significant $DS_{P}$ in the liver
cancer GWAS, including highly “overlapping” pathways. Pathway-length based
resampled $p$-values, denoted $p(DS_{P})$, are given for significant pathways,
along with the odds ratios and associated FDRs for a logistic regression
model.
|
arxiv-papers
| 2010-12-21T16:50:59 |
2024-09-04T02:49:15.897586
|
{
"license": "Public Domain",
"authors": "Rosemary Braun and Kenneth Buetow",
"submitter": "Rosemary Braun",
"url": "https://arxiv.org/abs/1012.4726"
}
|
1012.5022
|
# Numeric and symbolic evaluation of the pfaffian of general skew-symmetric
matrices
C. González-Ballestero L.M. Robledo G. F. Bertsch Departamento de Física
Teórica, Universidad Autónoma de Madrid, E-28049 Madrid, Spain Department of
Physics and Institute for Nuclear Theory, University of Washington, Seattle,
WA 98195–1560 USA
###### Abstract
Evaluation of pfaffians arises in a number of physics applications, and for
some of them a direct method is preferable to using the determinantal formula.
We discuss two methods for the numerical evaluation of pfaffians. The first is
tridiagonalization based on Householder transformations. The main advantage of
this method is its numerical stability that makes unnecessary the
implementation of a pivoting strategy. The second method considered is based
on Aitken’s block diagonalization formula. It yields to a kind of LU (similar
to Cholesky’s factorization) decomposition (under congruence) of arbitrary
skew-symmetric matrices that is well suited both for the numeric and symbolic
evaluations of the pfaffian. Fortran subroutines (FORTRAN 77 and 90)
implementing both methods are given. We also provide simple implementations in
Python and Mathematica for purpose of testing, or for exploratory studies of
methods that make use of pfaffians.
###### keywords:
Skew symmetric matrices , Pfaffian
††journal: Computer Physics Communications
PROGRAM SUMMARY
Manuscript Title: Numeric and symbolic evaluation of the pfaffian of general
skew-symmetric matrices
Authors: C. Gonzalez-Ballestero, L.M.Robledo and G.F. Bertsch
Program Title: Pfaffian
Journal Reference:
Catalogue identifier:
Licensing provisions:
Programming language: Fortran 77 and 90
Computer:
Operating system:
RAM: bytes
Number of processors used:
Supplementary material:
Keywords: Skew symmetric matrices, Pfaffian
Classification: 4.8 Linear Equations and Matrices
External routines/libraries: BLAS
Subprograms used:
Catalogue identifier of previous version:*
Journal reference of previous version:*
Does the new version supersede the previous version?:*
Nature of problem: Evaluation of the Pfaffian of a skew symmetric matrix.
Evaluation of pfaffians arises in a number of physics applications involving
fermionic mean field wave functions and their overlaps.
Solution method: Householder tridiagonalization. Aitken’s block
diagonalization formula.
Reasons for the new version:*
Summary of revisions:*
Restrictions:
Unusual features:
Additional comments: Python and Mathematica implementations are provided in
the main body of the paper
Running time: Depends on the size of the matrices. For matrices with 100 rows
and columns a few miliseconds are required.
## 1 Introduction
In a number of fields in physics, the formal equations derived from the theory
make use of the pfaffian of some skew-symmetric matrix appearing in the
theory. For example, the pfaffian arises in the treatment of electronic
structure with quantum Monte Carlo methods [1], the description of two-
dimensional Ising spin glasses [2], and the evaluation of entropy and its
relation to entanglement [3]. Pfaffians occur naturally in field theory and
nuclear physics in formalisms based on fermionic coherent states [4, 5, 6, 7].
A recent application is to the overlap of two Hartree Fock Bogoliubov (HFB)
product wave functions [8], needed for nuclear structure theory. While there
is a simple formula for the pfaffian of a skew-symmetric matrix $M$ in terms
of the determinant,
$\textrm{pf}(A)=\sqrt{{\rm det}(A)}$ (1)
the so-called “sign problem of the overlap” [9] associated with the square
root motivates the use of numerical algorithms that evaluate it directly. The
most straightforward method, the rule of “expanding in minors” [10], has bad
scaling with the size of the matrix and is prohibitive for large matrices. In
this paper we discuss two alternative methods that have the same scaling
property as the normal $N^{3}$ algorithms for the determinant. The methods are
implemented in the FORTRAN 77 and 90 subroutines provided in the accompanying
program library. We also comment on the practical implementation of the two
methods in Mathematica and in the Python programming language.
## 2 Evaluation of the Pfaffian
The Pfaffian $\textrm{pf}(A)$ is reduced to a simple form that is easily
evaluated by making repeated use of transformation formula given in A,
$\textrm{pf}(B^{t}AB)={\rm det(B)}\textrm{pf}(A).$ (2)
In order to perform the numerical evaluation of the Pfaffian of a complex
skew-symmetric matrix $A$ we reduce the skew-symmetric matrix to a tridiagonal
form $A_{TR}$ by using unitary matrices $U$. Once it is in this form, the
evaluation of the pfaffian is straightforward (see below).
### 2.1 Reduction to tridiagonal form by mean of Householder transformations
In this method, we will use the well-known Householder transformations [11] to
reduce $A$ to tridiagonal form. We present it in some detail because the
generalization to the complex number field is not entirely trivial.
Complex Householder transformations have the form
$P=\mathbb{I}-2\frac{u\otimes u^{+}}{\left|u\right|^{2}}$ (3)
where $u$ is an arbitrary complex complex row vector
$u=(u_{1},u_{2},\ldots,u_{N})$ and $\left(u\otimes
u^{+}\right)_{ij}=u_{i}u_{j}^{*}$. The vector $u$ must be chosen to zero all
the elements of a vector $x$ except a given one. If we take $u=x\mp
e^{i\arg(x_{j})}|x|e_{j}$, with $(e_{j})_{k}=\delta_{jk}$, it can be easily
proved that
$P_{u}x=\pm e^{i\arg(x_{j})}|x|e_{j}$
as required. The freedom on the sign in the expression defining the vector $u$
can be used to make sure that the vector $u$ is non zero. The rest of the
Householder tridiagonalization procedure follows exactly as in the real case.
Consider a skew-symmetric matrix of dimension N (even)
$A=\left(\begin{array}[]{c|ccc}0&a_{12}&a_{13}&\ldots\\\ \hline\cr-a_{12}\\\
-a_{13}&&{}^{(N-1)}A\\\ \vdots\end{array}\right)$ (4)
The Householder transformation matrix is
$P_{1}=\left(\begin{array}[]{c|ccc}1&0&0&\ldots\\\ \hline\cr 0\\\
0&&{}^{(N-1)}P_{1}\\\ \vdots\end{array}\right)$
where ${}^{(N-1)}P_{1}$ is built by using Eq. (3) and taking the vector $x$
(of dimension N-1) as $(a_{12},a_{13},\ldots)^{T}$. The resulting transformed
matrix is given by
$P_{1}AP_{1}^{T}=\left(\begin{array}[]{c|ccc}0&k_{1}&0&\ldots\\\ \hline\cr-
k_{1}\\\ 0&&{}^{(N-1)}\tilde{A}\\\ \vdots\end{array}\right)$
where $k_{1}=\pm e^{i\arg(a_{12})}|x|$ and the matrix ${}^{(N-1)}\tilde{A}$ is
skew-symmetric and given by
${}^{(N-1)}\tilde{A}={}^{(N-1)}P_{1}{}^{(N-1)}A{}^{(N-1)}P_{1}^{T}$.
Performing this procedure a total of N-2 times we end up with a tridiagonal
and skew-symmetric matrix
$P_{N-2}\ldots P_{2}P_{1}AP_{1}^{T}P_{2}^{T}\ldots
P_{N-2}^{T}=\left(\begin{array}[]{cccccc}0&k_{1}&0&0&\ldots&0\\\
-k_{1}&0&k_{2}&0&\ldots&0\\\ 0&-k_{2}&0&\ddots&\ldots&\vdots\\\
&&\ddots&0&\ddots\\\ 0&&&\ddots&0&k_{N-1}\\\
\vdots&&\vdots&&-k_{N-1}&0\end{array}\right)$ (5)
Using now a known property the Pfaffian (see A) we can deduce from the above
identity that
$\det(P_{1})\ldots\det(P_{N-2})\textrm{pf}(A)=\textrm{pf}(A_{TR})$ where
$A_{TR}$ is the triagonal and skew-symmetric matrix of the right hand side of
Eq. (5). Taking into account that the determinant of any Householder matrix is
-1 and that $N$ is even, we can express the Pfaffian of $A$ in terms of the
pfaffian of the tridiagonal $A_{TR}$
$\textrm{pf}(A)=\textrm{pf}(A_{TR})$
As will be shown below the Pfaffian of a tridiagonal skew-symmetric matrix is
simply given by $k_{1}k_{3}\ldots k_{N-1}=\prod_{i=1}^{N/2}k_{2i-1}$ (this
result can also be obtained using the “minor expansion” formula [10] ) and
finally we obtain
$\textrm{pf}(A)=\prod_{i=1}^{N/2}k_{2i-1}.$ (6)
In terms of numerical stability, the Householder transformation is very robust
and there is no need to consider any “pivoting” strategy common to other
methods. However, the presence of the square root of $x$ and the argument
$\arg(x_{j})$ of complex quantities prevents an easy implementation of the
Householder tridiagonalization procedure for symbolic computation. For this
purpose the second method described in the next section is far easier to
implement.
### 2.2 Aitken’s block diagonalization formula
There is an alternative method for the calculation of the pfaffian, which is
also well suited for a symbolic implementation and that relies on an
expression for the pfaffian of a bipartite skew-symmetric matrix. Let us start
with a general skew-symmetric matrix $A$ (dimension N, even) given by
$A=\left(\begin{array}[]{cc}R&Q\\\ -Q^{T}&S\end{array}\right)$ (7)
where $R$ and $S$ are square skew-symmetric matrices and $Q$ is a general
rectangular matrix (to account for the case where $R$ and $S$ have different
dimensions). Using Aitken’s block diagonalization formula (see [12] for an
early use of the formula and [13] for a recent and thorough reference) for a
bipartite matrix we obtain
$\displaystyle\left(\begin{array}[]{cc}\mathbb{I}&0\\\
Q^{T}R^{-1}&\mathbb{I}\end{array}\right)\left(\begin{array}[]{cc}R&Q\\\
-Q^{T}&S\end{array}\right)\left(\begin{array}[]{cc}\mathbb{I}&-R^{-1}Q\\\
0&\mathbb{I}\end{array}\right)$ $\displaystyle=$ (14)
$\displaystyle\left(\begin{array}[]{cc}R&0\\\
0&S+Q^{T}R^{-1}Q\end{array}\right)$ (17)
where the matrix $S+Q^{T}R^{-1}Q$ is referred to in the literature as the
Schur complement of the matrix $A$ (see, for instance, [13]). For the special
case of a skew-symmetric matrix $A$, the matrices $R$ and $S$ are also skew-
symmetric and the transformation of the matrix $A$ is a congruence (i.e. the
matrix acting on the left hand side of $A$ is the transpose of the one acting
on the right hand side). Denoting
$P_{1}=\left(\begin{array}[]{cc}\mathbb{I}&0\\\
Q^{T}R^{-1}&\mathbb{I}\end{array}\right)$ (18)
Eq. (17) becomes
$P_{1}AP_{1}^{T}=\left(\begin{array}[]{cc}R&0\\\
0&S+Q^{T}R^{-1}Q\end{array}\right)$
An equivalent expression involving $S^{-1}$ instead of $R^{-1}$ is easily
obtained
$P_{2}AP_{2}^{T}=\left(\begin{array}[]{cc}R+QS^{-1}Q^{T}&0\\\
0&S\end{array}\right)$
with
$P_{2}=\left(\begin{array}[]{cc}\mathbb{I}&-QS^{-1}\\\
0&\mathbb{I}\end{array}\right)$ (19)
By using the property $\textrm{pf}(P^{T}AP)=\textrm{det}(P)\textrm{pf}(A)$
(see A) and taking into account that $\det P_{1}=\det P_{2}$=1, we come to
$\displaystyle\textrm{pf}(A)$ $\displaystyle=$
$\displaystyle\textrm{pf}(R)\textrm{pf}(S+Q^{T}R^{-1}Q)$ (20) $\displaystyle=$
$\displaystyle\textrm{pf}(R+QS^{-1}Q^{T})\textrm{pf}(S)$ (21)
Another nice property of the matrices $P_{1}$ and $P_{2}$ is that their
inverses can be obtained very easily
$P_{1}^{-1}=\left(\begin{array}[]{cc}\mathbb{I}&0\\\
-Q^{T}R^{-1}&\mathbb{I}\end{array}\right)$ (22)
and
$P_{2}^{-1}=\left(\begin{array}[]{cc}\mathbb{I}&QS^{-1}\\\
0&\mathbb{I}\end{array}\right)$ (23)
These expressions of the inverses explicitly show that both $P_{1}$ and
$P_{2}$ are not orthogonal matrices.
Let us now apply the above result to an arbitrary skew-symmetric matrix of
dimension $N=2M$ which is written in block form as
$A=\left(\begin{array}[]{ccc}A^{(1)}&A_{N-1}&A_{N}\\\
-A_{N-1}^{T}&0&a_{N-1,N}\\\ -A_{N}^{T}&-a_{N-1,N}&0\end{array}\right)$ (24)
where $A^{(1)}$ is a skew-symmetric square matrix of dimension $N-2=2(M-1)$
and $A_{N-1}$ and $A_{N}$ are column vectors
$A_{N-1}=\\{A_{i,N-1},\,i=1,N-2\\}$ and $A_{N}=\\{A_{i,N},\,i=1,N-2\\}$ both
of dimension $(N-2)\times 1$. In the language of Eq (7) the matrix $R$ is the
matrix $A^{(1)}$, the matrix $Q$ is a rectangular matrix of dimension
$2\times(N-2)$ made of the two column vectors, $A_{N-1}$ and $A_{N}$ and
finally the matrix $S$ is the $2\times 2$ skew-symmetric matrix with matrix
element $S_{12}=a_{N-1,N}$. Using the ideas of Aitken’s block diagonalization
formula, it is easy to shows that the matrix $\tilde{A}=D_{1}^{T}AD_{1}$ is in
block diagonal form
$\tilde{A}=\left(\begin{array}[]{ccc}\mbox{$\tilde{A}$}^{(1)}&0&0\\\
0&0&a_{N-1,N}\\\ 0&-a_{N-1,N}&0\end{array}\right)$ (25)
with a matrix $D_{1}$ of the form
$D_{1}=\left(\begin{array}[]{ccc}\mathbb{I}_{N-2}&0&0\\\ X&1&0\\\
Y&0&1\end{array}\right)$ (26)
where $\mathbb{I}_{N-2}$ stands for the identity matrix of dimension $N-2$ and
both $X$ and $Y$ are row vectors of dimension $1\times(N-2)$ and given by
$X=-a_{N-1,N}^{-1}A_{N}^{T}$ and $Y=a_{N-1,N}^{-1}A_{N-1}^{T}$. In the above
equation 25, the skew-symmetric matrix $\tilde{A}^{(1)}$ is given by
$\tilde{A}^{(1)}=A^{(1)}+A_{N}(a_{N-1,N})^{-1}A_{N-1}^{T}-A_{N-1}a_{N-1,N}^{-1}A_{N}^{T}$
(27)
Taking into account that $\det D_{1}=1$ then
$\textrm{pf}(A)=\textrm{pf}(\tilde{A})=a_{N-1.N}\textrm{pf}(\tilde{A}^{(1)})$.
The algorithm can be applied recursively to $\tilde{A}^{(1)}$ to obtain
$\textrm{pf}(A)=a_{N-1,N}\tilde{a}_{N-3,N-2}^{(1)}\textrm{pf}(\tilde{A}^{(2)})$
so as to reduce, after $M-1$ iterations, the computation of the pfaffian to
the product of the corresponding elements.
This procedure can be easily implemented for a skew-symmetric tridiagonal
matrix, as the transformed matrices in Eq (27) coincide with the original
ones; for instance, $\tilde{A}^{(1)}=A^{(1)}$. As a consequence, the pfaffian
of a tridiagonal matrix is given by
$\textrm{pf }\left(\begin{array}[]{cccccc}0&d_{1}&0&0&\ldots&0\\\
-d_{1}&0&d_{2}&0&\ldots&0\\\ 0&-d_{2}&0&\ddots&\ldots&\vdots\\\
0&0&\ddots&0&\ddots&0\\\ \vdots&\vdots&\vdots&\ddots&0&d_{2N-1}\\\
0&0&\cdots&0&-d_{2N-1}&0\end{array}\right)=d_{1}d_{3}\ldots
d_{2N-1}=\prod_{i=1}^{N}d_{2i-1}$
#### 2.2.1 Pivoting
As a consequence of the division by matrix elements like $a_{N-1,N}$ in the
first iteration, the numerical stability of the algorithm requires the use of
pivoting strategy in the implementation of the method. Full pivoting amounts
to search the whole matrix for the matrix element with the largest modulus and
exchange it with the required matrix element. For instance, in the first
iteration of the procedure, the matrix element $a_{p,q}$ ($p<q)$ with the
largest modulus is searched for and exchanged with the matrix element
$a_{N-1,N}$. In this way we avoid dangerous divisions by small (or even zero)
matrix elements. We have to take into account that in the present case, the
exchange of both columns and rows is required to preserve the skew-symmetric
nature of the matrices involved. To carry out the exchange of rows and columns
we will use the exchange matrix $P(ij)$ that, when applied to the right of an
arbitrary matrix, exchanges columns $i$ and $j$. The exchange matrix is given
by the matrix elements
$\displaystyle P(ij)_{kl}$ $\displaystyle=$
$\displaystyle\delta_{kl}-\delta_{i,l}\delta_{i,k}-\delta_{j,l}\delta_{j,k}+\delta_{i,l}\delta_{j,k}+\delta_{j,l}\delta_{i,k}.$
(28)
To exchange the corresponding rows we have to apply $P(ij)^{T}$ to the left of
the matrix (notice that $P(ij)$ is symmetric). With the help of these matrices
we can write the matrix after pivoting $a_{p,q}$ with $a_{N-1,N}$ (and
$a_{q,p}$ with $a_{N,N-1})$ as
$A_{P}=P^{T}(N-1,p)P^{T}(N,q)\,A\,P(N-1,p)P(N,q)$
As a consequence of such exchange and taking into account that $\det P(ij)=-1$
we can conclude that the pfaffian of $A$ does not change by the pivoting
procedure. Finally we obtain
$A=P(N,q)P(N-1,p)\,A_{P}\,P^{T}(N-1,p)P^{T}(N,q)$
$=P(N,q)P(N-1,p)D_{1}^{T\,-1}\tilde{A}_{P}D_{1}^{-1}P^{T}(N-1,p)P^{T}(N,q)$
where $\tilde{A}_{P}$ has the same structure as $\tilde{S}$ in Eq. (25). As
before,
$\textrm{pf}(A)=\textrm{pf}(\tilde{A}_{P})=\left(A_{P}\right)_{N-1,N}\textrm{pf}(\tilde{A}_{P}^{(1)})$
and repeating recursively the whole procedure $M-1$ times we obtain the
pfaffian as the product of the corresponding matrix elements.
#### 2.2.2 Cholesky like decomposition of a skew-symmetric matrix
Although it is not necessary in order to compute the pfaffian, it can be
useful to show that even with pivoting we can write the matrix $A$ as
$A=PL^{T}\tilde{A}LP$ (29)
where $P$ is the product of exchange matrices as in Eq (28), $L$ is the
product of matrices of the $D^{-1}$ type, Eq (26), and therefore is a lower
triangular matrix with ones in the main diagonal and finally, $\tilde{A}$ is a
skew-symmetric matrix in canonical form, i.e. a block diagonal matrix with
skew-symmetric, $2\times 2$ blocks in the diagonal. This decomposition of a
general skew-symmetric matrix $A$ resembles the Cholesky decomposition of a
general matrix and can be useful in formal manipulations like, for instance,
the inversion of the matrix $A$. In order to show that Eq (29) holds the only
required property is that, when applying the pivoting procedure to
$\tilde{A}_{P}^{(1)}$ the exchange matrices required $P(N-2,s)P(N-3,r)$ have
the property of preserving the structure of the matrix $D_{1}$(and its
inverse). For instance,
$D_{1}^{T\,-1}P(N-2,s)P(N-3,r)=P(N-2,s)P(N-3,r)\tilde{D}_{1}^{T\,-1}$
with $\tilde{D}_{1}^{T\,-1}$ a matrix that is obtained from $D_{1}^{T\,-1}$ by
exchanging rows $N-2$ and $s$ and rows $N-3$ and $r$ and therefore has the
same upper triangular structure with ones in the diagonal as the original
matrix $D_{1}^{T\,-1}$ . Using this property we can move all the exchange
matrices to the right (or to the left) and the remaining matrix will be the
product of triangular matrices (lower for products involving $D^{-1}$) with
ones in the diagonal.
As mentioned earlier, Aitken’s method is better adapted to symbolic
evaluations. However, one must take care that in each step of the process some
specific matrix elements are non-zero.
## 3 Fortran implementation
The implementation of the algorithms considered in this paper in a high level
computer language is straightforward. However, specific code in FORTRAN (both
77 and 90, real and complex arithmetic) is provided along with this paper. The
algorithms are easy to follow and the comments included in the code are useful
guides. Just a few comments are in order: to implement the tridiagonalization
procedure in Fortran, it is advantageous to use the BLAS package [14] to
perform the required matrix by vector multiplication and rank two update.
Unfortunately there are no equivalent in the skew-symmetric case of the
routines SYM (to multiply a symmetric matrix by a vector) or SYR2 (to perform
a symmetric rank two update) but the general procedures GEMV and GERU can be
used instead. On output, both the pfaffian of the matrix and the set of
vectors required to bring it to tridiagonal form are returned. For the
implementation of the method based on Aitken’s block diagonalization formula a
pivoting strategy is required. We have used full pivoting in our
implementation due to its robustness. The routines provided only require the
upper part of the skew-symmetric matrix. The lower part is destroyed and
replaced with the tridiagonal transformation matrix that brings the skew-
symmetric matrix to canonical form upon congruence. An integer vector is also
returned to reconstruct the required exchange of rows and columns.
Perhaps the best test to check the validity of the two implementations is to
compute the pfaffian of a skew-symmetric matrix using both procedures in order
to compare the output. If it is the same up to a given accuracy then it is
very likely that the two implementations are correct. We have writen a test
program (also included in the distribution) that generates skew-symmetric
matrices of given dimension with random entries and compute the pfaffian using
both techniques. In our tests the pfaffians computed both ways coincide up to
one part in $10^{10}$ with dimensions of the matrices of one thousand. This
result also supports the adequacy of the implementation in terms of numerical
stability. Another possibility to test the numerical implementation is to use
the analytical formula given in B for a specific kind of $8\times 8$ matrices.
A test program implementing this approach has also been included in the
distribution.
To finish this section we will briefly comment on the timing of the FORTRAN
numerical implementations mentioned. In a modern personal computer under Linux
the computation of the pfaffian of a $100\times 100$ matrix takes a few
milliseconds in both implementations and the timing scales roughly as the cube
of the dimension of the matrix in such a way that for matrices of a
$1000\times 1000$ dimension the time is of the order of a few seconds.
## 4 A simple Python implementation
We provide here a simple implementation of the tridiagonal reduction method
(see [12] and [15]) in Python, which may be useful for testing purposes. It is
similar to the Householder, but it only use simple row and column operations
that have determinants of unity. The code is:
from numpy import *
def pfaff_py(m) :
mat=copy(m)
ndim = shape(mat)[0]
t1=1.0
for j in range(ndim/2) :
t1 *= mat[0,1]
print ’t1’, t1
if j <ndim/2-1 :
ndimr=shape(mat)[0]
for i in range(2,ndimr) :
if mat[0,1] != 0.0 :
tv=mat[1,:]*mat[i,0]/mat[1,0]
mat[i,: ] -= tv
tv=mat[:,1]*mat[0,i]/mat[0,1]
mat[:,i ] -= tv
else :
print ’need to pivot’
raise Exception
mat=mat[2:,2:]
return t1
The user should be cautioned that the algorithm is not guaranteed to be stable
without an additional pivot step. Also, the matrix is assumed to have been
constructed with the array function in the Numpy library.
## 5 A simple Mathematica implementation
We also provide a simple Mathematica implementation of the method based on
Aitken’s block diagonalization formula. As mentioned above, this method
requires pivoting to avoid divisions by small (or zero) numbers. In the
symbolic implementation, this issue is solved by replacing the denominator by
a variable (OO on the implementation below) in case it is zero and an
additional limit when the variable tends to zero is performed at the end. The
two Mathematica modules required are:
Aitken[M_,n_,OO_]:=
Module[{MM=M,i,j,p},
If[MM[[n-1,n]]==0,MM[[n-1,n]]=OO;MM[[n,n-1]]=-OO];
p=MM[[n-1,n]];
For[i=1,i<=n-2,i++,
For[j=1,j<=n-2,j++,
MM[[i,j]]=M[[i,j]]+(MM[[i,n]]*MM[[j,n-1]]-MM[[j,n]]*MM[[i,n-1]])/p
]
];
MM];
pfaffian[S_]:=
Module[{T=S,n,p},
n=Length[T]/2;
If[T[[2*n-1,2*n]]==0,T[[2*n-1,2*n]]=OO;T[[2*n,2*n-1]]=-OO];
For[n=Length[T]/2;p=T[[2*n-1,2*n]],n>1,n--,
T=Aitken[T,2*n,OO];
p=p*T[[2*(n-1)-1,2*(n-1)]]
];
Limit[p,OO->0]];
## 6 Conclusions
The issue of how to compute both numerically and symbolically the pfaffian of
a skew-symmetric matrix has been addressed using two different approaches.
Numerical stability issues are discussed and methods to assure the desired
accuracy are fully incorporated. A collection of subroutines and test programs
in FORTRAN (both 77 and 90, double precision and complex) are provided. A few
comments on the implementation of the algorithms in Mathematica and Python are
also given.
## 7 Acknowledgements
We acknowledge K. Roche for a careful reading of the manuscript and several
suggestions. This work was supported by MICINN (Spain) under research grants
FPA2009-08958, and FIS2009-07277, as well as by Consolider-Ingenio 2010
Programs CPAN CSD2007-00042 and MULTIDARK CSD2009-00064.
## Appendix A Definition and basic properties of the pfaffian
The pfaffian of a skew-symmetric matrix $R$ of dimension $2N$ and with matrix
elements $r_{ij}$ is defined as
$\textrm{pf}(R)=\frac{1}{2^{n}}\frac{1}{n!}\sum_{\textrm{Perm}}\epsilon(P)r_{i_{1}i_{2}}r_{i_{3}i_{4}}r_{i_{5}i_{6}}\ldots
r_{2n-1,2n}$
where the sum extends to all possible permutations of $i_{1},\ldots,i_{2n}$
and $\epsilon(P)$ is the parity of the permutation. For matrices of odd
dimension the pfaffian is by definition equal to zero. As an example, the
pfaffian of a $2\times 2$ matrix $R$ is $\textrm{pf}(R)=r_{12}$ and for a
$4\times 4$ one $\textrm{pf}(R)=r_{12}r_{34}-r_{13}r_{24}+r_{14}r_{23}$.
Useful properties of the pfaffian are
$\textrm{pf}(P^{T}RP)=\textrm{det}(P)\textrm{pf}(R),$ (30)
$\text{{pf}}\left(\begin{array}[]{cc}0&R\\\
-R^{T}&0\end{array}\right)=(-1)^{N(N-1)/2}\det(R)$
where the matrix $R$ is $N\times N$ and
$\text{{pf}}\left(\begin{array}[]{cc}R_{1}&0\\\
0&R_{2}\end{array}\right)=\text{{pf}}(R_{1})\text{{pf}}(R_{2})$
where $R_{1}$ and $R_{2}$ are skew-symmetric matrices. The matrices may be
defined on the real or on the complex number fields.
## Appendix B Pfaffian of a test matrix
In this appendix we give the expression of the pfaffian of a test matrix which
is big enough as not to be trivial but on the other hand is small enough as to
render the explicit expression of the pfaffian manageable. The expression
given below can be used to check both numerical and symbolic implementations
of the pfaffian.
Consider the two general skew-symmetric matrices of dimension 4
$M=\left(\begin{array}[]{cccc}0&f_{1}&m_{11}&m_{12}\\\
-f_{1}&0&m_{21}&m_{22}\\\ -m_{11}&-m_{21}&0&f_{2}\\\
-m_{12}&-m_{22}&-f_{2}&0\end{array}\right)$
and
$N=\left(\begin{array}[]{cccc}0&g_{1}&n_{11}&n_{12}\\\
-g_{1}&0&n_{21}&n_{22}\\\ -n_{11}&-n_{21}&0&g_{2}\\\
-n_{12}&-n_{22}&-g_{2}&0\end{array}\right)$
where the matrix elements can be complex numbers. With these two matrices and
the identity $4\times 4$ matrix we build the skew-symmetric matrix
$S=\left(\begin{array}[]{cc}N&-\mathbb{I}\\\
\mathbb{I}&-M^{*}\end{array}\right)$
of dimension $8\times 8$ (see Ref [8] for the physical context of this
matrix). It is relatively easy to compute its pfaffian
$\textrm{pf}[S]=1+f_{1}^{*}g_{1}+f_{2}^{*}g_{2}+m_{11}^{*}n_{11}+m_{22}^{*}n_{22}+m_{12}^{*}n_{12}+m_{21}^{*}n_{21}+$
$+(f_{1}^{*}f_{2}^{*}-m_{11}^{*}m_{22}^{*}+m_{12}^{*}m_{21}^{*})(g_{1}g_{2}-n_{11}n_{22}+n_{12}n_{21})$
## References
* [1] M. Bajdich, L. Mitas and L.K. Wagner, Phys. Rev B77, 115112 (2008)
* [2] C.K. Thomas and A. A. Middleton, Phys. Rev E80, 046708 (2009)
* [3] J.-M. Stéphan, S. Furukawa, G. Misguich, and V. Pasquier, Phys. Rev B80, 184421 (2009)
* [4] F.A. Berezin, _The Method of Second Quantization_ (Academic Press, New York, 1966)
* [5] Y. Ohnuki, and T. Kashiwa, Prog. Theor. Phys. 60, 548 (1978).
* [6] John R. Klauder, Bo-Sture Skagerstam, Coherent states: applications in physics and mathematical physics (World Scientific, Singapore, 1985)
* [7] G.H. Lang, C.W. Johnson, S.E. Koonin, and W.E. Ormand, Phys. Rev. C48, 1518 (1993)
* [8] L.M. Robledo, Phys. Rev. C79, 021302 (2009)
* [9] K. Neergard, and E. Wüst, Nucl. Phys. A402, 311 (1983)
* [10] E.R. Caianiello, _Combinatorics and renormalization in Quantum Field Theory_ (W.A. Benjamin, Massachusetts, 1973)
* [11] G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins University Press, Baltimore, 1996).
* [12] J. R. Bunch, Math of Comp. 38, 475 (1982)
* [13] F. Zhang Ed., The Schur Complement and Its Applications (Numerical Methods and Algorithms) (Springer, Berlin, 2005)
* [14] J. J. Dongarra, J. Du Croz, S. Hammarling, and R. J. Hanson, ACM Trans. Math. Soft. 14, 1 (1988),
* [15] J. O. Aasen, BIT 11, 233 (1971)
|
arxiv-papers
| 2010-12-22T16:05:07 |
2024-09-04T02:49:15.916797
|
{
"license": "Public Domain",
"authors": "C. Gonz\\'alez-Ballestero, L.M. Robledo and G. F. Bertsch",
"submitter": "Luis Robledo",
"url": "https://arxiv.org/abs/1012.5022"
}
|
1012.5043
|
# Near-Optimal and Explicit Bell Inequality Violations
Harry Buhrman CWI and University of Amsterdam, buhrman@cwi.nl. Supported by a
Vici grant from NWO, and EU-grant QCS. Oded Regev Blavatnik School of Computer
Science, Tel Aviv University, and CNRS, ENS Paris. Supported by the Israel
Science Foundation, by the Wolfson Family Charitable Trust, and by a European
Research Council (ERC) Starting Grant. Part of the work done while a DIGITEO
visitor in LRI, Orsay. Giannicola Scarpa CWI Amsterdam, g.scarpa@cwi.nl.
Supported by a Vidi grant from NWO, and EU-grant QCS. Ronald de Wolf CWI and
University of Amsterdam, rdewolf@cwi.nl. Supported by a Vidi grant from NWO,
and EU-grant QCS.
###### Abstract
Bell inequality violations correspond to behavior of entangled quantum systems
that cannot be simulated classically. We give two new two-player games with
Bell inequality violations that are stronger, fully explicit, and arguably
simpler than earlier work.
The first game is based on the Hidden Matching problem of quantum
communication complexity, introduced by Bar-Yossef, Jayram, and Kerenidis.
This game can be won with probability 1 by a quantum strategy using a
maximally entangled state with local dimension $n$ (e.g., $\log n$ EPR-pairs),
while we show that the winning probability of any classical strategy differs
from $\frac{1}{2}$ by at most $O(\log n/\sqrt{n})$.
The second game is based on the integrality gap for Unique Games by Khot and
Vishnoi and the quantum rounding procedure of Kempe, Regev, and Toner. Here
$n$-dimensional entanglement allows to win the game with probability $1/(\log
n)^{2}$, while the best winning probability without entanglement is $1/n$.
This near-linear ratio (“Bell inequality violation”) is near-optimal, both in
terms of the local dimension of the entangled state, and in terms of the
number of possible outputs of the two players.
## 1 Introduction
One of the most striking features of quantum mechanics is the fact that
_entangled_ particles can exhibit correlations that cannot be reproduced or
explained by classical physics, i.e., by “local hidden-variable theories.”
This was first noted by Bell [Bel64] in response to Einstein-Podolsky-Rosen’s
challenge to the completeness of quantum mechanics [EPR35]. Experimental
realization of such correlations is the strongest proof we have that nature
does not behave according to classical physics: nature cannot simultaneously
be “local” (meaning that information doesn’t travel faster than the speed of
light) and “realistic” (meaning that properties of particles such as its spin
always have a definite—if possibly unknown—value). Many such experiments have
been done. All behave in accordance with quantum predictions, though so far
none has closed all “loopholes” that would allow some (usually very contrived)
classical explanation of the observations based on imperfect behavior of, for
instance, the photon detectors used.
Here we study quantitatively how much such “quantum correlations” can deviate
from what is achievable classically. The setup for a game $G$ is as follows.
Two space-like separated parties, called Alice and Bob, receive inputs $x$ and
$y$ according to some fixed and known probability distribution $\pi$, and are
required to produce outputs $a$ and $b$, respectively. There is a predicate
specifying which outputs $a,b$ “win” the game on inputs $x,y$. The definition
of the game $G$ consists of this predicate and the distribution $\pi$. The
goal is to design games where entanglement-based strategies have much higher
winning probability than the best classical strategy. While this setting is
used to study non-locality in physics, the same set-up is also used
extensively to study the power of entanglement in computer science contexts
like multi-prover interactive proofs [KKM+08, KKMV08], parallel repetition
[CSUU07, KRT08], and cryptography.
Quantum strategies start out with an arbitrary fixed entangled state. No
communication takes place between Alice and Bob. For each input $x$ Alice has
a measurement, and for each input $y$ Bob has a measurement. They apply the
measurements corresponding to $x$ and $y$ to their halves of the entangled
state, producing classical outputs $a$ and $b$, respectively. Their goal is to
maximize the winning probability. The _entangled value_ $\omega^{*}(G)$ of the
game is the supremum of the expected winning probability, taken over all
entangled strategies. When restricting to strategies that use entanglement of
local dimension $n$, the value is denoted $\omega^{*}_{n}(G)$. This should be
contrasted with the _classical value_ $\omega(G)=\omega^{*}_{0}(G)$ of the
game, which is the maximum among all classical, non-entangled strategies.
Shared randomness between the two parties is allowed, but is easily seen not
to be beneficial.
The remarkable fact, alluded to above, that some “quantum correlations” cannot
be simulated classically, corresponds to the fact that there are games $G$
where the entangled value $\omega^{*}(G)$ is strictly larger than the
classical value $\omega(G)$. For reasons explained in Section 2, such examples
are known in the physics literature as “Bell inequality violations.” The CHSH
game is one particularly famous example [CHSH69]. Here the inputs
$x\in\\{0,1\\}$ and $y\in\\{0,1\\}$ are uniformly distributed, and Alice and
Bob win the game if their respective outputs $a\in\\{0,1\\}$ and
$b\in\\{0,1\\}$ satisfy $a\oplus b=x\wedge y$; in other words, $a$ should
equal $b$ unless $x=y=1$. The classical value of this game is $\omega(G)=3/4$,
while the entangled value is $\omega^{*}(G)=\cos(\pi/8)^{2}\approx 0.85$. The
entangled value is achieved already with 2-dimensional entanglement (i.e., one
EPR-pair), so $\omega^{*}(G)=\omega^{*}_{2}(G)$ for this game [Tsi87].
In the physics literature it is common to quantify the violation demonstrated
by a given game $G$ by the ratio of entangled and classical values. The larger
this ratio the better, both for philosophical reasons (to show the divergence
between classical and quantum worlds) and for practical reasons (a larger
violation is typically more noise-resistant and easier to realize in the noisy
circumstances of an actual laboratory). To be precise, Bell violations are
defined for a slight generalization of the notion of a game, which we call a
_Bell functional_ 111There is unfortunately no good name for this notion in
the literature. It is often called “Bell inequality,” but this is a misnomer
since (1) it is not an inequality, and (2) the term Bell inequality is used to
describe upper bounds on the classical value., where roughly speaking, some
outputs might lead to a loss (or a negative gain) to the players. Since this
discussion is not too relevant for our main results, we postpone it to Section
2. For now, suffice it to say that when considering the violation exhibited by
a game, instead of comparing the maximum winning probabilities as we did
above, one can also compare the maximum achievable deviation of the winning
probability from (say) $1/2$. This is exactly what we will do in our first
game in Section 3.
In two recent papers, Junge et al. [JPP+10, JP11] studied how large a Bell
inequality violation one can obtain. In terms of upper bounds, [JPP+10] proved
that the maximum Bell inequality violation $\omega^{*}_{n}(G)/\omega(G)$
obtainable with entangled strategies of local dimension $n$, is at most
$O(n)$, and [JP11, Theorem 6.8] proved that if Alice and Bob have at most $k$
possible outputs each, then the violation $\omega^{*}(G)/\omega(G)$ is at most
$O(k)$, irrespective of the amount of entanglement they can use. (This
improved an earlier $O(k^{2})$ upper bound due to Degorre et al. [DKLR09], and
was also obtained for the special case of games by Dukaric [Duk10, Theorem
4].) These upper bounds hold for all Bell functionals, and not just for games.
In terms of lower bounds, [JPP+10] showed the existence of a Bell inequality
violation of order $\sqrt{n}/(\log n)^{2}$, where $n$ is both the
entanglement-dimension and the number of outputs of Alice and Bob. This was
recently improved to $\sqrt{n}/\log n$ in [JP11]. Both constructions are
probabilistic, and the proofs show that with high probability the constructed
Bell functionals exhibit a large violation. Their proofs are heavily based on
connections to the mathematically beautiful areas of Banach spaces and
operator spaces, but as a result are arguably somewhat inaccessible to those
unfamiliar with these areas, and it is difficult to get a good intuition for
them. (It is actually possible to analyze their game and reprove many of their
results—often with improved parameters—using elementary probabilistic
techniques [Reg11].)
Our main result in this paper is to exhibit two stronger and fully explicit
Bell inequality violations. Interestingly, both of our games address a
question in theoretical physics but are inspired by theoretical computer
science (communication complexity and unique games, respectively), and the
tools used to analyze them are very much the tools from theoretical computer
science. In fact, one aim of this paper is to export our techniques to
mathematical physics.
In the remainder of this introduction we provide an overview of our two non-
local games, followed by some discussion and comparison.
### 1.1 The Hidden Matching game
The “Hidden Matching” problem was introduced in quantum communication
complexity by Bar-Yossef et al. [BJK08], and many variants of it were
subsequently studied [GKRW09, GKK+08, Gav09]. The original version is as
follows. Let $n$ be a power of 2. Alice is given input $x\in\\{0,1\\}^{n}$ and
Bob is given a perfect matching $M$ (i.e., a partition of $[n]$ into $n/2$
disjoint pairs $(i,j)$). Both inputs are uniformly distributed.222All our
results also hold with minor modifications for the case that Bob’s matching is
chosen uniformly from the set $\\{M_{k}~{}|~{}k\in\\{0,\ldots,n/2-1\\}\\}$,
where the matching $M_{k}$ consists of the pairs $(i,j)$ where $i\leq n/2$ and
$j=n/2+1+(i+k-1\mbox{ mod }n/2)$. This has the advantage of lowering the
number of possible inputs to Bob to $n/2$. We allow one-way communication from
Alice to Bob, and Bob is required to output a pair $(i,j)\in M$ and a bit
$v\in\\{0,1\\}$. They win if $v=x_{i}\oplus x_{j}$.
In Section 3.1 we show that if Alice sends Bob a $c$-bit message, then their
optimal winning probability is $\frac{1}{2}+\Theta(\frac{c}{\sqrt{n}})$. Bar-
Yossef et al. [BJK08] earlier proved this for $c=\Theta(\sqrt{n})$, using
information theory. However, their tools seem unable to give good bounds on
the success probability for much smaller $c$. Instead, the main mathematical
tool we use in our analysis is the so-called “KKL inequality” [KKL88] from
Fourier analysis of Boolean functions (see [O’D08, Wol08] for surveys of this
area). Roughly speaking, this inequality implies that if the message that
Alice sends about $x$ is short, then Bob will not be able to predict the
parity $x_{i}\oplus x_{j}$ well for many $(i,j)$ pairs. His matching $M$ is
uniformly distributed, independent of $x$, and contains only $n/2$ of all
${n\choose 2}$ possible $(i,j)$ pairs. Hence it is unlikely that he can
predict any one of those $n/2$ parities well. The KKL inequality was used
before to analyze another variant of Hidden Matching in [GKK+08], though their
analysis is different and more complicated because their variant is a promise
problem with a non-product input distribution.
The non-local game based on Hidden Matching is as follows: the inputs $x$ and
$M$ are the same as before, but now Alice and Bob don’t communicate. Instead,
Alice outputs an $a\in\\{0,1\\}^{\log n}$, Bob outputs $d\in\\{0,1\\}$ and
$(i,j)\in M$, and they win the game if the outputs satisfy the relation
$(a\cdot(i\oplus j))\oplus d=x_{i}\oplus x_{j}$, where the dot indicates inner
product (modulo 2) of two $\log n$-bit strings. Observe that Alice has $n$
possible outputs and Bob also has $2\cdot n/2=n$ possible outputs.
A classical strategy that wins this game induces a protocol for the original
Hidden Matching problem with communication $c=\log n$ bits and the same
winning probability $p$: Alice sends Bob the $\log n$-bit output $a$ from the
non-local strategy, allowing Bob to compute $v=(a\cdot(i\oplus j))\oplus d$.
Since $v=x_{i}\oplus x_{j}$ with probability $p$, Bob can now output
$(i,j),v$. Hence our bound for the original communication problem implies that
no classical strategy can win with probability that differs from $1/2$ by more
than $O(\frac{\log n}{\sqrt{n}})$.
In contrast, there is a strategy that wins with probability 1 using $\log n$
EPR-pairs, which shows $\omega^{*}_{n}(G)=1$.333The reader might be a bit
confused by the seeming overloading of the meaning of ‘$n$’. Formally, ‘$n$’
is a parameter in the specification of the game. As it happens, for both of
our games it’s also the number of possible outputs for each player, _and_ the
local dimension of the entangled state that our quantum protocol uses (though
we don’t claim that this entanglement-dimension $n$ is needed to achieve the
best-possible entangled value). This game therefore exhibits a Bell violation
of $\Omega(\sqrt{n}/\log n)$ (by measuring the maximal deviation of the
winning probability from $1/2$). This order is the same as that obtained by
Junge et al. [JPP+10, JP11], but our game is fully explicit and arguably
simpler (which would help any future experimental realization). One might feel
that our reduction to a communication complexity lower bound is responsible
for losing the $\log n$ factor; however in Theorem 6 we exhibit a classical
strategy with winning probability $1/2+\Omega(\sqrt{\log(n)/n})$. This shows
that at least the square root of the log-factor is really necessary.
### 1.2 The Khot-Vishnoi game
Our second non-local game derives from the work of Khot and Vishnoi [KV05] on
the famous _Unique Games Conjecture_ (UGC), which was introduced by Khot
[Kho02]. The UGC is a hardness-of-approximation assumption for a specific
graph labeling problem, the details of which need not concern us here. The
conjecture implies many other hardness-of-approximation results that do not
seem obtainable using the more standard techniques based on the PCP theorem.
Khot and Vishnoi exhibited a large integrality gap for the standard
semidefinite programming (SDP) relaxation of this labeling problem, showing
that at least SDP-solvers will not be able to efficiently approximate the
value of the optimal labeling. We use essentially the same set-up for our
game, though for our purposes we will not have to worry about SDPs or the UGC.
Kempe, Regev, and Toner [KRT08] already observed that they could combine their
“quantum rounding” technique with the game of [KV05] to get a Bell inequality
violation of $n^{\varepsilon}$ for some small constant $\varepsilon>0$, where
$n$ is the entanglement dimension and the number of possible outputs. Our main
contribution in the second part of this paper is a refined (and at the same
time simpler) analysis of both the Khot-Vishnoi game and of the quantum
rounding technique, showing that, somewhat surprisingly, nearly optimal
violations can be obtained using this method.
The game is parameterized by an integer $n$, which we assume to be a power of
2, and a “noise-parameter” $\eta\in[0,1/2]$. Consider the group
$\\{0,1\\}^{n}$ of all $n$-bit strings with ‘$\oplus$’ denoting bitwise
addition mod 2, and let $H$ be the subgroup containing the $n$ Hadamard
codewords. This subgroup partitions $\\{0,1\\}^{n}$ into $2^{n}/n$ cosets of
$n$ elements each. Alice receives a uniformly random coset $x$ as input, which
we can think of as $u\oplus H$ for uniformly random $u\in\\{0,1\\}^{n}$. Bob
receives a coset $y$ obtained from Alice’s by adding a string of low Hamming
weight, namely $y=x\oplus z=u\oplus z\oplus H$, where each bit of
$z\in\\{0,1\\}^{n}$ is set to 1 with probability $\eta$, independently of the
other bits. Notice that addition of $z$ gives a natural bijection between the
two cosets, mapping each element of the first coset to a relatively nearby
element of the second coset; namely, the distance between the two elements is
the Hamming weight of $z$, which is typically around $\eta n$. Each player is
supposed to output one element from its coset, and their goal is for their
elements to match under the bijection. In other words, Alice outputs an
element $a\in x$, Bob outputs $b\in y$, and they win the game iff $a\oplus
b=z$.444Note that the winning condition for this game is a “randomized
predicate”, as there are $n$ possible predicates (one for each $z$ in $x\oplus
y$) corresponding to each pair of inputs $x,y$. However, it is easy to see
that with very high probability exactly one of these $n$ constraints dominates
(namely, the one corresponding to a $z$ of Hamming weight around $\eta n$).
This allows one to modify the game in a straightforward manner, making it a
game with a deterministic predicate (although there is usually no reason to do
so). Notice that the number of possible inputs to each player is $2^{n}/n$ and
the number of possible outputs for each player is $n$.
Based on the integrality gap analysis of Khot and Vishnoi, we show that no
classical strategy can win this game with probability greater than
$1/n^{\eta/(1-\eta)}$. We also sketch a classical strategy that achieves
roughly this winning probability. In contrast, using a simplified version of
the “quantum rounding” technique of [KRT08], we exhibit a quantum strategy
that uses the $n$-dimensional maximally entangled state and wins with
probability at least $(1-2\eta)^{2}$. This strategy follows from the
observation that each coset of $H$ defines an orthonormal basis of
$\mathbb{R}^{n}$ in which we can do a measurement. Summarizing, we have
entangled value $\omega^{*}_{n}(G)\geq(1-2\eta)^{2}$ and classical value
$\omega(G)\leq 1/n^{\eta/(1-\eta)}$ for this game. Setting the noise-rate to
$\eta=1/2-1/\log n$, the entangled value is roughly $1/(\log n)^{2}$ while the
classical value is roughly $1/n$, leading to a Bell inequality violation
$\omega^{*}_{n}(G)/\omega(G)=\Omega(n/(\log n)^{2})$. Up to the
polylogarithmic factor, this is optimal both in terms of the local dimension,
and in terms of the number of possible outputs.
### 1.3 Discussion and open problems
The main advantage of the Khot-Vishnoi game is its strong, near-linear Bell
inequality violation of about $n/(\log n)^{2}$. This is quadratically stronger
than the violation of $\sqrt{n}/\log n$ given by Hidden Matching and by [JP11]
and almost matches the $O(n)$ bound proved in [JPP+10] for arbitrary Bell
inequality violations with entangled states of local dimension $n$. One open
question is to tweak the KV game (or define another game) to get rid of the
polylogarithmic term in the ratio, making it optimal up to a constant factor.
A second advantage of KV over HM is that the value is just the winning
probability rather than the bias from $1/2$. This might make the KV game more
relevant for physical non-locality experiments, as it is the probability of
winning that matters most there. This also means that the corresponding Bell
functional (see Section 2) is nonnegative, a case that has been investigated
in [JP11].
One advantage of HM over KV is that the entangled strategy wins the game with
probability 1. In contrast, the entangled strategy for KV wins only with
probability about $1/(\log n)^{2}$, which means any quantum experiment needs
to be repeated about $(\log n)^{2}$ times before we expect to see the first
win. Another advantage is that HM’s description is a bit simpler than KV’s.
Throughout this paper we considered the Bell violation as a function of the
number of outputs of the players and/or of the dimension of entanglement. One
can also analyze the violation in terms of the number of possible _inputs_. We
recall that in the KV game both players have inputs taken from an
exponentially large set, and that in the HM game (when modified as in Footnote
2) Bob has only $n/2$ possible inputs, but Alice still has an exponentially
large set of inputs. The Bell inequality violation of $\sqrt{n}/\log n$
presented by Junge and Palazuelos [JP11] has the advantage that the number of
inputs is only $O(n)$. Accordingly, another open question presents itself: can
we find a game with a (near-)linear Bell inequality violation, and linear
number of inputs and outputs for both Alice and Bob?
Finally, while this paper focuses on the two-party setting, obtaining stronger
Bell inequality violations for settings with three or more parties is also a
worthy goal. Pérez-García et al. [PWP+08] gave a randomized construction of a
three-party _XOR game_ (in such a game each party outputs a bit, and winning
or losing depends only on the XOR of those three bits) that gives a Bell
inequality violation of roughly $\sqrt{d}$ using an entangled state in
dimensions $d\times D\times D$ (with $D\gg d$).555They also showed that using
GHZ states cannot give a superconstant Bell inequality violation for XOR games
(see also [BBLV09]). In contrast, it is a known consequence of Grothendieck’s
inequality that such non-constant separations do not exist for _two_ -party
XOR games. We do not know how large Bell inequality violations can be for
arbitrary three-party games.
Note that it is easy to make a three-party version of Hidden Matching: Alice
gets input $x\in\\{0,1\\}^{n}$, Bob gets input $y\in\\{0,1\\}^{n}$, and
Charlie gets a matching $M$ as input (all uniformly distributed). The goal is
that Alice outputs $a\in\\{0,1\\}^{\log n}$, Bob outputs $b\in\\{0,1\\}^{\log
n}$, Charlie outputs $d\in\\{0,1\\}$ and $(i,j)\in M$, such that $((a\oplus
b)\cdot(i\oplus j))\oplus d=x_{i}\oplus x_{j}\oplus y_{i}\oplus y_{j}$. By
modifying the two-party proofs in this paper, it is not hard to show that the
winning probability using an $n$-dimensional GHZ state is 1, while the best
classical winning probability deviates from $1/2$ by at most $(\log n)^{2}/n$.
So going from two to three parties roughly squares the Bell inequality
violation for Hidden Matching. This improvement unfortunately does not scale
up with more than three parties, because one can show the classical winning
probability is always at least $1/2+\Omega(1/n)$.
## 2 A more formal look at Bell violations
Before we precisely analyze the two games mentioned above, let us first say
something more about the mathetical treatment of general Bell inequalities.
Readers who are happy with the above (more concrete) approach in terms of
winning probabilities of games, may safely skip this section.
Consider a game with $n$ possible inputs to each player and $k$ possible
outputs. The observed behavior of the players (whether they use a classical or
an entangled strategy) can be summarized in terms of $n^{2}$ probability
distributions, each on the set $[k]\times[k]$. We denote by $P(ab|xy)$ the
probability of producing outputs $a$ and $b$ when given inputs $x$ and $y$. As
described in the introduction, a game is defined by a probability distribution
$\pi$ on the input set $[n]\times[n]$, as well as a (possibly randomized)
predicate on $[k]\times[k]$ for each input pair $(x,y)$.The winning
probability of the players can be written as
$\langle{M},{P}\rangle=\sum_{abxy}M^{ab}_{xy}P(ab|xy).$
where $M_{xy}^{ab}$ is defined as the probability of the input pair $(x,y)$
multiplied by the probability that the output pair $(a,b)$ is accepted on this
input pair. We call $M=(M_{xy}^{ab})$ the _Bell functional corresponding to
the game_. More generally, a _Bell functional_ is an arbitrary tensor
$M=(M_{xy}^{ab})$ containing $n^{2}k^{2}$ real numbers.
We define the _classical value_ of a Bell functional $M$ as
$\omega(M)=\sup_{P}|\langle{M},{P}\rangle|,$
where the supremum is over all classical strategies. Similarly, the _entangled
value_ of $M$ is defined as
$\omega^{*}(M)=\sup_{P}|\langle{M},{P}\rangle|,$
where the supremum now is over all quantum strategies (using an entangled
state of arbitrary dimension). If the entangled state is restricted to local
dimension $n$, the value is denoted $\omega^{*}_{n}(M)$. We note that if $M$
is the Bell functional corresponding to a game, then these definitions
coincide with our definitions from the introduction (and in this case the
absolute value is unnecessary since $M$ is non-negative).
A _Bell inequality_ is an upper bound on $\omega(M)$ for some Bell functional
$M$; it shows a limitation of _classical_ strategies.666An upper bound on
$\omega^{*}(M)$ is known as a _Tsirelson inequality_ , and shows a limitation
of entangled strategies. The _Bell inequality violation_ demonstrated by a
Bell functional $M$ is defined as the ratio between the entangled and the
classical value
$\frac{\omega^{*}(M)}{\omega(M)}.$
This provides a convenient quantitative way to measure the extra power
provided by entangled strategies. This definition of Bell violation enjoys a
rich mathematical structure (as witnessed by the numerous connections found to
Banach space and operator space theory [JPP+10, JP11, Duk10]), and also has a
beautiful geometrical interpretation as the “distance” between the set of all
classical strategies and the set of all quantum strategies (see Section 6.1 in
[JP11]).
Clearly, any game $G$ for which $\omega^{*}(G)\geq K\omega(G)$ gives a Bell
violation of $K$ by just taking the functional corresponding to $G$. A more
interesting case is when we consider the largest deviation of the winning
probability from (say) $1/2$. We claim that if $G$ is a game for which the
winning probability of any classical strategy cannot deviate from $1/2$ by
more than $\delta_{1}$ and, moreover, there is a quantum strategy obtaining
winning probability at least $1/2+\delta_{2}$, then we obtain a Bell violation
of $\delta_{2}/\delta_{1}$. To see why, let $M$ be the functional
corresponding to the game, and let $M^{\prime}$ be the functional obtained by
subtracting from each $M_{xy}^{ab}$ half the probability of input pair
$(x,y)$. Then it is easy to see that for each strategy $P$,
$\langle{M^{\prime}},{P}\rangle=\langle{M},{P}\rangle-1/2$. Hence,
$\omega(M^{\prime})$ and $\omega^{*}(M^{\prime})$ measure the largest possible
deviation of the winning probability from $1/2$ of classical and entangled
strategies, respectively. The claim follows. The converse to this statement is
also true: any Bell functional can be converted to a game (by simply scaling
and adding a constant) in such a way that the Bell violation demonstrated by
the functional is equal to the ratio between the largest possible deviation of
winning probability from $1/2$ obtainable by classical and entangled
strategies.
## 3 Hidden Matching problem
In this section we define and analyze the Hidden Matching game.
### 3.1 The Hidden Matching problem in communication complexity
While our focus is non-locality, it will actually be useful to first study the
original version of the Hidden Matching problem in the context of protocols
where communication from Alice to Bob is allowed. Both the problem and the
efficient quantum protocol below come from [BJK08].
###### Definition 1 (Hidden Matching (HM)).
Let $n$ be a power of 2 and ${\cal M}_{n}$ be the set of all perfect matchings
on the set $[n]=\\{1,\ldots,n\\}$ (a perfect matching is a partition of $[n]$
into $n/2$ disjoint pairs $(i,j)$). Alice is given $x\in\\{0,1\\}^{n}$ and Bob
is given $M\in{\cal M}_{n}$, distributed according to the uniform
distribution. We allow one-way communication from Alice to Bob, and Bob
outputs an $(i,j)\in M$ and $v\in\\{0,1\\}$. They win if $v=x_{i}\oplus
x_{j}$.
###### Theorem 1.
There is a protocol for HM with $\log n$ qubits of one-way communication that
wins with probability 1 (i.e., $v=x_{i}\oplus x_{j}$ always holds).
###### Proof.
The protocol is the following:
1. 1.
Alice sends Bob the state
$|\psi\rangle=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}(-1)^{x_{i}}|i\rangle$.
2. 2.
Bob measures $|\psi\rangle$ in the $n$-element basis
$B=\\{\frac{1}{\sqrt{2}}(|i\rangle\pm|j\rangle)\mid(i,j)\in M\\}$. If the
outcome of the measurement is a state
$\frac{1}{\sqrt{2}}(|i\rangle+|j\rangle)$ then Bob outputs $(i,j)$ and $v=0$.
If the outcome of the measurement is a state
$\frac{1}{\sqrt{2}}(|i\rangle-|j\rangle)$, Bob outputs $(i,j)$ and $v=1$.
For each $(i,j)\in M$, the probability to get
$\frac{1}{\sqrt{2}}(|i\rangle+|j\rangle)$ is $2/n$ if $x_{i}\oplus x_{j}=0$
and 0 otherwise, and similarly for $\frac{1}{\sqrt{2}}(|i\rangle-|j\rangle)$.
Hence Bob’s output is always correct. ∎
#### 3.1.1 Limits of classical protocols for HM
Here we show that classical protocols with little communication cannot have
good success probability. To start, note that a protocol that uses shared
randomness is just a probability distribution over deterministic protocols,
hence the maximal winning probability is achieved by a deterministic protocol.
###### Theorem 2.
Every classical deterministic protocol for HM with $c$ bits of one-way
communication, where Bob outputs $(i,j),v$, has
$\Pr[v=x_{i}\oplus x_{j}]\leq\frac{1}{2}+O\left(\frac{c}{\sqrt{n}}\right).$
The intuition behind the proof is the following. If the communication $c$ is
small, the set $X_{m}$ of inputs $x$ for which Alice sends message $m$, will
typically be large (of size about $2^{n-c}$), meaning Bob has little knowledge
of most of the bits of $x$. By the KKL inequality, this implies that for most
of the ${n\choose 2}$ $(i,j)$-pairs, Bob cannot guess the parity $x_{i}\oplus
x_{j}$ well. Of course, Bob has some freedom in which $(i,j)$ he outputs, but
that freedom is limited to the $n/2$ $(i,j)$-pairs in his matching $M$, and it
turns out that on average he will not be able to guess any of those parities
well.
###### Proof.
Fix a classical deterministic protocol. For each $m\in\\{0,1\\}^{c}$, let
$X_{m}\subseteq\\{0,1\\}^{n}$ be the set of Alice’s inputs for which she sends
message $m$. These sets $X_{m}$ together partition Alice’s input space
$\\{0,1\\}^{n}$. Define $p_{m}=\frac{|X_{m}|}{2^{n}}$. Note that
$\sum_{m}p_{m}=1$, so $\\{p_{m}\\}$ is a probability distribution over the
$2^{c}$ messages $m$. Define $\varepsilon$ such that
$\Pr_{\mathcal{U}}[v=x_{i}\oplus x_{j}]=\frac{1}{2}+\varepsilon$, and
$\varepsilon_{m}$ such that $\Pr_{\mathcal{U}}[v=x_{i}\oplus
x_{j}\mid\mbox{Bob received }m]=\frac{1}{2}+\varepsilon_{m}$. Then
$\varepsilon=\sum_{m}p_{m}\varepsilon_{m}$.
For each $m$, define the following probability distribution over all
$(i,j)\in[n]^{2}$:
$q_{m}(i,j)=\Pr_{M\in{\cal M}_{n}}[\mbox{Bob outputs }(i,j)\mid\mbox{Bob
received }m].$
We have $q_{m}(i,j)\leq\frac{1}{n-1}$, because we assume Bob always outputs an
element from $M$ and for fixed $i\neq j$ we have $\Pr_{M}[(i,j)\in M]=1/(n-1)$
(each $j$ is equally likely to be paired up with $i$). The best Bob can do
when guessing $x_{i}\oplus x_{j}$ given message $m$, is to output the value of
$x_{i}\oplus x_{j}$ that occurs most often among the $x\in X_{m}$. Define
$\beta^{m}_{ij}=\mathop{\mathbb{E}}_{x\in
X_{m}}[(-1)^{x_{i}}\cdot(-1)^{x_{j}}]$. The fraction of $x\in X_{m}$ where
$x_{i}\oplus x_{j}=0$ is $1/2+\beta^{m}_{ij}/2$, hence Bob’s optimal success
probability when guessing $x_{i}\oplus x_{j}$ is $1/2+|\beta^{m}_{ij}|/2$.
This implies, for fixed $m$,
$\mathop{\mathbb{E}}_{(i,j)\sim
q_{m}}\left[\frac{1}{2}+\frac{|\beta^{m}_{ij}|}{2}\right]\geq\Pr_{\stackrel{{\scriptstyle
x\in X_{m}}}{{M\in{\cal M}_{n}}}}[v=x_{i}\oplus
x_{j}]=\frac{1}{2}+\varepsilon_{m},$
where the notation $x\sim Q$ stands for “$x$ chosen according to probability
distribution $Q$.”
As explained in [Wol08, Section 4.1], it follows from the KKL inequality
[KKL88] that
$\sum_{i,j:i\neq j}(\beta^{m}_{ij})^{2}\leq
O\left(\log\frac{1}{p_{m}}\right)^{2}.$ (1)
This allows us to upper bound $\varepsilon_{m}$:
$2\varepsilon_{m}\leq\mathop{\mathbb{E}}_{(i,j)\sim
q_{m}}[|\beta^{m}_{ij}|]=\sum_{i,j}q_{m}(i,j)|\beta^{m}_{ij}|\stackrel{{\scriptstyle(*)}}{{\leq}}\sqrt{\sum_{i,j}q_{m}(i,j)^{2}}\cdot\sqrt{\sum_{i,j}(\beta^{m}_{ij})^{2}}\stackrel{{\scriptstyle(**)}}{{\leq}}\frac{1}{\sqrt{n-1}}\cdot
O\left(\log\frac{1}{p_{m}}\right),$
where $(*)$ is Cauchy-Schwarz and $(**)$ follows from
$\sum_{i,j}q_{m}(i,j)^{2}\leq\max_{i,j}q_{m}(i,j)\cdot\sum_{i,j}q_{m}(i,j)\leq\max_{i,j}q_{m}(i,j)\leq\frac{1}{n-1}$
and Eq. (1). Now we can bound $\varepsilon$:
$\varepsilon=\sum_{m}p_{m}\varepsilon_{m}\leq\sum_{m}p_{m}\frac{O(\log(1/p_{m}))}{\sqrt{n-1}}=\frac{1}{\sqrt{n-1}}\sum_{m}p_{m}O(\log(1/p_{m}))=\frac{1}{\sqrt{n-1}}O(H(p))=O\left(\frac{c}{\sqrt{n}}\right)$
where $H$ denotes the binary entropy function, and $H(p)\leq c$ since the
distribution $\\{p_{m}\\}$ is on $2^{c}$ elements. ∎
### 3.2 Classical protocol for HM
Here we design a classical protocol that achieves the above upper bound on the
success probability. This protocol has no bearing on the large Bell inequality
violations that are our main goal in this paper, but it is nice to know the
previous upper bound on the maximal success probability is essentially tight.
###### Theorem 3.
For every positive integer $c\leq\sqrt{n}$, there exists a classical protocol
for HM with $c$ bits of one-way communication, such that for all inputs $x,M$,
$\Pr[v=x_{i}\oplus x_{j}]=\frac{1}{2}+\Omega\left(\frac{c}{\sqrt{n}}\right).$
###### Proof.
Assume for simplicity that $c$ is even and sufficiently large. Alice and Bob
use shared randomness to choose two disjoint subsets $S_{1},S_{2}$ of $[n]$ of
size $\sqrt{n}$ each. Let $y$ denote the bits of $x$ located in the indices
given by the first subset, and $z$ the bits located in the indices given by
the second subset. Alice and Bob use shared randomness to produce $2^{c/2}$
random $\sqrt{n}$-bit strings $y^{(1)},\ldots,y^{(2^{c/2})}$. For each $\ell$,
the distance $d(y,y^{(\ell)})$ is distributed binomially, as the sum of
$\sqrt{n}$ fair coin flips. There exists a constant $\delta>0$ such that
$\Pr[d(y,y^{(\ell)})\leq\sqrt{n}/2-\beta n^{1/4}]\geq 2^{-\delta\beta^{2}}$
(this can be seen for instance by estimating ${k\choose k/2-\beta\sqrt{k}}$
using Stirling’s approximation). Hence with probability close to 1, there will
be an $\ell$ such that $y$ and $y^{(\ell)}$ are at relative distance $\leq
1/2-\Omega(c^{1/2}/n^{1/4})$. If so, Alice sends Bob the first such $\ell$,
and otherwise she tells him there is no such $\ell$. This costs $c/2$ bits of
communication. Similarly, at the expense of another $c/2$ bits of
communication, Bob obtains an approximation of $z$ with relative distance at
most $\leq 1/2-\Omega(c^{1/2}/n^{1/4})$.
It is easy to see that with probability at least 1/2, Bob’s matching $M$
contains an $(i,j)$ with $i$ in $S_{1}$ and $j$ in $S_{2}$. Bob can predict
$x_{i}$ with success probability $1/2+\Omega(c^{1/2}/n^{1/4})$ from his
approximation of $y$, and can predict $x_{j}$ with success probability
$1/2+\Omega(c^{1/2}/n^{1/4})$ from his approximation of $z$. These success
probabilities are independent, hence he can predict $x_{i}\oplus x_{j}$ with
success probability $1/2+\Omega(c/\sqrt{n})$. If there is no such $(i,j)\in
M$, or if he didn’t get good approximations to $y$ or $z$, then Bob just
outputs any $(i,j)\in M$ and a random bit for $v$, giving success probability
$1/2$. Putting everything together, we have a protocol that wins with
probability $1/2+\Omega(c/\sqrt{n})$. ∎
### 3.3 Non-local version of Hidden Matching, and a quantum protocol
We now port our results to the non-local setting. The following non-local
version of HM and the subsequent protocol for it were originally due to
Buhrman, and related problems were studied in [GKRW09, Gav09].
###### Definition 2 (Non-Local Hidden Matching ($\mbox{\rm HM}_{nl}$)).
Let $n$ be a power of 2 and ${\cal M}_{n}$ be the set of all perfect matchings
on the set $[n]$. Alice is given $x\in\\{0,1\\}^{n}$ and Bob is given
$M\in{\cal M}_{n}$, distributed according to the uniform distribution. Alice’s
output is a string $a\in\\{0,1\\}^{\log n}$ and Bob’s output is an $(i,j)\in
M$ and $d\in\\{0,1\\}$. They win the game if and only if
$(a\cdot(i\oplus j))\oplus d=x_{i}\oplus x_{j}.$ (2)
###### Theorem 4.
There exists a quantum protocol for $\mbox{\rm HM}_{nl}$ using a maximally
entangled state with local dimension $n$, such that condition (2) is always
satisfied.
###### Proof.
The protocol is as follows. Alice and Bob share
$|\psi\rangle=\frac{1}{\sqrt{n}}\sum_{i\in\\{0,1\\}^{\log
n}}|i\rangle|i\rangle.$
1. 1.
Alice performs a phase-flip according to her input $x$. The state becomes
$\frac{1}{\sqrt{n}}\sum_{i\in\\{0,1\\}^{\log
n}}(-1)^{x_{i}}|i\rangle|i\rangle$.
2. 2.
Bob performs a projective measurement with projectors $P_{ij}=|i\rangle\langle
i|+|j\rangle\langle j|$, with $(i,j)\in M$. The state collapses to
$\frac{1}{\sqrt{2}}[(-1)^{x_{i}}|i\rangle|i\rangle+(-1)^{x_{j}}|j\rangle|j\rangle]$
for some $(i,j)\in M$ known to Bob.
3. 3.
Both players apply Hadamard transforms $H^{\otimes\log{n}}$, and the state
becomes
$\frac{1}{\sqrt{2}n}\sum_{a,b\in\\{0,1\\}^{\log n}}\left((-1)^{x_{i}+a\cdot
i+b\cdot i}+(-1)^{x_{j}+a\cdot j+b\cdot j}\right)|a\rangle|b\rangle.$
Notice that in the latter state, any pair $a,b$ with nonzero amplitude must
satisfy that
$(a\cdot(i\oplus j))\oplus(b\cdot(i\oplus j))=x_{i}\oplus x_{j}.$
Hence, if the players measure the state, Alice outputs $a$, and Bob outputs
$(i,j)$ and the bit $d=b\cdot(i\oplus j)$, they win the game with certainty. ∎
###### Theorem 5.
The winning probability of any classical protocol for $\mbox{\rm HM}_{nl}$
differs from $\frac{1}{2}$ by at most $O\left(\log{n}/{\sqrt{n}}\right)$.
###### Proof.
A protocol that wins $\mbox{\rm HM}_{nl}$ with success probability
$1/2+\varepsilon$ can be turned into a protocol for HM with $\log{n}$ bits of
communication and the same probability to win: the players play $\mbox{\rm
HM}_{nl}$, with Alice producing $a$ and Bob producing $i,j,d$; Alice then
sends $a$ to Bob, who outputs $i,j,(a\cdot(i\oplus j))\oplus d$. The latter
bit equals $x_{i}\oplus x_{j}$ with probability $1/2+\varepsilon$. This
requires $c=\log{n}$ bits of communication, so Theorem 2 gives the upper bound
on the winning probability. The lower bound follows similarly. ∎
### 3.4 Classical protocols for $\mbox{\rm HM}_{nl}$
Next we show that our upper bound on the success probability of classical
strategies for $\mbox{\rm HM}_{nl}$ is nearly optimal: we can achieve
advantage at least $\Omega(\sqrt{\log(n)/n})$. (In Appendix A we also give an
alternative protocol with a slightly weaker advantage $\Omega(1/\sqrt{n})$.)
###### Theorem 6.
There exists a classical deterministic protocol for $\mbox{\rm HM}_{nl}$ with
winning probability $\frac{1}{2}+\Omega\left(\sqrt{\frac{\log n}{n}}\right)$
(under the uniform input distribution).
###### Proof.
The protocol is as follows. Given $x$, Alice finds an $a\in\\{0,1\\}^{\log n}$
that maximizes $J_{ax}:=|\\{j\neq 1\mid a\cdot j=x_{1}\oplus x_{j}\\}|$. Bob
outputs $(1,j)$, where $j$ is the unique element matched to 1 by $M$, and
$d=0$. With these choices, and letting the number 1 correspond to the string
$0^{\log n}$, the winning condition $(a\cdot(i\oplus j))\oplus d=x_{i}\oplus
x_{j}$ is equivalent to $a\cdot j=x_{1}\oplus x_{j}$. Accordingly, for fixed
$x$ and uniformly distributed $M$ (and hence uniformly distributed
$j\in[n]\backslash\\{0^{\log n}\\}$), the winning probability equals
$p_{x}:=\max_{a}J_{ax}/(n-1)$. Below we use the second moment method to show
$\mathop{\mathbb{E}}_{x}[p_{x}]\geq 1/2+\Omega(\sqrt{\log(n)/n})$.
For a fixed $a$ and uniformly random $x\in\\{0,1\\}^{n}$, $J_{ax}$ behaves
like the sum of $n-1$ fair 0/1-valued coin flips. Let $Z_{a}$ be the indicator
random variable for the event that $J_{ax}\geq(n-1)/2+\beta\sqrt{n}$. Choosing
$\beta$ a sufficiently small constant multiple of $\sqrt{\log n}$, we have
$\mathop{\mathbb{E}}_{x}[Z_{a}]\geq 1/\sqrt{n}$ for each $a$ (this follows
from the $2^{-\delta\beta^{2}}$ probability lower bound mentioned in the proof
of Theorem 3). Let $Z=\sum_{a}Z_{a}$, then
$\mathop{\mathbb{E}}_{x}[Z]\geq\sqrt{n}$ by linearity of expectation. For
$a\neq b$, the covariance $\mbox{Cov}[Z_{a},Z_{b}]$ is non-positive,
informally because if $x$ is “well-aligned” with the string $(a\cdot j)_{j}$
then it’s less likely to be well-aligned with the orthogonal string $(b\cdot
j)_{j}$. Hence we can bound the variance of $Z$:
$\mbox{Var}[Z]=\sum_{a}\mbox{Var}[Z_{a}]+\sum_{a\neq
b}\mbox{Cov}[Z_{a},Z_{b}]\leq\sum_{a}\mbox{Var}[Z_{a}]\leq\sum_{a}\mathop{\mathbb{E}}_{x}[Z_{a}^{2}]=\sum_{a}\mathop{\mathbb{E}}_{x}[Z_{a}]=\mathop{\mathbb{E}}_{x}[Z].$
By Chebyshev’s inequality, we have
$\Pr[Z=0]\leq\Pr\left[|Z-\mathop{\mathbb{E}}_{x}[Z]|\geq\mathop{\mathbb{E}}_{x}[Z]\right]\leq\Pr\left[|Z-\mathop{\mathbb{E}}_{x}[Z]|\geq\sqrt{\mathop{\mathbb{E}}_{x}[Z]}\sqrt{\mbox{Var}[Z]}\right]\leq\frac{1}{\mathop{\mathbb{E}}_{x}[Z]}\leq\frac{1}{\sqrt{n}}.$
Hence with probability at least $1-1/\sqrt{n}$ over the choice of $x$, we have
$Z>0$, meaning there is at least one $a$ with
$J_{ax}\geq(n-1)/2+\beta\sqrt{n}$, and hence $p_{x}\geq
1/2+\Omega(\sqrt{\log(n)/n})$. This implies
$\mathop{\mathbb{E}}_{x}[p_{x}]\geq\left(1-\frac{1}{\sqrt{n}}\right)\left(1/2+\Omega(\sqrt{\log(n)/n})\right)=1/2+\Omega(\sqrt{\log(n)/n}).$
∎
## 4 The Khot-Vishnoi game
### 4.1 The classical value
In this section we analyze the classical value of the Khot-Vishnoi game. Our
main result is an upper bound on the classical value of $1/n^{\eta/(1-\eta)}$,
based on the analysis from [KV05].
Before we give that upper bound, let us first argue that it is essentially
tight, i.e., there exists a strategy whose winning probability is roughly
$1/n^{\eta/(1-\eta)}$. To get some intuition for this game, first think of
$\eta$ as some small constant (even though we will eventually choose it close
to $1/2$), and consider the following natural classical strategy:
> Alice and Bob each output the element of their coset that has highest
> Hamming weight.
The idea is that if $a$ is the element of highest Hamming weight in Alice’s
coset $x$, we expect $a\oplus z$ to also be of high Hamming weight (because it
is close to $a$ in Hamming distance), and so Bob is somewhat likely to pick
it. We now give a brief back-of-the-envelope calculation suggesting that the
winning probability of this strategy is of order $1/n^{\eta/(1-\eta)}$; since
it is not required for our main result, we will not attempt to make this
argument rigorous.
Let $t\geq 0$ be such that the probability that a binomial $B(n,1/2)$ variable
is greater than $(n+t)/2$ is $1/n$. Recalling that a binomial distribution
$B(n,p)$ can be approximated by the normal distribution $N(np,np(1-p))$, and
that the probability that a normal variable is greater than its mean by $s$
standard deviations is approximately $e^{-s^{2}/2}$, we can essentially choose
$t$ to be the solution to $e^{-t^{2}/(2n)}=1/n$ (so $t=\sqrt{2n\ln n}$). Then
we expect Alice’s $n$-element coset to contain exactly one element of Hamming
weight greater than $(n+t)/2$. Since the element $a$ that Alice picks is the
one of highest Hamming weight, we assume for simplicity that its Hamming
weight is $(n+t)/2$. The players win the game if and only if $a\oplus z$ has
the highest weight among Bob’s $n$ elements, which we heuristically
approximate by the event that $a\oplus z$ has Hamming weight at least
$(n+t)/2$. The Hamming weight of $a\oplus z$ is distributed as the sum of
$B((n+t)/2,1-\eta)$ and $B((n-t)/2,\eta)$, which can be approximated as above
by the normal distribution $N((n+t)/2-\eta t,n\eta(1-\eta))$. Hence for the
Hamming weight of $a\oplus z$ to be at least $(n+t)/2$, the normal variable
needs to be greater than its mean by $\eta t/\sqrt{n\eta(1-\eta)}$ standard
deviations, and the probability of this happening is approximately
$e^{-\eta^{2}t^{2}/(2n\eta(1-\eta))}=1/n^{\eta/(1-\eta)}$, as claimed.
Now we show that no classical strategy can be substantially better. The main
technical tool used in the proof is the so-called Bonami-Beckner
hypercontractive inequality, which is applicable to our setting because we
choose $u$ uniform and $u\oplus z$ may be viewed as a “noisy version” of $u$.
###### Theorem 7.
For any $n$ which is a power of 2, and any $\eta\in[0,1/2]$, every classical
strategy for the Khot-Vishnoi game (as defined in Section 1.2) has winning
probability at most $1/n^{\eta/(1-\eta)}$.
###### Proof.
Recall that the inputs are generated as follows: we choose a uniformly random
$u\in\\{0,1\\}^{n}$ and an $\eta$-biased $z\in\\{0,1\\}^{n}$, and define the
respective inputs to be the cosets $u\oplus H$ and $u\oplus z\oplus H$. We can
assume without loss of generality that Alice’s and Bob’s behavior is
deterministic. Define functions $A,B:\\{0,1\\}^{n}\to\\{0,1\\}$ by $A(u)=1$ if
and only if Alice’s output on $u\oplus H$ is $u$, and similarly for Bob.
Notice that by definition, these functions attain the value $1$ on exactly one
element of each coset. Recall that the players win if and only if the sum of
Alice’s output and Bob’s output equals $z$. Hence for all $u,z$, $\sum_{h\in
H}A(u\oplus h)B(u\oplus z\oplus h)$ is $1$ if the players win on input pair
$u\oplus H,u\oplus z\oplus H$ and $0$ otherwise. Therefore, the winning
probability is given by
$\displaystyle\mathop{\mathbb{E}}_{u,z}\left[\sum_{h\in H}A(u\oplus
h)B(u\oplus z\oplus h)\right]$ $\displaystyle=\sum_{h\in
H}\mathop{\mathbb{E}}_{u,z}\left[A(u\oplus h)B(u\oplus z\oplus h)\right]$
$\displaystyle=n\mathop{\mathbb{E}}_{u,z}[A(u)B(u\oplus z)],$
where the second equality uses the fact that for all $h$, $u\oplus h$ is
uniformly distributed.
We use the framework of hypercontractivity (see e.g. [O’D08, Wol08]), which we
briefly explain now. Specifically, for a function
$F:\\{0,1\\}^{n}\rightarrow\mathbb{R}$, define its $p$-norm by
${\left\|{F}\right\|}_{p}=(\mathop{\mathbb{E}}_{x}[|F(u)|^{p}])^{1/p}$, where
the expectation is uniform over all $u\in\\{0,1\\}^{n}$. The _noise-operator_
$T_{1-2\eta}$ adds “$\eta$-noise” to each of $F$’s input bits; more precisely,
$(T_{1-2\eta}F)(u)=\mathop{\mathbb{E}}_{z}[F(u\oplus z)]$, where $z$ is an
$\eta$-biased “noise string.” The linear operator $T_{\rho}$ is diagonal in
the Fourier basis: it just multiplies each character function $\chi_{S}$
($S\subseteq[n]$) by the factor $\rho^{|S|}$. It is easy to see that
$\mathop{\mathbb{E}}_{u}[F(u)\cdot(T_{\rho}G)(u)]=\mathop{\mathbb{E}}_{u}[(T_{\sqrt{\rho}}F)(u)\cdot(T_{\sqrt{\rho}}G)(u)]$.
The Bonami-Beckner inequality implies
${\left\|{T_{\rho}F}\right\|}_{2}\leq{\left\|{F}\right\|}_{1+\rho^{2}}$ for
all $\rho\in[0,1]$. We now have,
$\displaystyle\mathop{\mathbb{E}}_{u,z}[A(u)B(u\oplus z)]$
$\displaystyle=\mathop{\mathbb{E}}_{u}[A(u)(T_{1-2\eta}B)(u)]$
$\displaystyle=\mathop{\mathbb{E}}_{u}[(T_{\sqrt{1-2\eta}}A)(u)\cdot(T_{\sqrt{1-2\eta}}B)(u)]$
$\displaystyle\leq{\left\|{T_{\sqrt{1-2\eta}}A}\right\|}_{2}\cdot{\left\|{T_{\sqrt{1-2\eta}}B}\right\|}_{2}$
$\displaystyle\leq{\left\|{A}\right\|}_{2-2\eta}\cdot{\left\|{B}\right\|}_{2-2\eta}$
$\displaystyle=\left(\mathop{\mathbb{E}}_{u}[A(u)]\right)^{1/(2-2\eta)}\cdot\left(\mathop{\mathbb{E}}_{u}[B(u)]\right)^{1/(2-2\eta)}$
$\displaystyle=\frac{1}{n^{1/(1-\eta)}}.$
Here the first inequality is Cauchy-Schwarz, and the second is the
hypercontractive inequality. We complete the proof by noting that
$n/n^{1/(1-\eta)}=1/n^{\eta/(1-\eta)}$. ∎
### 4.2 Lower bound on the entangled value
In this section we describe a good quantum strategy for the Khot-Vishnoi game,
following the ideas of Kempe, Regev, and Toner [KRT08] and the SDP-solution of
[KV05].
###### Theorem 8.
For any $n$ which is a power of 2, and any $\eta\in[0,1/2]$, there exists a
quantum strategy that wins the Khot-Vishnoi game with probability at least
$(1-2\eta)^{2}$, using a maximally entangled state with local dimension $n$.
###### Proof.
For $a\in\\{0,1\\}^{n}$, let $v^{a}\in\mathbb{R}^{n}$ denote the unit vector
$((-1)^{a_{i}}/\sqrt{n})_{i\in[n]}$. Notice that for all $a,b$, we have
$\langle{v^{a}},{v^{b}}\rangle=1-2d(a,b)/n$, where $d(a,b)$ denotes the
Hamming distance between $a$ and $b$. In particular, the $n$ vectors $v^{a}$,
as $a$ ranges over a coset of $H$, form an orthonormal basis of
$\mathbb{R}^{n}$.
The quantum strategy is as follows. Alice and Bob start with the
$n$-dimensional maximally entangled state. Alice, given coset $x=u\oplus H$ as
input, performs a projective measurement in the orthonormal basis given by
$\\{v^{a}\mid a\in x\\}$ and outputs the value $a$ given by the measurement.
Bob proceeds similarly with the basis $\\{v^{b}\mid b\in y\\}$ induced by his
coset $y=x\oplus z\oplus H$. A standard calculation now shows that the
probability to obtain the pair of outputs $a,b$ is
$\langle{v^{a}},{v^{b}}\rangle^{2}/n$. Since the players win iff $b=a\oplus
z$, the winning probability on inputs $x,y$ is given by
$\frac{1}{n}\sum_{a\in x}\langle{v^{a}},{v^{a\oplus
z}}\rangle^{2}=\frac{1}{n}\sum_{a\in x}(1-2d(a,a\oplus
z)/n)^{2}=(1-2|z|/n)^{2},$
where $|z|$ denotes the Hamming weight (number of 1s) of the $\eta$-biased
string $z$. Taking expectation and using convexity, the overall winning
probability is
$\mathop{\mathbb{E}}_{z}[(1-2|z|/n)^{2}]\geq\left(\mathop{\mathbb{E}}_{z}[1-2|z|/n]\right)^{2}=(1-2\eta)^{2}.$
∎
##### Acknowledgements.
We thank Jop Briët, Dejan Dukaric, Carlos Palazuelos, and Thomas Vidick for
useful discussions and comments.
## References
* [BBLV09] J. Briët, H. Buhrman, T. Lee, and T. Vidick. Multiplayer XOR games and quantum communication complexity with clique-wise entanglement. arXiv:0911.4007, 2009.
* [Bel64] J. S. Bell. On the Einstein-Podolsky-Rosen paradox. Physics, 1:195–200, 1964.
* [BJK08] Z. Bar-Yossef, T. S. Jayram, and I. Kerenidis. Exponential separation of quantum and classical one-way communication complexity. SIAM Journal on Computing, 38(1):366–384, 2008. Earlier version in STOC’04.
* [CHSH69] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt. Proposed experiment to test local hidden-variable theories. Physical Review Letters, 23(15):880–884, 1969.
* [CSUU07] R. Cleve, W. Slofstra, F. Unger, and S. Upadhyay. Strong parallel repetition theorem for quantum XOR proof systems. In Proceedings of 22nd IEEE Conference on Computational Complexity, pages 282–299, 2007. quant-ph/0608146.
* [DKLR09] J. Degorre, M. Kaplan, S. Laplante, and J. Roland. The communication complexity of non-signaling distributions. In Proceedings of 34th MFCS, pages 270–281, 2009. arXiv:0804.4859.
* [Duk10] D. Dukaric. The Hilbertian tensor norm and its connection to quantum information theory. arXiv:1008.1948v2, 2010.
* [EPR35] A. Einstein, B. Podolsky, and N. Rosen. Can quantum-mechanical description of physical reality be considered complete? Physical Review, 47:777–780, 1935.
* [Gav09] D. Gavinsky. Classical interaction cannot replace quantum nonlocality. arXiv:0901.0956, 2009.
* [GKK+08] D. Gavinsky, J. Kempe, I. Kerenidis, R. Raz, and R. de Wolf. Exponential separation for one-way quantum communication complexity, with applications to cryptography. SIAM Journal on Computing, 38(5):1695–1708, 2008. Earlier version in STOC’07. quant-ph/0611209.
* [GKRW09] D. Gavinsky, J. Kempe, O. Regev, and R. de Wolf. Bounded-error quantum state identification and exponential separations in communication complexity. SIAM Journal on Computing, 39(1):1–24, 2009. Special issue on STOC’06. quant-ph/0511013.
* [JP11] M. Junge and C. Palazuelos. Large violation of Bell inequalities with low entanglement. Communications in Mathematical Physics, 2011. To appear. Preprint at arXiv:1007.3043v2.
* [JPP+10] M. Junge, C. Palazuelos, D. Pérez-García, I. Villanueva, and M. Wolf. Unbounded violations of bipartite Bell inequalities via Operator Space theory. Communications in Mathematical Physics, 300(3):715–739, 2010. arXiv:0910.4228. Shorter version appeared in PRL 104:170405, arXiv:0912.1941.
* [Kho02] S. Khot. On the power of unique 2-prover 1-round games. In Proceedings of 34th ACM STOC, pages 767––775, 2002.
* [KKL88] J. Kahn, G. Kalai, and N. Linial. The influence of variables on Boolean functions. In Proceedings of 29th IEEE FOCS, pages 68–80, 1988.
* [KKM+08] J. Kempe, H. Kobayashi, K. Matsumoto, B. Toner, and T. Vidick. Entangled games are hard to approximate. In Proceedings of 49th IEEE FOCS, pages 447–456, 2008.
* [KKMV08] J. Kempe, H. Kobayashi, K. Matsumoto, and T. Vidick. Using entanglement in quantum multi-prover interactive proofs. In Proceedings of 23th IEEE Complexity, pages 211–222, 2008.
* [KRT08] J. Kempe, O. Regev, and B. Toner. Unique games with entangled provers are easy. In Proceedings of 49th IEEE FOCS, pages 457–466, 2008. arXiv:0710.0655.
* [KV05] S. Khot and N. Vishnoi. The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into $\ell_{1}$. In Proceedings of 46th IEEE FOCS, pages 53–62, 2005.
* [O’D08] R. O’Donnell. Some topics in analysis of boolean functions. Technical report, ECCC Report TR08–055, 2008. Paper for an invited talk at STOC’08.
* [PWP+08] D. Pérez-García, M.M. Wolf, C. Palazuelos, I. Villanueva, and M. Junge. Unbounded violations of tripartite Bell inequalities. Communications of Mathematical Physics, 279:455, 2008. quant-ph/0702189.
* [Reg11] O. Regev. Bell violations through independent bases games. arXiv:1101.0576, 3 Jan 2011.
* [Tsi87] B. S. Tsirelson. Quantum analogues of the bell inequalities. the case of two spatially separated domains. Journal of Soviet Mathematics, 36:557–570, 1987.
* [Wol08] R. de Wolf. A brief introduction to Fourier analysis on the Boolean cube. Theory of Computing, 2008. ToC Library, Graduate Surveys 1.
## Appendix A An alternative strategy for $\mbox{\rm HM}_{nl}$
Here we give an alternative and slightly weaker version of Theorem 6, with
advantage $\Omega(1/\sqrt{n})$ instead of $\Omega(\sqrt{\log(n)/n})$.
###### Proof.
Fix arbitrary inputs $x,M$. Bob always outputs $i=1$ and $j$ is whatever is
matched to $i$ by $M$. Consider the following two unit vectors in
$\mathbb{R}^{n}$,
$u=\left((-1)^{x_{1}\oplus x_{k}}/\sqrt{n}\right)_{k=1}^{n}\qquad\qquad
v=e_{j}$
where $e_{j}$ is the vector with $1$ in the $j$th coordinate and zero
elsewhere. Notice that Alice knows $u$, Bob knows $v$, and that
$\langle{u},{v}\rangle=(-1)^{x_{1}\oplus x_{j}}/\sqrt{n}$. The players use
shared randomness to choose a random unit vector $w\in\mathbb{R}^{n}$. Bob
outputs $d=0$ if $\langle{w},{v}\rangle>0$, and $d=1$ otherwise. Alice outputs
$a=0^{\log n}$ if $\langle{w},{u}\rangle>0$, and a uniform
$a\in\\{0,1\\}^{\log n}$ otherwise.
We now analyze the success probability. Assume that $x_{1}\oplus x_{j}=0$ (the
other case being similar). It is easy to see that the probability of both
$\langle{w},{u}\rangle$ and $\langle{w},{v}\rangle$ being positive is
$\frac{1}{2}-\frac{1}{2\pi}\arccos\langle{u},{v}\rangle$, as this is
essentially a two-dimensional question. They have the same probability of both
being negative, and probability $\frac{1}{2\pi}\arccos\langle{u},{v}\rangle$
to be in each of the two remaining cases. In the two cases that
$\langle{w},{u}\rangle\leq 0$ (an event that happens with probability $1/2$),
$a\cdot(i\oplus j)$ is a uniform bit (since $i\neq j$) and the players win
with probability exactly $1/2$. Otherwise (i.e., if
$\langle{w},{u}\rangle>0$), the players win if and only if $d=0$ (i.e., if
also $\langle{w},{v}\rangle>0$). Hence, using that
$\arccos(z)=\pi/2-\Theta(z)$ for small $z$, the overall winning probability is
$\frac{1}{2}\cdot\frac{1}{2}+\frac{1}{2}-\frac{1}{2\pi}\arccos\langle{u},{v}\rangle=\frac{1}{2}+\Theta\left(\frac{1}{\sqrt{n}}\right).$
∎
|
arxiv-papers
| 2010-12-22T17:20:09 |
2024-09-04T02:49:15.923719
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Harry Buhrman, Oded Regev, Giannicola Scarpa, Ronald de Wolf",
"submitter": "Oded Regev",
"url": "https://arxiv.org/abs/1012.5043"
}
|
1012.5086
|
Current address: ]Advanced Photon Source, Argonne National Laboratory,
Argonne, IL 60439, USA
# Magnetic Structure in Fe/Sm-Co Exchange Spring Bilayers with Intermixed
Interfaces
Yaohua Liu yhliu@anl.gov S. G. E. te Velthuis tevelthuis@anl.gov J. S. Jiang
Y. Choi [ S. D. Bader Materials Science Division, Argonne National
Laboratory, Argonne, IL 60439, USA A. A. Parizzi H. Ambaye V. Lauter
Spallation Neutron Source Oak Ridge National Laboratory, Oak Ridge, TN 37831,
USA
###### Abstract
The depth profile of the intrinsic magnetic properties in an Fe/Sm-Co bilayer
fabricated under nearly optimal spring-magnet conditions was determined by
complementary studies of polarized neutron reflectometry and micromagnetic
simulations. We found that at the Fe/Sm-Co interface the magnetic properties
change gradually at the length scale of 8 nm. In this intermixed interfacial
region, the saturation magnetization and magnetic anisotropy are lower and the
exchange stiffness is higher than values estimated from the model based on a
mixture of Fe and Sm-Co phases. Therefore, the intermixed interface yields
superior exchange coupling between the Fe and Sm-Co layers, but at the cost of
average magnetization.
###### pacs:
75.70.Cn, 61.05.fj
## I Introduction
Exchange-coupled, high-magnetization (soft) and high-anisotropy (hard)
magnetic phases have potential applications as both ultra-strong permanent
magnet KnellerHawigIEEE1991 and ultra-high-density recording media.
VictoraIEEE2005 While the intrinsic properties of the two phases play the
most important role, optimization of the interface properties are also
important to achieve the best performance. For example, the interface
morphology need to be optimized for good exchange coupling between the soft
and hard phases. Micromagnetic simulations suggested that magnetically graded
interfaces, whose magnetic properties are gradually changed over a distance
$\sim 10$ nm, is more effective than sharp interfaces, in order to increase
the nucleation field in the soft phase and decrease the switching barrier of
the hard phase. JiangAPL2004 ; SuessAPL2006 Magnetically graded interfaces
can be fabricated from chemically intermixing phases via nanotechnology.
ChoiAPL2007 ; GollJAP2008 ; KirbyPRB2010 Nanoscale spatial resolution is
needed to experimentally determine the intrinsic magnetic properties of hard-
soft heterostructures. Interestingly, for Fe/Sm-Co spring magnets under
optimized fabrication conditions for the maximum energy product, the Fe layer
and the Sm-Co layer have an intermixed interface over a length scale $\sim
5-10$ nm. ChoiPRB2007 ; LiuAPL2008 Although the intrinsic magnetic properties
of individual Fe films and Sm-Co films have been well studied,
FullertonPRB1998 there lacks quantitative knowledge about the intrinsic
magnetic properties of the intermixed Fe/Sm-Co interface due to its complex
composition. ChoiPRB2007 ; LiuAPL2008
Polarized neutron reflectometry (PNR) is a sensitive tool to study the depth
profile of magnetic structures within multilayers with sub-nanometer depth
resolution, Fitz2005 ; Chatterji2006 and has been used to study exchange
coupling and anisotropy. DonovanPRL2002 ; FitzPRB2006 For example, in a
NiFe/FePt spring magnet, O’Donovan et al. have determined that the twisted
magnetic structure of the spring magnet is not confined to the magnetically
soft layer, but also penetrates into the hard magnetic phase. DonovanPRL2002
Recently, Kirby et al. showed qualitatively that the structural gradation
yields a graded anisotropy in Co/Pd multilayers. KirbyPRB2010
In the present work, we report PNR studies on a Fe/Sm-Co bilayer, which was
fabricated under nearly optimized spring magnet conditions. We first confirmed
that there is a structurally intermixed interface of $\sim 8$-nm wide between
the Fe and Sm-Co layers via combined X-ray reflectometry (XRR) and PNR
studies. We also determined the depth profile of the saturation magnetization
$M_{S}$. Furthermore, the profiles of the magnetic anisotropy $K$ and the
exchange stiffness $A$ were obtained by analyzing the PNR data with the aid of
micromagnetic simulations. The magnetic properties in the intermixed region
were compared to a model based on a mixture of Fe and Sm-Co phases. The
saturation magnetization is slightly lower than the value estimated from the
model, suggesting new compounds formed in the intermixed region. The intrinsic
anisotropy is also lower than the value from the model. However, the exchange
stiffness is higher, so that the interface efficiently couples the Fe and Sm-
Co layers.
## II Experimental Methods
The Fe/Sm-Co thin film was fabricated via _dc_ magnetron sputtering onto an
MgO (110) single-crystal substrate with a nominal structure of (10 nm Cr)/(10
nm Fe)/(20 nm Sm-Co)/(20 nm Cr)/MgO. The sample size is $10\times 10$ mm2. The
Cr (211) buffer layer and the Sm-Co layer were grown at 400 ∘C, and the Fe
layer was grown at 100 ∘C. ChoiAPL2007 The Sm-Co layer, nominally Sm2Co7
composition, has an uniaxial, in-plane magnetic easy axis. BenaissaIEEE1998
Magnetic hysteresis loops were obtained by means of vibrating sample
magnetometer (VSM). XRR studies were performed with a X-ray diffractometer
using Cu $K_{\alpha}$ radiation. PNR experiments were conducted at the SNS at
Oak Ridge National Laboratory, at the Magnetism Reflectometer.
LauterPhysicaB2009 This is a time-of-flight (TOF) instrument with a
wavelength band 2 - 5 Å and the polarization efficiency of $\sim 98\%$. All
experiments were conducted at room temperature.
Figure 1: (Color online) Easy-axis magnetic hysteresis loop. The film is
saturated at 1 T. The two circles label the magnetization states measured by
PNR, with applied fields of +1.11 and +0.39 T, respectively.
The external field was applied along the magnetic easy-axis for PNR
experiments. For convenience, we define the field direction as $+\hat{x}$ and
the sample surface normal as $-\hat{z}$. Figure 1 shows the easy-axis magnetic
hysteresis loop. The film saturates at fields above $+1$ T. There are two
major reversal processes along the increasing field branch, centered at
$+0.35$ and $+0.75$ T, respectively. A minor soft-phase was also observed,
which reverses below $+0.05$ T. PNR measurements were performed at $+1.11$ T
and in a demagnetized state at $+0.39$ T, after saturation with a $-1.11$ T
field, in order to determine the depth profiles of the saturation
magnetization, the exchange stiffness and the magnetic anisotropy.
Reflectometry is a non-destructive method to determine scattering length
density (SLD) profiles. In the specular condition, XRR yields the depth
profile of the electron density, which can be used to reconstruct the chemical
structures. Neutrons interact with both nuclei and the internal magnetic
field. Spin-up and spin-down neutrons feel the same nuclear scattering
potential, but an opposite magnetic scattering potential. From subsequent
measurement with oppositely polarized neutron beams, the two contributions can
be separated in order to reconstruct both depth profiles of the chemical
structure and the magnetization vector. Fitz2005 ; Chatterji2006 The contrast
of the neutron SLDs between Fe and Sm-Co is high, so that the PNR experiments
are sensitive to the interfacial structure of interest. There are four
reflectivities in PNR, including two non-spin-flip (NSF) reflectivities
$R^{++}$ and $R^{--}$, and two spin-flip (SF) reflectivities $R^{+-}$ and
$R^{-+}$. SF scattering occurs when the sample’s magnetization vector has a
non-zero component perpendicular to both the neutron’s polarization direction
and the momentum transfer direction. During magnetization reversal, the SF
scattering occurred in a sufficiently high magnetic field so that it was
necessary to take into account the Zeeman effect when analyzing the data.
FelcherPB1996 (See Appendix A.)
For PNR, Fredrikze’s formalism FredrikzePB2001 was used to determine the
optics’ polarization efficiency and the direct beam (DB) spectra from the
measured four reflectivities of the DB. The polarization corrections with the
error propagation were made following Wilder’s formalism. WildesNN2006
Simulations of the reflectivities were based on the Parratt formalism. Reflpak
A rough interface was modeled as a sequence of very thin slices whose SLDs
vary, following an error function so as to interpolate between adjacent
layers. Fitz2005 The instrumental resolution was handled by Gaussian
convolution. The GenCurvefit program GenCurvefit using the genetic algorithm
was employed for the model optimization.
## III Experimental Results and Analysis
### III.1 Depth profile of Saturated Magnetization
Figure 2: (Color online) (a) XRR data and (b) PNR data taken at saturation
field $+1.11$ T. (c) Depth profiles of neutron nuclear SLD (thin black line,
left), neutron magnetic SLD (thick red line, left) and X-ray SLD (dashed green
line, right). Only the real parts of the X-ray and the nuclear SLDs are shown.
The XRR data and the specular PNR data taken at a saturation field of $+1.11$
T are displayed in Figs. 2a and 2b, respectively. The reflectivities are shown
as functions of the momentum transfer perpendicular to the film plane
$Q_{z}=4\pi sin\theta/\lambda$, where $\theta$ is the incident angle and
$\lambda$ is the wavelength of the radiation source. Since there is no spin
flip scattering exists at saturation, we measured only $R^{+}$ (=$R^{++}$) and
$R^{-}$ (=$R^{--}$). Because of the sample’s high saturation magnetization,
$R^{+}\gg R^{-}$ for $Q_{z}$ above the critical edges ($Q_{z}>0.018$ Å-1).
There is an interesting feature in the PNR data below the critical edges: both
$R^{+}$ and $R^{-}$ showed frustrated total reflection. This is due to the
enhanced absorption at resonant conditions. MaazaPLA1996 Simulations show
that the $Q$ separation between the two dips strongly depends on the
magnetization along the field direction, especially that of the Sm-Co layer,
which is the locus of absorption. Therefore, this separation, as well as the
splitting between $R^{+}$ and $R^{-}$ above the critical edge, are direct
indications of the sample’s magnetization.
The chemical and the magnetic SLD profiles were determined by fitting the XRR
and the PNR data simultaneously with the same structural model.Model Since
there are well tabulated values for the neutron and the X-ray scattering
length and absorption length for each element, both the real and imaginary
parts of the X-ray and the neutron’s nuclear SLDs were calculated from the
chemical composition and the film density. Therefore a single parameter was
used to determine both the X-ray and the neutron nuclear SLDs for each layer
that has a well defined composition. This parameter is the mass density for
that particular layer. However, the intermixed interface has a complex
chemical composition that extends $>~{}5$ nm in depth. ChoiPRB2007 ;
LiuAPL2008 The X-ray and the neutron’s nuclear SLDs in the intermixed region
need to be determined separately, which was done by introducing an intermixed
layer between the Fe and Sm-Co layers. Therefore, both the nuclear and the
magnetic SLD profiles can vary in more sophisticated ways and they are less
correlated to each other over the intermixed region. Actually, the peak and
dip positions in $R^{-}$ between 0.025 and 0.045 Å can not be reproduced
without introducing this intermixed layer, which indicates that a single error
function is not sufficient to model the nuclear and/or the magnetic SLD
profiles at the Fe/Sm-Co interface.
Table 1: The layer thickness, _rms_ interface roughness, the nuclear and magnetic SLDs, saturated magnetization $M_{S}$, exchange stiffness $A$ and uniaxial anisotropy $K$ in the layers of interest from the best fits. Nuclear and magnetic SLDs and $M_{S}$ are from fitting the $+1.11$-T data (Sec. III.1). $A$ and $K$ are from fitting the $+0.39$-T data (Sec. III.2). Typical literature values are $M_{S}=1700~{}(550)$ emu/cc, $A=2.8~{}(1.2)$ ergs/cm, and $K=10^{3}~{}(5\times 10^{7}$) ergs/cc for the Fe (Sm-Co) layer. FullertonPRB1998 (∗Literature value of $K_{Fe}$ was used during the optimization.) | Fe | | mixed | | Sm-Co
---|---|---|---|---|---
Thickness (nm) | $7.4\pm 0.3$ | | $4.2\pm 0.3$ | | $20.1\pm 0.2$
Roughness (nm) | | $4.3\pm 0.3$ | | $2.6\pm 0.2$ |
$\rho_{nuc}$ ($10^{-6}$ Å-2) | $7.9\pm 0.2$ | | $5.2\pm 0.2$ | | $1.64\pm 0.04$
$\rho_{mag}$ ($10^{-6}$ Å-2) | $4.9\pm 0.1$ | | $2.5\pm 0.1$ | | $1.62\pm 0.03$
$M_{s}$ (emu/cc) | $1700\pm 50$ | | $890\pm 50$ | | $570\pm 10$
$M_{+0.39~{}T}$ (emu/cc) | $1540\pm 10$ | | $730\pm 30$ | | $420\pm 10$
$A$ (10-6 ergs/cm) | $2.6\pm 0.1$ | | $2.6\pm 0.1$ | | $1.2\pm 0.1$
$K$ (106 ergs/cc) | 0.001∗ | | $3.0\pm 0.2$ | | $27\pm 1$
The best-fit curves for the reflectivity data are overlaid on the data, and
the SLD depth profiles are shown in Fig. 2c. The SLD profiles show sharp
transitions at the Cr/MgO and the Sm-Co/Cr interfaces with _rms_ roughnesses
of $\sim$ 0.4 nm for both interfaces. However, the SLD profiles gradually
change between the Fe and Sm-Co layers over a distance of $\sim 8$ nm. Such a
large intermixed region is consistent with previous energy-dispersive-
spectroscopy (EDS) results from samples fabricated under the same conditions.
ChoiPRB2007 In the intermixed region, there appears a shoulder in the nuclear
SLD profile (Fig. 2c), showing that it is chemically rich in Fe. However, this
feature is not present in the magnetic SLD profile. This will be discussed
below. The parameters of interest are listed in Table 1, which shows that both
the Fe and the Sm-Co layers have their saturation magnetizations close to the
literature values FullertonPRB1998 despite the large intermixing at the
interface.
### III.2 Depth Profiles of Exchange Stiffness and Uniaxial Anisotropy
Figure 3: (Color online) (a) PNR data taken at $+0.39$ T after saturation in
$-1.11$ T. $R^{+-}$ and $R^{-+}$ are offset by a factor of 0.1 for clarity.
(b) The depth profiles of the magnetization vector $\vec{M}$ (obtained from
micromagnetic simulations) that yields the best fit to the PNR data. The field
direction is along $+\hat{x}$.
In order to get insight into the intrinsic magnetic properties, besides
$M_{S}$, in the intermixed region, we also studied the magnetization structure
in a demagnetized state. The PNR data were collected at $+0.39$ T after the
negative saturation in $-1.11$ T. The field is sufficiently high so that the
specular SF reflection showed at off-specular positions on the detector due to
the Zeeman effect, as shown in Fig. 6. Therefore, in the analysis the momentum
transfer $Q_{z}$ was modified according to Eq. 1. FelcherNature1995 ;
FelcherPB1996 It is worth noting that the off-specular scattering in the SF
reflectivity is dominated by the Zeeman effect in our data and possible off-
specular scattering due to in-plane magnetic domains is not distinguishable
from the background, and is therefore not considered in the data.
The PNR data are shown in Fig. 3a. In contrast to the saturation case,
$R^{--}$ is higher than $R^{++}$ for most $Q_{z}$ values above the critical
edge, which indicates that the average $M_{x}$ is still along the negative
applied field direction. Both the splitting between $R^{--}$ and $R^{++}$ and
the separation between the dips below the critical edge are smaller because
the average $M_{x}$ is much lower than $M_{s}$. At the same time, there is
significant SF scattering of the same amplitude as the NSF scattering,
indicating a large in-plane magnetization component perpendicular to the field
direction ($M_{y}$). This is caused by the magnetization spiral structure
observed in exchange-coupled bilayers during demagnetization. DonovanPRL2002
Figure 3b plots the depth profile of the magnetization vector $\vec{M}$ that
yields the best fit, which is obtained from micromagnetic simulations. Our PNR
studies do not reveal the chirality of $\vec{M}$ so that a positive $M_{y}$ is
used for convenience. As expected, the magnetic moment of the Sm-Co layer is
along the negative field direction and the magnetization vector rotates into
the intermixed region and the Fe layer.
Figure 4: (Color online) Top: The depth profile of the nuclear SLD in gray
scale in the unit of $10^{-6}$ Å-2. Bottom: The depth profile of the exchange
stiffness $A$ (left, red solid line) and the uniaxial anisotropy $K$ (right,
blue dashed line) which gives the best fit to the PNR data.
Rohlsberger _et al._ have shown that the depth profile of the magnetization
vector in exchange coupled bilayers agrees with the micromagnetic simulation
from the 1D spin-chain model. RohlsbergerPRL2002 Fitzsimmons _et al._ further
determined the micromagnetic parameters for individual layers in the exchange
coupled DyFe${}_{2}2$/YFe2 superlattice using combined studies of PNR and
micromagnetic simulations. FitzPRB2006 For the Fe/Sm-Co spring magnet studied
here, the Fe and the Sm-Co layers are largely intermixed for $\sim 8$ nm at
the interface and it is naturally to expect that the intrinsic magnetic
properties, i.e. uniaxial anisotropy $K$ and exchange stiffness $A$, vary
gradually due to the gradation of the chemical composition. Hence, we extended
Fitzsimmons’ approach and constructed depth profiles of the intrinsic magnetic
properties. Given a profile of the micromagnetic parameters, the equilibrium
magnetic structure can be simulated by energy minimization via the 1D spin-
chain model, FullertonPRB1998 from which the neutron’s magnetic scattering
potential can be directly calculated followed by the spin-dependent neutron
reflectivities. Therefore, the intrinsic magnetic properties can be optimized
to yield the best fit to the experimental PNR data.
These profiles of $K$ and $A$ are built up of layers identical to those of the
chemical structure as determined from the fits to the XRR and the PNR data in
saturations. The gradual changes between the layers are computed using the
interface roughness in the same way as is done for the SLDs. When fitting, the
chemical structure and the roughnesses are fixed to those already determined.
$M$ is allowed to be lower than $M_{s}$ to account for the effect from the
minor soft phase mentioned above. The model optimization is not sensitive to
the $K_{Fe}$ because the associated magnetic energy is negligible, and
therefore the literature value of $K_{Fe}$ is used. The error bars of the
parameters only reflect the statistical error and the accuracy of the values
depends on the model. Therefore it is important here to have the additional
layer between the Fe and the Sm-Co layers in the model because the exact
mapping function from the chemical structure to the micromagnetic properties
is unknown. The layer has its own free micromagnetic parameters so that $M$,
$A$ and $K$ in the intermixed region are able to vary more independently from
those in the Fe layer and the Sm-Co layer. We checked the robustness of the
model by allowing different interface roughnesses for $M$, $A$, and $K$ and
found that the optimized micromagnetic parameters have similar depth
dependencies.
The depth profiles of $A$ and $K$ from the best fit are shown in Fig. 4 and
the values of $M$, $A$ and $K$ for layers of interest are listed in Table 1.
Both the parameters $A$ and $K$ show monotonic depth dependence as expected.
$M_{Sm-Co}$ is $\sim 20\%$ lower than the saturation value, from which we
estimated that $\sim 10\%$ of the Sm-Co was reversed due to the minor phase.
Therefore, the 1D spin-chain model is approximately valid and yields fairly
good fits, which were overlaid on the data in Fig. 3a. As also listed in Table
1, in the intermixed region, $A$ is close to that of the Fe layer, but $K$ is
much lower than the average value between the Fe and the Sm-Co layers.
## IV Discussion and Summary
Figure 5: (Color online) Depth profiles of (a) the saturation magnetization,
and (b) the exchange stiffness (left) and the uniaxial anisotropy (right)
calculated by assuming that the intermixed region is a mixture of the Fe phase
and the Sm-Co phase (dashed lines). The experimentally determined profiles are
plotted as solid lines. Also shown in the top panel is the depth profile of
the nuclear SLD in gray scale. The unit is $10^{-6}$ Å-2.
By assuming that the intermixed region is composed of a mixture of the Fe and
Sm-Co phases, the saturation magnetization can be computed from the relative
volume of both phases via $M=f_{s}M_{s}+f_{h}M_{h}$, where $f_{s}$ ($M_{s}$)
and $f_{h}$ ($M_{h}$) are the relative volume (the saturation magnetization)
of the Fe phase and the Sm-Co phase, respectively. From the depth profile of
the nuclear SLD, the relative volume of the two phases can be determined.
Following this approach, the calculated saturation magnetization profile is
shown in Fig. 5a. Also shown is the saturation magnetization profile
calculated from the magnetic SLD profile. Clearly, the mixture model predicted
slightly larger saturation magnetization in the intermixed region. This agrees
with the suggestion that new compound(s) is (are) formed in the intermixed
region, JiangAPL2004 possibly due to interdiffusion between the Fe and Sm-Co
layers. The estimation shows that the average magnetization of the whole
sample is reduced by $\sim 4\%$ due to the intermixing of the two components.
The exchange stiffness and the anisotropy of a two phase mixture can also be
approximately estimated by the relative volume: $K=f_{s}K_{s}+f_{h}K_{h}$ and
$A=f_{s}A_{s}+f_{h}A_{h}$. SkomskiPRB1993 The results are shown in Fig. 5b.
In comparison to those determined from the PNR experiments, the model predicts
higher anisotropy but lower exchange stiffness in the intermixed region.
Therefore, we conclude that the optimized Fe/Sm-Co interface for a spring
magnet reduces the average magnetization.
Interfaces with graded magnetic properties are desirable for superior
performances in exchange coupling systems, such as spring magnets and magnetic
media. JiangAPL2004 ; VictoraIEEE2005 A straightforward approach to achieve
magnetically graded interfaces is to make structurally graded interfaces.
ChoiAPL2007 ; GollJAP2008 ; KirbyPRB2010 The micromagnetic properties could
be predicted from the composition. However, it is challenging if multiple
phases coexist even when the micromagnetic properties of each individual phase
are known, due to lacking of prior knowledge of the interphase coupling. As
shown above, the intrinsic magnetic properties in the intermixed Fe/Sm-Co
interface deviates from the results predicted by the mixture model. Therefore,
a complex correlation between the magnetic properties and the nominal
composition may exist at the chemically graded interfaces.
In summary, we determined the intrinsic magnetic properties of a Fe/Sm-Co
bilayer under the nearly optimized fabrication condition for spring magnets.
We determined that the intermixed interface between the Fe and Sm-Co layers
extends $\sim 8$ nm, where the magnetic properties changes gradually, as
expected. We compared the magnetic properties with the prediction of a mixture
model and found, in the intermixed region, the saturation magnetization is
slightly lower than that estimated from the model, but the exchange stiffness
is higher. This observation indicates that the intermixed interface is
efficient for magnetically coupling the Fe and Sm-Co layers but at the cost of
the average magnetization. The intrinsic anisotropy is also lower than the
value from the model. Overall, the intrinsic magnetic properties in the
structurally intermixed region may not be predicted correctly by the mixture
model of nominal compositions, which is worth keeping in mind when designing
the magnetically graded interfaces.
###### Acknowledgements.
We thank Gian P. Felcher for helpful discussions. Research at Argonne was
supported by the U.S. Department of Energy, Office of Basic Energy Sciences,
Division of Materials Sciences and Engineering under Award No.DE-
AC02-06CH11357. Research at Oak Ridge National Laboratory’s Spallation Neutron
Source was sponsored by the Scientific User Facilities Division, Office of
Basic Energy Sciences, U. S. Department of Energy.
## Appendix A Zeeman effect and off-specular scattering
The Zeeman effect in PNR was firstly clarified by Felcher _et. al_.
FelcherNature1995 ; FelcherPB1996 The effect shows up in the SF
reflectivities when the difference of the Zeeman energy for spin-up and spin-
down neutrons is not negligible in a sufficiently high magnetic field. Let
$k_{\perp}$ and $k_{//}$ label the perpendicular and the parallel components
of the wavevector with respect to the sample surface in the _vacuum_ region,
_i.e._ , before neutrons enter the magnetic-field region of interest.
$k_{\perp}=2\pi sin\theta/\lambda\ll k_{//}$ at glancing angles. There is
negligible energy exchange between the sample and the neutrons for elastic
scattering, therefore the Zeeman energy change after SF scattering accompanies
a change of the kinetic energy, which is associated with $\vec{k}$. The energy
change after spin-flip is twice the Zeeman energy. If the sample is
magnetically homogenous in the film plane as seen by the neutrons, the energy
change is totally associated with $k_{\perp}$. Since $k_{\perp}$’s are small
quantities, the outgoing angles of the spin-flipped neutrons are different
from the incoming angles. Therefore the SF reflections appear at off-specular
positions, which have a characteristic field dependence. FelcherNature1995 ;
FelcherPB1996 The Zeeman effect on the SF reflections becomes clear in a
magnetic field on $\sim 0.1$ T or higher. Figure 6 shows an example. It is
clear that the locus of the SF reflections deviates away from the specular
position, but are described by the prediction after considering the Zeeman
effect. Beside the reflection due to the Zeeman effect, there is no noticeable
off-specular scattering in the SF channels; therefore, the scattering from the
in-plane magnetic domains are not considered during the data analysis.
Consequently, the momentum transfer $Q_{z}$ are no longer equal to
$2k_{\perp}$ for spin-flipped neutrons, but follows
$\begin{array}[]{rcl}Q_{z}^{+-}&=&k_{\perp}+\sqrt{k_{\perp}^{2}+2CB}\\\
Q_{z}^{-+}&=&k_{\perp}+\sqrt{k_{\perp}^{2}-2CB},\end{array}$ (1)
where $C=|2m_{n}\mu_{n}/\hbar^{2}|=2.906\times 10^{-5}$ Å-2/T and $B=\mu_{0}H$
is the applied magnetic field. $R^{-+}$ is forbidden when
$k_{\perp}<\sqrt{2CB}$ because there is no enough kinetic energy to compensate
the Zeeman energy change. vandeKruijsPB2000 At the same time, the minimum
$Q_{z}^{+-}$ is also $\sqrt{2CB}$ for finite reflectivity. Therefore, there is
a cutoff $Q_{z}$ for non-zero SF scattering, $Q_{SFcutoff}=\sqrt{2CB}$.
Figure 6: (Color online) The coutour map of the neutron intensities measured
with the polarization analysis, which are presented as functions of the
neutron wavelengths $\lambda$. The data were collected from a Fe/Sm-Co spring
magnet sample at an incident angle $\theta_{i}=0.24^{o}$ in an external field
of +0.39 T after a negative saturation. The dashed lines indicate the specular
reflection positions without considering the Zeeman effect, i.e.
$\theta_{f}=\theta_{i}$. The solid lines in the SF channels indicate the off-
specular scattering following the the Zeeman effect, i.e.
$\sin^{2}\theta_{f}=\sin^{2}\theta_{i}\pm\frac{cB\lambda^{2}}{2\pi^{2}}$.
Actually, the kinetic energy also changes when neutrons enter and leave the
magnetic-field region of interest, but the change of $\vec{k}$ is essentially
in $k_{//}$ in these cases, Chatterji2006 which changes the time-of-flight.
However, the relative change of $k_{//}$ is so small in laboratory fields, and
TOF only changes $\sim 10$ ns, while the width of the neutron pulse is $\sim
0.2$ ms. Therefore, it does not result in any observable affect.
SF reflection is typically weak when the Zeeman effect matters, therefore the
Zeeman effect is usually not considered during data reduction and analysis.
Reflpak ; Fitz2005 However, we observed significant SF scattering at $+0.39$
T, which is sufficiently high so that the Zeeman effect needs to be
considered. We adopted the generalized algorithm of the GEPORE Chatterji2006
for the PNR simulations. $\vec{B}$ rather than $\vec{M}$ is used to calculate
the magnetic scattering potential since $H$ is now comparable to $M$ and the
spin eigenstates are aligned along $\vec{B}$ rather than $\vec{M}$.
Considering the problem along the normal direction of the sample surface (let
it be $\hat{z}$), it is reduced to a pair of coupled 1D differential wave
equations, Chatterji2006
$\begin{array}[]{rcl}\left[-\frac{\hbar^{2}}{2m}k_{0}^{2}+V_{++}(z)-E\right]\Psi_{+}(z)+V_{+-}\Psi_{-}(z)&=&0\\\
\left[-\frac{\hbar^{2}}{2m}k_{0}^{2}+V_{--}(z)-E\right]\Psi_{-}(z)+V_{-+}\Psi_{+}(z)&=&0.\end{array}$
(2)
$V$ is the potential operator. $\Psi_{\pm}(z)$’s are the wave functions for
spin-up and spin down states. $k_{0}$ is the wavevector in vacuum, which can
be either a real number or a pure imaginary number. States with
${k_{0}}^{2}\pm CB<0$ correspond to evanescent waves outside the sample. Due
to the Zeeman energy, the coupled eigenstates of $\Psi_{+}$ and $\Psi_{-}$
have different wavevectors outside the sample, namely $\sqrt{{k_{0}}^{2}-CB}$
and $\sqrt{{k_{0}}^{2}+CB}$, which has been considered to set up the boundary
conditions at the sample surface in the modified algorithm to take account the
Zeeman effect.
## References
* (1) E. Kneller and R. Hawig, Magnetics, IEEE Transactions on 27, 3588 (Jul. 1991)
* (2) R. Victora and X. Shen, Magnetics, IEEE Transactions on 41, 2828 (oct. 2005), ISSN 0018-9464
* (3) J. S. Jiang, J. E. Pearson, Z. Y. Liu, B. Kabius, S. Trasobares, D. J. Miller, S. D. Bader, D. R. Lee, D. Haskel, G. Srajer, and J. P. Liu, Appl. Phys. Lett. 85, 5293 (2004)
* (4) D. Suess, Appl. Phys. Lett. 89, 113105 (2006)
* (5) Y. Choi, J. S. Jiang, J. E. Pearson, S. D. Bader, and J. P. Liu, Appl. Phys. Lett. 91, 022502 (2007)
* (6) D. Goll, A. Breitling, L. Gu, P. A. van Aken, and W. Sigle, J. Appl. Phys. 104, 083903 (2008)
* (7) B. J. Kirby, J. E. Davies, K. Liu, S. M. Watson, G. T. Zimanyi, R. D. Shull, P. A. Kienzle, and J. A. Borchers, Phys. Rev. B 81, 100405 (2010)
* (8) Y. Choi, J. S. Jiang, Y. Ding, R. A. Rosenberg, J. E. Pearson, S. D. Bader, A. Zambano, M. Murakami, I. Takeuchi, Z. L. Wang, and J. P. Liu, Phys. Rev. B 75, 104432 (2007)
* (9) Y. Liu, Y. Q. Wu, M. J. Kramer, Y. Choi, J. S. Jiang, Z. L. Wang, and J. P. Liu, Appl. Phys. Lett. 93, 192502 (2008)
* (10) E. E. Fullerton, J. S. Jiang, M. Grimsditch, C. H. Sowers, and S. D. Bader, Phys. Rev. B 58, 12193 (1998)
* (11) M. R. Fitzsimmons and C. Majkrzak, in _Modern Techniques for Characterizing Magnetic Materials_, edited by Y. Zhu (Springer, US, 2005) pp. 107 – 155
* (12) C. Majkrzak, K. O’Donovan, and N. Berk, in _Neutron Scattering from Magnetic Materials_, edited by T. Chatterji (Elsevier Science, Amsterdam, 2006) pp. 397 – 471, both the REFLPAK Reflpak and the SPIN_FLIP Fitz2005 use the GEPORE’s algorithm to simulate PNR
* (13) K. V. O’Donovan, J. A. Borchers, C. F. Majkrzak, O. Hellwig, and E. E. Fullerton, Phys. Rev. Lett. 88, 067201 (2002)
* (14) M. R. Fitzsimmons, S. Park, K. Dumesnil, C. Dufour, R. Pynn, J. A. Borchers, J. J. Rhyne, and P. Mangin, Phys. Rev. B 73, 134413 (2006)
* (15) M. Benaissa, K. Krishnan, E. Fullerton, and J. Jiang, Magnetics, IEEE Transactions on 34, 1204 (1998)
* (16) V. Lauter, H. Ambaye, R. Goyette, W.-T. H. Lee, and A. Parizzi, Physica B 404, 2543 (2009)
* (17) G. Felcher, S. Adenwalla, V. D. Haan, and A. V. Well, Physica B 221, 494 (1996)
* (18) H. Fredrikze and R. W. E. van de Kruijs, Physica B 297, 143 (2001)
* (19) A. R. Wildes, Neutron News 17, 17 (2006), ISSN 1044-8632
* (20) P. Kienzle, K. V. O’Donovan, J. Ankner, N. Berk, and C. Majkrzak, http://www.ncnr.nist.gov/reflpak, 2000-2006
* (21) A. Nelson, Journal of Applied Crystallography 39, 273 (2006)
* (22) M. M aza, B. Pardo, J. P. Chauvineau, A. Raynal, A. Menelle, and F. Bridou, Phys. Lett. A 223, 145 (1996)
* (23) It was found that the PNR results suggest a larger surface roughness and a thicker top Cr layer than XRR. The thickness difference is about 0.9 nm and is most likely due to the increase of the oxide layer after $\sim 2$ years.
* (24) G. Felcher, S. Adenwalla, V. D. Haan, and A. V. Well, Nature 377, 409 (1995)
* (25) R. Röhlsberger, H. Thomas, K. Schlage, E. Burkel, O. Leupold, and R. Rüffer, Phys. Rev. Lett. 89, 237201 (2002)
* (26) R. Skomski and J. M. D. Coey, Phys. Rev. B 48, 15812 (Dec 1993)
* (27) R. W. E. van de Kruijs, H. Fredrikze, M. T. Rekveldt, A. A. van Well, Y. V. Nikitenko, and V. G. Syromyatnikov, Physica B 283, 189 (2000)
|
arxiv-papers
| 2010-12-22T20:18:14 |
2024-09-04T02:49:15.933981
|
{
"license": "Public Domain",
"authors": "Yaohua Liu, S. G. E. te Velthuis, J. S. Jiang, Y. Choi, S. D. Bader,\n A. A. Parizzi, H. Ambaye, and V. Lauter",
"submitter": "Yaohua Liu",
"url": "https://arxiv.org/abs/1012.5086"
}
|
1012.5293
|
# Phase Estimation with Non-Unitary Interferometers: Information as a Metric
Thomas B. Bahder Aviation and Missile Research, Development, and Engineering
Center,
US Army RDECOM, Redstone Arsenal, AL 35898, U.S.A.
###### Abstract
Determining the phase in one arm of a quantum interferometer is discussed
taking into account the three non-ideal aspects in real experiments: non-
deterministic state preparation, non-unitary state evolution due to losses
during state propagation, and imperfect state detection. A general expression
is written for the probability of a measurement outcome taking into account
these three non-ideal aspects. As an example of applying the formalism, the
classical Fisher information and fidelity (Shannon mutual information between
phase and measurements) are computed for few-photon Fock and N00N states input
into a lossy Mach-Zehnder interferometer. These three non-ideal aspects lead
to qualitative differences in phase estimation, such as a decrease in fidelity
and Fisher information that depends on the true value of the phase.
###### pacs:
PACS number 07.60.Ly, 03.75.Dg, 06.20.Dk, 07.07.Df
## I Introduction
Optical interferometers Hariharan (2003) and matter wave interferometers
Cronin et al. (2009) have been of great interest because of their practical
applications in metrology. Interferometers have been used to measure such
diverse quantities as electric, magnetic, and gravitational fields,
gravitational waves Cronin et al. (2009); Thorne (1980); Caves (1981), and
there are plans to use them to test the theory of general relativity
Dimopoulos et al. (2008). Classical optical interferometers Lefevre (1993)
have been routinely used for sensing rotation in gyroscopic applications based
on the Sagnac effect Sagnac (1913a, b, 1914); Post (1967); Chen et al. (2008)
and experiments with Sagnac interferometers have been done with single-photons
Bertocchi et al. (2006), with Bose-Einstein condensates(BEC) Gupta et al.
(2005); Wang et al. (2005); Tolstikhin et al. (2005), and schemes using
entangled particles have been proposed that are capable of Heisenberg limited
precision measurements that scale as $1/N$, where $N$ is the number of
particles Cooper et al. (2010).
On a more fundamental level, there is interest in interferometers because they
are a vehicle to study the limits of precision of quantum measurements Godun
et al. (2001); Giovannetti et al. (2006); Berry et al. (2009). Perhaps the
simplest generic measurement problem consists of determining the relative
phase shift between two arms of an interferometer from measurements made at
the output ports of the interferometer Combes and Wiseman (2005); Nagata et
al. (2007); Durkin and Dowling (2007); Pezze and Smerzi (2008); Dorner et al.
(2009); Cable and Durkin (2010). This phase shift may be related to a
classical external field incident on a phase shifter in one arm of the
interferometer, in which case the interferometer can be used as a sensor of
the field Bahder and Lopata (2006a). The determination of the phase shift is a
specific example of the more general problem of parameter estimation, whose
goal is to determine one or more parameters from measurements 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).
Recently, there have been experimental demonstrations using entangled states
to estimate the phase shift in one arm of a Mach-Zehnder interferometer
Walther et al. (2004); Mitchell et al. (2004); Nagata et al. (2007); Okamoto
et al. (2008). Even more recently, the effect of losses on phase determination
was studied experimentally Kacprowicz et al. (2010). In real experiments,
there are three non-ideal elements of the interferometer system: state
preparation Mitchell et al. (2004); Thomas-Peter et al. (2009), photon losses
in the interferometer Kacprowicz et al. (2010) and non-ideal photon-number
detection Nagata et al. (2007); Okamoto et al. (2008). Phase estimation has
been theoretically investigated by separately taking into account non-
deterministic state preparation Helstrom (1967, 1976); Holevo (1982);
Braunstein and Caves (1994); Braunstein et al. (1996); Barndorff-Nielsen and
Gill (2000); Barndorff-Nielsen et al. (2003), photon losses in the
interferometer itself Kim et al. (1998); Durkin et al. (2004); Rubin and
Kaushik (2007); Gilbert et al. (2008); Dorner et al. (2009); Demkowicz-
Dobrzanski et al. (2009); Cable and Durkin (2010); Ono and Hofmann (20010) and
photon-number counting efficiency Okamoto et al. (2008); D Ariano et al.
(2000); Cable and Durkin (2010).
In this work, I write down a formalism that simultaneously takes into account,
in a unified way, all three of the non-ideal elements in experiments: non-
deterministic state preparation, propagation through a lossy interferometer,
and imperfect state detection. Non-deterministic state preparation must be
described by a density matrix, rather than by a pure state, thereby allowing
for the finite probability of creating states other than intended. When the
optical state is created, it enters the interferometer, where propagation may
be non-ideal because photon absorption and scattering can occur. Finally, when
the optical state leaves the interferometer, it enters the detection system,
which may also be non-ideal: the state registered by the detection system may
not be the true state that entered the detection system.
Much of the previous work was focused on determining the optimum measurements
for determining phase and hence the quantum Fisher information was of primary
interest, because it gives a bound on the variance of the phase associated
with the optimum measurement Helstrom (1967, 1976); Holevo (1982); Braunstein
and Caves (1994); Braunstein et al. (1996); Barndorff-Nielsen and Gill (2000);
D Ariano et al. (2000); Monras (2006); Olivares and Paris (2009); Gaiba and
Paris (2009). In contrast, in this work I look at the information gain from
specific, simple, photon-number counting measurements that can be easily
implemented in the laboratory, and hence the classical Fisher information is
the quantity of interest because it depends on the particular measurement that
is performed.
In Section II, I briefly review the theory of phase determination based on
parameter estimation (Fisher information) and on fidelity (Shannon mutual
information between measurements and phase). In Section III, I introduce an
example of a non-ideal interferometer system, where state evolution is non-
unitary. I write a statistical expression for the probability of measurement
outcomes that takes into account the three non-ideal components of the
interferometer system described above. I use this probability in the classical
Fisher information in Eq. (2) and in the fidelity in Eq. (9) to analyze the
determination of phase in a non-ideal interferometer system. As simple
examples of the formalism, in sub-sections of Section III, I look at few
photon examples of non-deterministic state preparation, propagation through a
non-unitary (lossy) interferometer, and imperfect state detection. Finally, in
Section IV, I make some concluding remarks. My goal is to look at examples of
few-photon states that can be implemented experimentally, with the hope that
the examples and method described here can be helpful for analyzing real
experiments. Furthermore, in this work, I restrict myself to the simple case
of non-adaptive measurements Higgins et al. (2009), where the measurement is
fixed before phase estimation.
## II Theoretical background
The accuracy of estimating a (single) one-dimensional parameter, $\phi$, is
described in terms of the classical Cramer-Rao bound Cover and Thomas (2006),
which gives a lower bound on the variance $(\delta\phi)^{2}$ of an unbiased
estimator of the parameter $\phi$:
$\left({\delta\phi}\right)^{2}\geq\frac{1}{F_{cl}(\phi;M)}$ (1)
where $F_{cl}(\phi;M)$ is the classical Fisher information given by Cramér
(1958); Cover and Thomas (2006)
$F_{cl}(\phi;M)=\sum\limits_{\xi}{\frac{1}{{P(\xi|\phi,\rho)}}\,\left[{\frac{{\partial
P(\xi|\phi,\rho)}}{{\partial\phi}}}\right]^{2}}$ (2)
The classical Fisher information is described in terms of the conditional
probability distribution, $P(\xi|\phi,\rho)$, for measurement outcome, $\xi$,
which can take one or more continuous values, or, one or more discreet values.
If $\xi$ takes continuous values, the sum over $\xi$ in Eq. (2) is an
integral. For the case of quantum measurements, these probabilities are given
by
$P(\xi|\phi,\rho)={\rm{tr}}\left({\hat{\rho}_{\phi}\,\hat{\Pi}_{o}\left(\xi\right)}\right)={\rm{tr}}\left({\hat{\rho}_{o}\,\hat{\Pi}_{\phi}\left(\xi\right)}\right)$
(3)
where the state is specified by the Schrödinger picture density matrix,
$\hat{\rho}_{\phi}$, and the measurements by the positive-operator valued
measure (POVM) in the Schrödinger picture, $\hat{\Pi}_{o}(\xi)$. The POVM are
set a of non-negative Hermitian operators, $M=\\{\hat{\Pi}(\xi)\\}$,
representing a given physical measurement and so the expectation value of each
operator $\hat{\Pi}_{o}(\xi)$ is non-negative, and satisfies
$\sum\limits_{\xi}{\hat{\Pi}_{o}(\xi)}=\hat{I}$, where $\hat{I}$ is the
identity operator. Alternatively, in the Heisenberg picture, the state is
given by the density matrix, $\hat{\rho}_{o}$, and probabilities of
measurements by POVM, $\hat{\Pi}_{\phi}(\xi)$. Operators in the Schrodinger
and Heisenberg pictures, $\hat{O}_{S}$ and $\hat{O}_{H}(\phi)$, respectively
are related by
$\hat{O}_{H}(\phi)=\hat{U}^{\dagger}(\phi)\,\hat{O}_{S}\,\hat{U}(\phi)$, and
$\hat{U}(\phi)=\exp(-i\phi\hat{h})$, where $\hat{h}$ is the infinitesimal
displacement operator for the parameter $\phi$, satisfying
$i\frac{\partial}{\partial\phi}\left|\psi(\phi)\right\rangle=\hat{h}\,\left|\psi(\phi)\right\rangle$
(4)
The quantum Fisher information, $F_{Q}(\phi)$ is obtained by maximizing the
classical Fisher information, $F_{cl}(\phi;M)$, over all possible measurements
$M$, at a given value of $\phi$. Braunstein and Caves Braunstein and Caves
(1994); Braunstein et al. (1996) have shown that an improved lower bound is
possible for the variance, $\left({\delta\phi}\right)^{2}$, in terms of the
quantum Fisher information:
$\left({\delta\phi}\right)^{2}\geq\frac{1}{{F_{cl}(\phi;M)}}\geq\frac{1}{{F_{Q}(\phi)}}$
(5)
where $F_{Q}(\phi)$ is independent of the measurement $M$. The quantum Fisher
information is defined by
$F_{Q}\left(\phi\right)=\rm{tr}\left[\hat{\rho}_{\phi}\hat{\Lambda}_{\phi}^{2}\right]$
(6)
where the Hermitian operator, $\Lambda_{\phi}$, is the symmetric logarithmic
derivative (S.L.D.), defined implicitly by
$\frac{\partial\hat{\rho}_{\phi}}{\partial\phi}=\frac{1}{2}\left[\hat{\Lambda}_{\phi}\,\hat{\rho}_{\phi}+\hat{\rho}_{\phi}\hat{\Lambda}_{\phi}\right]$
(7)
The right side and left side of the inequality in Eq. (5) is sometimes called
the quantum Cramer-Rao bound, see also the work by Helstrom Helstrom (1967,
1976) and Holevo Holevo (1982), and discussion by Barndorff-Nielsen et al.
Barndorff-Nielsen and Gill (2000); Barndorff-Nielsen et al. (2003). The
expression in Eq. (5) provides a bound on the variance of an unbiased
estimator for an optimum measurement. However, the theory does not give a
procedure for determining the optimum measurement. For one-dimensional
parameter estimation and for simple (non-adaptive) measurements, Barndorff-
Nielsen and Gill have shown that in general the optimum measurement $M$ will
depend on the parameter $\phi$, which is unknown prior to estimation
Barndorff-Nielsen and Gill (2000). Consequently, Barndorff-Nielsen and Gill
have proposed a two-stage adaptive measurement procedure that will give
$F_{cl}(\phi;M)=F_{Q}\left(\phi\right)$ (8)
for optimum measurement $M$ for all $\phi$.
For the case of a pure state, $\left|{\psi_{o}}\right\rangle$, where the
density matrix is
$\rho_{o}=\left|{\psi_{o}}\right\rangle\,\left\langle{\psi_{o}}\right|$, and
where the path is generated by a unitary transformation, $\hat{U}(\phi)$, the
quantum Fisher information, $F_{Q}(\phi)$, does not depend on $\phi$
Braunstein et al. (1996); Olivares and Paris (2009); Gaiba and Paris (2009),
and is given by the fluctuations of the generator $\hat{h}$ by
$F_{Q}\left(\phi\right)=4(\Delta h)^{2}$. Furthermore, Hofman has shown that
for pure states having a path symmetry Hofmann (2009), the quantum Cramer-Rao
bound in Eq.(5) can be achieved at any value of phase $\phi$. The condition
for optimal measurements for the case of pure states has also been
investigated Durkin (2010).
In Section III below, I consider the case of a lossy Mach-Zehnder
interferometer, where the state evolution is effectively non-unitary, thereby
leading to a classical Fisher information that depends on the true value of
the phase $\phi$.
The above discussion of parameter estimation is based on classical and quantum
Fisher informations, which are local descriptions of phase estimation, because
they depend on the true value of $\phi$. Complementary to the above local
descriptions, is a global description given by the fidelity Bahder and Lopata
(2006a):
$\displaystyle H(M)$ $\displaystyle=$
$\displaystyle\sum_{\xi}\int_{-\pi}^{+\pi}d\,\phi\,\,P(\xi|\phi,\rho)\,p(\phi)\,\,\times\,$
(9)
$\displaystyle\log_{2}\left[\frac{P(\xi|\phi,\rho)\,}{\int_{-\pi}^{+\pi}\,\,\,P(\xi|\phi^{\prime},\rho)\,p(\phi^{\prime})\,\,d\,\phi^{\prime}}\right].$
where $P(\xi|\phi,\rho)$ is given in Eq. (3). The fidelity, $H(M)$, is the
Shannon mutual information Shannon (1948); Cover and Thomas (2006) between the
measurement $M$ and the unknown parameter $\phi$. The fidelity, $H(M)$, gives
the average amount of information (in bits) about the parameter $\phi$ that
can be obtained from the measurement $M$ for one use (one measurement cycle)
of the interferometer. The fidelity does not depend on $\phi$ because it is an
average over all possible phases $\phi$ and over all probabilities of
measurement outcomes for a given POVM $M$. (For an alternative discussion of
local versus global phase estimation, see Ref.Durkin and Dowling (2007).) The
fidelity also depends on the prior information about the parameter $\phi$
through the prior probability distribution $p(\phi)$. Consequently, the
fidelity characterizes the quality of the interferometer system as a whole, in
terms of mutual information between the measurement $M$ and the parameter
$\phi$. Note that the fidelity depends on the input state density matrix,
$\hat{\rho}$, and the measurement $M$, and therefore can be used to optimize
the system with respect to the input state and measurement. The fidelity is a
measure of the information that flows from the phase $\phi$ to the
measurements, which is analogous to a communication problem where Alice sends
messages to Bob. In the case of the measurement problem, quantum fluctuations
in the initial state, in the channel (interferometer), and the type of
measurement, determine the amount of information that is obtained about the
parameter $\phi$ from the measurements. The fidelity has been applied to
compare the use of Fock states and N00N states when no prior information is
present about the phase Bahder and Lopata (2006a) and when there is
significant prior information about the phase Bahder and Lopata (2006b).
The complimentary measures of fidelity and Fisher information may be
contrasted as follows. Assume that I want to shop to purchase the best
measurement device to determine the unknown parameter $\phi$. If I do not know
the true value of the parameter $\phi$, I would compare the overall
performance specifications of several devices and I would purchase the device
with the best over-all specifications for measuring $\phi$. The fidelity,
$H(M)$, is the over-all specification for the quality of the device, so I
would purchase the device with the largest fidelity. After I have purchased
the device, I want to use it to determine a specific value of the parameter
$\phi$ based on several measurements (data). This involves parameter
estimation, which requires the use of Fisher information, and depends on the
true value of the parameter $\phi$.
Historically, the variance, $(\delta\phi)^{2}$, of the estimated parameter
$\phi$ has been discussed in terms of the standard quantum limit,
$\delta\phi_{SL}=$ $1/\sqrt{N}$, and the Heisenberg limitCaves (1981); Ou
(1997); Giovannetti et al. (2004, 2006), $\delta\phi_{HL}=$ $1/N$, where $N$
is the number of particles or quanta that enter the interferometer during each
measurement cycle. The value $\delta\phi$ is presumably the width of some
probability distribution, $p(\phi|\xi,\rho)$, such as the distribution
calculated from Bayes’ rule, see Eq. (25). Detailed calculation of
$p(\phi|\xi,\rho)$ for a number of input states shows that these distributions
have multiple peaks Bahder and Lopata (2006a). Consequently, rather than using
the widths of these distributions as a metric for determining $\phi$, I use
the information measures, Fisher information and fidelity, which naturally
handle distributions with multiple peaks.
Figure 1: (Color) Interferometer system shown with three components: state
preparation, interferometer, and detection system.
## III Non-Ideal Optical System
As described in the introduction, an interferometer system can be divided into
three parts: state creation, state evolution through the optical
interferometer, and detection of the output state. In a real experiment, each
of these three parts can be non-ideal, see Fig. 1. For example, I may want to
create a quantum state $|\psi^{in}\rangle$ as input into the interferometer.
However, instead, the resulting state may be a mixture of states, each with
some probability, $P_{S}(\psi_{k}^{in})$, for $k=1,2,\cdots$. Such a quantum
state is described by the density matrix $\hat{\rho}$:
$\hat{\rho}=\sum\limits_{k}{\,\;P_{S}(\psi_{k}^{in})}\;\left|{\psi_{k}^{in}}\right\rangle\left\langle{\psi_{k}^{in}}\right|$
(10)
The state $\hat{\rho}$ is then input into the interferometer, where there may
be absorption and scattering of photons. For example, a two-photon state may
enter the interferometer and a one-photon state may exit the interferometer,
because one photon was absorbed inside the interferometer. Alternatively, a
two-photon state may enter the interferometer and a three-photon state may
exit the interferometer, due to light scattering into the interferometer from
the environment. I can describe these processes generally by a transfer
matrix, ${P_{I}(\psi_{j}^{out}|\psi_{k}^{in},\phi)}$, which gives the
conditional probability for state $\left|{\psi_{j}^{out}}\right\rangle$ to
exit the interferometer given that state $\left|{\psi_{k}^{in}}\right\rangle$
entered the interferometer. The transfer matrix,
${P_{I}(\psi_{j}^{out}|\psi_{k}^{in},\phi)}$, is general enough to describe
non-unitary propagation of the quantum state through the interferometer, and
so can take into account losses and scattering. Note that the transfer matrix
may depend on the state of the interferometer, which I specify here by single
parameter $\phi$. Finally, the detection of the quantum state that leaves the
interferometer can be non-ideal. For example, the detection system may
register a measurement $\xi$, when state $\left|{\psi_{i}^{out}}\right\rangle$
enters the detection system, whereas the true state that entered the detection
system was $\left|{\psi_{j}^{out}}\right\rangle$. I can represent such an
imperfect detection system by the conditional probability
$P_{D}(\xi|\psi_{j}^{out},\phi)$, which gives the probability for making a
measurement $\xi$ when state $\psi_{j}^{out}$ entered the detection system.
Note that in general this probability may or may not depend on $\phi$, a
parameter describing the state of the interferometer. For a non-ideal
interferometer system, the probability of obtaining a measurement $\xi$ is
given by Jaynes (2009)
$\small
P(\xi|\phi)=\sum\limits_{j}{P_{D}}(\xi|\psi_{j}^{out},\phi)\;\sum\limits_{k}{\,P_{I}(\psi_{j}^{out}|\psi_{k}^{in},\phi)\;P_{S}(\psi_{k}^{in})}$
(11)
where we must have each of the three probabilities sum to unity:
$\sum\limits_{k}{\,P_{S}(\psi_{k}^{in})}=1$ (12)
$\sum\limits_{j}{\,P_{I}(\psi_{j}^{out}|\psi_{k}^{in},\phi)}=1$ (13)
$\sum\limits_{\xi}{P_{D}(\xi|\psi_{j}^{out},\phi)}=1$ (14)
Equation (11) is a general statistical relation for the probability of
obtaining a measurement outcome $\xi$ for given phase shift $\phi$, taking
into account the three non-ideal aspects of interferometer systems. Note that
Eq. (11) is sufficiently general that it can be applied to the case where
states are represented by density matrices. In this case, in Eq. (11) we can
make the replacements $\psi_{k}^{in}\rightarrow\rho_{k}^{in}$ and
$\psi_{j}^{out}\rightarrow\rho_{j}^{out}$, where $\rho_{k}^{in}$ and
$\rho_{j}^{out}$ are a set of input and output density matrices labeled by
integers $j,k=1,2,\cdots$. In order to compute the probabilities of
measurement outcomes, $P(\xi|\phi)$, Eq. (11) must be augmented by a detailed
model of input and output states. I give several examples of applying Eq. (11)
in the sections that follow.
The probability of measurement outcome, given by Eq. (11), enters into the
Fisher information and into the Shannon mutual information, in Eq. (2) and Eq.
(9), respectively. In the next three subsections, A, B, and C, I give examples
of the effects of non-deterministic state preparation, state evolution through
an interferometer when absorption is present, and imperfect output state
detection, respectively, using Fisher and Shannon mutual informations as
metrics of performance of the interferometer.
### III.1 Non-Deterministic State Preparation
Consider an optical interferometer system that has non-deterministic state
preparation, but has no losses in the interferometer and has perfect state
detection. When I try to prepare a certain quantum state for input into the
interferometer, there is always a non-zero probability that another state than
intended will be prepared. This non-deterministic state preparation is
expressed by a density matrix for the input state, which assigns probabilities
for creating various quantum states, see Eq. (10). Since state detection is
assumed perfect, $P_{D}(\xi|\psi^{out},\phi)=1$ when the measurement $\xi$
corresponds to the true state that entered the detection system, $\psi^{out}$,
and otherwise $P_{D}(\xi|\psi^{out},\phi)=0$.
A general interferometer with no losses is characterized by a unitary
scattering matrix, $S_{ij}(\phi)$, that connects the $N_{p}$ input-mode field
operators, $\hat{\alpha}_{i}$, to the $N_{p}$ output field operators,
$\hat{\beta}_{i}$:
$\hat{\beta}_{i}=\sum_{j=1}^{N_{p}}S_{ij}(\phi)\,\hat{\alpha}_{j}=\hat{U}^{\dagger}(\phi)\hat{\alpha}_{i}\hat{U}(\phi)$
(15)
where $\hat{U}(\phi)$ is a unitary evolution operator, $i,j=1,2,\cdots,N_{p}$,
and $\phi$ is one or more parameters (e.g., phase shift) that describe the
state of the interferometer.
For simplicity, I consider a Mach-Zehnder interferometer, with no losses, with
input ports labeled, “a” and “b”, and output ports, “c” and “d”, having a
scattering matrix
$S_{ij}(\phi)=\frac{1}{2}\left(\begin{array}[]{cc}-i\left(1+e^{i\phi}\right)&\left(-1+e^{i\phi}\right)\\\
\left(-1+e^{i\phi}\right)&i\left(1+e^{i\phi}\right)\end{array}\right)$ (16)
where $\hat{\alpha}_{i}=\\{\hat{a},\hat{b}\\}$ and
$\hat{\beta}_{i}=\\{\hat{c},\hat{d}\\}$.
The probabilities, $P_{I}(\psi_{j}^{out}|\psi_{k}^{in},\phi)$, that relate the
input state $\psi_{k}^{in}$ to the output state $\psi_{j}^{out}$ of the
interferometer are given in terms of the projection operators
$\hat{\Pi}_{\phi}\left({n_{c},n_{d}}\right)$:
$P_{I}(\psi_{j}^{out}|\psi_{k}^{in},\phi)=\left\langle{\psi_{k}^{in}}\right|\,\hat{\Pi}_{\phi}\left({n_{c},n_{d}}\right)\,\left|{\psi_{k}^{in}}\right\rangle$
(17)
where the output state $\psi_{j}^{out}$ is specified by two integers,
$\left\\{n_{c},n_{d}\right\\}$, giving the photon numbers output in ports “c”
and “d”. In terms of the unitary evolution operator, $\hat{U}(\phi)$, the
output state in the Schrödinger picture is
$\left|\psi^{out}(\phi)\right\rangle=\hat{U}(\phi)\left|\psi^{in}\right\rangle$,
where $\left|\psi^{in}\right\rangle$ is the Heisenberg picture input state. In
Eq. (11), the sum over $k$ is now a double sum over all non-negative values of
the two integers $n_{c}$ and $n_{d}$. For a generic Mach-Zehnder
interferometer, with input ports “a” and “b”, and output ports “c” and “d”,
the projective operators are Bahder and Lopata (2006a)
$\hat{\Pi}_{\phi}\left({n_{c},n_{d}}\right)=\frac{1}{{n_{c}!\,n_{d}!}}\,\left({\hat{c}^{\dagger}}\right)^{n_{c}}\,({\hat{d}^{\dagger}})^{n_{d}}\,\,\left|0\right\rangle\,\left\langle
0\right|\,\,\left({\hat{c}}\right)^{n_{c}}\,({\hat{d}})^{n_{d}}$ (18)
where the vacuum state
$\left|0\right\rangle=\left|0\right\rangle_{a}\,\otimes\,\left|0\right\rangle_{b}$.
As an example, consider the simplest case of input given by a mixed state
represented by the density matrix
$\hat{\rho}=P_{0}\,\left|0\right\rangle\left\langle
0\right|+P_{1}\,\left|{10}\right\rangle\left\langle{10}\right|$ (19)
where $P_{1}$ is the probability for 1-photon input into port “a” and vacuum
input into port “b”, and $P_{0}$ is the probability of vacuum input into both
ports “a” and “b”, and $P_{0}+P_{1}=1$. In Eqs. (10) and (11),
$P_{S}(\psi^{in})=P_{o}$ when
$\left|\psi^{in}\right\rangle=\left|0\right\rangle$ and
$P_{S}(\psi^{in})=P_{1}$ when
$\left|\psi^{in}\right\rangle=\left|10\right\rangle$.
Figure 2: (Color) Lossy Mach-Zehnder interferometer is shown, with input modes
$a_{1}$, $a_{2}$, $v_{1}$, $v_{2}$, and output modes $b_{1}$, $b_{2}$, $d_{1}$
and $d_{2}$. Here modes $v_{1}$ and $v_{2}$ have vacuum input and the output
modes $d_{1}$ and $d_{2}$ are loss channels in each arm.
Equation (17) for the interferometer transfer matrix can then be written as
$P(n_{c},n_{d}|\phi,\rho)={\rm{tr}}\left({\hat{\rho}\,\hat{\Pi}_{\phi}\left({n_{c},n_{d}}\right)}\right)$
(20)
where the input state is represented by the density matrix $\hat{\rho}$. It
seems that there can be only two possible measurement outcomes,
$\xi=\\{n_{c},n_{d}\\}=\\{1,0\\}$ and $\xi=\\{n_{c},n_{d}\\}=\\{0,1\\}$.
However, the probabilities for the two measurement outcomes do not sum to
unity because, $P(10|\phi,\rho)+P(01|\phi,\rho)=P_{1}$. Therefore, there is a
non-zero probability of an inconclusive measurement outcome associated with
the probability $P_{0}$ of vacuum injected into both input ports “a” and “b”.
I introduce an inconclusive measurement operator,
$\hat{\Pi}_{\phi}\left(i\right)$, so that the sum of the three operators is
equal to the identity operator $\hat{I}$:
$\hat{\Pi}_{\phi}\left({10}\right)+\hat{\Pi}_{\phi}\left({01}\right)+\hat{\Pi}_{\phi}\left(i\right)=\hat{I}$
(21)
Using Eqs.(18)–(21), the probabilities given by Eq. (11), $P(\xi|\phi)$, for
measurement outcomes $\xi$ are given by
$\displaystyle P(10|\phi)$ $\displaystyle=$ $\displaystyle
P_{1}\cos^{2}\left(\frac{\phi}{2}\right)$ (22) $\displaystyle P(01|\phi)$
$\displaystyle=$ $\displaystyle P_{1}\sin^{2}\left(\frac{\phi}{2}\right)$ (23)
$\displaystyle P(i|\phi)$ $\displaystyle=$ $\displaystyle 1-P_{1}$ (24)
where $P(i|\phi)$ is the probability for an inconclusive measurement outcome.
Using Bayes’ rule and Eq. (11), we can write the conditional probability
distributions for the phase, $p(\phi|\xi)$, given measurement outcome, $\xi$,
as
$p(\phi|\xi)=\frac{{P(\xi|\phi)\,p(\phi)}}{{\int\limits_{-\pi}^{+\pi}{P(\xi|\phi^{\prime})\,p(\phi^{\prime})\,d\phi^{\prime}}}}$
(25)
where $p(\phi)$ is the prior probability distribution specifying our prior
information about the phase $\phi$. Assuming no prior information about the
phase, $p(\phi)=1/(2\pi)$, and using Bayes’ rule in Eq. (25), the conditional
probability distributions for the phase for a given measurement outcome are
given by
$\displaystyle p(\phi|10)$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\cos^{2}\left(\frac{\phi}{2}\right)$ (26)
$\displaystyle p(\phi|01)$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\sin^{2}\left(\frac{\phi}{2}\right)$ (27)
$\displaystyle p(\phi|i)$ $\displaystyle=$ $\displaystyle\frac{1}{{2\pi}}$
(28)
where $\xi=(n_{c},n_{d})$. As we would expect, for an inconclusive measurement
outcome the phase probability distribution, $p(\phi|i)$, is flat since an
inconclusive measurement result cannot be used to distinguish different values
of the phase. Note that the probabilities for the three measurement outcomes
in Eq. (26)–(28) sum to unity. When $P_{0}=0$, or equivalently $P_{1}=1$,
state preparation is deterministic, and the probability for an inconclusive
measurement outcome is zero. In this case, the probabilities for measurement
outcomes $(10)$ and $(01)$ reduce to the values for the case of a single-
photon input state created with probability unity. Note that this single
photon input case corresponds to the case of a classical interferometer fed by
a laser in one input port. Larger photon-number input states have measurement
outcome probabilities that differ from the probabilities for a classical
interferometer.
For the input state in Eq. (19), the classical Fisher information, defined in
Eq. (2), is given by
$F(\phi)=P_{1}$ (29)
where I dropped the subscript ${cl}$ on the classical Fisher information, a
convention that I follow in rest of this work. According to the Cramer-Rao
bound in Eq. (1), when the probability of creating a single photon approaches
zero, $P_{1}\rightarrow 0$, the variance $(\delta\phi)^{2}$ becomes
arbitrarily large, because the probability for inconclusive measurement
outcomes approaches unity.
Assuming no prior information about the phase, therefore taking
$p(\phi)=1/(2\pi)$, the fidelity (Shannon mutual information) of the system
defined in Eq. (9) is given by
$H(M)=P_{1}\,\left({\frac{1}{{\ln 2}}-1}\right)$ (30)
Similar to the Fisher information, the fidelity $H(M)$ also approaches zero
when the probability $P_{1}$ of having one photon in the input of each shot
approaches zero. The fidelity is the amount of information (in bits) that is
gained on average about the phase from a single use of the interferometer,
averaged over all possible phase values $\phi$.
### III.2 Lossy Mach-Zehnder Interferometer
Next, I consider an interferometer with absorption losses—so state evolution
is non-unitary. I assume that state preparation is deterministic (ideal) and
that state detection is perfect (no errors). Equation (11) is general enough
to describe processes other than losses in the interferometer, such as photons
scattering into the interferometer from the environment, in which case there
are more photons leaving the output ports than entering the input ports.
However, in what follows, I restrict myself to simple absorption in the
interferometer. I model losses in each arm of a Mach-Zehnder interferometer by
inserting two beam splitters, $S_{3}$ and $S_{4}$, one in each path, see Fig.
2. While a lossless Mach-Zehnder interferometer has two input and two output
ports, a general lossy Mach-Zehnder interferometer can be represented by four
input and four output ports, see Fig. 2. I label the input modes as $a_{1}$,
$a_{2}$, $v_{1}$, and $v_{2}$, where $v_{1}$, and $v_{2}$ have vacuum input
and I label the output modes as $b_{1}$, $b_{2}$, $d_{1}$, and $d_{2}$, where
$d_{1}$, and $d_{2}$ are the modes where probability amplitude is
“dissipated”. I take the phase shifts at the two mirrors, $M_{1}$ and $M_{2}$
to be equal to $\pi$. Furthermore, I assume that the interferometer is
balanced, so that path lengths satisfy,
$\displaystyle L$ $\displaystyle=$ $\displaystyle
l_{1}+l_{3}+l_{5}=l_{2}+l_{4}+l_{6}$ $\displaystyle l$ $\displaystyle=$
$\displaystyle l_{1}+l_{3}=l_{2}+l_{4}$ (31)
see Fig. 2. A calculation gives the input and output modes related by the
4$\times$4 unitary scattering matrix $S_{ij}(\phi)$
$\left({\begin{array}[]{*{20}c}{b_{1}}\\\ {b_{2}}\\\ {d_{1}}\\\ {d_{2}}\\\
\end{array}}\right)=\left[{\begin{array}[]{*{20}c}{}\hfil&{}\hfil&{}\hfil\\\
{}\hfil&{S_{ij}\left(\phi\right)}&{}\hfil\\\ {}\hfil&{}\hfil&{}\hfil\\\
\end{array}}\right]\cdot\left({\begin{array}[]{*{20}c}{a_{1}}\\\ {a_{2}}\\\
{v_{1}}\\\ {v_{1}}\\\ \end{array}}\right)$ (32)
where the phase-dependent scattering matrix is given by
$S_{ij}(\phi)=\left[{\begin{array}[]{*{20}c}{\frac{i}{2}e^{i\frac{{L\omega}}{c}}\left({\sqrt{1-r_{y}^{2}}-e^{i\phi}\sqrt{1-r_{x}^{2}}}\right)}&{-\frac{1}{2}e^{i\frac{{L\omega}}{c}}\left({\sqrt{1-r_{y}^{2}}+e^{i\phi}\sqrt{1-r_{x}^{2}}}\right)}&{\frac{i}{{\sqrt{2}}}r_{x}\,e^{i\left({L-l}\right)\frac{\omega}{c}}}&{\frac{1}{{\sqrt{2}}}r_{y}\,e^{i\left({L-l}\right)\frac{\omega}{c}}}\\\
{-\frac{1}{2}e^{i\frac{{L\omega}}{c}}\left({\sqrt{1-r_{y}^{2}}+e^{i\phi}\sqrt{1-r_{x}^{2}}}\right)}&{-\frac{i}{2}e^{i\frac{{L\omega}}{c}}\left({\sqrt{1-r_{y}^{2}}-e^{i\phi}\sqrt{1-r_{x}^{2}}}\right)}&{\frac{1}{{\sqrt{2}}}r_{x}\,e^{i\left({L-l}\right)\frac{\omega}{c}}}&{\frac{i}{{\sqrt{2}}}r_{y}\,e^{i\left({L-l}\right)\frac{\omega}{c}}}\\\
{-\frac{i}{{\sqrt{2}}}r_{x}\,e^{i\left({\phi+\frac{{l\omega}}{c}}\right)}}&{-\frac{1}{{\sqrt{2}}}r_{x}\,e^{i\left({\phi+\frac{{l\omega}}{c}}\right)}}&{-i\sqrt{1-r_{x}^{2}}}&0\\\
{-\frac{1}{{\sqrt{2}}}r_{y}\,e^{i\frac{{l\omega}}{c}}}&{-\frac{i}{{\sqrt{2}}}r_{y}\,e^{i\frac{{l\omega}}{c}}}&0&{-i\sqrt{1-r_{y}^{2}}}\\\
\end{array}}\right]$ (33)
It is easy to check that the scattering matrix is unitary, $S^{\dagger}\,S=I$
where $I$ is the 4$\times$4 unit matrix. The parameters, $r_{x}$ and $r_{y}$,
are the reflection amplitudes for beams splitters $S_{3}$ and $S_{4}$,
respectively, and they represent the strength of the loss or dissipation, see
Fig. 2. When the system is considered in terms of two input modes, $a_{1}$ and
$a_{2}$, and two output modes, $b_{1}$ and $b_{2}$, the evolution of the input
state in not unitary.
The 4$\times$4 unitary scattering matrix $S_{ij}(\phi)$ has some simple
properties. The case when $r_{x}=r_{y}=0$ corresponds to no dissipation. In
this case, the 4$\times$4 S-matrix reduces to two diagonal 2$\times$2 blocks.
The upper left 2$\times$2 block couple modes $a_{1}$ and $a_{2}$ to modes
$b_{1}$ and $b_{2}$, and this 2$\times$2 block (up to a phase) is given by Eq.
(16), which is the scattering matrix for the Mach-Zehnder interferometer with
no losses. For this case of no loss, in Eq. (33) the lower right 2$\times$2
block couples the dissipative modes, $d_{1}$, and $d_{2}$, to the vacuum
modes, $v_{1}$, and $v_{2}$.
The case $r_{x}=r_{y}=1$ corresponds to maximum dissipation, and the
4$\times$4 S-matrix again decouples, into two off-diagonal 2$\times$2 blocks.
The upper right 2$\times$2 block couples the two vacuum modes, $v_{1}$ and
$v_{2}$, to the two output modes, $b_{1}$ and $b_{2}$. The lower left
2$\times$2 block of this S-matrix couples the loss modes, $d_{1}$, and
$d_{2}$, to the input modes $a_{1}$ and $a_{2}$. For this case of maximum
dissipation, the input modes, $a_{1}$ and $a_{2}$, are decoupled from the
output modes, $b_{1}$ and $b_{2}$.
The probabilities for various measurement outcomes $\xi=(n,m)$ are given by
the analog of Eq. (3):
$P(n,m|\phi,\rho)={\rm{tr}}\left({\hat{\rho}_{o}\,\hat{\Pi}_{\phi}(n,m)}\right)$
(34)
where $n$ and $m$ are the number of photons leaving ports $b_{1}$ and $b_{2}$,
respectively. The trace is over the complete space of four direct product Fock
basis states,
${\left|n_{1}\right\rangle_{a_{1}}\otimes\left|n_{2}\right\rangle_{a_{2}}\otimes\left|n_{3}\right\rangle_{v_{1}}\otimes\left|n_{4}\right\rangle_{v_{2}}}$.
The input state density matrix, $\hat{\rho}_{o}$, is defined in terms of a sum
of products of creation operators $\hat{a}_{1}^{\dagger}$,
$\hat{a}_{2}^{\dagger}$, $\hat{v}_{1}^{\dagger}$ and $\hat{v}_{2}^{\dagger}$
acting on the vacuum
$\left|0\right\rangle=\left|0\right\rangle_{a_{1}}\otimes\left|0\right\rangle_{a_{2}}\otimes\left|0\right\rangle_{v_{1}}\otimes\left|0\right\rangle_{v_{2}}$
and has the form
$\hat{\rho}_{0}=\sum\limits_{n,m}{c_{nm}\,\left|{nm00}\right\rangle}\;\left\langle{nm00}\right|$
(35)
where I use the short-hand notation
${\left|{nm00}\right\rangle\equiv\left|n\right\rangle_{a_{1}}\otimes\left|m\right\rangle_{a_{2}}\otimes\left|0\right\rangle_{v_{1}}\otimes\left|0\right\rangle_{v_{2}}}$.
I am using the Heisenberg picture, so the input density matrix
$\hat{\rho}_{o}$ is independent of time (phase), while the operators
$\hat{\Pi}_{\phi}\left({n,m}\right)$ evolve in time (phase) and so they depend
on $\phi$. The projective measurement operators are given by
$\hat{\Pi}_{\phi}\left({n,m}\right)=\frac{1}{{n!\,m!}}\;\sum\limits_{k,l=0}^{\infty}{\frac{1}{{k!\,l!}}\left({\hat{b}_{1}^{\dagger}}\right)^{n}\left({\hat{b}_{2}^{\dagger}}\right)^{m}\left({\hat{d}_{1}^{\dagger}}\right)^{k}\left({\hat{d}_{2}^{\dagger}}\right)^{l}\left|0\right\rangle\left\langle
0\right|\left({\hat{b}_{1}}\right)^{n}\left({\hat{b}_{2}}\right)^{m}\left({\hat{d}_{1}}\right)^{k}\left({\hat{d}_{2}}\right)^{l}}$
(36)
where $\hat{b}_{1}^{\dagger}$, $\hat{b}_{2}^{\dagger}$,
$\hat{d}_{1}^{\dagger}$, and $\hat{d}_{2}^{\dagger}$ are creation operators
for the output modes $b_{1}$, $b_{2}$, $d_{1}$, and $d_{2}$, respectively. The
sums over $k$ and $l$ take into account the probabilities for losing photons
into ports $d_{1}$ and $d_{2}$. I use the short-hand notation for the input
modes $\\{\alpha_{i}\\}=\\{\hat{a}_{1},\hat{a}_{2},\hat{v}_{1},\hat{v}_{2}\\}$
and for the output modes
$\\{\beta_{i}\\}=\\{\hat{b}_{1},\hat{b}_{2},\hat{d}_{1},\hat{d}_{2}\\}$, see
Eq. (15).
Note that, even though dissipation is being modeled, it is easy to check that
there is global photon number conservation,
$\sum\limits_{i=0}^{4}{\hat{\alpha}_{i}^{\dagger}\hat{\alpha}_{i}=}\sum\limits_{i=0}^{4}{\hat{\beta}_{i}^{\dagger}\hat{\beta}_{i}}$
(37)
since the S-matrix in Eq. (33) is unitary. The measurement outcomes,
$\xi=(n,m)$, are specified by two integers, which label the number of photons
that are output at ports $b_{1}$ and $b_{2}$, see Fig. 2. So the probability
of output state, $\psi_{j}^{out}$, as given in Eq.(34), is expressed by the
two integers $n$ and $m$, see also Eq. (17). Note that in general the number
of photons $n+m$ in the output state, $\psi_{j}^{out}$, is not equal to the
number of photons in the input state, $\psi_{in}$, because
$\hat{a}_{1}^{\dagger}\hat{a}_{1}+\hat{a}_{2}^{\dagger}\hat{a}_{2}\neq\hat{b}_{1}^{\dagger}\hat{b}_{1}+\hat{b}_{2}^{\dagger}\hat{b}_{2}$
(38)
However, the sum of probabilities for all possible measurement outcomes is
unity:
$\sum\limits_{m,n=0}^{\infty}{P(n,m|\phi,\rho)}=1$ (39)
where $P(n,m|\phi,\rho)$ is given by Eq. (34), and is a result of
$\sum\limits_{n,m}{\hat{\Pi}_{\phi}\left({n,m}\right)}=\hat{I}$ (40)
For the case of a pure input state, $\left|{\psi_{in}}\right\rangle$, it is
convenient to define the operators
$\hat{N}\left({n,m,k,l}\right)=\frac{1}{{n!\,m!\,k!\,l!}}\left({\hat{b}_{1}}\right)^{n}\left({\hat{b}_{2}}\right)^{m}\left({\hat{d}_{1}}\right)^{k}\left({\hat{d}_{2}}\right)^{l}$
(41)
and the probabilities in Eq. (34) are then given by
$P(n,m|\phi,\psi_{in})=\sum\limits_{k,l=1}^{\infty}{\left|{\left\langle
0\right|\hat{N}\left({n,m,k,l}\right)\left|{\psi_{in}}\right\rangle}\right|}^{2}$
(42)
Equation (34), or Eq. (42) for the case of pure states, defines a unitary
mapping, $|nm00\rangle\rightarrow|n^{\prime}m^{\prime}kl\rangle$, between
interferometer input states, $|nm00\rangle$, and output states,
$|n^{\prime}m^{\prime}kl\rangle$, because the photon number is conserved:
${n+m=n^{\prime}+m^{\prime}+k+l}$, see Eq. (37).
If we restrict our attention to measurement outcomes projected onto the
Hilbert subspace with basis
${\left|n^{\prime}\right\rangle_{b_{1}}\otimes\left|m^{\prime}\right\rangle_{b_{2}}}$,
Eq. (34) or Eq. (42) defines the non-unitary mapping ${\cal E}$:
${\cal
E}\left[\left|n\right\rangle_{a_{1}}\otimes\left|m\right\rangle_{a_{2}}\right]\rightarrow\left|n^{\prime}\right\rangle_{b_{1}}\otimes\left|m^{\prime}\right\rangle_{b_{2}}$
(43)
since photon number is not conserved: $n+m\neq n^{\prime}+m^{\prime}$, see Eq.
(38), which represents losses in the Mach-Zehnder interferometer. Here the
Fock states ${|n^{\prime}\rangle_{b_{i}}}$, for $i=1,2$, are created from the
vacuum state, $|0\rangle_{b_{i}}$, by application of creation operators
$\hat{b}_{i}^{\dagger}$ in the usual way. The mapping ${\cal E}$ depends on
two parameters, $r_{x}$ and $r_{y}$, which specify the strength of the
dissipation or losses in each arm of the interferometer. In the limit of no
dissipation, when $r_{x}=0$ and $r_{y}=0$, the mapping ${\cal E}$ becomes a
unitary transformation and $n+m=n^{\prime}+m^{\prime}$.
In what follows, I use the short-hand notation $|nm\rangle$ for the input
state
$|nm00\rangle\equiv|n\rangle_{a_{1}}\otimes|m\rangle_{a_{2}}\otimes|0\rangle_{v_{1}}\otimes|0\rangle_{v_{2}}$.
In the next two subsections, III.3 and III.4, I discuss the Fisher information
and the fidelity (Shannon mutual information) for specific cases of few-photon
Fock state and N00N state input into the lossy Mach-Zehnder (MZ)
interferometer.
### III.3 Fock State Input into Lossy MZ Interferometer
Consider the $N$-photon Fock state
$\left|{\psi_{N}}\right\rangle=\frac{1}{{\sqrt{N!}}}\left({\hat{a}_{1}^{\dagger}}\right)^{N}\,\left|0\right\rangle=\left|{N000}\right\rangle\equiv\left|{N0}\right\rangle$
(44)
input into a lossy Mach-Zehnder interferometer given by the scattering matrix
in Eq. (33). The probabilities for measurement outcomes $\xi=(n,m)$ are given
by:
$\displaystyle
P(n,m|\phi,\psi_{N})=\frac{N!}{n!m!}\sum\limits_{k=0}^{N}\sum\limits_{l=0}^{N}\frac{1}{k!l!}\times$
$\displaystyle\left|S_{11}^{n}S_{21}^{m}S_{31}^{k}S_{41}^{l}\right|^{2}\,\delta_{n+m+k+l,N}$
(45)
where $S_{ij}$ are the matrix elements of Eq. (33) and $\delta_{m,n}$ is the
Kronecker delta function.
A direct calculation of the classical Fisher information for the $N$-photon
Fock state input, $F_{N}(\phi)$, gives
$F_{N}(\phi)=NF_{1}(\phi)$ (46)
where $F_{1}(\phi)$ is the classical Fisher information for one-photon input,
given in Eq. (53). This shows that for the lossy MZ interferometer with
$N$-photon Fock state input, the standard deviation $(\delta\phi)$ scales as
$1/\sqrt{N}$. From another point of view, since the Fisher information is
additive for independent events, the $N$-photon Fock state acts like $N$
independent 1-photon states. When $r_{x}=r_{y}$, then $F_{1}(\phi)=1$, and the
$N$-photon Fisher information becomes $F_{N}(\phi)=N$. This means that
dissipation in the (non-unitary) interferometer has the effect of introducing
a phase dependence into the Fisher information, see the discussion below.
#### III.3.1 1-Photon Fock State Input into Lossy MZ Interferometer
As the simplest example of the effect of dissipation, I consider the 1-photon
Fock state input into the lossy Mach-Zehnder interferometer with scattering
matrix given by Eq. (33)
$\left|\psi_{in}\right\rangle=\hat{a}_{1}^{{\dagger}}\left|0\right\rangle=\left|1000\right\rangle\equiv\left|10\right\rangle$
(47)
where I use the short-hand notation $\left|10\right\rangle$ for the input
state $\left|1000\right\rangle$. The probabilities for measurement outcomes
are given by $P(n,m|\phi,\psi_{in})$, where $n,m$ specify the photon numbers
output in port $b_{1}$ and $b_{2}$, respectively, see Fig. 2. The
probabilities $P(n,m|\phi,\psi_{in})$ for the three measurement outcomes are:
$\small\begin{array}[]{lllll}P(10|\phi,10)&=&\frac{1}{4}\left({2-r_{x}^{2}-r_{y}^{2}-2\sqrt{\left({1-r_{x}^{2}}\right)\left({1-r_{y}^{2}}\right)}\,\cos\phi}\right)\\\
P(01|\phi,10)&=&\frac{1}{4}\left({2-r_{x}^{2}-r_{y}^{2}+2\sqrt{\left({1-r_{x}^{2}}\right)\left({1-r_{y}^{2}}\right)}\,\cos\phi}\right)\\\
P(00|\phi,10)&=&\frac{1}{2}\left({r_{x}^{2}+r_{y}^{2}}\right)\\\ \end{array}$
(48)
The probability $P(00|\phi,10)$ is associated with an inconclusive measurement
outcome, since for this case zero photons leave the output ports, i.e., the
photon that entered in port “a” was absorbed in the interferometer, or more
precisely the photon was output in either port $d_{1}$ or $d_{2}$.
From Bayes’ rule in Eq. (25), the phase probability distributions,
$p(\phi|m\,n,\psi_{in})$, for input state $\left|\psi_{in}\right\rangle$ given
in Eq. (47), are given by
$\begin{array}[]{lllll}p(\phi|10,10)&=&\frac{1}{{2\pi}}\frac{{2-r_{x}^{2}-r_{y}^{2}-2\sqrt{\left({1-r_{x}^{2}}\right)\left({1-r_{y}^{2}}\right)}\,\cos\phi}}{{2-r_{x}^{2}-r_{y}^{2}}}\\\
p(\phi|01,10)&=&\frac{1}{{2\pi}}\frac{{2-r_{x}^{2}-r_{y}^{2}+2\sqrt{\left({1-r_{x}^{2}}\right)\left({1-r_{y}^{2}}\right)}\,\cos\phi}}{{2-r_{x}^{2}-r_{y}^{2}}}\\\
p(\phi|00,10)&=&\frac{1}{{2\pi}}\\\ \end{array}$ (49)
When $r_{x}\neq r_{y}$, there is a loss of contrast in the phase probability
distributions $p(\phi|m\,n,\psi_{in})$, see Fig. 3. When the absorption
probabilities are the same in both arms, in the limit $r_{x}=r_{y}$, the phase
probability distributions in Eq. (49) reduce to the case of a single photon
input without losses in the interferometer, which are given in Eq. (26)-(28),
with trivial phase change given by the replacements $\sin\rightarrow\cos$. The
phase probability density, $p(\phi|00,10)$, is associated with the
inconclusive outcome, $p(\phi|i)$. It is a remarkable feature that for equal
loss in both arms, $r_{x}=r_{y}$, the phase probability densities,
$p(\phi|m\,n,\psi_{in})$, in Eq.(49) do not depend on the size of the loss,
$r_{x}$. However, there is loss of information with increasing absorption,
$r_{x}$ and $r_{y}$, which is reflected in the information measures, see
below.
The effect of equal dissipation in both arms of the interferometer is the same
as the effect of non-deterministic state preparation, specified by input state
characterized by a density matrix in Eq. (19). However, when the dissipation
in both arms is not equal, say for $r_{y}=0$ and $r_{x}$ is finite, the phase
probability distributions show a loss of contrast, see Fig. 3. This feature
may be useful in applications to null-type measurements.
Figure 3: (Color) For the 1-photon input state $\left|10\right\rangle$, given
by Eq. (47), the probability distribution for the phase, $p(\phi|10,10)$, in
Eq. (49) is plotted for absorption $r_{y}=0$ and $r_{x}=$0.0, 0.90, 0.95,
0.98, and 1.0. As $r_{x}\rightarrow$1.0, the probability distribution becomes
flat and does not distinguish between different phase values. Figure 4:
(Color) For 1-photon Fock input state, $\left|10\right\rangle$, given by Eq.
(47), the fidelity (Shannon mutual information) $H(M)$ seems linear in
$r_{x}^{2}$ for $r_{x}=r_{y}$, however, this is not true for general values of
$r_{x}$ and $r_{y}$, see also Fig. 5. Figure 5: (Color) For 1-photon Fock
input state $\left|1000\right\rangle=\left|10\right\rangle$, given by Eq.
(47), the fidelity (Shannon mutual information) $H(M)$ is plotted as a
function of loss parameters $r_{x}^{2}$ and $r_{y}^{2}$. Figure 6: (Color)
The Fisher information $F(\phi)$ for 1-photon input state
$\left|1000\right\rangle$ (short-hand notation $\left|10\right\rangle$), is
plotted as a function of $r_{x}$ and $r_{y}$, for different values of
$\phi=0,0.125,\pi/2,3.0,3.1,\pi$, left to right in top row and bottom row.
In the discussion that follows, I assume no prior information on the phase, so
I take $p(\phi)=1/(2\pi)$. When there is no loss in the interferometer,
$r_{x}=r_{y}=0$, the Shannon mutual information (fidelity) as defined in Eq.
(9) is a constant:
$H(M)=\frac{1}{{\ln 2}}-1$ (50)
When the losses in both arms are equal, $r_{x}=r_{y}$, we have the exact
result
$H(M)=\left({\frac{1}{{\ln 2}}-1}\right)\left({1-r_{x}^{2}}\right)$ (51)
For general values of $r_{x}$ and $r_{y}$, the expression for $H(M)$ is large
and complicated, but for small ${r_{x}\ll 1}$ and ${r_{y}\ll 1}$, I can expand
it in a power series,
$\small H(M)=\left({\frac{1}{{\ln
2}}-1}\right)\left[{1-\frac{1}{2}\left({r_{x}^{2}+r_{y}^{2}}\right)}\right]+O(r_{x}^{4})+O(r_{y}^{4})$
(52)
where I dropped fourth order terms in $r_{x}$ and $r_{y}$. When $r_{x}=r_{y}$,
from Eq. (52), we may expect the fidelity (Shannon mutual information) $H(M)$
to be quadratic in $r_{x}$, however, this is not true for general values
$r_{x}$ and $r_{y}$, see Fig. 4, which shows $H(M)$ vs. $r_{x}^{2}$ for the
case $r_{x}=r_{y}$ and for $r_{y}=0$.
In Figure 5 the fidelity (Shannon mutual information) $H(M)$ is plotted as a
function of the dissipation parameters, $r_{x}$ and $r_{y}$. When the
dissipation in either arm is a maximum, $r_{x}=1$ or $r_{y}=1$, the fidelity
$H(M)=0$, indicating that we obtain zero information from each photon.
For the 1-photon Fock state (given in Eq. (47)) input into the lossy Mach-
Zehnder interferometer given in Eq. (33), the classical Fisher information
(defined by Eq. (2)) is given by:
$F_{1}(\phi)=\frac{2\left(1-r_{x}^{2}\right)\left(1-r_{y}^{2}\right)\left(2-r_{x}^{2}-r_{y}^{2}\right)\sin^{2}(\phi)}{\left(2-r_{x}^{2}-r_{y}^{2}\right)^{2}-4\left(1-r_{x}^{2}\right)\left(1-r_{y}^{2}\right)\cos^{2}(\phi)}$
(53)
see plots in Fig. 6. From these plots, it is clear that for a lossy
interferometer, the Fisher information depends strongly on the true value of
$\phi$. Through the Cramer-Rao bound in Eq. (1), this translates to a
dependence of the variance $(\delta\phi)^{2}$ on the true value of $\phi$.
Therefore, Fisher information $F_{1}(\phi)$ for 1-photon Fock state input into
a lossy interferometer is qualitatively different than for an ideal
interferometer without loss, see Eq. (53) for the case $r_{x}=r_{y}=0$. As a
consequence of Eq.(46), the Fisher information for $N$-photon Fock state input
into a lossy interferometer is qualitatively different than for a lossless
interferometer, where it is a constant given by $F_{N}(\phi)=N$. Specifically,
for a lossy interferometer the Fisher information depends on the value of the
true phase, see plots in Fig. 6. For values of the phase given by $\phi=0$ and
$\phi=\pm\pi$, the Fisher information vanishes for Fock state input,
independent of the value of the dissipation parameters, $r_{x}$ and $r_{y}$,
see comment 111This statement applies to a balanced interferometer, whose path
lengths satisfy Eq. (31).. According to the Cramer-Rao bound, the variance
$(\delta\phi)^{2}$, is large for values of true phase near $\phi=0$ and
$\phi=\pm\pi$. However, when there is no dissipation, $r_{x}=r_{y}=0$, the
Fisher information is independent of $\phi$ and so is the bound on the
variance $(\delta\phi)^{2}$. The presence of dissipation in the interferometer
introduces a dependence of the variance, $(\delta\phi)^{2}$, on true phase
$\phi$. The exception to this is when $r_{x}=r_{y}$, where $F_{1}(\phi)$
reduces to $F_{1}(\phi)=1-r_{x}^{2}$, and is independent of $\phi$.
#### III.3.2 2-Photon Fock State Input into Lossy MZ Interferometer
Consider now the 2-photon Fock state input into the lossy Mach-Zehnder
interferometer with S-matrix given by Eq. (33):
$\left|\psi_{in}\right\rangle=\frac{1}{{\sqrt{2}}}\left({a_{1}^{\dagger}}\right)^{2}\left|0\right\rangle=\left|2000\right\rangle\equiv\left|20\right\rangle$
(54)
where I use the short-hand notation $\left|20\right\rangle$ for the state
$\left|2000\right\rangle$. The probabilities $P(m,n|\phi,\psi_{in})$ for the
six measurement outcomes are given by
$\displaystyle P(20|\phi,20)$ $\displaystyle=$
$\displaystyle\frac{1}{{16}}\left[{6-6r_{x}^{2}-6r_{y}^{2}+4\sqrt{\left({r_{x}^{2}-1}\right)\left({r_{y}^{2}-1}\right)}\left({r_{x}^{2}+r_{y}^{2}-2}\right)\cos(\phi)+2\left({r_{x}^{2}-1}\right)\left({r_{y}^{2}-1}\right)\cos(2\phi)+4r_{x}^{2}r_{y}^{2}+r_{x}^{4}+r_{y}^{4}}\right]$
$\displaystyle P(02|\phi,20)$ $\displaystyle=$
$\displaystyle\frac{1}{{16}}\left[{6-6r_{x}^{2}-6r_{y}^{2}-4\sqrt{\left({r_{x}^{2}-1}\right)\left({r_{y}^{2}-1}\right)}\left({r_{x}^{2}+r_{y}^{2}-2}\right)\cos(\phi)+2\left({r_{x}^{2}-1}\right)\left({r_{y}^{2}-1}\right)\cos(2\phi)+4r_{x}^{2}r_{y}^{2}+r_{x}^{4}+r_{y}^{4}}\right]$
$\displaystyle P(11|\phi,20)$ $\displaystyle=$
$\displaystyle\frac{1}{8}\left[{2-2r_{x}^{2}-2r_{y}^{2}+r_{x}^{4}+r_{y}^{4}-2\left({1-r_{x}^{2}}\right)\left({1-r_{y}^{2}}\right)\cos\left({2\phi}\right)}\right]$
$\displaystyle P(10|\phi,20)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left({r_{x}^{2}+r_{y}^{2}}\right)\left[{2-r_{x}^{2}-r_{y}^{2}-2\sqrt{\left({1-r_{x}^{2}}\right)\left({1-r_{y}^{2}}\right)}\,\cos\left(\phi\right)}\right]$
$\displaystyle P(01|\phi,20)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left({r_{x}^{2}+r_{y}^{2}}\right)\left[{2-r_{x}^{2}-r_{y}^{2}+2\sqrt{\left({1-r_{x}^{2}}\right)\left({1-r_{y}^{2}}\right)}\,\cos\left(\phi\right)}\right]$
$\displaystyle P(00|\phi,20)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(r_{x}^{2}+r_{y}^{2}\right)^{2}$ (55)
The sum of the probabilities for the six possible measurement outcomes in Eq.
(55) is unity. Since a two-photon Fock state has been used as input, the
probability that no photon was absorbed is equal to the the sum
$P(20|\phi)+P(02|\phi)+P(11|\phi)=\frac{1}{4}\left({2-r_{x}^{2}-r_{y}^{2}}\right)^{2}$,
while the probability that exactly one photon was absorbed is
$P(10|\phi)+P(01|\phi)=\frac{1}{2}\left({2r_{x}^{2}+2r_{y}^{2}-2r_{x}^{2}r_{y}^{2}-r_{x}^{4}-r_{y}^{4}}\right)$.
The conditional probability distributions for the phase,
$p(\phi|m\,n,\psi_{in})$ , with input state $\left|\psi_{in}\right\rangle$ in
Eq. (54) and measurement outcome $\xi=(m,n)$ are given by
$\displaystyle p(\phi|20,20)$ $\displaystyle=$
$\displaystyle\frac{1}{{2\pi}}\left[{1+\frac{{4\sqrt{\left({1-r_{x}^{2}}\right)\left({1-r_{y}^{2}}\right)}\left({r_{x}^{2}+r_{y}^{2}-2}\right)\cos(\phi)+2\left({r_{x}^{2}-1}\right)\left({r_{y}^{2}-1}\right)\cos(2\phi)}}{{6-6(r_{x}^{2}+r_{y}^{2})+r_{x}^{4}+r_{y}^{4}+4r_{x}^{2}r_{y}^{2}}}}\right]$
$\displaystyle p(\phi|02,20)$ $\displaystyle=$
$\displaystyle\frac{1}{{2\pi}}\left[{1-\frac{{4\sqrt{\left({1-r_{x}^{2}}\right)\left({1-r_{y}^{2}}\right)}\left({r_{x}^{2}+r_{y}^{2}-2}\right)\cos(\phi)+2\left({r_{x}^{2}-1}\right)\left({r_{y}^{2}-1}\right)\cos(2\phi)}}{{6-6(r_{x}^{2}+r_{y}^{2})+r_{x}^{4}+r_{y}^{4}+4r_{x}^{2}r_{y}^{2}}}}\right]$
$\displaystyle p(\phi|11,20)$ $\displaystyle=$
$\displaystyle\frac{1}{{2\pi}}\left[{1-\frac{{2\left({1-r_{x}^{2}}\right)\left({1-r_{y}^{2}}\right)\cos(2\phi)}}{{2-2(r_{x}^{2}+r_{y}^{2})+r_{x}^{4}+r_{y}^{4}}}}\right]$
$\displaystyle p(\phi|10,20)$ $\displaystyle=$
$\displaystyle\frac{1}{{2\pi}}\left[{1-\frac{{2\sqrt{\left({1-r_{x}^{2}}\right)\left({1-r_{y}^{2}}\right)}\cos(\phi)}}{{2-r_{x}^{2}-r_{y}^{2}}}}\right]$
$\displaystyle p(\phi|01,20)$ $\displaystyle=$
$\displaystyle\frac{1}{{2\pi}}\left[{1+\frac{{2\sqrt{\left({1-r_{x}^{2}}\right)\left({1-r_{y}^{2}}\right)}\cos(\phi)}}{{2-r_{x}^{2}-r_{y}^{2}}}}\right]$
$\displaystyle p(\phi|00,20)$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi}$
In the limit of high photon absorption, $r_{x}=1$ and $r_{y}=1$, the phase
probability densities $p(\phi|m\,n,\,20)=1/(2\pi)$ for all measurement
outcomes, so that different phase values are not distinguishable because of
photon losses.
A remarkable property of the phase probability distributions in Eq.
(LABEL:2-PhotonPhaseProbs) (similar to that of Fock state input in Eq. (49))
is that for equal losses in both arms of the interferometer, $r_{x}=r_{y}$,
the phase probability distributions are independent of the magnitude of the
loss $r_{x}$, having values
$\tiny\frac{1}{\pi}\left\\{{\frac{4}{3}\sin^{4}\left({\frac{\phi}{2}}\right),\frac{4}{3}\cos^{4}\left({\frac{\phi}{2}}\right),\sin^{2}(\phi),\sin^{2}\left({\frac{\phi}{2}}\right),\cos^{2}\left({\frac{\phi}{2}}\right),\,\frac{1}{2}}\right\\}$
(57)
where I have written the values of the functions (from top to bottom) in Eq.
(LABEL:2-PhotonPhaseProbs) as components of a vector left to right in Eq.
(57). One may think that for $r_{x}=r_{y}$ the contrast in the photon number
measurements is lost, however, setting $r_{x}=r_{y}$ in Eq. (55) and expanding
for small $r_{x}\ll 1$ leads to
$\left({\begin{array}[]{*{20}c}{P(20|\phi,20)}\\\ {P(02|\phi,20)}\\\
{P(11|\phi,20)}\\\ {P(10|\phi,20)}\\\ {P(01|\phi,20)}\\\ {P(00|\phi,20)}\\\
\end{array}}\right)=\left({\begin{array}[]{*{20}c}{\sin^{4}\left({\frac{\phi}{2}}\right)-2r_{x}^{2}\sin^{4}\left({\frac{\phi}{2}}\right)+O\left({r_{x}^{4}}\right)}\\\
{\cos^{4}\left({\frac{\phi}{2}}\right)-2r_{x}^{2}\cos^{4}\left({\frac{\phi}{2}}\right)+O\left({r_{x}^{4}}\right)}\\\
{\frac{{\sin^{2}(\phi)}}{2}-r_{x}^{2}\sin^{2}(\phi)+O\left({r_{x}^{4}}\right)}\\\
{r_{x}^{2}(1-\cos(\phi))+O\left({r_{x}^{4}}\right)}\\\
{r_{x}^{2}(\cos(\phi)+1)+O\left({r_{x}^{4}}\right)}\\\
{O\left({r_{x}^{4}}\right)}\\\ \end{array}}\right)$ (58)
which shows that the probabilities become vanishingly small for the
measurement outcomes $P(10|\phi,20)$, $P(01|\phi,20)$, and $P(00|\phi,20)$,
which correspond to one or both photons being absorbed, but contrast for
different measurement outcomes is not lost.
For the case of equal loss in both arms, $r_{x}=r_{y}$, with no prior
information on the phase, $p(\phi)=1/(2\pi)$, the fidelity is given by
$\small H(M)=\frac{1}{{4\ln 2}}\left({1-r_{x}^{2}}\right)\left[{8-4\ln 2-3\ln
3+2r_{x}^{2}{\rm{arctanh}}\left({\frac{{11}}{{43}}}\right)}\right]$ (59)
For the case of small (but not equal) losses in both arms, $r_{x}\ll 1$ and
$r_{y}\ll 1$, the fidelity is given by
$\displaystyle H(M)=$ $\displaystyle\frac{{8-4\ln 2-3\ln 3}}{{4\ln
2}}+\left({r_{x}^{2}+r_{y}^{2}}\right)\left({\frac{{3\ln 3}}{{4\ln
2}}-\frac{1}{{\ln 2}}}\right)+$ (60) $\displaystyle
r_{x}^{2}r_{y}^{2}\left({\frac{{1+\ln 2-\ln 3}}{{2\ln
2}}}\right)+O\left({r_{x}}\right)^{4}+O\left({r_{y}}\right)^{4}$
where I have dropped terms of fourth order in $r_{x}$ and $r_{y}$. Equation
(60) gives the dependence on the dissipation parameters, $r_{x}$ and $r_{y}$,
of the information gain about $\phi$, for single use of the interferometer,
when there is no prior information about $\phi$.
### III.4 N00N State Input into Lossy MZ Interferometer
Next, I consider the $N$-photon N00N state input into the lossy Mach-Zehnder
interferometer with scattering matrix given by Eq. (33):
$\displaystyle\left|{\psi_{N00N}}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{{\sqrt{2N!}}}\left[{\left({\hat{a}_{1}^{\dagger}}\right)^{N}+\left({\hat{a}_{2}^{\dagger}}\right)^{N}}\right]\left|0\right\rangle$
(61) $\displaystyle=$
$\displaystyle\frac{1}{{\sqrt{2}}}\left[{\left|{N000}\right\rangle+\left|{0N00}\right\rangle}\right]$
(62)
For the input state in Eq. (62), the probability for measurement outcome
$\xi=(n,m)$ is given by
$\displaystyle
P(n,m|\phi,\psi_{N00N})=\frac{N!}{2n!m!}\sum\limits_{k=0}^{N}\sum\limits_{l=0}^{N}\frac{1}{k!l!}\times$
$\displaystyle\left|S_{11}^{n}S_{21}^{m}S_{31}^{k}S_{41}^{l}+S_{12}^{n}S_{22}^{m}S_{32}^{k}S_{42}^{l}\right|^{2}\,\delta_{n+m+k+l,N}$
(63)
where $n$ and $m$ are the number of photons output in ports $b_{1}$ and
$b_{2}$, respectively. Using the Fisher information, I compare how well Fock
states and N00N states perform in the presence of absorption losses. In Fig.
7, I plot the classical Fisher information for $N=$3, 4, and 5 photon Fock
states and N00N states, plotted vs. $r_{x}$ for the special case where
$r_{x}=r_{y}$. The plots show that, for equal dissipation in both arms, and
for equal photon number, Fock states perform better for phase estimation than
N00N states, for the same amount of dissipation $r_{x}$, see Eq. (1).
For N00N states, the Fisher information vanishes at the phase values:
$\phi=0,\pm\pi/2,\pm\pi$. While for Fock states, the Fisher information
vanished only at $\phi=0,\pm\pi$. For $\phi$ close to these values, phase
estimation may have large standard deviation, see Eq.(1).
Figure 8 shows the classical Fisher information for the case where Fock and
N00N states are input into a Mach-Zehnder interferometer with small
dissipation (losses) and when the losses are not equal in both arms.
Generally, Fock states perform better (have larger Fisher information) for all
values of dissipation $r_{x}$ except at the very highest values of $r_{x}\sim
0.95$. The comparison is made at a true value of $\phi=\pi/4$, where the
Fisher informations do not vanish.
Figure 9 shows a plot of the classical Fisher information for Fock and for
N00N states for $N=$ 3, 4 and 5 photons for the case of large dissipation in
one arm of the Mach-Zehnder interferometer, $r_{y}=0.9$. The comparison is
complicated, since for the $N=$3 photon case, Fock states perform better than
N00N states for large dissipation $r_{x}\sim 0.8$, whereas the situation is
reversed for small dissipation $r_{x}\sim 0.05$.
In Figures 8 and 9, the comparisons are made at a true value of phase
$\phi=\pi/4$, where the classical Fisher informations (for Fock and N00N
states) do not vanish. When dissipation is present, the classical Fisher
information has a complicated behavior as a function of the true phase $\phi$,
see Fig. 10. This shows that phase estimation using simple photon counting is
a sensitive procedure whose accuracy depends on the true value of phase.
The classical Fisher information for a lossy Mach-Zehnder interferometer
depends on the true value of the phase $\phi$. The fidelity (Shannon mutual
information) is an information measure that averages over all phases, for
prior information given by $p(\phi)$, see Eq. (9). Figure 11 shows a
comparison of the fidelity versus dissipation $r_{x}$ for Fock and N00N states
for equal dissipation in both arms, $r_{x}=r_{y}$. The fidelity of 1-photon
Fock and N00N states is equal, see the discussion below. For a given amount of
dissipation, $r_{x}$, for Fock states the fidelity increases with input photon
number $N$. The fidelity for 2-photon N00N state input is exactly zero for all
values of dissipation $r_{x}$ because this state carries no information about
the phase in a Mach-Zehnder interferometer, see the discussion below.
#### III.4.1 1-Photon N00N State Input into Lossy MZ Interferometer
Consider now the 1-photon entangled N00N state:
$\left|{\psi_{1}^{N00N}}\right\rangle=\frac{1}{{\sqrt{2}}}\left({a_{1}^{\dagger}+a_{2}^{\dagger}}\right)\left|0\right\rangle=\frac{1}{\sqrt{2}}\left[\left|10\right\rangle+\left|01\right\rangle\right]$
(64)
where again, I use the short-hand notation $\left|10\right\rangle$ for
$\left|1000\right\rangle$ and $\left|01\right\rangle$ for
$\left|0100\right\rangle$. The probabilities for the measurement outcomes for
this input state are given by Eq. (48) with the replacement
$\cos\phi\rightarrow\sin\phi$. Similarly, the phase probability distributions,
assuming no prior information, $p(\phi)=1/(2\pi)$, are given by Eq. (49) with
the replacement $\cos\phi\rightarrow\sin\phi$. The fidelity for this input
state is the same as for the 1-photon Fock state, given by Eq. (50)–(52).
Therefore, according to Shannon mutual information (fidelity), the presence of
entanglement in the 1-photon N00N state has not improved the information on
the phase.
The Fisher information for this entangled state is given by the 1-photon Fock
state Fisher information in Eq. (53) with the replacements
$\phi\rightarrow\frac{\pi}{2}-\phi$. Therefore, the entanglement simply has
the effect of changing the phase of the classical Fisher information. This
phase change changes the places where $F(\phi)=0$, which, for this entangled
state, is now $\phi=\pm\pi/2$. Comparison of the 1-photon Fock state to the
the 1-photon entangled N00N state shows that the introduction of entanglement
does not remove the $\phi$ dependence of the Fisher information when arbitrary
losses $r_{x}$ and $r_{y}$ are present. However, when $r_{x}=r_{y}$”, the
Fisher information, $F_{1}(\pi/2-\phi)$, is independent of $\phi$. This is in
agreement with the result of Chen and Jiang, who derived the Fisher
information for N00N state input using a master equation for a quantum
continuous variable system for the case of symmetrical losses Chen and Jiang
(2007). When losses are absent, $r_{x}=r_{y}=0$, the Fisher information for
input state given by Eq. (64) reduces to $F(\phi)=1$, independent of $\phi$,
as in the 1-photon Fock state without losses.
Figure 7: (Color) The Fisher information is plotted vs. $r_{x}$ for
$r_{x}=r_{y}$, for Fock states (red) and N00N states (blue), for $\phi=\pi/4$,
where the Fisher information is non-zero for both Fock states and N00N states.
Figure 8: (Color) The Fisher information is plotted vs. $r_{x}$ for small
dissipation in one arm, $r_{y}=0.10$, for Fock states (red) and N00N states
(blue), for $\phi=\pi/4$. Figure 9: (Color) The Fisher information is plotted
vs. $r_{x}$ for large dissipation in one arm, $r_{y}=0.90$, for Fock states
(red) and N00N states (blue), for $\phi=\pi/4$. This example of high
dissipation shows that the situation is complicated at high values of the
dissipation in one arm and small values of dissipation in the other arm.
Figure 10: (Color) The Fisher information is plotted vs. $\phi$ for loss
parameters $r_{x}=0.3$ and $r_{y}=0.4$, for N00N state input into a Mach-
Zehnder interferometer for $N=$1, 2, 3, 4, and 5 photons. Figure 11: (Color)
The fidelity (Shannon mutual information) is plotted as a function of
dissipation $r_{x}$, for $r_{x}=r_{y}$, for one-, two- and three-photon Fock
state input, and for one- and two-photon N00N state input.
#### III.4.2 2-Photon N00N State Input into Lossy MZ Interferometer
The 2-photon N00N state,
$\left|{\psi_{2}^{N00N}}\right\rangle=\frac{1}{2}\left[{\left({\hat{a}_{1}^{\dagger}}\right)^{2}+\left({\hat{a}_{2}^{\dagger}}\right)^{2}}\right]\,\left|0\right\rangle=\frac{1}{\sqrt{2}}\left[\left|20\
\right\rangle+\left|02\ \right\rangle\right]$ (65)
has a peculiar behavior when input into a Mach-Zehnder interferometer with
losses. The probabilities distributions, given by Eqs. (42) and (63), for the
six measurement outcomes are independent of $\phi$ and are given by
$\displaystyle P(20|\phi,20)$ $\displaystyle=$ $\displaystyle
P(02|\phi,20)=\frac{1}{2}\left(1-r_{x}^{2}-r_{y}^{2}+r_{x}^{2}r_{y}^{2}\right)$
$\displaystyle P(11|\phi,20)$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle P(10|\phi,20)$ $\displaystyle=$ $\displaystyle
P(01|\phi,20)=\frac{1}{2}\left(r_{x}^{2}+r_{y}^{2}-2r_{x}^{2}r_{y}^{2}\right)$
$\displaystyle P(00|\phi,20)$ $\displaystyle=$ $\displaystyle
r_{x}^{2}r_{y}^{2}$
The measurement outcomes in Eq.(66) are independent of $\phi$ because of the
Hilbert space geometry of the measurement operators, $\hat{\Pi}_{\phi}(n,m)$,
and input state vector in Eq.(65), see Eq.(34). For no prior information on
the phase, $p(\phi)=1/(2\pi)$, using Bayes’ rule in Eq. (25), the phase
probability densities are independent of $\phi$, and are given by
$p(\phi|m\,n,20)=1/(2\pi)$, for all measurement outcomes $\xi=(m,n)$.
Therefore, the 2-photon N00N state cannot be used in a Mach-Zehnder
interferometer for determining the phase $\phi$. However, such an arrangement
can be useful in applications that require phase in-sensitive interferometry
to be performed. In the limit of no loss, $r_{x}=r_{y}=0$, the interferometer
acts as a beam splitter and both photons come out the same port.
Figure 12: (Color) The conditional phase probability density, $p(\phi|10)$,
is plotted as a function of $\phi$ for increasing detector error probabilities
$p_{x}=0,\,0.2,\,0.4,\,0.5$, showing a loss of phase distinguishability. At
$p_{x}=0.5$, all phases $\phi$ are equally probable. Figure 13: (Color) The
fidelity (Shannon mutual information) between the measurements and the phase
is plotted vs. the probability of incorrect detection, $p_{x}$. Consistent
with the phase probability density plotted in Fig. 12, the fidelity decreases
to zero at $p_{x}=0.5$ because there is no information on the phase in the
measurements, so there is no discrimination between different phases.
### III.5 Imperfect Photon Number Detection
Next, I consider the simplest example of imperfect photon-number detection. I
assume that state preparation is deterministic and that the interferometer is
ideal, so there are no losses. I also assume that the input state is a pure
state, in Eq. (11) taking
$P_{S}(\psi^{in})=\left\\{\begin{array}[]{l}1,\quad{\rm{if}}\;\left|{\psi^{in}}\right\rangle=\left|{10}\right\rangle\\\
0,\quad{\rm{otherwise}}\\\ \end{array}\right.$ (67)
where again I use the short-hand notation $\left|{10}\right\rangle$ for
$\left|1000\right\rangle$. For this 1-photon input state,
$\left|{\psi^{in}}\right\rangle=\left|{10}\right\rangle$, the no-loss Mach-
Zehnder interferometer transfer matrix is given by
$\small
P_{I}(\psi^{out}|\psi^{in},\phi)=\left\\{\begin{array}[]{l}\sin^{2}\phi,\;{\rm{for}}\;\left|{\psi^{in}}\right\rangle=\left|{10}\right\rangle{\rm{and}}\;\left|{\psi^{out}}\right\rangle=\left|{10}\right\rangle\\\
\cos^{2}\phi,\;{\rm{for}}\;\left|{\psi^{in}}\right\rangle=\left|{10}\right\rangle{\rm{and}}\;\left|{\psi^{out}}\right\rangle=\left|{01}\right\rangle\\\
0,\quad{\rm{otherwise}}\\\ \end{array}\right.$ (68)
For the detection system, I assume that there is a probability $p_{d}$ to
detect the state correctly and a probability $p_{x}$ to detect the state
incorrectly, where $p_{d}+p_{x}=1$. I am neglecting the possibility of an
inconclusive measurement outcome. The matrix, $P_{D}(\xi|\psi^{out},\phi)$, in
Eq. (11) describing the state detection is then given by
$P_{D}(\xi|\psi^{out},\phi)=\left\\{{\begin{array}[]{*{20}c}{p_{d},\quad\xi=\psi^{out}}\\\
{p_{x},\quad\xi\neq\psi^{out}}\\\ \end{array}}\right.$ (69)
From Eq. (11), the probabilities $P(\xi|\phi)$ for measurement outcomes are
$\begin{array}[]{l}P(10|\phi)=p_{d}\sin^{2}\phi+p_{x}\cos^{2}\phi\\\
P(01|\phi)=p_{x}\sin^{2}\phi+p_{d}\cos^{2}\phi\\\ \end{array}$ (70)
where $\xi=(m,n)$ specifies that $m$ and $n$ photons are detected in output
ports “c” and “d”, respectively, see Eq. (15)–(20).
From Bayes’ rule in Eq. (25), I find the conditional probability density,
$p(\phi|\xi)$, for the phase shift $\phi$ for a given measurement outcome
$\xi$ to be
$\begin{array}[]{l}p(\phi|10)=\frac{1}{\pi}\left[{\left({1-p_{x}}\right)\sin^{2}\phi+p_{x}\cos^{2}\phi}\right]\\\
p(\phi|01)=\frac{1}{\pi}\left[{p_{x}\sin^{2}\phi+\left({1-p_{x}}\right)\cos^{2}\phi}\right]\\\
\end{array}$ (71)
Figure 12 shows a plot of the phase probability density, $p(\phi|10)$ vs.
$\phi$, for different values of detector error probability $p_{x}$. With
increasing probability $p_{x}$ of detecting the state incorrectly, the
constrast in the phase probability decreases. Note that this contrast is not a
“visibility” because $p(\phi|10)$ is a probability, and not an optical
intensity.
The fidelity (Shannon mutual information) defined in Eq. (9) is plotted in
Fig. 13. As expected for this simple model, the fidelity $H(M)$ decreases with
increasing probability of incorrect detection, $p_{x}$, reaching zero at
$p_{x}=0.5$. Note that the fidelity is symmetric about $p_{x}=0.5$.
## IV Conclusion
I considered the experimentally relevant problem of determining the phase
shift in one arm of a quantum interferometer when state creation is not
perfectly deterministic, state propagation through the interferometer is non-
unitary due to absorption losses in the interferometer, and state detection is
not ideal. In Section II, I have argued that two types of information are
useful for evaluating the quality of a parameter estimation device, such as a
quantum optical system used to determine phase shifts. First, fidelity
(Shannon mutual information between measurements and parameter) is useful for
deciding the overall quality of the optical system. The fidelity represents an
average over probabilities of all possible measurements and parameter values
(phases). The fidelity is the metric to use when choosing or designing a
system and the prior parameter (phase) distribution is unknown. Once a system
is chosen, it is to be used in estimating the parameter based on measurements
(data), which is an estimation problem. At this point, the (classical or
quantum) Fisher information can be exploited, using the classical or quantum
Cramer-Rao theorem, to estimate the variance of the parameter associated with
its unbiased estimator.
In Eq. (11), I have written down a general statistical expression for the
probability of a measurement outcome that simultaneously takes into account
the three non-ideal aspects of real experiments: non-deterministic state
preparation, losses in the interferometer, and non-ideal quantum state
detection. This expression requires detailed models for each of the three non-
ideal elements. In Section III, using simple, few-photon Fock states and N00N
states, I give examples of applying Eq. (11). In subsection, A, of Section
III, I look at a simple example of the effect of non-deterministic state
creation, where there is a probability of creating one photon and a
probability of creating vacuum, as input into a Mach-Zehnder interferometer.
As expected, the non-zero probability of creating a vacuum input leads to a
probability for an inconclusive measurement outcome, which in turn reduces the
information on phase, as measured by Fisher information and fidelity (Shannon
mutual information).
In subsection B, of Section III, I have constructed a scattering matrix for a
lossy (non-unitary) Mach-Zehnder interferometer. I find that for simple photon
counting measurements, losses introduce a strong phase dependence in the
classical Fisher information, making accuracy of phase estimation dependent on
the unknown true phase. In subsection, C and D, of Section III, I use the
classical Fisher information and fidelity to examine in detail the propagation
of Fock and N00N states, respectively, through the lossy Mach-Zehnder
interferometer. Finally, in subsection E of Section III, I use Eq. (11), to
look at the effect of imperfect photon number detection on determining the
phase in a Mach-Zehnder interferometer with no losses, using the simplest
model that takes into account a probability for incorrect detection of photon
number.
The examples that I have used have been simple to illustrate the application
of Eq. (11). The theory can be applied to more complicated cases so that real
experiments can be analyzed.
###### Acknowledgements.
The author acknowledges stimulating discussions with Paul Lopata on Bayesian
statistics and data analysis.
## References
* Hariharan (2003) P. Hariharan, _Optical interferometry_ (Academic Press, New York, 2003), second edition ed.
* Cronin et al. (2009) A. D. Cronin, J. Schmiedmayer , and D. E. Pritchard, Rev. Mod. Phys. 81, 1051 (2009).
* Thorne (1980) K. Thorne, Rev. Mod. Phys. 52, 285 (1980).
* Caves (1981) C. M. Caves, Phys. Rev. D 23, 1693 (1981).
* Dimopoulos et al. (2008) S. Dimopoulos, P. W. Graham, J. M. Hogan, and M. A. Kasevich, Phys. Rev. D 78, 042003 (2008).
* Lefevre (1993) H. Lefevre, _The fiber-optic gyroscope_ (Artech House, Boston, USA, 1993).
* Sagnac (1913a) G. Sagnac, Compt. Rend. 157, 708 (1913a).
* Sagnac (1913b) G. Sagnac, Compt. Rend. 157, 1410 (1913b).
* Sagnac (1914) G. Sagnac, J. Phys. Radium 5th Series 4, 177 (1914).
* Post (1967) E. J. Post, Rev. Mod. Phys. 39, 475 (1967).
* Chen et al. (2008) J. Chen, J. B. Altepeter, and P. Kumar, New J. Phys. 10, 123019 (2008).
* Bertocchi et al. (2006) G. Bertocchi, O. Alibart, D. B. Ostrowsky, S. Tanzilli, and P. Baldi, J. Phys. B 39, 1011 (2006).
* Gupta et al. (2005) S. Gupta, K. W. Murch, K. L. Moore, T. P. Purdy, and D. M. Stamper-Kurn, Phys. Rev. Lett. 95, 143201 (2005).
* Wang et al. (2005) Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, Phys. Rev. Lett. 94, 090405 (2005).
* Tolstikhin et al. (2005) O. I. Tolstikhin, T. Morishita, and S. Watanabe, Phys. Rev. A 72, 051603(R) (2005).
* Cooper et al. (2010) J. J. Cooper, D. W. Hallwood, and J. A. Dunningham, Phys. Rev. A 81, 043624 (2010).
* Godun et al. (2001) R. M. Godun, M. B. d’Arcy, G. S. Summy, and K. Burnett, Contemporary Physics 42, 77 (2001).
* Giovannetti et al. (2006) V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett. 96, 010401 (2006).
* Berry et al. (2009) D. W. Berry, B. L. Higgins, S. D. Bartlett, M. W. Mitchell, G. J. Pryde, and H. M. Wiseman, Phys. Rev. A 80, 052114 (2009).
* Combes and Wiseman (2005) J. Combes and H. M. Wiseman, J. Opt. B: Quantum Semiclass 7, 14 (2005).
* Nagata et al. (2007) T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, Science 316, 726 (2007).
* Durkin and Dowling (2007) G. A. Durkin and J. P. Dowling, Phys. Rev. Lett. 99, 070801 (2007).
* Pezze and Smerzi (2008) L. Pezze and A. Smerzi, Phys. Rev. Lett. 100, 073601 (2008).
* Dorner et al. (2009) U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, Phys. Rev. Lett. 102, 040403 (2009).
* Cable and Durkin (2010) H. Cable and G. A. Durkin, Phys. Rev. Lett. 105, 013603 (2010).
* 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.
* 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.
* Walther et al. (2004) P. Walther, J. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, Nature 429, 158 (2004).
* Mitchell et al. (2004) M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, Nature 429, 161 (2004).
* Okamoto et al. (2008) R. Okamoto, H. F. Hofmann, T. Nagata, J. L. O’Brien, K. Sasaki, and S. Takeuchi, New J. Phys. 10, 073033 (2008).
* Kacprowicz et al. (2010) M. Kacprowicz, R. Demkowicz-Dobrzanski, W. Wasilewski, K. Banaszek, and I. A. Walmsley, Nature Photonics 4, 357 (2010), URL http://lanl.arxiv.org/abs/0906.3511.
* Thomas-Peter et al. (2009) N. Thomas-Peter, B. J. Smith, and I. A. Walmsley, in _Lasers and Electro-Optics, 2009 and 2009 Conference on Quantum electronics and Laser Science Conference. CLEO/QELS 2009._ (Baltimore, MD, 2009), pp. 978–1–55752–869–8.
* Kim et al. (1998) T. Kim, O. Pfister, M. J. Holland, J. Noh, and J. L. Hall, Phys. Rev. A 57, 4004 (1998).
* Durkin et al. (2004) G. A. Durkin, C. Simon, J. Eisert, and D. Bouwmeester, Phys. Rev. A 70, 062305 (2004).
* Rubin and Kaushik (2007) M. A. Rubin and S. Kaushik, Phys. Rev. A 75, 053805 (2007).
* Gilbert et al. (2008) G. Gilbert, M. Hamrick, and Y. S. Weinstein, J. Opt. Soc. Am. B 25, 1336 (2008).
* Demkowicz-Dobrzanski et al. (2009) R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, Phys. Rev. A 80, 013825 (2009).
* Ono and Hofmann (20010) T. Ono and H. F. Hofmann, Phys. Rev. A 81, 033819 (20010).
* D Ariano et al. (2000) G. M. D Ariano, M. G. A. Paris, and M. F. Sacchi, Phys. Rev. A 62, 023815 (2000).
* Monras (2006) A. Monras, Phys. Rev. A 73, 033821 (2006).
* 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).
* Higgins et al. (2009) B. L. Higgins, D. W. Berry, S. D. Bartlett, M. W. Mitchell, H. M. Wiseman, and G. J. Pryde, New J. Phys. 11, 073023 (2009).
* 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.
* Hofmann (2009) H. F. Hofmann, Phys. Rev. A 79, 033822 (2009).
* Durkin (2010) G. A. Durkin, New J. Phys. 12, 023010 (2010).
* Shannon (1948) C. E. Shannon, The Bell System Technical Journal 27, 379 (1948).
* 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.
* Ou (1997) Z. Y. Ou, Phys. Rev. A 55, 2598 (1997).
* Giovannetti et al. (2004) V. Giovannetti, S. Lloyd, and L. Maccone, Science 306, 1330 (2004).
* Jaynes (2009) E. T. Jaynes, _Probability Theory the Logic of Science_ (Cambridge Press, Cambridge, UK, 2009), sixth printing.
* Note (1) Note1, this statement applies to a balanced interferometer, whose path lengths satisfy Eq. (31).
* Chen and Jiang (2007) X. Chen and L. Jiang, J. Phys. B: At. Mol. Phys. 40, 2799 (2007).
|
arxiv-papers
| 2010-12-23T20:40:57 |
2024-09-04T02:49:15.947040
|
{
"license": "Public Domain",
"authors": "Thomas B. Bahder",
"submitter": "Thomas B. Bahder",
"url": "https://arxiv.org/abs/1012.5293"
}
|
1012.5494
|
¡html¿ ¡head¿ ¡title¿COMP6411 Summer 2010¡/title¿ ¡/head¿ ¡body¿
¡a href=”http://users.encs.concordia.ca/ mokhov”¿Serguei A. Mokhov¡/a¿¡br /¿
mokhov@cse.concordia.ca
¡h1¿TOC¡/h1¿
¡ul¿ ¡li¿¡a href=”#abstract”¿Abstract¡/a¿¡/li¿ ¡li¿¡a href=”#lecture-
notes”¿Lecture Notes¡/a¿¡/li¿ ¡!–li¿¡a href=”#project-brief”¿Brief Project
Overview¡/a¿¡/li–¿ ¡li¿¡a href=”#reports”¿Reports¡/a¿¡/li¿ ¡/ul¿
¡h1¿Abstract¡/h1¿ ¡a id=”abstract” name=”abstract” /¿
¡p¿This index covers the lecture notes and the final course project reports
for COMP6411 Summer 2010 at Concordia University, Montreal, Canada,
Comparative Study of Programming Languages by 4 teams trying compare a set of
common criteria and their applicability to about 10 distinct programming
languages, where 5 language choices were provided by the instructor and five
were picked by each team and each student individually compared two of the 10
and then the team did a summary synthesis across all 10 languages. Their
findings are posted here for further reference, comparative studies, and
analysis. ¡/p¿
¡hr /¿
¡h1¿Lecture Notes¡/h1¿ ¡a id=”lecture-notes” name=”lecture-notes” /¿
¡ul¿ ¡li¿Current COMP6411 Lecture Notes ¡ul¿ ¡li¿ LIST:arXiv:1007.2123 ¡/li¿
¡/ul¿ ¡/li¿ ¡/ul¿
¡hr /¿
¡!– ¡h1¿Brief Project Overview¡/h1¿ ¡a id=”project-brief” name=”project-brief”
/¿
¡hr /¿ –¿
¡h1¿COMP6411 Summer 2010 Select Final Project Reports¡/h1¿ ¡a id=”reports”
name=”reports” /¿ ¡ul¿ ¡li¿Team 5’s Approach ¡ul¿ ¡li¿ LIST:arXiv:1008.3434
¡/li¿ ¡/ul¿ ¡/li¿ ¡li¿Team 7’s Approach ¡ul¿ ¡li¿ LIST:arXiv:1009.0305 ¡/li¿
¡/ul¿ ¡/li¿ ¡li¿Team 10’s Approach ¡ul¿ ¡li¿ LIST:arXiv:1008.3561 ¡/li¿ ¡/ul¿
¡/li¿ ¡li¿Team 11’s Approach ¡ul¿ ¡li¿ LIST:arXiv:1008.3431 ¡/li¿ ¡/ul¿ ¡/li¿
¡/ul¿
¡hr /¿
¡/body¿ ¡/html¿
|
arxiv-papers
| 2010-12-26T03:33:09 |
2024-09-04T02:49:15.961285
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serguei A. Mokhov",
"submitter": "Serguei Mokhov",
"url": "https://arxiv.org/abs/1012.5494"
}
|
1012.5508
|
# A new approach in modeling the behavior of RPC detectors
L. Benussi S. Bianco S.Colafranceschi F.L. Fabbri M. Giardoni L.
Passamonti D. Piccolo D. Pierluigi A. Russo G. Saviano S. Buontempo A.
Cimmino M. de Gruttola F. Fabozzi A.O.M. Iorio L. Lista P. Paolucci P.
Baesso G. Belli D. Pagano S.P. Ratti A. Vicini P. Vitulo C. Viviani A.
Sharma A. K. Bhattacharyya INFN Laboratori Nazionali di Frascati, Via E.
Fermi 40, I-00044 Frascati, Italy Sapienza Università degli Studi di Roma “La
Sapienza”, Piazzale A. Moro, Roma, Italy CERN CH-1211 Genéve 23 F-01631
Switzerland INFN Sezione di Napoli, Complesso Universitario di Monte
Sant’Angelo, edificio 6, 80126 Napoli, Italy Università di Napoli Federico
II, Complesso Universitario di Monte Sant’Angelo, edificio 6, 80126 Napoli,
Italy INFN Sezione di Pavia and Università degli studi di Pavia, Via Bassi 6,
27100 Pavia, Italy
###### Abstract
The behavior of RPC detectors is highly sensitive to environmental variables.
A novel approach is presented to model the behavior of RPC detectors in a
variety of experimental conditions. The algorithm, based on Artificial Neural
Networks, has been developed and tested on the CMS RPC gas gain monitoring
system during commissioning.
###### keywords:
RPC , CMS , Neural Network , muon detectors HEP
, , , , , , , , , , , , , , , , , , , , , , , , , ††thanks: Corresponding
author: Stefano Colafranceschi
E-mail address: stefano.colafranceschi@cern.ch
## 1 Introduction
Resistive Plate Chamber (RPC) detectors [1] are widely used in HEP experiments
for muon detection and triggering at high-energy, high-luminosity hadron
colliders [2, 3], in astroparticle physics experiments for the detection of
extended air showers [4], as well as in medical and imaging applications [5].
At the LHC, the muon system of the CMS experiment[6] relies on drift tubes,
cathode strip chambers and RPCs[7].
In this paper a new approach is proposed to model the behavior of an RPC
detector via a multivariate strategy. Full details on the developed algorithm
and results can be found in Ref.[8]. The algorithm, based on Artificial Neural
Networks (ANN), allows one to predict the behavior of RPCs as a function of a
set of variables, once enough data is available to provide a training to the
ANN. At the present stage only environmental variables (temperature $T$,
atmospheric pressure $p$ and relative humidity $H$) have been considered.
Further studies including radiation dose are underway and will be the subject
of a forthcoming paper. In a preliminary phase we trained a neural network
with just one variable and we found out, as expected, that the predictions are
improved after adding more variables into the network. The agreement found
between data and prediction has to be considered a pessimistic evaluation of
the validity of the algorithm, since it also depends on the presence of
unknown variables not considered for training.
The data for this study have been collected utilizing the gas gain monitoring
(GGM) system [9][10][11] of the CMS RPC muon detector during the commissioning
with cosmic rays in the ISR test area at CERN.
The GGM system is composed by the same type of RPC used in the CMS detector (2
mm-thick Bakelite gaps) but of smaller size (50$\times$50 cm2). Twelve gaps
are arranged in a stack. The trigger is provided by four out of twelve gaps of
the stack, while the remaining eight gaps are used to monitor the working
point by means of a cosmic ray telescope based on RPC detectors.
In this study, the GGM was operated in open loop mode with a Freon 95.5%,
Isobutane 4.2%, SF6 0.3% gas mixture. Six out of eight monitoring gaps were
used, two out of eight monitoring gaps failed during the study and were
therefore excluded from the analysis. The monitoring is performed by measuring
the charge distributions of each chamber. The six gaps are operated at
different high voltages, fixed for each chamber, in order to monitor the total
range of operating modes of the gaps (Table 1). The operation mode of the RPC
changes as a function of the voltage applied, in particular the chamber will
change from avalanche mode to streamer mode when increasing HV.
Table 1: Applied high voltage for power supplies for GGM RPC detectors used in this study | CH1 | CH2 | CH3 | CH6 | CH7 | CH8
---|---|---|---|---|---|---
Applied high voltage (kV) | 10.2 | 9.8 | 10.0 | 10.4 | 10.2 | 10.4
## 2 The Artificial Neural Network simulation code
An Artificial Neural Network (ANN) is an information processing paradigm that
is inspired by the way biological nervous systems, such as the brain, process
information[12]. The most common type of artificial neural network (Fig. 1)
consists of three groups, or layers, of units: a layer of input units is
connected to a layer of hidden units, which is connected to a layer of output
unit. The activity of the input units represents the raw information that is
fed into the network.
Figure 1: Example of a simple Neural Network configuration.
The activity of each hidden unit is determined by the activities of the input
units and the weights on the connections between the inputs and the hidden
units. The behavior of the output units depends on the activity of the hidden
units and the weights between the hidden and output units. For this study
temperature, humidity and pressure have been selected as inputs and anodic
charge as output variable. It was demonstrated[13] that the number of layers
is not critical for the network performance, so we decided to go with 3 layers
and give to the neural network a sufficient number of hidden units
automatically optimized by a genetic algorithm that can take into account
several configurations.
For each configuration a genetic algorithm performs the training process with
an estimation of the global error; then the configuration is stored and the
genetic algorithm continues to evaluate a slightly different configuration.
Once the algorithm has taken into account all the possible configurations the
best one in terms of global error is chosen.
During the training phase the network is taught with environmental data as
input, the output depends on the neuronal weights, that at the very beginning
are initialized with random numbers. The network output is compared to the
experimental data we want to model, then the network estimates the error and
modifies the neurons weights in order to minimize the estimated error.
The training phase consists of determining both weights and configuration
(nunber of neurons and number of layers) by minimizing the error, i.e., the
difference between data and output.
## 3 Environmental variables and datasets
The environmental variables are monitored by an Oregon Scientific weather
station WMR100. The DAQ has been modified in order to acquire via USB the
environmental informations and merge environmental variables with output
variables. The accuracy of the temperature sensor is $\pm 1^{o}$C in the range
$0-40^{o}$C and the resolution is $0.1^{o}$C. The relative humidity sensor has
an operating range from 2% to 98% with a 1% resolution, $\pm 7\%$ absolute
accuracy from $25\%$ to $40\%$, and $\pm 5\%$ from $40\%$ to $80\%$. The
barometer operational range is between 700 mbar and 1050 mbar with a 1 mbar
resolution and a $\pm 10$ mbar accuracy.
The online monitoring system records the ambient temperature, pressure and
humidity of the GGM box that contains the RPC stack. Pressure and temperature
are mainly responsible of different detector behavior as well as the humidity
for the bakelite and gas properties.
The used dataset is composed of four periods, each period composed of runs
(about 270 each). Each run contains $10^{4}$ cosmic ray events where
environmental variables and GGM anodic output charges (Q) are collected. The
acquisition rate is typically 9.5 Hz.
## 4 Results
Typical ANN outputs show generally good agreement between data and prediction
during training phase. (Fig. 2 $(a)$). In periods where the prediction is not
accurate, the discrepancy is typically concentrated in narrow regions
(“spikes”). Fig 2 $(b)$ shows the prediction on period 3 using the period 1 as
training, the discrepancy around run 137 and run 256 are due to a set of
environmental variables not available in the training period as shown in Fig.
2 $(c)$ and Fig. 3.
Figure 2: $(a)$ Gap 7 trained on the period 3 - prediction on period 3; the
prediction is performed on the same period used as training with very good
agreement between experimental data and prediction. $(b)$ Gap 7 trained on the
period 1 - prediction on period 3, the prediction is performed on a period
different from the training one, the agreement depends on dispersion of
environmental variables. $(c)$ Environmental variables during the period 3.
Figure 3: Environmental variables during the period 1
The comparison between data and prediction is shown in Fig. 4 where the
quantity
$\frac{\Delta Q}{Q}\equiv\frac{Q_{EXP}-Q_{ANN}}{Q_{EXP}}$ (1)
is plotted for all four periods both for training (top) and predictions
(bottom), divided for training and prediction respectively. The error
distribution for the predictions is much wider than for the training, as
expected.
Figure 4: Error for training (top) and prediction (bottom) for all runs.
Gaussian fit superimposed. The quantity $\hat{\sigma}$ is the width of the
gaussian fit to the data in a reduced range which excludes the nongaussian
tails.
The gaussian fit superimposed (Fig. 4) is not able to fit the data properly
due to the presence of large nongaussian tails, which are caused by runs with
very large discrepancy between data and prediction. To evaluate the width
$\hat{\sigma}$ of the error distribution we perform a gaussian fit in a
reduced range which does not take into account the nongaussian tails. The
distribution of the error for the predictions shows a $\hat{\sigma}$ =
$6.7\%$. In the Table 3 there is a summary with error for training and
predictions. The cases with very large discrepancy were studied in detail, and
found to be characterized by a $(p,T,H)$ value at the edges of the variables
space.
To determine the measure of the dispersion of the environmental variables
considering all the runs ($N$) we computed the:
$\frac{\Delta
X}{X}\equiv\sqrt{\sum_{j=1,3}\biggr{[}\frac{(x_{j}-X_{j})}{X_{j}}\biggr{]}^{2}}$
(2)
$X_{j}\equiv\sum_{i=1,N}(x_{j})_{i}\quad;\quad{\bf x}\equiv(p,T,H)\quad$ (3)
The distribution of the $\frac{\Delta Q}{Q}$ error as a function of the
dispersion of environmental variables $\frac{\Delta X}{X}$ (Fig. 5) shows
three distinct structures. The satellite bands with very large error were
studied in detail. All data point in such bands belong to period four and gap
six for which problems were detected. Period four and gap six therefore were
excluded in the analysis. The distribution of the error as a function of
dispersion of environmental variables after this selection has a
$\hat{\sigma}\sim 4\%$ width and nongaussian tails extending up to
$\frac{\Delta Q}{Q}=200\%$.
Figure 5: Distribution of $\frac{\Delta Q}{Q}$ as a function of the dispersion
of environmental variables $\frac{\Delta X}{X}$ for all periods, six gaps and
both training and prediction. Each training period is included once, each
prediction is included 4 times, due to different training period chosen.
A selection on the fiducial volume in the x variables space (Table 2) was
applied in order to exclude from the analysis data with $(p,T,H)$ close to the
edges of the variable space. After the selection cuts, prediction on two
periods based on training on the third period were performed. The nongaussian
(NG) tails were defined as the fractional area outside the region $\pm
4\hat{\sigma}$. The selection cuts slightly reduce the width
($\hat{\sigma}<3.7\%$), while drastically reducing the nongaussian tails
(Table 3).
Table 2: Synopsis of the selection cuts for fiducial volume applied to predicted data. $(958<p<968){\rm mbar}$ | $(19.4<T<20.4)^{o}$C | $(34<H<44)\%$
---|---|---
Table 3: Summary of errors $\hat{\sigma}$ and nongaussian (NG) tails for various selection cuts and samples. Data sets | $\hat{\sigma}$ | NG tail
---|---|---
| % | %
All six chambers, all four periods training | $2.7$ | $2.26$
All six chambers, all four periods prediction | $6.7$ | $6.60$
Chamber six and period four excluded prediction | $3.0$ | $4.63$
Predict. on per. 2 and 3, train. on per. 1 | $4.0$ | $3.52$
Predict. on per. 3 and 1, train. on per. 2 | $3.4$ | $2.95$
Predict. on per. 1 and 2, train. on per. 3 | $3.8$ | $1.63$
Predict. on per. 2 and 3, train. on per. 1, fiducial cuts | $3.7$ | $0.49$
Predict. on per. 3 and 1, train. on per. 2, fiducial cuts | $2.9$ | $0.98$
Predict. on per. 1 and 2, train. on per. 3, fiducial cuts | $3.3$ | $0.29$
## 5 Discussion
In this study the GGM is the system used to train the neural network with
anode charge as output variable and $(p,T,H)$ as input variable. The addition
of the dark current as a output variable and dose as input variable is
expected to improve predictions and will be implemented. The main advantage of
this approach is that several variables can be used together in order to
predict chamber behavior without the needs of studying the surface corrosion,
environmental/radiation dependence and bakelite aging due to chemical
reactions and deposits; also in the ANN analysis, given enough data, it is
possible to decouple the effect of the chosen variables used as output. This
approach, once properly trained, could spot immediately and online
pathological chambers whose behavior is shifting from the normal one. Further
studies are in progress to determine and cure the residual nongaussian tails
of the $\frac{\Delta Q}{Q}$ errors distributions, to deal with training and
prediction on detectors with different high voltage supply, to widen the
sample of environmental conditions, and in adding new dimensions to the
variables space such as radiation levels.
## 6 Conclusions
A new approach for modeling the RPC behavior, based on ANN, has been
introduced and preliminary results obtained using data from the CMS RPC GGM
system. The ANN was trained for predicting the behavior of the anode charge Q
(output variables) as function of the environmental variables $(p,T,H)$ (input
variables), resulting in a prediction error $\frac{\Delta Q}{Q}=4\%$. In a
forthcoming work we plan to include the dose as input variable and the dark
current as output variable, aiming at a further improvement on the
predictions.
Acknowledgements
The skills of M. Giardoni, L. Passamonti, D. Pierluigi, B. Ponzio and A. Russo
(Frascati) in setting up the experimental setup are gratefully acknowledged.
The technical support of the CERN gas group is gratefully acknowledged. Thanks
are due to R. Guida (CERN Gas Group), Nadeesha M. Wickramage, Yasser Assran
for discussions and help. This research was supported in part by the Italian
Istituto Nazionale di Fisica Nucleare and Ministero dell’ Istruzione,
Università e Ricerca.
## References
* [1] R. Santonico and R. Cardarelli, Nucl. Instrum. Meth. 187 (1981) 377.
* [2] CMS Collaboration, JINST 0803 (2008) S08004. doi:10.1088/1748-0221/3/08/S08004.
* [3] The ATLAS Collaboration, G. Aad et al., CERN Large Hadron Collider , JINST 3 (2008) S08003.
* [4] G. D’Ali Staiti [ARGO-YBJ Collaboration], Nucl. Instrum. Meth. A 588 (2008) 7.
* [5] P. Fonte, IEEE Transactions on Nuclear Science, vol. 49, no. 3, June 2002.
* [6] CMS Collaboration, JINST 3 (2008) S08004.
* [7] CMS Collaboration,CERN-LHCC-97-032 ; CMS-TDR-003. Geneva, CERN, 1997.
* [8] L. Benussi et al., CMS NOTE 2010/076.
* [9] M. Abbrescia et al., LNF-06-34-P, LNF-04-25-P, Jan 2007. 9pp. Presented by S. Bianco on behalf of the CMS RPC Collaboration at the 2006 IEEE Nuclear Science Symposium (NSS), Medical Imaging Conference (MIC) and 15th International Room Temperature Semiconductor Detector Workshop, San Diego, California, 29 Oct - 4 Nov 2006. arXiv:physics/0701014.
* [10] L. Benussi et al., Nucl. Instrum. Meth. A 602 (2009) 805 [arXiv:0812.1108 [physics.ins-det]].
* [11] L. Benussi et al., JINST 4 (2009) P08006 [arXiv:0812.1710 [physics.ins-det]].
* [12] W. S. Mc Culloch, W. Pitts, Bulletin of Mathematical Biophysics 5 (1943) 115\.
* [13] K. Hornik, M. Stinchcombe and H. White, Neural Networks, vol. 2, pp. 359, 1989.
|
arxiv-papers
| 2010-12-26T11:55:58 |
2024-09-04T02:49:15.965545
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "L. Benussi (1), S. Bianco (1), S.Colafranceschi (1 and 2 and 3), F.L.\n Fabbri (1), M. Giardoni (1), L. Passamonti (1), D. Piccolo (1), D. Pierluigi\n (1), A. Russo (1), G. Saviano (1 and 2), S. Buontempo (4), A. Cimmino (4 and\n 5), M. de Gruttola (4 and 5), F Fabozzi (4), A.O.M. Iorio (4 and 5), L. Lista\n (4), P. Paolucci (4), P. Baesso (6), G. Belli (6), D. Pagano (6), S.P. Ratti\n (6), A. Vicini (6), P. Vitulo (6), C. Viviani (6), A. Sharma (3), A. K.\n Bhattacharyya (3) ((1) INFN Laboratori Nazionali di Frascati Via E. Fermi,\n Italy, (2) Sapienza Universit`a degli Studi di Roma \"La Sapienza\" Piazzale A.\n Moro Roma Italy, (3) CERN CH-1211 Gen\\'eve 23 F-01631 Switzerland, (4) INFN\n Sezione di Napoli Complesso Universitario di Monte Sant'Angelo, Italy, (5)\n Universit`a di Napoli Federico II Complesso Universitario di Monte\n Sant'Angelo, Italy, (6) INFN Sezione di Pavia and Universit`degli studi di\n Pavia Via Bassi, Italy)",
"submitter": "Stefano Colafranceschi",
"url": "https://arxiv.org/abs/1012.5508"
}
|
1012.5511
|
# STUDY OF GAS PURIFIERS FOR THE CMS RPC DETECTOR
L. Benussi S. Bianco S.Colafranceschi F.L. Fabbri F. Felli M. Ferrini M.
Giardoni T. Greci A. Paolozzi L. Passamonti D. Piccolo D. Pierluigi A.
Russo G. Saviano S. Buontempo A. Cimmino M. de Gruttola F Fabozzi A.O.M.
Iorio L. Lista P. Paolucci P. Baesso G. Belli D. Pagano S.P. Ratti A.
Vicini P. Vitulo C. Viviani R. Guida A. Sharma INFN Laboratori Nazionali
di Frascati, Via E. Fermi 40, I-00044 Fr ascati, Italy Sapienza Università
degli Studi di Roma ”La Sapienza”, Piazzale A. Moro, Roma, Italy CERN CH-1211
Genéve 23 F-01631 Switzerland INFN Sezione di Napoli, Complesso Universitario
di Monte Sant’Angelo, edificio 6, 80126 Napoli, Italy Università di Napoli
Federico II, Complesso Universitario di Monte Sant’Angelo, edificio 6, 80126
Napoli, Italy INFN Sezione di Pavia and Università degli studi di Pavia, Via
Bassi 6, 27100 Pavia, Italy
###### Abstract
The CMS RPC muon detector utilizes a gas recirculation system called closed
loop (CL) to cope with large gas mixture volumes and costs. A systematic study
of CL gas purifiers has been carried out over 400 days between July 2008 and
August 2009 at CERN in a low-radiation test area, with the use of RPC chambers
with currents monitoring, and gas analysis sampling points. The study aimed to
fully clarify the presence of pollutants, the chemistry of purifiers used in
the CL, and the regeneration procedure. Preliminary results on contaminants
release and purifier characterization are reported.
###### keywords:
RPC , CMS , gas , purifier detectors HEP muon
††thanks: Corresponding author: Giovanna Saviano
E-mail address: giovanna.saviano@uniroma1.it
## 1 Introduction
The Resistive Plate Chamber (RPC) [1] muon detector of the Compact Muon
Solenoid (CMS) experiment[2] utilizes a gas recirculation system called closed
loop (CL) [3], [4] to cope with large gas mixture volumes and costs. A
systematic study of Closed Loop gas purifiers has been carried out in 2008 and
2009 at the ISR experimental area of CERN with the use of RPC chambers exposed
to cosmic rays with currents monitoring and gas analysis sampling points.
Goals of the study [5] were to observe the release of contaminants in
correlation with the dark current increase in RPC detectors, to measure the
purifier lifetime, to observe the presence of pollutants and to study the
regeneration procedure. Previous results had shown the presence of metallic
contaminants, and an incomplete regeneration of purifiers [6],[7].
The basic function of the CMS CL system is to mix and purify the gas
components in the appropriate proportions and to distribute the mixture to the
individual chambers. The gas mixture used is 95.2% of C2H2F4 in its
environmental-friendly version R137a, 4.5% of $i$C4H10, and 0.3% SF6 to
suppress streamers and operate in saturated avalanche mode. Gas mixture is
humidified at the 45% RH (Relative Humidity) level typically to balance
ambient humidity, which affects the resistivity of highly hygroscopic
bakelite, and to improve efficiency at lower operating voltage. The CL is
operated with a fraction of fresh mixture continuously injected into the
system. Baseline design fresh mixture fraction for CMS is 2%, the test CL
system was operated at 10% fresh mixture. The fresh mixture fraction is the
fraction of the total gas content continuously replaced in the CL system with
fresh mixture. The filter configuration is identical to the CMS experiment.
## 2 Experimental setup and data sample
In the CL system gas purity is guaranteed by a multistage purifier system:
* •
The purifier-1 consisting of a cartridge filled with 5Å (10%) and 3Å (90%)
Type LINDE[12] molecular sieve[11] based on Zeolite manufactured by
ZEOCHEM[8];
* •
The purifier-2 consisting of a cartridge filled with 50% Cu-Zn filter type R12
manufactured by BASF[9] and 50% Cu filter type R3-11G manufactured by BASF;
* •
The purifier-3 consisting of a cartridge filled with Ni AlO3 filter type 6525
manufactured by LEUNA[10].
The experimental setup (Fig. 1) is composed of a CL system and an open mode
gas system. A detailed description of the CL, the experimental setup, and the
filters studied can be found in [7]. The CL is composed of mixer, purifiers
(in the subunit called “filters“ in the Fig. 1) , recirculation pump and
distribution to the RPC detectors. Eleven double-gap RPC detectors are
installed, nine in CL and two in open mode. Each RPC detector has two gaps
(upstream and downstream) whose gas lines are serially connected. The the gas
flows first in the upstream gap and then in the downstream gap. The detectors
are operated at a 9.2 kV power supply. The anode dark current drawn because of
the high bakelite resistivity is approximately 1-2 $\mu$A. Gas sampling points
before and after each filter in the closed loop allow gas sampling for
chemical and gaschromatograph analysis. The system is located in a temperature
and humidity controlled hut, with online monitoring of environmental
parameters.
Figure 1: Test setup with CL and open loop.
Chemical analyses have been performed in order to study the dynamical
behaviour of dark currents increase in the double-gap experimental setup and
correlate to the presence of contaminants, measure lifetime of unused
purifiers, and identify contaminant(s) in correlation with the increase of
currents. In the chemical analysis set-up (Fig. 2) the gas is sampled before
and after each CL purifier, and bubbled into a set of PVC flasks. The first
flask is empty and acts as a buffer, the second and third flasks contain 250
ml solution of LiOH (0.001 mole/l corresponding to 0.024 g/l, optimized to
keep the pH of the solution at 11). The bubbling of gas mixture into the two
flasks allows one to capture a wide range of elements that are likely to be
released by the system, such as Ca, Na, K, Cu, Zn, Cu, Ni, F. At the end of
each sampling line the flow is measured in order to have the total gas amount
for the whole period of sampling. Tygon filters ($0.45\mu$m) have been
installed upstream the flasks.
Figure 2: Chemical setup
The sampling points (Fig. 3) are located before the whole filters unit at
position HV61, after purifier-1 (Zeolite) at HV62, after purifier-2 (Cu/Zn
filter) at HV64 and after the Ni filter at position HV66. RPC are very
sensitive to environmental parameters (atmospheric pressure, humidity,
temperature), this study has been performed in environmentally controlled hut
with pressure, temperature and relative humidity online monitoring. The
comparison of temperature and humidity inside and outside the hut is displayed
in Fig. 4 and Fig. 5, respectively, over the whole time range of the test. The
inside temperature shows a variation of less than $\pm 0.5\,^{\circ}{\rm C}$;
the inside humidity still reveals seasonal structures between 35% and 50%, it
is, however, much smaller than the variation outside.
Gas mixture composition was monitored twice a day by gaschromatography, which
also provided the amount of air contamination, stable over the entire data
taking run and below 300 (100) ppm in closed (open) loop. Purifiers were
operated with unused filter material.
Figure 3: Chemical setup sampling points
Figure 4: Temperature distribution inside and outside the experimental hut.
Figure 5: Relative humidity inside and outside the experimental hut.
## 3 Results and discussion
The data-taking run was divided into cycles where different phenomena were
expected. We have four cycles (Fig. 6), i.e., initial stable currents (cycles
1 and 2), at the onset of the raise of currents (cycle 3), in the full
increase of currents (cycle 4). Cycle 4 was terminated in order not to damage
permanently the RPC detectors. The currents of all RPC detectors in open loop
were found stable over the four cycles. Fig. 6 shows the typical behaviour of
one RPC detector in CL. While the current of the downstream gap is stable
throughout the run, the current of the upstream gap starts increasing after
about seven months. Such behaviour is suggestive of the formation of
contaminants in the CL which are retained in the upstream gap, thus causing
its current to increase, and leaving the downstream gap undisturbed. While the
production of F- is constant during the run period, significant excess of K
and Ca is found in the gas mixture in cycles 3 and 4. The production of F- is
efficiently depressed by the zeolite purifier (Fig. 7). The observed excess of
K and Ca could be explained by a damaging effect of HF (continuously produced
by the system) on the zeolite filter whose structure contains such elements.
Further studies are in progress to confirm this model.
Figure 6: Dark currents increase in the upstream gap and not in the downstream
gap, correlated to the detection of contaminants in gas.
Figure 7: Concentration of F- in gas as measured by the chemical setup.
Sampling point HV61 is before the zeolite purifier, the others after each
purifier.
## 4 Conclusions
Preliminary results show that the lifetime of purifiers using unused material
is approximately seven months. Contaminants (K, Ca) are released in the gas in
correlation with the dark currents increase. The currents increase is observed
only in the upstream gap. The study suggests that contaminants produced in the
system stop in the upstream gap and affect its noise behaviour, leaving the
downstream gap undisturbed. The presence of an excess production of K and Ca
in coincidence with the currents increase suggests a damaging effect of HF
produced in the system on the framework of zeolites which is based on K and
Ca. Further studies are in progress to fully characterize the system over the
four cycles from the physical and the chemical point of view. The main goal is
to better schedule the operation and maintenance of filters for the CMS
experiment, where for a safe and reliable operation the filter regeneration is
presently performed several times per week. A second run is being started with
regenerated filter materials to measure their lifetime and confirm the
observation of contaminants. Finally, studies in high-radiation environment at
the CERN gamma irradiation facility are being planned.
Acknowledgements
The technical support of the CERN gas group is gratefully acknowledged. Thanks
are due to F. Hahn for discussions, and to Nadeesha M. Wickramage, Yasser
Assran for help in data taking shifts. This research was supported in part by
the Italian Istituto Nazionale di Fisica Nucleare and Ministero dell’
Istruzione, Università e Ricerca.
## References
* [1] R. Santonico and R. Cardarelli, “Development Of Resistive Plate Counters,” Nucl. Instrum. Meth. 187 (1981) 377.
* [2] CMS Collaboration, “The CMS experiment at the CERN LHC”, JINST 3 (2008) S08004.
* [3] M. Bosteels et al., “CMS Gas System Proposal”, CMS Note 1999/018.
* [4] L. Besset et al., “Experimental Tests with a Standard Closed Loop Gas Circulation System”, CMS Note 2000/040.
* [5] M. Abbrescia et al., “Proposal for a Systematic Study of the CERN Closed Loop Gas System Used by the RPC Muon Detectors in CMS”, Frascati preprint LNF-06/27(IR), available at http://www.lnf.infn.it/sis/preprint/ .
* [6] G. Saviano et al., “Materials studies for the RPC detector in CMS ”, presented at the RPC07 Conference, Mumbai (India), January 2008.
* [7] S.Bianco et al., “Chemical analyses of materials used in the CMS RPC muon detector”, CMS NOTE 2010/006.
* [8] Manufactured by ZEOCHEM, 8708 Uetikon (Switzerland).
* [9] BASF Technical Bulletin.
* [10] LEUNA Data Sheet September 9, 2003, Catalyst KL6526-T.
* [11] GRACE Davison Molecular Sieves data sheet.
* [12] LINDE Technical Bullettin.
* [13] L. Benussi et al., “Sensitivity and environmental response of the CMS RPC gas gain monitoring system,” JINST 4 (2009) DOI:10.1088 1748-0221 4 08 P08006 [arXiv:0812.1710 [physics.ins-det]].
* [14] M. Abbrescia et al., “HF Production In Cms-Resistive Plate Chambers,” Nucl. Phys. Proc. Suppl. 158 (2006) 30. NUPHZ,158,30;
* [15] G. Aielli et al., “Fluoride production in RPCs operated with F-compound gases”, 8th Workshop on Resistive Plate Chambers and Related Detectors, Seoul, Korea, 10-12 Oct 2005. Published in Nucl.Phys.Proc.Suppl. 158 (2006) 143.
|
arxiv-papers
| 2010-12-26T12:20:14 |
2024-09-04T02:49:15.970286
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "L. Benussi (1), S. Bianco (1), S.Colafranceschi (1 and 2 and3), F.L.\n Fabbri (1), F. Felli (1 and 2), M. Ferrini (1 and 2), M. Giardoni (1), T.\n Greci (1 and 2), A. Paolozzi (1 and 2), L. Passamonti (1), D. Piccolo (1), D.\n Pierluigi (1), A. Russo (1), G. Saviano (1 and 2), S. Buontempo (4), A.\n Cimmino (4 and 5), M. de Gruttola (4 and 5), F Fabozzi d A.O.M. Iorio (4 and\n 5), L. Lista (4), P. Paolucci (4), P. Baesso (6), G. Belli (6), D. Pagano\n (6), S.P. Ratti (6), A. Vicini (6), P. Vitulo (6), C. Viviani (6), R. Guida\n (3), A. Sharma (3) ((1) INFN Laboratori Nazionali di Frascati Via E. Fermi,\n Italy, (2) Sapienza Universit\\'a degli Studi di Roma \"La Sapienza\" Piazzale\n A. Moro Roma Italy, (3) CERN CH-1211 Gen\\'eve 23 F-01631 Switzerland, (4)\n INFN Sezione di Napol Complesso Universitario di Monte Sant'Angelo edificio 6\n 80126 Napoli Italy, (5) Universit\\'a di Napoli Federico II Complesso\n Universitario di Monte Sant'Angelo edificio 6 80126 Napoli Italy, (6) INFN\n Sezione di Pavia and Universit\\'a degli studi di Pavia Via Bassi, Italy)",
"submitter": "Stefano Colafranceschi",
"url": "https://arxiv.org/abs/1012.5511"
}
|
1012.5538
|
Generating functions for the Bernstein polynomials: A unified approach to
deriving identities for the Bernstein basis functions
Yilmaz Simsek
Department of Mathematics, Faculty of Science University of Akdeniz TR-07058
Antalya, Turkey
E-mail: ysimsek@akdeniz.edu.tr
Abstract
> The main aim of this paper is to provide a unified approach to deriving
> identities for the Bernstein polynomials using a novel generating function.
> We derive various functional equations and differential equations using this
> generating function. Using these equations, we give new proofs both for a
> recursive definition of the Bernstein basis functions and for derivatives of
> the $n$th degree Bernstein polynomials. We also find some new identities and
> properties for the Bernstein basis functions. Furthermore, we discuss
> analytic representations for the generalized Bernstein polynomials through
> the binomial or Newton distribution and Poisson distribution with mean and
> variance. Using this novel generating function, we also derive an identity
> which represents a pointwise orthogonality relation for the Bernstein basis
> functions. Finally, by using the mean and the variance, we generalize Szasz-
> Mirakjan type basis functions.
2010 Mathematics Subject Classification. 14F10, 12D10, 26C05, 26C10, 30B40,
30C15.
Key Words and Phrases. Bernstein polynomials; generating function; Szasz-
Mirakjan basis functions; Bezier curves; Binomial distribution; Poisson
distribution.
## 1\. Introduction and main definition
The Bernstein polynomials have many applications in approximations of
functions, in statistics, in numerical analysis, in $p$-adic analysis and in
the solution of differential equations. It is also well-known that in Computer
Aided Geometric Design polynomials are often expressed in terms of the
Bernstein basis functions.
Many of the known identities for the Bernstein basis functions are currently
derived in an ad hoc fashion, using either the binomial theorem, the binomial
distribution, tricky algebraic manipulations or blossoming. The main purpose
of this work is to construct novel generating functions for the Bernstein
polynomials. Using these novel generating functions, we develop a unify
approach both to standard and to new identities for the Bernstein polynomials.
The following definition gives us generating functions for the Bernstein basis
functions:
###### Definition 1.
Let $a$ and $b$ be nonnegative real parameters with $a\neq b$. Let $m$ a be
positive integer and let $x\in\left[a,b\right]$. Then the Bernstein basis
functions $\mathbb{Y}_{k}^{n}(x;a,b,m)$ are defined by means of the following
generating function:
$\displaystyle f_{\mathbb{Y},k}(x,t;a,b,m)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}\sum_{l=0}^{k}\left(\begin{array}[]{c}j+m-1\\\
j\end{array}\right)(-1)^{k-l}\frac{t^{k}x^{l}a^{j+k-l}b^{-m-j}e^{(b-x)t}}{l!(k-l)!}$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\mathbb{Y}_{k}^{n}(x;a,b,m)\frac{t^{n}}{n!},$
where $t\in\mathbb{C}$ and $0^{j}=\left\\{\begin{array}[]{cc}0&\text{if }j\neq
0,\\\ 1&\text{if }j=0.\end{array}\right.$
The remainder of this study is organized as follows:
Section 2: We find many functional equations and differential equations of
this novel generating function. Using these equations, many properties of the
Bernstein basis functions can be determined. For instance, we give new proofs
of the recursive definition of the Bernstein basis functions as well as a
novel derivation for the two term formula for the derivatives of the $n$th
degree Bernstein basis functions. We also prove many other properties of the
Bernstein basis functions via functional equations.
Jetter and Stöckler [9] proved an identity for multivariate Bernstein
polynomials on a simplex, which is considered a pointwise orthogonality
relation. The integral version of this identity provides a new representation
for the polynomial basis dual to the Bernstein basis. An identity for the
reproducing kernel is used to define quasi-interpolants of arbitrary order. As
an application of the identity of Jetter and Stöckler, Abel and Li [1] gave
Proposition 1, in Section 3. Their method is based on generating functions,
which reveals the general structure of the identity. As an applications of
Proposition 1 they derive generating functions for the Baskakov basis
functions and the Szasz-Mirakjan basis functions. Using Eq-(2.7) in Section 2,
they exhibit a special case of the identity of Jetter and Stöckler for the
Bernstein basis functions. In Section 3; we give relations between the
Bernstein basis functions, the binomial distribution and the Poisson
distribution. Using the Poisson distribution, we give generating functions for
the Szasz-Mirakjan type basis functions. By using Abel and Li’s [1] method,
and applying our generating functions to Proposition 1, we derive identities
which give pointwise orthogonality relations for the Bernstein polynomials and
the Szasz-Mirakjan type basis functions.
## 2\. Unified approach to deriving new proofs of the identities and
properties for the Bernstein polynomials
The Bernstein polynomials and related polynomials have been studied and
defined in many different ways, for examples by $q$-series, complex functions,
$p$-adic Volkenborn integrals and many algorithms.
In this section, we provide fundamental properties of the Bernstein basis
functions and their generating functions. We introduce some functional
equations and differential equations of the novel generating functions for the
Bernstein basis functions. We also give new proofs of some well known
properties of the Bernstein basis functions via functional equations and
differential equations.
### 2.1. Generating Functions
We now modify (1) as follows:
By the negative binomial theorem, we have
$\frac{1}{b^{m}(1-\frac{a}{b})^{m}}=\frac{1}{b^{m}}\sum_{j=0}^{\infty}\left(\begin{array}[]{c}j+m-1\\\
j\end{array}\right)a^{j}b^{-m-j}.$ (2.1)
Substituting (2.1) into (1), we get
$\displaystyle f_{\mathbb{Y},k}(x,t;a,b,m)$ $\displaystyle=$
$\displaystyle\frac{t^{k}e^{(b-x)t}}{(b-a)^{m}k!}\sum_{l=0}^{k}\left(\begin{array}[]{c}k\\\
l\end{array}\right)(-1)^{k-l}x^{l}a^{k-l}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\mathbb{Y}_{k}^{n}(x;a,b,m)\frac{t^{n}}{n!}.$
Thus we obtain the following novel generating function, which is a
modification of (1):
$\displaystyle f_{\mathbb{Y},k}(x,t;a,b,m)$ $\displaystyle=$
$\displaystyle\frac{t^{k}\left(x-a\right)^{k}e^{(b-x)t}}{(b-a)^{m}k!}$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\mathbb{Y}_{k}^{n}(x;a,b,m)\frac{t^{n}}{n!}.$
###### Remark 1.
If we set $a=0$ and $b=1$ in (2.1), we obtain a result given by Simsek and
Acikgoz [13] and Acikgoz and Arici [2]:
$\frac{(xt)^{k}}{k!}e^{(1-x)t}=\sum_{n=0}^{\infty}B_{k}^{n}(x)\frac{t^{n}}{n!},$
so that, obviously;
$\mathbb{Y}_{k}^{n}(x;0,1,m)=B_{k}^{n}(x),$
where $B_{k}^{n}(x)$ denote the Bernstein polynomials.
By using the Taylor series for $e^{(b-x)t}$ in (2.1), we get
$\frac{(x-a)^{k}}{(b-a)^{m}k!}\sum_{n=0}^{\infty}\mathbb{(}b-x\mathbb{)}^{n}\frac{t^{n+k}}{n!}=\sum_{n=0}^{\infty}\mathbb{Y}_{k}^{n}(x;a,b,m)\frac{t^{n}}{n!}.$
Comparing the coefficients of $t^{k}$ on the both sides of the above equation,
we arrive at the following theorem:
###### Theorem 1.
Let $a$ and $b$ be nonnegative real parameters with $a\neq b$. Let $m$ be a
positive integer and let $x\in\left[a,b\right]$. Let $k$ and $n$ be non-
negative integers with $n\geq k$. Then
$\mathbb{Y}_{k}^{n}(x;a,b,m)=\left(\begin{array}[]{c}n\\\
k\end{array}\right)\frac{\left(x-a\right)^{k}(b-x)^{n-k}}{(b-a)^{m}},$ (2.4)
where $k=0$, $1$,$\cdots$, $n$, and $\left(\begin{array}[]{c}n\\\
k\end{array}\right)=\frac{n!}{k!(n-k)!}$.
###### Remark 2.
For $m=n$, the Bernstein basis functions of degree $n$ are defined by (2.4).
###### Remark 3.
In the special case when $m=n$, Theorem 1 immediately yields the corresponding
well known results concerning the Bernstein basis functions $B_{k}^{n}(x)$
that appears for example in Goldman [5, p. 384, Eq.(24.6)] and cf. [3]:
$\mathbb{Y}_{k}^{n}(x;a,b,n)=B_{k}^{n}(x;a,b)=\left(\begin{array}[]{c}n\\\
k\end{array}\right)\frac{\left(x-a\right)^{k}(b-x)^{n-k}}{(b-a)^{n}},$
where $k=0$, $1$,$\cdots$, $n$ and $x\in[a,b]$. One can easily see that
$B_{k}^{n}(x)=\left(\begin{array}[]{c}n\\\
k\end{array}\right)x^{k}(1-x)^{n-k},$ (2.5)
where $k=0,1,\cdots,n$ and $x\in[0,1]$ cf. [1]-[13]. In [5], Goldman gives
many properties of the Bernstein polynomials $B_{k}^{n}(x,a,b)$. The functions
$B_{0}^{n}(x,a,b),\cdots,B_{n}^{n}(x,a,b)$ are called the Bernstein basis
functions. Goldman [5], in Chapter 26, shows that the Bernstein basis
functions form a basis for the polynomials of degree $n$. The Bezier curve
$B(t)$ with control points $P_{0}$,$\cdots$, $P_{n}$ is defined as follows:
$B(t)=\sum_{k=0}^{n}P_{k}B_{k}^{n}(x,a,b)\text{ cf.
\cite[cite]{[\@@bibref{}{GoldmanBOOK}{}{}]}.}$
###### Remark 4.
By using (2.4), we have
$\sum_{n=0}^{\infty}\mathbb{Y}_{k}^{n}(x;a,b,m)\frac{t^{n}}{n!}=\sum_{n=0}^{\infty}\left(\begin{array}[]{c}n\\\
k\end{array}\right)\frac{\left(x-a\right)^{k}(b-x)^{n-k}}{(b-a)^{m}}\frac{t^{n}}{n!}.$
From this equation, we obtain
$\sum_{n=0}^{\infty}\mathbb{Y}_{k}^{n}(x;a,b,m)\frac{t^{n}}{n!}=\frac{\left(x-a\right)^{k}t^{k}}{k!(b-a)^{m}}\sum_{n=k}^{\infty}(b-x)^{n-k}\frac{t^{n-k}}{\left(n-k\right)!}.$
The series on the right hand side is the Taylor series for $e^{(b-x)t}$; thus
we arrive at (2.1).
Substituting $m=n$ in (2.4), we now give another well-known generating
function for the Bernstein basis functions:
$\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}\mathbb{Y}_{k}^{n}(x;a,b,n)t^{k}\right)\frac{z^{n}}{n!}=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}\left(\begin{array}[]{c}n\\\
k\end{array}\right)t^{k}\left(\frac{x-a}{b-a}\right)^{k}\left(\frac{b-x}{b-a}\right)^{n-k}\right)\frac{z^{n}}{n!}.$
By using the Cauchy product in the above equation, we have
$\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}\mathbb{Y}_{k}^{n}(x;a,b,n)t^{k}\right)\frac{z^{n}}{n!}=\sum_{n=0}^{\infty}\left(t\frac{x-a}{b-a}\right)\frac{z^{n}}{n!}\sum_{n=0}^{\infty}\left(\frac{b-x}{b-a}\right)\frac{z^{n}}{n!}.$
From this equation, we find that
$\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}\mathbb{Y}_{k}^{n}(x;a,b,n)t^{k}\right)\frac{z^{n}}{n!}=e^{z\left(\frac{b-x}{b-a}+t\frac{x-a}{b-a}\right)}.$
After some elementary calculations in the above relation, we arrive at the
following generating function for the Bernstein basis functions:
$\sum_{k=0}^{n}\mathbb{Y}_{k}^{n}(x;a,b,n)t^{k}=\left(\frac{b-x}{b-a}+t\frac{x-a}{b-a}\right)^{n}.$
(2.6)
###### Remark 5.
If we set $a=0$, $b=1$ and $m=n$ in (2.6), then we have
$\sum_{k=0}^{n}B_{k}^{n}(x)t^{k}=\left(\left(1-x\right)+tx\right)^{n}.$ (2.7)
This generating functions is given by Goldman [7]-[6, Chapter 5, pages
299-306]. Goldman [7]-[6, Chapter 5, pages 299-306] also constructs the
following generating functions the univariate and bivariate Bernstein basis
functions:
$\sum_{k=0}^{n}B_{k}^{n}(x)e^{ky}=\left(\left(1-x\right)+te^{y}\right)^{n},$
$\sum_{i+j+k=n}B_{i,j,k}^{n}(s,t)x^{i}y^{j}=\left(\left(1-s-t\right)+sx+ty\right)^{n},$
where
$B_{i,j,k}^{n}(s,t)=\left(\begin{array}[]{c}n\\\
ijk\end{array}\right)s^{i}t^{j}\left(1-s-t\right)^{k}\text{ and
}\left(\begin{array}[]{c}n\\\ ijk\end{array}\right)=\frac{n!}{i!j!k!}$
and
$\sum_{i+j+k=n}B_{i,j,k}^{n}(s,t)e^{ix}e^{jy}=\left(\left(1-s-t\right)+se^{x}+te^{y}\right)^{n}.$
Below are some well-known properties of the Bernstein basis functions:
Non-negative property:
$\mathbb{Y}_{k}^{n}(x;a,b,m)\geq 0\text{, for }0\leq a\leq x\leq b.$ (2.8)
Symmetry property:
$\mathbb{Y}_{k}^{n}(x;a,b,m)=\mathbb{Y}_{n-k}^{n}(b+a-x;a,b,m).$ (2.9)
Corner values:
$\mathbb{Y}_{k}^{n}(a;a,b,m)=\left\\{\begin{array}[]{cc}0&\text{if }k\neq
0,\\\ 1&\text{if }k=0,\end{array}\right.$ (2.10)
and
$\mathbb{Y}_{k}^{n}(b;a,b,m)=\left\\{\begin{array}[]{cc}0&\text{if }k\neq
n,\\\ 1&\text{if }k=n.\end{array}\right.$ (2.11)
Alternating sum:
Substituting $m=n$ in (2.4), we get
$\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}(-1)^{k}\mathbb{Y}_{k}^{n}(x;a,b,n)\right)\frac{t^{n}}{n!}=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}\frac{\left(\frac{a-x}{b-a}\right)^{k}\left(\frac{b-x}{b-a}\right)^{n-k}}{k!(n-k)!}\right)t^{n}.$
By using the Cauchy product in the above equation, we have
$\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}(-1)^{k}\mathbb{Y}_{k}^{n}(x;a,b,n)\right)\frac{t^{n}}{n!}=e^{\left(\frac{a+b-2x}{b-a}\right)t}.$
From this relation, we arrive at the following formula for the alternating
sum.
$\sum_{k=0}^{n}(-1)^{k}\mathbb{Y}_{k}^{n}(x;a,b,n)=\left(\frac{a+b-2x}{b-a}\right)^{n}.$
(2.12)
###### Remark 6.
If we set $a=0$, $b=1$ and $m=n$, then Eq-(2.8)-Eq-(2.12) reduce to Goldman’s
results [7]-[6, Chapter 5, pages 299-306]. In [7] and [6, Chapter 5, pages
299-306], Goldman also gives many identities and properties for the univariate
and bivariate Bernstein basis functions, for example boundary values, maximum
values, partitions of unity, representation of monomials, representation in
terms of monomials, conversion to monomial form, linear independence,
Descartes’ law of sign, discrete convolution, unimodality, subdivision,
directional derivatives, integrals, Marsden identities, De Boor-Fix formulas,
and the other properties.
A Bernstein polynomial $\mathcal{P}(x,a,b,m)$ is a polynomial represented in
the Bernstein basis functions:
$\mathcal{P}(x,a,b,m)=\sum_{k=0}^{n}c_{k}^{n}\mathbb{Y}_{k}^{n}(x;a,b,m).$
(2.13)
###### Remark 7.
If we set $a=0$, $b=1$ and $m=n$ (2.13), then we have
$P(x)=\sum_{k=0}^{n}c_{k}^{n}B_{k}^{n}(x)$
cf. [4].
By using (2.1), we obtain the following functional equation:
$f_{\mathbb{Y},k_{1}}(x,t;a,b,m_{1})f_{\mathbb{Y},k_{2}}(x,t;a,b,m_{2})=\frac{\left(\begin{array}[]{c}k_{1}+k_{2}\\\
k_{1}\end{array}\right)}{2^{k_{1}+k_{2}}}f_{\mathbb{Y},k_{1}+k_{2}}(x,2t;a,b,m_{1}+m_{2}),$
where
$\left(\begin{array}[]{c}k_{1}+k_{2}\\\
k_{1}\end{array}\right)=\left(\begin{array}[]{c}k_{1}+k_{2}\\\
k_{2}\end{array}\right)=\frac{\left(k_{1}+k_{2}\right)!}{k_{1}!k_{2}!}.$
By using the definition of the novel generating function
$f_{\mathbb{Y},k}(x,t;a,b,m)$ in the preceding equation, we get
$\displaystyle\sum_{n=0}^{\infty}\mathbb{Y}_{k_{1}}^{n}(x;a,b,m_{1})\frac{t^{n}}{n!}\sum_{n=0}^{\infty}\mathbb{Y}_{k_{2}}^{n}(x;a,b,m_{2})\frac{t^{n}}{n!}$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\mathbb{Y}_{k_{1}+k_{2}}^{n}(x;a,b,m_{1}+m_{2})\frac{2^{n-k_{1}-k_{2}}\left(k_{1}+k_{2}\right)!t^{n}}{n!k_{1}!k_{2}!}.$
And using the Cauchy product in this equation, we have
$\displaystyle\sum_{n=0}^{\infty}\left(\mathop{\displaystyle\sum}\limits_{j=0}^{n}\left(\begin{array}[]{c}n\\\
j\end{array}\right)\mathbb{Y}_{k_{1}}^{j}(x;a,b,m_{1})\mathbb{Y}_{k_{2}}^{n-j}(x;a,b,m_{2})\right)\frac{t^{n}}{n!}$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\mathbb{Y}_{k_{1}+k_{2}}^{n}(x;a,b,m_{1}+m_{2})\frac{2^{n-k_{1}-k_{2}}\left(k_{1}+k_{2}\right)!t^{n}}{n!k_{1}!k_{2}!}.$
Comparing the coefficients of $\frac{t^{n}}{n!}$ on the both sides of the
above equation, we arrive at the following theorem:
###### Theorem 2.
Let $m_{1}$ and $m_{2}$ be integers. Then the following identity holds:
$\mathbb{Y}_{k_{1}+k_{2}}^{n}(x;a,b,m_{1}+m_{2})=\frac{2^{k_{1}+k_{2}-n}k_{1}!k_{2}!}{\left(k_{1}+k_{2}\right)!}\mathop{\displaystyle\sum}\limits_{j=0}^{n}\left(\begin{array}[]{c}n\\\
j\end{array}\right)\mathbb{Y}_{k_{1}}^{j}(x;a,b,m_{1})\mathbb{Y}_{k_{2}}^{n-j}(x;a,b,m_{2}).$
Observe that if we set $a=0$ and $b=1$, then we have
$B_{k_{1}+k_{2}}^{n}(x)=\frac{2^{k_{1}+k_{2}-n}k_{1}!k_{2}!}{\left(k_{1}+k_{2}\right)!}\mathop{\displaystyle\sum}\limits_{j=0}^{n}\left(\begin{array}[]{c}n\\\
j\end{array}\right)B_{k_{1}}^{j}(x)B_{k_{2}}^{n-j}(x).$
Note that many new identities can be found via functional equations for the
novel generating functions of the Bernstein basis functions. We derive some
functional equations and identities related to the generating functions and
the Bernstein basis functions in the remainder of this section.
### 2.2. Subdivision property
The following functional equation of the novel generating functions is
fundamental to driving the subdivision property for the Bernstein basis
functions.
Let we us define
$f_{\mathbb{Y},j}(xy,t;a,b,n)=f_{\mathbb{Y},j}\left(x,t\left(\frac{y-a}{b-a}\right);a,b,n\right)e^{t\left(\frac{b-y}{b-a}\right)}.$
(2.15)
From this generating function, we have the following theorem:
###### Theorem 3.
Let $a\leq yx\leq b$. Then the following identity holds:
$\mathbb{Y}_{j}^{n}(xy;a,b,n)=\mathop{\displaystyle\sum}\limits_{k=j}^{n}\mathbb{Y}_{j}^{k}(x;a,b,k)\mathbb{Y}_{k}^{n}(y;a,b,n-k).$
###### Proof.
By equations (2.1) and (2.15), we obtain
$\displaystyle\sum_{n=j}^{\infty}\mathbb{Y}_{j}^{n}(xy;a,b,n)\frac{t^{n}}{n!}$
$\displaystyle=$
$\displaystyle\left(\sum_{n=0}^{\infty}\mathbb{Y}_{j}^{n}(x;a,b,n)\left(\frac{y-a}{b-a}\right)^{n}\frac{t^{n}}{n!}\right)\left(\sum_{n=0}^{\infty}\frac{\left(\frac{b-y}{b-a}\right)^{n}t^{n}}{n!}\right).$
Using the Cauchy product in this equation, we get
$\sum_{n=j}^{\infty}\mathbb{Y}_{j}^{n}(xy;a,b,m)\frac{t^{n}}{n!}=\sum_{n=j}^{\infty}\left(\mathop{\displaystyle\sum}\limits_{k=j}^{n}\mathbb{Y}_{j}^{n}(x;a,b,k)\frac{\left(\frac{y-a}{b-a}\right)^{k}\left(\frac{b-y}{b-a}\right)^{n-k}}{k!\left(n-k\right)!}\right)t^{n}.$
Substituting (2.4) into the above equation then after some elementary
manipulations, we arrive at the desired result.
###### Remark 8.
Substituting $a=0$, $b=1$ and $m=n$ into Theorem 3, we have
$B_{j}^{n}(xy)=\mathop{\displaystyle\sum}\limits_{k=j}^{n}B_{j}^{k}(x)B_{k}^{n}(y).$
(2.16)
The above identity is essentially the subdivision property for the Bernstein
basis functions. This identity is a bit tricky to prove with algebraic
manipulations.
###### Remark 9.
Goldman [7]-[6, Chapter 5, pages 299-306] proves equation (2.16) with
algebraic manipulations. He also proves the following subdivision properties:
$B_{j}^{n}(\left(1-y\right)x+y)=\mathop{\displaystyle\sum}\limits_{k=0}^{j}B_{j-k}^{n-k}(x)B_{k}^{n}(y),$
and
$B_{j}^{n}(\left(1-y\right)x+yz)=\mathop{\displaystyle\sum}\limits_{k=0}^{n}\left(\mathop{\displaystyle\sum}\limits_{p+q=j}B_{p}^{n-k}(x)B_{q}^{k}(z)\right)B_{k}^{n}(y)$
for the others see cf. [7]-[6, Chapter 5, pages 299-306].
### 2.3. Differentiating the generating function
In this section we give higher order derivatives of the Bernstein basis
functions by differentiating the generating function in (2.1) with respect to
$x$. Using Leibnitz’s formula for the $l$th derivative, with respect to $x$,
of the product $f_{\mathbb{Y},k}(x,t;a,b,m)$ of two functions
$g(t,x;a,b)=\frac{t^{k}\left(x-a\right)^{k}}{(b-a)^{m}k!}$ with $a\neq b$ and
$h(t,x;b)=e^{(b-x)t}$, we obtain the following higher order partial derivative
equation:
$\frac{\partial^{l}f_{\mathbb{Y},k}(x,t;a,b,m)}{\partial
x^{l}}=\mathop{\displaystyle\sum}\limits_{j=0}^{l}\left(\begin{array}[]{c}l\\\
j\end{array}\right)\left(\frac{\partial^{j}g(t,x;a,b)}{\partial
x^{j}}\right)\left(\frac{\partial^{l-j}h(t,x;b)}{\partial x^{l-j}}\right).$
From this equation, we arrive at the following theorem:
###### Theorem 4.
Let $l$ be a non-negative integer. Then
$\frac{\partial^{l}f_{\mathbb{Y},k}(x,t;a,b,m)}{\partial
x^{l}}=\mathop{\displaystyle\sum}\limits_{j=0}^{l}\left(\begin{array}[]{c}l\\\
j\end{array}\right)(-1)^{l-j}\frac{t^{l}}{(b-a)^{j}}f_{\mathbb{Y},k-j}(x,t;a,b,m-j).$
By using Theorem 4, we obtain higher order derivatives of the Bernstein basis
functions by the following theorem:
###### Theorem 5.
Let $a$ and $b$ be nonnegative real parameters with $a\neq b$. Let $m$ be a
positive integer and let $x\in\left[a,b\right]$. Let $k$, $l$ and $n$ be
nonnegative integers with $n\geq k$. Then
$\frac{d^{l}\mathbb{Y}_{k}^{n}(x;a,b,m)}{dx^{l}}=\mathop{\displaystyle\sum}\limits_{j=0}^{l}(-1)^{l-j}\left(\begin{array}[]{c}n\\\
n-l,l-j,j\end{array}\right)\frac{l!}{(b-a)^{j}}\mathbb{Y}_{k-j}^{n-l}(x;a,b,m-j),$
where
$\left(\begin{array}[]{c}n\\\ x,y,z\end{array}\right)=\frac{n!}{x!y!z!}\text{,
with }n=x+y+z.$
###### Remark 10.
Substituting $a=0$, $b=1$ and $m=n$ into Theorem 5, we have
$\frac{d^{l}B_{k}^{n}(x)}{dx^{l}}=\mathop{\displaystyle\sum}\limits_{j=0}^{l}(-1)^{l-j}\left(\begin{array}[]{c}n\\\
n-l,l-j,j\end{array}\right)l!B_{k-j}^{n-l}(x),$
or
$\frac{d^{l}B_{k}^{n}(x)}{dx^{l}}=\frac{n!}{(n-l)!}\mathop{\displaystyle\sum}\limits_{j=0}^{l}(-1)^{l-j}\left(\begin{array}[]{c}n\\\
j\end{array}\right)B_{k-j}^{n-l}(x),$
cf. ([7], [6, Chapter 5, pages 299-306]).
Substituting $l=1$ into Theorem 5, we arrive at the following corollary:
###### Corollary 1.
Let $a$ and $b$ be nonnegative real parameters with $a\neq b$. Let $m$ be a
positive integer and let $x\in\left[a,b\right]$. Let $k$ and $n$ be
nonnegative integers with $n\geq k$. Then
$\frac{d}{dx}\mathbb{Y}_{k}^{n}(x;a,b,m)=n\left(\frac{\mathbb{Y}_{k-1}^{n-1}(x;a,b,m-1)-\mathbb{Y}_{k}^{n-1}(x;a,b,m-1)}{b-a}\right).$
###### Remark 11.
By setting $m=n$ in Corollary 1, we arrive at the known known result recorded
by Goldman [5]:
$\frac{d}{dx}B_{k}^{n}(x;a,b)=n\left(\frac{B_{k-1}^{n-1}(x;a,b)-B_{k}^{n-1}(x;a,b)}{b-a}\right).$
###### Remark 12.
One can also see the following special case of Theorem 1 when $a=0$ and $b=1$:
$\frac{d}{dx}B_{k}^{n}(x)=n\left(B_{k-1}^{n-1}(x)-B_{k}^{n-1}(x)\right)$
cf. [1]-[13].
### 2.4. Recurrence Relation
In this section by using higher order derivatives of the novel generating
function with respect to $t$, we derive a partial differential equation. Using
this equation, we shall give a new proof of the recurrence relation for the
Bernstein basis functions.
Differentiating Eq-(1) with respect to $t$, we prove a recurrence relation for
the polynomials $\mathbb{Y}_{k}^{n}(x;a,b,m)$. This recurrence relation can
also be obtained from Eq-(2.4). By using Leibnitz’s formula for the $v$th
derivative, with respect to $t$, of the product $f_{\mathbb{Y},k}(x,t;a,b,m)$
of two function $g(t,x;a,b)=\frac{t^{k}\left(x-a\right)^{k}}{(b-a)^{m}k!}$
with $a\neq b$ and $h(t,x;b)=e^{(b-x)t}$, we obtain another higher order
partial differential equation as follows:
$\frac{\partial^{v}f_{\mathbb{Y},k}(x,t;a,b,m)}{\partial
t^{v}}=\mathop{\displaystyle\sum}\limits_{j=0}^{v}\left(\begin{array}[]{c}v\\\
j\end{array}\right)\left(\frac{\partial^{j}g(t,x;a,b)}{\partial
t^{j}}\right)\left(\frac{\partial^{v-j}h(t,x;b)}{\partial t^{v-j}}\right).$
From the above equation, we have the following theorem:
###### Theorem 6.
Let $v$ be an integer number. Then
$\frac{\partial^{v}f_{\mathbb{Y},k}(x,t;a,b,m)}{\partial
t^{v}}=\mathop{\displaystyle\sum}\limits_{j=0}^{v}(b-a)^{v-j}\mathbb{Y}_{j}^{v}(x;a,b,v)f_{\mathbb{Y},k-j}(x,t;a,b,m-j),$
where $f_{\mathbb{Y},k}(x,t;a,b,m)$ and $\mathbb{Y}_{j}^{v}(x;a,b,v)$ are
defined in (2.1) and (2.4), respectively.
Using definition (2.1) and (2.4) in Theorem 6, we obtain a recurrence relation
for the Bernstein basis functions by the following theorem:
###### Theorem 7.
Let $a$ and $b$ be nonnegative real parameters with $a\neq b$. Let $m$ be a
positive integer and let $x\in\left[a,b\right]$. Let $k$, $v$ and $n$ be
nonnegative integers with $n\geq k$. Then
$\mathbb{Y}_{k}^{n}(x;a,b,m)=\mathop{\displaystyle\sum}\limits_{j=0}^{v}(b-a)^{v-j}\mathbb{Y}_{j}^{v}(x;a,b,v)\mathbb{Y}_{k-j}^{n-v}(x;a,b,m-j).$
###### Remark 13.
Substituting $a=0$ and $b=1$ into Theorem 7, we obtain the following result:
$B_{k}^{n}(x)=\mathop{\displaystyle\sum}\limits_{j=0}^{v}B_{j}^{v}(x)B_{k-j}^{n-v}(x).$
Substituting $v=1$ into Theorem 7, we arrive at the following corollary:
###### Corollary 2.
(Recurrence Relation) Let $a$ and $b$ be nonnegative real parameters with
$a\neq b$. Let $m$ be a positive integer and let $x\in\left[a,b\right]$. Let
$k$ and $n$ be nonnegative integers with $n\geq k$. Then
$\displaystyle\mathbb{Y}_{k}^{n}(x;a,b,m)$ $\displaystyle=$
$\displaystyle\frac{x-a}{b-a}\mathbb{Y}_{k-1}^{n-1}(x;a,b,m-1)$
$\displaystyle+\frac{b-x}{b-a}\mathbb{Y}_{k}^{n-1}(x;a,b,m-1).$
###### Remark 14.
Differentiating equation (1) with respect to $t$, we also get
$\displaystyle\frac{x-a}{b-a}f_{\mathbb{Y},k-1}(x,t;a,b,m-1)+\frac{b-x}{b-a}f_{\mathbb{Y},k}(x,t;a,b,m-1)$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\mathbb{Y}_{k}^{n}(x;a,b,m)\frac{t^{n-1}}{\left(n-1\right)!}.$
From this equation, one can also obtain Corollary 2.
###### Remark 15.
By setting $a=0$ and $b=1$ in (2), one obtains the following relation:
$B_{k}^{n}(x)=(1-x)B_{k}^{n-1}(x)+xB_{k-1}^{n-1}(x).$
### 2.5. Multiplication and division by powers of $(\frac{x-a}{b-a})^{d}$ and
$(\frac{b-x}{b-a})^{d}$
In [4], Buse and Goldman present much background material on computations with
Bernstein polynomials. They provide formulas for multiplication and division
of Bernstein polynomials by powers of $x$ and $1-x$ and for degree elevation
of Bernstein polynomials. Our method is similar to that of Buse and Goldman’s
[4]. In this section we find two functional equations. Using these equations,
we also give new proofs of both the multiplication and division properties for
the Bernstein polynomials.
By using the generating function in (1), we provide formulas for multiplying
Bernstein polynomials by powers of $(\frac{x-a}{b-a})^{d}$ and
$(\frac{b-x}{b-a})^{d}$ and for degree elevation of the Bernstein polynomials.
Using (2.1), we obtain the following functional equation:
$(\frac{x-a}{b-a})^{d}f_{\mathbb{Y},k}(x,t;a,b,n)=\frac{(k+d)!}{k!t^{d}}f_{\mathbb{Y},k}(x,t;a,b,n).$
After elementary manipulations in this equation, we get
$(\frac{x-a}{b-a})^{d}\mathbb{Y}_{k}^{n}(x;a,b,n)=\frac{n!(k+d)!}{k!(n+d)!}\mathbb{Y}_{k+d}^{n+d}(x;a,b,n+d).$
(2.18)
Substituting $d=1$, we have
$(\frac{x-a}{b-a})\mathbb{Y}_{k}^{n}(x;a,b,n)=\frac{k+1}{n+1}\mathbb{Y}_{k+1}^{n+1}(x;a,b,n+1).$
(2.19)
###### Remark 16.
Substituting $a=0$ and $b=1$ into (2.19), we have
$xB_{k}^{n}(x)=\frac{k+1}{n+1}B_{k+1}^{n+1}(x).$
The above relation can also be proved by (2.5) cf. [4].
Similarly, using (2.4), we obtain
$(\frac{b-x}{b-a})^{d}\mathbb{Y}_{k}^{n}(x;a,b,n)=\frac{n!(n+d-k)!}{\left(n+d\right)!(n-k)!}\mathbb{Y}_{k}^{n+d}(x;a,b,n+d).$
Substituting $d=1$ into the above equation, we have
$(\frac{b-x}{b-a})\mathbb{Y}_{k}^{n}(x;a,b,n)=\frac{n+1-k}{n+1}\mathbb{Y}_{k}^{n+1}(x;a,b,n+1).$
(2.20)
Consequently, by the same method as in [4], if we have (2.13), then
$(\frac{x-a}{b-a})^{d}\mathcal{P}(x,a,b)=\sum_{k=0}^{n}c_{k}^{n}\frac{n!(k+d)!}{k!(n+d)!}\mathbb{Y}_{k+d}^{n+d}(x;a,b,n+d),$
(2.21)
and
$(\frac{b-x}{b-a})^{d}\mathcal{P}(x,a,b)=\sum_{k=0}^{n}c_{k}^{n}\frac{n!(n+d-k)!}{\left(n+d\right)!(n-k)!}\mathbb{Y}_{k}^{n+d}(x;a,b,n+d).$
(2.22)
We now consider division properties. We assume that (2.13) holds and that we
are given an integer $j>0$. Since $(\frac{x-a}{b-a})^{j}$ divides
$\mathbb{Y}_{k}^{n}(x;a,b,n)$ for all $k\geq j$, it follows that
$(\frac{x-a}{b-a})^{j}$ divides $\mathcal{P}(x,a,b)$. Similarly, using (2.1),
we obtain the following functional equation:
$\frac{f_{\mathbb{Y},k}(x,t;a,b,n)}{(\frac{x-a}{b-a})^{j}}=\frac{(k-f)!t^{j}}{k!}f_{\mathbb{Y},k-j}(x,t;a,b,n-j).$
For $k\geq j$, from the above equation, we have
$\frac{\mathbb{Y}_{k}^{n}(x;a,b,n)}{(\frac{x-a}{b-a})^{j}}=\frac{n!(k-j)!}{k!(n-j)!}\mathbb{Y}_{k-j}^{n-j}(x;a,b,n-j).$
By a calculation similar to the calculation in [4], for $j\leq n-k$, we have
$\frac{\mathbb{Y}_{k}^{n}(x;a,b,n)}{(\frac{b-x}{b-a})^{j}}=\frac{n!(n-j-k)!}{\left(n-k\right)!(n-j)!}\mathbb{Y}_{k}^{n-j}(x;a,b,n-j).$
Therefore
$\frac{\mathcal{P}(x,a,b)}{(\frac{x-a}{b-a})^{j}}=\sum_{k=j}^{n}c_{k}^{n}\frac{n!(k-j)!}{k!(n-j)!}\mathbb{Y}_{k-j}^{n-j}(x;a,b,n-j),$
(2.23)
and
$\frac{\mathcal{P}(x,a,b)}{(\frac{b-x}{b-a})^{j}}=\sum_{k=0}^{n-j}c_{k}^{n}\frac{n!(n-j-k)!}{\left(n-k\right)!(n-j)!}\mathbb{Y}_{k}^{n-j}(x;a,b,n-j).$
(2.24)
### 2.6. Degree elevation
According to Buse and Goldman [4], given a polynomial represented in the
univariate Bernstein basis of degree $n$, degree elevation computes
representations of the same polynomial in the univariate Bernstein bases of
degree greater than $n$. Degree elevation allows us to add two or more
Bernstein polynomials which are not represented in the same degree Bernstein
basis functions.
Adding (2.19) and (2.20), we obtain the degree elevation formula for the
Bernstein basis functions:
$\mathbb{Y}_{k}^{n}(x;a,b,n)=\frac{k+1}{n+1}\mathbb{Y}_{k+1}^{n+1}(x;a,b,n+1)+\frac{n+1-k}{n+1}\mathbb{Y}_{k}^{n+1}(x;a,b,n+1).$
Substituting $d=1$ into (2.22), and adding these two equations gives the
following degree elevation formula for the Bernstein polynomials:
$\mathcal{P}(x,a,b)=\sum_{k=0}^{n}\left(\frac{k}{n+1}c_{k-1}^{n}+\frac{n+1-k}{\left(n+1\right)}c_{k}^{n}\right)\mathbb{Y}_{k}^{n+1}(x;a,b,n+1),$
(2.25)
where
$c_{k}^{n+1}=\frac{k}{n+1}c_{k-1}^{n}+\frac{n+1-k}{\left(n+1\right)}c_{k}^{n}.$
###### Remark 17.
If we set $a=0$ and $b=1$, then (2.25) reduces to Eq-(2.5) in [4, p. 853].
## 3\. Relation between the generating functions
$f_{\mathbb{Y},k}(x,t;a,b,m)$, Poisson distribution and Szasz-Mirakjan type
basis functions
The identity of Jetter and Stöckler represents a pointwise orthogonality
relation for the multivariate Bernstein polynomials on a simplex. This
identity give us a new representation for the dual basis which can be used to
construct general quasi-interpolant operators cf. (See, for details, [9],
[1]). As an application of the generating functions for the basis functions to
the identity of Jetter and Stöckler, Abel and Li [1] proved Proposition 1,
which is given in this section. Applying our generating functions to
Proposition 1, we give pointwise orthogonality relations for the Bernstein
polynomials and the Szasz-Mirakjan basis functions.
In this section, we give relations between the Bernstein basis functions, the
binomial distribution and the Poisson distribution. First we we consider the
generalized binomial or Newton distribution (probability function). Suppose
that $0\leq\frac{x-a}{b-a}\leq 1$ and $0\leq\frac{b-x}{b-a}\leq 1$. Set
$\mathbb{Y}_{k}^{n}(x;a,b,n)=\left(\begin{array}[]{c}n\\\
k\end{array}\right)\left(\frac{x-a}{b-a}\right)^{k}\left(\frac{b-x}{b-a}\right)^{n-k}.$
(3.1)
From the above definition, one can see that
$\sum_{k=0}^{n}\mathbb{Y}_{k}^{n}(x;a,b,n)=1.$
###### Remark 18.
If we set $a=0$ and $b=1$, then (3.1) reduces to
$\mathbb{Y}_{k}^{n}(x;0,1,n)=\left(\begin{array}[]{c}n\\\
k\end{array}\right)x^{k}(1-x)^{n-k}$
which is the binomial or Newton distribution (probabilities) function. If
$0\leq x\leq 1$ is the probability of an event $E$, then
$\mathbb{Y}_{k}^{n}(x;0,1,n)$ is the probability that $E$ will occur exactly
$k$ times in $n$ independent trials cf. [11].
Expected value or mean and variance of $\mathbb{Y}_{k}^{n}(x;a,b,n)$ are given
by
$\mu=\sum_{k=0}^{n}k\mathbb{Y}_{k}^{n}(x;a,b,n)=n\left(\frac{x-a}{b-a}\right),$
and
$\sigma^{2}=\sum_{k=0}^{n}k^{2}\mathbb{Y}_{k}^{n}(x;a,b,n)-\mu^{2}=\frac{n\left(x-a\right)\left(b-x\right)}{\left(b-a\right)^{2}}.$
If we let $n\rightarrow\infty$ in (3.1), then we arrive at the well-known
Poisson distribution function:
$\mathbb{Y}_{k}^{n}(\frac{b-a}{n}\mu+a;a,b,n)\rightarrow\frac{\mu^{k}e^{-\mu}}{k!}.$
(3.2)
The following proposition is proved by Abel and Li [1, p. 300, Proposition 3]:
###### Proposition 1.
Let the system $\\{f_{n}(x)\\}$ of functions be defined by the generating
function
$A_{t}(x)=\mathop{\displaystyle\sum}\limits_{n=0}^{\infty}f_{n}(x)t^{n}.$
If there exists a sequence $w_{k}=w_{k}(x)$ such that
$\mathop{\displaystyle\sum}\limits_{k=0}^{\infty}w_{k}\mathcal{D}^{k}A_{t}(x)\mathcal{D}^{k}A_{z}(x)=A_{tz}(x)$
with $\mathcal{D}=\frac{d}{dx}$, then for $i,j=0,1,\cdots$,
$\mathop{\displaystyle\sum}\limits_{k=0}^{\infty}w_{k}\mathcal{D}^{k}f_{i}(x)\mathcal{D}^{k}f_{j}(x)=\delta_{i,j}f_{i}(x).$
As an application of Proposition 1, Abel and Li [1] use the generating
function in Eq-(2.7) for the Bernstein basis functions. They also use
generating functions for the Szasz-Mirakjan basis functions and Baskakov basis
functions.
In this section, we apply our novel generating functions to Proposition 1,
which give pointwise orthogonality relations for the Bernstein polynomials and
the Szasz-Mirakjan type basis functions, respectively.
As applications of Proposition 1, we give the following examples:
###### Example 1.
For given $n$ and $k$, the Bernstein basis functions
$f_{i}(x,n;a,b)=\mathbb{Y}_{i}^{n}(x;a,b,n)=\left(\begin{array}[]{c}n\\\
i\end{array}\right)\left(\frac{x-a}{b-a}\right)^{k}(\frac{b-x}{b-a})^{n-k}$
are generated by the function in (2.1), that is
$A_{t}(x)=\frac{t^{k}\left(x-a\right)^{k}e^{(b-x)t}}{(b-a)^{n}k!}=\mathop{\displaystyle\sum}\limits_{i=0}^{\infty}\frac{f_{i}(x,n;a,b)}{i!}t^{i}.$
It is easy to check that Proposition 1 holds with
$w_{k}=w_{k}(x)=\mathbb{Y}_{k}^{n}(x;a,b,n)$.
###### Example 2.
Using (3.2), for $i\geq 0$, we generalize the Szasz-Mirakjan type basis
functions as follows
$f_{i}(x,n;a,b)=\frac{(n\frac{x-a}{b-a})^{i}e^{-n\frac{x-a}{b-a}}}{i!},$
where $a$ and $b$ are nonnegative real parameters with $a\neq b$, $n$ is a
positive integer and $x\in\left[a,b\right]$. The functions $f_{i}(x,n;a,b)$
are generated by
$A_{t}(x)=\exp\left((t-1)n\left(\frac{x-a}{b-a}\right)\right)=\mathop{\displaystyle\sum}\limits_{i=0}^{\infty}f_{i}(x,n;a,b)t^{i},$
where $\exp(x)=e^{x}$. In this case, Proposition 1 holds with
$w_{k}=w_{k}(x)=\frac{\left(\frac{x-a}{b-a}\right)^{k}}{n^{k}k!}$. Therefore,
we have
$\mathop{\displaystyle\sum}\limits_{k=0}^{\infty}\frac{\left(\frac{x-a}{b-a}\right)^{k}}{n^{k}k!}\mathcal{D}^{k}f_{i}(x,n;a,b)\mathcal{D}^{k}f_{i}(x,n;a,b)=\delta_{i,j}f_{i}(x,n;a,b).$
###### Remark 19.
If $a=0$ and $b=1$ in Example 2, then we arrive at the Szasz-Mirakjan basis
functions which are given in [1, p. 300, Example 2].
###### Acknowledgement 1.
The author would like to thank Professor Ronald Goldman (Rice University,
Houston, USA) for his very valuable comments, criticisms and for his very
useful suggestions on this present paper.
The present investigation was supported by the Scientific Research Project
Administration of Akdeniz University.
## References
* [1] U. Abel and Z. Li, A new proof of an identity of Jetter and Stöckler for multivariate Bernstein polynomials, Computer Aided Geometric Design, 23(3) (2006), 297-301.
* [2] M. Acikgoz and S. Araci, On generating function of the Bernstein polynomials, Numerical Analysis and Applied Mathematics, Amer. Inst. Phys. Conf. Proc. CP1281, (2010) 1141-1143.
* [3] S. N. Bernstein, Démonstration du théorème de Weierstrass fondée sur la calcul des probabilités. Comm. Soc. Math. Charkow Sér. 2 t. 13, 1-2 (1912-1913).
* [4] L. Busé and R. Goldman, Division algorithms for Bernstein polynomials, Computer Aided Geometric Design, 25(9) (2008), 850-865.
* [5] R. Goldman, An Integrated Introduction to Computer Graphics and Geometric Modeling, CRC Press, Taylor and Francis, New York, 2009.
* [6] R. Goldman, Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling, Morgan Kaufmann Publishers, Academic Press, San Diego, 2002.
* [7] R. Goldman, Identities for the Univariate and Bivariate Bernstein Basis Functions, Graphics Gems V, edited by Alan Paeth, Academic Press, (1995), 149-162.
* [8] L. C. Jang, W.-J. Kim and Y. Simsek, A study on the $p$-adic integral representation on $\mathbb{Z}p$ associated with Bernstein and Bernoulli polynomials, Advances in Difference Equations 2010 (2010), Article ID 163217, 6pp.
* [9] K. Jetter and J. Stöckler, An identity for multivariate Bernstein poynomials, Computer Aided Geometric Design, 20 (2003), 563-577.
* [10] M. S. Kim, D. Kim, and T. Kim, On the $q$-Euler numbers related to modified $q$-Bernstein polynomials, Abstr. Appl. Anal. 2010, Art. ID 952384, 15 pages, doi:10.1155/2010/952384, arXiv:1007.3317v1.
* [11] G. G. Lorentz, Bernstein Polynomials, Chelsea Pub. Comp. New York, N. Y. 1986.
* [12] G. M. Phillips, Interpolation and approximation by polynomials, CMS Books in Mathematics/ Ouvrages de Mathématiques de la SMC, 14. Springer-Verlag, New York, (2003).
* [13] Y. Simsek and M. Acikgoz, A new generating function of ($q$-) Bernstein-type polynomials and their interpolation function, Abstract and Applied Analysis, vol. 2010, Article ID 769095, 12 pages, 2010. doi:10.1155/2010/769095.
|
arxiv-papers
| 2010-12-21T10:01:40 |
2024-09-04T02:49:15.975820
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yilmaz Simsek",
"submitter": "Yilmaz Simsek",
"url": "https://arxiv.org/abs/1012.5538"
}
|
1012.5636
|
Alexandrov meets Kirszbraun
S. Alexander, V. Kapovitch, A. Petrunin
We give a simplified proof of the generalized Kirszbraun theorem for Alexandrov spaces,
which is due to Lang and Schroeder.
We also discuss related questions, both solved and open.
§ INTRODUCTION
Kirszbraun's theorem states that any short map (i.e. 1-Lipschitz map) from a subset of Euclidean space to another in Euclidean space can be extended as a short map to the whole space.
This theorem was proved first by Kirszbraun in <cit.>. Later it was reproved by Valentine in <cit.> and <cit.>, where he also generalized it to pairs of Hilbert spaces of arbitrary dimension
as well as pairs of spheres of the same dimension
and pairs of hyperbolic spaces with the same curvature.
J. Isbel in <cit.> studied target spaces that satisfy the above condition for any source space.
Valentine was also interested in pairs of metric spaces,
say $\spc{U}$ and $\spc{L}$,
which satisfy the above property, namely, given a subset $Q\subset\spc{U}$,
any short map $Q\to\spc{L}$
can be extended to a short map $\spc{U}\to \spc{L}$.
It turns out that this property has a lot in common with the definition of Alexandrov spaces
(see theorems <ref>, <ref> and <ref>).
Surprisingly, this relationship was first discovered only in the 1990's;
it was first published by Lang and Schroeder in <cit.>.
(The third author of this paper came to similar conclusions a couple of years earlier, and told it to the first author, but did not publish the result.)
We slightly improve the results of Lang and Schroeder.
Our proof is based on the barycentric maps introduced by Kleiner in <cit.>.
The material of this paper will be included in the book on Alexandrov geometry that we are currently writing, but it seems useful to publish it now.
Structure of the paper.
We introduce notations in Section <ref>.
In section <ref> we give altternative definitions of Alexandrov spaces
based on the Kirszbraun property for 4-point sets.
The generalized Kirszbraun theorem is proved in Section <ref>.
In the sections <ref> and <ref> we describe some comparison properties of finite subsets of Alexandrov spaces.
In Section <ref> we discuss related open problems.
Appendices <ref> and <ref> describe Kleiner's barycentric map and an analog of Helly's theorem for Alexandrov spaces.
Historical remark.
Not much is known about the author of this remarkable theorem.
The theorem appears in Kirszbraun's master's thesis which he defended in Warsaw University in 1930.
His name is Mojżesz and his second name is likely to be Dawid but is uncertain.
He was born either in 1903 or 1904 and died in a ghetto in 1942.
After university,
he worked as an actuary in an insurance company;
<cit.> seems to be his only publication in mathematics.
We want to thank S. Ivanov, N. Lebedeva and A. Lytchak
for useful comments
and pointing out misprints.
Also we want to thank L. Grabowski for bringing to our attention the entry about Kirszbraun in the Polish Biographical Dictionary.
§ PRELIMINARIES
In this section we mainly introduce our notations.
Metric spaces. Let $\spc{X}$ be a metric space. The distance between two points $x,y\in\spc{X}$ will be denoted as $\dist{x}{y}{}$ or $\dist{x}{y}{\spc{X}}$.
Given $R\in[0,\infty]$ and $x\in \spc{X}$, the sets
\begin{align*}
\oBall(x,R)&=\{y\in \spc{X}\mid \dist{x}{y}{}<R\},
\\
\cBall[x,R]&=\{y\in \spc{X}\mid \dist{x}{y}{}\le R\}.
\end{align*}
are called respectively the open and closed ball of radius $R$ with center at $x$.
A metric space $\spc{X}$ is called
if for any $\eps>0$ and any two points $x,y\in \spc{X}$ with $\dist{x}{y}{}<\infty$ there is an $\eps$-midpoint for $x$ and $y$;
i.e. there is a point $z\in \spc{X}$ such that $\dist{x}{z}{},\dist{z}{y}{}<\tfrac{1}{2}\cdot \dist[{{}}]{x}{y}{}+\eps$.
Model space.
$\Lob{m}{\kappa}$ denotes $m$-dimensional model space with curvature $\kappa$;
i.e. the simply connected $m$-dimensional Riemannian manifold with constant sectional curvature $\kappa$.
Set $\varpi\kappa=\diam\Lob2\kappa$$\varpi\kappa$, so
$\varpi\kappa=\infty$ if $\kappa\le0$ and $\varpi\kappa=\pi/\sqrt{\kappa}$ if $\kappa>0$.
(The letter $\varpi{}$ is a glyph variant of lower case $\pi$,
but is usually pronounced as pomega.)
Ghost of Euclid. Let $\spc{X}$ be a metric space
and $\II$ be a real interval.
A globally isometric map $\gamma\:\II\to \spc{X}$ will be called a unitspeed geodesic.
A unitspeed geodesic between $p$ and $q$ will be denoted by $\geod_{[p q]}$.
We consider $\geod_{[p q]}$ with parametrization starting at $p$;
i.e. $\geod_{[p q]}(0)=p$ and $\geod_{[p q]}(\dist{p}{q}{})=q$.
The image of $\geod_{[p q]}$ will be denoted by $[p q]$ and called a geodesicgeodesic.
Also we will use the following short-cut notation:
\begin{align*}
\l] p q \r[&=[p q]\backslash\{p,q\},
\l] p q \r]&=[p q]\backslash\{p\},
\l[ p q \r[&=[p q]\backslash\{q\}.
\end{align*}
A metric space $\spc{X}$ is called
if for any two points $x,y\in \spc{X}$ there is a geodesic $[x y]$ in $\spc{X}$.
Given a geodesic $[p q]$, we denote by $\dir{p}{q}$ its direction at $p$.
We may think of $\dir{p}{q}$ as belonging to the space of directions $\Sigma_p$ at $p$,
which in turn can be identified with the unit sphere in the tangent space $\T_p$ at $p$.
Further we set $\ddir{p}{q}=\dist[{{}}]{p}{q}{}\cdot\dir{p}{q}$;
it is a tangent vector at $p$, that is, an element of $\T_p$.
For a triple of points $p,q,r\in \spc{X}$, a choice of triple of geodesics $([q r], [r p], [p q])$ will be called a triangle and we will use the notation
$\trig p q r=([q r], [r p], [p q])$.
If $p$ is distinct from $x$ and $y$, a pair of geodesics $([p x],[p y])$ will be called a hingehinge, and denoted by
$\hinge p x y=([p x],[p y])$.
A locally Lipschitz function $f$ on a metric space $\spc{X}$ is called $\lambda$-convex ($\lambda$-concave)
if for any geodesic $\geod_{[p q]}$ in $\spc{X}$ the real-to-real function
$$t\mapsto f\circ\geod_{[p q]}(t)-\tfrac\lambda2\cdot t^2$$
is convex (respectively concave).
In this case we write $f''\ge \lambda$ (respectively $f''\le \lambda$).
A function $f$ is called strongly convex (strongly concave)
if $f''\ge \delta$ (respectively $f''\le -\delta$) for some $\delta>0$.
Model angles and triangles.
Let $\spc{X}$ be a metric space,
$p,q,r\in \spc{X}$
and $\kappa\in\RR$.
Let us define a model triangle $\trig{\~p}{\~q}{\~r}$
$\trig{\~p}{\~q}{\~r}=\modtrig\kappa(p q r)$) to be a triangle in the model plane $\Lob2\kappa$ such that
\ \ \dist{\~q}{\~r}{}=\dist{q}{r}{},
\ \ \dist{\~r}{\~p}{}=\dist{r}{p}{}.$$
If $\kappa\le 0$, the model triangle is said to be defined, since such a triangle always exists and is unique up to an isometry of $\Lob2\kappa$.
If $\kappa>0$, the model triangle is said to be defined if in addition
$$\dist{p}{q}{}+\dist{q}{r}{}+\dist{r}{p}{}< 2\cdot\varpi\kappa.$$
In this case the triangle also exists and is unique up to an isometry of $\Lob2\kappa$.
If for $p,q,r\in \spc{X}$, the model triangle
$\trig{\~p}{\~q}{\~r}=\modtrig\kappa(p q r)$ is defined
and $\dist{p}{q}{},\dist{p}{r}{}>0$, then the angle measure of
$\trig{\~p}{\~q}{\~r}$ at $\~p$ will be called the model angle of the triple $p$, $q$, $r$, and will be denoted by
$\angk\kappa p q r$.
Curvature bounded below.
We will denote by $\CBB{}{\kappa}$, complete intrinsic spaces $\spc{L}$ with curvature $\ge\kappa$ in the sense of Alexandrov.
Specifically, $\spc{L}\in \CBB{}{\kappa}$ if for any quadruple of points $p,x^1,x^2,x^3\in \spc{U}$ , we have
$$\angk\kappa p{x^1}{x^2}
+\angk\kappa p{x^2}{x^3}
+\angk\kappa p{x^3}{x^1}\le 2\cdot\pi.\eqlbl{Yup-kappa}$$
or at least one of the model angles $\angk\kappa p{x^i}{x^j}$ is not defined.
Condition <ref> will be called (1+3)-point comparison.
According to Plaut's theorem <cit.>,
any space $\spc{L}\in \CBB{}{}$ is $G_\delta$-geodesic;
that is, for any point $p\in \spc{L}$ there is a dense $G_\delta$-set $W_p\subset\spc{L}$ such that for any $q\in W_p$ there is a geodesic $[p q]$.
We will use two more equivalent definitions of $\CBB{}{}$ spaces (see <cit.>).
Namely, a complete $G_\delta$-geodesic space is in $\CBB{}{}$
if and only if it satisfies either of following conditions:
* (point-on-side comparison)
For any geodesic $[x y]$ and $z\in \l]x y\r[$, we have
$$\angk\kappa x p y\le\angk\kappa x p z; \eqlbl{POS-CBB}$$
or, equivalently,
$$\dist{\~p}{\~z}{}\le \dist{p}{z}{},$$
where $\trig{\~p}{\~x}{\~y}=\modtrig\kappa(p x y)$, $\~z\in\l] \~x\~y\r[$, $\dist{\~x}{\~z}{}=\dist{x}{z}{}$.
* (hinge comparison)
For any hinge $\hinge x p y$, the angle
$\mangle\hinge x p y$ is defined and
$$\mangle\hinge x p y\ge\angk\kappa x p y.$$
Moreover, if $z\in\l]x y\r[$, $z\not=p$ then for any two hinges $\hinge z p y$ and $\hinge z p x$ with common side $[z p]$
$$\mangle\hinge z p y + \mangle\hinge z p x\le\pi.$$
We also use the following standard result in Alexandrov geometry,
which follows from the discussion in the survey of Plaut <cit.>.
Let $\spc{L}\in \CBB{}{}$.
Given an array of points $(x^1,x^2\dots,x^n)$ in $\spc{L}$,
there is a dense $G_\delta$-set $W\subset\spc{L}$ such that
for any $p\in W$, all the directions $\dir{p}{x^i}$ lie in
an isometric copy of a unit sphere in $\Sigma_p$.
(Or, equivaletntly, all the vectors $\ddir{p}{x^i}$ lie in
a subcone of the tangent space $\T_p$ which is isometric to Euclidean space.)
Curvature bounded above.
We will denote by $\Cat{}{\kappa}$
the class of metric spaces $\spc{U}$ in which any two points at distance $<\varpi\kappa$ are joined by a geodesic,
and which have curvature $\le\kappa$ in the following global sense of Alexandrov: namely, for any quadruple of points $p^1,p^2,x^1,x^2\in \spc{U}$, we have
\angk{\kappa}{p^1}{x^1}{x^2}
\le
\angk{\kappa}{p^1}{p^2}{x^1}+\angk{\kappa}{p^1}{p^2}{x^2},
\ \t{or}\
\angk{\kappa} {p^2}{x^1}{x^2}\le \angk{\kappa} {p^2}{p^1}{x^1} + \angk{\kappa} {p^2}{p^1}{x^2},
\eqlbl{gokova:eq:2+2}$$
one of the six model angles above
is undefined.
The condition <ref> will be called (2+2)-point comparison (or (2+2)-point $\kappa$-comparison
if a confusion may arise).
We denote the complete $\Cat{}{\kappa}$ spaces by $\cCat{}{\kappa}$.
The following lemma is a direct consequence of the definition:
Any complete intrinsic
space $\spc{U}$ in which every quadruple $p^1,p^2,x^1,x^2$ satisfies
the (2+2)-point $\kappa$-
is a $\cCat{}{\kappa}$ space (that is, any two points at distance $<\varpi\kappa$ are joined by a geodesic).
In particular, the completion of a
$\Cat{}{\kappa}$ space again lies in $\Cat{}{\kappa}$.
We have the following basic facts (see [1]):
In a $\Cat{}{\kappa}$ space, geodesics of length $<\varpi\kappa$ are uniquely determined by, and continuously dependent on, their endpoint pairs.
In a $\Cat{}{\kappa}$ space, any open ball $\oBall(x,R)$ of radius $R\le\varpi\kappa/2$ is convex, that is, $\oBall(x,R)$ contains every geodesic whose endpoints it contains.
We also use an equivalent definition of $\Cat{}{\kappa}$ spaces (see <cit.>).
Namely, a metric space $\spc{U}$ in which any two points at distance $<\varpi\kappa$ are joined by a geodesic is a $\Cat{}{\kappa}$ space if and only if it satisfies the following condition:
* (point-on-side comparison)
for any geodesic $[x y]$ and $z\in \l]x y\r[$, we have
$$\angk\kappa x p y\ge\angk\kappa x p z,$$
or equivalently,
$$\dist{\~p}{\~z}{}\ge \dist{p}{z}{}, \eqlbl{POS-CAT}$$
where $\trig{\~p}{\~x}{\~y}=\modtrig\kappa(p x y)$, $\~z\in\l] \~x\~y\r[$, $\dist{\~x}{\~z}{}=\dist{x}{z}{}$.
We also use Reshetnyak's majorization theorem <cit.>.
Suppose $\~\alpha$ is a simple closed curve of finite length in $\Lob2{\kappa}$,
and $D\subset\Lob2{\kappa}$ is a closed region bounded by $\~\alpha$. If $\spc{X}$ is a metric space, a length-nonincreasing map $F\:D\to\spc{X}$ is called majorizing if it is length-preserving on $\~\alpha$.
In this case, we say that $D$ majorizes the curve $\alpha=F\circ\~\alpha$ under the map $F$.
Reshetnyak's majorization theorem
Any closed curve $\alpha$ of length $<2\cdot \varpi\kappa$ in $\spc{U}\in\Cat{}{\kappa}$ is majorized by a convex region in $\Lob2\kappa$.
Ultralimit of metric spaces.
Given a metric space $\spc{X}$, its ultrapower
(i.e. ultralimit of constant sequence $\spc{X}_n=\spc{X}$) will be denoted as $\spc{X}^\o$;
here $\o$ denotes a fixed nonprinciple ultrafilter.
For definitions and properties of ultrapowers,
we refer to a paper of Kleiner and Leeb <cit.>.
We use the following facts about ultrapowers which easily follow from the definitions (see <cit.> for details):
* $\spc{X}\in\cCat{}{\kappa}\ \Longleftrightarrow\ \spc{X}^\o\in\cCat{}{\kappa}$.
* $\spc{X}\in\CBB{}{\kappa}\ \Longleftrightarrow\ \spc{X}^\o\in\CBB{}{\kappa}$.
* $\spc{X}$ is intrinsic if and only if $\spc{X}^\o$ is geodesic.
Note that if $\spc{X}$ is proper (namely, bounded closed sets are compact), then $\spc{X}$ and $\spc{X}^\o$ coincide.
Thus a reader interested only in proper spaces may ignore everything related to ultrapower in this article.
section.thm. #1.
|
arxiv-papers
| 2010-12-27T16:54:34 |
2024-09-04T02:49:15.984552
|
{
"license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/",
"authors": "Stephanie Alexander, Vitali Kapovitch and Anton Petrunin",
"submitter": "Anton Petrunin",
"url": "https://arxiv.org/abs/1012.5636"
}
|
1012.5709
|
Hawking temperature for constant curvature black bole and its analogue in de
Sitter Space
Rong-Gen Cai1,***Email address: cairg@itp.ac.cn and Yun Soo Myung2,†††Email
address: ysmyung@inje.ac.kr
1Key Laboratory of Frontiers in Theoretical Physics, Institute of Theoretical
Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China
2Institute of Basic Science and School of Computer Aided Science, Inje
University, Gimhae 621-749, Korea
Abstract
The constant curvature (CC) black holes are higher dimensional generalizations
of BTZ black holes. It is known that these black holes have the unusual
topology of ${\cal M}_{D-1}\times S^{1}$, where $D$ is the spacetime dimension
and ${\cal M}_{D-1}$ stands for a conformal Minkowski spacetime in $D-1$
dimensions. The unusual topology and time-dependence for the exterior of these
black holes cause some difficulties to derive their thermodynamic quantities.
In this work, by using globally embedding approach, we obtain the Hawking
temperature of the CC black holes. We find that the Hawking temperature takes
the same form when using both the static and global coordinates. Also it is
identical to the Gibbons-Hawking temperature of the boundary de Sitter spaces
of these CC black holes. Employing the same approach, we obtain the Hawking
temperature for the counterparts of CC black holes in de Sitter spaces.
## 1 Introduction
Over the past years, the so-called BTZ (Banados-Teitelboim-Zanelli) [1] black
hole solutions have played the important role in understanding microscopic
degrees of freedom of black hole. The BTZ black hole is an exact solution of
Einstein field equations with a negative cosmological constant in three
dimensions. It is well known the BTZ black hole can be constructed by
identifying points along the orbit of a Killing vector in a three dimensional
anti-de Sitter (AdS) space.
The BTZ black hole has a topology of ${\cal M}_{2}\times S^{1}$, where $M_{2}$
denotes a conformal Minkowski space in two dimensions. Following the same way
as done in three dimensions, one can construct analogues of the BTZ solution,
the so-called constant curvature (CC) black holes in higher $(D\geq 4$)
dimensional AdS spaces [2, 3, 4]. However, such black holes have topology of
${\cal M}_{D-1}\times S^{1}$ in $D$ dimensions, which is quite different from
the known topology of ${\cal M}_{2}\times S^{D-2}$ for the usual black holes
in $D$ dimensions. In addition, the exterior region of these CC black holes is
time-dependent and thus, there is no global time-like Killing vector [2].
Because of this, it is difficult to discuss Hawking radiation and
thermodynamics associated with these black holes. For example, see [5, 6, 7]
and references therein.
On the other hand, these spacetimes are interesting examples of smooth time-
dependent solutions. Particularly, they are consistent background spacetimes
for string theory at least to leading order since they are vacuum solutions to
Einstein field equations with a negative cosmological constant too. Further we
note that these spacetimes are time-dependent, the boundary metric is also
time-dependent, and it is asymptotically AdS. Therefore, it might open a
window to investigate dual strong coupling field theory in the time-dependent
backgrounds through the AdS/CFT correspondence [8]. Especially, the
$D$-dimensional CC black holes have the boundary topology of $dS_{D-2}\times
S^{1}$, where $dS_{D-2}$ denotes a $(D-2)$-dimensional de Sitter (dS) space.
Resorting the AdS/CFT correspondence, these CC black holes are gravity duals
to strong coupling conformal field theories living on $dS_{D-2}\times S^{1}$.
Finally it is observed in [9] that these CC black holes have a close
connection to the so-called “bubbles of nothing” in AdS space [10, 11]. The
bubbles of nothing were constructed by analytically continuing (Schwarzschild,
Reissner-Nordström, and Kerr) black holes in AdS spaces. The stress-energy
tensor for dual conformal field theories to these CC black holes was
calculated in [9, 11].
It is well known that there is the Gibbons-Hawking temperature $T_{\rm GH}$
for a comoving observer in a dS space [12]. This temperature may be viewed as
the Hawking temperature $T_{\rm HK}$ associated with cosmological horizon of
dS space. A $D$-dimensional dS space can be embedded as a hypersurface into a
$(D+1)$-dimensional Minkowski space. Then, the comoving observer in dS space
is identical to an observer with a constant acceleration in Minkowski space.
According to Davies [13] and Unruh [14], an observer with a constant
acceleration in Minkowski space will see a hot bath with the Davies-Unruh
temperature $T_{\rm DU}=a/2\pi$ where $a$ is the acceleration of the observer.
It turns out that the Gibbons-Hawking temperature of dS space is equivalent to
the Davies-Unruh temperature of the corresponding observer in Minkowski space.
One decade ago, it was shown that an observer with a constant acceleration $a$
in dS space will detect a temperature given by $\sqrt{a^{2}+1/l^{2}}/2\pi$,
where $l$ is the radius of the dS space [15]. This was soon generalized to the
cases of dS/AdS space by Deser and Levin [16] with temperatures of
$\sqrt{a^{2}\pm 1/l^{2}}/2\pi$. Further, Deser and Levin have shown that the
temperature is equivalent to the Davies-Unruh temperature for the
corresponding observer in Minkowski space. Further examples for the
equivalence have been shown by globally embedding curved spaces including BTZ,
Schwarzschild, Schwarzschild-AdS (dS), and Reissner-Nordström solutions into
higher dimensional Minkowski spaces in Ref.[17]. For more examples on the
equivalence, see [18] and references therein.
In this work, the “globally embedding approach” will be employed to determine
the Hawking temperature of CC black holes and positive CC spaces. This
approach shows a clear way to compute the Hawking temperature, in comparison
to other methods with ambiguity to calculate it.
In the next section, we show that the Hawking temperature of the constant
curvature black holes is given by $T_{\rm HK}=r_{+}/(2\pi l)$ using both the
static and global coordinates in AdS spaces. Further it is shown that the
Hawking temperature is identical to the Gibbons-Hawking temperature of the
boundary dS space. In section 3 we consider the counterparts of the CC black
holes in dS spaces. These are constant curvature (CC) spaces with the
cosmological horizon. We find that the Hawking temperature for these spaces
are given by $T_{\rm HK}=r_{+}/(2\pi l)$, where $r_{+}$ and $l$ are the
cosmological horizon radius and the radius of dS spaces, respectively. We give
our conclusions and discussions in section 4\. In this paper, we confine
ourselves to the five dimensional space. The generalization to other
dimensions is straightforward.
## 2 Hawking temperature of CC black holes
A five dimensional AdS space is defined as the universal covering space of a
surface obeying
$-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}-x_{5}^{2}=-l^{2},$ (2.1)
where $l$ is the AdS radius. This surface has fifteen Killing vectors of seven
rotations and eight boosts. We consider the boost
$\xi=(r_{+}/l)(x_{4}\partial_{5}+x_{5}\partial_{4})$ with its norm
$\xi^{2}=r_{+}^{2}(-x_{4}^{2}+x_{5}^{2})/l^{2}$ where $r_{+}$ is an arbitrary
real constant. The so-called CC black hole is constructed by identifying
points along the orbit of the Killing vector $\xi$. Since the starting point
is the AdS, the resulting black hole has a constant curvature as the AdS does
show. The topology of the black holes is changed to be ${\cal M}_{4}\times
S^{1}$, which is quite different from the usual topology of ${\cal
M}_{2}\times S^{3}$ for five dimensional black holes. Here ${\cal M}_{n}$
denotes a conformal Minkowski space in $n$ dimensions. For more details for
the construction of the black hole, see [3, 4].
The CC black holes can be nicely described by using Kruskal coordinates. For
this purpose, a set of coordinates on the AdS for the region of $\xi^{2}>0$
has been introduced in Ref.[3]. The six dimensionless local coordinates
$(y_{i},\varphi)$ are given by
$\displaystyle x_{i}=\frac{2ly_{i}}{1-y^{2}},\ \ \ i=0,1,2,3$ $\displaystyle
x_{4}=\frac{lr}{r_{+}}\sinh\left(\frac{r_{+}\varphi}{l}\right),$
$\displaystyle
x_{5}=\frac{lr}{r_{+}}\cosh\left(\frac{r_{+}\varphi}{l}\right),$ (2.2)
with
$r=r_{+}\frac{1+y^{2}}{1-y^{2}},\ \ \
y^{2}=-y_{0}^{2}+y_{1}^{2}+y_{2}^{2}+y_{3}^{2}.$ (2.3)
Here the allowed regions are $-\infty<y_{i}<\infty$ and
$-\infty<\varphi<\infty$ with the restriction $-1<y^{2}<1$. In these
coordinates, the boundary at $r\to\infty$ corresponds to the hyperbolic “ball”
which satisfies $y^{2}=1$. The induced line element can be written down
$ds^{2}=\frac{l^{2}(r+r_{+})^{2}}{r_{+}^{2}}(-dy_{0}^{2}+dy_{1}^{2}+dy_{2}^{2}+dy_{3}^{2})+r^{2}d\varphi^{2}.$
(2.4)
Obviously, the Killing vector is given by $\xi=\partial_{\varphi}$ with its
norm $\xi^{2}=r^{2}$. The black hole spacetime could be obtained by
identifying $\varphi\sim\varphi+2\pi n$, and the topology of the black hole
takes the form of ${\cal M}_{4}\times S^{1}$ clearly.
On the other hand, the CC black holes can also be described by introducing
Schwarzschild coordinates. The local “spherical” coordinates
($t,r,\theta,\chi$) in the hyperplane $y_{i}$ are
$\displaystyle y_{0}=f\sin\theta\sinh(r_{+}t/l),\ \ \
y_{1}=f\sin\theta\cosh(r_{+}t/l),$ $\displaystyle y_{2}=f\cos\theta\sin\chi,\
\ \ \ \ \ y_{3}=f\cos\theta\cos\chi,$ (2.5)
where $f=[(r-r_{+})/(r+r_{+})]^{1/2}$, $0\leq\theta\leq\pi/2$, $0\leq\chi\leq
2\pi$, and $r_{+}\leq r<\infty$. One finds that the solution (2.4) can be
rewritten as
$ds^{2}=l^{2}N^{2}d\Omega_{3}^{2}+N^{-2}dr^{2}+r^{2}d\varphi^{2},$ (2.6)
where
$N^{2}=\frac{r^{2}-r_{+}^{2}}{l^{2}},\ \ \ d\Omega_{3}^{2}=-\sin^{2}\theta
dt^{2}+\frac{l^{2}}{r_{+}^{2}}(d\theta^{2}+\cos^{2}\theta d\chi^{2}).$ (2.7)
This is the black hole solution expressed in terms of Schwarzschild
coordinates. Here $r=r_{+}$ is the location of black hole horizon. In these
coordinates the solution seems static. However, we observe from (2.6) that the
form (2.7) does not cover the whole exterior region of black hole since the
difference of $y^{2}_{1}-y_{0}^{2}$ is required to be positive in the region
covered by these coordinates. Indeed, it has been proved that there is no
globally timelike Killing vector in this geometry [2].
Now we consider a static observer with constant
$(r>r_{+},\theta,\chi,\varphi)$ in the black hole background (2.6). To this
observer, we find that an associated acceleration $a_{5}$ is given by
$a_{5}^{2}=\frac{1}{l^{2}(r^{2}-r_{+}^{2})}\frac{1}{\sin^{2}\theta}\left(r^{2}\sin^{2}\theta+r_{+}^{2}\cos^{2}\theta\right).$
(2.8)
On the other hand, the acceleration of $a_{6}$ for the corresponding observer
in six dimensional embedding Minkowski space is given by
$a_{6}^{-2}=x_{1}^{2}-x_{0}^{2}=\frac{l^{2}(r^{2}-r_{+}^{2})}{r_{+}^{2}}\sin^{2}\theta.$
(2.9)
It is easy to check that these two accelerations obey the relation
$a_{6}^{2}=-\frac{1}{l^{2}}+a^{2}_{5}.$ (2.10)
This shows that the Davies-Unruh temperature for the local observer in six
dimensional Minkowski space is
$T_{\rm DU}=\frac{a_{6}}{2\pi}=\frac{r_{+}}{2\pi
l\sqrt{r^{2}-r_{+}^{2}}}\frac{1}{\sin\theta}.$ (2.11)
We note that the redshift factor of $\sqrt{-g_{00}}=lN\sin\theta$ for the
black hole (2.6) is necessary to define the Hawking temperature. Hence we
conclude that the Hawking temperature of the CC black hole is
$T_{\rm HK}=\sqrt{-g_{00}}~{}T_{\rm DU}=\frac{r_{+}}{2\pi l}.$ (2.12)
We notice that the Hawking temperature $T_{\rm HK}$ is consistent with the
inverse period of the Euclidean time derived from the solution (2.7).
As the case in four dimensions [4], there is another set of coordinates
covering the whole exterior of the Minkowskian black hole geometry as [9]
$\displaystyle y_{0}=f\sinh(r_{+}t/l),\ \ \ y_{1}=f\cos\theta\cosh(r_{+}t/l),$
$\displaystyle y_{2}=f\sin\theta\cos\chi\cosh(r_{+}t/l),\ \ \
y_{3}=f\sin\theta\sin\chi\cosh(r_{+}t/l),$ (2.13)
where $f$ is given by (2) and the allowed regions are $0\leq\theta\leq\pi$,
$r_{+}\leq r<\infty$, and $0\leq\chi\leq 2\pi$. In these coordinates, the
solution can be expressed as
$ds^{2}=N^{2}l^{2}d\Omega_{3}^{2}+N^{-2}dr^{2}+r^{2}d\varphi^{2},$ (2.14)
where $N^{2}=(r^{2}-r_{+}^{2})/l^{2}$ and
$d\Omega_{3}^{2}=-dt^{2}+\frac{l^{2}}{r_{+}^{2}}\cosh^{2}(r_{+}t/l)(d\theta^{2}+\sin^{2}\theta
d\chi^{2}).$ (2.15)
The time-dependence of the solution is manifest in this coordinate system. We
introduce a static observer located at constant $r>r_{+}$, $\varphi$ and
$\chi$, but $\theta=0$ due to the spherical symmetry of the solution [16].
Here, we find that the acceleration $a_{5}$ associated with the observer is
$a_{5}^{2}=\frac{r^{2}}{l^{2}(r^{2}-r_{+}^{2})},$ (2.16)
while the acceleration $a_{6}$ of the corresponding observer in six
dimensional Minkowski space is given by
$a^{-2}_{6}=x_{1}^{2}-x_{0}^{2}=\frac{r_{+}^{2}}{l^{2}(r^{2}-r_{+}^{2})}.$
(2.17)
They satisfy the relation (2.10) too. In this case, the Davies-Unruh
temperature is given by
$T_{\rm DU}=\frac{a_{6}}{2\pi}=\frac{r_{+}}{2\pi l\sqrt{r^{2}-r_{+}^{2}}}.$
(2.18)
Considering the redshift factor of $\sqrt{-g_{00}}$, we get the Hawking
temperature of the black hole in the line element of (2.14) as
$T_{\rm HK}=\frac{r_{+}}{2\pi l}.$ (2.19)
Thus we have obtained the Hawking temperature of the CC black hole by
employing globally embedding approach combined with the Davies-Unruh
temperature in six dimensional Minkowski space. The Hawking temperatures
(2.12) and (2.19) are our main results.
Here some remarks are in order. First, in general, Hawking temperature of
black hole depends on coordinates used to calculate it. That is, the Hawking
temperature may be different when using different coordinates, even for the
same black hole. In our case, we obtained the same Hawking temperature for the
CC black hole even when used the different coordinate systems (2.6) and
(2.14). Second, we mention that the Hawking temperature (2.12) is the same as
the inverse period of the Euclidean time for the Euclidean sector of the
solution (2.6). However, when used the coordinates (2.15), the Hawking
temperature is no longer the same as the inverse period of the Euclidean time.
In order to see this, let us consider carefully the Euclidean sector of the
black hole solution which can be obtained by replacing the time $t$ by
$-i(\tau+\pi l/(2r_{+}))$ in (2.15). In this case, $d\Omega_{3}^{2}$ becomes
$d\Omega_{3}^{2}=d\tau^{2}+\frac{l^{2}}{r_{+}^{2}}\sin^{2}(r_{+}\tau/l)(d\theta^{2}+\sin^{2}\theta
d\chi^{2}).$ (2.20)
In order that $d\Omega_{3}^{2}$ be a regular three-sphere, $\tau$ must have
the period of $\tau\sim\tau+\tilde{\beta}$ with
$\tilde{\beta}=\frac{\pi l}{r_{+}}.$ (2.21)
Clearly this is not the inverse of Hawking temperature. This shows that the
Euclidean method does not always provide a correct Hawking temperature of CC
black holes. However, using both coordinates (2) and (2), the Euclidean time
$\tau=it$ obtained by Wick rotation leads to the fact that it has a
periodicity with period $2\pi l/r_{+}$, which gives a correct Hawking
temperature of the black hole. This may be related to the issue of the factor
2 in [19].
Finally, we observe from (2.14) that the black hole solution has a boundary
topology $dS_{3}\times S^{1}$ at $r=\infty$. The three dimensional de Sitter
space $dS_{3}$ has a Hubble constant $H=r_{+}/l$. It is well known that for a
de Sitter space with a Hubble constant $H$, there is the Gibbons-Hawking
temperature $T=H/2\pi$ for a comving observer. We find that the Gibbons-
Hawking temperature in our case is identical to the Hawking temperature of the
CC black hole
$T_{\rm GH}=\frac{H}{2\pi}=\frac{r_{+}}{2\pi l}=T_{\rm HK}.$ (2.22)
## 3 Hawking temperature of a positive CC space
In this section we consider the analogue of the CC black hole in dS space.
This space is constructed by identifying points along the orbit of a Killing
vector in dS space. In fact, this space is a generalization of the three-
dimensional Schwarzschild-de Sitter solution in higher dimensions. This space
has a cosmological event horizon, and its topology is ${\cal M}_{D-1}\times
S^{1}$ where ${\cal M}_{D-1}$ denotes a $(D-1)$-dimensional conformal
Minkowski spacetime. Such space was constructed in Ref.[20].
As the case with a negative cosmological constant, we consider a five
dimensional de Sitter space, which can be viewed as a hypersurface embedded
into a six dimensional Minkowski space, satisfying
$-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}=l^{2}$ (3.1)
with $l$ the radius of the dS space. This dS space has fifteen Killing vectors
of five boosts and ten rotations. We consider a rotational Killing vector
$\xi=(r_{+}/l)(x_{4}\partial_{5}-x_{5}\partial_{4})$ with its norm
$\xi^{2}=r_{+}^{2}/l^{2}(x_{4}^{2}+x_{5}^{2})$ where $r_{+}$ is an arbitrary
real constant. Identifying points along the orbit of a Killing vector $\xi$,
another one-dimensional manifold becomes compact and it is isomorphic to
$S^{1}$. Thus, we obtain a spacetime of topology ${\cal M}^{4}\times S^{1}$
with cosmological horizon. For details of the construction of the space, see
[20].
We can describe the spacetime in the region with $0\leq\xi^{2}\leq r_{+}^{2}$
by introducing six dimensionless local coordinates $(y_{i},\phi)$,
$\displaystyle x_{i}=\frac{2ly_{i}}{1+y^{2}},\ \ \ i=0,1,2,3$ $\displaystyle
x_{4}=\frac{lr}{r_{+}}\sin\left(\frac{r_{+}\phi}{l}\right),$ $\displaystyle
x_{5}=\frac{lr}{r_{+}}\cos\left(\frac{r_{+}\phi}{l}\right),$ (3.2)
where
$r=r_{+}\frac{1-y^{2}}{1+y^{2}},\ \ \
y^{2}=-y_{0}^{2}+y_{1}^{2}+y_{2}^{2}+y_{3}^{2}.$ (3.3)
Here the allowed regions are $-\infty<y_{i}<+\infty$ and
$-\infty<\phi<+\infty$ with the restriction $-1<y^{2}<1$ to have a positive
$r$. In the coordinates (3), the induced line element is
$ds^{2}=\frac{l^{2}(r+r_{+})^{2}}{r_{+}^{2}}(-dy_{0}^{2}+dy_{1}^{2}+dy_{2}^{2}+dy_{3}^{2})+r^{2}d\phi^{2},$
(3.4)
which is the same form as the case of a negative constant curvature [3].
However, it is noted that the coordinates (3) and the definition of $r$ differ
from those in the CC black holes. In this coordinate system, it is evident
that the Killing vector is $\xi=\partial_{\phi}$ with norm $\xi^{2}=r^{2}$.
Imposing the identification $\phi\sim\phi+2\pi n$, the solution has the
topology ${\cal M}_{4}\times S^{1}$.
We now introduce the Schwarzschild coordinates to describe the solution. Using
local “spherical” coordinates $(t,r,\theta,\chi)$ defined as
$\displaystyle y_{0}=f\sin\theta\sinh(r_{+}t/l),\
~{}~{}~{}~{}~{}y_{1}=f\sin\theta\cosh(r_{+}t/l),$ (3.5) $\displaystyle
y_{2}=f\cos\theta\sin\chi,\ ~{}~{}~{}~{}~{}y_{3}=f\cos\theta\cos\chi,$
where $f=[(r_{+}-r)/(r+r_{+})]^{1/2}$, and the allowed coordinate ranges are
$0<\theta<\pi/2$, $0<\chi<2\pi$, and $0<r<r_{+}$. The line element can be
expressed as
$ds^{2}=l^{2}N^{2}d\Omega_{3}^{2}+N^{-2}dr^{2}+r^{2}d\phi^{2}.$ (3.6)
Here $N^{2}=(r_{+}^{2}-r^{2})/l^{2}$ and
$d\Omega_{3}^{2}=-\sin^{2}\theta
dt^{2}+\frac{l^{2}}{r_{+}^{2}}(d\theta^{2}+\cos^{2}\theta d\chi^{2}).$ (3.7)
Clearly the location of $r=r_{+}$ represents a cosmological horizon. This
solution is the counterpart of a five dimensional CC black hole described in
the previous section. The only difference is that
$N^{2}=(r^{2}-r_{+}^{2})/l^{2}$ is replaced by $N^{2}=(r_{+}^{2}-r^{2})/l^{2}$
here. Further, in three dimensions, the corresponding induced line element
takes the form
$ds^{2}=-(r_{+}^{2}-r^{2})dt^{2}+\frac{l^{2}}{r_{+}^{2}-r^{2}}dr^{2}+r^{2}d\phi^{2},$
(3.8)
After a suitable rescaling of coordinates, it can be transformed to three
dimensional Schwarzschild-de Sitter solution [21]. In this sense, the solution
(3.6) can be viewed as an analogue of the three dimensional Schwarzschild-de
Sitter solution in five dimensions.
The solution (3.6) seems to be static, but it does not cover the whole region
within the cosmological horizon. It can be seen from the definition of
coordinates (3.5) because they must obey the constraint:
$y_{1}^{2}-y_{0}^{2}=f^{2}\cos^{2}\theta\geq 0$. Considering a static observer
located at constant $(r<r_{+},\theta,\chi$) in the background (3.6), we find
that the static observer has a constant acceleration $a_{5}$ as
$a_{5}^{2}=\frac{1}{l^{2}(r_{+}^{2}-r^{2})\sin^{2}\theta}\left(r^{2}\sin^{2}\theta+r_{+}^{2}\cos^{2}\theta\right),$
(3.9)
while the observer in six dimensional Minkowski space has a constant
acceleration $a_{6}$ as
$a_{6}^{-2}=x_{1}^{2}-x_{0}^{2}=\frac{l^{2}(r_{+}^{2}-r^{2})}{r_{+}^{2}}\sin^{2}\theta.$
(3.10)
These two accelerations are related to each other as
$a_{6}^{2}=1/l^{2}+a_{5}^{2}.$ (3.11)
According to Davies and Unruh, the observer has a temperature as
$T_{\rm DU}=\frac{a_{6}}{2\pi}=\frac{r_{+}}{2\pi
l\sqrt{r_{+}^{2}-r^{2}}\sin\theta}.$ (3.12)
Taking into account the redshift factor $\sqrt{-g_{00}}$ of the observer, one
has the Hawking temperature as
$T_{\rm HK}=\frac{r_{+}}{2\pi l}.$ (3.13)
On the other hand, one has another set of coordinates which covers the whole
region within the cosmological horizon,
$\displaystyle y_{0}=f\sinh(r_{+}t/l),\
~{}~{}~{}~{}~{}~{}y_{1}=f\cos\theta\cosh(r_{+}t/l),$ $\displaystyle
y_{2}=f\sin\theta\cos\chi\cosh(r_{+}t/l),\
~{}~{}~{}~{}~{}~{}y_{3}=f\sin\theta\sin\chi\cosh(r_{+}t/l).$ (3.14)
In this case, the line element is described by
$ds^{2}=l^{2}N^{2}\tilde{d\Omega_{3}^{2}}+N^{-2}dr^{2}+r^{2}d\phi^{2},$ (3.15)
where $N^{2}=(r_{+}^{2}-r^{2})/l^{2}$ and
$\tilde{d\Omega_{3}^{2}}=-dt^{2}+\frac{l^{2}}{r_{+}^{2}}\cosh^{2}(r_{+}t/l)(d\theta^{2}+\sin^{2}\theta
d\chi^{2}).$ (3.16)
We consider a static observer located at constant position of
$(r<r_{+},\chi,\varphi)$ and $\theta=0$. For such an observer, we have a
constant acceleration $a_{5}$ as
$a_{5}^{2}=\frac{r^{2}}{l^{2}(r^{2}_{+}-r^{2})}.$ (3.17)
In six dimensional Minkowski space, the acceleration $a_{6}$ associated with
the corresponding observer takes the form
$a_{6}^{-2}=x_{1}^{2}-x_{0}^{2}=\frac{l^{2}(r_{+}^{2}-r^{2})}{r_{+}^{2}}.$
(3.18)
We check that they obey the relation $a_{6}^{2}=a_{5}^{2}+1/l^{2}$. We
conclude that the observer has the Davies-Unruh temperature
$T_{\rm DU}=\frac{a_{6}}{2\pi}=\frac{r_{+}}{2\pi l\sqrt{r_{+}^{2}-r^{2}}}.$
(3.19)
Considering the redshift factor for the observer in the background (3.15), we
have the Hawking temperature observed as
$T_{\rm HK}=\frac{r_{+}}{2\pi l}.$ (3.20)
Consequently, we find the same Hawking temperature as (3.13) obtained when
using the coordinates (3.6).
## 4 Conclusions and Discussions
A $D$-dimensional CC black hole has unusual topological structure ${\cal
M}_{D-1}\times S^{1}$ and there is no globally timelike Killing vector in the
geometry of the black hole. Hence it was quite difficult to discuss
thermodynamic properties and Hawking temperature associated with this black
hole.
For example, Banados has considered a five dimensional rotating CC black hole
and embedded it into a Chern-Simons supergravity theory [3]. By computing
related conserved charges, it was shown that the black hole mass is
proportional to the product of outer horizon $r_{+}$ and inner horizon
$r_{-}$, while the angular momentum is proportional to the sum of two
horizons. In this case, the entropy of black hole is found to be proportional
not to the outer horizon $r_{+}$ but the inner horizon $r_{-}$. This approach
has two drawbacks. One is that the result cannot be degenerated to the non-
rotating case. The other is that it cannot be generalized to other dimensions.
Creighton and Mann have considered the quasilocal thermodynamics of a four
dimensional CC black hole in general relativity by computing thermodynamic
quantities at a finite boundary which encloses the black hole [5]. They have
shown that the entropy is not associated with the event horizon, but the
Killing horizon of a static observer which is tangent to the event horizon of
the black hole. The quasilocal energy density [see (11) of [5]] is negative.
In this work, we have derived Hawking temperature of CC black holes by
employing the globally embedding approach since these black holes can be
embedded into higher dimensional Minkowski space. We found that the Hawking
temperature of CC black holes is given by $r_{+}/2\pi l$ when using both
static and global coordinates. Here $r_{+}$ and $l$ are black hole horizon and
the radius of AdS space. Furthermore we found that the Hawking temperature is
also identical to the Gibbons-Hawking temperature of the boundary dS space of
the CC black holes. Importantly, we mention that the Hawking temperature
obtained in this work is the same as that obtained from semi-classical
tunneling method [19]. It turns out that the globally embedding technique is
powerful to determine the Hawking temperature of CC black hole without any
ambiguity. Using the same approach, we also obtained Hawking temperature of a
positive CC space which is counterpart of CC black hole in dS space.
Finally, we comment that those solutions including CC black holes and positive
CC spaces depend on an arbitrary real constant $r_{+}$. The $r_{+}$-dependence
can be made disappear by rescaling coordinates. In this case, the Hawking
temperature is given by $1/2\pi l$.
## Acknowledgments
This work was initiated during the APCTP joint focus program: frontiers of
black hole physics, Dec. 6-17, Pohang, Korea and Inje workshop on gravity and
numerical relativity during Dec. 16-18, Busan, Korea, the warm hospitality
extended to the authors in both places are grateful. RGC thanks S.P. Kim for
useful discussions during APCTP focus program. This work was supported in part
by a grant from Chinese Academy of Sciences and in part by the National
Natural Science Foundation of China under Grant Nos. 10821504, 10975168 and
11035008, and by the Ministry of Science and Technology of China under Grant
No. 2010CB833004. YSM was supported by the National Research Foundation of
Korea (NRF) grant funded by the Korea government (MEST) (No.2009-0062869).
## References
* [1] M. Banados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. 69, 1849 (1992) [arXiv:hep-th/9204099]; M. Banados, M. Henneaux, C. Teitelboim and J. Zanelli, Phys. Rev. D 48, 1506 (1993) [arXiv:gr-qc/9302012].
* [2] S. Holst and P. Peldan, Class. Quant. Grav. 14, 3433 (1997) [arXiv:gr-qc/9705067].
* [3] M. Banados, Phys. Rev. D 57, 1068 (1998) [arXiv:gr-qc/9703040].
* [4] M. Banados, A. Gomberoff and C. Martinez, Class. Quant. Grav. 15, 3575 (1998) [arXiv:hep-th/9805087].
* [5] J. D. Creighton and R. B. Mann, Phys. Rev. D 58, 024013 (1998) [arXiv:gr-qc/9710042].
* [6] S. F. Ross and G. Titchener, JHEP 0502, 021 (2005) [arXiv:hep-th/0411128].
* [7] J. A. Hutasoit, S. P. Kumar and J. Rafferty, JHEP 0904, 063 (2009) [arXiv:0902.1658 [hep-th]].
* [8] J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)] [arXiv:hep-th/9711200]; S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B 428, 105 (1998) [arXiv:hep-th/9802109]; E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998) [arXiv:hep-th/9802150].
* [9] R. G. Cai, Phys. Lett. B 544, 176 (2002) [arXiv:hep-th/0206223].
* [10] D. Birmingham and M. Rinaldi, Phys. Lett. B 544, 316 (2002) [arXiv:hep-th/0205246].
* [11] V. Balasubramanian and S. F. Ross, Phys. Rev. D 66, 086002 (2002) [arXiv:hep-th/0205290].
* [12] G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2738 (1977).
* [13] P. C. W. Davies, J. Phys. A 8, 609 (1975).
* [14] W. G. Unruh, Phys. Rev. D 14, 870 (1976).
* [15] H. Narnhofer, I. Peter and W. E. Thirring, Int. J. Mod. Phys. B 10, 1507 (1996).
* [16] S. Deser and O. Levin, Class. Quant. Grav. 14, L163 (1997) [arXiv:gr-qc/9706018]; S. Deser and O. Levin, Class. Quant. Grav. 15, L85 (1998) [arXiv:hep-th/9806223].
* [17] S. Deser and O. Levin, Phys. Rev. D 59, 064004 (1999) [arXiv:hep-th/9809159].
* [18] Y. W. Kim, J. Choi and Y. J. Park, Int. J. Mod. Phys. A 25, 3107 (2010) [arXiv:0909.3176 [gr-qc]].
* [19] A. Yale, arXiv:1012.2114 [gr-qc].
* [20] R. G. Cai, Phys. Lett. B 552, 66 (2003) [arXiv:hep-th/0207053].
* [21] S. Deser and R. Jackiw, Annals Phys. 153, 405 (1984); M. I. Park, Phys. Lett. B 440, 275 (1998) [arXiv:hep-th/9806119].
|
arxiv-papers
| 2010-12-28T07:44:51 |
2024-09-04T02:49:15.991087
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rong-Gen Cai and Yun Soo Myung",
"submitter": "Rong-Gen Cai",
"url": "https://arxiv.org/abs/1012.5709"
}
|
1012.5832
|
# On the Existence of Bertrand-Nash Equilibrium Prices Under Logit Demand
W. Ross Morrow and Steven J. Skerlos Iowa State University
wrmorrow@iastate.edu The University of Michigan, Department of Mechanical
Engineering skerlos@umich.edu
###### Abstract.
This article proves the existence of equilibrium prices in Bertrand
competition with multi-product firms using the Logit model of demand. The most
general proof, an application of the Poincare-Hopf Theorem, does not rely on
restrictive assumptions such as single-product firms, firm homogeneity or
symmetry, homogeneous product costs, or even concavity of the utility function
with respect to prices. This proof relies on new conditions for the indirect
utility function, along with fixed-point equations derived from the first-
order conditions and a direct analysis of the second-order conditions that
proves the uniqueness of profit-maximizing prices. The degree to which our
conditions are as weak as possible is discussed. Models with finite purchasing
power and convex total costs are also addressed. Analysis of equilibrium
prices for multi-product firms with constant unit costs suggests that
Bertrand-Nash equilibrium cannot adequately describe multi-product pricing in
differentiated product markets.
Significant portions of this research were undertaken while W. Ross Morrow was
a Ph.D. student in Mechanical Engineering at the University of Michigan. The
National Science Foundation, the University of Michigan Transportation
Research Institute’s Doctoral Studies Program, and Iowa State University
provided support for this research. The authors wish to thank Fred Feinberg,
Erin MacDonald, Jong-Shi Pang, Che-Lin Su, and Norman Shiau for helpful
suggestions.
## 1\. Introduction
Bertrand competiton has been a prominent paradigm for the empirical study of
differentiated product markets for over twenty years [14, 10, 19, 51, 24, 20,
49, 43, 11, 48, 1]. Most of these empirical applications have been undertaken
without theoretical assurances of the existence, uniqueness, and even
“plausibility” of Bertrand-Nash equilibrium prices. This article proves the
existence of equilibrium prices for differentiated product market models based
on the Logit model under weak conditions on the (indirect) utility function
and convex total costs (i.e., unit costs that increase with volume). Further
analysis reveals some counter-intuitive properties of equilibrium prices that
suggest more complex models than Logit are required to adequately model
differentiated product markets in price competition.
Most existing theoretical analyses of Bertrand competition are based on
assumptions too restrictive to suit empirical applications of Bertrand
competition and that obscure potentially counterintuitive properties of
equilibrium. For example, there are few theoretical studies that consider
multi-product firms (see, e.g., [46, 4, 6]), but real firms in differentiated
product markets almost always offer more than one product. Theoretical
analyses of Bertrand-Nash equilibrium prices have also typically relied on
homogeneity or “symmetry” between firms with respect to the costs and
“values”111Authors in the theoretical literature use the term “quality” to
describe the non-price utility of a product [35]. This use of the term is
confounded with the way it would be interpreted by many engineers, marketers,
operations researchers, or laypeople as a measure of reliability. of the
products offered. Real markets are heterogeneous with respect to the number of
products offered, the values consumers derive from these products, and the
costs with which these products are produced. In one analysis, Anderson &
dePalma [6] state that
> “empirical application[s] would have to relax the symmetry assumptions and
> allow firms to produce products of different qualities, allow for
> heterogeneity across firms, and differing costs to introducing products.”
> [6, pg. 98]
Thus, existing theoretical analyses currently offer limited support to
empirical applications of differentiated product market models or other models
in which Logit models might be useful, such as those described by Gallego et.
al [18].
A theoretical understanding of Bertrand-Nash equilibrium prices begins with
the conditions under which equilibrium prices exist. Perloff [42] provided an
early existence proof for Bertrand-Nash equilibrium under a general Random
Utility Maximization (RUM) model. Firms in Perloff’s model are ‘systematically
homogeneous’ in that product differentiation exists only through random brand
preference, rather than differentiated product characteristics and unit costs.
Anderson & dePalma [3] undertake an analysis of equilibrium with single-
product firms focusing on the linear-in-price utility Logit model. They
characterize equilibrium prices with a closed-form expression when there is no
outside good, and as solutions to a fixed-point equation when an outside good
exists. Milgrom & Roberts [28]m Caplin & Nalebuff [15], and Gallego et. al
[18] have also provided equilibrium existence proofs for Bertrand competition
between single-product firms that apply to the Logit RUM, assuming utility is
linear in price. Such results have been used to ensure that empirical single-
product firm models based on Bertrand competition are well-posed [40, 7, 21,
5]. More recently, Sandor [45] and Konovalov & Sandor [27] have proven the
existence and uniqueness of equilibrium prices with multi-product firms and
the linear-in-price utility Logit model. Beyond models with single-product
firms and linear utility functions, the literature lacks general conditions
under which equilibrium exists. Without this understanding, it is not known if
empirical examples cannot have equilibrium prices. This article provides one
example where this has already occurred.
Unfortunately the mathematical methods employed in these works cannot be
extended to establish the existence of equilibrium prices for models with
multi-product firms and general non-linear utility functions. Perloff’s [42]
and Anderson & dePalma’s [3] analyses are specific to symmetric equilibrium
between homogeneous single-product firms. While Milgrom & Roberts [28] apply a
general property $-$ “supermodularity” $-$ to prove the existence (and
uniqueness) of single-product firm equilibrium prices under Logit, Sandor [45]
has shown that multi-product firm profit functions under linear-in-price
utility Logit fail to be supermodular arbitrarily near equilibrium prices,
ruling out the application of this property for multi-product firms; Appendix
C extends Sandor’s proof to any Logit model within the class studied here.
Similarly, the proofs from Caplin & Nalebuff [15] and Gallego et. al [18] rely
on quasi-concavity of the firms’ profit functions. Hanson & Martin [22],
however, have observed that multi-product firm profits are not quasi-concave
under the Logit model. Thus new mathematical tools are needed to prove the
existence of equilibrium prices for Bertrand competition between multi-product
firms.
The remainder of this article proceeds as follows. Section 2 presents a
framework for Bertrand competition under an arbitrary RUM model assuming unit
costs are constant (equivalently, total costs depend linearly on quantity
sold). Special attention is paid to the interpretation of an RUM model as the
generator of a stochastic choice process leading to random demand,
highlighting some implicit assumptions that may not be commonly acknowledged.
Section 3 specializes this framework to the Logit RUM model, equivalent to the
“attraction demand model” being used by some researchers in revenue management
(e.g., [18]). Specifically, Section 3 presents a new set of utility
specifications, defines the Logit choice probabilities, and identifies when
firm profits under the Logit model are bounded.
Section 4 proves the existence of Bertrand-Nash equilibrium prices using
fixed-point equations derived from the first-order or “Simultaneous
Stationarity” condition when unit costs are constant. Most existing analyses
of equilibrium prices also rely on the first-order condition. Moreover, fixed-
point expressions have already been utilized to characterize equilibrium under
linear-in-price utility Logit models [3, 10, 12, 45, 27] and even for more
complex Mixed Logit models [10, 31, 33, 32]. Here three fixed-point equations
for Logit models are derived, two of which generalize to Mixed Logit models;
see [31, 33, 32]. As is common in analyses of equilibrium, the existence proof
has two parts: proving that (i) there exist simultaneously stationary prices
and (ii) simultaneously stationary prices are in fact equilibria. Existence
proofs using both Brouwer’s fixed-point theorem and the Poincare-Hopf Theorem
[29, Chapter 6] are given. The Poincare-Hopf approach to this problem was
first taken in [31], and has also been used by Konovalov & Sandor [27],
restricting the utility function to be linear-in-price. Simsek et. al [47]
provide several references to applications of the Poincare-Hopf Theorem in
general equilibrium models. Identifying simultaneously stationary prices with
equilibria requires a direct analysis of the second-order conditions, combined
with a second application of the Poincare-Hopf Theorem to prove the uniqueness
of profit-maximizing prices, circumventing the lack of quasi-concavity in
Logit profits for multi-product firms. These results are based on new, general
conditions on the utility function that are weaker than most assumptions
currently applied in theoretical economics, econometrics, and marketing.
Sections 5 and 6 generalize the analysis in Section 4 to address non-constant
unit costs and populations with finite purchasing power, respectively. The
treatment of non-constant unit costs is a fairly straightforward extension of
the analysis for the constant unit cost case, at least when total costs are
convex (that is, unit costs are increasing in volume). Limits on purchasing
power qualitatively change the behavior that may occur in equilibrium: While
every product has a non-zero (if small) probability of being purchased in
equilibrium with no such limit, some products can be profit-optimally “priced
out of the market” when there is a finite limit on purchasing power. The
fixed-point approach from the traditional, no-limit case is extended to
characterize equilibrium prices when there is a limit, and existence is proved
with essentially the same methods.
Finally, Section 7 identifies several structural properties of equilibrium
prices under the Logit model. First, if the consumer population systematically
values some product’s characteristics more than the same firm’s other
offerings, that product must be given lower profit-optimal markup by the firm
in equilibrium when unit costs are constant. This counterintuitive result
cannot be observed in analyses with single-product firms, and relies only on
the common assumptions that (i) utility is concave in price and separable in
price and characteristics and (ii) unit costs, though constant, increase with
the value of product characteristics [35, 14]. As a consequence, Bertand
competition under Logit with conventional utility specifications and constant
unit costs cannot have fixed percentage markups as an equilibrium outcome;
i.e. “cost plus pricing” [36] is not rationalized by Bertrand competition
under conventional Logit models. Second, there exists a portfolio effect for
Bertrand-Nash equilibrium under Logit with constant unit costs: equilibrium
prices for an identical product offered at the same unit cost by two distinct
firms depends on the profitability of the entire portfolio of products offered
by these firms. In other words, heterogeneous portfolios can lead to distinct
equilibrium prices for otherwise identical products. This property would not
be observed from analyses that assume firms are homogeneous.
The most limiting assumption in this article is the absence of consumer
heterogeneity in the choice model. The Logit RUM model does allows some random
variance in the utilities individuals in the population derive from the
differentiated products, and thus contains some degree of population
heterogeneity. However, the degree to which this is expressed with the Logit
model has long been known to generate patterns of substitution that are
unrealistic form many empirical applications [10, 50]. The techniques used in
this paper to prove the existence of simultaneously stationary prices can be
extended to a large class of Mixed Logit models with some ease; see [33, 32].
Moreover, the fixed-point equations used here are very useful in computations
of equilibrium prices under such models [33]. However, the central element of
the existence results established here is a generalization of quasi-concavity
property that ensures profits under Logit have unique profit-maximizing
prices. The conditions under which such a condition holds for Mixed Logit
models are not obvious and will be nontrivial [33].
A review of mathematical notation, several examples, and additional results
are provided in the appendices.
## 2\. Bertrand Competition Under an Arbitrary Random Utility Model
This section presents a mathematical framework for Bertrand competition under
an arbitrary Random Utility Maximization (RUM) model. This generalizes the
discussion in [8] to multi-product firms and RUM demand. Conceptually, a fixed
number of firms decide on prices for a fixed set of products prior to some
time period in which these prices must remain fixed. During this purchasing
period, a fixed number of individuals independently choose to purchase one of
the products offered by these firms, or to forgo purchase of any of these
products, following a given RUM model. Verboven [51] describes this as a two-
stage stochastic game, where in the first stage the firms choose prices and in
the second stage individuals choose products to maximize their own utility
after sampling, or “drawing,” from the distribution of random utilities.222In
the second stage, all consumers have dominant strategies.
### 2.1. Random Utility Models and Demand for Products
RUM models provide a means to describe selection from a choice set, a
collection of $J\in\mathbb{N}$ products that individuals may choose to
purchase along with a no-purchase option (or “outside good”) indexed by 0.
Each product $j\in\mathbb{N}(J)$ is characterized by its price
$p_{j}\in[0,\infty)$ and vector of characteristics
$\mathbf{y}_{j}\in\mathcal{Y}$, where $\mathcal{Y}\subset\mathbb{R}^{K}$ for
some $K\in\mathbb{N}$.
The random variable $U_{i,j}(\mathbf{y}_{j},p_{j})$ gives the utility
individual $i$ receives by purchasing product $j\in\mathbb{N}(J)$, while the
random variable $U_{i,0}$ gives the utility received by not purchasing any of
the products (i.e. “purchasing the outside good”). Conditional on the values
of $\\{U_{i,0}\\}\cup\\{U_{i,j}(\mathbf{y}_{j},p_{j})\\}_{j\in\mathbb{N}(J)}$,
individual $i$ chooses the option $j\in\\{0\\}\cup\mathbb{N}(J)$ with the
highest utility. The choice variable $C_{i}(\mathbf{Y},\mathbf{p})$
encapsulates this selection, taking values in $\\{0\\}\cup\mathbb{N}(J)$
following the distribution
$\mathbb{P}(C_{i}(\mathbf{Y},\mathbf{p})=j)=\left\\{\begin{aligned}
&\mathbb{P}\left(U_{i,j}(\mathbf{y}_{j},p_{j})=\max\left\\{\;U_{i,0}\;,\;\max_{k\in\mathbb{N}(J)}U_{i,k}(\mathbf{y}_{k},p_{k})\;\right\\}\right)&&\quad\text{if
}j\in\mathbb{N}(J)\\\
&\mathbb{P}\left(U_{i,0}=\max\left\\{\;U_{i,0}\;,\;\max_{k\in\mathbb{N}(J)}U_{i,k}(\mathbf{y}_{k},p_{k})\;\right\\}\right)&&\quad\text{if
}j=0\end{aligned}\right.$
The distribution of these random utilities assures that “ties” occur with
probability zero. Let
$\mathbf{U}_{i}(\mathbf{Y},\mathbf{p})=(U_{i,0},U_{i,1}(\mathbf{Y},\mathbf{p}),\dotsc,U_{i,J}(\mathbf{Y},\mathbf{p})),$
and make the following assumption:
###### Assumption 2.1.
For any $(\mathbf{Y},\mathbf{p})\in\mathcal{Y}^{J}\times\mathbb{R}_{+}^{J}$
and $i,i^{\prime}\in\mathbb{N}(I)$, $\mathbf{U}_{i}(\mathbf{Y},\mathbf{p})$
and $\mathbf{U}_{i^{\prime}}(\mathbf{Y},\mathbf{p})$ are independent and
identically distributed.
Under this common assumption, the individual index on utilities and the choice
variable can be dropped. Note also that this does not imply that
$U_{j}(\mathbf{y}_{j},p_{j})$ and $U_{k}(\mathbf{y}_{k},p_{k})$ are
independent.
In practice, models often take the form
$U_{0}=\vartheta+\mathcal{E}_{0}\quad\quad\text{and}\quad\quad
U_{j}(\mathbf{y}_{j},p_{j})=u(\mathbf{y}_{j},p_{j})+\mathcal{E}_{j}\text{ for
all }j\in\mathbb{N}(J)$
for some (conditional, indirect) utility function
$u:\mathcal{Y}\times[0,\infty)\to\mathbb{R}$, $\vartheta\in[-\infty,\infty)$
and “error” vector $\boldsymbol{\mathcal{E}}=\\{\mathcal{E}_{j}\\}_{j=0}^{J}$.
When $\boldsymbol{\mathcal{E}}$ is given an i.i.d. extreme value distribution,
we have the Logit RUM [3, 50].333This independence assumption on
$\boldsymbol{\mathcal{E}}$ is distinct from our independence assumption on the
random utilities in that now it is independence across products in the choice
set, rather than across individuals in the population. Letting
$\boldsymbol{\mathcal{E}}$ have a Generalized Extreme Value distribution we
have a GEV RUM like the Nested Logit model [50, 13], or taking
$\boldsymbol{\mathcal{E}}$ multivariate normal gives the Probit RUM [50].
Either of these latter two forms can have a $\boldsymbol{\mathcal{E}}$ with
correlated components.444More generally, we can let $\mathcal{T}$ be a space
of individual characteristics or “demographics” and define
$U_{0}=\vartheta(\boldsymbol{\Theta})+\mathcal{E}_{0}\quad\quad\text{and}\quad\quad
U_{j}(\mathbf{y}_{j},p_{j})=u(\boldsymbol{\Theta},\mathbf{y}_{j},p_{j})+\mathcal{E}_{j}\text{
for all }j\in\mathbb{N}(J)$ where
$u:\mathcal{T}\times\mathcal{Y}\times[0,\infty)\to\mathbb{R}$,
$\vartheta:\mathcal{T}\to[-\infty,\infty)$ where $\boldsymbol{\Theta}$ is a
$\mathcal{T}$-valued random variable with the distribution $\mu$, ostensibly
representing the distribution of demographic variables over the population.
With the same error distribution, we obtain a “mixed” RUM. Particularly,
taking $\boldsymbol{\mathcal{E}}$ i.i.d. extreme value gives the “random
coefficients” or Mixed Logit RUM class [50, Chapter 6].
Demands, the total quantity of each product purchased during the purchasing
period, must be defined to define firms’ profits. Extrapolating demands from
the stochastic choice model above requires the following assumption.
###### Assumption 2.2.
Every individual $i\in\mathbb{N}(I)$ observes the same choice set,
$\\{0\\}\cup\mathbb{N}(J)$, during the purchase period.
Under this assumption, the demand $Q_{j}(\mathbf{Y},\mathbf{p})$ for each
product $j\in\mathbb{N}(J)$ can be expressed simply as
$Q_{j}(\mathbf{Y},\mathbf{p})=\sum_{i=1}^{I}1_{\\{C_{i}(\mathbf{Y},\mathbf{p})=j\\}}$,
where here $\\{C_{i}(\mathbf{Y},\mathbf{p})\\}_{i\in\mathbb{N}(I)}$ are $I$
i.i.d. “copies” of $C(\mathbf{Y},\mathbf{p})$. The primary benefit of
Assumption 2.2 is that
$\\{Q_{0}(\mathbf{Y},\mathbf{p})\\}\cup\\{Q_{j}(\mathbf{Y},\mathbf{p})\\}_{j\in\mathbb{N}(J)}$
is a multinomial family of variables with parameter $I$ and probabilities
$\\{P_{j}(\mathbf{Y},\mathbf{p})\\}_{j=0}^{J}$, and thus expected demands for
each product are given simply by
$\mathbb{E}[Q_{j}(\mathbf{Y},\mathbf{p})]=IP_{j}(\mathbf{Y},\mathbf{p})$ [17].
A more serious implication of Assumption 2.2 is there must be at least $I$
units of every product available for the individuals to choose during the
purchasing period.555We could alternatively interpret this condition in terms
of consumers “ordering” products during the purchasing period and assuming
delivery schedules do not impact demand. Specifically, no product can “sell
out,” or if it does, “backordering” does not impact utilities. If any firm
commits (or is forced by capacity constraints) to only produce $I^{\prime}<I$
units of some product they offer during the purchasing period and individuals
do not “backorder”, then with some positive probability Assumption 2.2 will be
violated.
### 2.2. Firms, Product Portfolios, Costs, and Profits
Let $F\in\mathbb{N}$ denote the number of firms. For each $f\in\mathbb{N}(F)$,
there exists a set $\mathcal{J}_{f}\subset\mathbb{N}(J)$ of indices that
corresponds to the $J_{f}=\left\lvert\mathcal{J}_{f}\right\rvert$ products
offered by firm $f$. The collection of all these sets,
$\\{\mathcal{J}_{f}\\}_{f=1}^{F}$, forms a partition of $\mathbb{N}(J)$.
Subsequently, in writing “$f(j)$” for some $j\in\mathbb{N}(J)$, we mean the
unique $f\in\mathbb{N}(F)$ such that $j\in\mathcal{J}_{f}$. The vector
$\mathbf{p}_{f}\in\mathbb{R}^{J_{f}}$ refers to the vector of prices of the
products offered by firm $f$. Negative subscripts denote competitor’s
variables as in, for instance, $\mathbf{p}_{-f}\in\mathbb{R}^{J_{-f}}$, where
$J_{-f}=\sum_{g\neq f}J_{g}$, is the vector of prices for products offered by
all of firm $f$’s competitors. Firm-specific choice probability functions are
denoted by $\mathbf{P}_{f}(\mathbf{p})$.
Additional assumptions concerning costs and production are required to
complete the definition of firms’ profits.
###### Assumption 2.3.
There exists a unit cost function $c_{f}^{U}:\mathcal{Y}\to\mathbb{R}_{+}$ and
a fixed cost function $c_{f}^{F}:\mathfrak{F}(\mathcal{Y})\to\mathbb{R}_{+}$
for all $f\in\mathbb{N}(F)$ that depend only on the collection of product
characteristics chosen by the firm.
Particularly, unit and fixed costs are independent of the quantity sold,
ruling out dependence of unit and fixed costs on production volumes. This
assumption is relaxed in Section 5 below. Bertrand competition also entails
the following “comittment” assumption on the quantities produced [8].
###### Assumption 2.4 (Bertrand Production Assumption).
Each firm commits to producing exactly $Q_{j}(\mathbf{Y},\mathbf{p})$ units of
each product $j\in\mathcal{J}_{f}$ during the purchasing period.
Again, this implies that the firm has no production capacity constraints that
limit a firm’s ability to meet any demands that arise during the purchase
period. The random variable
$\sum_{j\in\mathcal{J}_{f}}c_{f}^{U}(\mathbf{y}_{j})Q_{j}(\mathbf{Y},\mathbf{p})+c_{f}^{F}(\mathbf{Y}_{f})$
gives the total cost firm $f$ incurs in producing
$Q_{j}(\mathbf{Y},\mathbf{p})$ units of product $j$, for all
$j\in\mathcal{J}_{f}$, during the purchasing period. We let
$\mathbf{c}_{f}^{U}(\mathbf{Y}_{f})$ be the vector of these unit costs for the
products offered by firm $f$.
Under Assumption 2.4, the random variable
$\Pi_{f}(\mathbf{Y},\mathbf{p})=\mathbf{Q}_{f}(\mathbf{Y},\mathbf{p})^{\top}(\mathbf{p}_{f}-\mathbf{c}_{f}^{U}(\mathbf{Y}_{f}))-c_{f}^{F}(\mathbf{Y}_{f})$
gives firm $f$’s profits for the production period as a function of product
characteristics and prices. Following most of the theoretical and empirical
literature in both marketing and economics, we assume that firms take expected
profits,
(1)
$\pi_{f}(\mathbf{Y},\mathbf{p})=I\hat{\pi}_{f}(\mathbf{Y},\mathbf{p})-c_{f}^{F}(\mathbf{Y}_{f})\quad\text{where}\quad\hat{\pi}_{f}(\mathbf{Y},\mathbf{p})=\mathbf{P}_{f}(\mathbf{Y},\mathbf{p})^{\top}(\mathbf{p}_{f}-\mathbf{c}_{f}^{U}(\mathbf{Y}_{f})),$
as the metric by which they optimize their pricing decisions in this
stochastic optimization problem.
Eqn. (1) demonstrates that neither the total firm fixed costs $c_{f}^{F}$ nor
the population size $I$ play a role in determining the prices that maximize
expected profits. Therefore we only consider the “population-normalized gross
expected profits” $\hat{\pi}_{f}(\mathbf{p})$, referred to below as simply
“profits”. We also consider $\mathbf{Y}$ fixed, and cease to include this
characteristic matrix as an argument. Finally, we write
$\mathbf{c}_{f}=\mathbf{c}_{f}^{U}$ as these are the only relevant costs for
the price equilibrium problem. Henceforth we write simply
$\hat{\pi}_{f}(\mathbf{p})=\mathbf{P}_{f}(\mathbf{p})^{\top}(\mathbf{p}_{f}-\mathbf{c}_{f}).$
The following adaptation of well-known necessary conditions for the local
maximization of an unconstrained, continuously differentiable function (e.g.,
[34]) informs our derivation of the Simultaneous Stationarity Condition.
###### Lemma 2.1.
Suppose $\mathbf{P}_{f}(\cdot,\mathbf{p}_{-f})$ is continuously differentiable
on some open
$\mathcal{A}\subset(\mathbf{0},\boldsymbol{\infty})\subset\mathbb{R}^{J_{f}}$.
If $\mathbf{p}_{f}\in\mathcal{A}$ is a local maximizer of
$\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$, then
(2)
$(\nabla_{f}\hat{\pi}_{f})(\mathbf{p})=(D_{f}\mathbf{P}_{f})(\mathbf{p})^{\top}(\mathbf{p}_{f}-\mathbf{c}_{f})+\mathbf{P}_{f}(\mathbf{p})=\mathbf{0}.$
### 2.3. Local Equilibrium and the Simultaneous Stationarity Conditions
As in much of the existing literature, our analysis relies on local conditions
for optimality of prices and thus must rely on the following local definition
of equilibrium.
###### Definition 2.1.
A price vector $\mathbf{p}\in[\mathbf{0},\boldsymbol{\infty}]$ is called a
local equilibrium if $\mathbf{p}_{f}$ is a local maximizer of
$\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$ for all $f\in\mathbb{N}(F)$. A price
vector $\mathbf{p}\in[\mathbf{0},\boldsymbol{\infty}]$ is called an
equilibrium if $\mathbf{p}_{f}$ is a maximizer of
$\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$ for all $f\in\mathbb{N}(F)$.
Finally, the following Simultaneous Stationarity Condition is a generic
necessary condition for local equilibrium if the RUM choice probabilities are
continuously differentiable in prices.
###### Definition 2.2.
Let $(\tilde{\nabla}\hat{\pi})(\mathbf{p})$ denote the “combined gradient”
with components
$((\tilde{\nabla}\hat{\pi})(\mathbf{p}))_{j}=(D_{j}\hat{\pi}_{f(j)})(\mathbf{p})$.
Let $(\tilde{D}\mathbf{P})(\mathbf{p})$ be the sparse matrix corresponding to
the intra-firm price derivatives of choice probabilities; that is,
$\big{(}(\tilde{D}\mathbf{P})(\mathbf{p})\big{)}_{j,k}=\left\\{\begin{aligned}
&(D_{k}P_{j})(\mathbf{p})&&\quad\text{if }f(j)=f(k)\\\ &\quad\quad
0&&\quad\text{if }f(j)\neq f(k)\end{aligned}\right..$
###### Lemma 2.2 (Simultaneous Stationarity Condition).
Suppose $\mathbf{P}$ is continuously differentiable on some open
$\mathcal{A}\subset(\mathbf{0},\boldsymbol{\infty})$. If
$\mathbf{p}\in\mathcal{A}$ is a local equilibrium, then
(3)
$(\tilde{\nabla}\hat{\pi})(\mathbf{p})=(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}(\mathbf{p}-\mathbf{c})+\mathbf{P}(\mathbf{p})=\mathbf{0}.$
Prices satisfying Eqn. (3) are called “simultaneously stationary.” In
principle, simultaneously stationary prices need not be equilibria. Additional
analysis is required to link stationarity with local optimality of profits
with respect to changes to the prices of a firm’s own products.
The necessity of the Simultaneous Stationarity Condition does not depend on
the RUM type, but only on the continuous differentiability of the choice
probabilities (with respect to price) and the cost assumption. Furthermore,
much of this development is the same for an arbitrary demand function, rather
than a RUM; see, e.g., [14, 24]. Thus, Eqn. (3) has appeared in many different
studies using alternative RUM specifications such as Logit models [12],
Generalized Extreme Value models [19, 20, 12, 52], and Mixed Logit models [10,
11, 37, 38, 49, 43, 48, 1, 9, 33]. In most of these studies, Eqn. (3) has not
been investigated far beyond Lemma 2.2.
Eqn. (3) has been consistently used through the corresponding BLP markup
equation $\mathbf{p}=\mathbf{c}+\boldsymbol{\eta}(\mathbf{p})$ where
(4)
$\boldsymbol{\eta}(\mathbf{p})=-(\tilde{D}\mathbf{P})(\mathbf{p})^{-\top}\mathbf{P}(\mathbf{p})$
assuming $(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}$ is nonsingular.666For
competing single-product firms, this reduces to the famous “negative
reciprocal of elasticity” form for the Lerner index (i.e. percent markups);
see [41]. This equation is typically used to estimate costs assuming prices
are in equilibrium. However, the markup equation
$\mathbf{p}=\mathbf{c}+\boldsymbol{\eta}(\mathbf{p})$ is a fixed-point
equation satisfied by all simultaneously stationary prices. In Section 4, we
give a specific form for $\boldsymbol{\eta}$ under the Logit model and derive
a new fixed-point equation for simultaneously stationary prices by factoring
out the gradient of the inclusive value from
$(\tilde{\nabla}\hat{\pi})(\mathbf{p})$; this fixed-point equation is a
specialization of the $\boldsymbol{\zeta}$-markup equation used by Morrow &
Skerlos for large-scale computations of equilibrium prices [33, 32].
## 3\. Logit Models
This section reviews the Logit model, providing the groundwork for the
analysis in later sections. Section 3.1 defines a new class of nonlinear
utility functions for which equilibrium prices can be shown to exist. Section
3.2 derives the corresponding Logit choice probabilities and their
derivatives. Finally, Section 3.3 provides conditions under which profit-
maximizing prices are finite, a pre-requisite for the existence of (finite)
equilibrium prices.
### 3.1. Systematic Utility Specifications
The random utility any individual receives by purchasing any particular
product is parameterized by its characteristic vector and price through some
function $u:\mathcal{Y}\times[0,\infty)\to[-\infty,\infty)$. We consider
specifications of the following form.
###### Assumption 3.1.
There are functions $w:\mathcal{Y}\times[0,\infty)\to(-\infty,\infty)$ and
$v:\mathcal{Y}\to(-\infty,\infty)$ such that utility can be written
$u(\mathbf{y},p)=w(\mathbf{y},p)+v(\mathbf{y})$. Concerning the behavior of
$w$, we assume that, for all $\mathbf{y}\in\mathcal{Y}$,
$w(\mathbf{y},\cdot):[0,\infty)\to(-\infty,\infty)$ is (a) strictly
decreasing, and (b) continuously differentiable on $(0,\infty)$. We also
assume that (c) $\lim_{p\uparrow\infty}w(\mathbf{y},p)=-\infty$, and
subsequently set $w(\mathbf{y},\infty)=-\infty$.
Writing $u(\mathbf{y},p)=w(\mathbf{y},p)+v(\mathbf{y})$ is completely general
so long as utility is defined for all $p\in[0,\infty)$. This form is
convenient to define the “value” of a product as that component of utility
that does not vary with price, and to define “separable” utilities, the most
common class of utility functions used in practice.
###### Definition 3.1.
We say $v(\mathbf{y})$ is the value of any product with characteristic vector
$\mathbf{y}$, and that utility is separable in price and characteristics (or
simply separable) if $w(\mathbf{y},p)=w(p)$ for all
$\mathbf{y}\in\mathcal{Y}$. We call
$\left\lvert(Dw)(\mathbf{y},p)\right\rvert^{-1}$ the (local) willingness to
pay (for product value).
The class formed by Assm. 3.1 encompasses the majority of utility functions
used in the theoretical and empirical literature. Assm. (a) is required of a
suitable indirect utility function, and (b) is required for an analysis of
equilibrium based on the first-order conditions. The assumption (c) is a
natural condition that ensures that the choice probabilities vanish as prices
increase without bound. In fact, by including utility functions that are not
concave-in-price, this class is larger than that typically studied. A number
of examples are given in Appendix B.
Concave-in-price utilities are certainly an important special case often
considered in economics. However, concavity turns out to be a stronger
assumption than is required to ensure the existence of finite equilibrium
prices under Logit. Instead, the following weaker property of the utility
price derivatives is sufficient.
###### Definition 3.2.
$w$ eventually decreases sufficiently quickly at $\mathbf{y}\in\mathcal{Y}$ if
there exists some $r(\mathbf{y})>1$ and some
$\bar{p}(\mathbf{y})\in[0,\infty)$ such that
$(Dw)(\mathbf{y},p)\leq-r(\mathbf{y})/p=-r(\mathbf{y})D[\log p]$ for all
$p>\bar{p}(\mathbf{y})$. $w$ itself eventually decreases sufficiently quickly
if $w$ eventually decreases sufficiently quickly at all
$\mathbf{y}\in\mathcal{Y}$.
The most commonly used finite utility functions, particularly strictly
decreasing and concave in price utility functions, satisfy
$\lim_{p\to\infty}(Dw)(\mathbf{y},p)<0$, and hence eventually decrease
sufficiently quickly with any $r$. Note also that if $w$ does not eventually
decrease sufficiently quickly at $\mathbf{y}$, then necessarily
$\left\lvert(Dw)(\mathbf{y},p)\right\rvert\to 0$ as $p\to\infty$. The example
$w(\mathbf{y},p)=-\alpha(\mathbf{y})\log p$ ($\alpha(\mathbf{y})>0$) shows
that this does not contradict Assm. 3.1, (c).
A distinct requirement on the second derivatives of utility is synonymous with
the sufficiency of stationarity under Logit.
###### Definition 3.3.
Suppose $w(\mathbf{y},\cdot)$ is twice continuously differentiable for all
$\mathbf{y}\in\mathcal{Y}$. We say that $w$ has sub-quadratic second
derivatives at $(\mathbf{y},p)\in\mathcal{Y}\times[0,\infty)$ if
$\omega(\mathbf{y},p)=(D^{2}w)(\mathbf{y},p)/(Dw)(\mathbf{y},p)^{2}<1$. We say
that $w$ itself has sub-quadratic second derivatives if $w$ has sub-quadratic
second derivatives at all $(\mathbf{y},p)\in\mathcal{Y}\times[0,\infty)$.
Note that if $w(\mathbf{y},\cdot)$ is concave, then $w$ trivially has sub-
quadratic second derivatives. However, $w(\mathbf{y},\cdot)$ can be convex and
still have sub-quadratic second derivatives. For example,
$w(\mathbf{y},p)=-\alpha(\mathbf{y})\log p$ for $\alpha(\mathbf{y})>1$ has
$\omega(\mathbf{y},p)=1/\alpha<1$.
With any collection of fixed product characteristic vectors
$\\{\mathbf{y}_{j}\\}_{j=1}^{J}$, we set $w_{j}(p)=w(\mathbf{y}_{j},p)$ and
$v_{j}=v(\mathbf{y}_{j})$ and thus generate a collection of product-specific
utility functions, $u_{j}(p)=w_{j}(p)+v_{j}$, that depend on price alone.
Vector functions $\mathbf{w}:[0,\infty]^{J}\to[-\infty,\infty)^{J}$ and
$\mathbf{u}:[0,\infty]^{J}\to[-\infty,\infty)^{J}$ are constructed from these
product-specific components by taking
$(\mathbf{w}(\mathbf{p}))_{j}=w_{j}(p_{j})$ and
$(\mathbf{u}(\mathbf{p}))_{j}=u_{j}(p_{j})$. In particular,
$\mathbf{u}(\mathbf{p})=\mathbf{w}(\mathbf{p})+\mathbf{v}$. Firm-specific
product values $\mathbf{v}_{f}$ and utilities
$\mathbf{u}_{f}(\mathbf{p}_{f})=\mathbf{w}_{f}(\mathbf{p}_{f})+\mathbf{v}_{f}$
are also defined in the natural way.
### 3.2. Logit Choice Probabilities
The Logit model [50, Chapter 3] takes the utility any individual receives when
purchasing product $j$ to be the random variable
$U_{j}(\mathbf{y}_{j},p_{j})=u(\mathbf{y}_{j},p_{j})+\mathcal{E}_{j}$ and the
utility of the outside good to be the random variable
$U_{0}=\vartheta+\mathcal{E}_{0}$, where
$\boldsymbol{\mathcal{E}}=\\{\mathcal{E}_{j}\\}_{j=0}^{J}$ is a family of
i.i.d. standard extreme value variables and $\vartheta\in[-\infty,\infty)$ is
a number representing the utility of the outside good.
The i.i.d. standard extreme value specification for $\boldsymbol{\mathcal{E}}$
generates the following choice probabilities (see, e.g., [50]):
(5)
$P_{j}^{L}(\mathbf{p})=\frac{e^{u_{j}(p_{j})}}{e^{\vartheta}+\sum_{k=1}^{J}e^{u_{k}(p_{k})}}$
The equivalent formula
$P_{j}^{L}(\mathbf{p})=\frac{e^{(u_{j}(p_{j})-\vartheta)}}{1+\sum_{k=1}^{J}e^{(u_{k}(p_{k})-\vartheta)}}$
corresponding to setting $\vartheta=0$ is often seen in the literature, but
offers no substantial advantage to the analysis in this article. When
$\vartheta=-\infty$,
$P_{j}^{L}(\mathbf{p})=\frac{e^{u_{j}(p_{j})}}{\sum_{k=1}^{J}e^{u_{k}(p_{k})}}.$
The following basic properties of the Logit choice probabilities are used
throughout.
###### Lemma 3.1.
The following hold under Assumption 3.1, for any $j$ and $f$: (i)
$0<P_{j}^{L}(\mathbf{p})<1$ and
$\mathbf{P}_{f}^{L}(\mathbf{p})^{\top}\mathbf{1}<1$ for all
$\mathbf{p}\in[0,\infty)^{J}$. (iii) If $\vartheta>-\infty$ and
$\mathbf{q}\in[0,\infty]^{J}$,
$\lim_{\mathbf{p}\to\mathbf{q}}P_{j}^{L}(\mathbf{p})$ exists. Moreover,
$\lim_{\mathbf{p}\to\mathbf{q}}P_{j}^{L}(\mathbf{p})=0$ if $q_{j}=\infty$, and
$\mathbf{P}_{f}^{L}(\mathbf{p})^{\top}\mathbf{1}<1$ for all
$\mathbf{p}\in[0,\infty]^{J}$. (iv) If $\vartheta=-\infty$, then for any
$\mathbf{x}\in[0,1]^{J}$, $\sum_{j=1}^{J}x_{j}=1$, there exists some sequence
$\\{\mathbf{p}^{(n)}\\}_{n\in\mathbb{N}}\subset[0,\infty)^{J}$ with
$\mathbf{p}^{(n)}\to\boldsymbol{\infty}$ such that
$\lim_{n\to\infty}\mathbf{P}^{L}(\mathbf{p}^{(n)})=\mathbf{x}$.
###### Proof.
(i), (ii), and (iii) follow easily from Eqn. 5. To prove (iv), first note that
$\mathbf{P}:[0,\infty)^{J}\to\triangle(J)$ is onto when $\vartheta=-\infty$,
where $\triangle(J)=\\{\mathbf{x}\in[0,1]^{J}:\sum_{j=1}^{J}x_{j}=1\\}$. Let
$\mathbf{x}\in\triangle(J)$. It suffices to solve $u_{j}(p_{j})=\log x_{j}$
for $p_{j}$, for all $j$, for then
$\displaystyle
P_{j}^{L}(\mathbf{p})=\frac{e^{u_{j}(p_{j})}}{\sum_{k=1}^{J}e^{u_{k}(p_{k})}}=\frac{e^{\log
x_{j}}}{\sum_{k=1}^{J}e^{\log
x_{k}}}=\frac{x_{j}}{\sum_{k=1}^{J}x_{k}}=x_{j}.$
So long as $\log x_{j}\geq u_{j}(0)$, such a $p_{j}$ exists and is unique.
Assuming, without loss of generality, that $u_{j}(0)\geq 0$ for all $j$
ensures that this condition holds for all $x_{j}\in[0,1]$. The existence of a
sequence tending to infinity with
$\lim_{n\to\infty}\mathbf{P}^{L}(\mathbf{p}^{(n)})=\mathbf{x}$ then follows
from the invariance result in Lemma 3.2 below. ∎
Claim (iv) amounts to the fact that the Logit choice probabilities without an
outside good cannot be both continuous and single valued on $[0,\infty]^{J}$,
and suggests that the presence of an outside good “purchased” with positive
probability is very important to optimization and equilibrium problems under
Logit. As noted in the proof, this claim is a consequence of the following
generalization of the “invariance of uniform price shifts” property of the
linear in price utility Logit model to the class of utility functions
specified by Assumption 3.1:
###### Lemma 3.2.
Suppose $w$ satisfies Assumption 3.1. For any $p\in(0,\infty)$ and each
$j\in\mathbb{N}(J)$, define $\chi_{j,p}:[1,\infty)\to[p,\infty)$ by
$\chi_{j,p}(\lambda)=w_{j}^{-1}(w_{j}(p)-\log\lambda)$, and define
$\boldsymbol{\chi}_{\mathbf{p}}:[1,\infty)\to[\mathbf{p},\boldsymbol{\infty})$
componentwise by
$(\boldsymbol{\chi}_{\mathbf{p}}(\lambda))_{j}=\chi_{j,p_{j}}(\lambda)$. (i)
$\boldsymbol{\chi}_{\mathbf{p}}(\lambda)$ is well-defined, strictly
increasing, and
$\lim_{\lambda\to\infty}\boldsymbol{\chi}_{\mathbf{p}}(\lambda)=\boldsymbol{\infty}$.
(ii) If $\vartheta=-\infty$, $\mathbf{P}^{L}$ is invariant on
$\boldsymbol{\chi}_{\mathbf{p}}([1,\infty))$; i.e.,
$\mathbf{P}^{L}(\boldsymbol{\chi}_{\mathbf{p}}(\lambda))\equiv\mathbf{P}^{L}(\mathbf{p})$.
(iii) If $\vartheta>-\infty$,
$\mathbf{P}^{L}(\boldsymbol{\chi}_{\mathbf{p}}(\lambda))$ is strictly
decreasing in $\lambda$, and
$\mathbf{P}^{L}(\boldsymbol{\chi}_{\mathbf{p}}(\lambda))\to\mathbf{0}$ as
$\lambda\to\infty$.
###### Proof.
(i): By definition, $w_{j}(\chi_{j,p}(\lambda))=w_{j}(p)-\log\lambda$. Because
$w_{j}$ is strictly decreasing and
$w_{j}(\cdot):[p,\infty)\to(-\infty,w_{j}(p)]$ is onto, $\chi_{j,p}(\lambda)$
is uniquely defined for all $\lambda\geq 1$ and strictly increasing. Because
$w_{j}(p)-\log\lambda\downarrow-\infty$ as $\lambda\uparrow\infty$,
$\lim_{\lambda\uparrow\infty}\chi_{j,p}(\lambda)=\infty$.
(ii): Note that
$e^{u_{j}(\chi_{j,p}(\lambda))}=e^{w_{j}(\chi_{j,p}(\lambda))+v_{j}}=e^{w_{j}(p)-\log\lambda+v_{j}}=\lambda^{-1}\big{(}e^{w_{j}(p)+v_{j}}\big{)}.$
Thus if $\vartheta=-\infty$,
$P_{j}^{L}(\boldsymbol{\chi}_{\mathbf{p}}(\lambda))=\frac{e^{u_{j}(\chi_{j,p_{k}}(\lambda))}}{\sum_{k=1}^{J}e^{u_{k}(\chi_{k,p_{k}}(\lambda))}}=\frac{\lambda^{-1}e^{w_{j}(p_{j})+v_{j}}}{\lambda^{-1}\sum_{k=1}^{J}e^{w_{k}(p_{k})+v_{k}}}=P_{j}^{L}(\mathbf{p}).$
(iii): Similarly, if $\vartheta>-\infty$,
$P_{j}^{L}(\boldsymbol{\chi}_{\mathbf{p}}(\lambda))=\frac{e^{u_{j}(\chi_{j,p_{k}}(\lambda))}}{e^{\vartheta}+\sum_{k=1}^{J}e^{u_{k}(\chi_{k,p_{k}}(\lambda))}}=\frac{e^{w_{j}(p_{j})+v_{j}}}{\lambda
e^{\vartheta}+\sum_{k=1}^{J}e^{w_{k}(p_{k})+v_{k}}}<P_{j}^{L}(\mathbf{p})$
for all $\lambda>1$ and $P_{j}^{L}(\boldsymbol{\chi}_{\mathbf{p}}(\lambda))\to
0$ as $\lambda\to\infty$. ∎
The invariance of the Logit choice probabilities over sequences of prices that
tend to infinity should be viewed as an unacceptable property for realistic
market models.777Mizuno [30] makes explicit use of this unrealistic property
in proving the existence and uniqueness of equilibrium prices under Logit with
single-product firms and linear in price utilities. Individuals are sure to
make purchasing decisions based on the absolute value of product prices,
rather than just the relative value. It is easy also fairly easy to see that
Lemma 3.2, (iv) extends beyond Logit to any Generalized Extreme Value model
without an outside good.
The following form for the price derivatives of the Logit choice probabilities
is also required.
###### Lemma 3.3.
If $w$ satisfies Assm. 3.1 (b), then $\mathbf{P}^{L}$ is continuously
differentiable for all $\mathbf{p}\in(0,\infty)^{J}$ with
(6)
$\displaystyle(D_{k}P_{j}^{L})(\mathbf{p})=P_{j}^{L}(\mathbf{p})(\delta_{j,k}-P_{k}^{L}(\mathbf{p}))(Dw_{k})(p_{k})=(\delta_{j,k}-P_{j}^{L}(\mathbf{p}))\lambda_{k}(\mathbf{p})$
where $\lambda_{k}(\mathbf{p})=(Dw_{k})(p_{k})P_{k}^{L}(\mathbf{p})$. In other
words,
(7)
$(D_{f}\mathbf{P}^{L}_{f})(\mathbf{p})=\left(\mathbf{I}-\mathbf{P}^{L}_{f}(\mathbf{p})\mathbf{1}^{\top}\right)\boldsymbol{\Lambda}_{f}(\mathbf{p})\quad\text{and}\quad(D\mathbf{P}^{L})(\mathbf{p})=\left(\mathbf{I}-\mathbf{P}^{L}(\mathbf{p})\mathbf{1}^{\top}\right)\boldsymbol{\Lambda}(\mathbf{p})$
where
$\boldsymbol{\Lambda}_{f}(\mathbf{p})=\mathrm{diag}(\boldsymbol{\lambda}_{f}(\mathbf{p}))$
and
$\boldsymbol{\Lambda}(\mathbf{p})=\mathrm{diag}(\boldsymbol{\lambda}(\mathbf{p}))$.
When $w$ is twice differentiable on $(0,\infty)$, $\mathbf{P}^{L}$ is as well
and the second derivatives of the Logit choice probabilities are given by
(8) $\displaystyle(D_{l}D_{k}P_{j}^{L})(\mathbf{p})$
$\displaystyle=\delta_{k,l}\big{(}(D^{2}w_{k})(p_{k})+(Dw_{k})(p_{k})^{2}\big{)}P_{k}^{L}(\mathbf{p})\big{(}\delta_{j,k}-P_{j}^{L}(\mathbf{p})\big{)}$
$\displaystyle\quad\quad\quad\quad+\lambda_{k}(\mathbf{p})\big{(}2P_{j}^{L}(\mathbf{p})-\delta_{j,k}-\delta_{j,l}\big{)}\lambda_{l}(\mathbf{p}).$
###### Proof.
These follow directly from Eqn. (5). ∎
### 3.3. Bounded and Vanishing Logit Profits
An understanding of when profits are bounded over the set of all non-negative
prices is a pre-requisite to a general analysis of profit-optimal prices and
corresponding price equilibrium. One might expect that because Assumption 3.1
(c) implies that the choice probabilities vanish as prices increase without
bound that profits should also, but this is not true:
$w(\mathbf{y},p)=-\alpha\log p$, a specification derived by Allenby & Rossi to
represent “asymmetric brand switching under price changes” [2], can generate
unbounded profits even though the choice probabilities vanish. The following
property of utility functions guarantees not only the finiteness of Logit
profits, but that these profits vanish as prices increase without bound.888The
constant $\kappa(\mathbf{y})$ is convenient, but not necessary; it is easy to
show that $w$ is eventually log bounded with
$(r(\mathbf{y}),\bar{p}(\mathbf{y}),\kappa(\mathbf{y}))$ where
$\kappa(\mathbf{y})\neq 0$ if and only if it is so with some
$(r^{\prime}(\mathbf{y}),\bar{p}^{\prime}(\mathbf{y}),0)$.
###### Definition 3.4.
$w$ is eventually log bounded at $\mathbf{y}\in\mathcal{Y}$ if there exists
some $r(\mathbf{y})>1$, $\kappa(\mathbf{y})$, and some
$\bar{p}(\mathbf{y})\in[0,\infty)$ such that
$w(\mathbf{y},p)\leq-r(\mathbf{y})\log p+\kappa(\mathbf{y})$ for all
$p>\bar{p}(\mathbf{y})$. $w$ itself is eventually log bounded if $w$ is
eventually log bounded at all $\mathbf{y}\in\mathcal{Y}$.
Note that if $w$ is eventually log bounded then Assumption 3.1 (c) necessarily
holds. Furthermore, if $w$ eventually decreases sufficiently quickly then the
fundamental theorem of calculus implies that $w$ is also eventually log
bounded. Appendix B contains a somewhat pathological example demonstrating
that the converse need not hold.
###### Lemma 3.4.
Suppose $w$ satisfies Assumption 3.1. (i) Let $\vartheta>-\infty$,
$\mathbf{q}\in[0,\infty]^{J}$, and suppose that there exists
$r:\mathcal{Y}\to[1,\infty)$, $\bar{p}:\mathcal{Y}\to[0,\infty)$, and
$\kappa:\mathcal{Y}\to\mathbb{R}$ such that
$w(\mathbf{y},p)\leq-r(\mathbf{y})\log p+\kappa(\mathbf{y})$ for all
$p>\bar{p}(\mathbf{y})$. Then
$\lim_{\mathbf{p}\to\mathbf{q}}\hat{\pi}_{f}(\mathbf{p})<\infty$. (ii) If in
fact $w$ is eventually log bounded, i.e.
$r(\mathbf{y}):\mathcal{Y}\to(1,\infty)$, then
$\lim_{\mathbf{p}\to\mathbf{q}}\hat{\pi}_{f}(\mathbf{p})=0$ if
$\mathbf{q}_{f}=\boldsymbol{\infty}$.
###### Proof.
The following inequality always holds:
$P_{j}^{L}(\mathbf{p})p_{j}\leq\frac{e^{u_{j}(p_{j})}}{e^{\vartheta}+e^{u_{j}(p_{j})}}p_{j}=\frac{e^{w_{j}(p_{j})+v_{j}-\vartheta}}{1+e^{w_{j}(p_{j})+v_{j}-\vartheta}}p_{j}=p_{j}e^{w_{j}(p_{j})+v_{j}-\vartheta}.$
Under the hypothesis of (i),
$P_{j}^{L}(\mathbf{p})p_{j}\leq
p_{j}^{1-r_{j}}e^{\kappa_{j}+v_{j}-\vartheta}\leq
e^{\kappa_{j}+v_{j}-\vartheta}$
for all $p_{j}$ sufficiently large. Claim (i) is a consequence of this bound.
Moreover, $r_{j}<1$ for all $j$, then $P_{j}^{L}(\mathbf{p})p_{j}\downarrow 0$
as $p_{j}\uparrow\infty$. Claim (ii) is a consequence. ∎
Appendix B contains an example demonstrating that the converse to the second
claim is false. That is, bounded and vanishing Logit profits need not imply
that $w$ is eventually log bounded. If eventual log boundedness is strongly
violated in the sense of the hypothesis in the following lemma, then profits
must increase without bound as prices do.
###### Lemma 3.5.
Let $\vartheta>-\infty$ and Assumption 3.1 hold. Suppose that for some
$\mathbf{y}_{*}\in\mathcal{Y}$ there exists $r(\mathbf{y}_{*})\in(0,1)$,
$\kappa(\mathbf{y}_{*})$, and $\bar{p}\in[0,\infty)$ such that for all
$p>\bar{p}$, $w(\mathbf{y}_{*},p)\geq-r\log p+\kappa(\mathbf{y}_{*})$. Suppose
further that $\mathbf{y}_{j}=\mathbf{y}_{*}$ for some $j\in\mathcal{J}_{f}$.
Then $\lim_{\mathbf{p}\to\mathbf{q}}\hat{\pi}_{f}(\mathbf{p})=\infty$ for any
$\mathbf{q}\in[0,\infty]^{J}$ with $q_{j}=\infty$.
###### Proof.
Under the hypothesis, $p_{j}e^{u_{j}(p_{j})}\geq(p_{j})^{1-r}e^{\kappa+v_{j}}$
for all sufficiently large $p_{j}$. Thus $p_{j}e^{u_{j}(p_{j})}\to\infty$ as
$p_{j}\uparrow\infty$ because $r<1$. Clearly then
$P_{j}^{L}(p_{j},\mathbf{p}_{-j})p_{j}\to\infty$ as $p_{j}\uparrow\infty$. The
claim follows. ∎
The results above establish when optimal profits are positive and finite, and
when profit-optimal prices are not all infinite. Showing that profit
maximizing prices are all finite is proved in Section 4 with the slightly
strengthened hypothesis that $w$ eventually decreases sufficiently quickly.
###### Lemma 3.6.
Suppose $w$ satisfies Assumption 3.1 and is eventually log bounded. (i) If
$\vartheta>-\infty$ and $\mathbf{p}_{f}\in[0,\infty]^{J_{f}}$ locally
maximizes $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$ for any
$\mathbf{p}_{-f}\in[0,\infty]^{J_{-f}}$, then
$\mathbf{p}_{f}\neq\boldsymbol{\infty}$. (ii) If $\vartheta=-\infty$ and
$\mathbf{p}_{f}\in[0,\infty]^{J_{f}}$ locally maximizes
$\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$ for any
$\mathbf{p}_{-f}\in[0,\infty]^{J_{-f}}\setminus\\{\boldsymbol{\infty}\\}$,
then $\mathbf{p}_{f}\neq\boldsymbol{\infty}$. However,
$\hat{\pi}_{f}(\cdot,\boldsymbol{\infty})$ is maximized only by
$\mathbf{p}_{f}=\boldsymbol{\infty}$, and thus $\boldsymbol{\infty}$ is always
an equilibrium.
###### Proof.
(i): Profit maximizing prices $\mathbf{p}_{f}$ are not all infinite because
$\hat{\pi}_{f}(\boldsymbol{\infty},\mathbf{p}_{-f})=0$, and any prices
$\mathbf{p}_{f}>\mathbf{c}_{f}$ give $\hat{\pi}_{f}(\mathbf{p})>0$. (ii): The
same holds when $\vartheta=-\infty$ and
$\mathbf{p}_{-f}\neq\boldsymbol{\infty}$, because some product’s utility is
finite. However, if $\mathbf{p}_{-f}=\boldsymbol{\infty}$, Lemma 3.2 proves
that
$\displaystyle\hat{\pi}_{f}(\boldsymbol{\chi}_{f,\mathbf{p}_{f}}(\lambda),\boldsymbol{\infty})=\mathbf{P}_{f}^{L}(\boldsymbol{\chi}_{f,\mathbf{p}}(\lambda),\boldsymbol{\infty})^{\top}(\boldsymbol{\chi}_{f,\mathbf{p}_{f}}(\lambda)-\mathbf{c}_{f})=\mathbf{P}_{f}^{L}(\mathbf{p}_{f},\boldsymbol{\infty})^{\top}(\boldsymbol{\chi}_{f,\mathbf{p}_{f}}(\lambda)-\mathbf{c}_{f})\to\infty$
as $\lambda\uparrow\infty$, for any finite $\mathbf{p}_{f}$. ∎
## 4\. Equilibrium Prices Under Logit Models
This section proves the following theorem regarding equilibrium prices for
Bertrand competition under the Logit model as described in Sections 2-3:
###### Theorem 4.1.
Suppose that $\vartheta>-\infty$, Assumption 3.1 holds, $w$ eventually
decreases sufficiently quickly (Defn. 3.2) and $w$ has sub-quadratic second
derivatives (Defn. 3.3). There is at least one equilibrium $\mathbf{p}$, and
any equilibrium satisfies $\mathbf{c}<\mathbf{p}<\boldsymbol{\infty}$.
For further clarity, the key results for both profit maximization and
equilibrium problems are outlined with their assumptions in Tables 2 and 2.
Three fixed-point characterizations are applied to prove Theorem 4.1. One
fixed-point characterization is a generalization of existing results, while
two are apparently novel. The first of these novel characterizations states
that markups are equal to profits plus the (local) willingness to pay for
product value. In essence, this equation is derived by factoring out the
gradient of the “inclusive value,” or expected maximum utility, from
$(\tilde{\nabla}\hat{\pi})(\mathbf{p})$. This new fixed-point equation also
proves that multi-product firm optimal pricing problems under Logit are always
“one-parameter” problems. Specifically, all profit-maximizing prices are
determined uniquely from a knowledge of profits a single price. This
observation yields our second novel fixed-point characterization, a “reduced-
form” characterization in terms of equilibrium profits alone.
As is common in theoretical economics, the proof has two parts: First, the
existence of simultaneously stationary prices is proved, followed by a proof
that simultaneously stationary prices are equilibria. The most general proof
of the existence of finite simultaneously stationary prices is accomplished
using the Poincare-Hopf theorem [29, 47]. Brouwer’s theorem can also be
applied under stronger assumptions. Proving that simultaneously stationary
prices are in fact equilibria is somewhat more involved. In the past appeals
to quasi-concavity have been used to prove that profits have unique maximizers
(see, e.g., [15]). While the multi-product firm Logit profit functions are not
quasi-concave [22], under utilities with sub-quadratic second derivatives
first-order stationarity of profits in fact implies local concavity, the
second-order sufficiency condition. A distinct application of the Poincare-
Hopf theorem then implies that Logit profits have unique stationary points
which must be unique global profit maximizers for fixed competitor’s prices,
effectively circumventing the difficulties with profits that are not quasi-
concave. Note that while this argument establishes that simultaneously
stationary prices are equilibria, it does not necessarily imply that
equilibria are unique. The analysis in this section concerning models with
constant unit costs and no finite limit on purchasing power serves as a
prototype for the analysis of the more general cases presented in Sections 5
and 6.
Table 1. Assumptions required for important profit maximizations results.
Stationarity is necessary | Assumption 3.1
---|---
Stationarity is sufficient | Assumption 3.1, Defn. 3.3
Optimal profits are finite | Defn. 3.4
Profit-maximizing prices are finite | $\vartheta>-\infty$, Assumption 3.1, Defn. 3.2
Profit-maximizing prices are unique | $\vartheta>-\infty$, Assumption 3.1, Defn. 3.2, Defn. 3.3
Table 2. Assumptions required for important equilibrium results.
Simultaneous stationarity is necessary | Assumption 3.1
---|---
Simultaneously stationary prices are local equilibria | Assumption 3.1, Defn. 3.3
Simultaneously stationary prices exist | $\vartheta>-\infty$, Assumption 3.1, Defn. 3.2
Local equilibria are equilibria | Assumption 3.1, Defn. 3.3
### 4.1. Fixed-Point Characterizations of Price Equilibrium
This section characterizes simultaneous stationarity in terms of fixed-point
equations.
#### 4.1.1. The BLP Markup Equation
One fixed-point characterization is derived by noting that
$(D_{f}\mathbf{P}_{f})(\mathbf{p})^{\top}=\boldsymbol{\Lambda}_{f}(\mathbf{p})(\mathbf{I}-\mathbf{1}\mathbf{P}_{f}(\mathbf{p})^{\top})$
and hence $(\nabla_{f}\hat{\pi}_{f})(\mathbf{p})=\mathbf{0}$ for
$\mathbf{p}\in(0,\infty)^{J}$ if, and only if,
(9)
$(\mathbf{I}-\mathbf{1}\mathbf{P}_{f}(\mathbf{p})^{\top})(\mathbf{p}_{f}-\mathbf{c}_{f})=-(D_{f}\mathbf{w}_{f})(\mathbf{p}_{f})^{-1}\mathbf{1}.$
This is a direct generalization of the fixed-point equations derived under
“constant coefficient” linear in price utility (i.e., $w(\mathbf{y},p)=-\alpha
p$ for some $\alpha>0$) for single-product firms by Anderson & de Palma [3]
and for multi-product firms by Besanko et al [12].
The following statements formalize this result.
###### Lemma 4.2.
Suppose $w$ satisfies Assumption 3.1. (i)
$(\mathbf{I}-\mathbf{1}\mathbf{P}^{L}_{f}(\mathbf{p})^{\top})^{-1}$ exists
whenever $\vartheta>-\infty$ or, if $\vartheta=-\infty$, when
$\mathbf{p}_{-f}\neq\boldsymbol{\infty}$, but not otherwise. (ii) If
$\mathbf{p}_{f}\in(0,\infty)^{J_{f}}$ locally maximizes
$\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$, then
$\mathbf{p}_{f}=\mathbf{c}_{f}+\boldsymbol{\eta}_{f}(\mathbf{p})$ where
(10)
$\boldsymbol{\eta}_{f}(\mathbf{p})=-(\mathbf{I}-\mathbf{1}\mathbf{P}^{L}_{f}(\mathbf{p})^{\top})^{-1}(D\mathbf{w}_{f})(\mathbf{p}_{f})^{-1}\mathbf{1}.$
(iii) If $\mathbf{p}\in(0,\infty)^{J}$ is a local equilibrium, then
$\mathbf{p}=\mathbf{c}+\boldsymbol{\eta}(\mathbf{p})$ where
(11)
$\boldsymbol{\eta}_{f}(\mathbf{p})=-(D\mathbf{w}_{f})(\mathbf{p}_{f})^{-1}\mathbf{1}-\left(\frac{\mathbf{P}^{L}_{f}(\mathbf{p})^{\top}(D\mathbf{w}_{f})(\mathbf{p}_{f})^{-1}\mathbf{1}}{1-\mathbf{P}^{L}_{f}(\mathbf{p})^{\top}\mathbf{1}}\right)\mathbf{1}$
(iv) For any $\mathbf{p}\in(0,\infty)^{J}$,
$\boldsymbol{\eta}_{f}(\mathbf{p})>\mathbf{0}$ and
$\boldsymbol{\eta}_{f}(\mathbf{p})>\mathbf{0}$. As a consequence, equilibrium
prices have positive markups.
The fixed-point equation in (iii) is a specialization of the “markup” equation
Eqn. (LABEL:MarkupEqn) popularized for Mixed Logit models by Berry, Levinsohn,
& Pakes [10]; see also [31, 33].
###### Proof.
(i): The Sherman-Morrison-Woodbury formula for the inverse of a rank-one
perturbation of the identity [39, Chapter 2, pg. 50] implies that
(12)
$\left(\mathbf{I}-\mathbf{1}\mathbf{P}_{f}^{L}(\mathbf{p})^{\top}\right)^{-1}=\mathbf{I}+\left(\frac{1}{1-\mathbf{P}_{f}^{L}(\mathbf{p})^{\top}\mathbf{1}}\right)\mathbf{1}\mathbf{P}_{f}^{L}(\mathbf{p})^{\top};$
so long as $\mathbf{P}_{f}^{L}(\mathbf{p})^{\top}\mathbf{1}<1$. This last
condition will hold if either $\vartheta>-\infty$ or, if $\vartheta=-\infty$,
$\mathbf{p}_{-f}\neq\boldsymbol{\infty}$.
(ii): We can write
$\displaystyle(\nabla_{f}\hat{\pi}_{f})(\mathbf{p})$
$\displaystyle=\boldsymbol{\Lambda}_{f}(\mathbf{p})\Big{(}(\mathbf{I}-\mathbf{1P}_{f}^{L}(\mathbf{p})^{\top})(\mathbf{p}_{f}-\mathbf{c}_{f})+(D_{f}\mathbf{w}_{f})(\mathbf{p}_{f})^{-1}\mathbf{1}\Big{)}.$
Stationarity then requires
$(\mathbf{I}-\mathbf{1P}_{f}^{L}(\mathbf{p})^{\top})(\mathbf{p}_{f}-\mathbf{c}_{f})+(D_{f}\mathbf{w}_{f})(\mathbf{p}_{f})^{-1}\mathbf{1}=\mathbf{0}$.
(iii) is a consequence of (ii).
(iv): Eqn. (12) proves that $\boldsymbol{\eta}_{f}(\mathbf{p})>\mathbf{0}$ so
long as $(Dw_{j})(p_{j})<0$. ∎
#### 4.1.2. A New Equation
Another fixed-point characterization follows by multiplying
$\mathbf{p}_{f}-\mathbf{c}_{f}$ through
$\mathbf{I}-\mathbf{1}\mathbf{P}_{f}(\mathbf{p})^{\top}$, instead of inverting
$\mathbf{I}-\mathbf{1}\mathbf{P}_{f}(\mathbf{p})^{\top}$ as a whole, yielding
$\mathbf{p}_{f}-\mathbf{c}_{f}=\hat{\pi}_{f}(\mathbf{p})\mathbf{1}-(D_{f}\mathbf{w}_{f})(\mathbf{p}_{f})^{-1}\mathbf{1}$
$\mathbf{1}\mathbf{P}_{f}(\mathbf{p})^{\top}$ could be considered the
“contractive” part of
$(\mathbf{I}-\mathbf{1}\mathbf{P}_{f}(\mathbf{p})^{\top})$ because
$\lvert\lvert\mathbf{1}\mathbf{P}_{f}(\mathbf{p})^{\top}\rvert\rvert_{\infty}=\lvert\lvert\mathbf{P}_{f}(\mathbf{p})\rvert\rvert_{1}<1$.
This derivation proves the following result.
###### Lemma 4.3.
Suppose $w$ satisfies Assumption 3.1 (a) and (b). Define
$\boldsymbol{\zeta}:(0,\infty)^{J}\to\mathbb{R}^{J}$ by
$\boldsymbol{\zeta}(\mathbf{p})=\tilde{\boldsymbol{\pi}}(\mathbf{p})-(D\mathbf{w})(\mathbf{p})^{-1}\mathbf{1}$
where $\tilde{\boldsymbol{\pi}}(\mathbf{p})\in\mathbb{R}^{J}$ is the vector
with components
$(\tilde{\boldsymbol{\pi}}(\mathbf{p}))_{j}=\hat{\pi}_{f(j)}(\mathbf{p})$.
$\boldsymbol{\zeta}$ has components
$\zeta_{j}(\mathbf{p})=\hat{\pi}_{f}(\mathbf{p})-(Dw_{j})(p_{j})^{-1}$ where
$j\in\mathcal{J}_{f}$, and “intra-firm” components
$\boldsymbol{\zeta}_{f}(\mathbf{p})=\hat{\pi}_{f}(\mathbf{p})\mathbf{1}-(D\mathbf{w}_{f})(\mathbf{p}_{f})^{-1}\mathbf{1}$.
(i) For any $\mathbf{p}\in(0,\infty)^{J}$,
$(\nabla_{f}\hat{\pi}_{f})(\mathbf{p})=\boldsymbol{\Lambda}_{f}(\mathbf{p})\boldsymbol{\varphi}_{f}(\mathbf{p})$
and
$(\tilde{\nabla}\hat{\pi})(\mathbf{p})=\boldsymbol{\Lambda}(\mathbf{p})\boldsymbol{\varphi}(\mathbf{p})$
where
(13)
$\boldsymbol{\varphi}_{f}(\mathbf{p})=\mathbf{p}_{f}-\mathbf{c}_{f}-\boldsymbol{\zeta}_{f}(\mathbf{p})\quad\quad\text{and}\quad\quad\boldsymbol{\varphi}(\mathbf{p})=\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p}).$
(ii) If $\mathbf{p}_{f}\in(0,\infty)^{J_{f}}$ locally maximizes
$\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$, then
$\boldsymbol{\varphi}_{f}(\mathbf{p})=\mathbf{0}$; i.e.,
$\mathbf{p}_{f}=\mathbf{c}_{f}+\boldsymbol{\zeta}_{f}(\mathbf{p}_{f},\mathbf{p}_{-f})$.
(iii) If $\mathbf{p}\in(0,\infty)^{J}$ is a local equilibrium, then
$\boldsymbol{\varphi}(\mathbf{p})$; i.e.
$\mathbf{p}=\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$.
In Appendix B, Eqn. (13) is used to show that profits under Logit with
$w(\mathbf{y},p)=-\alpha\log p$, a model first posed by Allenby & Rossi [2],
have no finite profit-maximizing prices when $\alpha\leq 1$. Sandor [45] has
also made this observation. Notably, Allenby & Rossi do undertake price
optimization exercises. Such exercises thus rely on estimating a coefficient
$\alpha$ that is statistically significantly strictly greater than one, a
question not addressed in [2].
Positivity and finiteness of equilibrium prices can be considered important
regularity properties, and follow from the $\boldsymbol{\zeta}$
characterization.
###### Lemma 4.4.
Suppose $w$ satisfies Assumption 3.1, $w$ eventually decreases sufficiently
quickly, and either $\vartheta>-\infty$ or
$\mathbf{p}_{-f}\neq\boldsymbol{\infty}$ if $\vartheta=-\infty$. Then no
$\mathbf{p}_{f}$ with some $p_{j}<c_{j}$ or $p_{j}=\infty$ maximizes
$\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$.
###### Proof.
Lemma 4.2, (iv), proves that $p_{j}>c_{j}$ if $p_{j}\neq 0$. We complete the
claim by proving that no price $p_{j}=0$ in equilibrium. For $p_{j}>0$, the
profit derivatives are
$\displaystyle(D_{j}\hat{\pi}_{f})(\mathbf{p})=\lambda_{j}(\mathbf{p})(p_{j}-c_{j}-\hat{\pi}_{f}(\mathbf{p}))+P_{j}^{L}(\mathbf{p})=\lambda_{j}(\mathbf{p})(p_{j}-c_{j}-\zeta_{j}(\mathbf{p}))$
If $\lim_{p\downarrow 0}(Dw_{j})(p)=0$, then the first equation here proves
that $\lim_{p_{j}\downarrow
0}(D_{j}\hat{\pi}_{f})(\mathbf{p})=P_{j}^{L}(\mathbf{p})>0$, and thus
$p_{j}=0$ cannot be profit-maximizing. If $\lim_{p\downarrow 0}(Dw_{j})(p)>0$,
$(D_{j}\hat{\pi}_{f})(\mathbf{p})\leq 0$ if, and only if,
$p_{j}-c_{j}-\zeta_{j}(\mathbf{p})\geq 0$. As $p_{j}\downarrow 0$,
$p_{j}-c_{j}-\zeta_{j}(\mathbf{p})\geq 0$ if, and only if,
$\displaystyle-\hat{\pi}_{f}(\mathbf{p})\geq(1-P_{j}^{L}(\mathbf{p}))c_{j}+\frac{1}{\left\lvert(Dw_{j})(p_{j})\right\rvert}.$
Because profit-optimal prices are positive, the left hand side is negative
while the right hand side is positive. By contradiction, $p_{j}=0$ cannot be
profit-maximizing. The finiteness of equilibrium prices follows from Lemma
4.10. ∎
#### 4.1.3. A Single-Parameter Equation
The $\boldsymbol{\zeta}$ characterization also illustrates that price
equilibrium problems with the Logit model and constant unit costs are “single-
parameter problems.” Define $\psi_{j}(p)=p-c_{j}+(Dw_{j})(p)^{-1}$, and write
$p_{j}=c_{j}+\zeta_{j}(\mathbf{p})$ as
$\psi_{j}(p_{j})=\hat{\pi}_{f}(\mathbf{p})$. Note that for fixed $f$, the
right hand side of this equation is the same for all $j\in\mathcal{J}_{f}$.
Thus if the right hand side is known and $\psi_{j}$ is invertible, all prices
are uniquely defined. Conversely if a single price is known the right hand
side can be computed, thus generating all prices.
The following characteristics of the maps
$\psi_{j}:[c_{j},\infty)\to[0,\infty)$ formalize this logic.
###### Lemma 4.5.
Suppose Assumption 3.1 holds. (i) $\psi_{j}(c_{j})<0$. (ii) If $w$ also
eventually decreases sufficiently quickly, then $\psi_{j}(p)\to\infty$ as
$p\to\infty$. (iii) Finally, $\psi_{j}$ is differentiable and strictly
increasing if, and only if, $w$ is twice differentiable and has sub-quadratic
second derivatives.
###### Proof.
(i): $\psi_{j}(c_{j})=(Dw_{j})(c_{j})^{-1}<0$. (ii): By assumption, there
exists some $r_{j}>1$ and $\bar{p}_{j}>0$ such that $(Dw_{j})(p_{j})^{-1}\geq-
r_{j}/p_{j}$ for all $p_{j}\geq\bar{p}_{j}$. Then
$\psi_{j}(p_{j})=p_{j}-c_{j}+\frac{1}{(Dw_{j})(p_{j})}\geq\left(1-\frac{1}{r_{j}}\right)p_{j}-c_{j}\to
0\quad\text{as}\quad p_{j}\uparrow\infty.$
(iii): $\psi_{j}$ is continuously differentiable if $w_{j}$ is twice
continuously differentiable and strictly decreasing. Specifically,
$(D\psi_{j})(p)=1-\omega_{j}(p)$, and $\psi_{j}$ is increasing if, and only
if, $w$ has sub-quadratic second derivatives. ∎
###### Corollary 4.6.
Let $w$ satisfy Assumption 3.1, eventually decrease sufficiently quickly, and
be twice continuously differentiable with sub-quadratic second derivatives.
Then for all $j$ the equation $\psi_{j}(p)=\pi$ has a unique solution
$\Psi_{j}(\pi)>c_{j}$ for any $\pi>0$.
Equilibrium prices under Logit models can thus be characterized in terms of a
fixed-point equation for equilibrium profits alone. Let
$\boldsymbol{\Psi}:\mathcal{P}^{F}\to\prod_{j=1}^{J}[c_{j},\infty)$ be defined
component-wise by
$(\boldsymbol{\Psi}(\boldsymbol{\pi}))_{j}=\Psi_{j}(\pi_{f(j)})$, where
$\boldsymbol{\pi}\in\mathcal{P}^{F}$ and $f(j)\in\\{1,\dotsc,F\\}$ denotes the
(unique) index of the firm offering product $j$. Next, let
$\hat{\boldsymbol{\pi}}:\prod_{j=1}^{J}[c_{j},\infty)\to\mathcal{P}^{F}$ have
profits $\hat{\pi}_{f}$ as component functions; i.e.
$\hat{\pi}_{f}:\mathcal{P}^{J}\to\mathbb{R}$. Equilibrium profits satisfy the
fixed-point equation
$\boldsymbol{\pi}=\hat{\boldsymbol{\pi}}(\boldsymbol{\Psi}(\boldsymbol{\pi}))=(\hat{\boldsymbol{\pi}}\circ\boldsymbol{\Psi})(\boldsymbol{\pi})=\boldsymbol{\phi}(\boldsymbol{\pi}).$
Given any such fixed-point $\boldsymbol{\pi}$, all equilibrium prices can be
recovered by evaluating $\boldsymbol{\Psi}$.
### 4.2. Existence of Simultaneously Stationary Prices
This section provides three proofs of the existence of simultaneously
stationary prices using each of the fixed-point characterizations given above.
Brouwer’s fixed-point theorem is the typical tool, and is used for proofs
based on the $\boldsymbol{\eta}$ and $\boldsymbol{\phi}$ characterizations.
However the most general result applies the $\boldsymbol{\zeta}$
characterization and the Poincare-Hopf theorem. Throughout this section it is
assumed that $\vartheta>-\infty$.
#### 4.2.1. A proof based on the $\boldsymbol{\eta}$ map.
Brouwer’s theorem can be applied to the $\boldsymbol{\eta}$ characterization.
First recall that $\boldsymbol{\eta}(\mathbf{p})\geq\mathbf{0}$. Next, assume
that
$\tau=\sup_{\mathbf{p}\in(\mathbf{0},\boldsymbol{\infty})}\lvert\lvert\boldsymbol{\eta}(\mathbf{p})\rvert\rvert_{\infty}<\infty$;
conditions for this are given below. It then follows that
$\mathbf{c}+\boldsymbol{\eta}(\mathbf{p})$ maps
$[\mathbf{c},\tau\mathbf{1}]\subset\mathbb{R}^{J}$, a compact convex set, into
itself. Since $\mathbf{c}+\boldsymbol{\eta}(\cdot)$ is also continuous,
Brouwer’s fixed-point theorem implies the existence of a fixed-point
$\mathbf{p}=\mathbf{c}+\boldsymbol{\eta}(\mathbf{p})$.
###### Lemma 4.7.
Suppose $\vartheta>-\infty$, Assumption 3.1 holds, and $w$ is concave in
price. Then $\tau<\infty$.
###### Proof.
Because
$\lvert\lvert(\mathbf{I}-\mathbf{1P}_{f}^{L}(\mathbf{p})^{\top})^{-1}\rvert\rvert_{\infty}\leq(1-\lvert\lvert\mathbf{P}_{f}^{L}(\mathbf{p})\rvert\rvert_{1})^{-1}$,
the bound
$\displaystyle\lvert\lvert\boldsymbol{\eta}_{f}(\mathbf{p})\rvert\rvert_{\infty}$
$\displaystyle=\frac{\max_{j\in\mathcal{J}_{f}}\left\lvert(Dw_{j})(p_{j})\right\rvert^{-1}}{1-\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\mathbf{p})}$
controls the growth in $\boldsymbol{\eta}$. If $w$ satisfies Assumption 3.1
and is concave in price, then
$\left\lvert(Dw_{j})(p_{j})\right\rvert^{-1}\leq\left\lvert(Dw_{j})(c_{j})\right\rvert^{-1}$
for all $p_{j}\geq c_{j}$. This implies that
$\max_{j\in\mathcal{J}_{f}}\left\lvert(Dw_{j})(p_{j})\right\rvert^{-1}$ is
bounded over
$\mathbf{p}_{f}\in[\mathbf{c}_{f},\boldsymbol{\infty})\subset\mathbb{R}^{J_{f}}$.
If $\vartheta>-\infty$,
$\sup_{\mathbf{p}\in(\mathbf{0},\boldsymbol{\infty})}(\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\mathbf{p}))<1$
for all $f$. Under these assumptions, $\tau<\infty$. ∎
###### Corollary 4.8.
If $\vartheta>-\infty$ and $w$ satisfies Assumption 3.1 and is concave in
price, then there exists a fixed-point
$\mathbf{p}=\mathbf{c}+\boldsymbol{\eta}(\mathbf{p})$.
#### 4.2.2. A proof based on $\boldsymbol{\phi}$.
Section 4.1 defined simultaneously stationary profits as a fixed-point of the
map $\boldsymbol{\phi}=\hat{\boldsymbol{\pi}}\circ\boldsymbol{\Psi}$. This
characterization and Brouwer’s theorem can be used to prove the existence of
simultaneously stationary prices.
###### Lemma 4.9.
Suppose $w$ satisfies Assumption 3.1, eventually decreases sufficiently
quickly, has sub-quadratic second derivatives and $\vartheta>-\infty$. Then
there exists at least one fixed-point
$\boldsymbol{\pi}=\boldsymbol{\phi}(\boldsymbol{\pi})$.
###### Proof.
$\Psi_{j}$ can be continuously extended to $[0,\infty]$ as
$\Psi_{j}(\infty)=\infty$ under the condition that $w$ eventually decreases
sufficiently quickly. Because $w$ is then also eventually log bounded, letting
$\vartheta>-\infty$ ensures that for any $f$,
$\hat{\pi}_{f}(\mathbf{p})<\infty$ for all $\mathbf{p}\in[0,\infty]^{J}$. Thus
$\boldsymbol{\phi}$ is continuous and maps the compact, convex set
$[0,\infty]^{J}$ strictly into itself. By Brouwer’s fixed-point theorem, there
exists a fixed-point $\boldsymbol{\pi}$ on $[0,\infty]^{F}$. Furthermore, this
fixed-point has no infinite components, by Lemma 3.6. ∎
While the restriction $\vartheta>-\infty$ could be removed through an
application of Kakutani’s extension of Brouwer’s theorem [25, 15], the fact
that $\boldsymbol{\infty}$ is always an equilibrium makes this approach
uninformative.
#### 4.2.3. A proof based on the $\boldsymbol{\zeta}$ map.
The Poincare-Hopf theorem requires that sum of the indices of the vector field
$\boldsymbol{\varphi}(\mathbf{p})=\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})$
over all zeros of $\boldsymbol{\varphi}$ equals one, so long as
$\boldsymbol{\varphi}$ points outward on the boundary of some compact hyper-
rectangle $[\mathbf{c},\bar{\mathbf{p}}]\subset\mathcal{P}^{J}$ for some
$\bar{\mathbf{p}}<\boldsymbol{\infty}$. Particularly, this sum of indices
cannot be empty and hence there must be at least one zero of
$\boldsymbol{\varphi}$, and thus at least one simultaneously stationary point.
The hypotheses required in the Poincare-Hopf Theorem follow from the next
lemma.
###### Lemma 4.10.
Suppose $\vartheta>-\infty$ and $w$ satisfies Assumption 3.1. (i) If
$\mathbf{p}\geq\mathbf{c}$ and $p_{j}=c_{j}$, $\varphi_{j}(\mathbf{p})<0$.
(ii) If $w$ eventually decreases sufficiently quickly, there exists some
$\bar{\mathbf{p}}\in(\mathbf{c},\boldsymbol{\infty})$ such that
$\varphi_{j}(\mathbf{p})>0$ whenever $p_{j}\geq\bar{p}_{j}$, regardless of
$\mathbf{p}_{-j}$.
###### Proof.
(i): When $\mathbf{p}\geq\mathbf{c}$ and $p_{j}=c_{j}$,
$\displaystyle
p_{j}-c_{j}-\zeta_{j}(\mathbf{p})=-\hat{\pi}_{f}(\mathbf{p}_{f},\mathbf{p}_{-f})-\frac{1}{\left\lvert(Dw_{j})(c_{j})\right\rvert}<0.$
(ii): Observe that the following bound is valid for large enough $p_{j}$
because $w$ eventually decreases sufficiently quickly:
$\displaystyle
p_{j}-c_{j}-\zeta_{j}(\mathbf{p})\geq\left(1-\frac{1}{r_{j}}\right)p_{j}-(c_{j}+\hat{\pi}_{f}(\mathbf{p}_{f},\mathbf{p}_{-f})).$
Because $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$ is bounded ($w$ is eventually
log bounded), and $r_{j}>1$, $(1-1/r_{j})p_{j}\to\infty$ as $p_{j}\to\infty$,
$p_{j}$ can always be chosen large enough to make
$p_{j}-c_{j}-\zeta_{j}(\mathbf{p})>0$. When $\vartheta>-\infty$,
$\hat{\pi}_{f}(\cdot)$ itself is bounded and hence $\bar{\mathbf{p}}_{f}$ can
be chosen independently of $\mathbf{p}_{-f}$. ∎
###### Theorem 4.11.
Suppose $\vartheta>-\infty$, $w$ satisfies Assumption 3.1 and eventually
decreases sufficiently quickly. There exists at least one
$\mathbf{p}\in(\mathbf{c},\boldsymbol{\infty})$ such that
$\mathbf{p}=\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$.
###### Proof.
Let $\bar{\mathbf{p}}$ be as in Lemma 4.10. $\boldsymbol{\varphi}(\mathbf{p})$
is a vector field on $[\mathbf{c},\bar{\mathbf{p}}]$ that points outward on
the boundary of $[\mathbf{c},\bar{\mathbf{p}}]$. Let the set of zeros of
$\boldsymbol{\varphi}$ be denoted by
$\mathfrak{Z}=\\{\mathbf{p}\in(\mathbf{c},\bar{\mathbf{p}}):\boldsymbol{\varphi}(\mathbf{p})=\mathbf{0}\\}$
and let $\mathrm{index}_{\mathbf{p}}(\boldsymbol{\varphi})$ denote the index
of $\boldsymbol{\varphi}$ at $\mathbf{p}\in\mathfrak{Z}$. The Poincare-Hopf
Theorem states that
$\sum_{\mathbf{p}\in\mathfrak{Z}}\mathrm{index}_{\mathbf{p}}(\boldsymbol{\varphi})=1$,
where the value of the sum on the left is taken to be $0$ if
$\mathfrak{Z}=\\{\emptyset\\}$. Hence there is at least one zero of
$\boldsymbol{\varphi}$. ∎
Note that it is not required that $w$ be concave in price or have sub-
quadratic second derivatives in order for simultaneously stationary prices to
exist.
Lemma 4.10 also shows that $\mathbf{c}+\boldsymbol{\zeta}(\cdot)$ maps
$[\mathbf{c},\bar{\mathbf{p}}+\boldsymbol{\epsilon}]$ into itself, for any
$\boldsymbol{\epsilon}\geq\mathbf{0}$. Thus Brouwer’s Theorem could be applied
just as easily to achieve this existence result. This is not the case in
Section 5 below, however.
### 4.3. Sufficiency of Stationarity
A general approach to multi-product firm equilibrium problems can rely on
quasi-concavity to establish the uniqueness of profit-maximizing prices [22].
However, something like quasi-concavity is required to be able to connect
fixed-points
$\mathbf{p}_{f}=\mathbf{c}_{f}+\boldsymbol{\zeta}_{f}(\mathbf{p})$ to profit
maximizers, and thus fixed-points
$\mathbf{p}=\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$ to local equilibria.
Furthermore, uniqueness of profit maximizing prices is required to ensure the
existence of equilibria proper.
Logit profits have the surprising property that stationarity, the first-order
necessary condition, implies local concavity, the second-order sufficient
condition when $w$ has sub-quadratic second derivatives. The Poincare-Hopf
theorem again serves to commute this local result on the second derivatives of
profits to a global property, the uniqueness of profit-maximizing prices.
###### Lemma 4.12.
Suppose $\vartheta>-\infty$ and $w$ satisfies Assumption 3.1 and has sub-
quadratic second derivatives. (i) Satisfaction of the first-order condition
$(\nabla_{f}\hat{\pi}_{f})(\mathbf{p}_{f},\mathbf{p}_{-f})=\mathbf{0}$ is
sufficient for
$\mathbf{p}_{f}\in(\mathbf{0},\boldsymbol{\infty})\subset\mathbb{R}^{J_{f}}$
to be a local maximizer of $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$. (ii)
Satisfaction of the simultaneous stationarity condition
$(\tilde{\nabla}\hat{\pi})(\mathbf{p})=\mathbf{0}$ is sufficient for
$\mathbf{p}\in(\mathbf{0},\boldsymbol{\infty})\subset\mathbb{R}^{J}$ to be a
local equilibrium. (iii) If, in addition, $w$ also eventually decreases
sufficiently quickly then there is a unique stationary point that is a finite
maximizer of $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$. (iv) When $w$ eventually
decreases sufficiently quickly, the simultaneous stationarity condition
$(\tilde{\nabla}\hat{\pi})(\mathbf{p})=\mathbf{0}$ is sufficient for
$\mathbf{p}\in(\mathbf{0},\boldsymbol{\infty})\subset\mathbb{R}^{J}$ to be an
equilibrium.
###### Proof.
Claim (ii) is an obvious corollary to (i), and claim (iv) is an obvious
corollary to (iii).
Claim (i) is a consequence of the following componentwise formula for the
intra-firm profit price-Hessians $(D_{f}\nabla_{f}\hat{\pi}_{f})(\mathbf{p})$
when $\mathbf{p}_{f}$ makes $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$ stationary:
for $k,l\in\mathcal{J}_{f}$,
$\displaystyle(D_{l}D_{k}\hat{\pi}_{f}^{L})(\mathbf{p})$
$\displaystyle=(D_{k}\lambda_{k})(\mathbf{p})\varphi_{k}(\mathbf{p})+\lambda_{k}(\mathbf{p})(D_{l}\varphi_{k})(\mathbf{p})$
$\displaystyle=\lambda_{k}(\mathbf{p})\Bigg{(}\delta_{k,l}-(D_{l}\hat{\pi}_{f})(\mathbf{p})-\omega_{k}(p_{k})\delta_{k,l}\Bigg{)}$
$\displaystyle=\lambda_{k}(\mathbf{p})(1-\omega_{k}(p_{k}))\delta_{k,l}$
In matrix form,
$(D_{f}\nabla_{f}\hat{\pi}_{f})(\mathbf{p})=\boldsymbol{\Lambda}_{f}(\mathbf{p})(\mathbf{I}-\boldsymbol{\Omega}_{f}(\mathbf{p}_{f}))$
where $\boldsymbol{\Omega}_{f}(\mathbf{p}_{f})$ is a diagonal matrix with
entries $\omega_{j}(p_{j})=(D^{2}w_{j})(p_{j})/(Dw_{j})(p_{j})^{2}$. Thus, the
Hessians are diagonal matrices with negative diagonal entries when $w$ has
sub-quadratic second derivatives and $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$ is
locally concave at any stationary prices.
Claim (iii) is a consequence of the Poincare-Hopf Theorem: Because
$\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$ is maximized at
$\mathbf{p}_{f}=\mathbf{c}_{f}+\boldsymbol{\zeta}_{f}(\mathbf{p})$,
$\displaystyle(-1)^{J_{f}}$
$\displaystyle=\mathrm{index}_{\mathbf{p}_{f}}((\nabla_{f}\hat{\pi}_{f})(\cdot,\mathbf{p}_{-f}))=\mathrm{sign}\det(D_{f}\nabla_{f}\hat{\pi}_{f})(\mathbf{p})$
$\displaystyle=\mathrm{sign}\det\boldsymbol{\Lambda}_{f}(\mathbf{p})\cdot\mathrm{sign}\det(D_{f}\boldsymbol{\varphi}_{f})(\mathbf{p})=(-1)^{J_{f}}\cdot\mathrm{sign}\det(D_{f}\boldsymbol{\varphi}_{f})(\mathbf{p});$
see Chapter 6 in [29] for some of the basic results invoked here. Because of
these equalities,
$\mathrm{index}_{\mathbf{p}_{f}}(\boldsymbol{\varphi}_{f}(\cdot,\mathbf{p}_{-f}))=\mathrm{sign}\det(D_{f}\boldsymbol{\varphi}_{f})(\mathbf{p})=1.$
But the Poincare-Hopf theorem require the sum of indices of all zeros of
$\boldsymbol{\varphi}_{f}(\cdot,\mathbf{p}_{-f})$ over
$[\mathbf{c}_{f},\bar{\mathbf{p}}_{f}]$ to be 1. The zero must, therefore, be
unique. ∎
## 5\. Quantity-Dependent Unit Costs
Much of the theoretical literature allows unit costs to depend on sales
volumes. This section extends the techniques used in the previous section to
this case.
Specifically, the analysis in this section proves the following theorem:
###### Theorem 5.1.
Let $\vartheta>-\infty$, Assumption 3.1 holds with a $w$ that eventually
decreases sufficiently quickly with sub-quadratic second derivatives, and
total costs satisfy Assumption 5.1. Then there exists a vector of equilibrium
prices $\mathbf{p}$ satisfying
$\mathbf{c}(\mathbf{P}(\mathbf{p}))<\mathbf{p}<\boldsymbol{\infty}$, and no
equilibrium prices that do not satisfy these bounds.
### 5.1. Assumptions
Assumption 2.3 restricted attention to constant unit costs for the following
reason: Suppose that unit costs did depend on the quantity sold, and let
$c_{f}^{U}(\mathbf{y}_{j},Q_{j}(\mathbf{Y},\mathbf{p}))$ give the unit costs
to firm $f$ for offering product $\mathbf{y}_{j}$ in the market of products
with characteristics $\mathbf{Y}$, prices $\mathbf{p}$, and resultant demand
$Q_{j}(\mathbf{Y},\mathbf{p})$. Then firm $f$’s random profits are
$\Pi_{f}(\mathbf{Y},\mathbf{p})=\mathbf{Q}_{f}(\mathbf{Y},\mathbf{p})^{\top}(\mathbf{p}_{f}-\mathbf{c}_{f}^{U}(\mathbf{Y}_{f},\mathbf{Q}_{f}(\mathbf{Y},\mathbf{p})))-c_{f}^{F}(\mathbf{Y}_{f}).$
Notice that profits are no longer a linear function of the demands, and thus
expected profits need not be a linear function of the expected demands.
Computing these expected costs could be quite difficult, as this would involve
sums over the space of realizable demands.
One way to relax the assumption that unit costs do not depend on production
volume without overly complicating the resulting equilibrium conditions is to
suppose that firms decide on pricing based on the total costs corresponding to
expected demands, as opposed to the actual expected total costs. The following
assumption then generalizes Assumption 2.3 to unit costs that may depend on
the quantity sold:
###### Assumption 5.1.
Firm $f$ has a normalized cost function
$C_{f}:\mathcal{Y}\times[0,1]\to[0,\infty)$ so that the total cost of offering
a product with characteristics $\mathbf{y}\in\mathcal{Y}$ for any demand $q$
is $IC_{f}(\mathbf{y},q/I)$, where again $I$ is the market size. Assume also
that for any $\mathbf{y}\in\mathcal{Y}$,
$C_{f}(\mathbf{y},\cdot):[0,1]\to\mathbb{R}$ is twice continuously
differentiable, strictly increasing and convex on $[0,1)$,
$C_{f}(\mathbf{y},0)=0$, and $\sup_{P\in[0,1]}c_{f}(\mathbf{y},P)<\infty$
where $c_{f}(\mathbf{y},P)=(D^{P}C_{f})(\mathbf{y},P)$.
The simplest example is, of course,
$C_{f}(\mathbf{y},P)=c_{f}^{U}(\mathbf{y})P$ as studied in Sections 2-4. Given
$\mathbf{Y}_{f}\in\mathcal{Y}^{J_{f}}$, define $C_{j}:[0,1]\to[0,\infty)$ by
$C_{j}(P)=C_{f}(\mathbf{y}_{j},P)$ and $c_{j}(P)=(DC_{f})(\mathbf{y}_{j},P)$
where the derivative is with respect to $P$.
Under Assumption 5.1, firms choose prices by solving
$\text{maximize}\quad\mathbb{E}\Pi_{f}(\mathbf{p})=I\sum_{j\in\mathcal{J}_{f}}\Big{(}P_{j}^{L}(\mathbf{p})p_{j}-C_{j}(\mathbb{E}[Q_{j}(\mathbf{p})]/I)\Big{)}\quad\mathrm{with\;respect\;to}\quad
p_{j}\in\mathcal{J}_{f}$
equivalent to
(14)
$\text{maximize}\quad\hat{\pi}_{f}(\mathbf{p})=\sum_{j\in\mathcal{J}_{f}}\big{(}P_{j}^{L}(\mathbf{p})p_{j}-C_{j}(P_{j}^{L}(\mathbf{p}))\big{)}\quad\mathrm{with\;respect\;to}\quad
p_{j}\in\mathcal{J}_{f}$
The following observation is an extension of Lemma 3.6.
###### Lemma 5.2.
Suppose $\vartheta>-\infty$, Assumption 3.1 holds with a $w$ that is
eventually log bounded, and Assumption 5.1 holds. For any $\mathbf{p}_{-f}$,
the optimal profits for Prob. (14) are positive and finite.
###### Proof.
Note that, for any $j$,
$\lim_{p_{j}\uparrow\infty}\left(\frac{C_{j}(P_{j}^{L}(\mathbf{p}))}{P_{j}^{L}(\mathbf{p})}\right)=\lim_{p_{j}\uparrow\infty}\left(\frac{c_{j}(P_{j}^{L}(\mathbf{p}))(D_{j}P_{j}^{L})(\mathbf{p})}{(D_{j}P_{j}^{L})(\mathbf{p})}\right)=c_{j}(0)<\infty$
by L’Hopital’s rule. Thus, as $p_{j}\uparrow\infty$,
$p_{j}-C_{j}(P_{j}^{L}(\mathbf{p}))/P_{j}^{L}(\mathbf{p})\to\infty$. Thus for
all $j$, $p_{j}$ can be chosen large enough so that
$p_{j}-C_{j}(P_{j}^{L}(\mathbf{p}))/P_{j}^{L}(\mathbf{p})>0$. Writing
$\hat{\pi}_{f}(\mathbf{p})=\sum_{j\in\mathcal{J}_{f}}\big{(}P_{j}^{L}(\mathbf{p})p_{j}-C_{j}(P_{j}^{L}(\mathbf{p}))\big{)}=\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\mathbf{p})\left(p_{j}-\frac{C_{j}(P_{j}^{L}(\mathbf{p}))}{P_{j}^{L}(\mathbf{p})}\right)$
proves that there exists $\mathbf{p}_{f}$, for any $\mathbf{p}_{-f}$ such that
$\hat{\pi}_{f}(\mathbf{p}_{f},\mathbf{p}_{-f})>0$. If $w$ is eventually log
bounded, then
$P_{j}(\mathbf{p})(p_{j}-C_{j}(P_{j}^{L}(\mathbf{p}))/P_{j}^{L}(\mathbf{p}))\to
0$ as $p_{j}\uparrow\infty$. Because
$\hat{\pi}_{f}(\mathbf{p}_{f},\mathbf{p}_{-f})\to 0$ as
$\mathbf{p}_{f}\to\boldsymbol{\infty}$, and
$\hat{\pi}_{f}(\mathbf{p}_{f},\mathbf{p}_{-f})$ is finite if $p_{j}<\infty$
for any $j\in\mathcal{J}_{f}$, optimal profits are finite. ∎
Note that this proof does not say that all prices are finite; this is proved
below using similar techniques to those used in Section 4.
### 5.2. Stationarity
The stationarity conditions for Prob. (14) are
$\displaystyle(D_{k}\hat{\pi}_{f})(\mathbf{p})=\sum_{j\in\mathcal{J}_{f}}(D_{k}P_{j}^{L})(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))+P_{k}^{L}(\mathbf{p})=0$
for all $k\in\mathcal{J}_{f}$ and all $f$, as can be checked.
As before, stationarity can be written in either of two fixed-point forms:
###### Lemma 5.3.
Suppose $\vartheta>-\infty$ and Assumptions 3.1 and 5.1 hold. At any
simultaneously stationary prices $\mathbf{p}\in(0,\infty)^{J}$,
$\mathbf{p}=\mathbf{c}(\mathbf{P}(\mathbf{p}))+\boldsymbol{\eta}(\mathbf{p})$
where $\boldsymbol{\eta}$ is as defined above and
$\mathbf{p}=\mathbf{c}(\mathbf{P}(\mathbf{p}))+\boldsymbol{\zeta}(\mathbf{p})$
where $\boldsymbol{\zeta}:[0,\infty)^{J}\to\mathbb{R}^{J}$ is defined
componentwise by
$\displaystyle\zeta_{k}(\mathbf{p})=\sum_{j\in\mathcal{J}_{f(k)}}P_{j}^{L}(\mathbf{p})\big{(}p_{j}-c_{j}(P_{j}^{L}(\mathbf{p}))\big{)}+\frac{1}{\left\lvert(Dw_{k})(p_{k})\right\rvert}$
As in Section 4, these characterizations immediately establishes the
positivity of markups in equilibrium:
###### Lemma 5.4.
Suppose $\vartheta>-\infty$, Assumption 3.1 holds with $w$ eventually
decreasing sufficiently quickly, and Assumption 5.1 holds. If
$\mathbf{p}\in[0,\infty]^{J}$ is a vector of equilibrium prices, then
$\mathbf{c}(\mathbf{P}(\mathbf{p}))<\mathbf{p}<\boldsymbol{\infty}$.
###### Proof.
Lemma 5.6 below proves that there exists some $\bar{p}_{j}$ such that
$\hat{\pi}_{f}(\mathbf{p})<0$ whenever $p_{j}>\bar{p}_{j}$. Thus no price can
be infinite in equilibrium. Moreover, no price can be zero: Suppose
$\lim_{p_{k}\downarrow 0}(Dw_{k})(p_{k})=0$. Then for $p_{k}>0$,
$(D_{k}\hat{\pi}_{f})(\mathbf{p})=(Dw_{k})(p_{k})P_{k}^{L}(\mathbf{p})\left(p_{k}-c_{k}-\sum_{j\in\mathcal{J}_{f(k)}}P_{j}^{L}(\mathbf{p})\big{(}p_{j}-c_{j}(P_{j}^{L}(\mathbf{p}))\right)+P_{k}^{L}(\mathbf{p})\to
P_{k}^{L}(\mathbf{p})>0$
as $p_{k}\downarrow 0$. Now suppose $\lim_{p_{k}\downarrow
0}(Dw_{k})(p_{k})<0$. For $\mathbf{p}$ with $0<p_{k}<\infty$,
$(D_{k}\hat{\pi}_{f})(\mathbf{p})\leq 0$ if, and only if,
$\displaystyle
p_{k}-c_{k}(P_{k}^{L}(\mathbf{p})))-\sum_{j\in\mathcal{J}_{f(k)}}P_{j}^{L}(\mathbf{p})\big{(}p_{j}-c_{j}(P_{j}^{L}(\mathbf{p}))\big{)}-\frac{1}{\left\lvert(Dw_{k})(p_{k})\right\rvert}\geq
0.$
Taking the limit as $p_{k}\downarrow 0$ yields
$\displaystyle-\sum_{j\in\mathcal{J}_{f(k)}\setminus
k}P_{j}^{L}(\mathbf{p})\big{(}p_{j}-c_{j}(P_{j}^{L}(\mathbf{p}))\big{)}\geq(1-P_{k}^{L}(\mathbf{p}))c_{k}(P_{k}^{L}(\mathbf{p})))+\frac{1}{\left\lvert(Dw_{k})(0)\right\rvert}.$
The right hand side is positive, but if profits are optimal, the left hand
side is negative. Thus $(D_{k}\hat{\pi}_{f})(\mathbf{p})>0$ for all $p_{k}$
sufficiently close to zero and all positive profits. Hence, $p_{k}=0$ cannot
be profit-optimal.
Knowing then that $\mathbf{p}\in(0,\infty)^{J}$, the equation
$\mathbf{p}=\mathbf{c}(\mathbf{P}(\mathbf{p}))+\boldsymbol{\eta}(\mathbf{p})$
applies. Because $\boldsymbol{\eta}(\mathbf{p})$ is positive valued for all
$\mathbf{p}\in(0,\infty)^{J}$,
$\mathbf{p}>\mathbf{c}(\mathbf{P}(\mathbf{p}))$. ∎
The remainder of the proof of equilibrium existence is analogous to the
process for constant unit costs: First simultaneously stationary prices are
shown to exist, followed by a proof that such prices are in fact always
equilibrium prices.
### 5.3. Existence of Stationary Prices
The approach to establishing the existence of simultaneously stationary points
in the following lemmas is as follows: First, a homeomorphism,
$\boldsymbol{\rho}$, between
$\\{\mathbf{p}:\mathbf{p}\geq\mathbf{c}(\mathbf{P}(\mathbf{p}))\\}$ and
$[\mathbf{0},\boldsymbol{\infty})$ is constructed. Second, the vector field
$\boldsymbol{\varphi}:[\mathbf{0},\boldsymbol{\infty})$ defined componentwise
by
$\displaystyle\varphi_{k}(\mathbf{p})=p_{k}-c_{k}(P_{k}^{L}(\mathbf{p}))-\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))-\frac{1}{\left\lvert(Dw_{k})(p_{k})\right\rvert}$
is transported from
$\\{\mathbf{p}:\mathbf{p}\geq(D\mathbf{C})(\mathbf{P}(\mathbf{p}))\\}$ to
$[\mathbf{0},\boldsymbol{\infty})$ by defining
$\boldsymbol{\psi}:[\mathbf{0},\boldsymbol{\infty})\to\mathbb{R}^{J}$ by
$\boldsymbol{\psi}(\boldsymbol{\epsilon})=\boldsymbol{\varphi}(\boldsymbol{\rho}(\boldsymbol{\epsilon}))$
for any $\boldsymbol{\epsilon}\in[\mathbf{0},\boldsymbol{\infty})$. Third, a
compact rectangle $[\mathbf{0},\bar{\boldsymbol{\epsilon}}]$ is constructed,
on which $\boldsymbol{\psi}$ is continuous and points outward on the boundary.
The Poincare-Hopf theorem then proves the existence of a zero
$\boldsymbol{\epsilon}_{0}\in(\mathbf{0},\bar{\boldsymbol{\epsilon}})$ for
$\boldsymbol{\psi}$, which maps to a zero
$\mathbf{p}_{0}=\boldsymbol{\rho}(\boldsymbol{\epsilon}_{0})$ of
$\boldsymbol{\varphi}$ such that
$\mathbf{p}_{0}>(D\mathbf{C})(\mathbf{P}(\mathbf{p}_{0}))$. Such a point is
necessarily simultaneously stationary because
$(D_{k}\hat{\pi}_{f})(\mathbf{p})=\lambda_{k}(\mathbf{p})\varphi_{k}(\mathbf{p})$.
###### Lemma 5.5.
Suppose $\vartheta>-\infty$, Assumption 3.1 holds with $w$ eventually
decreasing sufficiently quickly, and Assumption 5.1 holds. The fixed-point
problem $\mathbf{p}=\mathbf{F}_{\boldsymbol{\epsilon}}(\mathbf{p})$ where
$\mathbf{F}_{\boldsymbol{\epsilon}}(\mathbf{p})=\mathbf{c}(\mathbf{P}(\mathbf{p}))+\boldsymbol{\epsilon}$
has a unique solution
$\boldsymbol{\rho}(\boldsymbol{\epsilon})>\boldsymbol{\epsilon}$ for every
$\boldsymbol{\epsilon}\in[\mathbf{0},\boldsymbol{\infty})\subset\mathbb{R}^{J}$.
Moreover, the corresponding solution map,
$\boldsymbol{\rho}:[\mathbf{0},\boldsymbol{\infty})\to\\{\mathbf{p}:\mathbf{p}\geq\mathbf{c}(\mathbf{P}(\mathbf{p}))\\}$
is a homeomorphism between $[\mathbf{0},\boldsymbol{\infty})$ and
$\\{\mathbf{p}:\mathbf{p}\geq\mathbf{c}(\mathbf{P}(\mathbf{p}))\\}$.
###### Proof.
Suppose that $\mathbf{c}(\mathbf{P}(\mathbf{p}))$ depends on $\mathbf{p}$,
otherwise the claims are trivial.
Assume the fixed-point $\boldsymbol{\rho}(\boldsymbol{\epsilon})$ is unique.
$\boldsymbol{\rho}(\boldsymbol{\epsilon})>\boldsymbol{\epsilon}$ because
$c_{j}(q)>0$ for all $q<1$ and $P_{j}^{L}(\mathbf{p})<1$ for all $\mathbf{p}$
when there is an outside good. The implicit function theorem guarantees the
continuity of $\boldsymbol{\rho}(\boldsymbol{\epsilon})$, and the inverse map
$\mathbf{p}\mapsto\boldsymbol{\epsilon}=\mathbf{p}-\mathbf{c}(\mathbf{P}(\mathbf{p}))$
is continuous because $\mathbf{C}$ and $\mathbf{P}$ are continuously
differentiable. $\boldsymbol{\rho}(\boldsymbol{\epsilon})$ is thus a
homeomorphism.
We now show that $\boldsymbol{\rho}(\boldsymbol{\epsilon})$ does indeed exist,
as claimed. Because $\sup_{P\in[0,1]}c_{j}(P)=\kappa_{j}<\infty$, there are no
fixed points with $p_{j}\geq\bar{p}_{j}=\kappa_{j}+\epsilon_{j}+\delta$ for
any fixed $\delta>0$. Moreover, there are no fixed points with $p_{j}=0$ when
$\inf_{\mathbf{p}}c_{j}(P_{j}^{L}(\mathbf{p}))>0$; this will hold when there
is an outside good, even if $c_{j}(1)=0$, so long as $c_{j}(P)>0$ for all
$P\in[0,1)$. Let $\mathcal{D}=[\mathbf{0},\bar{\mathbf{p}}]$, and consider
$(D\mathbf{F}_{\boldsymbol{\epsilon}})(\mathbf{p})=(D^{2}\mathbf{C})(\mathbf{P}(\mathbf{p}))(D\mathbf{P})(\mathbf{p})=(D^{2}\mathbf{C})(\mathbf{P}(\mathbf{p}))(\mathbf{I}-\mathbf{P}(\mathbf{p})\mathbf{1}^{\top})\boldsymbol{\Lambda}(\mathbf{p})$
$(D\mathbf{F}_{\boldsymbol{\epsilon}})(\mathbf{p})$ has one as an eigenvalue
only if there exists $\mathbf{x}\neq\mathbf{0}$ such that
$\displaystyle(D^{2}\mathbf{C})(\mathbf{P}(\mathbf{p}))(\mathbf{I}-\mathbf{P}(\mathbf{p})\mathbf{1}^{\top})\boldsymbol{\Lambda}(\mathbf{p})\mathbf{x}$
$\displaystyle=\mathbf{x}.$
When
$\left\lvert\lambda_{j}(\mathbf{p})\right\rvert(D^{2}C_{j})(P_{j}^{L}(\mathbf{p}))\neq-1$
for all $j$, this holds only if
$\displaystyle\sum_{j=1}^{J}\beta_{j}(\mathbf{p})P_{j}^{L}(\mathbf{p})=1\quad\text{where}\quad\beta_{j}(\mathbf{p})=\frac{\left\lvert\lambda_{j}(\mathbf{p})\right\rvert(D^{2}C_{j})(P_{j}^{L}(\mathbf{p}))}{\left\lvert\lambda_{j}(\mathbf{p})\right\rvert(D^{2}C_{j})(P_{j}^{L}(\mathbf{p}))+1}$
as can be checked. Clearly
$\left\lvert\lambda_{j}(\mathbf{p})\right\rvert(D^{2}C_{j})(P_{j}^{L}(\mathbf{p}))\neq-1$
and $0\leq\beta_{j}(\mathbf{p})<1$ for all $j$ when $C_{j}$ is convex. Thus,
when $\vartheta\neq-\infty$,
$\displaystyle\sum_{j=1}^{J}\beta_{j}(\mathbf{p})P_{j}^{L}(\mathbf{p})<\sum_{j=1}^{J}\beta_{j}(\mathbf{p})\left(\frac{e^{u_{j}(p_{j})}}{\sum_{k=1}^{J}e^{u_{k}(p_{k})}}\right)\leq\max_{j=1,\dotsc,J}\beta_{j}(\mathbf{p})<1.$
Thus $(D\mathbf{F}_{\boldsymbol{\epsilon}})(\mathbf{p})$ cannot have 1 as an
eigenvalue. By Kellogg’s Uniqueness Theorem [26],
$\boldsymbol{\rho}(\boldsymbol{\epsilon})$ is well-defined. ∎
The assumption of convex costs $C_{j}$ would be difficult to relax in this
proof: $\beta_{j}(\mathbf{p})$ cannot always be non-negative for concave costs
such as $C_{j}(P_{j}^{L}(\mathbf{p}))=-\kappa_{j}P_{j}^{L}(\mathbf{p})^{2}$
because $\left\lvert\lambda_{j}(\mathbf{p})\right\rvert\to 0$ as
$p_{j}\uparrow\infty$, and thus
$\left\lvert\lambda_{j}(\mathbf{p})\right\rvert\kappa_{j}\leq 1$ for large
enough $p_{j}$. Bounding the sums of $\beta$’s in a similar way can be done if
we ensure that
$\left\lvert\lambda_{j}(\mathbf{p})\right\rvert\left\lvert(D^{2}C_{j})(P_{j}^{L}(\mathbf{p}))\right\rvert>1$,
but this suggests assumptions that simultaneously restrict the behavior
allowed in the utilities and costs.
###### Lemma 5.6.
Suppose $\vartheta>-\infty$ and Assumptions 3.1 and 5.1 hold. (i) If
$\mathbf{p}\geq\mathbf{c}(\mathbf{P}(\mathbf{p}))$ and
$p_{k}=c_{k}(P_{k}^{L}(\mathbf{p}))$, then $\varphi_{k}(\mathbf{p})<0$. (ii)
When $w$ also eventually decreases sufficiently quickly, there exists
$\bar{p}_{k}>0$ such that $\varphi_{k}(\mathbf{p})>0$ for all
$p_{k}\geq\bar{p}_{k}$ regardless of $\mathbf{p}_{-k}$.
###### Proof.
(i) If $\mathbf{p}\geq\mathbf{c}(\mathbf{P}(\mathbf{p}))$ and
$p_{k}=c_{k}(P_{k}^{L}(\mathbf{p}))$,
$\displaystyle\varphi_{k}(\mathbf{p})$
$\displaystyle=-\sum_{j\in\mathcal{J}_{f}\setminus\\{k\\}}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))-\frac{1}{\left\lvert(Dw_{k})(p_{k})\right\rvert}$
$\displaystyle<-\sum_{j\in\mathcal{J}_{f}\setminus\\{k\\}}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))\leq
0.$
(ii) Write $\varphi_{k}(\mathbf{p})>0$ as
(15) $\displaystyle
p_{k}-\frac{1}{\left\lvert(Dw_{k})(p_{k})\right\rvert}>c_{k}(P_{k}^{L}(\mathbf{p}))+\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))$
Because there exists $r_{k}>1$ and $\bar{p}_{k}$ such that
$\displaystyle
p_{k}-\frac{1}{\left\lvert(Dw_{k})(p_{k})\right\rvert}\geq\left(1-\frac{1}{r_{k}}\right)p_{k}$
for all $p_{k}>\bar{p}_{k}$, the left-hand-side in (15) can be made as large
as desired. Similarly, because $\sup_{q\in[0,1]}c_{j}(q)<\infty$ and $w_{k}$
is necessarily eventually log-bounded, the right-hand-side
$\displaystyle
c_{k}(P_{k}^{L}(\mathbf{p}))+\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))\leq
c_{k}(P_{k}^{L}(\mathbf{p}))+\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\mathbf{p})p_{j}$
is bounded over all $\mathbf{p}$. Thus there must exist $\bar{p}_{k}$ so large
to make (15) hold for all $p_{k}\geq\bar{p}_{k}$. ∎
###### Corollary 5.7.
Suppose $\vartheta>-\infty$, Assumption 3.1 holds with $w$ eventually
decreasing sufficiently quickly, and Assumption 5.1 holds.
$\boldsymbol{\psi}=\boldsymbol{\varphi}\circ\boldsymbol{\rho}:[\mathbf{0},\bar{\mathbf{p}}]\to\mathbb{R}^{J}$
(i.e.
$\boldsymbol{\psi}(\boldsymbol{\epsilon})=\boldsymbol{\varphi}(\boldsymbol{\rho}(\boldsymbol{\epsilon}))$)
points outward on the boundary of $[\mathbf{0},\bar{\mathbf{p}}]$.
###### Proof.
This follows immediately from Lemma 5.6, recalling that
$\rho_{k}(\boldsymbol{\epsilon})>\bar{p}_{k}$ when $\epsilon_{k}=\bar{p}_{k}$.
∎
###### Theorem 5.8.
There exists a zero, $\mathbf{p}$, of $\boldsymbol{\varphi}$ such that
$\mathbf{c}(\mathbf{P}(\mathbf{p}))<\mathbf{p}<\boldsymbol{\infty}$.
###### Proof.
Apply the Poincare-Hopf theorem to $\boldsymbol{\psi}$, as described above, to
establish the existence of
$\boldsymbol{\epsilon}\in(\mathbf{0},\bar{\mathbf{p}})$ such that
$\boldsymbol{\psi}(\boldsymbol{\epsilon})=\mathbf{0}$. Define
$\mathbf{p}=\boldsymbol{\rho}(\boldsymbol{\epsilon})>\mathbf{c}(\mathbf{P}(\mathbf{p}))$,
observing that
$\boldsymbol{\varphi}(\mathbf{p})=\boldsymbol{\psi}(\boldsymbol{\epsilon})=\mathbf{0}$.
∎
Note that the Poincare-Hopf Theorem is very useful here, relative to Brouwer’s
Theorem. Specifically, there is no obvious fixed-point equation for
$\boldsymbol{\epsilon}$, and it is not obvious when
$\\{\mathbf{p}:\mathbf{p}\geq\mathbf{c}(\mathbf{P}(\mathbf{p}))\\}$ (and thus
$\\{\mathbf{p}:\mathbf{p}\geq\mathbf{c}(\mathbf{P}(\mathbf{p}))\\}\cap[\mathbf{0},\bar{\mathbf{p}}]$)
would be convex.
### 5.4. Existence of Equilibrium
The second component of the proof that equilibrium exists requires
demonstrating the “sufficiency of stationarity” and the uniqueness of profit-
maximizing prices. As in Section 4, this is accomplished by proving that
$(D_{f}\nabla_{f}\hat{\pi}_{f})(\mathbf{p})$ is negative definite at any
stationary prices and then applying the Poincare-Hopf Theorem.
###### Lemma 5.9.
Suppose $\vartheta>-\infty$, Assumption 3.1 holds with $w$ twice continuously
differentiable, and Assumption 5.1 holds. At any stationary point,
$(D_{f}\nabla_{f}\hat{\pi}_{f})(\mathbf{p})=\boldsymbol{\Lambda}_{f}(\mathbf{p})(\mathbf{I}-\boldsymbol{\Omega}_{f}(\mathbf{p}))-\boldsymbol{\Lambda}_{f}(\mathbf{p})\mathbf{H}_{f}(\mathbf{p})\boldsymbol{\Lambda}_{f}(\mathbf{p})$
where
$\mathbf{H}_{f}(\mathbf{p})=\mathbf{K}_{f}(\mathbf{p})-\mathbf{K}_{f}(\mathbf{p})\mathbf{P}_{f}^{L}(\mathbf{p})\mathbf{1}^{\top}-\mathbf{1}\mathbf{P}_{f}^{L}(\mathbf{p})^{\top}\mathbf{K}_{f}(\mathbf{p})+\left(\mathbf{P}_{f}^{L}(\mathbf{p})^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{P}_{f}^{L}(\mathbf{p})\right)\mathbf{1}\mathbf{1}^{\top}$
and
$\mathbf{K}_{f}(\mathbf{p})=(D^{2}\mathbf{C}_{f})(\mathbf{P}_{f}^{L}(\mathbf{p}))$.
###### Proof.
Because
$(D_{k}\hat{\pi}_{f})(\mathbf{p})=\lambda_{k}(\mathbf{p})\varphi_{k}(\mathbf{p})$
and $\varphi_{k}(\mathbf{p})=0$ at any stationary point,
$(D_{l}D_{k}\hat{\pi}_{f})(\mathbf{p})=\lambda_{k}(\mathbf{p})(D_{l}\varphi_{k})(\mathbf{p})$
for any $k,l\in\mathcal{J}_{f}$. Moreover,
$\displaystyle(D_{l}\varphi_{k})(\mathbf{p})$
$\displaystyle=(1-\omega_{k}(p_{k}))\delta_{k,l}-(D^{2}C_{k})(P_{k}^{L}(\mathbf{p}))(D_{l}P_{k}^{L})(\mathbf{p})$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad-\sum_{j\in\mathcal{J}_{f}}(D_{l}P_{j}^{L})(\mathbf{p}))(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p}))-P_{l}^{L}(\mathbf{p})$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\mathbf{p})(D^{2}C_{j})(P_{j}^{L}(\mathbf{p}))(D_{l}P_{j}^{L})(\mathbf{p})$
$\displaystyle=(1-\omega_{k}(p_{k}))\delta_{k,l}-(D^{2}C_{k})(P_{k}^{L}(\mathbf{p}))(D_{l}P_{k}^{L})(\mathbf{p})$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\mathbf{p})(D^{2}C_{j})(P_{j}^{L}(\mathbf{p}))(D_{l}P_{j}^{L})(\mathbf{p})$
because
$(D_{l}\hat{\pi}_{f})(\mathbf{p})=\sum_{j\in\mathcal{J}_{f}}(D_{l}P_{j}^{L})(\mathbf{p}))(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p}))-P_{l}^{L}(\mathbf{p})=0$.
The result follows by substituting the definition of
$(D_{l}P_{k}^{L})(\mathbf{p})$ into this last equation and re-arranging terms.
∎
A sufficient condition for $\hat{\pi}_{f}(\mathbf{p})$ to be locally concave
at any stationary point follows:
###### Lemma 5.10.
Suppose $\vartheta>-\infty$, Assumptions 3.1 and 5.1 hold, and $w$ has sub-
quadratic second derivatives. (i)
$(D_{f}\nabla_{f}\hat{\pi}_{f})(\cdot,\mathbf{p}_{-f})$ is negative definite
at any stationary prices $\mathbf{p}_{f}$. (ii) As a consequence, profit-
maximizing prices $\mathbf{p}_{f}$ are unique for any competitor’s prices
$\mathbf{p}_{-f}$ and (iii) any simultaneously stationary point is an
equilibrium.
###### Proof.
First note that if $w$ has sub-quadratic second derivatives and
$\mathbf{H}_{f}(\mathbf{p})$ is positive semi-definite (at any stationary
prices $\mathbf{p}_{f}$) then $(D_{f}\nabla_{f}\hat{\pi}_{f})(\mathbf{p})$ is
negative definite (at any stationary prices $\mathbf{p}_{f}$). For if $w$ has
sub-quadratic second derivatives, then
$\boldsymbol{\Lambda}_{f}(\mathbf{p})(\mathbf{I}-\boldsymbol{\Omega}_{f}(\mathbf{p}))$
is negative definite, and if $\mathbf{H}_{f}(\mathbf{p})$ is positive semi-
definite, then
$-\boldsymbol{\Lambda}_{f}(\mathbf{p})\mathbf{H}_{f}(\mathbf{p})\boldsymbol{\Lambda}_{f}(\mathbf{p})$
is negative semi-definite.
We now show that when $C_{j}$ is convex, $\mathbf{H}_{f}(\mathbf{p})$ is
positive semi-definite. Because $\mathbf{K}_{f}(\mathbf{p})$ is positive
definite, define an inner product
$\langle\mathbf{x},\mathbf{y}\rangle_{f}=\mathbf{x}^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{y}$
on $\mathbb{R}^{J_{f}}$ with
$\lvert\lvert\mathbf{x}\rvert\rvert_{f}=\sqrt{\langle\mathbf{x},\mathbf{x}\rangle_{f}}$
the corresponding norm. The Cauchy-Schwartz inequality states that
$\big{(}\mathbf{P}_{f}(\mathbf{p})^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{x}\big{)}^{2}=\left\lvert\langle\mathbf{P}_{f}(\mathbf{p}),\mathbf{x}\rangle_{f}\right\rvert^{2}\leq\lvert\lvert\mathbf{P}_{f}(\mathbf{p})\rvert\rvert_{f}^{2}\lvert\lvert\mathbf{x}\rvert\rvert_{f}^{2}=\big{(}\mathbf{P}_{f}(\mathbf{p})^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{P}_{f}(\mathbf{p})\big{)}\big{(}\mathbf{x}^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{x}\big{)}$
for any vector $\mathbf{x}\in\mathbb{R}^{J_{f}}$. Note that
$\mathbf{x}^{\top}\mathbf{H}_{f}(\mathbf{p})\mathbf{x}=\big{(}\mathbf{P}_{f}(\mathbf{p})^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{P}_{f}(\mathbf{p})\big{)}\big{(}\mathbf{1}^{\top}\mathbf{x}\big{)}^{2}-2\big{(}\mathbf{P}_{f}(\mathbf{p})^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{x}\big{)}\big{(}\mathbf{1}^{\top}\mathbf{x}\big{)}+\mathbf{x}^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{x}$
Because any convex quadratic $q(\xi)=a\xi^{2}-2b\xi+c$ (i.e., where $a>0$) is
minimized at $\xi_{*}=b/a$ with value $q(\xi_{*})=c-b^{2}/a$,
$\mathbf{x}^{\top}\mathbf{H}_{f}(\mathbf{p})\mathbf{x}\geq 0$ for all
$\mathbf{x}$ if
$\big{(}\mathbf{P}_{f}(\mathbf{p})^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{x}\big{)}^{2}\leq\big{(}\mathbf{P}_{f}(\mathbf{p})^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{P}_{f}(\mathbf{p})\big{)}\big{(}\mathbf{x}^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{x}\big{)}$
for all $\mathbf{x}$, which follows from the Cauchy-Schwartz inequality.
Strictly speaking, this inequality is only required for $\mathbf{x}$
satisfying
$\mathbf{1}^{\top}\mathbf{x}=\mathbf{P}_{f}(\mathbf{p})^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{x}/\mathbf{x}^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{x}$
to prove the positive semi-definiteness of $\mathbf{H}_{f}(\mathbf{p})$;
however the Cauchy-Schwartz inequality requires this to hold for all
$\mathbf{x}$.
The discussion above proves that any vector of stationary prices
$\mathbf{p}_{f}$ is in fact a local maximizer of firm $f$’s profits; it
remains to prove that this is fact a unique, global maximizer of firm $f$’s
profits. The homotopic construction in Lemma 5.5 applies equally well to a
single firm, given fixed competitor prices $\mathbf{p}_{-f}$. By the analogue
of Lemma 5.6, the vector field
$\boldsymbol{\psi}_{f}=\boldsymbol{\varphi}_{f}\circ\boldsymbol{\rho}_{f}:[\mathbf{0},\bar{\mathbf{p}}_{f}]\to\mathbb{R}^{J_{f}}$
points outward on the boundary of $[\mathbf{0},\bar{\mathbf{p}}_{f}]$ and has
at least one zero $\boldsymbol{\epsilon}_{f}$. Assume that the index of any
such zero is one. Then there can be only one such zero, because the Poincare-
Hopf Theorem requires the sum of the indices of all zeros to be one.
The proof that the index of any zero $\boldsymbol{\epsilon}_{f}$ of
$\boldsymbol{\psi}_{f}$ is one begins with the standard index formula:
$\mathrm{index}_{\boldsymbol{\epsilon}_{f}}(\boldsymbol{\psi}_{f})=\mathrm{sign}\det(D_{f}\boldsymbol{\psi}_{f})(\boldsymbol{\epsilon}_{f})=\mathrm{sign}\det(D_{f}\boldsymbol{\varphi}_{f})(\boldsymbol{\rho}_{f}(\boldsymbol{\epsilon}_{f}))\cdot\mathrm{sign}\det(D_{f}\boldsymbol{\rho}_{f})(\boldsymbol{\epsilon}_{f}).$
Both determinants on the right-hand-side are positive, as we now prove.
First, consider
$\mathrm{sign}\det(D_{f}\boldsymbol{\varphi}_{f})(\boldsymbol{\rho}_{f}(\boldsymbol{\epsilon}_{f}))$.
The zero $\mathbf{p}_{f}=\boldsymbol{\rho}_{f}(\boldsymbol{\epsilon}_{f})$ of
$\boldsymbol{\varphi}_{f}(\cdot,\mathbf{p}_{-f})$ is a local maximizer of
$\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$, and thus the index of the gradient
vector field $(\nabla_{f}\hat{\pi}_{f})(\cdot,\mathbf{p}_{-f})$ at
$\mathbf{p}_{f}$ is $(-1)^{J_{f}}$. But then
$(-1)^{J_{f}}=\mathrm{sign}\det(D_{f}\nabla_{f}\hat{\pi}_{f})=\mathrm{sign}\det\Big{(}\boldsymbol{\Lambda}_{f}(D_{f}\boldsymbol{\varphi}_{f})\Big{)}=(-1)^{J_{f}}\mathrm{sign}\det(D_{f}\boldsymbol{\varphi}_{f}),$
and
$\mathrm{sign}\det(D_{f}\boldsymbol{\varphi}_{f})(\boldsymbol{\rho}_{f}(\boldsymbol{\epsilon}_{f}),\mathbf{p}_{-f})=1$
as claimed.
Next, consider
$\mathrm{sign}\det(D_{f}\boldsymbol{\rho}_{f})(\boldsymbol{\epsilon}_{f})$.
The Jacobian of $(D_{f}\boldsymbol{\rho}_{f})(\boldsymbol{\epsilon}_{f})$ is
given by
$(D\boldsymbol{\rho}_{f})(\boldsymbol{\epsilon}_{f})=(\mathbf{I}-(D^{2}\mathbf{C}_{f})(D_{f}\mathbf{P}_{f}^{L}))^{-1}$
(where we neglect the arguments for simplicity); this inverse is well defined
because $(D^{2}\mathbf{C}_{f})(D_{f}\mathbf{P}_{f}^{L})$ does not have one as
an eigenvalue, as proved in Lemma 5.5. Thus zero is not an eigenvalue of
$(D_{f}\boldsymbol{\rho}_{f})(\boldsymbol{\epsilon}_{f})$, and any eigenvalue
$\mu$ ($\neq 0$) satisfies
$\Big{(}\mathbf{I}-(D^{2}\mathbf{C}_{f})(D_{f}\mathbf{P}_{f}^{L})\Big{)}\mathbf{x}=\left(\frac{1}{\mu}\right)\mathbf{x}.$
Rearranging this equation yields
$\left[\left(\frac{1}{\mu}-1\right)\mathbf{I}+(D^{2}\mathbf{C}_{f})\boldsymbol{\Lambda}_{f}\right]\mathbf{x}=\big{(}\mathbf{1}^{\top}\boldsymbol{\Lambda}_{f}\mathbf{x}\big{)}(D^{2}\mathbf{C}_{f})\mathbf{P}_{f}^{L}.$
It is straightforward to see that this can hold only if
$1=\theta\left(1-\frac{1}{\mu}\right)\quad\text{where}\quad\theta(\alpha)=\sum_{j\in\mathcal{J}_{f}}\left(\frac{\left\lvert\lambda_{j}\right\rvert(D^{2}C_{j})}{\alpha+\left\lvert\lambda_{j}\right\rvert(D^{2}C_{j})}\right)P_{j}^{L}.$
Without loss of generality, suppose $\mathcal{J}_{f}=\\{1,\dotsc,J_{f}\\}$,
$\left\lvert\lambda_{1}\right\rvert(D^{2}C_{1})\leq\dotsb\leq\left\lvert\lambda_{J_{f}}\right\rvert(D^{2}C_{J_{f}}),$
let $N$ be the number of distinct values of
$\left\lvert\lambda_{j}\right\rvert(D^{2}C_{j})$,
$\kappa_{1}<\dotsb<\kappa_{N}$ these values, and $M_{n}$ be the number of
$j\in\mathcal{J}_{f}$ for which
$\left\lvert\lambda_{j}\right\rvert(D^{2}C_{j})=\kappa_{n}$.
Note that $\theta$ is not defined for
$\alpha\in\mathcal{S}_{f}=\\{-\left\lvert\lambda_{j}\right\rvert(D^{2}C_{j}):j\in\mathcal{J}_{f}\\}=\\{-\kappa_{n}\\}_{n=1}^{N}$,
and
$D\theta(\alpha)=-\sum_{j\in\mathcal{J}_{f}}\left(\frac{\left\lvert\lambda_{j}\right\rvert(D^{2}C_{j})}{(\alpha+\left\lvert\lambda_{j}\right\rvert(D^{2}C_{j}))^{2}}\right)P_{j}^{L}<0,$
when defined. Moreover, $\theta(\alpha)<0<1$ for all
$\alpha\in(-\infty,\kappa_{N})$; thus no solutions to $\theta(\alpha)=1$ are
less than $-\kappa_{N}$. Consider any interval $(-\kappa_{n+1},-\kappa_{n})$,
$n\in\\{1,\dotsc,N-1\\}$. In such an interval $\theta$ is strictly decreasing,
$\theta(\alpha)\uparrow\infty$ as $\alpha\downarrow-\kappa_{n+1}$, and
$\theta(\alpha)\downarrow-\infty$ as $\alpha\uparrow-\kappa_{n}$. There is
thus a unique $\alpha_{n}\in(-\kappa_{n+1},-\kappa_{n})$ such that
$\theta(\alpha_{n})=1$. Finally, $\theta(\alpha)>0$ for all
$\alpha>-\kappa_{1}$, $D\theta(\alpha)<0$ for all $\alpha>-\kappa_{1}$, and
$\theta(0)=\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}<1$ imply that there is a unique
$\alpha_{N}\in(-\kappa_{1},0)$ such that $\theta(\alpha_{N})=1$.
The $N$ solutions to this equation map to the $N$ distinct eigenvalues of
$(D_{f}\boldsymbol{\rho}_{f})(\boldsymbol{\epsilon}_{f})$ (with multiplicities
$M_{n}$) via $\alpha_{n}=1-1/\mu_{n}$; that is, $\mu=1/(1-\alpha_{n})$.
Because each $\alpha_{n}<0$, each distinct eigenvalue of
$(D_{f}\boldsymbol{\rho}_{f})(\boldsymbol{\epsilon}_{f})$ is positive. Thus
$\mathrm{sign}\det(D_{f}\boldsymbol{\rho}_{f})(\boldsymbol{\epsilon}_{f})=1$
as claimed. ∎
Consider this proof from the perspective of concave costs $C_{j}$.
$\mathbf{x}^{\top}\mathbf{H}_{f}(\mathbf{p})\mathbf{x}=-\big{(}\mathbf{P}_{f}(\mathbf{p})^{\top}\left\lvert\mathbf{K}_{f}(\mathbf{p})\right\rvert\mathbf{P}_{f}(\mathbf{p})\big{)}\big{(}\mathbf{1}^{\top}\mathbf{x}\big{)}^{2}+2\big{(}\mathbf{P}_{f}(\mathbf{p})^{\top}\left\lvert\mathbf{K}_{f}(\mathbf{p})\right\rvert\mathbf{x}\big{)}\big{(}\mathbf{1}^{\top}\mathbf{x}\big{)}-\mathbf{x}^{\top}\left\lvert\mathbf{K}_{f}(\mathbf{p})\right\rvert\mathbf{x}$
is now a concave quadratic in $\mathbf{1}^{\top}\mathbf{x}$,
$q(\xi)=-a\xi^{2}+2b\xi-c$ (where $a,c>0$), maximized at $\xi_{*}=b/a$ with
value $q(\xi_{*})=b^{2}/a-c$. However, the Cauchy-Schwartz inequality then
requires $q(\xi)\leq q(\xi_{*})\leq 0$, and $\mathbf{H}_{f}(\mathbf{p})$ is
negative semi-definite. $(D_{f}\nabla_{f}\hat{\pi}_{f})(\mathbf{p})$ is
negative definite then only if
$\boldsymbol{\Lambda}_{f}(\mathbf{p})(\mathbf{I}-\boldsymbol{\Omega}_{f}(\mathbf{p}))$
is “more” negative-definite than
$\boldsymbol{\Lambda}_{f}(\mathbf{p})\mathbf{H}_{f}(\mathbf{p})\boldsymbol{\Lambda}_{f}(\mathbf{p})$
is, in the sense that
$\mathbf{x}^{\top}\boldsymbol{\Lambda}_{f}(\mathbf{p})(\mathbf{I}-\boldsymbol{\Omega}_{f}(\mathbf{p}))\mathbf{x}<\mathbf{x}^{\top}\boldsymbol{\Lambda}_{f}(\mathbf{p})\mathbf{H}_{f}(\mathbf{p})\boldsymbol{\Lambda}_{f}(\mathbf{p})\mathbf{x}.$
## 6\. Finite Purchasing Power
This section considers models in which there exists some limit on the
population’s purchasing power. That is, there exists $\varsigma\in[0,\infty)$
such that if $p_{j}\geq\varsigma$, no individual can purchase product $j$.
However, important empirical examples of Mixed Logit models have finite
purchasing power; see, e.g., [10, 43]. Thus it is important to consider this
case in the analysis of existence and uniqueness of equilibrium prices.
Section 4 above proves that if $\varsigma=\infty$ (no purchasing power limit)
and there is an outside good, it is not possible for any product’s price to be
$\infty$ in equilibrium. However, when $\varsigma<\infty$, it is possible for
some$-$but not all$-$prices to be equal to $\varsigma$; in other words, firms
may “price some products out of the market”. Theoretical and computational
treatments must be specially adapted to this case to account for this
qualitatively different behavior.
The analysis in this section proves the following theorem:
###### Theorem 6.1.
Suppose $\vartheta>-\infty$, costs satisfy Assumption 5.1 with
$(D^{2}C)(\mathbf{y},P)$ finite as $P\downarrow 0$ for any
$\mathbf{y}\in\mathcal{Y}$, and Assumption 6.1 holds with $\varsigma<\infty$
and a $w$ that eventually decreases sufficiently quickly, has finite sub-
quadratic second derivatives, and
$\lim_{p\uparrow\varsigma}(D^{2}w)(\mathbf{y},p)$ exists for any
$\mathbf{y}\in\mathcal{Y}$. Then there exists at least one equilibrium
$\mathbf{p}\in[0,\varsigma]^{J}$, and any equilibrium satisfies
$\mathbf{p}>\mathbf{c}(\mathbf{P}(\mathbf{p}))$.
### 6.1. Assumptions
Assumption 3.1 must be revised to account for finite $\varsigma$:
###### Assumption 6.1.
There exists $\varsigma\in(c_{*},\infty]$, where
$c_{*}=\max_{j}\\{\sup_{P\in[0,1]}c_{j}(P)\\}<\infty$, and functions
$w:\mathcal{Y}\times[0,\varsigma)\to(-\infty,\infty)$ and
$v:\mathcal{Y}\to(-\infty,\infty)$ such that the utility
$u:\mathcal{Y}\times[0,\infty)\to\mathbb{R}$ can be written
$u(\mathbf{y},p)=w(\mathbf{y},p)+v(\mathbf{y})$ for all $p<\varsigma$ and
$u(\mathbf{y},p)=-\infty$ for all $p\geq\varsigma$. Moreover, assume that, for
all $\mathbf{y}\in\mathcal{Y}$,
$w(\mathbf{y},\cdot):[0,\varsigma)\to(-\infty,\infty)$ is (a) strictly
decreasing, and (b) continuously differentiable, and (c)
$\lim_{p\uparrow\varsigma}w(\mathbf{y},p)=-\infty$.
The basic properties so useful in the case of infinite purchasing power must
also be generalized:
###### Definition 6.1.
If $\varsigma<\infty$, $w(\mathbf{y},\cdot)$ eventually decreases sufficiently
quickly if there exists $\delta(\mathbf{y})>0$ and $\alpha(\mathbf{y})>1$ such
that $(Dw)(\mathbf{y},p)\leq-\alpha(\mathbf{y})/(\varsigma-p)$ for all
$p\in[\varsigma-\delta(\mathbf{y}),\varsigma)$.
###### Definition 6.2.
$w(\mathbf{y},\cdot)$ has sub-quadratic second derivatives if
$\omega(\mathbf{y},p)<1$ for all $p\in(0,\varsigma)$.
Gallego et. al [18] consider equilibrium pricing under the attraction demand
model, equivalent to the Logit model with nonlinear utilities, and allow
$\varsigma<\infty$. Specifically, Gallego et. al formulate their model in
terms of the “attraction function” $a_{j}(p)=e^{u_{j}(p)}$ and make what
amount to the following assumptions on $u_{j}$:
$u_{j}:[0,\varsigma)\to\mathbb{R}$ is continuously differentiable, strictly
decreasing, and $p(D^{2}u_{j})(p)\geq(Du_{j})(p)$. This last assumption is
violated by the “BLP”-type utility $u_{j}(p)=\alpha\log(\varsigma-p)+v_{j}$
[10, 31, 33, 32], as can be easily checked. Assumption 6.1 and Defs. 6.1 and
6.2 above are weak enough to allow an analysis of this important case.
The following basic observations, stated without proof, are needed:
###### Lemma 6.2.
Under Assumption 6.1 with $\vartheta>-\infty$ and $\varsigma<\infty$,
* •
$P_{j}^{L}$ can be continuously extended to $[0,\varsigma]^{J}$, with
$P_{j}^{L}(\mathbf{p})\downarrow 0$ as $p_{j}\uparrow\varsigma$. Specifically,
$P_{j}^{L}(\mathbf{p})=\frac{e^{u_{j}(p_{j})}}{e^{\vartheta}+\sum_{k:p_{k}<\varsigma}e^{u_{k}(p_{k})}}$
* •
$\hat{\pi}_{f}(\mathbf{p})$ can be continuously extended to
$[0,\varsigma]^{J}$, with $\hat{\pi}_{f}(\mathbf{p})\downarrow 0$ as
$\mathbf{p}_{f}\uparrow\varsigma\mathbf{1}$ with
$\hat{\pi}_{f}(\mathbf{p})=\sum_{j\in\mathcal{J}_{f},p_{j}<\varsigma}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))\quad\text{and}\quad
0<\sup_{\mathbf{p}_{f}\in[0,\varsigma]^{J_{f}}}\hat{\pi}_{f}(\mathbf{p})<\infty$
* •
If $w$ eventually decreases sufficiently quickly,
$\lambda_{j}(\mathbf{p})=(Dw_{j})(p_{j})P_{j}^{L}(\mathbf{p})$ can be
continuously extended to $[0,\varsigma]^{J}$ with
$\lambda_{j}(\mathbf{p})\uparrow 0$ as $p_{j}\uparrow\varsigma$.
If $\vartheta=-\infty$, $P_{j}^{L}$ cannot be continuously extended to
$[0,\varsigma]^{J}$, for reasons analogous to those discussed in Section 3.
The following consequences of Assumption 6.1 and Defs. 6.1 and 6.2 are also
used below.
###### Lemma 6.3.
Suppose Assumption 6.1 holds with $\varsigma<\infty$. (i) If
$\lim_{p\uparrow\varsigma}(Dw)(\mathbf{y},p)$ exists, then
$(Dw)(\mathbf{y},p)=-\infty$ as $p\uparrow\varsigma$, and thus
$(Dw)(\mathbf{y},p)^{-1}\uparrow 0$ as $p\uparrow\varsigma$. (ii) If $w$
eventually decreases sufficiently quickly, then $(Dw)(\mathbf{y},p)=-\infty$
as $p\uparrow\varsigma$. (iii) If $w(\mathbf{y},\cdot)$ is twice continuously
differentiable and $\lim_{p\uparrow\varsigma}(D^{2}w)(\mathbf{y},p)$ exists,
then it is “eventually concave” in the sense that there exists some
$\epsilon\in(0,\varsigma)$ such that $(D^{2}w)(\mathbf{y},\cdot)<0$ on
$(\varsigma-\epsilon,\varsigma)$. (iv) If $w$ is twice continuously
differentiable and $\lim_{p\uparrow\varsigma}(D^{2}w)(\mathbf{y},p)$ exists,
then $\limsup_{p\uparrow\varsigma}\omega(\mathbf{y},p)\leq 0$. (v) If $w$ is
twice continuously differentiable and eventually decreases sufficiently
quickly, then $\liminf_{p\uparrow\varsigma}\omega(\mathbf{y},p)>-1$.
###### Proof.
(ii) is trivial. (i) and (iii) are both consequences of the following
technical result: If $f:(0,1)\to(-\infty,0)$ is continuously differentiable,
$\lim_{x\uparrow 1}f(x)=-\infty$, and $\lim_{x\uparrow 1}(Df)(x)$ exists, then
$\lim_{x\uparrow 1}(Df)(x)=-\infty$. For proof, note that the fundamental
theorem of calculus requires that
$\lim_{\delta\downarrow 0}\int_{x}^{1-\delta}(Df)(y)dy=-\infty.$
If $\lim_{x\uparrow 1}(Df)(x)$ exists and were finite, then
$\limsup_{x\uparrow 1}(Df)(x)>-\infty$ and this integral would be finite. Thus
if $\lim_{x\uparrow 1}(Df)(x)$ exists then $\lim_{x\uparrow
1}(Df)(x)=-\infty$. The assumption that the limit exists is required because
$(Df)$ could be highly oscillatory (in a manner similar to $\sin(x^{-1})$) and
still generate $\lim_{x\uparrow 1}f(x)=-\infty$.
(iv): By (iii), $(D^{2}w)(\mathbf{y},p)$ is negative for all $p$ sufficiently
close to $\varsigma$. Thus $\omega(\mathbf{y},p)<0$ for all $p$ sufficiently
close to $\varsigma$.
(v): Note that $-\omega(\mathbf{y},p)=D[(Dw)(\mathbf{y},p)^{-1}]$. The mean
value theorem then states that for all $\delta\in(0,\varsigma)$, there exists
some $\epsilon\in(0,\delta]$ such that
$\frac{1}{(Dw)(\mathbf{y},\varsigma-\delta)}=-\left(\frac{1}{(Dw)(\mathbf{y},\varsigma)}-\frac{1}{(Dw)(\mathbf{y},\varsigma-\delta)}\right)=-\Big{(}-\omega(\mathbf{y},\varsigma-\epsilon)\delta\Big{)}=\omega(\mathbf{y},\varsigma-\epsilon)\delta.$
Because $w$ eventually decreases sufficiently quickly, there exists $\gamma>0$
and $\alpha>1$ such that
$\omega(\mathbf{y},\varsigma-\epsilon)\delta=\frac{1}{(Dw)(\mathbf{y},\varsigma-\delta)}\geq-\left(\frac{\delta}{\alpha}\right).$
Thus $\omega(\mathbf{y},\varsigma-\epsilon)\geq-\alpha^{-1}>-1$, and thus
$\liminf_{p\uparrow\varsigma}\omega(\mathbf{y},p)>-1$. Note that we have not
assumed $\lim_{p\uparrow\varsigma}\omega(\mathbf{y},p)$ exists in proving that
$\liminf_{p\uparrow\varsigma}\omega(\mathbf{y},p)>-1$. ∎
### 6.2. A Variational Approach
The natural approach to characterizing profit-maximizing prices when
$\varsigma<\infty$ would be to assume firms solve the bound-constrained
optimization problem
$\displaystyle\text{maximize}\quad\hat{\pi}_{f}(\mathbf{p})\quad\mathrm{with\;respect\;to}\quad\mathbf{p}_{f}\in[0,\varsigma]^{J_{f}}.$
The KKT conditions are the Variational Inequality (VI)
(16)
$(\nabla_{f}\hat{\pi}_{f})(\mathbf{p})^{\top}(\mathbf{p}_{f}-\mathbf{q}_{f})\geq\mathbf{0}\quad\text{for
all}\quad\mathbf{q}_{f}\in[0,\varsigma]^{J_{f}}$
This approach is, unfortunately, not useful because of the following result:
###### Lemma 6.4.
Suppose $\vartheta>-\infty$, costs satisfy Assumption 5.1, Assumption 6.1
holds with $\varsigma<\infty$, and $w$ eventually decreases sufficiently
quickly. Then $\hat{\pi}_{f}(\mathbf{p})$ can be continuously differentiably
extended to $[0,\infty)^{J}$, where $(D_{k}\hat{\pi}_{f})(\mathbf{p})=0$
whenever $p_{k}\geq\varsigma$, $k\in\mathcal{J}_{f}$.
###### Proof.
Suppose $\mathbf{q}\in(\mathbf{0},\varsigma\mathbf{1})$. Then
$(D_{k}\hat{\pi}_{f})(\mathbf{q})=\lambda_{k}(\mathbf{q})\left(q_{k}-c_{k}(P_{k}(\mathbf{q}))-\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\mathbf{q})(q_{j}-c_{j}(P_{j}(\mathbf{q})))-\frac{1}{\left\lvert(Dw_{k})(q_{k})\right\rvert}\right).$
Each quantity can be extended, continuously, to $[0,\varsigma]^{J}$, and thus
so can $(D_{k}\hat{\pi}_{f})$. Note in particular that the term in parentheses
tends to
$\varsigma-
c_{k}(0)-\sum_{j\in\mathcal{J}_{f},p_{j}<\varsigma}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))$
which is finite, and $\lambda_{k}(\mathbf{p})\uparrow 0$ as
$p_{k}\uparrow\varsigma$. Thus $(D_{k}\hat{\pi}_{f})(\mathbf{p})=0$ when
$p_{k}=\varsigma$. ∎
As a result, the standard VI (16) does not provide any information about
profit-optimal prices or equilibria that have some prices equal to
$\varsigma$. For an extreme illustration of this fact, note that
$\mathbf{p}_{f}=\varsigma\mathbf{1}$ (trivially) solves (16). However,
$\varsigma\mathbf{1}$ cannot possible be profit-maximizing, because
$\hat{\pi}_{f}(\varsigma,\mathbf{1},\mathbf{p}_{-f})=0$ while
$\hat{\pi}_{f}(\mathbf{p}_{f},\mathbf{p}_{-f})>0$ for any $\mathbf{p}_{f}$
satisfying
$\mathbf{c}_{f}(\mathbf{P}_{f}(\mathbf{p}))<\mathbf{p}_{f}<\varsigma\mathbf{1}$.
The $\boldsymbol{\zeta}$ fixed-point form introduced above provides a
convenient solution to this problem.
###### Theorem 6.5.
Suppose $\vartheta>-\infty$, Assumptions 5.1 and 6.1 hold, $w$ eventually
decreases sufficiently quickly and has sub-quadratic second derivatives. For
any $\mathbf{p}\in[0,\varsigma]^{J}$, let
$\mathcal{J}_{f}^{\circ}=\\{j\in\mathcal{J}_{f}:p_{j}<\varsigma\\}$,
$\mathcal{J}_{f}^{*}=\\{j\in\mathcal{J}_{f}:p_{j}=\varsigma\\}$ and if
$j\in\mathcal{J}_{f}^{*}$, define
$\zeta_{j}(\mathbf{p})=\lim_{p_{j}\uparrow\varsigma}\zeta_{j}(\mathbf{p})=\sum_{j\in\mathcal{J}_{f}^{\circ}}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p}))).$
(i) If $\mathbf{p}_{f}\in[0,\varsigma]^{J_{f}}$ locally maximizes
$\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$ then
$p_{j}=c_{j}(P_{j}(\mathbf{p}))+\zeta_{j}(\mathbf{p})$ for all
$j\in\mathcal{J}_{f}^{\circ}$ and $\varsigma-
c_{j}(0)-\zeta_{j}(\mathbf{p})\leq 0$ for all $j\in\mathcal{J}_{f}^{*}$. (ii)
If, in addition, $\lim_{p\uparrow\varsigma}(D^{2}w)(\mathbf{y},\cdot)$ exists
and $(D^{2}C)(\mathbf{y},P)$ is finite as $P\downarrow 0$ for all
$\mathbf{y}\in\mathcal{Y}$, then
$p_{j}=c_{j}(P_{j}(\mathbf{p}))+\zeta_{j}(\mathbf{p})$ for all
$j\in\mathcal{J}_{f}^{\circ}$ and $\varsigma-
c_{j}(0)-\zeta_{j}(\mathbf{p})\leq 0$ for all $j\in\mathcal{J}_{f}^{*}$ is
sufficient $\mathbf{p}_{f}\in[0,\varsigma]^{J_{f}}$ to maximize
$\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$.
###### Proof.
Note that $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$ is continuously
differentiable on $(0,\infty)^{J_{f}}$, and thus we can apply the vector mean
value theorem to $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$ on
$(0,\varsigma]^{J_{f}}$.
We also note that there exist neighborhoods $\mathcal{U}_{f}^{*}$ and
$\mathcal{U}_{f}^{\circ}$, of $\mathbf{p}_{f}^{*}$ and
$\mathbf{p}_{f}^{\circ}$ respectively, and a map
$\mathbf{p}_{f}^{\circ}:(\mathcal{U}_{f}^{*}\cap[0,\varsigma]^{J_{f}^{*}})\to\mathcal{U}_{f}^{\circ}$
such that
$(\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{q}_{f}^{*})\in\mathcal{U}_{f}^{\circ}\times(\mathcal{U}_{f}^{*}\cap[0,\varsigma]^{J_{f}^{*}})$
and $\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*})$ is the unique solution to
$(\nabla_{f}^{\circ}\hat{\pi}_{f})\big{(}\mathbf{q}_{f}^{\circ},\mathbf{q}_{f}^{*},\mathbf{p}_{-f})=\mathbf{0}$
as a problem in $\mathbf{q}_{f}^{\circ}$ only. This actually follows from the
Implicit Function Theorem, applied to the continuously differentiable
extension of $\boldsymbol{\varphi}$ to all of $[0,\varsigma]^{J}$ given below
in Lemma 6.7. Because of the sufficiency of stationarity, when $w$ has sub-
quadratic second derivatives, $\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*})$ is,
in fact, the unique local maximizer of
$\hat{\pi}_{f}(\cdot,\mathbf{q}_{f}^{*},\mathbf{p}_{-f})$ on
$\mathcal{U}_{f}^{\circ}$. Thus
$\hat{\pi}_{f}\big{(}\mathbf{q}_{f}^{\circ},\mathbf{q}_{f}^{*},\mathbf{p}_{-f})<\hat{\pi}_{f}\big{(}\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{q}_{f}^{*},\mathbf{p}_{-f})$
for all
$\mathbf{q}_{f}^{*}\in\mathcal{U}_{f}^{*}\cap[0,\varsigma]^{J_{f}^{*}}$ and
$\mathbf{q}_{f}^{\circ}\in\mathcal{U}_{f}^{\circ}$. Furthermore,
$(D_{f}^{*}\mathbf{p}_{f}^{\circ})(\mathbf{q}_{f}^{*})\to\mathbf{0}$ as
$\mathbf{q}_{f}^{*}\uparrow\varsigma\mathbf{1}$. For
$(D_{f}^{*}\mathbf{p}_{f}^{\circ})(\mathbf{q}_{f}^{*})$ solves
$(D_{f}^{\circ}\boldsymbol{\varphi}_{f}^{\circ})(\mathbf{q}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{p}_{-f})(D_{f}^{*}\mathbf{p}_{f}^{\circ})(\mathbf{q}_{f}^{*})=(D_{f}^{*}\boldsymbol{\varphi}_{f}^{\circ})(\mathbf{q}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{p}_{-f})$
while
$(D_{f}^{*}\boldsymbol{\varphi}_{f}^{\circ})(\mathbf{p})=\lim_{\mathbf{q}_{f}^{*}\uparrow\varsigma\mathbf{1}}(D_{f}^{*}\boldsymbol{\varphi}_{f}^{\circ})(\mathbf{q}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{p}_{-f})\to\mathbf{0}$,
and $(D_{f}^{\circ}\boldsymbol{\varphi}_{f}^{\circ})(\mathbf{p})$ is
nonsingular.
(i): The necessity of
$\varphi_{j}(\mathbf{p})=p_{j}-c_{j}(P_{j}(\mathbf{p}))-\zeta_{j}(\mathbf{p})=0$
for $j\in\mathcal{J}_{f}^{\circ}$ is obvious. Suppose then that there is some
$j\in\mathcal{J}_{f}^{*}$ such that $\varsigma-
c_{j}(0)-\zeta_{j}(\mathbf{p})>0$. We can choose the neighborhood
$\mathcal{U}_{f}^{*}$ above so that
$\varphi_{j}(\mathbf{q}_{f},\mathbf{p}_{-f})>0$ for all
$\mathbf{q}_{f}=(\mathbf{q}_{f}^{\circ},\mathbf{q}_{f}^{*})\in\mathcal{U}_{f}^{\circ}\times(\mathcal{U}_{f}^{*}\cap[0,\varsigma]^{J_{f}^{*}})$.
Letting $\mathbf{q}_{f}^{*}=\varsigma\mathbf{1}-(\varsigma-
q_{j})\mathbf{e}_{j}$ for some $q_{j}<\varsigma$ (that is, changing only the
$j^{\text{th}}$ products’ price), the vector mean value theorem states that
there exists some
$\mathbf{r}_{f}^{*}=\varsigma\mathbf{1}-(\varsigma-\tau)\mathbf{e}_{j}$,
$\tau\in(q_{j},\varsigma)$, such that
$\displaystyle\hat{\pi}_{f}(\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{q}_{f}^{*},\mathbf{p}_{-f})=\hat{\pi}_{f}(\mathbf{p})+(D_{j}\hat{\pi}_{f})(\mathbf{r}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{r}_{f}^{*}),\mathbf{p}_{-f})(\varsigma-
q_{j})>\hat{\pi}_{f}(\mathbf{p}).$
Thus $\mathbf{p}_{f}$ is not locally profit-maximizing for
$\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$. By contraposition, (i) holds.
(ii): Define
$\hat{\pi}_{f}^{*}(\mathbf{q}_{f}^{*})=\hat{\pi}_{f}(\mathbf{q}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{p}_{-f})$.
Also let $\nabla_{f}^{*}\hat{\pi}_{f}$ and $\nabla_{f}^{\circ}\hat{\pi}_{f}$
denote the derivatives of firm $f$’s profits with respect to the prices of
products in $\mathcal{J}_{f}^{*}$ and $\mathcal{J}_{f}^{\circ}$, respectively.
Note that
$\displaystyle(\nabla^{*}\hat{\pi}_{f}^{*})(\mathbf{q}_{f}^{*})$
$\displaystyle=(\nabla^{*}\hat{\pi}_{f})(\mathbf{q}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{p}_{-f})+(D_{f}^{*}\mathbf{p}_{f}^{\circ})(\mathbf{q}_{f}^{*})^{\top}(\nabla^{\circ}\hat{\pi}_{f})(\mathbf{q}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{p}_{-f})$
$\displaystyle=(\nabla^{*}\hat{\pi}_{f})(\mathbf{q}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{p}_{-f})$
because
$(\nabla^{\circ}\hat{\pi}_{f})(\mathbf{q}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{p}_{-f})=\mathbf{0}$,
by definition. Let
$\mathbf{q}_{f}^{*}=\varsigma\mathbf{1}-\boldsymbol{\delta}$ for some
$\boldsymbol{\delta}\geq 0$, $\boldsymbol{\delta}\neq\mathbf{0}$. Then the
vector mean value theorem states that there exists
$\mathbf{r}_{f}^{*}=\varsigma\mathbf{1}-\tau\boldsymbol{\delta}$,
$\tau\in(0,1)$, such that
$\hat{\pi}_{f}(\mathbf{q}_{f}^{*},\mathbf{q}_{f}^{\circ},\mathbf{p}_{-f})\leq\hat{\pi}_{f}^{*}(\mathbf{q}_{f}^{*})=\hat{\pi}_{f}^{*}(\mathbf{p}_{f}^{*})-(\nabla_{f}^{*}\hat{\pi}_{f}^{*})(\mathbf{r}_{f}^{*})^{\top}\boldsymbol{\delta}=\hat{\pi}_{f}(\mathbf{p})-(\nabla_{f}^{*}\hat{\pi}_{f})(\mathbf{r}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{r}_{f}^{*}),\mathbf{p}_{-f})^{\top}\boldsymbol{\delta}$
Note also that
$\displaystyle\boldsymbol{\varphi}_{f}^{*}(\mathbf{q}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{p}_{-f})$
$\displaystyle=\boldsymbol{\varphi}_{f}^{*}(\mathbf{p})-\Big{(}(D_{f}^{*}\boldsymbol{\varphi}_{f}^{*})(\mathbf{p})+(D_{f}^{\circ}\boldsymbol{\varphi}_{f}^{*})(\mathbf{p})(D_{f}^{*}\mathbf{p}_{f}^{\circ})(\varsigma\mathbf{1})\Big{)}\boldsymbol{\delta}+\mathcal{O}(\lvert\lvert\boldsymbol{\delta}\rvert\rvert^{2})$
$\displaystyle=\boldsymbol{\varphi}_{f}^{*}(\mathbf{p})-(D_{f}^{*}\boldsymbol{\varphi}_{f}^{*})(\mathbf{p})\boldsymbol{\delta}+\mathcal{O}(\lvert\lvert\boldsymbol{\delta}\rvert\rvert^{2})$
$\displaystyle=\boldsymbol{\varphi}_{f}^{*}(\mathbf{p})-(\mathbf{I}-\boldsymbol{\Omega}_{f}^{*}(\varsigma\mathbf{1}))\boldsymbol{\delta}+\mathcal{O}(\lvert\lvert\boldsymbol{\delta}\rvert\rvert^{2})$
Because $\boldsymbol{\varphi}_{f}^{*}(\mathbf{p})\leq\mathbf{0}$ and
$1-\omega_{j}(\varsigma)>0$ for all $j\in\mathcal{J}_{f}^{*}$,
$\boldsymbol{\varphi}_{f}^{*}(\mathbf{q}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{p}_{-f})<\mathbf{0}$
for all $\boldsymbol{\delta}\neq\mathbf{0}$ sufficiently small.
For such $\boldsymbol{\delta}\geq 0$, $\boldsymbol{\delta}\neq\mathbf{0}$,
also satisfying
$\mathbf{q}_{f}^{*}=\varsigma\mathbf{1}-\boldsymbol{\delta}\in\mathcal{U}_{f}^{*}\cap[0,\varsigma]^{J_{f}^{*}}$,
$(\nabla_{f}^{*}\hat{\pi}_{f})(\mathbf{r}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{r}_{f}^{*}),\mathbf{p}_{-f})=\boldsymbol{\Lambda}_{f}^{*}(\mathbf{r}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{r}_{f}^{*}),\mathbf{p}_{-f})\boldsymbol{\varphi}_{f}^{*}(\mathbf{r}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{r}_{f}^{*}),\mathbf{p}_{-f})\geq\mathbf{0}$
with at least one positive component. Thus
$(\nabla_{f}^{*}\hat{\pi}_{f})(\mathbf{r}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{r}_{f}^{*}),\mathbf{p}_{-f})^{\top}\boldsymbol{\delta}>0$,
and
$\displaystyle\hat{\pi}_{f}(\mathbf{q}_{f}^{*},\mathbf{q}_{f}^{\circ},\mathbf{p}_{-f})\leq\hat{\pi}_{f}(\mathbf{p})-(\nabla_{f}^{*}\hat{\pi}_{f})(\mathbf{r}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{r}_{f}^{*}),\mathbf{p}_{-f})^{\top}\boldsymbol{\delta}<\hat{\pi}_{f}(\mathbf{p}).$
$\mathbf{p}_{-f}$ is thus a local maximizer of
$\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$. ∎
###### Corollary 6.6.
Suppose $\vartheta>-\infty$, Assumption 5.1 holds with $D^{2}C$ finite as
$P\downarrow 0$, and 6.1 holds with $w$ that eventually decreases sufficiently
quickly, has sub-quadratic second derivatives, and
$\lim_{p\uparrow\varsigma}(D^{2}w)(\mathbf{y},p)$ exists. (i)
$\mathbf{p}_{-f}$ locally maximizes $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$,
for any $\mathbf{p}_{-f}\in[0,\varsigma]^{J_{-f}}$, if, and only if,
$\mathbf{p}_{-f}$ solves the VI
(17)
$\boldsymbol{\varphi}_{f}(\mathbf{p})^{\top}(\mathbf{p}_{f}-\mathbf{q}_{f})\leq\mathbf{0}\quad\text{for
all}\quad\mathbf{q}_{f}\in[0,\varsigma]^{J_{f}}.$
(ii) $\mathbf{p}$ is a local equilibrium if, and only if, $\mathbf{p}$ solves
the VI
(18)
$\boldsymbol{\varphi}(\mathbf{p})^{\top}(\mathbf{p}-\mathbf{q})\leq\mathbf{0}\quad\text{for
all}\quad\mathbf{q}\in[0,\varsigma]^{J}.$
Theorem 6.5 also suggests the following general demonstration that it is
possible for prices to be equal to $\varsigma$ in equilibrium if unit costs to
be constant and differ within firms. Let $c_{j}=\varsigma-\kappa$ be constant
unit costs for some $\kappa>0$ and observe that $\varsigma-
c_{j}-\hat{\pi}_{f}(\mathbf{p})<0$ if, and only if,
$\kappa<\hat{\pi}_{f}(\mathbf{p})$. Because $\hat{\pi}_{f}(\mathbf{p})$ is
independent of $c_{j}$ when $p_{j}=\varsigma$, $\kappa$ can be made small
enough so that it is less than any lower bound on the optimal profits for firm
$f$ excluding product $j$ from their set of offerings. Setting
$c_{j}=\varsigma-\kappa$ with such a value of $\kappa$ then ensures that
$p_{j}=\varsigma$ is locally profit-optimal for the original problem including
product $j$.
### 6.3. Existence of Equilibrium
To prove the existence of equilibrium, it remains to show that profit-
maximizing prices are unique. While the modified VI (18) can be used to
characterize profit-maximizing prices, smooth nonlinear systems are often
easier to analyze. Particularly, establishing the uniqueness of profit-
maximizing prices with (17) would traditionally require strict monotonicity of
$\boldsymbol{\varphi}_{f}$ [23], a property that may be difficult to verify.
These obstacles can be overcome by continuously extending the
$\boldsymbol{\zeta}$ map, and thus $\boldsymbol{\varphi}$, to all of
$[0,\infty)^{J}$ in such a way that solutions of the nonlinear system with the
extended $\boldsymbol{\varphi}$ are solutions to the VIs (17) and (18). This
enables an existence and uniqueness proofs using the same process applied
above. Another approach, enabled by the analysis below, is to apply a VI
uniqueness theorem due to Simsek et. al [47, Proposition 5.1] also based on
the Poincare-Hopf Theorem.
###### Lemma 6.7.
Suppose $\vartheta>-\infty$, Assumption 5.1 holds with $D^{2}C$ finite as
$P\downarrow 0$, and 6.1 holds with $w$ that eventually decreases sufficiently
quickly, has sub-quadratic second derivatives, and
$\lim_{p\uparrow\varsigma}(D^{2}w)(\mathbf{y},p)$ exists. Define the map
$\mathbf{z}:[0,\infty)^{J}\to\mathbb{R}^{J}$ componentwise by
$z_{k}(\mathbf{p})=\left\\{\begin{aligned}
&\sum_{j\in\mathcal{J}_{f(j)}^{\circ}}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))+\frac{1}{\left\lvert(Dw_{j})(p_{j})\right\rvert}&&\quad\text{if
}p_{k}<\varsigma\\\
&\omega_{k}(\varsigma)(p_{k}-\varsigma)+\sum_{j\in\mathcal{J}_{f(k)}^{\circ}}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))&&\quad\text{if
}p_{k}\geq\varsigma\end{aligned}\right.$
and the map $\boldsymbol{\Phi}:[0,\infty)^{J}\to\mathbb{R}^{J}$ by
$\boldsymbol{\Phi}(\mathbf{p})=\mathbf{p}-\mathbf{c}(\mathbf{P}(\mathbf{p}))-\mathbf{z}(\mathbf{p})$.
(i) $\mathbf{z}$ (or $\boldsymbol{\Phi}$) is a continuously differentiable
extension of $\boldsymbol{\zeta}$ (or $\boldsymbol{\varphi}$) from
$[0,\varsigma]^{J}$ to $[0,\infty)^{J}$. (ii) For all $j$,
$\Phi_{j}(\mathbf{p})<0$ when
$\mathbf{p}\geq\mathbf{c}(\mathbf{P}(\mathbf{p}))$ and
$p_{j}=c_{j}(P_{j}(\mathbf{p}))$ and there exists $\bar{p}_{j}$ such that
$\Phi_{j}(\mathbf{p})>0$ for all $p_{j}>\bar{p}_{j}$, regardless of
$\mathbf{p}_{-j}$. (iii)
$\boldsymbol{\Phi}_{f}(\mathbf{p}_{f},\mathbf{p}_{-f})=\mathbf{0}$ if, and
only if, $\mathrm{proj}_{[0,\varsigma]}(\mathbf{p}_{f})$ solves the VI (17),
where “$\mathrm{proj}_{[0,\varsigma]}$” denotes the projection onto
$[0,\varsigma]^{J_{f}}$. (iv) $\boldsymbol{\Phi}(\mathbf{p})=\mathbf{0}$ if,
and only if, $\mathrm{proj}_{[0,\varsigma]}(\mathbf{p})$ solves the VI (18),
where “$\mathrm{proj}_{[0,\varsigma]}$” denotes the projection onto
$[0,\varsigma]^{J}$.
###### Proof.
(i): The claim concerning $\mathbf{z}$ and $\boldsymbol{\zeta}$ follow by
taking derivatives for any prices in $(0,\varsigma)^{J}$ and taking limits.
Specifically,
$(D_{l}\zeta_{k})(\mathbf{p})=\omega_{k}(p_{k})\delta_{k,l}+(D_{l}\bar{\pi}_{f})(\mathbf{p})$.
Now, $(D_{l}\bar{\pi}_{f})(\mathbf{p})\to 0$ as $p_{l}\uparrow\varsigma$, from
which we can deduce the following:
* •
$(D_{l}\zeta_{k})(\mathbf{p})=(D_{l}z_{k})(\mathbf{p})$ when
$k,l\in\mathcal{J}^{\circ}$,
* •
$(D_{l}\zeta_{k})(\mathbf{p})\to\omega_{k}(\varsigma)\delta_{k,l}=(D_{l}z_{k})(\mathbf{p})$
as $p_{k},p_{l}\uparrow\varsigma$,
* •
$(D_{l}\zeta_{k})(\mathbf{p})\to 0=(D_{l}z_{k})(\mathbf{p})$ when
$p_{k}<\varsigma$ but $p_{l}\uparrow\varsigma$, and
* •
$(D_{l}\zeta_{k})(\mathbf{p})\to(D_{l}\bar{\pi}_{f(k)})(\mathbf{p})=(D_{l}z_{k})(\mathbf{p})$
when $p_{k}\uparrow\varsigma$ but $p_{l}<\varsigma$.
The claim concerning $\boldsymbol{\Phi}$ and $\boldsymbol{\varphi}$ is an
obvious consequence, noting that $D_{l}[c_{k}(P_{k}(\mathbf{p}))]\downarrow 0$
as $p_{k}$ or $p_{l}\uparrow\varsigma$, and thus
$\mathbf{c}(\mathbf{P}(\mathbf{p}))$ is continuously differentiable on
$[0,\infty)^{J}$.
(ii): The first part of this claim follows from the corresponding result for
$\varphi_{j}$. To prove the second part, note that, by definition,
$\Phi_{k}(\mathbf{p})=(1-\omega_{k}(\varsigma))(p_{k}-\varsigma)+\varsigma-
c_{k}(0)-\sum_{j\in\mathcal{J}_{f(k)}^{\circ}}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))$
for all $p_{j}\geq\varsigma$. Because
$\sum_{j\in\mathcal{J}_{f(k)}^{\circ}}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))$
is bounded over $[0,\infty)^{J}$ and $w_{k}$ has finite sub-quadratic second
derivatives, $\bar{p}_{k}\geq\varsigma$ can be chosen large enough so that
$\Phi_{k}(\mathbf{p})>0$ for all $p_{k}\geq\bar{p}_{k}$, regardless of
$\mathbf{p}_{-k}$.
(iii) and (iv): We prove (iv), the proof for $(iii)$ being nearly identical.
Let $\boldsymbol{\Phi}(\mathbf{p})=\mathbf{0}$. Then
$\varphi_{j}(\mathbf{p})=0$ for all $j\in\mathcal{J}^{\circ}$ and
$p_{k}-c_{k}(0)-\omega_{k}(\varsigma)(p_{k}-\varsigma)-\sum_{j\in\mathcal{J}_{f(k)}^{\circ}}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))=0.$
Therefore
$\varsigma-
c_{k}(0)-\sum_{j\in\mathcal{J}_{f(k)}^{\circ}}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))=-(1-\omega_{k})(p_{k}-\varsigma)\leq
0.$
This implies that $\mathbf{p}$ solves the VI (18). Conversely, suppose
$\mathbf{q}\in[0,\varsigma]^{J}$ solves the VI (18). Define $p_{k}=q_{k}$ for
all $k\in\mathcal{J}^{\circ}$ and
$p_{k}=\varsigma-\varphi_{k}(\mathbf{q})/(1-\omega_{k}(\varsigma))$ for all
$k\in\mathcal{J}^{*}$. Note that
$\Phi_{k}(\mathbf{p})=\varphi_{k}(\mathbf{q})=0$, $p_{k}\geq\varsigma$ for all
$k\in\mathcal{J}^{*}$ (because $\varphi_{k}(\mathbf{q})\leq 0$ for all such
$k$), and, for all $k\in\mathcal{J}^{*}$,
$\displaystyle\Phi_{k}(\mathbf{p})$
$\displaystyle=p_{k}-c_{k}(0)-\omega_{k}(\varsigma)(p_{k}-\varsigma)-\sum_{j\in\mathcal{J}_{f(k)}^{\circ}}P_{j}^{L}(\mathbf{q})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{q})))$
$\displaystyle=(1-\omega_{k}(\varsigma))(p_{k}-\varsigma)+\varphi_{k}(\mathbf{q})=0.$
Thus $\mathbf{q}=\mathrm{proj}_{[0,\varsigma]}(\mathbf{p})$ and
$\boldsymbol{\Phi}(\mathbf{p})=\mathbf{0}$. ∎
The obvious corollary is as follows:
###### Corollary 6.8.
Assume the hypotheses of Lemma 6.7, and let $\mathbf{p}\in[0,\infty)^{J}$. (i)
$\mathrm{proj}_{[0,\varsigma]}(\mathbf{p}_{f})$ locally maximizes
$\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$ if, and only if,
$\boldsymbol{\Phi}_{f}(\mathbf{p})=\mathbf{0}$. (ii)
$\mathrm{proj}_{[0,\varsigma]}(\mathbf{p})$ is a local equilibrium if, and
only if, $\boldsymbol{\Phi}(\mathbf{p})=\mathbf{0}$.
An adaptation of the techniques in Sections 4 and 5 again establishes the
uniqueness of profit-maximizing prices.
###### Lemma 6.9.
Assume the hypotheses of Lemma 6.7. For any $\mathbf{p}_{-f}$, the nonlinear
system $\boldsymbol{\Phi}_{f}(\mathbf{p}_{f},\mathbf{p}_{-f})=\mathbf{0}$ has
a unique solution $\mathbf{p}_{f}\in[0,\infty)^{J_{f}}$ satisfying
$c_{j}(P_{j}(\mathbf{p}))<p_{j}$ for all $j\in\mathcal{J}_{f}$ with
$p_{j}<\varsigma$ for at least one $j\in\mathcal{J}_{f}$.
###### Proof.
For any $\mathbf{p}_{-f}$, $\boldsymbol{\Phi}_{f}(\cdot,\mathbf{p}_{-f})$ has
a zero in the interior of
$\\{\mathbf{p}_{f}:\mathbf{p}_{f}\geq\mathbf{c}_{f}(\mathbf{P}_{f}(\mathbf{p}))\\}$;
the proof is exactly analogous to the proof of existence in the case
$\varsigma=\infty$: We first show that the homotopy $\boldsymbol{\rho}_{f}$
between $[0,\infty)^{J_{f}}$ and
$\\{\mathbf{p}_{f}:\mathbf{p}_{f}\geq\mathbf{c}_{f}(\mathbf{P}_{f}(\mathbf{p}))\\}$
still exists. The inverse map
$\mathbf{p}_{f}\mapsto\boldsymbol{\epsilon}_{f}=\mathbf{p}_{f}-\mathbf{c}_{f}(\mathbf{P}_{f}(\mathbf{p}))$
is again well-defined and continuous, trivially.
$\boldsymbol{\rho}_{f}(\boldsymbol{\epsilon}_{f})$ is ostensibly defined by
$\mathbf{p}_{f}=\mathbf{c}_{f}(\mathbf{P}_{f}^{L}(\mathbf{p}))+\boldsymbol{\epsilon}_{f}$.
For $\epsilon_{k}=\varsigma-c_{k}(0)+\delta$, $\delta\geq 0$,
$\rho_{k}(\boldsymbol{\epsilon})=\varsigma+\delta$ solves this fixed-point
equation regardless of $\epsilon_{j}$, $j\neq k$:
$p_{k}=\varsigma+\delta=c_{k}(0)+(\varsigma-
c_{k}(0)+\delta)=c_{k}(P_{k}(\mathbf{p}))+\epsilon_{k}$
Note that $(D_{k}^{\epsilon}\rho_{k})(\boldsymbol{\epsilon}_{f})=\delta_{k,l}$
for $\epsilon_{k}>\varsigma$.
Supposing $\mathbf{p}_{f}<\varsigma\mathbf{1}$,
$D_{f}\Big{[}\mathbf{c}_{f}(\mathbf{P}_{f}^{L}(\mathbf{p}))+\boldsymbol{\epsilon}_{f}\Big{]}=(D^{2}\mathbf{C}_{f})(\mathbf{P}_{f}^{L}(\mathbf{p}))\boldsymbol{\Lambda}_{f}(\mathbf{p})-(D^{2}\mathbf{C}_{f})(\mathbf{P}_{f}^{L}(\mathbf{p}))\mathbf{P}_{f}^{L}(\mathbf{p})\boldsymbol{\lambda}_{f}(\mathbf{p})^{\top}.$
As $p_{k}\uparrow\varsigma$, the $k^{\text{th}}$ row and $k^{\text{th}}$
column of this matrix vanish because $\lim_{P\downarrow
0}(D^{2}C_{j})(P)<\infty$. In particular, the proof that the spectrum of the
fixed-point map does not contain 1 given before holds on $[0,\infty)^{J_{f}}$.
Constructing an upper bound on the magnitude of the fixed point for any
$\boldsymbol{\epsilon}$ then proves the fixed point is unique, again by
Kellogg’s uniqueness theorem.
The vanishing of the derivatives also proves that
$(D_{l}\rho_{k})(\boldsymbol{\epsilon}_{f})\to\delta_{k,l}$ as either $p_{k}$
or $p_{l}$ $\uparrow\varsigma$, $k,l\in\mathcal{J}_{f}$, and thus
$\boldsymbol{\rho}_{f}$ is continuously differentiable on
$[0,\infty)^{J_{f}}$. $\boldsymbol{\rho}_{f}$ must then be continuous on
$[0,\infty)^{J_{f}}$.
As before, consider the vector field
$\boldsymbol{\Psi}_{f}=\boldsymbol{\Phi}_{f}\circ\boldsymbol{\rho}_{f}:[0,\infty)^{J_{f}}\to\mathbb{R}^{J_{f}}$.
This vector field points outward on the boundary of
$[\mathbf{0},\bar{\mathbf{p}}_{f}]$, where the existence of
$\bar{\mathbf{p}}_{f}$ was established in Lemma 6.7. By the Poincare-Hopf
Theorem, $\boldsymbol{\Psi}_{f}$ has a zero
$\boldsymbol{\epsilon}_{f}\in(\mathbf{0},\bar{\mathbf{p}}_{f})$, which is
uniquely related to a zero
$\mathbf{p}_{f}=\boldsymbol{\rho}_{f}(\boldsymbol{\epsilon}_{f})$ satisfying
$c_{j}(P_{j}(\mathbf{p}))<p_{j}$ for all $j\in\mathcal{J}_{f}$. Note that this
zero cannot have $p_{j}\geq\varsigma$ for all $j\in\mathcal{J}_{f}$: For if
that were true, then
$\varsigma\leq\left(\frac{1}{1-\omega_{j}(\varsigma)}\right)c_{j}(0)-\left(\frac{\omega_{j}(\varsigma)}{1-\omega_{j}(\varsigma)}\right)\varsigma=-\left(\frac{1}{1-\omega_{j}(\varsigma)}\right)(\varsigma-
c_{j}(0))+\varsigma<\varsigma.$
By contradiction, there must exist some $j\in\mathcal{J}_{f}$ such that
$p_{j}<\varsigma$.
Uniqueness of profit-maximizing prices follows from a very similar approach to
that used previously. Here we show that any zero of $\boldsymbol{\Phi}_{f}$
has index one, and the rest of the proof proceeds exactly the same way. Note
that
* •
$(D_{l}\Phi_{k})(\mathbf{p})=(D_{l}\varphi_{k})(\mathbf{p})$ when
$k,l\in\mathcal{J}_{f}^{\circ}$,
* •
$(D_{l}\Phi_{k})(\mathbf{p})=(1-\omega_{k}(\varsigma))\delta_{k,l}$ when
$k,l\in\mathcal{J}_{f}^{*}$
* •
$(D_{l}\Phi_{k})(\mathbf{p})=0$ when $k\in\mathcal{J}_{f}^{\circ}$ but
$l\in\mathcal{J}_{f}^{*}$
These relations imply that there exists a symmetric permutation
$\mathbf{T}_{f}$ to make $(D_{f}\boldsymbol{\Phi}_{f})(\mathbf{p})$ block-
triangular:
$(D_{f}\boldsymbol{\Phi}_{f})(\mathbf{p})=\mathbf{T}_{f}\begin{bmatrix}(D_{f}^{\circ}\boldsymbol{\varphi}_{f}^{\circ})(\mathbf{p})&\mathbf{0}\\\
\mathbf{A}&\mathbf{I}-\boldsymbol{\Omega}_{f}^{*}(\varsigma)\end{bmatrix}\mathbf{T}_{f}$
where $\mathbf{A}$ is some matrix. Thus
$\det(D_{f}\boldsymbol{\Phi}_{f})(\mathbf{p})=(\det\mathbf{T}_{f})^{2}\det\Big{(}(D_{f}^{\circ}\boldsymbol{\varphi}_{f}^{\circ})(\mathbf{p})\Big{)}\det\Big{(}\mathbf{I}-\boldsymbol{\Omega}_{f}^{*}(\varsigma)\Big{)}$
and, because $1-\omega_{k}(\varsigma)>0$,
$\mathrm{sign}\det(D_{f}\boldsymbol{\Phi}_{f})(\mathbf{p})=\mathrm{sign}\det\Big{(}(D_{f}^{\circ}\boldsymbol{\varphi}_{f}^{\circ})(\mathbf{p})\Big{)}.$
But $\boldsymbol{\varphi}_{f}^{\circ}$ is identical to the
$\boldsymbol{\varphi}_{f}$ map if we exclude the products
$\mathcal{J}_{f}^{*}$ from $\mathcal{J}_{f}$, and $\mathbf{p}_{f}^{\circ}$ are
profit-maximizing prices strictly in the interior of
$[0,\varsigma]^{J_{f}^{\circ}}$ where
$J_{f}^{\circ}=|\mathcal{J}_{f}^{\circ}|$. Then
$(-1)^{J_{f}^{\circ}}=\mathrm{sign}\det(D_{f}^{\circ}\hat{\pi}_{f}^{\circ})(\mathbf{p})=\mathrm{sign}\det\boldsymbol{\Lambda}_{f}^{\circ}(\mathbf{p})\mathrm{sign}\det\Big{(}(D_{f}^{\circ}\boldsymbol{\varphi}_{f}^{\circ})(\mathbf{p})\Big{)}=(-1)^{J_{f}^{\circ}}\mathrm{sign}\det\Big{(}(D_{f}^{\circ}\boldsymbol{\varphi}_{f}^{\circ})(\mathbf{p})\Big{)}$
implies
$\mathrm{sign}\det(D_{f}\boldsymbol{\Phi}_{f})(\mathbf{p})=\mathrm{sign}\det\Big{(}(D_{f}^{\circ}\boldsymbol{\varphi}_{f}^{\circ})(\mathbf{p})\Big{)}=1.$
∎
This result can also be proved using the uniqueness theorem for VI’s given by
Simsek et. al [47, Proposition 5.1].
## 7\. Properties of Equilibrium Prices
This section establishes properties that the finite prices and markups of any
equilibrium must satisfy based on properties of $\boldsymbol{\zeta}$. The most
general result is Corollary 7.1, which states that the difference between
profit optimal markups for two products offered by the same firm with prices
less than $\varsigma$ depends only on the prices and characteristics of those
two products when unit costs are constant. This property is very similar to
the embodiment of the “Independence of Irrelevant Alternatives” (IIA) property
in Logit models: the ratio of choice probabilities depends only on the
characteristics and prices of those two products [50]. In Corollaries 7.5
through 7.8 this result is applied to concave-in-price utility functions under
hypotheses on the unit cost and value functions to illuminate some
counterintuitive properties of equilibrium prices under Logit. This is the
only section of the article that focuses somewhat on concave-in-price utility
functions.
### 7.1. Properties of Profit-Optimal Prices
For this subsection, we focus on a single firm $f\in\mathbb{N}(F)$ and derive
our results as properties of locally profit-optimal prices. Naturally, these
properties will be manifest in locally equilibrium prices as well. This
section is also the only portion of this article in which we focus heavily on
concave in price utilities, which will satisfy our existence conditions.
The basic observation is as follows.
###### Corollary 7.1.
Suppose Assumptions 5.1 and 6.1 hold. Let $\mathbf{p}_{f}$ be profit
maximizing. For any $j,k\in\mathcal{J}_{f}^{\circ}$,
(19)
$\Big{(}p_{j}-c_{j}(P_{j}(\mathbf{p}))\Big{)}-\Big{(}p_{k}-c_{k}(P_{k}(\mathbf{p}))\Big{)}=-\left(\frac{1}{(Dw_{j})(p_{j})}-\frac{1}{(Dw_{k})(p_{k})}\right).$
That is, the difference between profit optimal markups for any two products
offered by a single firm depends only on the corresponding utility
derivatives. If unit costs are constant, this implies that the difference
between profit optimal markups for any two products offered by a single firm
depends only on the characteristics and prices of those products.
###### Proof.
Eqn. (19) follows immediately from the fixed-point equation
$p_{j}=c_{j}(P_{j}(\mathbf{p}))+\bar{\pi}_{f}(\mathbf{p})+\left\lvert(Dw_{j})(p_{j})\right\rvert^{-1}$
for all $j\in\mathcal{J}_{f}^{\circ}$. ∎
One application is motivated by the frequent application of constant
coefficient linear in price utility functions.
###### Corollary 7.2.
Suppose Assumption 5.1 holds. If $w(\mathbf{y},p)\equiv-\alpha p$ for some
$\alpha>0$, $p_{j}-c_{j}(P_{j}(\mathbf{p}))=p_{k}-c_{k}(P_{k}(\mathbf{p}))$
for all $j,k\in\mathcal{J}_{f}$ when $\mathbf{p}_{f}$ is profit-maximizing.
In other words, profit-optimal markups are constant regardless of product
costs or the value of product characteristics. Constant intra-firm markups
have appeared as an assumption [44, 16], but not often proven to be an
equilibrium outcome.
The following example motivates the more general propositions on profit-
optimal markups given below. Consider the quadratic in price utility
$w(\mathbf{y},p)\equiv w(p)=-\alpha p^{2}$ and constant unit costs. Then
$(p_{j}-c_{j})-(p_{k}-c_{k})=\left(\frac{1}{2\alpha}\right)\left(\frac{1}{p_{j}}-\frac{1}{p_{k}}\right),$
demonstrating that locally profit optimal markups decrease with the
corresponding prices (i.e., $p_{j}-c_{j}>p_{k}-c_{k}$ if and only if
$p_{j}<p_{k}$). Rearranging and setting $\lambda=1/(2\alpha)$, we obtain
$\left(p_{j}-\frac{\lambda}{p_{j}}\right)-\left(p_{k}-\frac{\lambda}{p_{k}}\right)=c_{j}-c_{k}.$
The function $\eta_{\lambda}(p)=p-\lambda/p$ is strictly increasing in $p$ for
non-negative $\lambda$, and thus $c_{j}>c_{k}$ implies $p_{j}>p_{k}$. Thus,
locally profit optimal prices increase with costs while the corresponding
markups decrease with costs. Additionally, we note that if $c_{j}=c_{k}$ then
$p_{j}=p_{k}$, even if $\mathbf{y}_{j}\neq\mathbf{y}_{k}$; that is, profit-
optimal prices reflect only product costs, not value.
We first generalize the counterintuitive property that differences in
characteristics that do not impact costs or (local) willingness to pay do not
impact prices, even if they impact product value.
###### Corollary 7.3.
Suppose unit costs are constant, 6.1 holds and $w$ has sub-quadratic second
derivatives. Let $\mathbf{p}_{f}$ be profit-maximizing, and suppose that
$c_{j}=c_{k}$ and $(Dw_{j})(p)=(Dw_{k})(p)$ for all $p\in(0,\varsigma)$ for
some $j,k\in\mathcal{J}_{f}^{\circ}$, even if
$\mathbf{y}_{j}\neq\mathbf{y}_{k}$. Then $p_{j}=p_{k}$.
In other words, for any separable utility with sub-quadratic second
derivatives, profit-optimal prices are determined by costs, not value. One
would expect that real firms would not follow this rule, charging higher
markups for the more valued product.
###### Proof.
Corollary 7.1 implies
$p_{j}-\frac{1}{\left\lvert(Dw_{j})(p_{j})\right\rvert}=\theta(p_{j})=\theta(p_{k})=p_{k}-\frac{1}{\left\lvert(Dw_{k})(p_{j})\right\rvert}.$
Because the map
$\theta(p)=p-1/\left\lvert(Dw_{j})(p)\right\rvert=p-1/\left\lvert(Dw_{k})(p)\right\rvert$
is strictly increasing when $w$ has sub-quadratic second derivatives,
$p_{j}=p_{k}$. ∎
This proposition, as stated, must be restricted to constant unit costs. A
weaker result applies for non-constant unit costs:
###### Corollary 7.4.
Assume Assumption 5.1 holds, unit costs are strictly convex, 6.1 holds, $w$
has sub-quadratic second derivatives. Let $\mathbf{p}_{f}$ be profit-optimal,
and suppose that $c_{j}(P)=c_{k}(P)=c(P)$ for all $P\in[0,1]$ and
$(Dw_{j})(p)=(Dw_{k})(p)$ for all $p\in[0,\varsigma)$ for some
$j,k\in\mathcal{J}_{f}^{\circ}$, even if $\mathbf{y}_{j}\neq\mathbf{y}_{k}$.
Then $p_{j}>p_{k}$ if, and only if, $P_{j}(\mathbf{p})>P_{k}(\mathbf{p})$,
$p_{j}=p_{k}$ if, and only if, $P_{j}(\mathbf{p})=P_{k}(\mathbf{p})$, and
$p_{j}<p_{k}$ if, and only if, $P_{j}(\mathbf{p})<P_{k}(\mathbf{p})$.
This result may also be seen as slightly counterintuitive, as higher-priced
products are intuitively associated with lower choice probabilities.
###### Proof.
Corollary 7.1 implies
$\theta(p_{j})-\theta(p_{k})=c(P_{j}(\mathbf{p}))-c(P_{k}(\mathbf{p}))$
Because total costs are strictly convex, unit costs are strictly increasing in
$P$. Thus,
$\displaystyle\begin{matrix}p_{j}>p_{k}&\iff&\theta(p_{j})>\theta(p_{k})&\iff&c(P_{j}(\mathbf{p}))>c(P_{k}(\mathbf{p}))&\iff&P_{j}(\mathbf{p})>P_{k}(\mathbf{p})\\\
p_{j}=p_{k}&\iff&\theta(p_{j})=\theta(p_{k})&\iff&c(P_{j}(\mathbf{p}))=c(P_{k}(\mathbf{p}))&\iff&P_{j}(\mathbf{p})=P_{k}(\mathbf{p})\\\
p_{j}<p_{k}&\iff&\theta(p_{j})<\theta(p_{k})&\iff&c(P_{j}(\mathbf{p}))<c(P_{k}(\mathbf{p}))&\iff&P_{j}(\mathbf{p})<P_{k}(\mathbf{p})\end{matrix}.$
∎
Corollary 7.1 also implies the second counterintuitive property of locally
profit optimal markups $-$ that they decrease with costs $-$ under Logit with
any utility function that is both strictly concave in price and separable in
price and characteristics.
###### Corollary 7.5.
Suppose that $w$ is separable in price and characteristics and strictly
concave in price. Then firm $f$’s higher unit cost products (at optimality)
have lower locally profit optimal markups. That is, if
$j,k\in\mathcal{J}_{f}^{\circ}$ and
$c_{j}(P_{j}(\mathbf{p}))>c_{k}(P_{k}(\mathbf{p}))$, then
$p_{j}-c_{j}(P_{j}(\mathbf{p}))<p_{k}-c_{k}(P_{k}(\mathbf{p}))$.
###### Proof.
We prove that $p_{j}-c_{j}(P_{j}(\mathbf{p}))\geq
p_{k}-c_{k}(P_{k}(\mathbf{p}))$ implies $c_{j}(P_{j}(\mathbf{p}))\leq
c_{k}(P_{k}(\mathbf{p}))$. By Corollary 7.1,
$p_{j}-c_{j}(P_{k}(\mathbf{p}))\geq p_{k}-c_{k}(P_{k}(\mathbf{p}))$ implies
$(Dw)(p_{j})^{-1}\leq(Dw)(p_{k})^{-1}$ or, equivalently,
$(Dw)(p_{j})\geq(Dw)(p_{k})$. By strict concavity, this implies that
$p_{j}\leq p_{k}$. But then $p_{j}-c_{j}(P_{j}(\mathbf{p}))\geq
p_{k}-c_{k}(P_{k}(\mathbf{p}))$ implies that
$c_{j}(P_{j}(\mathbf{p}))-c_{k}(P_{k}(\mathbf{p}))\leq p_{j}-p_{k}\leq 0$
∎
When unit costs are constant, these propositions can be easily connected to
value. Intuition holds that both locally profit optimal markups and costs
should increase with value, if not costs. The following assumption makes this
connection explicit.
###### Assumption 7.1 (Value Costs Hypothesis).
More valued products cost more per unit to offer; that is,
$v(\mathbf{y})>v(\mathbf{y}^{\prime})$ implies that
$c(\mathbf{y})>c(\mathbf{y}^{\prime})$ for all
$\mathbf{y},\mathbf{y}^{\prime}\in\mathcal{Y}$.
Mussa & Rosen [35] include this as a basic feature of cost functions.
Bresnahan [14] has also remarked that this is a natural condition. When
considering equilibrium prices, this assumption need only be applied within
firms. That is, there may be firm-specific cost functions each independently
satisfying the value costs hypothesis, while the value costs hypothesis is
violated across firms. This states that two distinct firms can produce a
value-equivalent product at distinct unit costs without violating the results
that apply this hypothesis. With this definition, we provide the following
restatement of Corollary 7.5.
###### Corollary 7.6.
Suppose unit costs are constant, Assumption 6.1 holds, $w$ is separable in
price and characteristics, strictly concave in price, and that the value costs
hypothesis holds. Then firm $f$’s higher value products have lower locally
profit optimal markups. That is, if $j,k\in\mathcal{J}_{f}$ and $v_{j}>v_{k}$,
then $p_{j}-c_{j}<p_{k}-c_{k}$.
###### Proof.
$v_{j}>v_{k}$ implies $c_{j}>c_{k}$, and the result follows from Corollary
7.5. ∎
Markups can increase with value when $w$ is convex in price. Consider
$w(\mathbf{y},p)\equiv w(p)=-\alpha\log p$ and constant unit costs, for which
$(p_{j}-c_{j})-(p_{k}-c_{k})=\left(\frac{1}{\alpha}\right)\left(p_{j}-p_{k}\right).$
Thus, locally profit optimal markups increase with the corresponding prices.
This implies
$\displaystyle p_{j}-p_{k}$
$\displaystyle=\left(\frac{1}{\alpha-1}\right)(c_{j}-c_{k}).$
Hence if $\alpha>1$, locally profit optimal prices increase with costs, and
locally profit optimal markups increase with costs. While this is a more
intuitive outcome, it comes from a less intuitive utility specification.
Another assumption, the “unique value hypothesis,” further connects value with
profit-optimal prices. As defined by Nagle [36], the unique value hypothesis
postulates that as a product’s combination of characteristics becomes more
valued, individuals are less sensitive to price changes. This is transcribed
to our framework as follows.
###### Assumption 7.2 (Unique Value Hypothesis).
For any $\mathbf{y},\mathbf{y}^{\prime}\in\mathcal{Y}$,
$v(\mathbf{y})>v(\mathbf{y}^{\prime})$ implies
$\left\lvert(Dw)(\mathbf{y},p)\right\rvert\leq\left\lvert(Dw)(\mathbf{y}^{\prime},p)\right\rvert\quad\text{
for all }\quad p\in(0,\varsigma).$
This definition suggests an example of a non-separable but convex in price
utility for which markups increase with value. Consider
$w(\mathbf{y},p)=-\alpha(\mathbf{y})p$, where
$\alpha:\mathcal{Y}\to(0,\infty)$, and assume unit costs are constant. Then
$(p_{j}-c(\mathbf{y}_{j}))-(p_{k}-c(\mathbf{y}_{k}))=\left(\frac{1}{\alpha(\mathbf{y}_{j})}-\frac{1}{\alpha(\mathbf{y}_{k})}\right),$
and $(p_{j}-c(\mathbf{y}_{j}))\geq(p_{k}-c(\mathbf{y}_{k}))$ if and only if
$\alpha(\mathbf{y}_{j})\leq\alpha(\mathbf{y}_{k})$. The unique value
hypothesis mandates that $v(\mathbf{y}_{j})>v(\mathbf{y}_{k})$ implies
$\alpha(\mathbf{y}_{j})\leq\alpha(\mathbf{y}_{k})$, and hence markups do not
decrease with value if this hypothesis holds. Note that this is consistent
with our previous result for constant coefficient linear in price utility
where $\alpha(\mathbf{y})\equiv\alpha\in(0,\infty)$. Whenever
$\alpha(\mathbf{y}_{j})<\alpha(\mathbf{y}_{k})$, that is whenever the unique
value hypothesis holds in a non-trivial way, markups can strictly increase
with value.
A related and important question is whether higher value products have higher
locally profit optimal prices. By Corollary 7.4, this cannot hold without an
additional hypothesis.
###### Corollary 7.7.
Suppose unit costs are constant, Assumption 6.1 holds, $w$ has sub-quadratic
second derivatives, satisfies the unique value hypothesis, and the value costs
hypothesis holds. Then firm $f$’s higher value products have higher locally
profit optimal prices. That is, for any $j,k\in\mathcal{J}_{f}^{\circ}$,
$v_{j}>v_{k}$ implies that $p_{j}>p_{k}$ when $\mathbf{p}_{f}$ are profit-
maximizing.
###### Proof.
The unique value hypothesis implies that when
$v(\mathbf{y})>v(\mathbf{y}^{\prime})$,
$\theta(\mathbf{y},p)\leq\theta(\mathbf{y}^{\prime},p)$ for all
$p\in(0,\varsigma)$, where
$\theta(\mathbf{y},p)=p-\left\lvert(Dw)(\mathbf{y},p)\right\rvert^{-1}$.
Specifically, if $v(\mathbf{y}_{j})>v(\mathbf{y}_{k})$ then
$\theta_{j}(p_{k})\leq\theta_{k}(p_{k})$. Suppose that
$v(\mathbf{y}_{j})>v(\mathbf{y}_{k})$ while $p_{j}\leq p_{k}$. Because
$\theta_{j}(p)$ is a strictly increasing function of $p$, we have
$\theta_{j}(p_{j})\leq\theta_{j}(p_{k})\leq\theta_{k}(p_{k}).$
Thus Eqn. (19) implies that
$c_{j}-c_{k}=\theta_{j}(p_{j})-\theta_{k}(p_{k})\leq 0$, in contradiction to
the value costs hypothesis. ∎
Because any separable utility trivially satisfies the unique value hypotheses,
the following is a direct consequence of Corollary 7.7.
###### Corollary 7.8.
Suppose unit costs are constant, Assumption 6.1 holds, $w$ is separable in
price and characteristics, has sub-quadratic second derivatives, and that the
value costs hypothesis holds. Then firm $f$’s higher value products have
higher locally profit optimal prices. That is, $v_{j}>v_{k}$ implies that
$p_{j}>p_{k}$ for any $j,k\in\mathcal{J}_{f}^{\circ}$.
### 7.2. An Inter-Firm Property of Equilibrium Prices
Eqn. (19) is a special case of the following:
###### Corollary 7.9.
Suppose Assumptions 5.1 and 6.1 hold, and let $\mathbf{p}\in(0,\infty)^{J}$ be
equilibrium prices. For any $f,g\in\mathbb{N}(F)$,
$j\in\mathcal{J}_{f}^{\circ}$, and $k\in\mathcal{J}_{g}^{\circ}$,
(20)
$(p_{j}-c_{j}(P_{j}(\mathbf{p})))-(p_{k}-c_{k}(P_{k}(\mathbf{p})))=\left(\bar{\pi}_{f}(\mathbf{p})-\frac{1}{(Dw_{j})(p_{j})}\right)-\left(\bar{\pi}_{g}(\mathbf{p})-\frac{1}{(Dw_{k})(p_{k})}\right).$
This equation expresses the existence of a portfolio effect present in
equilibrium pricing with multi-product firms, constant unit costs, and even
the simplest Logit model.
###### Corollary 7.10.
Assume unit costs are constant, Assumption 6.1 holds, $w$ have sub-quadratic
second derivatives, and $\mathbf{p}\in(0,\infty)^{J}$ are equilibrium prices.
Suppose that $\mathbf{y}_{j}=\mathbf{y}_{k}$ and
$c_{f}(\mathbf{y}_{j})=c_{g}(\mathbf{y}_{k})$ for some
$j\in\mathcal{J}_{f}^{\circ}$ and $k\in\mathcal{J}_{g}^{\circ}$. Then
$p_{j}>p_{k}$ if, and only if,
$\hat{\pi}_{f}(\mathbf{p})>\hat{\pi}_{g}(\mathbf{p})$, $p_{j}<p_{k}$ if, and
only if, $\hat{\pi}_{f}(\mathbf{p})<\hat{\pi}_{g}(\mathbf{p})$, and
$p_{j}=p_{k}$ if, and only if,
$\hat{\pi}_{f}(\mathbf{p})=\hat{\pi}_{g}(\mathbf{p})$.
That is, equilibrium prices for the same product offered at the same cost but
by different firms are influenced by the profitability of other products in
these firms’ portfolios. Stated another way, if the other products offered by
a particular firm did not matter in determining equilibrium prices, then we
would expect $\mathbf{y}_{j}=\mathbf{y}_{k}$ and
$c_{f}(\mathbf{y}_{j})=c_{g}(\mathbf{y}_{k})$ for some $j\in\mathcal{J}_{f}$
and $k\in\mathcal{J}_{g}$ to imply that $p_{j}=p_{k}$.
###### Proof.
The proof follows by observing that Eqn. (20) can be written
$\theta_{j}(p_{j})-\theta_{k}(p_{k})=\hat{\pi}_{f}(\mathbf{p})-\hat{\pi}_{g}(\mathbf{p})$,
and $\theta_{j}=\theta_{k}$. The result follows. ∎
## 8\. Conclusions
This article has proved the existence of equilibrium prices for Bertrand
competition with multi-product firms under the Logit model without restrictive
assumptions on the firms or their products. Instead of studying a particular
utility function, general conditions on the utility function are identified
under which existence holds. The proofs circumvents fundamental obstacles to
the extension of existing equilibrium proofs for single-product firms by
applying the Poincare-Hopf theorem. One of the fixed-point equations
explicitly demonstrates that Logit price equilibrium problems are “single-
parameter problems” when unit costs are constant, even when firms offer many
products. By invoking the conventional assumption that utility is concave in
price and separable in price and characteristics along with the reasonable
assumption that more valued products always cost more to make per unit, a
counterintuitive result is obtained: the more the population values a
product’s characteristics, the lower its profit-optimal markup.
There are at least two important areas for future research. One is
establishing the uniqueness of equilibrium prices. Kellogg’s uniqueness
condition for Brouwer-Schauder fixed-point theorem [26], used in Section 5,
can be applied to show that equilibrium prices under linear-in-price utility
Logit models are unique, a result already known for both single-product [28,
15] and multi-product [45, 27] firms. Generalizing this analysis to nonlinear
utility functions and non-constant costs may be a promising direction. As
suggested in the introduction, another important area is the extension of this
analysis to non-Logit RUMs, especially those with heterogeneity. Formally the
$\boldsymbol{\eta}$ and $\boldsymbol{\zeta}$ characterizations presented in
this article extend to both any GEV and Mixed Logit models; see [31, 33, 32]
for the extension to Mixed-Logit models, subsequent analysis, and application
in large-scale computations of equilibrium prices. Establishing the existence
of simultaneously stationary prices using these characterizations is
straightforward, but not enough to ensure the existence of equilibrium [32].
## Appendix A Mathematical Notation
Sets. $\mathbb{N}$ denotes the natural numbers $\\{1,2,\dotsc\\}$, and
$\mathbb{N}(N)$ denotes the natural numbers up to $N$, that is,
$\mathbb{N}(N)=\\{1,\dotsc,N\\}$. $\mathbb{R}$ denotes the set of real numbers
$(-\infty,\infty)$, $[0,\infty)$ denotes the non-negative real numbers, and
$[0,\infty]$ denotes the extended non-negative half-line. We denote the
$(J-1)$-dimensional simplex
$\\{(x_{1},\dotsc,x_{N})\in[0,1]^{N}:\sum_{n=1}^{N}x_{n}=1\\}$ by
$\mathbb{S}(N)$, and the $J$-dimensional “pyramid”
$\\{(x_{1},\dotsc,x_{N})\in[0,1]^{N}:\sum_{n=1}^{N}x_{n}\leq 1\\}$ by
$\triangle(J)$. Hyper-rectangles in $\mathbb{R}^{N}$, i.e. sets of the form
$[a_{1},b_{1}]\times\dotsb\times[a_{N},b_{N}]$ for some
$a_{n},b_{n}\in\mathbb{R}$ with $a_{n}<b_{n}$ for all $n\in\mathbb{N}(N)$, are
denoted by $[\mathbf{a},\mathbf{b}]$ where $\mathbf{a}=(a_{1},\dotsc,a_{N})$
and $\mathbf{b}=(b_{1},\dotsc,b_{N})$. For other sets, we typically use
calligraphic upper case letters such as “$\mathcal{A}$”. For any set
$\mathcal{A}$, $\left\lvert\mathcal{A}\right\rvert$ denotes its cardinality.
For any $\mathcal{B}\subset\mathcal{A}$, $\mathcal{A}\setminus\mathcal{B}$
denotes the set $\\{b\in\mathcal{A}:b\notin\mathcal{B}\\}$. For any set
$\mathcal{A}$, $\mathfrak{F}(\mathcal{A})$ denotes the collection of finite
subsets of $\mathcal{A}$.
Symbols. Bold, un-italicized symbols (e.g., “$\mathbf{x}$”) denote vectors and
matrices; typically we reserve lower case letters to refer to vectors and use
upper case letters to refer to matrices; the vector of choice probabilities
“$\mathbf{P}$” is an exception made to conform with existing notation of these
quantities. Throughout we use $\mathbf{1}$ to denote a vector of ones of the
appropriate size for the context in which it appears. $\mathbf{I}$ always
denotes the identity matrix of a size appropriate for the context. For any
$\mathbf{x}\in\mathbb{R}^{N}$, $\mathrm{diag}(\mathbf{x})$ denotes the
$N\times N$ diagonal matrix whose diagonal is $\mathbf{x}$. Any vector
inequalities between vectors are to be taken componentwise: for example,
$\mathbf{x}<\mathbf{y}$ means $x_{n}<y_{n}$ for all $n$. Random variables are
denoted with capital letters “$X$”, with random vectors being denoted with
bold capital letters (e.g., “$\mathbf{Q}$”). While this overlaps with our
notation for matrices, it should not cause any confusion. $\mathbb{P}$ denotes
a probability and $\mathbb{E}$ denotes an expectation. “$\log$” always denotes
the natural (base $e$) logarithm. “$\mathrm{ess}\sup$” denotes the essential
supremum of a measurable function, where the measure on measurable subsets of
the domain should always be clear.
Differentiation. Our conventions for denoting differentiation follow [34]. We
use the symbol “$D$” to denote differentiation using subscripts to invoke
additional specificity. Letting $\mathbf{f}:\mathbb{R}^{M}\to\mathbb{R}^{N}$,
$(D_{m}f_{n})(\mathbf{x})$ denotes the derivative of the $n^{\text{th}}$
component function with respect to the $m^{\text{th}}$ variable and
$(D\mathbf{f})(\mathbf{x})$ is the $N\times M$ derivative matrix of
$\mathbf{f}$ at $\mathbf{x}$ with components
$((D\mathbf{f})(\mathbf{x}))_{n,m}=(D_{m}f_{n})(\mathbf{x})$. Thus for
$f:\mathbb{R}^{M}\to\mathbb{R}$, $(Df)(\mathbf{x})$ is a row vector. If
$f:\mathbb{R}^{M}\to\mathbb{R}$, we define the gradient $(\nabla
f)(\mathbf{x})\in\mathbb{R}^{M}$ as the transposed derivative: $(\nabla
f)(\mathbf{x})=(Df)(\mathbf{x})^{\top}$.
Other Definitions. Let $\mathcal{X}$ be any topological space and let
$f:\mathcal{X}\to\mathbb{R}$. We say $\mathbf{x}^{*}\in\mathcal{X}$ is a local
maximizer (over $\mathcal{X}$) of $f$ if there exists a neighborhood of
$\mathbf{x}^{*}$, say $\mathcal{U}$, such that $f(\mathbf{x}^{*})\geq
f(\mathbf{x})$ for all $\mathbf{x}\in\mathcal{U}$. We say
$\mathbf{x}^{*}\in\mathcal{X}$ is a maximizer (over $\mathcal{X}$) of $f$ if
$f(\mathbf{x}^{*})\geq f(\mathbf{x})$ for all $\mathbf{x}\in\mathcal{X}$.
## Appendix B Examples for the Logit Model
We first provide some examples of indirect utilities to illustrate properties
(a-c). A linear in price utility, given by
$w(\mathbf{y},p)=-\alpha(\mathbf{y})p$ for some
$\alpha:\mathcal{Y}\to(0,\infty)$, satisfies (a-c). More generally, any “Cobb-
Douglas” in price utility, given by
$w(\mathbf{y},p)=-\alpha(\mathbf{y})p^{\beta(\mathbf{y})}$ with
$\alpha,\beta:\mathcal{Y}\to(0,\infty)$, satisfies (a-c). A “Cobb-Douglas”
specification for “remaining income,”
$w(\mathbf{y},p)=\alpha(\mathbf{y})(\varsigma-p)^{\beta(\mathbf{y})}$ is a bit
more complicated, being a function finite for all finite prices and satisfying
(a-c) only for $\beta:\mathcal{Y}\to(2\mathbb{N}+1)$, where $2\mathbb{N}+1$
denotes the set of odd positive integers: if
$\beta(\mathbf{y}):\mathcal{Y}\to(-\infty,0)$ then $w$ is not finite for all
finite $p$; clearly $w$ violates (a) if $\beta(\mathbf{y})=0$; if
$\beta(\mathbf{y})>0$ is not an integer, then $w$ is complex for
$p>\varsigma$; finally, if $\beta(\mathbf{y})\in\mathbb{N}$ is not an odd
positive integer then $w$ violates (a). The common “log-transformed” Cobb-
Douglas in “remaining income” utility
$w(\mathbf{y},p)=\alpha(\mathbf{y})\log(\varsigma-p)$ for
$p<\varsigma<\infty$, $\alpha:\mathcal{Y}\to(0,\infty)$,999This log
transformation usually occurs (see [10], [44]) based on the observation that
choices are invariant over increasing utility transformations, so that
$u^{\prime}(\mathbf{y},p)=e^{w(\mathbf{y},p)}e^{v(\mathbf{y})}$ yields the
same random choices as the specification introduced in the text, with the
caveat that the additive errors introduced in the text are taken as
multiplicative errors (with a related distribution) in the former
specification. In a Cobb-Douglas specification for the former,
$u^{\prime}(\mathbf{y},p)\propto(\varsigma-p)^{\alpha(\mathbf{y})}=e^{\alpha(\mathbf{y})\log(\varsigma-p)}$,
illustrating that the logarithm of this utility has the log-transformed
specification for the price component. is not finite for all finite prices.
Allenby & Rossi’s negative log of price utility, given by
$w(\mathbf{y},p)=-\alpha(\mathbf{y})\log p$ for
$\alpha:\mathcal{Y}\to(0,\infty)$ satisfies (a-c) [2]. Finally, the utility
$w(p)=-\alpha(\log p-\varepsilon\sin\log p)$, where $\alpha>1$ and
$\varepsilon\in(0,1)$, satisfies (a-c).
We now demonstrate which of these utility functions is eventually log bounded
and/or eventually decreases sufficiently quickly. Any linear in price or Cobb-
Douglas in price utility is both eventually log bounded and eventually
decreases sufficiently quickly. For if $\beta(\mathbf{y})\geq 1$,
$(Dw)(\mathbf{y},p)=-\alpha(\mathbf{y})\beta(\mathbf{y})p^{\beta(\mathbf{y})-1}\downarrow-\infty$
as $p\to\infty$. If $\beta(\mathbf{y})<1$, then although
$(Dw)(\mathbf{y},p)=-\alpha(\mathbf{y})\beta(\mathbf{y})p^{-(1-\beta(\mathbf{y}))}\uparrow
0$ as $p\to\infty$,
$(Dw)(\mathbf{y},p)-\frac{r}{p}=-\alpha(\mathbf{y})\beta(\mathbf{y})\frac{1}{p^{1-\beta(\mathbf{y})}}+\frac{r}{p}=\left(\frac{1}{p}\right)\left[r-\alpha(\mathbf{y})\beta(\mathbf{y})p^{\beta(\mathbf{y})}\right]\leq
0$
if $p\geq\sqrt[\beta(\mathbf{y})]{\alpha(\mathbf{y})\beta(\mathbf{y})/r}$ and
hence $w(\mathbf{y},p)$ eventually decreases sufficiently quickly for any $r$.
The class of negative log of price utility functions contain the most obvious
examples of utilities that are neither eventually log bounded nor eventually
decrease sufficiently quickly; particularly
$w(\mathbf{y},p)\leq-\alpha(\mathbf{y})\log p$ with $\alpha(\mathbf{y})\leq
1$. If $\alpha(\mathbf{y})<1$ there are no finite profit maximizing prices
under this utility.
In the text we defined utilities with sub-quadratic second derivatives. Any
linear in price utility has sub-quadratic second derivatives, since
$(D^{2}w)(\mathbf{y},p)\equiv 0$. More generally, under any Cobb-Douglas in
price utility
$\frac{(D^{2}w)(\mathbf{y},p)}{(Dw)(\mathbf{y},p)^{2}}=-\left(\frac{1}{\alpha(\mathbf{y})}\right)\left(\frac{\beta(\mathbf{y})-1}{\beta(\mathbf{y})}\right)\left(\frac{1}{p^{\beta(\mathbf{y})}}\right)\\\
=\left(\frac{\beta(\mathbf{y})-1}{\beta(\mathbf{y})}\right)\left(\frac{1}{w(\mathbf{y},p)}\right),$
and hence $w$ has sub-quadratic second derivatives if $\beta(\mathbf{y})\geq
1$. If $\beta(\mathbf{y})<1$, then $w$ has sub-quadratic second derivatives
only at $(\mathbf{y},p)$ such that $\left\lvert
w(\mathbf{y},p)\right\rvert>(1-\beta(\mathbf{y}))/\beta(\mathbf{y})$, i.e.
$p>\sqrt[\beta(\mathbf{y})]{(1-\beta(\mathbf{y}))/(\alpha(\mathbf{y})\beta(\mathbf{y}))}$.
Finally, if $w(\mathbf{y},p)=-\alpha(\mathbf{y})\log p$ then
$(D^{2}w)(\mathbf{y},p)/(Dw)(\mathbf{y},p)^{2}\equiv 1/\alpha(\mathbf{y})$ and
hence $w$ has sub-quadratic second derivatives if $\alpha(\mathbf{y})>1$.
Hence far from requiring concavity, some convex utility functions have sub-
quadratic second derivatives.
Let $\alpha(\mathbf{y})\equiv\alpha>0$. For the linear-in-price utility,
$\zeta_{j}(\mathbf{p})=\hat{\pi}_{f}(\mathbf{p})+1/\alpha$ with the fixed-
point equation being $p_{j}=c_{j}+\hat{\pi}_{f}(\mathbf{p})+1/\alpha$. For any
Cobb-Douglas in price utility,
$\zeta_{j}(\mathbf{p})=\hat{\pi}_{f}(\mathbf{p})+(1/(\alpha\beta))p_{j}^{1-\beta}$
with the fixed-point equation being
$p_{j}=c_{j}+\hat{\pi}_{f}(\mathbf{p})+(1/(\alpha\beta))p_{j}^{1-\beta}$. For
negative log of price,
$\zeta_{j}(\mathbf{p})=\hat{\pi}_{f}(\mathbf{p})+(1/\alpha)p_{j}$ with the
fixed-point equation being
$p_{j}=c_{j}+\hat{\pi}_{f}(\mathbf{p})+(1/\alpha)p_{j}$.
Our proof that the negative log of price utility has no finite profit
maximizing prices can be strengthened using the relationship between
$\boldsymbol{\zeta}$ and the profit gradients. We already know that
$w(\mathbf{y},p)/1=-(\alpha(\mathbf{y})/1)\log p$ does not eventually decrease
sufficiently quickly when $\alpha(\mathbf{y})\leq 1$. We have also observed
that $\zeta_{j}(\mathbf{p})=\hat{\pi}_{f}(\mathbf{p})+(1/\alpha_{j})p_{j}$,
which implies that
$\zeta_{j}(\mathbf{p})-(p_{j}-c_{j})=(\hat{\pi}_{f}(\mathbf{p})+c_{j})+\left(\frac{1}{\alpha_{j}}-1\right)p_{j}=(\hat{\pi}_{f}(\mathbf{p})+c_{j})+\left(\frac{1-\alpha_{j}}{\alpha_{j}}\right)p_{j}.$
Thus, if $\alpha_{j}=\alpha(\mathbf{y}_{j})\leq 1$, the $j^{\text{th}}$ price
derivative of profit is always positive. While we have already shown that only
infinite prices maximize profits under this utility when $\alpha_{j}<1$, this
shows the same holds for $\alpha_{j}=1$ as well even though the corresponding
maximal profits are finite.
We now present an example of a utility function for which has finite profit-
maximizing prices but for which a “local” criterion restricting profit
maximization at infinity fails. This local criterion is simply that profits
decrease for all sufficiently large prices. Let $w(p)=-\alpha(\log
p-\varepsilon\sin\log p)$ with $\alpha>1$ and
$\varepsilon\in[1-\alpha^{-1},1)$. Then $p_{j}-c_{j}-\zeta_{j}(\mathbf{p})\geq
0$ if and only if
(21) $p_{j}\left(1-\frac{1}{\alpha(1-\varepsilon\cos\log p_{j})}\right)\geq
c_{j}+\hat{\pi}_{f}(\mathbf{p}).$
But based on our choice of $\varepsilon$, there exist arbitrarily large
$p_{j}$ such that the left hand side above is non-positive: For all $\bar{p}$
there exists some $p_{j}>\bar{p}$ such that $\alpha(1-\varepsilon\cos\log
p_{j})=\alpha(1-\varepsilon)\leq 1$, which implies the claim. Since
$c_{j}+\hat{\pi}_{f}(\mathbf{p})$ is positive (or rather is for all
$\mathbf{p}$ that matter), the inequality (21) is violated and there exist
arbitrarily large $p_{j}$ such that profits increase, locally, with $p_{j}$,
despite the fact that profits must vanish as
$\mathbf{p}_{f}\to\boldsymbol{\infty}$ since this utility is eventually log
bounded. That is, the local criterion for finite profit maximizing prices is
violated.
## Appendix C Inapplicability of Supermodularity
This appendix states a generalization of Sandor’s [45] result that profits are
neither supermodular nor log-supermodular arbitrarily close to equilibrium
prices under Logit with linear in price utility [45, Chapter 4]. Such a result
rules out the applicability of the approach developed by Milgrom & Roberts
[28] to proving existence of equilibrium prices in the multi-product firm
setting by implying that there cannot exist a compact set with non-empty
interior containing any equilibrium on which Logit profits are supermodular or
log-supermodular.
###### Lemma C.1.
Let $\vartheta>-\infty$, unit costs be constant, and Assumption 3.1 hold with
a $w$ with sub-quadratic second derivatives. Suppose
$\mathbf{p}_{f}^{*}\in(0,\infty)^{J_{f}}$ maximizes
$\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$. Then for any $\varepsilon>0$, there
exists a $\mathbf{p}_{f}$ such that
$\lvert\lvert\mathbf{p}_{f}-\mathbf{p}_{f}^{*}\rvert\rvert<\varepsilon$, and
$(D_{l}D_{k}\hat{\pi}_{f})(\mathbf{p})<0$, and
$(D_{l}D_{k}\log\hat{\pi}_{f})(\mathbf{p})<0$ for all $k,l\in\mathcal{J}_{f}$,
$k\neq l$, where $\mathbf{p}=(\mathbf{p}_{f},\mathbf{p}_{-f})$.
Naturally, because supermodularity has been used to prove the existence of
equilibrium prices under Logit for single-product firms, the proof relies on
the fact that firms produce more than one product.
###### Proof.
It can be shown that when $k,l\in\mathcal{J}_{f}$ and $k\neq l$, the second
derivatives of profits are given by
$\displaystyle(D_{l}D_{k}\hat{\pi}_{f})(\mathbf{q})$
$\displaystyle\quad\quad=\left\lvert(Dw_{k})(q_{k})\right\rvert
P_{k}^{L}(\mathbf{q})\big{(}\hat{\pi}_{f}(\mathbf{q})-(q_{k}-c_{k})-(Dw_{k})(q_{k})^{-1}\big{)}P_{l}^{L}(\mathbf{q})\left\lvert(Dw_{l})(q_{l})\right\rvert$
$\displaystyle\quad\quad\quad\quad+\left\lvert(Dw_{k})(q_{k})\right\rvert
P_{k}^{L}(\mathbf{q})\big{(}\hat{\pi}_{f}(\mathbf{q})-(q_{l}-c_{l})-(Dw_{l})(q_{l})^{-1}\big{)}P_{l}^{L}(\mathbf{q})\left\lvert(Dw_{l})(q_{l})\right\rvert$
for any $\mathbf{q}$. The goal is to choose $\mathbf{q}$,
$\lvert\lvert\mathbf{q}-\mathbf{p}\rvert\rvert<\varepsilon$, so that
$\hat{\pi}_{f}(\mathbf{q})-(q_{k}-c_{k})-(Dw_{k})(q_{k})^{-1}<0$ and
$\hat{\pi}_{f}(\mathbf{q})-(q_{l}-c_{l})-(Dw_{l})(q_{l})^{-1}<0$ for any
$k,l\in\mathcal{J}_{f}$, $k\neq l$.
By the $\boldsymbol{\zeta}$ fixed-point characterization,
$\displaystyle\hat{\pi}_{f}(\mathbf{p}_{f},\mathbf{p}_{-f})-(p_{k}-c_{k})-(Dw_{k})(p_{k})^{-1}$
$\displaystyle\quad\quad\quad\quad<\hat{\pi}_{f}(\mathbf{p}_{f}^{*},\mathbf{p}_{-f})-(p_{k}-c_{k})-(Dw_{k})(p_{k})^{-1}$
$\displaystyle\quad\quad\quad\quad=\hat{\pi}_{f}(\mathbf{p}_{f}^{*},\mathbf{p}_{-f})-(p_{k}^{*}-c_{k})-(p_{k}-p_{k}^{*})-(Dw_{k})(p_{k})^{-1}$
$\displaystyle\quad\quad\quad\quad=(Dw_{k})(p_{k}^{*})^{-1}-(Dw_{k})(p_{k})^{-1}-(p_{k}-p_{k}^{*}).$
Thus,
$\hat{\pi}_{f}(\mathbf{p}_{f},\mathbf{p}_{-f})-(p_{k}-c_{k})-(Dw_{k})(p_{k})^{-1}<0$
if $\theta_{k}(p_{k})\leq\theta_{k}(p_{k}^{*})$. Because $w$ has sub-quadratic
second derivatives, $\theta_{k}$ is strictly increasing and any
$p_{k}<p_{k}^{*}-\varepsilon$ will do. The same logic goes for
$l\in\mathcal{J}_{f}$, and the claim follows.
For the second claim, note that
$\displaystyle(D_{l}D_{k}\log\hat{\pi}_{f})(\mathbf{p})$
$\displaystyle=\frac{(D_{l}D_{k}\hat{\pi}_{f})(\mathbf{p})\hat{\pi}_{f}(\mathbf{p})-(D_{k}\hat{\pi}_{f})(\mathbf{p})(D_{l}\hat{\pi}_{f})(\mathbf{p})}{\hat{\pi}_{f}(\mathbf{p})^{2}}.$
We have already established that the first term in the numerator is negative
at $\mathbf{p}$ as defined above. Furthermore,
$(D_{k}\hat{\pi}_{f})(\mathbf{p})=\left\lvert(Dw_{k})(p_{k})\right\rvert
P_{k}^{L}(\mathbf{p})(\hat{\pi}_{f}(\mathbf{p})-(p_{k}-c_{k})-(Dw_{k})(p_{k})^{-1})<0$
by the same argument and hence
$(D_{k}\hat{\pi}_{f})(\mathbf{p})(D_{l}\hat{\pi}_{f})(\mathbf{p})>0$, making
the second term in the numerator also negative. This completes the proof. ∎
## References
* [1] Victor Aguirregabiria and Gustavo Vicentini, _Dynamic spatial competition between multi-store firms_ , Working Paper, University of Toronto, August 2006\.
* [2] Greg M Allenby and Peter E Rossi, _Quality perceptions and asymmetric switching between brands_ , Marketing Science 10 (1991), no. 3, 185–204.
* [3] Simon P. Anderson and Andre de Palma, _The logit as a model of product differentiation_ , Oxford Economic Papers 44 (1992), no. 1, 51–67.
* [4] Simon P Anderson and Andre de Palma, _Multiproduct firms: A nested logit approach_ , The Journal of Industrial Economics 40 (1992), no. 3, 271–276.
* [5] Simon P. Anderson and Andre de Palma, _Product diversity in asymmetric oligopoly: Is the quality of consumer goods too low?_ , The Journal of Industrial Economics 49 (2001), no. 2, 113–135.
* [6] by same author, _Market performance of multi-product firms_ , The Journal of Industrial Economics 54 (2006), no. 1, 95–124.
* [7] Simon P. Anderson, Andre de Palma, and Yurii Nesterov, _Oligopolistic competition and the optimal provision of products_ , Econometrica 63 (1995), no. 6, 1281–1301.
* [8] Michael R. Baye and Dan Kovenock, _Bertrand competition_ , The New Pagrave Dictionary of Economics, Second Edition (Steven N. Durlaf and Lawrence E. Blume, eds.), Palgrave Macmillian, 2008.
* [9] Arie Beresteanu and Shanjun Li, _Gasoline prices, government support, and the demand for hybrid vehicles in the u.s._ , Working Paper, Duke University, 2008\.
* [10] Steven Berry, James Levinsohn, and Ariel Pakes, _Automobile prices in market equilibrium_ , Econometrica 63 (1995), no. 4, 841–890.
* [11] by same author, _Differentiated products demand systems from a combination of micro and macro data: The new car market_ , Journal of Political Economy 112 (2004), no. 1, 68–105.
* [12] David Besanko, Sachin Gupta, and Dipak Jain, _Logit demand estimation under competitive pricing behavior: An equilibrium framework_ , Management Science 44 (1998), no. 11, 1533–1547.
* [13] Michel Bierlaire, Denis Bolduc, and Daniel McFadden, _Characteristics of generalized extreme value distributions_ , Working Paper, University of California Berkeley, March 2003.
* [14] Timothy Bresnahan, _Competition and collusion in the american automobile industry: The 1955 price war_ , The Journal of Industrial Economics 35 (1987), no. 4, 457–482.
* [15] Andrew Caplin and Barry Nalebuff, _Aggregation and imperfect competition: On the existence of equilibrium_ , Econometrica 59 (1991), no. 1, 25–59.
* [16] Ulrich Doraszelski and Michaela Draganska, _Market segmentation strategies of multiproduct firms_ , Journal of Industrial Economics 54 (2006), no. 1, 125–149.
* [17] William Feller, _Introduction to probability and its applications, volume i_ , Wiley, 1968.
* [18] Guillermo Gallego, Woonghee Tim Huh, Wanmo Kang, and Robert Phillips, _Price Competition with the Attraction Demand Model_ , Manufacturing and Service Operations Management 8 (2006), no. 4, 359–375.
* [19] Pinelopi K. Goldberg, _Product differentiation and oligopoly in international markets: The case of the u.s. automobile industry_ , Econometrica 63 (1995), no. 4, 891–951.
* [20] by same author, _The effects of the corporate average fuel efficiency standards in the us_ , The Journal of Industrial Economics 46 (1998), no. 1, 1–33.
* [21] Gautam Gowrisankaran and Robert J. Town, _Dynamic equilibrium in the hospital industry_ , Journal of Economics and Management Strategy 6 (1997), no. 1, 45–74.
* [22] Ward Hanson and Kipp Martin, _Optimizing multinomial logit profit functions_ , Management Science 42 (1996), no. 7, 992–1003.
* [23] Patrick T Harker and Jong-Shi Pang, _Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms, and applications_ , Mathematical Programming 48 (1990), 161–220.
* [24] Jerry A. Hausman and Gregory K. Leonard, _The competitive effects of a new product introduction_ , The Journal of Industrial Economics 50 (2002), no. 3, 237–263.
* [25] Shizuo Kakutani, _A generalization of brouwer’s fixed point theorem_ , Duke Mathematics Journal 8 (1941), no. 3, 457–459.
* [26] R. B. Kellogg, _Uniqueness in the schauder fixed point theorem_ , Proceedings of the American Mathematical Society 60 (1976), no. 1, 207–210.
* [27] Alexander Konovalov and Zsolt Sandor, _On price equilibrium with multi-product firms_ , Economic Theory 44 (2010), 271–292.
* [28] Paul Milgrom and John Roberts, _Rationalizability, learning, and equilibrium in games with strategic complementarities_ , Econometrica 58 (1990), no. 6, 1255–1277.
* [29] John W. Milnor, _Topology from the differentiable viewpoint_ , Princeton Landmarks in Mathematics, Princeton University Press, 1965.
* [30] Toshihide Mizuno, _On the existence of a unique price equilibrium for models of product differentiation_ , International Journal of Industrial Organization 21 (2003), 761–793.
* [31] W. Ross Morrow and Steven J. Skerlos, _Fixed-point approaches for computing bertrand-nash equilibrium prices under mixed logit demand_ , Manuscript, submitted to operations research, University of Michigan, 2008\.
* [32] by same author, _Fixed-Point Approaches to Computing Bertrand-Nash Equilibrium Prices Under Mixed-Logit Demand: A Technical Framework for Analysis and Efficient Computational Methods_ , Tech. report, Iowa State University, 2010.
* [33] by same author, _Fixed-Point Approaches to Computing Bertrand-Nash Equilibrium Prices Under Mixed-Logit Demand_ , Operations Research 59 (2011), no. 2, 328–345.
* [34] James R. Munkres, _Analysis on manifolds_ , Westview Press, 1991.
* [35] Micheal Mussa and Sherwin Rosen, _Monopoly and product quality_ , Journal of Economic Theory 18 (1978), 301–317.
* [36] Thomas Nagle and Reed Holden, _The strategy and tactics of pricing_ , Prentice Hall, 1987.
* [37] Aviv Nevo, _Mergers with differentiated products: The case of the ready-to-eat cereal industry_ , The RAND Journal of Economics 31 (2000), no. 3, 395–421.
* [38] by same author, _Measuring market power in the ready-to-eat cereal industry_ , Econometrica 69 (2001), no. 2, 307–342.
* [39] J. M. Ortega and W. C. Rheinboldt, _Iterative solution of nonlinear equations in several variables_ , Society for Industrial and Applied Mathematics, 1970.
* [40] Ariel Pakes and Paul McGuire, _Computing markov-perfect nash equilibria: Numerical implications of a dynamic differentiated product model_ , RAND Journal of Economics 25 (1994), no. 4, 555–589.
* [41] Jeffrey M. Perloff, Larry S. Karp, and Amos Golan, _Estimating market power and strategies_ , Cambridge University Press, 2007.
* [42] Jeffrey M. Perloff and Steven C. Salop, _Equilibrium with product differentiation_ , The Review of Economic Studies 52 (1985), no. 1, 107–120.
* [43] Amil Petrin, _Quantifying the benefits of new products: The case of the minivan_ , Journal of Political Economy 110 (2002), no. 4, 705–729.
* [44] Peter Rossi, Greg M Allenby, and Robert McCulloch, _Bayesian statistics and marketing_ , Wiley, 2006.
* [45] Zsolt Sandor, _Computation, efficiency and endogeneity in discrete choice models_ , Ph.D. thesis, University if Groningen, 2001.
* [46] Avner Shaked and John Sutton, _Multiproduct firms and market structure_ , The RAND Journal of Economics 21 (1990), no. 1, 45–62.
* [47] Alp Simsek, Asuman Ozdaglar, and Daron Acemoglu, _Generalized poincare-hopf theorem for compact nonsmooth regions_ , Mathematics of Operations Research 32 (2007), no. 1, 193–214.
* [48] Howard Smith, _Supermarket choice and supermarket competition in market equilibrium_ , The Review of Economic Studies 71 (2004), 235–263.
* [49] K Sudhir, _Competitive pricing behavior in the auto market: A structural analysis_ , Marketing Science 20 (2001), no. 1, 42–60.
* [50] Kenneth Train, _Discrete choice methods with simulation_ , Cambridge University Press, 2003.
* [51] Frank Verboven, _Product line rivalry and market segmentation - with an application to automobile optional engine pricing_ , Journal if Industrial Economics 47 (1999), no. 4, 399–425.
* [52] J Miguel Villas-Boas and Ying Zhao, _Retailers, manufacturers, and individual consumers: Modeling the supply side in the ketchup marketplace_ , Journal of Marketing Research 41 (2005), no. 1, 83–95.
|
arxiv-papers
| 2010-12-28T20:22:42 |
2024-09-04T02:49:16.003021
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "W. Ross Morrow and Steven J. Skerlos",
"submitter": "William Morrow",
"url": "https://arxiv.org/abs/1012.5832"
}
|
1012.5836
|
# Fixed-Point Approaches to Computing Bertrand-Nash Equilibrium Prices Under
Mixed-Logit Demand: A Technical Framework for Analysis and Efficient
Computational Methods.
W. Ross Morrow Departments of Mechanical Engineering and Economics, Iowa
State University, Ames IA 50011 and Steven J. Skerlos Department of
Mechanical Engineering, University of Michigan, Ann Arbor MI 48104
###### Contents
1. 1 Introduction
2. 2 A Technical Framework
1. 2.1 Mathematical Notation
1. 2.1.1 Sets.
2. 2.1.2 Symbols.
3. 2.1.3 Differentiation.
2. 2.2 Consumers, Products, and Choice Probabilities
3. 2.3 Utility Specification
4. 2.4 Profits
1. 2.4.1 Quantity-Dependent Costs
2. 2.4.2 Bounded and Vanishing Profits
5. 2.5 Local Equilibrium and the Simultaneous Stationarity Conditions
6. 2.6 Choice Probability Derivatives
7. 2.7 The BLP-Markup Equation
8. 2.8 The $\zeta$-Markup function
9. 2.9 Existence of Simultaneously Stationary Prices
3. 3 Computational Methods
1. 3.1 Newton’s Method
2. 3.2 Newton’s Method on the Combined Gradient
3. 3.3 Newton’s Method and the Markup Equations
4. 3.4 Fixed-Point Iteration
1. 3.4.1 $\zeta$ Fixed-Point Iteration
2. 3.4.2 $\eta$ Fixed-Point Iteration
5. 3.5 Practical Considerations
1. 3.5.1 Simulation
2. 3.5.2 Truncation of Low Purchase Probability Products
3. 3.5.3 Termination Conditions
4. 3.5.4 Second-Order Conditions.
5. 3.5.5 Computational Burden
6. 3.6 Computing Jacobian Matrices for Newton’s Method
1. 3.6.1 Jacobian of the Combined Gradient
2. 3.6.2 The $\boldsymbol{\eta}$ map.
3. 3.6.3 The $\boldsymbol{\zeta}$ map.
4. 4 The GMRES-Newton Hookstep Method
1. 4.1 Inexact Newton Methods
2. 4.2 GMRES
1. 4.2.1 Householder GMRES
2. 4.2.2 Preconditioning
3. 4.3 The GMRES Hookstep
1. 4.3.1 Model Trust Region Methods.
2. 4.3.2 Model Trust Region Methods on a Subspace
3. 4.3.3 The GMRES-Newton Hookstep
4. 4.3.4 Directional Finite Differences
5. 5 Other Methods
1. 5.1 Variational Methods
1. 5.1.1 The VI formulation is poorly posed
2. 5.1.2 General Results.
3. 5.1.3 The Resolution of Equilibria with $\boldsymbol{\zeta}$
2. 5.2 Tatonnement
3. 5.3 Least-Squares Minimization and the Gauss-Newton Method
6. 6 Acknowledgements
## 1\. Introduction
Bertrand competiton has been a prominent paradigm for the empirical study of
differentiated product markets for at least twenty years. Firms engaged in
Bertrand competition maximize profits by choosing prices for portfolios of
differentiated products, and Bertrand-Nash equilibrium prices simultaneously
maximize profits for all firms. Models combining Bertrand competition with the
Mixed Logit discrete choice model of consumer demand have been used to study
the automotive industry, electronics, entertainment, and food products and
services; see Dube et al. (2002).
Many applications of Bertrand competition rely on counterfactual experiments:
exercises in which hypothetical market conditions are simulated with an
estimated model. Such experiments have been used to study corporate mergers
(Nevo, 2000a), novel products and services (Petrin, 2002; Goolsbee and Petrin,
2004; Beresteanu and Li, 2008), store locations (Thomadsen, 2005), and
regulatory policy changes (Goldberg, 1995, 1998; Beresteanu and Li, 2008). By
definition, simulating market outcomes in counterfactual experiments requires
computing equilibrium prices after changing the values of exogenous variables
such as the number of firms or the products offered. Numerical methods for
computing equilibrium prices have not yet received a thorough treatment in the
literature, which currently focuses on model specification and estimation; see
Knittel and Metaxoglou (2008); Dube et al. (2008); Su and Judd (2008) for
recent developments in estimation. Morrow and Skerlos (2010) fills this gap
with a detailed investigation of four approaches for computing Bertrand-Nash
equilibrium prices in single-period, multi-firm models with Mixed Logit
demand. This working paper provides most of the technical background for that
investigation.
Applying Newton’s method to some form of the first-order or “simultaneous
stationarity” condition is currently the de facto approach for computing
equilibrium prices; see, for example, Nevo (1997, 2000a); Petrin (2002); Smith
(2004); Doraszelski and Draganska (2006); Jacobsen (2006). Newton’s method
applied directly to the first-order condition may converge when started at
observed prices if changes in exogenous variables have a marginal impact on
equilibrium prices. However, when the changes to exogenous variables imply
significant changes in product prices Newton’s method applied directly to the
first-order conditions may fail to compute equilibrium prices. Furthermore
analyses that do not have observed prices to use as an initial guess will
require methods with greater reliability.
Morrow and Skerlos (2010) demonstrate that solving fixed-point equations
equivalent to the first-order condition for equilibrium is more reliable and
efficient than solving the first-order condition itself. One fixed-point
equation equivalent to the first-order conditions is the BLP-markup equation
popularized by Berry et al. (1995). A second fixed-point equation, here termed
the $\boldsymbol{\zeta}$-markup equation, is a novel way to write the same
condition on markups. Both markup equations lead to more robust numerical
methods than found with a simple application of Newton’s method to the first-
order condition. Using the fixed-point expressions in this way can be
considered “nonlinearly” or “analytically” pre-conditioning the first-order
condition satisfied by equilibrium prices, a technique well-known in applied
mathematics (Brown and Saad, 1990; Cai and Keyes, 2002).
The existence of fixed-point equations for equilibrium suggests applying
fixed-point iteration (Judd, 1998) to compute equilibrium prices, instead of
Newton’s method. The BLP-markup equation does not appear to be well-suited to
fixed-point iteration. Example 7 in Morrow and Skerlos (2010) provides a case
in which iterating on the BLP-markup equation is not necessarily locally
convergent, while iterating on the $\boldsymbol{\zeta}$-markup equation is
superlinearly locally convergent. Iterating on the $\boldsymbol{\zeta}$-markup
equation also eliminates the need to solve linear systems, required to
implement Newton’s method and to iterate on the BLP-markup equation. This
property makes fixed-point steps based on the $\zeta$-markup equation very
inexpensive relative to Newton steps, an essential property to obtaining fast
computations from generally linearly convergent fixed-point iterations.
Besides Newton’s method and fixed-point iteration, few other practical
approaches to the computation of equilibrium prices exist. Variational
formulations, widely applied in economic and engineering problems (Ferris and
Pang, 1997), contain many solutions that need not be equilibria of the
original problem. Explicit least-square minimization or Gauss-Newton methods
can also be implemented, but are computational disadvantages relative to
applications of standard Newton-type methods for nonlinear systems. Some
authors apply tattonement $-$ iterating on a game’s best response
correspondence $-$ to compute equilibrium in prices or other strategic
variables including product mix (Choi et al., 1990), product characteristics
(CBO, 2003; Austin and Dinan, 2005; Bento et al., 2005), and engineering
variables (Michalek et al., 2004). Tattonement, however, has three issues: it
requires the iterative computation of profit-optimal prices (a special case of
the problem discussed in this article), should be inefficient relative to
direct methods whenever optimal strategies are coupled, and lacks the global
convergence guarantees of contemporary Newton solvers. Section 5 reviews these
conclusions in more detail.
This article should be viewed as a companion to Morrow and Skerlos (2010);
some of our notation and text may seem out of place without first reviewing
that article. In several places, text from Morrow and Skerlos (2010) is
repeated.
## 2\. A Technical Framework
This section describes the mathematical framework employed in Morrow and
Skerlos (2010). Several key assumptions are introduced and summarized in Table
1.
Table 1. List of important assumptions used in this section. Assumption | Purpose
---|---
2.1 | To provide a general form for utility functions
2.2 | To ensure profits are bounded and vanish as prices increase without bound
2.3 | To ensure the Leibniz Rule holds, validating Eqn. (9)
2.4 | To ensure that $\boldsymbol{\eta}$ is bounded. Implies the coercivity of $\mathbf{F}_{\eta},\mathbf{F}_{\zeta}$
| and the existence of simultaneously stationary prices.
2.5 | To ensure that $\boldsymbol{\zeta}$ is bounded. Implies the coercivity of $\mathbf{F}_{\zeta}$
| and the existence of simultaneously stationary prices.
3.1 | To ensure that the derivatives of profit vanish as prices increase without bound
3.2 | To ensure the coercivity of $\mathbf{F}_{\eta},\mathbf{F}_{\zeta}$ under weaker conditions than
| Assumption 2.4.
### 2.1. Mathematical Notation
#### 2.1.1. Sets.
Table 2 lists some important sets and the symbols used for them. $\mathbb{N}$
denotes the natural numbers $\\{1,2,\dotsc\\}$, and $\mathbb{N}(N)$ denotes
the natural numbers up to $N$, that is, $\mathbb{N}(N)=\\{1,\dotsc,N\\}$.
$\mathbb{R}$ denotes the set of real numbers $(-\infty,\infty)$, $[0,\infty)$
denotes the non-negative real numbers, and $[0,\infty]$ denotes the extended
non-negative half-line. We denote the $(J-1)$-dimensional simplex
$\\{(x_{1},\dotsc,x_{N})\in[0,1]^{N}:\sum_{n=1}^{N}x_{n}=1\\}$ by
$\mathbb{S}(N)$, and the $J$-dimensional “pyramid”
$\\{(x_{1},\dotsc,x_{N})\in[0,1]^{N}:\sum_{n=1}^{N}x_{n}\leq 1\\}$ by
$\triangle(J)$. Hyper-rectangles in $\mathbb{R}^{N}$, i.e. sets of the form
$[a_{1},b_{1}]\times\dotsb\times[a_{N},b_{N}]$ for some
$a_{n},b_{n}\in\mathbb{R}$ with $a_{n}<b_{n}$ for all $n\in\mathbb{N}(N)$, are
denoted by $[\mathbf{a},\mathbf{b}]$ where $\mathbf{a}=(a_{1},\dotsc,a_{N})$
and $\mathbf{b}=(b_{1},\dotsc,b_{N})$. $\mathcal{P}$ always denotes the non-
negative numbers: $\mathcal{P}=[0,\infty)$. For other sets, we typically use
calligraphic upper case letters such as “$\mathcal{A}$”. For any set
$\mathcal{A}$, $\left\lvert\mathcal{A}\right\rvert$ denotes its cardinality.
For any $\mathcal{B}\subset\mathcal{A}$, $\mathcal{A}\setminus\mathcal{B}$
denotes the set $\\{b\in\mathcal{A}:b\notin\mathcal{B}\\}$.
Table 2. Important sets. Symbol | Description
---|---
$\mathbb{N}$ | $=$ | $\\{1,2,\dotsc\\}$ | Natural numbers
$\mathbb{R}$ | $=$ | $(-\infty,\infty)$ | Real numbers
$\mathcal{P}$ | $=$ | $[0,\infty)$ | Non-negative real numbers
$\mathcal{J}$ | $=$ | $\\{1,\dotsc,J\\}$ | Set of product indices
$\mathcal{X}$ | $\subset$ | $\mathbb{R}^{K}$ | Set of product characteristics
$\mathcal{T}$ | $\subset$ | $\mathbb{R}^{L}$ | Set of individual characteristics
#### 2.1.2. Symbols.
Table 3 itemizes specific symbols used in the text.
Table 3. Summary of important symbols. Symbol | Description | Defined in
---|---|---
Products (see Section 2.2)
$J$ | $\in$ | $\mathbb{N}$ | number of products |
$K$ | $\in$ | $\mathbb{N}$ | number of non-price product characteristics |
$\mathbf{x}_{j}$ | $\in$ | $\mathcal{X}$ | non-price characteristics of product $j$ |
$p_{j}$ | $\in$ | $\mathcal{P}$ | price of product $j$ |
$\mathbf{p}$ | $\in$ | $\mathcal{P}^{J}$ | vector of all product prices |
Individual Characteristics (see Section 2.2)
$\boldsymbol{\theta}$ | $\in$ | $\mathcal{T}$ | individual characteristics, including observed |
| | | demographics and “random coefficients” |
$\mu$ | $-$ | $-$ | distribution of individual characteristics |
Choice Probabilities (see Section 2.2)
$u_{j}(\boldsymbol{\theta},p_{j})$ | $\in$ | $[-\infty,\infty)$ | utility of product $j$ |
$\vartheta(\boldsymbol{\theta})$ | $\in$ | $[-\infty,\infty)$ | utility of the outside good |
$P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})$ | $\in$ | [0,1] | Logit choice probability for product $j$ | Eqn. (1)
$P_{j}(\mathbf{p})$ | $\in$ | [0,1] | Mixed Logit choice probability for product $j$ |
$\mathbf{P}(\mathbf{p})$ | $\in$ | $[0,1]^{J}$ | vector of Mixed Logit choice probabilities for all |
| | | products |
Firms, Costs, Profits, and Stationarity (see Section 2.4, 2.5)
$F$ | $\in$ | $\mathbb{N}$ | number of firms |
$\mathcal{J}_{f}$ | $\subset$ | $\mathcal{J}$ | indices of the products offered by firm $f$ |
$c_{j}$ | $\in$ | $\mathcal{P}$ | (fixed) unit cost of product $j$ |
$\mathbf{c}$ | $\in$ | $\mathcal{P}^{J}$ | vector of all (fixed) unit costs |
$\hat{\pi}_{f}(\mathbf{p})$ | $\in$ | $\mathbb{R}$ | expected profits for firm $f$ | Eqn. (2)
$(D_{k}\hat{\pi}_{f})(\mathbf{p})$ | $\in$ | $\mathbb{R}$ | derivative of firm $f$’s profits, with respect to the | Eqn. (6)
| | | price of product $k$ |
$(\tilde{\nabla}\hat{\pi})(\mathbf{p})$ | $\in$ | $\mathbb{R}^{J}$ | Combined Gradient of profits | Prop. 2.2, Eqn. (7)
Choice Probability Derivatives (see Sections 2.5, 2.8)
$(D_{k}P_{j})(\mathbf{p})$ | $\in$ | $\mathbb{R}$ | derivative of product $j$’s choice probability |
| | | with respect to the price of product $k$ |
$(\tilde{D}\mathbf{P})(\mathbf{p})$ | $\in$ | $\mathbb{R}^{J\times J}$ | “intra-firm” Jacobian matrix of the choice | Eqn. (8)
| | | probability vector |
$\boldsymbol{\Lambda}(\mathbf{p})$, $\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$ | $\in$ | $\mathbb{R}^{J\times J}$ | matrices appearing in our decomposition of $(\tilde{D}\mathbf{P})(\mathbf{p})$ | Eqn. (9),
Fixed-Point Equations (see Sections 2.7, 2.8)
$\boldsymbol{\eta}(\mathbf{p})$ | $\in$ | $\mathbb{R}^{J}$ | the BLP-markup function (Berry et al., 1995) | Eqn. (13)
$\boldsymbol{\zeta}(\mathbf{p})$ | $\in$ | $\mathbb{R}^{J}$ | our $\boldsymbol{\zeta}$-markup function | Eqn. (18)
Bold, un-italicized symbols (e.g., “$\mathbf{x}$”) denote vectors and
matrices; typically we reserve lower case letters to refer to vectors and use
upper case letters to refer to matrices; the vector of choice probabilities
“$\mathbf{P}$” is an exception. Throughout we use $\mathbf{1}$ to denote a
vector of ones of the appropriate size for the context in which it appears.
$\mathbf{I}$ always denotes the identity matrix of a size appropriate for the
context. For any $\mathbf{x}\in\mathbb{R}^{N}$, $\mathrm{diag}(\mathbf{x})$
denotes the $N\times N$ diagonal matrix whose diagonal is $\mathbf{x}$. Any
vector inequalities between vectors are to be taken componentwise: for
example, $\mathbf{x}<\mathbf{y}$ means $x_{n}<y_{n}$ for all $n$.
Random variables are denoted with capital letters “$X$”, with random vectors
being denoted with bold capital letters (e.g., “$\mathbf{Q}$”). While this
overlaps with our notation for matrices, it should not cause any confusion.
$\mathbb{P}$ denotes a probability and $\mathbb{E}$ denotes an expectation.
$\mathrm{ess}\sup_{\mu}f$ denotes the essential supremum of the (measurable)
function $f$ over $\mathcal{T}$, with respect to the measure $\mu$; see, e.g.,
Bartle (1966).
$\log$ always denotes the natural (base $e$) logarithm. We use the “Big-O”
notation $\mathcal{O}(g)$ as follows: If there exists some $M<\infty$ such
that $\lim_{p\to q}[f(p)/g(p)]\leq M$, we say $f\in\mathcal{O}(g)$; the point
$q$ is left implicit.
#### 2.1.3. Differentiation.
Our conventions for denoting differentiation follow Munkres (1991). We use the
symbol “$D$” to denote differentiation using subscripts to invoke additional
specificity. Letting $\mathbf{f}:\mathbb{R}^{M}\to\mathbb{R}^{N}$,
$(D_{m}f_{n})(\mathbf{x})$ denotes the derivative of the $n^{\text{th}}$
component function with respect to the $m^{\text{th}}$ variable and
$(D\mathbf{f})(\mathbf{x})$ is the $N\times M$ derivative matrix of
$\mathbf{f}$ at $\mathbf{x}$ with components
$((D\mathbf{f})(\mathbf{x}))_{n,m}=(D_{m}f_{n})(\mathbf{x})$. Thus for
$f:\mathbb{R}^{M}\to\mathbb{R}$, $(Df)(\mathbf{x})$ is a row vector. If
$f:\mathbb{R}^{M}\to\mathbb{R}$, we define the gradient $(\nabla
f)(\mathbf{x})\in\mathbb{R}^{M}$ as the transposed derivative: $(\nabla
f)(\mathbf{x})=(Df)(\mathbf{x})^{\top}$.
### 2.2. Consumers, Products, and Choice Probabilities
A collection of $F\in\mathbb{N}$ firms offer a total of $J\in\mathbb{N}$
products to a population of individuals (or households). Each product
$j\in\mathcal{J}=\\{1,\dotsc,J\\}$ is defined by a price,
$p_{j}\in\mathcal{P}=[0,\infty)$, and a vector of $K\in\mathbb{N}$ product
“characteristics” $\mathbf{x}_{j}\in\mathcal{X}\subset\mathbb{R}^{K}$.
Individuals are identified by a vector of characteristics
$\boldsymbol{\theta}$ from some set $\mathcal{T}$. These individual
characteristics can include both observed demographics and “random
coefficients” (Berry et al., 1995; Nevo, 2000b; Train, 2003) that characterize
unobserved individual-specific heterogeneity with respect to preference for
product characteristics. The relative density of individual characteristic
vectors in the population is described by a probability distribution $\mu$
over $\mathcal{T}$.
An individual identified by $\boldsymbol{\theta}\in\mathcal{T}$ receives the
(random) utility
$U_{j}(\boldsymbol{\theta},\mathbf{x}_{j},p_{j})=u(\boldsymbol{\theta},\mathbf{x}_{j},p_{j})+\mathcal{E}_{j}$
from purchasing product $j\in\mathcal{J}$, and
$U_{0}(\boldsymbol{\theta})=\vartheta(\boldsymbol{\theta})+\mathcal{E}_{0}$
for forgoing purchase of any of these products; i.e. “purchasing the outside
good.” Individuals choose the “product” $j\in\\{0,\dotsc,J\\}$ with maximum
utility. Here
$u:\mathcal{T}\times\mathcal{X}\times\mathcal{P}\to[-\infty,\infty)$ is a
systematic utility function, $\vartheta:\mathcal{T}\to\mathbb{R}$ is a
valuation of the no-purchase option or “outside good,” and
$\boldsymbol{\mathcal{E}}=\\{\mathcal{E}_{j}\\}_{j=0}^{J}$ is a random vector
of i.i.d. standard extreme value variables. Section 2.3 below gives a general
specification of utility functions appropriate for equilibrium pricing. The
basic requirements are that $u$ is continuously differentiable and strictly
decreasing in price, and without lower bound as prices increase.
Demand for each product $j$ is characterized by choice probabilities
$P_{j}:\mathcal{P}^{J}\to[0,1]$ derived from (random) utility maximization.
Given the distributional assumption on $\boldsymbol{\mathcal{E}}$, the choice
probabilities for an individual characterized by
$\boldsymbol{\theta}\in\mathcal{T}$ are those of the Logit model (Train, 2003,
Chapter 3):
(1)
$P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})=\frac{e^{u_{j}(\boldsymbol{\theta},p_{j})}}{e^{\vartheta(\boldsymbol{\theta})}+\sum_{k=1}^{J}e^{u_{k}(\boldsymbol{\theta},p_{k})}}.$
The vector $\mathbf{p}\in\mathcal{P}^{J}$ denotes the vector of all product
prices. Product-specific utility functions
$u_{j}:\mathcal{T}\times\mathcal{P}\to[-\infty,\infty)$ for all $j$, defined
by $u_{j}(\boldsymbol{\theta},p)=u(\boldsymbol{\theta},\mathbf{x}_{j},p)$ for
all $(\boldsymbol{\theta},p)\in\mathcal{T}\times\mathcal{P}$, are used in Eqn.
(1) and in the following sections. The Mixed Logit choice probabilities
$P_{j}(\mathbf{p})=\int
P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})d\mu(\boldsymbol{\theta})$ follow
from integrating over the distribution of individual characteristics (Train,
2003, Chapter 6). The vector of Mixed Logit choice probabilities for all
products is denoted by $\mathbf{P}(\mathbf{p})\in[0,1]^{J}$.
The examples below review several instances of this choice model. Examples 1
and 2 are used in Morrow and Skerlos (2010). Example 3 illustrates the type
general specifications used in estimation. Example 4 describes one kind of
“simulation” of a Mixed Logit model (Train, 2003).
###### Example 1.
(Boyd and Mellman, 1980) Take $\mathcal{T}=\mathcal{P}\times\mathbb{R}^{K}$,
denoting $\boldsymbol{\theta}=(\alpha,\boldsymbol{\beta})$ for
$\alpha\in\mathcal{P}$ and $\boldsymbol{\beta}\in\mathbb{R}^{K}$. Set
$u(\alpha,\boldsymbol{\beta},\mathbf{x},p)=-\alpha
p+\boldsymbol{\beta}^{\top}\mathbf{x}$ and
$\vartheta(\alpha,\boldsymbol{\beta})=-\infty$ for all
$(\alpha,\boldsymbol{\beta})\in\mathcal{P}\times\mathbb{R}^{K}$. $\mu$ is
defined by specifying that $\alpha$ and $\boldsymbol{\beta}$ are independently
lognormally distributed (with appropriately chosen signs, means, and
variances).
###### Example 2.
(Berry et al., 1995) Take
$\mathcal{T}=\mathcal{P}\times\mathbb{R}^{K}\times\mathbb{R}$, denoting
$\boldsymbol{\theta}=(\phi,\boldsymbol{\beta},\beta_{0})$ for
$\phi\in\mathcal{P}$, $\boldsymbol{\beta}\in\mathbb{R}^{K}$, and
$\beta_{0}\in\mathbb{R}$. Set
$u(\phi,\boldsymbol{\beta},\mathbf{x},p)=\left\\{\begin{aligned}
&\alpha\log(\phi-p)+\boldsymbol{\beta}^{\top}\mathbf{x}&&\quad\text{if
}p<\phi\\\
&-\infty&&\quad\text{otherwise}\end{aligned}\right.\quad\quad\text{and}\quad\quad\vartheta(\phi,\beta_{0})=\alpha\log\phi+\beta_{0}$
for some fixed coefficient $\alpha>0$. $\phi$ represents income and is given a
lognormal distribution, while the random coefficients
$\boldsymbol{\beta},\beta_{0}$ are independently normally distributed with
some mean and variance. Note that income ($\phi$) serves as an upper bound on
the price an individual can pay for any product.
###### Example 3.
(Nevo, 2000b) Take
$\mathcal{T}=\mathcal{P}\times\mathbb{R}^{D}\times\mathbb{R}^{K+2}$, denoting
$\boldsymbol{\theta}=(\phi,\mathbf{d},\boldsymbol{\nu})$ for
$\phi\in\mathcal{P}$, $\mathbf{d}\in\mathbb{R}^{D}$, and
$\boldsymbol{\nu}\in\mathbb{R}^{K+2}$. Again, $\phi$ represents income;
$\mathbf{d}\in\mathbb{R}^{D}$ represents a vector of $D$ observed demographic
variables (which may include income); $\boldsymbol{\nu}\in\mathbb{R}^{K+2}$
represents a vector of $K+2$ random coefficients: one for each product
characteristic, one for price, and one for the outside good. Set
$\displaystyle u(\phi,\mathbf{d},\boldsymbol{\nu},\mathbf{x},p)$
$\displaystyle=(\alpha+\boldsymbol{\pi}_{p}^{\top}\mathbf{d}+\boldsymbol{\sigma}_{p}^{\top}\boldsymbol{\nu})(\phi-p)+\left(\boldsymbol{\beta}+\boldsymbol{\Pi}\mathbf{d}+\boldsymbol{\Sigma}\boldsymbol{\nu}\right)^{\top}\mathbf{x}$
$\displaystyle\vartheta(\phi,\mathbf{d},\boldsymbol{\nu})$
$\displaystyle=(\alpha+\boldsymbol{\pi}_{p}^{\top}\mathbf{d}+\boldsymbol{\sigma}_{p}^{\top}\boldsymbol{\nu})\phi+\boldsymbol{\pi}_{0}^{\top}\mathbf{d}+\boldsymbol{\sigma}_{0}^{\top}\boldsymbol{\nu}$
where $\alpha\in\mathbb{R}$, $\boldsymbol{\beta}\in\mathbb{R}^{K}$,
$\boldsymbol{\pi}_{p},\boldsymbol{\pi}_{0}\in\mathbb{R}^{D}$,
$\boldsymbol{\Pi}\in\mathbb{R}^{K\times D}$,
$\boldsymbol{\sigma}_{p},\boldsymbol{\sigma}_{0}\in\mathbb{R}^{K+2}$, and
$\boldsymbol{\Sigma}\in\mathbb{R}^{K\times(K+2)}$ are coefficients. The
distribution of $\mathbf{d}$ is estimated from available data (e.g., Census
data) and $\boldsymbol{\nu}$ is assumed to be standard independent
multivariate normal. When
$\alpha+\boldsymbol{\pi}_{p}^{\top}\mathbf{d}+\boldsymbol{\sigma}_{p}^{\top}\boldsymbol{\nu}$,
the coefficient on price, is positive, an individual prefers higher prices.
Petrin (2002) and Berry et al. (2004) adopt similar specifications that
eliminate this counterintuitive property. Petrin (2002) takes the price
component of utility to be $\alpha(\phi)\log(\phi-p)$, where
$\alpha:\mathcal{P}\to\mathcal{P}$ is a step function. Berry et al. (2004)
take the price component of utility to be $\alpha p$, but define
$\alpha=-e^{-(\alpha+\boldsymbol{\pi}_{p}^{\top}\mathbf{d}+\boldsymbol{\sigma}_{p}^{\top}\boldsymbol{\nu})}$.
###### Example 4.
(Simulation). Take any of the examples above, and draw $S\in\mathbb{N}$
vectors $\boldsymbol{\theta}_{s}\in\mathcal{T}$ according to the distribution
$\mu$. Let $\mathcal{T}^{\prime}=\\{\boldsymbol{\theta}_{s}\\}_{s=1}^{S}$ and
define a probability measure $\mu^{\prime}$ over $\mathcal{T}^{\prime}$ by
$\mu^{\prime}(\boldsymbol{\theta}_{s})=1/S$ for all $s$. Then
$(u,\vartheta,\mathcal{T}^{\prime},\mu^{\prime})$ defines a simulator of the
“full” Mixed Logit model with $(u,\vartheta,\mathcal{T},\mu)$; see Train
(2003). These approximations are essential in estimation of Mixed Logit models
and in computations of equilibrium prices.
### 2.3. Utility Specification
This section presents a generalization of the systematic utility functions
used in the examples given in the text, a specification closely related to the
one introduced by Caplin and Nalebuff (1991). Morrow (2008); Morrow and
Skerlos (2008) use a similar specification to analyze equilibrium prices in
simple Logit models.
###### Assumption 2.1.
For all $j$, there exist functions
$w_{j}:\mathcal{T}\times\mathcal{P}\to[-\infty,\infty)$ and
$v_{j}:\mathcal{T}\to(-\infty,\infty)$ such that the systematic utility
function $u_{j}:\mathcal{T}\times\mathcal{P}\to[-\infty,\infty)$ can be
written
$u_{j}(\boldsymbol{\theta},p)=w_{j}(\boldsymbol{\theta},p)+v_{j}(\boldsymbol{\theta})$.
Furthermore there exists $\varsigma:\mathcal{T}\to(0,\infty]$ such that
$w_{j}:\mathcal{T}\times[0,\infty]\to[-\infty,\infty)$ satisfies, for all $j$
and $\mu$-almost every (a.e.) $\boldsymbol{\theta}\in\mathcal{T}$,
* (a)
$w_{j}(\boldsymbol{\theta},\cdot):(0,\varsigma(\boldsymbol{\theta}))\to[-\infty,\infty)$
is continuously differentiable, strictly decreasing, and finite
* (b)
$w_{j}(\boldsymbol{\theta},p)=-\infty$ for all
$p\geq\varsigma(\boldsymbol{\theta})$, and
* (c)
$w_{j}(\boldsymbol{\theta},p)\downarrow-\infty$ as
$p\uparrow\varsigma(\boldsymbol{\theta})$.
$v_{j}:\mathcal{T}\to(-\infty,\infty)$ is arbitrary.
Note that we have not restricted $\mu$, the distribution of individual
characteristics, with Assumption 2.1. Important examples of $\mu$ from the
econometrics and marketing literature include finitely supported distributions
(often empirical frequency distributions for integral observed demographic
variables), standard continuous distributions (e.g. normal, lognormal and
$\chi^{2}$), truncated standard continuous distributions, finite mixtures of
standard continuous distributions, and independent products of any of these
types of distributions. This generality allows us to address a wide variety of
otherwise disparate examples with a single notation. In particular, this
generality allows us to use a single framework to treat both “full” Mixed
Logit models defined by some $\mu$ with uncountable support and simulation-
based approximations to such models.
Some existing empirical specifications violate Assumption 2.1 by admiting
positive price coefficients for
$\boldsymbol{\theta}\in\mathcal{T}^{\prime}\subset\mathcal{T}$, where
$\mathcal{T}^{\prime}\subset\mathcal{T}$ has nonzero $\mu$-measure. See, for
example, Nevo (2000a) (Example 3) or Brownstone et al. (2000). This implies
that $w(\boldsymbol{\theta},\cdot)$ is increasing on $\mathcal{T}^{\prime}$.
If $w(\boldsymbol{\theta},\cdot)$ is not decreasing for $\mu$-a.e.
$\boldsymbol{\theta}$, or at least eventually decreasing for $\mu$-a.e.
$\boldsymbol{\theta}$ in the sense that there are always prices large enough
to ensure that $w(\boldsymbol{\theta},\cdot)$ is decreasing for $\mu$-a.e.
$\boldsymbol{\theta}$, then profit-optimal pricing is not a well-posed problem
and finite equilibrium prices will not exist.
The variable $\Sigma=\varsigma(\boldsymbol{\theta})$ represents an individual-
specific reservation price. As in the Berry et al. (1995) model of Example 2,
this reservation price is most often derived from household or individual
income. Correspondingly, $\Sigma$ is often given a lognormal distribution to
(roughly) fit empirical income data. In principle, this reservation price
could be related to purchasing power derived from observed demographic
variables other than income, or unobserved demographic variables such as
family wealth. Thus we allow this reservation price to be specified as a
function of all “demographic” characteristics, $\boldsymbol{\theta}$.
Conditions (b) and (c) in Assumption 2.1 imply that the probability an
individual characterized by $\boldsymbol{\theta}$ will purchase a product is
zero for any price above $\varsigma(\boldsymbol{\theta})$ and vanishes as the
price approaches $\varsigma(\boldsymbol{\theta})$. We set
$\varsigma_{*}=\mathrm{ess}\sup\varsigma$ and allow, but do not require,
$\varsigma_{*}=\infty$. For example, simulation-based approximations to the
Berry et al. demand model have $\varsigma_{*}<\infty$, as can be easily
checked.
Note also that Condition (c) in Assumption 2.1 ensures the continuity of
$P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})$ at any vector of prices with some
component equal to $\varsigma(\boldsymbol{\theta})$. We must require this of
the Logit choice probabilities to obtain Mixed Logit choice probabilities that
are continuous on $(0,\varsigma_{*})^{J}$ for the important class of
simulation-based approximations with finitely supported $\mu$. Continuous
Logit choice probabilities also imply continuous Mixed Logit choice
probabilities, by the Dominated Convergence Theorem.
### 2.4. Profits
To describe the optimal pricing problems faced by each firm we use the
following notation. Let $F\in\mathbb{N}$ denote the number of firms. For each
$f\in\\{1,\dotsc,F\\}$, there exists a set $\mathcal{J}_{f}\subset\mathcal{J}$
of indices that corresponds to the
$J_{f}=\left\lvert\mathcal{J}_{f}\right\rvert$ products offered by firm $f$.
The collection of all these sets, $\\{\mathcal{J}_{f}\\}_{f=1}^{F}$, forms a
partition of $\mathcal{J}$. Subsequently, in writing “$f(j)$” for some
$j\in\mathcal{J}$, we mean the unique $f\in\\{1,\dotsc,F\\}$ such that
$j\in\mathcal{J}_{f}$. The vector $\mathbf{p}_{f}\in\mathbb{R}^{J_{f}}$ refers
to the vector of prices of the products offered by firm $f$. Negative
subscripts denote competitor’s variables as in, for instance,
$\mathbf{p}_{-f}\in\mathbb{R}^{J_{-f}}$, where $J_{-f}=\sum_{g\neq f}J_{g}$,
is the vector of prices for products offered by all of firm $f$’s competitors.
Firm-specific choice probability functions are denoted by
$\mathbf{P}_{f}(\mathbf{p})\in\mathbb{R}^{J_{f}}$.
Two additional assumptions are required to complete the definition of firms’
profits in a manner consistent with empirical applications of Bertrand
competition. First, we must specify unit and fixed costs: for each product $j$
there exists a unit cost $c_{j}\in\mathcal{P}$ and for each firm there exists
a fixed cost $c_{f}^{F}\in\mathcal{P}$. Both $c_{j}$ and $c_{f}^{F}$ depend
only on the collection of product characteristics chosen by the firm, and not
on the quantity sold by the firm during the purchasing period for the reasons
discussed below. We let $\mathbf{c}_{f}\in\mathcal{P}^{J_{f}}$ denote the
vector of unit costs for the products offered by firm $f$, and
$\mathbf{c}\in\mathcal{P}^{J}$ denote the vector of unit costs for all
products.
Second, Bertrand competition entails the following “comittment” assumption on
the quantities produced (Baye and Kovenock, 2008). Let $Q_{j}(\mathbf{p})$
denote the (random) quantity of product $j$ that the population will demand
during the purchasing period, given prices for all products $\mathbf{p}$.
These random demands are derived from random utility maximization. We assume
each firm commits to producing exactly $Q_{j}(\mathbf{p})$ units of each
product $j\in\mathcal{J}_{f}$ during the purchasing period. This implies
either that there are no production capacity constraints that limit a firm’s
ability to meet any demands that arise during the purchase period, or that
production backlogs do not affect demand.
With the commitment and constant costs assumptions, the total cost firm $f$
incurs in producing (and selling) $Q_{j}(\mathbf{p})$ units of product $j$
during the purchasing period are given by the random variable
$\sum_{j\in\mathcal{J}_{f}}c_{j}Q_{j}(\mathbf{p})+c_{f}^{F}.$
Random revenues are, of course, given by
$\sum_{j\in\mathcal{J}_{f}}Q_{j}(\mathbf{p})p_{j}$. The random variable
$\Pi_{f}(\mathbf{p})=\sum_{j\in\mathcal{J}_{f}}Q_{j}(\mathbf{p})(p_{j}-c_{j})-c_{f}^{F}$
then gives firm $f$’s (random) profits for the purchasing period as a function
of all product prices. Following most of the theoretical and empirical
literature in both marketing and economics, we assume that firms take expected
profits,
(2)
$\mathbb{E}[\Pi_{f}(\mathbf{p})]=I\hat{\pi}_{f}(\mathbf{p})-c_{f}^{F}\quad\text{where}\quad\hat{\pi}_{f}(\mathbf{p})=\sum_{j\in\mathcal{J}_{f}}P_{j}(\mathbf{p})(p_{j}-c_{j})$
as the metric by which they optimize their pricing decisions in this
stochastic optimization problem. Here $I\in\mathbb{N}$ denotes the number of
individuals in the population.
Eqn. (2) demonstrates that neither the total firm fixed costs $c_{f}^{F}$ nor
the population size $I$ play a role in determining the prices that maximize
expected profits under the assumptions described above. Henceforth we focus on
the “population-normalized gross expected profits”
$\hat{\pi}_{f}(\mathbf{p})$, referred to in the text and below as simply
“profits”. Firms thus solve
(3) maximize
$\displaystyle\quad\hat{\pi}_{f}(\mathbf{p})=\sum_{j\in\mathcal{J}_{f}}P_{j}(\mathbf{p})(p_{j}-c_{j})$
$\displaystyle\mathrm{with\;respect\;to}$
$\displaystyle\quad\mathbf{p}_{f}\in\mathcal{P}^{J_{f}}$
Before continuing with our framework, we discuss quantity-dependent costs and
clarify when profits are bounded.
#### 2.4.1. Quantity-Dependent Costs
Including costs that depend on quantities produced is certainly possible,
though this should introduce extra terms into the first-order equations
presented below (Eqn. (7)). Generally speaking, unit costs that depend on the
quantity produced would be expressed as $c_{j}:\mathbb{Z}_{+}\to\mathcal{P}$,
and unit costs that depended on the expected quantity produced would be
expressed as $c_{j}:\mathcal{P}\to\mathcal{P}$. If unit costs depend on the
quantity produced, then product $j$’s unit costs for the purchasing period (i)
are random and (ii) depend on prices. To see this, simply note that product
$j$’s unit costs for the purchasing period are $c_{j}(Q_{j}(\mathbf{p}))$.
Assuming quantity-dependent costs also obscures expected profits, since there
are now nonlinear terms $Q_{j}(\mathbf{p})c_{j}(Q_{j}(\mathbf{p}))$ in the
formula for random profits. If unit costs depend only on the expected quantity
produced, then unit costs are not random but still depend on prices:
$c_{j}(\mathbb{E}[Q_{j}(\mathbf{p})])=c_{j}(IP_{j}(\mathbf{p}))$. In either
case the derivatives of unit costs with respect to prices should appear in the
first-order conditions. This is acknowledged in the theoretical literature. As
these terms have not yet been included in the empirical literature, even when
costs are assumed to depend on quantities produced (Berry et al., 1995;
Petrin, 2002), we focus on costs that are independent of the quantity
produced.
#### 2.4.2. Bounded and Vanishing Profits
Here we present a technical assumption that ensures that profits are not only
bounded, but vanish as all prices approach $\varsigma_{*}$.
###### Assumption 2.2.
For all $j$ there exists some $r_{j}:\mathcal{T}\to(1,\infty)$ and some
$\bar{p}_{j}:\mathcal{T}\to\mathcal{P}$ satisfying
(4)
$\sup\Big{\\{}\;p\mu(\\{\boldsymbol{\theta}:\bar{p}(\boldsymbol{\theta})>p\\})\;:\;p\in(0,\varsigma_{*})\;\Big{\\}}<\infty.$
such that
(5) $u_{j}(\boldsymbol{\theta},p)\leq-r_{j}(\boldsymbol{\theta})\log
p+\vartheta(\boldsymbol{\theta})$
for all $p\geq\bar{p}_{j}(\boldsymbol{\theta})$, $\mu$-a.e.
###### Lemma 2.1.
If Assumption 2.2 holds, then $\hat{\pi}_{f}(\mathbf{p})$ is bounded in
$\mathbf{p}$ and vanishes as
$\mathbf{p}_{f}\to\varsigma_{*}\mathbf{1}\in\mathbb{R}^{J_{f}}$.
###### Proof.
We use the Dominated Convergence Theorem. Eqn. (5) ensures that
$p_{j}P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})$ vanishes $\mu$-a.e. as
$p_{j}\uparrow\varsigma_{*}$; see also Morrow and Skerlos (2008). Eqn. (4)
ensures that
$\hat{\pi}_{f}(\mathbf{p})=\sum_{j\in\mathcal{J}_{f}}\int
p_{j}P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})d\mu(\boldsymbol{\theta})-\sum_{j\in\mathcal{J}_{f}}c_{j}\int
P_{j}(\mathbf{p})$
is bounded as prices approach $\varsigma_{*}$, as we now show.
The key quantities in this integral are
$\int
p_{j}P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})d\mu(\boldsymbol{\theta})\leq\int
p_{j}\left(\frac{e^{u_{j}(\boldsymbol{\theta},p_{j})-\vartheta(\boldsymbol{\theta})}}{1+e^{u_{j}(\boldsymbol{\theta},p_{j})-\vartheta(\boldsymbol{\theta})}}\right)d\mu(\boldsymbol{\theta});$
the $c_{j}P_{j}(\mathbf{p})$ terms vanish if $p_{j}\uparrow\varsigma_{*}$
since $P_{j}(\mathbf{p})$ vanishes. We must show that these terms are bounded
as $p_{j}\uparrow\varsigma_{*}$. By assumption,
$p_{j}\left(\frac{e^{u_{j}(\boldsymbol{\theta},p_{j})-\vartheta(\boldsymbol{\theta})}}{1+e^{u_{j}(\boldsymbol{\theta},p_{j})-\vartheta(\boldsymbol{\theta})}}\right)\leq\left(\frac{1}{p_{j}}\right)^{r_{j}(\boldsymbol{\theta})-1}\left(\frac{1}{1+e^{u_{j}(\boldsymbol{\theta},p_{j})-\vartheta(\boldsymbol{\theta})}}\right)$
for all $p_{j}\geq\bar{p}_{j}(\boldsymbol{\theta})$. Thus we write
$\displaystyle\int
p_{j}\left(\frac{e^{u_{j}(\boldsymbol{\theta},p_{j})-\vartheta(\boldsymbol{\theta})}}{1+e^{u_{j}(\boldsymbol{\theta},p_{j})-\vartheta(\boldsymbol{\theta})}}\right)d\mu(\boldsymbol{\theta})$
$\displaystyle=\int_{\\{\boldsymbol{\theta}:p_{j}<\bar{p}_{j}(\boldsymbol{\theta})\\}}p_{j}\left(\frac{e^{u_{j}(\boldsymbol{\theta},p_{j})-\vartheta(\boldsymbol{\theta})}}{1+e^{u_{j}(\boldsymbol{\theta},p_{j})-\vartheta(\boldsymbol{\theta})}}\right)d\mu(\boldsymbol{\theta})$
$\displaystyle\quad\quad\quad\quad\quad\quad+\int_{\\{\boldsymbol{\theta}:p_{j}\geq\bar{p}_{j}(\boldsymbol{\theta})\\}}p_{j}\left(\frac{e^{u_{j}(\boldsymbol{\theta},p_{j})-\vartheta(\boldsymbol{\theta})}}{1+e^{u_{j}(\boldsymbol{\theta},p_{j})-\vartheta(\boldsymbol{\theta})}}\right)d\mu(\boldsymbol{\theta})$
$\displaystyle\leq
p_{j}\mu{\\{\boldsymbol{\theta}:p_{j}<\bar{p}_{j}(\boldsymbol{\theta})\\}}$
$\displaystyle\quad\quad\quad\quad\quad\quad+\int_{\\{\boldsymbol{\theta}:p_{j}\geq\bar{p}_{j}(\boldsymbol{\theta})\\}}\left(\frac{1}{p_{j}}\right)^{r_{j}(\boldsymbol{\theta})-1}d\mu(\boldsymbol{\theta}).$
By Eqn. (4), the first term is bounded. We take $p_{j}>1$, without loss of
generality, so that $1/p^{r_{j}(\boldsymbol{\theta})-1}\leq 1$ for $\mu$-a.e.
$\boldsymbol{\theta}$ and the second term is bounded. ∎
We now make some remarks regarding Assumption 2.2.
Note that if $\varsigma(\boldsymbol{\theta})<\infty$ then Eqn. (5) holds for
any $r(\boldsymbol{\theta})>1$ by taking
$\bar{p}(\boldsymbol{\theta})=\varsigma(\boldsymbol{\theta})$. If
$\varsigma(\boldsymbol{\theta})=\infty$, Eqn. (5) admits any utility function
$u(\boldsymbol{\theta},\cdot)$ that is (eventually) concave in price.
If $\varsigma_{*}<\infty$, then $\varsigma(\boldsymbol{\theta})<\infty$ for
$\mu$-a.e. $\boldsymbol{\theta}$. Furthermore, Eqn. (4) is trivial.
To further analyze Eqn. (4), we assume $\varsigma_{*}=\infty$. We define
$Z=\bar{p}(\boldsymbol{\Theta})$, where $\boldsymbol{\Theta}$ is the
$\mathcal{T}$-valued random variable with
$\mathbb{P}(\boldsymbol{\Theta}\in\mathcal{A})=\mu(\mathcal{A})=\int_{\mathcal{A}}d\mu(\boldsymbol{\theta})$.
If $\varsigma(\boldsymbol{\theta})<\infty$ for $\mu$-a.e.
$\boldsymbol{\theta}$, then we can take
$Z=\Sigma=\varsigma(\boldsymbol{\Theta})$. Eqn. (4) can be re-written as
$\sup\\{p\mathbb{P}(Z>p):p\in(0,\infty)\\}<\infty$, or equivalently
$\lim_{p\to\infty}[p\mathbb{P}(Z>p)]<\infty$. Eqn. (4) admits any $Z$ with
finite expectation, and even admits any $Z$ with a “fat-tailed” distribution
satisfying $p^{1+\beta}\mathbb{P}(Z>p)\to 1$ as $p\to\infty$ for some
$\beta>0$. Eqn. (4) can be written $\mathbb{P}(Z>p)=\mathcal{O}(1/p)$.
### 2.5. Local Equilibrium and the Simultaneous Stationarity Conditions
Assuming that the choice probabilities are continuously differentiable in
prices, at equilibrium each firm’s prices satisfy the stationarity condition
(6)
$(D_{k}\hat{\pi}_{f})(\mathbf{p})=\sum_{j\in\mathcal{J}_{f}}(D_{k}P_{j})(\mathbf{p})(p_{j}-c_{j})+P_{k}(\mathbf{p})\quad\quad\text{for
all }k\in\mathcal{J}_{f}.$
Combining the stationarity condition for each firm we obtain the Simultaneous
Stationarity Condition, a first-order (necessary) condition for local
equilibrium prices.
###### Proposition 2.2 (Simultaneous Stationarity Condition).
Suppose $\mathbf{P}$ is continuously differentiable. Let
$(\tilde{\nabla}\hat{\pi})(\mathbf{p})\in\mathbb{R}^{J}$ denote the “combined
gradient” with components
$((\tilde{\nabla}\hat{\pi})(\mathbf{p}))_{j}=(D_{j}\hat{\pi}_{f(j)})(\mathbf{p})$
where $f(j)$ denotes the index of the firm offering product $j$. If
$\mathbf{p}$ is a local equilibrium, then
(7)
$(\tilde{\nabla}\hat{\pi})(\mathbf{p})=(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}(\mathbf{p}-\mathbf{c})+\mathbf{P}(\mathbf{p})=\mathbf{0}.$
where $(\tilde{D}\mathbf{P})(\mathbf{p})\in\mathbb{R}^{J\times J}$ is the
“intra-firm” Jacobian matrix of price derivatives of the choice probabilities
defined by
(8)
$\big{(}(\tilde{D}\mathbf{P})(\mathbf{p})\big{)}_{j,k}=\left\\{\begin{aligned}
&(D_{k}P_{j})(\mathbf{p})&&\quad\text{if products $j$ and $k$ are offered by
the same firm }\\\ &\quad\quad 0&&\quad\text{otherwise }\end{aligned}\right.$
Prices $\mathbf{p}$ satisfying Eqn. (7) are called “simultaneously
stationary.”
The matrix $-(\tilde{D}\mathbf{P})(\mathbf{p})$ has previously been denoted by
“$\triangle$” (Berry et al., 1995; Petrin, 2002; Beresteanu and Li, 2008),
“$\boldsymbol{\Omega}$” (Nevo, 2000a), and “$\boldsymbol{\Phi}$” (Dube et al.,
2002). We prefer the “$D$” notation to emphasize the relationship of
$(\tilde{D}\mathbf{P})(\mathbf{p})$ to the Jacobian matrix of the choice
probabilities $\mathbf{P}$, while using the superscript “$\sim$” to denote the
intra-firm sparsity structure.
A set of simultaneously stationary prices are a local equilibrium only if
every firm’s profits are locally maximized at those prices; this can be
verified by confirming that every firm’s profits are locally concave (Section
LABEL:SUBSECSufficiency). Note that there is no convenient condition to verify
that every firm’s profits are globally maximized at a particular local
equilibrium. That is, there is no convenient condition to ensure that certain
prices are a proper equilibrium.
### 2.6. Choice Probability Derivatives
In this section we examine the price derivatives of Mixed Logit choice
probabilities. In what follows, $(Dw_{j})(p_{j})$ denotes the derivative of
the price component of utility, $w_{j}$, with respect to price.
###### Proposition 2.3.
Fix $\mathbf{p}\in(0,\varsigma_{*})^{J}$, let $u_{j}$ be given as in
Assumption 2.1 for all $j$, and suppose the Leibniz Rule holds for the Mixed
Logit choice probabilities $P_{j}(\mathbf{p})=\int
P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})d\mu(\boldsymbol{\theta})$; that is,
$(D_{k}P_{j})(\mathbf{p})=\int(D_{j}P_{k}^{L})(\boldsymbol{\theta},\mathbf{p})d\mu(\boldsymbol{\theta})$.
Then the Jacobian matrix of $\mathbf{P}$ is given by
(9)
$(D\mathbf{P})(\mathbf{p})=\boldsymbol{\Lambda}(\mathbf{p})-\boldsymbol{\Gamma}(\mathbf{p})$
where $\boldsymbol{\Lambda}(\mathbf{p})\in\mathbb{R}^{J\times J}$ is the
diagonal matrix with diagonal entries
$\lambda_{j}(\mathbf{p})=\int_{\mathcal{L}(p_{j})}(Dw_{j})(\boldsymbol{\theta},p_{j})P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})d\mu(\boldsymbol{\theta}),\quad\quad\mathcal{L}(p)=\\{\boldsymbol{\theta}:\varsigma(\boldsymbol{\theta})>p\\}$
and $\boldsymbol{\Gamma}(\mathbf{p})$ is the full $J\times J$ matrix with
entries
$\gamma_{j,k}(\mathbf{p})=\int_{\mathcal{G}(p_{j},p_{k})}P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})P_{k}^{L}(\boldsymbol{\theta},\mathbf{p})(Dw_{k})(\boldsymbol{\theta},p_{k})d\mu(\boldsymbol{\theta}),\quad\quad\mathcal{G}(p,q)=\mathcal{L}(p)\cap\mathcal{L}(q).$
The intra-firm price derivatives of the Mixed Logit choice probabilities are
given by
$(\tilde{D}\mathbf{P})(\mathbf{p})=\boldsymbol{\Lambda}(\mathbf{p})-\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$
where
$\big{(}\tilde{\boldsymbol{\Gamma}}(\mathbf{p})\big{)}_{j,k}=\gamma_{j,k}(\mathbf{p})$
if $f(j)=f(k)$ and
$\big{(}\tilde{\boldsymbol{\Gamma}}(\mathbf{p})\big{)}_{j,k}=0$ otherwise.
###### Proof.
We first characterize the Logit choice probabilities. For all $j,k$ we have
$\displaystyle(D_{k}P_{j}^{L})(\boldsymbol{\theta},\mathbf{p})$
$\displaystyle=P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})(\delta_{j,k}-P_{k}^{L}(\boldsymbol{\theta},\mathbf{p}))(Dw_{k})(\boldsymbol{\theta},p_{k})$
$\displaystyle=\delta_{j,k}P_{k}^{L}(\boldsymbol{\theta},\mathbf{p})(Dw_{k})(\boldsymbol{\theta},p_{k})-P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})P_{k}^{L}(\boldsymbol{\theta},\mathbf{p})(Dw_{k})(\boldsymbol{\theta},p_{k})$
for any $\boldsymbol{\theta}\in\mathcal{L}(p_{k})$ and
$(D_{k}P_{j}^{L})(\boldsymbol{\theta},\mathbf{p})=0$ for any
$\boldsymbol{\theta}\in\\{\boldsymbol{\theta}^{\prime}\in\mathcal{T}:p_{k}>\varsigma(\boldsymbol{\theta}^{\prime})\\}$
(because $P_{j}^{L}(\boldsymbol{\theta},\cdot)$ is identically zero in a
neighborhood of $\mathbf{p}$). Neglecting values
$\boldsymbol{\theta}\in\varsigma^{-1}(p_{k})$ for the moment, we observe that
these formulae and the Leibniz rule generate the desired expression for the
Mixed Logit choice probabilities.
We complete the proof by considering
$\boldsymbol{\theta}\in\varsigma^{-1}(p_{k})$. If $\varsigma^{-1}(p_{k})$ has
$\mu$-measure zero for any $p_{k}$, then we do not need to worry about Logit
choice probability derivatives at
$\boldsymbol{\theta}\in\varsigma^{-1}(p_{k})$. On the other hand if
$\varsigma^{-1}(p_{k})$ has positive $\mu$-measure for some $p_{k}$, we must
assume continuity of the Logit choice probability derivatives: i.e.
$(D_{k}P_{j}^{L})(\boldsymbol{\theta},\mathbf{p})\to 0$ as
$p_{k}\uparrow\varsigma(\boldsymbol{\theta})$. Otherwise, the Logit choice
probability derivative is not defined on a set of demographics with positive
measure. ∎
$\boldsymbol{\lambda}$ is closely related to a familiar economic quantity.
Recall that the “inclusive value,” or expected maximum utility, conditional on
demographics is given by (Small and Rosen, 1981; Train, 2003)
$\iota^{L}(\boldsymbol{\theta},\mathbf{p})=\log\left(e^{\vartheta(\boldsymbol{\theta})}+\sum_{j=1}^{J}e^{u_{j}(\boldsymbol{\theta},p_{j})}\right)$
It is easy to see that $\lambda_{k}$ is the derivative of the “aggregate
inclusive value”
$\iota(\mathbf{p})=\int\iota^{L}(\boldsymbol{\theta},\mathbf{p})d\mu(\boldsymbol{\theta})$
with respect to the $k^{\text{th}}$ price:
$\lambda_{k}(\mathbf{p})=(D_{k}\iota)(\mathbf{p})=\int(D_{k}\iota^{L})(\boldsymbol{\theta},\mathbf{p})d\mu(\boldsymbol{\theta})$.
Note that $\boldsymbol{\Gamma}(\mathbf{p})$ and
$\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$ are not necessarily symmetric for
all $\mathbf{p}$. If $(Dw_{k})(\boldsymbol{\theta},p)$ is independent of both
$k$ and $p$, as in the case of the Boyd and Mellman (1980) model presented in
Example 1 above, then $\boldsymbol{\Gamma}(\mathbf{p})$ (and thus
$\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$) is symmetric for all $\mathbf{p}$.
On the other hand if $(Dw_{k})(\boldsymbol{\theta},\cdot)$ is independent of
$k$ and strictly monotone in $p$, as is the case of the strictly concave in
price utility from Berry et al. (1995), then
$\gamma_{j,k}(\mathbf{p})=\gamma_{k,j}(\mathbf{p})$ if and only if
$p_{j}=p_{k}$.
The following assumption gives a simple, abstract condition on
$(\mathbf{u},\vartheta,\mu)$ that guarantees the Leibniz Rule holds and
defines continuously differentiable choice probabilities.
###### Assumption 2.3.
Let $k$ be arbitrary and define
$\psi_{k}:\mathcal{T}\times\mathcal{P}\to\mathcal{P}$ by
$\psi_{k}(\boldsymbol{\theta},p)=\left\\{\begin{aligned}
&\left\lvert(Dw_{k})(\boldsymbol{\theta},p)\right\rvert\left(\frac{e^{u_{k}(\boldsymbol{\theta},p)}}{e^{\vartheta(\boldsymbol{\theta})}+e^{u_{k}(\boldsymbol{\theta},p)}}\right)&&\quad\text{if
}p<\varsigma(\boldsymbol{\theta})\\\ &\quad\quad\quad\quad\quad
0&&\quad\text{if }p\geq\varsigma(\boldsymbol{\theta})\\\ \end{aligned}\right.$
Assume (i)
$\psi_{k}(\boldsymbol{\theta},\cdot):(0,\varsigma_{*})\to\mathcal{P}$ is
continuous for $\mu$-a.e. $\boldsymbol{\theta}\in\mathcal{T}$; that is,
$\psi_{k}(\boldsymbol{\theta},q)\to\psi_{k}(\boldsymbol{\theta},p)$ as $q\to
p$ for any $p\in(0,\varsigma_{*})$. (ii)
$\psi_{k}(\cdot,p^{\prime}):\mathcal{T}\to\mathcal{P}$ is uniformly
$\mu$-integrable for all $p^{\prime}$ in some neighborhood of any
$p\in(0,\varsigma_{*})$; that is, there exists some
$\varphi:\mathcal{T}\to[0,\infty)$ with
$\int\varphi(\boldsymbol{\theta})d\mu(\boldsymbol{\theta})<\infty$ (that may
depend on $k$ and $p$), such that
$\psi_{k}(\boldsymbol{\theta},p^{\prime})\leq\varphi(\boldsymbol{\theta})$ for
all $p^{\prime}$ in some neighborhood of $p$.
Note that under Assumption 2.1, (i) requires only that
$\psi_{k}(\boldsymbol{\theta},p)\to 0$ as
$p\uparrow\varsigma(\boldsymbol{\theta})$ for $\mu$-a.e.
$\boldsymbol{\theta}$.
###### Proposition 2.4.
If Assumption 2.3 holds, then the Leibniz Rule holds for the Mixed Logit
choice probabilities which are also continuously differentiable on
$(0,\varsigma_{*})^{J}$.
###### Proof.
Taking for granted that $(D_{k}P_{j}^{L})(\boldsymbol{\theta},\cdot)$ is
continuous at $\mathbf{p}$ and the differences
(10)
$h^{-1}\big{(}P_{j}^{L}(\boldsymbol{\theta},\mathbf{p}+h\mathbf{e}_{k})-P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})\big{)}$
are uniformly $\mu$-integrable for small enough $h$, the Dominated Convergence
Theorem implies that
$\displaystyle\lim_{h\to 0}h^{-1}\Big{(}\int
P_{j}^{L}(\boldsymbol{\theta},\mathbf{p}+h\mathbf{e}_{k})d\mu(\boldsymbol{\theta})-\int
P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})d\mu(\boldsymbol{\theta})\Big{)}$
$\displaystyle\quad\quad\quad\quad=\lim_{h\to 0}\int
h^{-1}\big{(}P_{j}^{L}(\boldsymbol{\theta},\mathbf{p}+h\mathbf{e}_{k})-P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})\big{)}d\mu(\boldsymbol{\theta})$
$\displaystyle\quad\quad\quad\quad=\int\lim_{h\to
0}h^{-1}\big{(}P_{j}^{L}(\boldsymbol{\theta},\mathbf{p}+h\mathbf{e}_{k})-P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})\big{)}d\mu(\boldsymbol{\theta})$
$\displaystyle\quad\quad\quad\quad=\int(D_{k}P_{j}^{L})(\boldsymbol{\theta},\mathbf{p})d\mu(\boldsymbol{\theta}).$
This validates the Leibniz Rule. This proof is essentially that given in a
general setting by (Bartle, 1966, Chapter 5, pg. 46).
To complete the proof we must validate that
$(D_{k}P_{j}^{L})(\boldsymbol{\theta},\cdot)$ is continuous in $p_{k}$ and the
differences in Eqn. (10) are uniformly $\mu$-integrable in a neighborhood of
$p_{k}$. It is easy to see that the desired continuity follows from Assumption
2.1 and Assumption 2.3, Condition (i). Specifically, note that
$(D_{k}P_{j}^{L})(\boldsymbol{\theta},\mathbf{p})=0=\psi_{k}(\boldsymbol{\theta},p_{k})$
for
$\boldsymbol{\theta}\in\\{\boldsymbol{\theta}^{\prime}\in\mathcal{T}:p_{k}>\varsigma(\boldsymbol{\theta}^{\prime})\\}$
and
$\displaystyle(D_{k}P_{j}^{L})(\boldsymbol{\theta},\mathbf{p})$
$\displaystyle=\big{(}\delta_{j,k}-P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})\big{)}P_{k}^{L}(\boldsymbol{\theta},\mathbf{p})(Dw_{k})(\boldsymbol{\theta},p_{k})$
$\displaystyle=\big{(}\delta_{j,k}-P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})\big{)}\left(\frac{e^{\vartheta(\boldsymbol{\theta})}+e^{u_{k}(\boldsymbol{\theta},p_{k})}}{e^{\vartheta(\boldsymbol{\theta})}+\sum_{i=1}^{J}e^{u_{i}(\boldsymbol{\theta},p_{i})}}\right)\left(\frac{e^{u_{k}(\boldsymbol{\theta},p_{k})}}{e^{\vartheta(\boldsymbol{\theta})}+e^{u_{k}(\boldsymbol{\theta},p_{k})}}\right)(Dw_{k})(\boldsymbol{\theta},p_{k})$
$\displaystyle=\big{(}\delta_{j,k}-P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})\big{)}\left(\frac{e^{\vartheta(\boldsymbol{\theta})}+e^{u_{k}(\boldsymbol{\theta},p_{k})}}{e^{\vartheta(\boldsymbol{\theta})}+\sum_{i=1}^{J}e^{u_{i}(\boldsymbol{\theta},p_{i})}}\right)\psi_{k}(\boldsymbol{\theta},p_{k})$
for $\boldsymbol{\theta}\in\mathcal{L}(p_{k})$. Suppose
$p_{k}=\varsigma(\boldsymbol{\theta})$. By Assumption 2.1 (a) and (b), the
first two terms are continuous. By Assumption 2.1 (c),
$\displaystyle\lim_{\mathbf{q}\to\mathbf{p},q_{k}<\varsigma(\boldsymbol{\theta})}(D_{k}P_{j}^{L})(\boldsymbol{\theta},\mathbf{p})$
$\displaystyle=\left(\delta_{j,k}-\frac{e^{u_{j}(\boldsymbol{\theta},p_{j})}}{e^{\vartheta(\boldsymbol{\theta})}+\sum_{i\neq
k}e^{u_{i}(\boldsymbol{\theta},p_{i})}}\right)\left(\frac{e^{\vartheta(\boldsymbol{\theta})}}{e^{\vartheta(\boldsymbol{\theta})}+\sum_{i\neq
k}e^{u_{i}(\boldsymbol{\theta},p_{i})}}\right)\lim_{q_{k}\uparrow\varsigma(\boldsymbol{\theta})}\psi_{k}(\boldsymbol{\theta},q_{k})$
Assumption 2.3, Condition (i) is then necessary for the continuity of
$(D_{k}P_{j}^{L})(\boldsymbol{\theta},\mathbf{p})$ for all $j,k$ and
$\mathbf{p}\in(\mathbf{0},\varsigma_{*}\mathbf{1})$. Specifically if
$\psi_{k}(\boldsymbol{\theta},\cdot)$ is discontinuous at
$\varsigma(\boldsymbol{\theta})$, then
$\displaystyle\lim_{\mathbf{q}\to\mathbf{p},q_{k}<\varsigma(\boldsymbol{\theta})}(D_{k}P_{k}^{L})(\boldsymbol{\theta},\mathbf{p})$
$\displaystyle=\left(\frac{e^{\vartheta(\boldsymbol{\theta})}}{e^{\vartheta(\boldsymbol{\theta})}+\sum_{i\neq
k}e^{u_{i}(\boldsymbol{\theta},p_{i})}}\right)\lim_{q_{k}\uparrow\varsigma(\boldsymbol{\theta})}\psi_{k}(\boldsymbol{\theta},q_{k})$
To prove the integrability, we first note that for all $j,k$ and $\mathbf{p}$
we have
$\left\lvert(D_{k}P_{j}^{L})(\boldsymbol{\theta},\mathbf{p})\right\rvert\leq\psi_{k}(\boldsymbol{\theta},p_{k})$.
This bound is a consequence of the formula above, and is tight as
$\mathbf{p}_{-k}$ varies. The mean value theorem for functions of a single
real variable states that
$\displaystyle
h^{-1}(P_{j}^{L}(\boldsymbol{\theta},\mathbf{p}+h\mathbf{e}_{k})-P_{j}^{L}(\boldsymbol{\theta},\mathbf{p}))=(D_{k}P_{j}^{L})(\boldsymbol{\theta},\mathbf{p}+\eta\mathbf{e}_{k})$
for some $\eta$ such that $\left\lvert\eta\right\rvert<\left\lvert
h\right\rvert$, and thus
$\displaystyle\left\lvert h\right\rvert^{-1}\left\lvert
P_{j}^{L}(\boldsymbol{\theta},\mathbf{p}+h\mathbf{e}_{k})-P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})\right\rvert\leq\psi_{k}(\boldsymbol{\theta},p_{k}+\eta)\leq\varphi(\boldsymbol{\theta})$
for $\mu$-a.e. $\boldsymbol{\theta}\in\mathcal{T}$ and small enough $h$. Thus,
the desired uniform $\mu$-integrability follows from Assumption 2.3, Condition
(ii). ∎
An “easier” bound is simply
$\left\lvert(D_{k}P_{j}^{L})(\boldsymbol{\theta},\mathbf{p})\right\rvert\leq\left\lvert(Dw_{k})(\boldsymbol{\theta},p_{k})\right\rvert$,
and thus we might consider changing the statement of Proposition 2.4 to
hypothesize only the uniform $\mu$-integrability of the utility price
derivatives. In fact, this bound can be used to validate the Leibniz Rule for
the Boyd and Mellman model of Example 1 that lacks an outside good. However,
this bound fails to be useful for the Berry et al. model of Example 2, since
$w(p)=\alpha\log(\varsigma(\boldsymbol{\theta})-p)$ and
$\left\lvert(Dw_{k})(\boldsymbol{\theta},p_{k})\right\rvert=\alpha/(\varsigma(\boldsymbol{\theta})-p)$
is singular on $\varsigma^{-1}(p)$. In empirical applications, $\varsigma$ is
onto, generating a singularity somewhere in $\mathcal{T}$ for all $p$; this
singularity cannot be “controlled” for all $p$ by choosing the measure $\mu$.
In this case, a hypothesis only about the price derivatives of utility is not
useful.
We close this section by stating some basic results concerning
$(\tilde{D}\mathbf{P})(\mathbf{p})$ that are used below.
###### Lemma 2.5.
Under Assumption 2.1, $P_{j}(\mathbf{p})$ and $\lambda_{j}(\mathbf{p})$ are
never zero on $(0,\varsigma_{*})^{J}$. Thus $\boldsymbol{\Lambda}(\mathbf{p})$
is nonsingular for all $\mathbf{p}\in(0,\varsigma_{*})^{J}$.
###### Proof.
Note that $\mathcal{L}(p_{j})$ is nonempty and has positive $\mu$-measure,
$P_{j}^{L}(\cdot,\mathbf{p})$ is strictly positive on $\mathcal{L}(p_{j})$,
and $(Dw_{j})(\cdot,p_{j})P_{j}^{L}(\cdot,\mathbf{p})$ is strictly negative on
$\mathcal{L}(p_{j})$. It follows that $P_{j}(\mathbf{p})$ and
$\lambda_{j}(\mathbf{p})$ are nonzero. ∎
###### Lemma 2.6.
Let $\mathbf{p}\in(0,\varsigma_{*})^{J}$, suppose
$\vartheta:\mathcal{T}\to(-\infty,\infty)$, and define
(11) $\displaystyle\boldsymbol{\Omega}_{f}(\mathbf{p})$
$\displaystyle=\boldsymbol{\Lambda}_{f}(\mathbf{p})^{-1}\boldsymbol{\Gamma}_{f}(\mathbf{p})^{\top}\in\mathbb{R}^{J_{f}\times
J_{f}}\text{ for all }f$ (12)
$\displaystyle\tilde{\boldsymbol{\Omega}}(\mathbf{p})$
$\displaystyle=\boldsymbol{\Lambda}(\mathbf{p})^{-1}\tilde{\boldsymbol{\Gamma}}(\mathbf{p})^{\top}\in\mathbb{R}^{J\times
J}.$
These matrices are well-defined by Lemma 2.5, and have the following
properties:
* (i)
$(D_{f}\mathbf{P}_{f})(\mathbf{p})^{\top}=\boldsymbol{\Lambda}_{f}(\mathbf{p})(\mathbf{I}-\boldsymbol{\Omega}_{f}(\mathbf{p}))$
and
$(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}=\boldsymbol{\Lambda}(\mathbf{p})(\mathbf{I}-\tilde{\boldsymbol{\Omega}}(\mathbf{p}))$.
* (ii)
$\lvert\lvert\boldsymbol{\Omega}_{f}(\mathbf{p})\rvert\rvert_{\infty}<1$ and
$\lvert\lvert\tilde{\boldsymbol{\Omega}}(\mathbf{p})\rvert\rvert_{\infty}<1$.
* (iii)
$\mathbf{I}-\boldsymbol{\Omega}_{f}(\mathbf{p})\in\mathbb{R}^{J_{f}\times
J_{f}}$ and
$\mathbf{I}-\tilde{\boldsymbol{\Omega}}(\mathbf{p})\in\mathbb{R}^{J\times J}$
are strictly diagonally dominant and nonsingular.
* (iv)
$(\mathbf{I}-\boldsymbol{\Omega}_{f}(\mathbf{p}))^{-1}\in\mathbb{R}^{J_{f}\times
J_{f}}$ and
$(\mathbf{I}-\tilde{\boldsymbol{\Omega}}(\mathbf{p}))^{-1}\in\mathbb{R}^{J\times
J}$ map positive vectors to positive vectors.
###### Proof.
* (i)
This follows immediately from Prop. 2.3.
* (ii)
We note that
$\displaystyle\omega_{k,l}(\mathbf{p})=\frac{\gamma_{l,k}(\mathbf{p})}{\lambda_{k}(\mathbf{p})}=\int_{\mathcal{L}(p_{k})}P_{l}^{L}(\boldsymbol{\theta},\mathbf{p})d\mu_{k,\mathbf{p}}(\boldsymbol{\theta})$
where $\mu_{k,\mathbf{p}}$ is the probability distribution with density, with
respect to $\mu$, given by
$d\mu_{k,\mathbf{p}}(\boldsymbol{\theta})=\frac{P_{k}^{L}(\boldsymbol{\theta},\mathbf{p})\left\lvert(Dw_{k})(\boldsymbol{\theta},p_{k})\right\rvert
d\mu(\boldsymbol{\theta})}{\int_{\mathcal{L}(p_{k})}P_{k}^{L}(\boldsymbol{\phi},\mathbf{p})\left\lvert(Dw_{k})(\boldsymbol{\phi},p_{k})\right\rvert
d\mu(\boldsymbol{\phi})}.$
Thus
$\boldsymbol{\Lambda}_{f}(\mathbf{p})^{-1}\boldsymbol{\Gamma}_{f}(\mathbf{p})^{\top}$
has row sums
$\displaystyle\int\left(\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})\right)d\mu_{k,\mathbf{p}}(\boldsymbol{\theta})<1.$
The additional assumption that $\vartheta:\mathcal{T}\to(-\infty,\infty)$
plays a role in establishing this inequality because then there is always a
set $\mathcal{T}_{k}^{\prime}\subset\mathcal{T}$ with
$\mu_{k,\mathbf{p}}(\mathcal{T}_{k}^{\prime})>0$ on which
$\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})<1$.
* (iii)
The inequality
$\displaystyle
1>\int\left(\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})\right)d\mu_{k,\mathbf{p}}(\boldsymbol{\theta})$
is equivalent to
$\displaystyle\left\lvert 1-\omega_{k,k}(\mathbf{p})\right\rvert$
$\displaystyle=1-\int
P_{k}^{L}(\boldsymbol{\theta},\mathbf{p})d\mu_{k,\mathbf{p}}(\boldsymbol{\theta})$
$\displaystyle>\int\left(\sum_{j\in\mathcal{J}_{f}\setminus
k}P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})\right)d\mu_{k,\mathbf{p}}(\boldsymbol{\theta})=\sum_{l\neq
k}\omega_{k,l}(\mathbf{p}).$
The claim follows.
* (iii)
Because $\boldsymbol{\Omega}_{f}(\mathbf{p})$ maps positive vectors to
positive vectors, so does its power series
$\sum_{n=1}^{\infty}\boldsymbol{\Omega}_{f}(\mathbf{p})^{n}=(\mathbf{I}-\boldsymbol{\Omega}_{f}(\mathbf{p}))^{-1}.$
∎
###### Corollary 2.7.
$(D_{f}\mathbf{P}_{f})(\mathbf{p})^{\top}$ and
$(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}$ are strictly diagonally dominant
and nonsingular for $\mathbf{p}\in(\mathbf{0},\varsigma_{*}\mathbf{1})$.
###### Proof.
This follows directly from Lemma 2.6, claims (i) and (iii). ∎
### 2.7. The BLP-Markup Equation
A prominent form of the first-order conditions Eqn. (7) is the BLP-markup
equation:
(13)
$\mathbf{p}=\mathbf{c}+\boldsymbol{\eta}(\mathbf{p})\quad\text{where}\quad\boldsymbol{\eta}(\mathbf{p})=-(\tilde{D}\mathbf{P})(\mathbf{p})^{-\top}\mathbf{P}(\mathbf{p}).$
Note that $\boldsymbol{\eta}$ is defined for any continuously differentiable
choice probabilities with nonsingular
$(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}$. We have shown above that this
applies to certain Mixed Logit models (Section 2.6). Eqn. (13) and Corollary
2.7 show that $\boldsymbol{\eta}$ is well-defined and continuous, at least for
$\mathbf{p}\in(0,\varsigma_{*})^{J}$.
Traditionally, the BLP-markup equation (13) has been used to estimate costs
assuming observed prices are in equilibrium via the formula
$\mathbf{c}=\mathbf{p}-\boldsymbol{\eta}(\mathbf{p})$; see, e.g., Berry et al.
(1995) or Nevo (2000a). These cost estimates form the basis of counterfactual
experiments with an estimated demand model. Beresteanu and Li (2008) have
recently suggested that the BLP-markup equation is also useful for computing
equilibrium prices, a suggestion we explore further below. Note that the BLP-
markup equation must be interpreted as a nonlinear fixed-point equation when
applied to compute equilibrium prices.
We now derive several important properties of $\boldsymbol{\eta}$ from an
alternative form of Eqn. (13) based on Lemma 2.6, valid when
$\mathbf{p}\in(0,\varsigma_{*})^{J}$:
(14)
$\Big{(}\mathbf{I}-\tilde{\boldsymbol{\Omega}}(\mathbf{p})\Big{)}\boldsymbol{\eta}(\mathbf{p})=-\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p}).$
First, Eqn. (14) proves that profit-optimal markups are positive for the class
of Mixed Logit models we consider, thanks to Lemma 2.6, claim (iv).
###### Corollary 2.8.
Suppose Assumptions 2.1-2.3 hold. Then
$\boldsymbol{\eta}(\mathbf{p})>\mathbf{0}$ for all
$\mathbf{p}\in(0,\varsigma_{*})^{J}$. Hence if
$\mathbf{p}\in(0,\varsigma_{*})^{J}$ is a local equilibrium, then
$\mathbf{p}>\mathbf{c}$.
Second, Eqn. (14), rather than Eqn. (13), should be used to actually compute
$\boldsymbol{\eta}$. Recall that $\kappa_{2}(\mathbf{A})$ denotes the 2-norm
condition number of the matrix $\mathbf{A}$ (Trefethen and Bau, 1997).
###### Lemma 2.9.
Suppose Assumptions 2.1-2.3 hold. Eqn. (14) is better conditioned than Eqn.
(13), in the sense that
$\kappa_{2}\big{(}\mathbf{I}-\tilde{\boldsymbol{\Omega}}(\mathbf{p})\big{)}\leq\left(\frac{\min_{j}\left\lvert\lambda_{j}(\mathbf{p})\right\rvert}{\max_{j}\left\lvert\lambda_{j}(\mathbf{p})\right\rvert}\right)\kappa_{2}\big{(}(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}\big{)}$
for all $\mathbf{p}\in(\mathbf{0},\varsigma_{*}\mathbf{1})$.
###### Proof.
This follows from Lemma 2.6, claim (i), the inequality
$\kappa_{2}(\mathbf{AB})\leq\kappa_{2}(\mathbf{A})\kappa_{2}(\mathbf{B})$
valid for any matrices $\mathbf{A}$ and $\mathbf{B}$, and the formula
$\kappa_{2}(\boldsymbol{\Lambda}(\mathbf{p})^{-1})=\frac{\min_{j}\left\lvert\lambda_{j}(\mathbf{p})\right\rvert}{\max_{j}\left\lvert\lambda_{j}(\mathbf{p})\right\rvert}.$
∎
Lemma 2.9 states that the greater the variation in aggregate absolute rate of
change in inclusive values, the more poorly conditioned
$(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}$ is relative to
$\mathbf{I}-\tilde{\boldsymbol{\Omega}}(\mathbf{p})$. Because
$\boldsymbol{\Lambda}(\mathbf{p})$ is diagonal,
$\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})\rvert\rvert_{1}=\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})\rvert\rvert_{2}=\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})\rvert\rvert_{\infty}$
and thus the same bound applies for condition numbers in norms other than the
2-norm.
Third, Eqn. (14) also provides bounds on the magnitude of values taken by
$\boldsymbol{\eta}$:
###### Lemma 2.10.
Suppose Assumptions 2.1-2.3 hold. For all
$\mathbf{p}\in(0,\varsigma_{*})^{J}$, $\boldsymbol{\eta}$ satisfies
(15)
$\frac{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})\rvert\rvert_{\infty}}{1+\lvert\lvert\tilde{\boldsymbol{\Omega}}(\mathbf{p})\rvert\rvert_{\infty}}\leq\lvert\lvert\boldsymbol{\eta}(\mathbf{p})\rvert\rvert_{\infty}\leq\frac{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})\rvert\rvert_{\infty}}{1-\lvert\lvert\tilde{\boldsymbol{\Omega}}(\mathbf{p})\rvert\rvert_{\infty}}.$
###### Proof.
This follows immediately from Eqn. (14), using the triangle inequality. ∎
The upper bound suggests the following assumptions to ensure that
$\boldsymbol{\eta}$ itself is bounded:
###### Assumption 2.4.
Suppose there exist $M\in(0,\infty)$ and $\varepsilon\in(0,1)$ such that
(16)
$\displaystyle\sup\big{\\{}\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})\rvert\rvert_{\infty}\;:\;\mathbf{p}\in(0,\varsigma_{*})^{J}\big{\\}}$
$\displaystyle=M<\infty$ (17)
$\displaystyle\sup\big{\\{}\lvert\lvert\tilde{\boldsymbol{\Omega}}(\mathbf{p})\rvert\rvert_{\infty}\;:\;\mathbf{p}\in(0,\varsigma_{*})^{J}\big{\\}}$
$\displaystyle=1-\varepsilon<1.$
Under simple Logit,
$P_{k}^{L}(\mathbf{p})/\left\lvert\lambda_{k}(\mathbf{p})\right\rvert=\left\lvert(Dw_{k})(p_{k})\right\rvert^{-1}$
and
$\boldsymbol{\Omega}_{f}(\mathbf{p})=\mathbf{1}\mathbf{P}_{f}^{L}(\mathbf{p})^{\top}$.
Thus Eqn. (16) is akin to concavity of $w_{k}$ for all sufficiently large
$p_{k}$, and Eqn. (17) is implied by $\vartheta>-\infty$, i.e. the existence
of an outside good with positive purchase probability.
###### Lemma 2.11.
Suppose Assumptions 2.1-2.3 hold.
* (i)
If Assumption 2.4 also holds,
$N=\sup\\{\lvert\lvert\boldsymbol{\eta}(\mathbf{p})\rvert\rvert_{\infty}:\mathbf{p}\in(0,\varsigma_{*})^{J}\\}<\infty$.
* (ii)
Moreover Eqn. (16) in Assumption 2.4 is necessary for $N$ to be finite.
Unfortunately some simple models do not satisfy Assumption 2.4. A simple Logit
model with $w(p)=-\alpha\log p$ for some $\alpha>0$ violates Eqn. (16). More
generally, the Boyd and Mellman (1980) model of Example 1 does not satisfy
Eqn. (16). This is most easily seen by noting that finite-sample
approximations to this model have
$\lim_{p_{k}\to\infty}\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})\rvert\rvert_{\infty}=\max_{s=1,\dotsc,S}\left\\{\frac{1}{\alpha_{s}}\right\\}$
where $\\{\alpha_{s}\\}_{s=1}^{S}$ are the sampled price coefficients. Of
course, as $S\to\infty$, $\min_{s=1,\dotsc,S}\\{\alpha_{s}\\}\to 0$, and thus
$\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})\rvert\rvert_{\infty}\to\infty$.
### 2.8. The $\zeta$-Markup function
Substituting Eqn. (9) into Eqn. (7) yields the $\boldsymbol{\zeta}$-markup
equation introduced in Morrow and Skerlos (2010):
(18)
$\mathbf{p}=\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})\quad\text{where}\quad\boldsymbol{\zeta}(\mathbf{p})=\boldsymbol{\Lambda}(\mathbf{p})^{-1}\tilde{\boldsymbol{\Gamma}}(\mathbf{p})^{\top}(\mathbf{p}-\mathbf{c})-\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})$
when $\boldsymbol{\Lambda}(\mathbf{p})$ is nonsingular (Section 2.6, 2.8).
Thus the $\boldsymbol{\zeta}$-markup equation is specific to Mixed Logit
models, unlike the BLP-markup equation.
We observe a relationship between the maps $\boldsymbol{\zeta}$ and
$\boldsymbol{\eta}$.
###### Proposition 2.12.
Suppose Assumption 2.1-2.3 hold. For any $\mathbf{p}\in(0,\varsigma_{*})^{J}$,
$\boldsymbol{\zeta}(\mathbf{p})=\tilde{\boldsymbol{\Omega}}(\mathbf{p})(\mathbf{p}-\mathbf{c})+(\mathbf{I}-\tilde{\boldsymbol{\Omega}}(\mathbf{p}))\boldsymbol{\eta}(\mathbf{p})$.
###### Proof.
This follows directly from Eqns. (14) and (18). ∎
In so far as $\boldsymbol{\eta}$ and $\boldsymbol{\zeta}$ are distinct maps,
they can generate numerical methods for the computation of equilibrium prices
with entirely different properties. The equation above implies that
$\boldsymbol{\zeta}(\mathbf{p})=\boldsymbol{\eta}(\mathbf{p})$ if, and only
if,
$\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p})=\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})$
lies in the null space of $\tilde{\boldsymbol{\Omega}}(\mathbf{p})$. Thus if
$\tilde{\boldsymbol{\Omega}}(\mathbf{p})$ is full-rank, $\boldsymbol{\zeta}$
and $\boldsymbol{\eta}$ coincide only at simultaneously stationary prices.
Simple examples of Mixed Logit models can be constructed that always have
$\mathrm{rank}(\tilde{\boldsymbol{\Omega}}(\mathbf{p}))=J$. For Logit,
$\boldsymbol{\Omega}_{f}(\mathbf{p})=\mathbf{1}\mathbf{P}_{f}^{L}(\mathbf{p})^{\top}$
for all $f$ and $\tilde{\boldsymbol{\Omega}}(\mathbf{p})$ always has rank
$F\leq J$. However the analysis in Morrow and Skerlos (2008) can be used to
show that $\boldsymbol{\zeta}$ and $\boldsymbol{\eta}$ coincide only at
simultaneously stationary prices.
We now explore $\boldsymbol{\zeta}$’s asymptotic properties.
###### Lemma 2.13.
Under Assumption 2.4
$\lvert\lvert\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}<\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}$
whenever
$\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}>M/\varepsilon$.
Moreover
$\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}-\lvert\lvert\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}\to\infty$
as $\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}\to\infty$.
###### Proof.
We simply note that
$\displaystyle\lvert\lvert\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}$
$\displaystyle\leq\lvert\lvert\tilde{\boldsymbol{\Omega}}(\mathbf{p})\rvert\rvert_{\infty}\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}+\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})\rvert\rvert_{\infty}$
$\displaystyle\leq(1-\varepsilon)\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}+M$
$\displaystyle\leq\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}-\big{(}\varepsilon\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}-M\big{)}$
$\displaystyle\leq\left[1-\varepsilon+\frac{M}{\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}}\right]\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}$
Now if $\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}>M/\varepsilon$,
then $M/\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}<\varepsilon$.
Thus
$\displaystyle\lvert\lvert\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}<\left[1-\varepsilon+\varepsilon\right]\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}=\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}.$
To prove that
$\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}-\lvert\lvert\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}\to\infty$,
note that
$\displaystyle\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}-\lvert\lvert\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}\geq\left(\varepsilon-\frac{M}{\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}}\right)\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}$
For all
$\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}>M/\varepsilon$, the
term in parentheses is positive. Furthermore, this term approaches
$\varepsilon$ as
$\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}\to\infty$. Thus
$\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}-\lvert\lvert\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}\to\infty$
as $\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}\to\infty$. ∎
A slightly different assumption concerning
$\tilde{\boldsymbol{\Omega}}(\mathbf{p})$ is useful when analyzing the
$\boldsymbol{\zeta}$ map.
###### Assumption 2.5.
Suppose
(19)
$\sup\Big{\\{}\lvert\lvert\tilde{\boldsymbol{\Omega}}(\mathbf{p})(\mathbf{p}-\mathbf{c})\rvert\rvert_{\infty}:\mathbf{p}\in(0,\varsigma_{*})^{J}\Big{\\}}<\infty,$
###### Lemma 2.14.
Suppose Assumption 2.1-2.3 holds and $\varsigma_{*}=\infty$. Then
$\boldsymbol{\zeta}$ is bounded if, and only if, Eqn. (16) and Eqn. (19) hold.
###### Proof.
This follows directly from the triangle inequality and the non-negativity of
$\tilde{\boldsymbol{\Omega}}(\mathbf{p})(\mathbf{p}-\mathbf{c})$ and
$-\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})$ for all
$\mathbf{p}\geq\mathbf{c}$. ∎
For future reference, we prove that Eqn. (19) strengthens Eqn. (17).
###### Lemma 2.15.
If Eqn. (19) holds, then Eqn. (17) holds.
###### Proof.
Note that Eqn. (19) implies that for any $k$,
$\lim_{p_{j}\to\infty}\big{(}\omega_{k,j}(\mathbf{p})(p_{j}-c_{j})\big{)}<\infty\quad\text{for
all}\quad j\in\mathcal{J}_{f(k)}.$
This, in turn, implies that $\omega_{k,j}(\mathbf{p})\to 0$ as
$p_{j}\to\infty$.
Now Eqn. (17) fails only if
$\lim_{\mathbf{p}\to\mathbf{q}}\lvert\lvert\tilde{\boldsymbol{\Omega}}(\mathbf{p})\rvert\rvert_{\infty}=1$
where $\mathbf{q}$ has some $q_{j}=\infty$. But the row sums of
$\tilde{\boldsymbol{\Omega}}(\mathbf{p})$ satisfy
$\lim_{\mathbf{p}\to\mathbf{q}}\left[\sum_{j\in\mathcal{J}_{f(k)}}\omega_{k,j}(\mathbf{p})\right]=\sum_{j\in\mathcal{J}_{f(k)}\cup\\{j:q_{j}<\infty\\}}\omega_{k,j}(\mathbf{q})<1.$
Thus if Eqn. (19) holds, Eqn. (17) cannot fail. ∎
### 2.9. Existence of Simultaneously Stationary Prices
This section provides two existence results. Neither establish the existence
of a local equilibrium, or the uniqueness of simultaneously stationary points.
To address the existence of local equilibrium will require additional
conditions to ensure that each firm’s profits are locally concave at the
simultaneously stationary prices whose existence can be ensured (Morrow,
2008). Little is known about how to address the uniqueness of simultaneously
stationary points. Indeed, Morrow and Skerlos (2010) provide an example of a
Mixed Logit model with 9 simultaneously stationary prices, 4 of which are
local equilibria and 2 of which are proper equilibria.
Assumption 2.4 ensures the existence of finite simultaneously stationary
prices when $\varsigma_{*}=\infty$.
###### Corollary 2.16.
Suppose $\varsigma_{*}=\infty$ and Assumptions 2.1-2.3, 2.4 hold. Then there
exists at least one vector of finite simultaneously stationary prices.
###### Proof.
This is a direct consequence of Brouwer’s fixed-point theorem.
$\mathbf{c}+\boldsymbol{\eta}(\cdot)$ is a continuous map that takes the
compact, convex set $[0,M/\varepsilon]^{J}$ into itself, and thus there is at
least one fixed-point
$\mathbf{p}=\mathbf{c}+\boldsymbol{\eta}(\mathbf{p})\in[0,M/\varepsilon]^{J}$.
∎
To apply Corollary 2.16 to cases when $\varsigma_{*}<\infty$,
$\boldsymbol{\eta}$ must be extended from $(0,\varsigma_{*})$ to all of
$\mathcal{P}^{J}$ preserving the bounds (15). This is easy for many of the
simulation-based approximations encountered in practice, but difficult for the
general case.
We can extend this existence result using Eqn. (22) and the
$\boldsymbol{\zeta}$ map.
###### Lemma 2.17.
Suppose $\varsigma_{*}=\infty$, Assumptions 2.1-2.3, Eqn. (19), and Eqn. (22)
hold. Then there exists some $\bar{q}_{k}>c_{k}$ such that
$p_{k}-c_{k}-\zeta_{k}(\mathbf{p})>0$ for all $\mathbf{p}$ with
$p_{k}\geq\bar{q}_{k}$.
###### Proof.
The assumed bound implies
$\left\lvert\sum_{j\in\mathcal{J}_{f}}\omega_{k,j}(\mathbf{p})(p_{j}-c_{j})\right\rvert\leq
L<\infty$
for any $k$ and any $\mathbf{p}\in(0,\infty)^{J}$. Consider
$\displaystyle p_{k}-c_{k}-\zeta_{k}(\mathbf{p})$
$\displaystyle=(p_{k}-c_{k})-\sum_{j\in\mathcal{J}_{f}}\omega_{k,j}(\mathbf{p})(p_{j}-c_{j})+\frac{P_{k}(\mathbf{p})}{\lambda_{k}(\mathbf{p})}$
$\displaystyle=\left(1-\frac{P_{k}(\mathbf{p})}{\left\lvert\lambda_{k}(\mathbf{p})\right\rvert(p_{k}-c_{k})}\right)(p_{k}-c_{k})-\sum_{j\in\mathcal{J}_{f}}\omega_{k,j}(\mathbf{p})(p_{j}-c_{j})$
If Eqn. (22) holds, then
$0\leq\lim_{p_{k}\to\infty}\left[\frac{P_{k}(\mathbf{p})}{\left\lvert\lambda_{k}(\mathbf{p})\right\rvert(p_{k}-c_{k})}\right]\leq\delta<1.$
Thus for any $\epsilon>0$, there exists some $\bar{p}_{k}>0$ and
$\triangle(\mathbf{p})$ with
$\left\lvert\triangle(\mathbf{p})\right\rvert<\epsilon$ such that
$\frac{P_{k}(\mathbf{p})}{\lambda_{k}(\mathbf{p})(p_{k}-c_{k})}\leq\delta+\triangle(\mathbf{p})\quad\quad\text{for
all}\quad\quad p_{k}>\bar{p}_{k}.$
Thus
$\displaystyle
p_{k}-c_{k}-\zeta_{k}(\mathbf{p})\geq(1-\delta+\triangle(\mathbf{p}))(p_{k}-c_{k})-L\quad\quad\text{for
all}\quad\quad p_{k}>\bar{p}_{k}.$
In particular, if we choose $\epsilon\leq(1-\delta)/2$ we have
$\displaystyle
p_{k}-c_{k}-\zeta_{k}(\mathbf{p})\geq\left(\frac{1-\delta}{2}\right)(p_{k}-c_{k})-L=\left(\frac{1-\delta}{2}\right)\left(p_{k}-c_{k}-\frac{2L}{1-\delta}\right)>0$
for all $p_{k}\geq\bar{q}_{k}=\max\\{c_{k}+2L/(1-\delta),\bar{p}_{k}\\}$. ∎
One consequence of this lemma is that infinite prices are never an
equilibrium.
###### Corollary 2.18.
Under the assumptions of Lemma 2.17, any profit derivative is eventually
negative.
###### Proof.
Note that
$(D_{k}\hat{\pi}_{f(k)})(\mathbf{p})=-\left\lvert\lambda_{k}(\mathbf{p})\right\rvert(p_{k}-c_{k}-\zeta_{k}(\mathbf{p})).$
Since $p_{k}-c_{k}-\zeta_{k}(\mathbf{p})$ is positive for all large enough
$p_{k}$, $(D_{k}\hat{\pi}_{f(k)})(\mathbf{p})$ is negative for all large
enough $p_{k}$, regardless of $\mathbf{p}_{-k}$. ∎
Another consequence of Lemma 2.17 is an alternative existence result.
###### Corollary 2.19.
Under the assumptions of Lemma 2.17 there exists at least one simultaneously
stationary point.
###### Proof.
Following Morrow and Skerlos (2008), we prove this proposition using the
Poincare-Hopf Theorem (Milnor, 1965). The logic is simple: We will consider
the vector field $\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})$ on a
hyper-rectangle $[\mathbf{c},\bar{\mathbf{q}}]$ whose critical points are
simultaneously stationary; $\bar{\mathbf{q}}$ has components $\bar{q}_{k}$
defined in Lemma 2.17. The Poincare-Hopf Theorem then states that the sum of
the indices of all the critical points of this vector field equals one, the
Euler characteristic of $[\mathbf{c},\bar{\mathbf{q}}]$. Thus it is not
possible that the vector field have no critical points, for then the sum of
indices would be zero.
We must only prove one property of
$\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})$: that this vector field
points outward on the boundary of the chosen hyper-rectangle. Half of this
proof is Lemma 2.17, in which we prove that
$p_{k}-c_{k}-\zeta_{k}(\mathbf{p})>0$ if
$\mathbf{p}\in[\mathbf{c},\bar{\mathbf{q}}]$ with $p_{k}=\bar{q}_{k}$. We must
also show that $p_{k}-c_{k}-\zeta_{k}(\mathbf{p})<0$ if
$\mathbf{p}\in[\mathbf{c},\bar{\mathbf{q}}]$ with $p_{k}=c_{k}$. But
$\displaystyle
c_{k}-c_{k}-\zeta_{k}(\mathbf{p})=-\left(\sum_{j\in\mathcal{J}_{f}}\omega_{k,j}(\mathbf{p})(p_{j}-c_{j})+\frac{P_{k}(\mathbf{p})}{\left\lvert\lambda_{k}(\mathbf{p})\right\rvert}\right)<0.$
∎
This proof does not need to make any claims about the number of critical
points, or of their indices. If it can be shown that any critical point of
$\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})$ cannot have a zero or
negative index, then the simultaneously stationary point is unique.
## 3\. Computational Methods
This section provides details for the four approaches to computing equilibrium
prices described in Morrow and Skerlos (2010); see Table 4. Section 3.1
briefly reviews Newton’s method, followed by application of Newton’s method to
solve Eqn. (7) in Section 3.2. Newton’s method applied directly to Eqn. (7)
may compute “spurious” solutions with infinite prices because the combined
gradient vanishes as prices increase without bound. Section 3.3 avoids this
difficulty by applying Newton’s method to the two markup equations instead of
Eqn. (7) itself. Section 3.4 discusses fixed-point iterations based on the
BLP- and $\boldsymbol{\zeta}$-markup equations, and Section 3.5 reviews a
number of practical considerations.
Table 4. Summary of the numerical methods examined in this article. Newton
Methods (NM)
---
Abbr. | Method | Section | Advantage | Our Experience${}^{\text{(a)}}$
CG-NM | Solve $\mathbf{F}_{\pi}(\mathbf{p})=(\tilde{\nabla}\hat{\pi})(\mathbf{p})=\mathbf{0}$ | 3.2 | $-$ | Unreliable, slow
$\boldsymbol{\eta}$-NM | Solve $\mathbf{F}_{\eta}(\mathbf{p})=\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p})=\mathbf{0}$ | 3.3 | Coercive | Reliable, slow
$\boldsymbol{\zeta}$-NM | Solve $\mathbf{F}_{\zeta}(\mathbf{p})=\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})=\mathbf{0}$ | 3.3 | Coercive | Reliable, slow
Fixed-Point Iterations (FPI)
Abbr. | Method | Section | Advantage | Our Experience
$\boldsymbol{\zeta}$-FPI | Iterate $\mathbf{p}\leftarrow\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$ | 3.4 | Easy to evaluate | Reliable, fast
$\boldsymbol{\eta}$-FPI | Iterate $\mathbf{p}\leftarrow\mathbf{c}+\boldsymbol{\eta}(\mathbf{p})$ | 3.4 | $-$ | Not convergent
(a) Conclusions on behavior of these methods is based on the numerical
experiments
described in Morrow and Skerlos (2010), using a novel GMRES-Newton method with
Levenberg-Marquardt style trust-region global convergence strategy.
### 3.1. Newton’s Method
Newton’s method, a classical technique to compute a zero of an arbitrary
function $\mathbf{F}:\mathbb{R}^{J}\to\mathbb{R}^{J}$, is now a portfolio of
related approaches to solve nonlinear systems (Ortega and Rheinboldt, 1970;
Kelley, 1995; Dennis and Schnabel, 1996; Judd, 1998; Kelley, 2003). Generally
speaking, Newton-type methods are differentiated in two relatively independent
directions: (i) the technique used to approximate the Jacobian matrices
$(D\mathbf{F})$ and solve for the Newton step and (ii) the technique used to
enforce convergence from arbitrary initial conditions. See Dennis and Schnabel
(1996), Judd (1998), or Kelley (2003) for good treatments of these issues.
Choosing the right variant of Newton’s method determines the reliability and
efficiency of equilibrium price computations.
Problem formulation also determines the reliability and efficiency of
equilibrium price computations using Newton’s method. Scalings of the
variables and function values are one prominent example of a problem
transformation that improves the performance of Newton’s method (Dennis and
Schnabel, 1996). Nonlinear problem preconditioning can also be important (Cai
and Keyes, 2002), as the following example demonstrates.
###### Example 5.
Let $\mathbf{F}:\mathbb{R}^{N}\to\mathbb{R}^{N}$ be defined by
$\mathbf{F}(\mathbf{x})=\mathbf{x}/(1+\lvert\lvert\mathbf{x}\rvert\rvert_{2}^{2})$.
Iterating Newton steps converges to the unique (finite) zero
$\mathbf{x}_{*}=\mathbf{0}$ only from initial conditions $\mathbf{x}_{0}$ with
$\lvert\lvert\mathbf{x}_{0}\rvert\rvert_{2}<1/\sqrt{3}$. Newton’s method
diverges or fails for all other starting points. Standard global convergence
strategies for Newton’s method (line search, trust region methods) cannot
improve this poor global convergence behavior because
$\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}$ has unbounded level sets;
see Morrow and Skerlos (2010) for details.
A simple nonlinear transformation overcomes this poor global convergence
behavior. Note that
$\mathbf{F}(\mathbf{x})=\mathbf{A}(\mathbf{x})\mathbf{f}(\mathbf{x})$ where
$\mathbf{A}(\mathbf{x})=(1+\lvert\lvert\mathbf{x}\rvert\rvert_{2}^{2})^{-1}\mathbf{I}$
and $\mathbf{f}(\mathbf{x})=\mathbf{x}$. Because $\mathbf{A}(\mathbf{x})$ is
nonsingular for all $\mathbf{x}$, the problems
$\mathbf{F}(\mathbf{x})=\mathbf{0}$ and $\mathbf{f}(\mathbf{x})=\mathbf{0}$
have identical solution sets. However applying Newton’s method to the problem
$\mathbf{f}(\mathbf{x})=\mathbf{0}$ trivially converges in a single step from
any initial condition without a global convergence strategy.
Example 5 illustrates why computing equilibrium prices based on the markup
equations is more reliable and efficient than using Eqn. (7) directly. The
following two sections echo the pattern of this example to provide the
details.
### 3.2. Newton’s Method on the Combined Gradient
The most direct approach to compute equilibrium prices using Newton’s method
is to solve
$\mathbf{F}_{\pi}(\mathbf{p})=(\tilde{\nabla}\hat{\pi})(\mathbf{p})=\mathbf{0}$,
abbreviated CG-NM in Table 4. This approach works well when the initial
condition is near an equilibrium, as required by theory (Ortega and
Rheinboldt, 1970; Kelley, 1995; Dennis and Schnabel, 1996). In practice,
computing counterfactual equilibrium prices starting with the observed prices
may exploit this local convergence if changes to exogeneous variables have a
relatively small impact on equilibrium prices. On the other hand, CG-NM can be
unreliable when started “far” from equilibrium.
The challenge is the tendency for the derivatives of profits to vanish as
prices become large Morrow and Skerlos (2010), as demonstrated in Example 6
below.
###### Example 6.
Consider a simple Logit model with linear in price utility and an outside
good: $u(p)=-\alpha p+v$ for some $\alpha>0$ and any $v\in\mathbb{R}$, and
$\vartheta>-\infty$. The derivative of firm $f$’s profit function with respect
to the price of product $k\in\mathcal{J}_{f}$ is
$(D_{k}\hat{\pi}_{f})(\mathbf{p})=-\alpha
P_{k}^{L}(\mathbf{p})(p_{k}-c_{k})+\alpha
P_{k}^{L}(\mathbf{p})\hat{\pi}_{f}(\mathbf{p})+P_{k}^{L}(\mathbf{p}).$
Since $P_{k}^{L}(\mathbf{p})$ and $P_{k}^{L}(\mathbf{p})(p_{k}-c_{k})$ both
vanish as $p_{k}\to\infty$ (as is easily checked), $\hat{\pi}_{f}(\mathbf{p})$
is bounded in $\mathbf{p}$. Thus $(D_{k}\hat{\pi}_{f})(\mathbf{p})\to 0$ as
$p_{k}\to\infty$.
We now provide a general assumption under which
$(D_{k}\hat{\pi}_{f(k)})(\mathbf{p})\to 0$ as $p_{k}\uparrow\varsigma_{*}$.
###### Assumption 3.1.
Let $\psi_{k}$ be defined as in Assumption 2.3. Assume: (i)
$p_{k}\psi_{k}(\boldsymbol{\theta},p_{k})\to 0$ as
$p_{k}\uparrow\varsigma(\boldsymbol{\theta})$ for $\mu$-a.e.
$\boldsymbol{\theta}$. (ii) There exists $M<\infty$ and
$\bar{p}_{k}\in[0,\varsigma_{*})$ such that
$p_{k}\int\psi_{k}(\boldsymbol{\theta},p_{k})d\mu(\boldsymbol{\theta})\leq M$
for all $p_{k}\in(\bar{p}_{k},\varsigma_{*})$.
As with Assumption 2.3 above, (i) and (ii) are essentially conditions for the
Dominated Convergence Theorem.
Assumption 3.1 (i) extends Assumption 2.3 (i) to include a neighborhood of
$\varsigma_{*}$. Note that if $\varsigma(\boldsymbol{\theta})<\infty$ then (i)
holds if, and only if, $\psi_{k}(\boldsymbol{\theta},p_{k})\to 0$ as
$p_{k}\uparrow\varsigma(\boldsymbol{\theta})$; i.e.
$\psi_{k}(\boldsymbol{\theta},\cdot)$ is continuous at
$\varsigma(\boldsymbol{\theta})$. Thus if
$\varsigma(\boldsymbol{\theta})<\infty$ Assumption 3.1 (i) and Assumption 2.3
(i) are the same. If $\varsigma(\boldsymbol{\theta})=\infty$ and
$p_{k}\psi_{k}(\boldsymbol{\theta},p_{k})\to 0$ as $p_{k}\uparrow\infty$, then
necessarily $\psi_{k}(\boldsymbol{\theta},p_{k})\to 0$ as
$p_{k}\uparrow\infty$. The converse, however, need not hold.
If $\varsigma_{*}<\infty$, Assumption 3.1 (ii) simply says that
$\int\psi_{k}(\boldsymbol{\theta},p_{k})d\mu(\boldsymbol{\theta})$ is bounded
as $p_{k}\uparrow\varsigma_{*}$. This is not implied by Assumption 2.3 (ii),
but is a natural extension of it.
###### Lemma 3.1.
Suppose Assumptions 2.1-3.1 hold. Then
$p_{k}\left\lvert\lambda_{k}(\mathbf{p})\right\rvert\to 0$ as
$p_{k}\uparrow\varsigma_{*}$ for all $k$. Additionally,
$p_{k}\left\lvert\gamma_{j,k}(\mathbf{p})\right\rvert\to 0$ as
$p_{k}\uparrow\varsigma_{*}$ for all $j$. Subsequently,
$(D_{k}\hat{\pi}_{f(k)})(\mathbf{p})\to 0$ as $p_{k}\uparrow\varsigma_{*}$.
###### Proof.
Let $\\{p_{k}^{(n)}\\}_{n\in\mathbb{N}}\subset(0,\varsigma_{*})$ be any
sequence converging to $\varsigma_{*}$. Define
$\Psi_{k}^{(n)}:\mathcal{T}\to\mathcal{P}$ by
$\Psi_{k}^{(n)}(\boldsymbol{\theta})=p_{k}^{(n)}\psi_{k}(\boldsymbol{\theta},p_{k}^{(n)})$.
The functions $\\{\Psi_{k}^{(n)}\\}_{n\in\mathbb{N}}$ converge pointwise to
zero and have integrals uniformly bounded by the constant $M$. By the
Dominated Convergence Theorem
$\lim_{n\to\infty}\int\Psi_{k}^{(n)}(\boldsymbol{\theta})d\mu(\boldsymbol{\theta})=\int\Big{(}\lim_{n\to\infty}\Psi_{k}^{(n)}(\boldsymbol{\theta})\Big{)}d\mu(\boldsymbol{\theta})=0.$
∎
In other words, under Assumption 3.1 the components of $\mathbf{F}_{\pi}$
vanish as the corresponding price tends to $\varsigma_{*}$ even though this
may not mean that $\varsigma_{*}$ maximizes profits. Because of this, CG-NM
may converge to a zero of $\mathbf{F}_{\pi}$ at $\varsigma_{*}\mathbf{1}$, or
with some components equal to $\varsigma_{*}$, that is not an equilibrium.
Note that even though the price derivatives vanish at infinity, this does not
mean that infinite prices maximize profits. Nonetheless, CG-NM may converge to
a zero of $\mathbf{F}_{\pi}$ with some components equal to infinity that is
not an equilibrium. Moreover, because the components of
$\mathbf{F}_{\pi}(\mathbf{p})$ can vanish over some divergent sequences,
standard global convergence strategies based on minimizing
$\lvert\lvert\mathbf{F}_{\pi}(\mathbf{p})\rvert\rvert_{2}$ will not be
effective ways of avoiding this behavior. As in Example 5, we must reformulate
the problem to obtain reliable and efficient approaches for computing
equilibrium prices.
### 3.3. Newton’s Method and the Markup Equations
Reliable and efficient implementations of Newton’s method are found by
observing that the combined gradient, $\mathbf{F}_{\pi}$, can be written as
follows:
(20) $\displaystyle\mathbf{F}_{\pi}(\mathbf{p})$
$\displaystyle=(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}\mathbf{F}_{\eta}(\mathbf{p})$
where $\displaystyle\mathbf{F}_{\eta}(\mathbf{p})$
$\displaystyle=\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p})$ (21)
$\displaystyle\mathbf{F}_{\pi}(\mathbf{p})$
$\displaystyle=\boldsymbol{\Lambda}(\mathbf{p})\mathbf{F}_{\zeta}(\mathbf{p})$
where $\displaystyle\mathbf{F}_{\zeta}(\mathbf{p})$
$\displaystyle=\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p}).$
Either $\mathbf{F}_{\eta}$ or $\mathbf{F}_{\zeta}$ can be used to compute
simultaneously stationary prices when
$(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}$ and
$\boldsymbol{\Lambda}(\mathbf{p})$, respectively, are nonsingular (Morrow and
Skerlos, 2010). Of course, $\mathbf{F}_{\eta}$ and $\mathbf{F}_{\zeta}$ recast
the first-order condition as a fixed-point problem: $\mathbf{F}_{\eta}$ is
zero if and only if the BLP-markup equation holds, and $\mathbf{F}_{\zeta}$ is
zero if and only if the $\boldsymbol{\zeta}$-markup equation holds.
Solving $\mathbf{F}_{\eta}(\mathbf{p})=\mathbf{0}$ or
$\mathbf{F}_{\zeta}(\mathbf{p})=\mathbf{0}$, abbreviated
$\boldsymbol{\eta}$-NM and $\boldsymbol{\zeta}$-NM respectively in Table 4,
requires the solution of nontrivial nonlinear systems with Newton’s method.
$\boldsymbol{\eta}$-NM and $\boldsymbol{\zeta}$-NM, however, are less likely
to have the computational problems that CG-NM exhibits because they exploit
norm-coercivity of the maps $\mathbf{F}_{\eta}$ and $\mathbf{F}_{\zeta}$
(Morrow and Skerlos, 2010). A norm-coercive map has a norm that tends to
infinity with the norm of its argument (Ortega and Rheinboldt, 1970; Harker
and Pang, 1990). Globally convergent implementations of Newton’s method that
decrease the value of $\lvert\lvert\mathbf{F}(\mathbf{p})\rvert\rvert_{2}$ in
each step produce bounded sequences of iterates when $\mathbf{F}$ is norm-
coercive. Thus, solving the BLP- or $\boldsymbol{\zeta}$-markup equation
instead of the literal first-order condition removes the tendency for
applications of Newton’s method to compute “spurious” solutions at infinity.
We now prove that the maps $\mathbf{F}_{\eta}$ and $\mathbf{F}_{\zeta}$ are
indeed coercive.
###### Lemma 3.2.
Suppose $\varsigma_{*}=\infty$ and Assumption 2.1-2.3 hold.
* (i)
Norm-coercivity of $\mathbf{F}_{\zeta}(\mathbf{p})$ implies that of
$\mathbf{F}_{\eta}(\mathbf{p})$.
* (ii)
If Eqn. (17) holds, then norm-coercivity of $\mathbf{F}_{\eta}(\mathbf{p})$
implies that of $\mathbf{F}_{\zeta}(\mathbf{p})$.
###### Proof.
Proposition 2.12 implies that
$\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})=\big{(}\mathbf{I}-\tilde{\boldsymbol{\Omega}}(\mathbf{p})\big{)}(\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p})).$
To prove (i), note that
$\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p})\rvert\rvert_{\infty}\geq\left(\frac{1}{1+\lvert\lvert\tilde{\boldsymbol{\Omega}}(\mathbf{p})\rvert\rvert_{\infty}}\right)\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}\geq\left(\frac{1}{2}\right)\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}.$
To prove (ii), note that if Eqn. (17) holds,
$\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}\geq\big{(}1-\lvert\lvert\boldsymbol{\Omega}(\mathbf{p})\rvert\rvert_{\infty}\big{)}\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p})\rvert\rvert_{\infty}\geq\varepsilon\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p})\rvert\rvert_{\infty}.$
∎
###### Lemma 3.3.
Suppose $\varsigma_{*}=\infty$ and Assumption 2.1-2.3 and 2.4 hold. Then
$\lim_{\lvert\lvert\mathbf{p}\rvert\rvert_{\infty}\to\infty}\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p})\rvert\rvert_{\infty}=\infty=\lim_{\lvert\lvert\mathbf{p}\rvert\rvert_{\infty}\to\infty}\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}.$
###### Proof.
The norm-coercivity of $\boldsymbol{\eta}$ is a trivial consequence of the
boundedness of $\boldsymbol{\eta}$ under Assumption 2.4. The norm-coercivity
of $\boldsymbol{\zeta}$ then follows from Lemma 3.2. ∎
We now weaken Assumption 2.4’s Eqn. (16).
###### Assumption 3.2.
Suppose that $\varsigma_{*}=\infty$ and
(22)
$\displaystyle\lim_{M\uparrow\infty}\sup\left\\{\frac{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})\rvert\rvert_{\infty}}{\lvert\lvert\mathbf{p}\rvert\rvert_{\infty}}:\mathbf{p}\in\mathcal{P}^{J},\lvert\lvert\mathbf{p}\rvert\rvert_{\infty}\geq
M\right\\}=\delta\in[0,1).$
Note that the limit is of a non-increasing sequence of non-negative numbers,
and thus exists.
###### Lemma 3.4.
Assuming Eqn. (22) is equivalent to assuming that for any sequence
$\mathbf{p}_{n}$ with
$\lvert\lvert\mathbf{p}_{n}\rvert\rvert_{\infty}\to\infty$,
$\lim_{n\to\infty}\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p}_{n})^{-1}\mathbf{P}(\mathbf{p}_{n})\rvert\rvert_{\infty}/\lvert\lvert\mathbf{p}_{n}\rvert\rvert_{\infty}\leq\delta$.
###### Proof.
If Eqn. (22) holds, then for any $\varepsilon>0$ there exists an $M>0$ such
that
$\sup\left\\{\frac{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})\rvert\rvert_{\infty}}{\lvert\lvert\mathbf{p}\rvert\rvert_{\infty}}:\mathbf{p}\in\mathcal{P}^{J},\lvert\lvert\mathbf{p}\rvert\rvert_{\infty}\geq
M\right\\}<\delta+\varepsilon.$
If $\lvert\lvert\mathbf{p}_{n}\rvert\rvert_{\infty}\to\infty$, then there is
also an $N_{\epsilon}$ such that
$\lvert\lvert\mathbf{p}_{n}\rvert\rvert_{\infty}\geq M$ for all
$n>N_{\epsilon}$. Thus
$\frac{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p}_{n})^{-1}\mathbf{P}(\mathbf{p}_{n})\rvert\rvert_{\infty}}{\lvert\lvert\mathbf{p}_{n}\rvert\rvert_{\infty}}<\delta+\varepsilon$
for all $n>N_{\epsilon}$, and thus
$\lim_{n\to\infty}\left[\frac{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p}_{n})^{-1}\mathbf{P}(\mathbf{p}_{n})\rvert\rvert_{\infty}}{\lvert\lvert\mathbf{p}_{n}\rvert\rvert_{\infty}}\right]\leq\delta.$
Conversely, if Eqn. (22) fails, then there is a $\bar{M}>0$ such that
$S(M)=\sup\left\\{\frac{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})\rvert\rvert_{\infty}}{\lvert\lvert\mathbf{p}\rvert\rvert_{\infty}}:\mathbf{p}\in\mathcal{P}^{J},\lvert\lvert\mathbf{p}\rvert\rvert_{\infty}\geq
M\right\\}\geq 1$
for all $M\geq\bar{M}$. We can thus choose $\mathbf{p}_{M}$ with
$\lvert\lvert\mathbf{p}_{M}\rvert\rvert_{\infty}\geq M$ satisfying
$1-\frac{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p}_{M})^{-1}\mathbf{P}(\mathbf{p}_{M})\rvert\rvert_{\infty}}{\lvert\lvert\mathbf{p}_{M}\rvert\rvert_{\infty}}\leq
S(M)-\frac{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p}_{M})^{-1}\mathbf{P}(\mathbf{p}_{M})\rvert\rvert_{\infty}}{\lvert\lvert\mathbf{p}_{M}\rvert\rvert_{\infty}}\leq\frac{1}{M}.$
In other words,
$\frac{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p}_{M})^{-1}\mathbf{P}(\mathbf{p}_{M})\rvert\rvert_{\infty}}{\lvert\lvert\mathbf{p}_{M}\rvert\rvert_{\infty}}\geq
1-\frac{1}{M}$
for all $M\geq\bar{M}$, and thus
$\lim_{M\to\infty}\left[\frac{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p}_{M})^{-1}\mathbf{P}(\mathbf{p}_{M})\rvert\rvert_{\infty}}{\lvert\lvert\mathbf{p}_{M}\rvert\rvert_{\infty}}\right]\geq
1.$
Hence the “sequence version” of Eqn. (22) fails, and thus by contraposition
the sequence version and Eqn. (22) are identical. ∎
Next we note that Eqn. (22) weakens Eqn. (16).
###### Lemma 3.5.
If Eqn. (16) holds, then Eqn. (22) holds.
###### Proof.
If
$\sup\\{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})\rvert\rvert_{\infty}:\mathbf{p}\in\mathcal{P}^{J}\\}\leq
M$, then
$S(L)=\sup\left\\{\frac{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})\rvert\rvert_{\infty}}{\lvert\lvert\mathbf{p}\rvert\rvert_{\infty}}:\mathbf{p}\in\mathcal{P}^{J},\lvert\lvert\mathbf{p}\rvert\rvert_{\infty}\geq
L\right\\}\leq\frac{M}{L}.$
Thus $\lim_{L\to\infty}S(L)=0$, a special case of Eqn. (22). ∎
Now we prove the alternative coercivity result.
###### Lemma 3.6.
Suppose $\varsigma_{*}=\infty$ and Assumptions 2.1-2.3, 2.5 and 3.2 hold. Then
$\lim_{\lvert\lvert\mathbf{p}\rvert\rvert_{\infty}\to\infty}\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p})\rvert\rvert_{\infty}=\infty=\lim_{\lvert\lvert\mathbf{p}\rvert\rvert_{\infty}\to\infty}\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}.$
###### Proof.
We prove the claim for $\boldsymbol{\zeta}$; the result for
$\boldsymbol{\eta}$ then follows from Lemma 2.15. Note that
$\displaystyle\left\lvert
p_{k}-c_{k}-\sum_{j\in\mathcal{J}_{f(k)}}\omega_{k,j}(\mathbf{p})(p_{j}-c_{j})-\frac{P_{k}(\mathbf{p})}{\lambda_{k}(\mathbf{p})}\right\rvert$
$\displaystyle\quad\quad\quad\quad\quad\quad=\left\lvert\left(1-\frac{P_{k}(\mathbf{p})}{p_{k}\lambda_{k}(\mathbf{p})}\right)p_{k}-c_{k}-\sum_{j\in\mathcal{J}_{f(k)}}\omega_{k,j}(\mathbf{p})(p_{j}-c_{j})\right\rvert$
$\displaystyle\quad\quad\quad\quad\quad\quad\geq\left\lvert\;\left\lvert
1-\frac{P_{k}(\mathbf{p})}{p_{k}\lambda_{k}(\mathbf{p})}\right\rvert
p_{k}-\Bigg{|}c_{k}+\sum_{j\in\mathcal{J}_{f(k)}}\omega_{k,j}(\mathbf{p})(p_{j}-c_{j})\Bigg{|}\;\right\rvert$
Suppose that $p_{k}\to\infty$. By assumption,
$\lim_{n\to\infty}\left[1-\frac{P_{k}(\mathbf{p})}{p_{k}\lambda_{k}(\mathbf{p})}\right]\geq
1-\delta>0$
while the second term is bounded. Thus
$\displaystyle\left\lvert
p_{k}-c_{k}-\sum_{j\in\mathcal{J}_{f(k)}}\omega_{k,j}(\mathbf{p})(p_{j}-c_{j})-\frac{P_{k}(\mathbf{p})}{\lambda_{k}(\mathbf{p})}\right\rvert\to\infty.$
∎
Note that since we did not require that Eqn. (17) held, $\boldsymbol{\zeta}$
need not be bounded for $\mathbf{F}_{\eta}$ and $\mathbf{F}_{\zeta}$ to be
coercive.
### 3.4. Fixed-Point Iteration
In addition to applications of Newton’s method, the BLP- and
$\boldsymbol{\zeta}$-markup equations suggest applying fixed-point iteration
to solve for equilibrium prices. We examine fixed-point iterations based on
both equations.
#### 3.4.1. $\zeta$ Fixed-Point Iteration
The fixed-point iteration
$\mathbf{p}\leftarrow\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$ based on the
$\boldsymbol{\zeta}$-markup equation, here abbreviated
$\boldsymbol{\zeta}$-FPI, can efficiently compute equilibrium prices for some
problems. $\boldsymbol{\zeta}$-FPI has relatively efficient steps because no
linear systems need to be solved, unlike every other method listed in Table 4.
While we are not aware of a general convergence proof for
$\boldsymbol{\zeta}$-FPI, this iteration has converged reliably on test
problems including the examples in Morrow and Skerlos (2010).
The first observation we make is that the $\boldsymbol{\zeta}$-FPI steps
always point in directions of “myopic gradient ascent.”
###### Lemma 3.7.
Let $\mathbf{p}\in(0,\varsigma_{*})^{J}$, and let
$\boldsymbol{\delta}\mathbf{p}=\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})-\mathbf{p}$
denote the $\boldsymbol{\zeta}$-FPI step. Then
$\frac{1}{\max_{j}\left\lvert\lambda_{j}(\mathbf{p})\right\rvert}\leq\frac{(\tilde{\nabla}\hat{\pi})(\mathbf{p})^{\top}\boldsymbol{\delta}\mathbf{p}}{(\tilde{\nabla}\hat{\pi})(\mathbf{p})^{\top}(\tilde{\nabla}\hat{\pi})(\mathbf{p})}\leq\frac{1}{\min_{j}\left\lvert\lambda_{j}(\mathbf{p})\right\rvert}.$
Similarly, let $\theta(\mathbf{p})$ denote the angle between
$\boldsymbol{\delta}\mathbf{p}$ and $(\tilde{\nabla}\hat{\pi})(\mathbf{p})$,
and suppose $\mathbf{p}$ is not simultaneously stationary. Then
$\cos\theta(\mathbf{p})\geq\frac{\min_{j}\left\lvert\lambda_{j}(\mathbf{p})\right\rvert}{\max_{j}\left\lvert\lambda_{j}(\mathbf{p})\right\rvert}.$
###### Proof.
Both results follows directly from the equation
$(\tilde{\nabla}\hat{\pi})(\mathbf{p})=\left\lvert\boldsymbol{\Lambda}(\mathbf{p})\right\rvert\boldsymbol{\delta}\mathbf{p}$
where $\left\lvert\boldsymbol{\Lambda}(\mathbf{p})\right\rvert$ denotes the
absolute value of the components of $\boldsymbol{\Lambda}(\mathbf{p})$. ∎
Specifically, the $\boldsymbol{\zeta}$-FPI steps have a positive projection
onto the combined gradient, and cannot become orthogonal to the combined
gradient over any sequence of non-simultaneously stationary prices that stay
in $(0,\varsigma_{*})^{J}$.
If $F=1$, and the equilibrium problem is an optimization problem, this implies
$\boldsymbol{\zeta}$-FPI has steps that point in gradient ascent directions
and, when properly scaled, converge to local maximizers of profit. More
specifically, $\boldsymbol{\zeta}$-FPI cannot converge to minimizers of
profits. This may generate the properties of $\boldsymbol{\zeta}$-FPI observed
in Example 10 from Morrow and Skerlos (2010).
###### Corollary 3.8.
Let Assumptions 2.1-2.4 hold, and suppose
$\\{\mathbf{p}^{(n)}\\}_{n=1}^{\infty}$ is the $\boldsymbol{\zeta}$-FPI
sequence. Then $\\{\mathbf{p}^{(n)}\\}_{n=1}^{\infty}$ is bounded.
###### Proof.
By Lemma 2.13, for any sufficiently large $M>0$ we can find some $L>0$ such
that
$\lvert\lvert\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}<\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}-M\quad\text{for
all}\quad\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}>L.$
If the $\boldsymbol{\zeta}$-FPI sequence diverges, then for any such $L$ there
is an $N$ such that
$\lvert\lvert\mathbf{p}^{(n)}-\mathbf{c}\rvert\rvert_{\infty}>L\quad\text{for
all}\quad n>N.$
But then
$\lvert\lvert\mathbf{p}^{(n+1)}-\mathbf{c}\rvert\rvert_{\infty}=\lvert\lvert\boldsymbol{\zeta}(\mathbf{p}^{(n)})\rvert\rvert_{\infty}<\lvert\lvert\mathbf{p}^{(n)}-\mathbf{c}\rvert\rvert_{\infty}-M<\lvert\lvert\mathbf{p}^{(n)}-\mathbf{c}\rvert\rvert_{\infty}\quad\text{for
all}\quad n>N,$
which states that the $\boldsymbol{\zeta}$-FPI sequence is decreasing. This is
a contradiction of the hypothesis that the $\boldsymbol{\zeta}$-FPI sequence
diverges. ∎
To implement $\boldsymbol{\zeta}$-FPI, one simply needs to iterate the
assignment $\mathbf{p}\leftarrow\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$
where Eqn. (18) defines $\boldsymbol{\zeta}(\mathbf{p})$. As shown in Table 5
below, integral approximations, rather than the actual computation of the
step, drive the computational burden. Given a price vector, utilities, and
utility derivatives, computing $\mathbf{P}(\mathbf{p})$,
$\boldsymbol{\Lambda}(\mathbf{p})$, and
$\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$ for a set of $S$ samples requires
$\mathcal{O}(S\sum_{f=1}^{F}J_{f}^{2})$ floating point operations (flops),
while the fixed-point step itself only requires
$\mathcal{O}(\sum_{f=1}^{F}J_{f}^{2})$ flops. Note that computing the fixed-
point step $\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$ requires an equivalent
amount of work as computing the combined gradient
$(\tilde{\nabla}\hat{\pi})(\mathbf{p})$. Furthermore, because
$\boldsymbol{\Lambda}(\mathbf{p})$ is a diagonal matrix, no serious obstacles
to computing the fixed point step arise as $J$ becomes large.
#### 3.4.2. $\eta$ Fixed-Point Iteration
The fixed-point iteration
$\mathbf{p}\leftarrow\mathbf{c}+\boldsymbol{\eta}(\mathbf{p})$, abbreviated
$\boldsymbol{\eta}$-FPI, based on the BLP-markup equation need not converge.
Example 7 below, repeated from Morrow and Skerlos (2010), gives a case in
which $\boldsymbol{\eta}$ can fail to be even locally convergent.
###### Example 7.
Consider multi-product monopoly pricing with a simple Logit model having
$u_{j}(p)=-\alpha p+v_{j}$ for some $\alpha>0$, any $v_{j}\in\mathbb{R}$, and
$\vartheta>-\infty$. It is well known that for a single-product firm, unique
profit-maximizing prices exist (Anderson and de Palma, 1988; Milgrom and
Roberts, 1990; Caplin and Nalebuff, 1991). Morrow (2008) proves that profit-
optimal prices $\mathbf{p}_{*}$ are unique for the multi-product case $-$ and
even so with multiple firms $-$ even though profits are not quasi-concave
(Hanson and Martin, 1996).
In this example, $\boldsymbol{\eta}$-FPI is not always locally convergent near
$\mathbf{p}_{*}$, while $\boldsymbol{\zeta}$-FPI is always superlinearly
locally convergent. For an arbitrary continuously differentiable function
$\mathbf{F}$ and $\mathbf{p}_{*}=\mathbf{F}(\mathbf{p}_{*})$, $\mathbf{F}$ is
contractive on some neighborhood of $\mathbf{p}_{*}$ in some norm
$\lvert\lvert\cdot\rvert\rvert$ if $\rho((D\mathbf{F})(\mathbf{p}_{*}))<1$
where $\rho(\mathbf{A})$ (Ortega and Rheinboldt, 1970). We show that
$\rho((D\boldsymbol{\eta})(\mathbf{p}_{*}))>1$ may hold while
$\rho((D\boldsymbol{\zeta})(\mathbf{p}_{*}))=0$, where $\rho(\mathbf{A})$
denotes the spectral radius of the matrix $\mathbf{A}$.
The components of the BLP-markup function $\boldsymbol{\eta}$ are given by
$\eta_{k}(\mathbf{p})=\alpha^{-1}(1-\sum_{j=1}^{J}P_{j}^{L}(\mathbf{p}))^{-1}$
for all $k$. From this formula the equation
$\rho((D\boldsymbol{\eta})(\mathbf{p}_{*}))=\frac{\sum_{j=1}^{J}P_{j}^{L}(\mathbf{p}_{*})}{1-\sum_{j=1}^{J}P_{j}^{L}(\mathbf{p}_{*})}=\sum_{j=1}^{J}e^{u_{j}(p_{j,*})-\vartheta}$
can be derived. For valuations of the outside good, $\vartheta$, sufficiently
close to $-\infty$, $\rho((D\boldsymbol{\eta})(\mathbf{p}_{*}))>1$ can hold;
see Morrow and Skerlos (2010) for details.
To prove the claim regarding $\rho((D\boldsymbol{\zeta})(\mathbf{p}_{*}))$,
note that $\zeta_{k}(\mathbf{p})=\hat{\pi}(\mathbf{p})+1/\alpha$, and thus
$(D_{l}\zeta_{k})(\mathbf{p}_{*})=(D_{l}\hat{\pi})(\mathbf{p}_{*})=0$ for all
$k,l$.
Even if the BLP-markup equation does generate a convergent fixed-point
iteration, evaluating $\boldsymbol{\eta}$ involves the solution of $F$ linear
systems that grow in size with the number of products offered by the firms.
The work required to evaluate $\boldsymbol{\eta}$ using a direct method like
PLU or QR factorization is $\mathcal{O}([\max_{f}J_{f}]^{3})$, given values of
$\mathbf{P}(\mathbf{p})$, $\boldsymbol{\Lambda}(\mathbf{p})$, and
$\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$ as approximated using simulation.
The work to evaluate $\boldsymbol{\zeta}$ is only
$\mathcal{O}([\max_{f}J_{f}]^{2})$ given $\mathbf{P}(\mathbf{p})$,
$\boldsymbol{\Lambda}(\mathbf{p})$, and
$\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$ (Table 5). Generally speaking,
function evaluations must be cheap for the linear convergence of fixed-point
iterations to result in faster computations than the superlinearly or
quadratically convergent variants of Newton’s method.
### 3.5. Practical Considerations
This section addresses several practical considerations.
#### 3.5.1. Simulation
Any method for computing equilibrium prices under Mixed Logit models faces a
common obstacle: the integrals that define the choice probabilities
($\mathbf{P}$) and their derivatives
($\boldsymbol{\Lambda},\tilde{\boldsymbol{\Gamma}}$) cannot be computed
exactly. We employ finite-sample versions of the methods discussed below by
drawing $S\in\mathbb{N}$ samples from the demographic distribution and
applying the method to the finite-sample model thus generated. Particularly,
these samples are used to compute approximate $\mathbf{P}(\mathbf{p})$,
$\boldsymbol{\Lambda}(\mathbf{p})$, and
$\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$; see Table 5. These samples are kept
fixed for all steps of the method and, in principle, can be generated in any
way. We draw directly from the demographic distribution, although importance
and quasi-random sampling (e.g., see Train (2003)) can also be employed. The
Law of Large Numbers motivates this widely-used approach to econometric
analysis (e.g., see McFadden (1989) and Draganska and Jain (2004)). While all
numerical approaches for computing equilibrium prices described here rely on a
Law of Large Numbers for simultaneously stationary prices, we do not provide a
formal convergence theorem. We do provide numerical evidence that computed
equilibrium prices based on the fixed-point iteration for our examples do
indeed follow such a law.
#### 3.5.2. Truncation of Low Purchase Probability Products
All of the methods we implement can be built to ignore products with
excessively low choice probabilities. That is, one can ignore price updates
for all products with $P_{j}(\mathbf{p})\leq\varepsilon_{P}$, where
$\varepsilon_{P}$ is some small value (say $10^{-10}$). Products with a choice
probability this small (or smaller) need not be considered a part of the
market in the price equilibrium computations. For example, Wards (2007)
reports total sales of cars and light trucks during 2005 as $N=16,947,754$.
Particularly, 7,667,066 cars and 9,280,688 light trucks. Because expected
demand is defined by $\mathbb{E}[Q_{j}(\mathbf{p})]=NP_{j}(\mathbf{p})$, any
$\varepsilon_{P}\leq 0.5*N^{-1}\approx 3\times 10^{-8}$ ignores any vehicle
that, as priced, is not expected to have a single customer out of the millions
of customers that bought or considered buying new vehicles. There are also
technical reasons for this truncation. Particularly,
$\boldsymbol{\Lambda}(\mathbf{p})$ and
$(D\tilde{\nabla}\hat{\pi})(\mathbf{p})$ become singular as
$P_{j}(\mathbf{p})\to 0$, for any $j$. Truncating avoids this non-singularity
and hopefully helps conditioning.
#### 3.5.3. Termination Conditions
We terminate all iterations with the numerical simultaneous stationarity
condition
$\lvert\lvert(\tilde{\nabla}\hat{\pi})(\mathbf{p})\rvert\rvert_{\infty}\leq\varepsilon_{T}$
where $\varepsilon_{T}$ is some small number (e.g., $10^{-6}$). Note that a
standard application of Newton’s method to solve
$\mathbf{F}_{\eta}(\mathbf{p})=\mathbf{0}$ or
$\mathbf{F}_{\zeta}(\mathbf{p})=\mathbf{0}$ would terminate when either
(23)
$\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p})\rvert\rvert_{\infty}\leq\varepsilon_{T}\quad\quad\text{or}\quad\quad\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}\leq\varepsilon_{T},$
respectively. For example, Aguirregabiria and Vicentini (2006) use the
condition
$\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p})\rvert\rvert_{\infty}\leq\varepsilon_{T}$.
Ensuring that Eqn. (23) holds does not necessarily imply that
$\lvert\lvert(\tilde{\nabla}\hat{\pi})(\mathbf{p})\rvert\rvert_{\infty}\leq\varepsilon_{T}$,
the strictly interpreted first-order condition.
Because
$(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}(\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p}))=(\tilde{\nabla}\hat{\pi})(\mathbf{p})=\boldsymbol{\Lambda}(\mathbf{p})(\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})),$
it is easy to terminate all methods, CG-NM, $\boldsymbol{\eta}$-NM,
$\boldsymbol{\zeta}$-NM, and $\boldsymbol{\zeta}$-FPI, when
$\lvert\lvert(\tilde{\nabla}\hat{\pi})(\mathbf{p})\rvert\rvert_{\infty}\leq\varepsilon_{T}$.
While this is done here to ensure consistency in our comparisons of different
methods,
$\lvert\lvert(\tilde{\nabla}\hat{\pi})(\mathbf{p})\rvert\rvert_{\infty}\leq\varepsilon_{T}$
should always be the termination condition for price equilibrium computations.
Three other standard termination conditions are used (Brown and Saad, 1990;
Dennis and Schnabel, 1996). We terminate the iteration if the (relative) step
length becomes too small, if a maximum number of iterations is exceeded, or if
an exceptional event occurs (e.g. division by zero). These three conditions
are considered “failure” as the iteration has failed to compute a numerically
simultaneously stationary point in the sense of the first termination
condition.
#### 3.5.4. Second-Order Conditions.
Each method in Table 4 finds simultaneously stationary points, rather than
local equilibria. Unlike in optimization, there is no a priori assurance that
first-order iterative methods for equilibrium problems will converge to
certain types of stationary points. Thus in computing equilibria it is vitally
important to check the second-order sufficient conditions to verify that a
local equilibrium has indeed been found.
In local equilibrium every firm’s profit Hessian,
$(D_{f}\nabla_{f}\hat{\pi}_{f})(\mathbf{p})$, should also be negative
definite. The formulas given in Proposition 3.9 below provide an expression
for $(D_{f}\nabla_{f}\hat{\pi}_{f})(\mathbf{p})$ that we use to check the
second-order sufficient condition. Cholesky factorization, rather than direct
approximation of the spectrum, is used to test the negative definiteness of
$(D_{f}\nabla_{f}\hat{\pi}_{f})(\mathbf{p})$ (Golub and Loan, 1996).
#### 3.5.5. Computational Burden
Table 5 reviews the formulae and computational burden of computing
$(\tilde{\nabla}\hat{\pi})$, $\boldsymbol{\eta}$, and $\boldsymbol{\zeta}$.
Table 5. Work required to evaluate $(\tilde{\nabla}\hat{\pi})$, $\boldsymbol{\eta}$, and $\boldsymbol{\zeta}$ given $S$ samples $\\{\boldsymbol{\theta}_{s}\\}_{s=1}^{S}\subset\mathcal{T}$, an $S\times J$ matrix $\mathbf{L}(\mathbf{p})$ of Logit choice probabilities ($(\mathbf{L}(\mathbf{p}))_{s,j}=P_{j}^{L}(\boldsymbol{\theta}_{s},\mathbf{p})$), and an $S\times J$ matrix of utility derivatives $\mathbf{D}(\mathbf{p})$ ($(\mathbf{D}(\mathbf{p}))_{s,j}=(Dw_{j})(\boldsymbol{\theta}_{s},p_{j})$). The first section gives work required for sample-average approximations to $\mathbf{P}(\mathbf{p})$, $\boldsymbol{\Lambda}(\mathbf{p})$, and $\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$. The second section takes $\mathbf{P}(\mathbf{p})$, $\boldsymbol{\Lambda}(\mathbf{p})$, and $\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$ as given. Quantity | Formula | flops
---|---|---
$\mathbf{P}(\mathbf{p})$ | $S^{-1}\mathbf{L}(\mathbf{p})^{\top}\mathbf{1}$ | $SJ$
$\mathbf{V}(\mathbf{p})$ | $\mathbf{L}(\mathbf{p})\cdot\mathbf{D}(\mathbf{p})$${}^{\text{(a)}}$ | $SJ$
$\boldsymbol{\Lambda}(\mathbf{p})$ | $S^{-1}\mathbf{V}(\mathbf{p})^{\top}\mathbf{1}$ | $SJ$
$\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$ | $S^{-1}\mathbf{L}(\mathbf{p})^{\top}\mathbf{V}(\mathbf{p})$ | $2S\sum_{f=1}^{F}J_{f}^{2}$
Total work to compute $\mathbf{P}(\mathbf{p})$, $\boldsymbol{\Lambda}(\mathbf{p})$, and $\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$ | $S\left(3J+2\sum_{f=1}^{F}J_{f}^{2}\right)$
$\boldsymbol{\zeta}(\mathbf{p})$ | $\tilde{\boldsymbol{\Omega}}(\mathbf{p})(\mathbf{p}-\mathbf{c})-\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})$ | $2\sum_{f=1}^{F}J_{f}^{2}+4J$
$\boldsymbol{\eta}(\mathbf{p})$ | $(\mathbf{I}-\tilde{\boldsymbol{\Omega}}(\mathbf{p}))\boldsymbol{\eta}(\mathbf{p})=-\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})$ | $\left(\frac{4}{3}\right)\sum_{f=1}^{F}J_{f}^{3}+\left(\frac{7}{2}\right)\left(\sum_{f=1}^{F}J_{f}^{2}+J\right)-2$
$(\tilde{\nabla}\hat{\pi})(\mathbf{p})$ | $(\boldsymbol{\Lambda}(\mathbf{p})-\tilde{\boldsymbol{\Gamma}}(\mathbf{p})^{\top})(\mathbf{p}-\mathbf{c})+\mathbf{P}(\mathbf{p})$ | $2\sum_{f=1}^{F}J_{f}^{2}+5J$
| $=\boldsymbol{\Lambda}(\mathbf{p})(\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p}))$ | $2\sum_{f=1}^{F}J_{f}^{2}+6J$
(a) “$\cdot$” here denotes element-by-element multiplication.
Computing $\boldsymbol{\eta}$ and applying Newton’s method to
$\mathbf{F}_{\eta}$ requires solving linear systems. We give some more details
regarding these computations here. As stated above, the linear system
$(\mathbf{I}-\tilde{\boldsymbol{\Omega}}(\mathbf{p}))\boldsymbol{\eta}(\mathbf{p})=-\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})$
should be used to solve for $\boldsymbol{\eta}(\mathbf{p})$. Note also that
only the systems
$(\mathbf{I}-\boldsymbol{\Omega}_{f}(\mathbf{p}))\boldsymbol{\eta}_{f}(\mathbf{p})=-\boldsymbol{\Lambda}_{f}(\mathbf{p})^{-1}\mathbf{P}_{f}(\mathbf{p})$
for all $f$ need be solved. Of course, our condition bound applies within
firms as well:
$\kappa_{2}\big{(}(D_{f}\mathbf{P}_{f})(\mathbf{p})^{\top}\big{)}\geq\left(\frac{\max_{j\in\mathcal{J}_{f}}\left\lvert\lambda_{j}(\mathbf{p})\right\rvert}{\min_{j\in\mathcal{J}_{f}}\left\lvert\lambda_{j}(\mathbf{p})\right\rvert}\right)\kappa_{2}\big{(}\mathbf{I}-\boldsymbol{\Omega}_{f}(\mathbf{p})\big{)}.$
If Householder QR factorization is used to solve these systems, then computing
$\boldsymbol{\eta}(\mathbf{p})$ from $\mathbf{P}(\mathbf{p})$,
$\boldsymbol{\Lambda}(\mathbf{p})$, and
$\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$ requires
$\mathcal{O}(\sum_{f=1}^{F}J_{f}^{3})$ flops (Table 5).
This is a significant increase in computational effort relative to computing
$\boldsymbol{\zeta}(\mathbf{p})$ or $(\tilde{\nabla}\hat{\pi})(\mathbf{p})$.
The diagonal dominance of
$\mathbf{I}-\tilde{\boldsymbol{\Omega}}(\mathbf{p})$, indeed of
$(\tilde{D}\mathbf{P})(\mathbf{p})$ itself, suggests that Jacobi, Gauss-
Seidel, and Successive Over-Relaxation (SOR) iterations (Golub and Loan, 1996)
may be a relatively efficient way to compute $\boldsymbol{\eta}$.
Additional work is required to compute $(D\boldsymbol{\eta})(\mathbf{p})$, if
this is to be used in Newton’s method. Though it requires solving a matrix-
linear system of the type
$(\tilde{D}\mathbf{P})(\mathbf{p})(D\boldsymbol{\eta})(\mathbf{p})=\mathbf{B}(\mathbf{p})$,
the required matrix factorizations of
$\mathbf{I}-\boldsymbol{\Omega}_{f}(\mathbf{p})$ need only be computed once to
compute both $\boldsymbol{\eta}$ and $(D\boldsymbol{\eta})$, but must be
updated for each vector of prices.
### 3.6. Computing Jacobian Matrices for Newton’s Method
Standard “exact” or Quasi-Newton methods to solve
$\mathbf{F}(\mathbf{x})=\mathbf{0}$ either always or periodically require the
Jacobian matrix $(D\mathbf{F})(\mathbf{x})$. Using finite differences to
approximate Jacobian matrices requires $J$ evaluations of the function
$\mathbf{F}$, an unacceptable workload. In the 993 vehicle example from Morrow
and Skerlos (2010), approximating $(D\mathbf{F})(\mathbf{x})$ once with finite
differences would take roughly 993 evaluations of $\mathbf{F}$, when the work
of less than 50 evaluations appears to sufficient to converge to equilibrium
prices using the $\boldsymbol{\zeta}$-FPI.
We recommend directly approximating $(D\mathbf{F})(\mathbf{x})$ using integral
expressions for $(D\tilde{\nabla}\hat{\pi})(\mathbf{p})$,
$(D\boldsymbol{\eta})(\mathbf{p})$, and $(D\boldsymbol{\zeta})(\mathbf{p})$
provided below. An alternative is to use automatic differentiation, but we are
skeptical that this would in fact be faster than the direct formulae provided
here.
#### 3.6.1. Jacobian of the Combined Gradient
Assuming a second application of the Leibniz Rule holds, we can derive
integral expressions for the second derivatives
$(D_{l}D_{k}\hat{\pi}_{f(k)})(\mathbf{p})$ through
$\big{(}(D\tilde{\nabla}\hat{\pi})(\mathbf{p})\big{)}_{k,l}=(D_{l}D_{k}\hat{\pi}_{f(k)})(\mathbf{p})=\int(D_{l}D_{k}\hat{\pi}_{f(k)}^{L})(\boldsymbol{\theta},\mathbf{p})d\mu(\boldsymbol{\theta}).$
###### Proposition 3.9.
Let $w$ be twice continuously differentiable in $p$ and suppose a second
application of the Leibniz Rule holds for the Mixed Logit choice probabilities
at $\mathbf{p}$. Set
$\displaystyle\phi_{k,l}(\mathbf{p})$
$\displaystyle=\int(Dw_{k})(\boldsymbol{\theta},p_{k})P_{k}^{L}(\boldsymbol{\theta},\mathbf{p})P_{l}^{L}(\boldsymbol{\theta},\mathbf{p})(Dw_{l})(\boldsymbol{\theta},p_{l})d\mu(\boldsymbol{\theta})$
$\displaystyle\psi_{k,l}(\mathbf{p})$
$\displaystyle=\int(Dw_{k})(\boldsymbol{\theta},p_{k})P_{k}^{L}(\boldsymbol{\theta},\mathbf{p})\hat{\pi}_{f(k)}^{L}(\boldsymbol{\theta},\mathbf{p})P_{l}^{L}(\boldsymbol{\theta},\mathbf{p})(Dw_{l})(\boldsymbol{\theta},p_{l})d\mu(\boldsymbol{\theta})$
$\displaystyle\chi_{k}(\mathbf{p})$
$\displaystyle=\left(\frac{1}{2}\right)\int\big{(}(D^{2}w_{k})(\boldsymbol{\theta},p_{k})+(Dw_{k})(\boldsymbol{\theta},p_{k})^{2}\big{)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\times
P_{k}^{L}(\boldsymbol{\theta},\mathbf{p})\big{(}(p_{k}-c_{k})-\hat{\pi}_{f(k)}^{L}(\boldsymbol{\theta},\mathbf{p})\big{)}d\mu(\boldsymbol{\theta})$
* (i)
Component form: Setting
$\xi_{k,l}(\mathbf{p})=\delta_{k,l}(\lambda_{k}(\mathbf{p})+\chi_{k}(\mathbf{p}))-\gamma_{k,l}(\mathbf{p})-(p_{k}-c_{k})\varphi_{k,l}(\mathbf{p})$
we have
$\displaystyle(D_{l}D_{k}\hat{\pi}_{f(k)})(\mathbf{p})=\xi_{k,l}(\mathbf{p})+2\psi_{k,l}(\mathbf{p})+\delta_{f(k),f(l)}\xi_{l,k}(\mathbf{p})$
* (ii)
Matrix form: Let $\boldsymbol{\Phi}(\mathbf{p})$,
$\boldsymbol{\Psi}(\mathbf{p})$ and
$\mathbf{X}(\mathbf{p})=\mathrm{diag}(\boldsymbol{\chi}(\mathbf{p}))$ be the
matrices of these quantities. Also set
$\boldsymbol{\Xi}(\mathbf{p})=\boldsymbol{\Lambda}(\mathbf{p})-\boldsymbol{\Gamma}(\mathbf{p})-\mathrm{diag}(\mathbf{p}-\mathbf{c})\boldsymbol{\Phi}(\mathbf{p})+\mathbf{X}(\mathbf{p}).$
and
$(\tilde{\boldsymbol{\Xi}}(\mathbf{p}))_{k,l}=\left\\{\begin{aligned}
&\xi_{k,l}(\mathbf{p})&&\quad\text{if }f(k)=f(l)\\\ &\quad 0&&\quad\text{if
}f(k)\neq f(l)\end{aligned}\right.$
Then
(24)
$(D\tilde{\nabla}\hat{\pi})(\mathbf{p})=\boldsymbol{\Xi}(\mathbf{p})+2\boldsymbol{\Psi}(\mathbf{p})+\tilde{\boldsymbol{\Xi}}(\mathbf{p})^{\top}.$
###### Proof.
To see that this only relies on a second application of the Leibniz Rule to
the choice probabilities, note that
$(D_{l}D_{k}\hat{\pi}_{f(k)})(\mathbf{p})=\sum_{j\in\mathcal{J}_{f(k)}}(D_{l}D_{k}P_{j})(\mathbf{p})(p_{j}-c_{j})+\delta_{f(k),f(l)}(D_{k}P_{l})(\mathbf{p})+(D_{l}P_{k})(\mathbf{p})$
and thus the continuous second-order differentiability of
$\hat{\pi}_{f}(\mathbf{p})$ depends only on the second-order continuous
differentiability of $\mathbf{P}_{f}$. This result is then an immediate
consequence of the validity of the Leibniz Rule, if a bit tedious to derive. ∎
The validity of a second application of the Leibniz Rule to the choice
probabilities is ensured by the following condition.
###### Proposition 3.10.
Let $(u,\vartheta,\mu)=(w+v,\vartheta,\mu)$ be such that
* (i)
$w(\boldsymbol{\theta},\mathbf{y},\cdot):(0,\varsigma_{*})\to\mathbb{R}$ is
twice continuously differentiable for all $\mathbf{y}\in\mathcal{Y}$ and
$\mu$-a.e. $\boldsymbol{\theta}\in\mathcal{T}$
* (ii)
for all $(\mathbf{y},p)\in\mathcal{Y}\times(0,\varsigma_{*})$,
$\left\lvert(D^{2}w)(\cdot,\mathbf{y},q)+(Dw)(\cdot,\mathbf{y},q)^{2}\right\rvert
e^{u(\cdot,\mathbf{y},q)-\vartheta(\cdot)}:\mathcal{T}\to[0,\infty)$ is
uniformly $\mu$-integrable for all $q$ in some neighborhood of $p$.
* (iii)
for all
$(\mathbf{y},p),(\mathbf{y}^{\prime},p^{\prime})\in\mathcal{Y}\times(0,\varsigma_{*})$,
$\left\lvert(Dw)(\cdot,\mathbf{y},q)\right\rvert
e^{u(\cdot,\mathbf{y},q)-\vartheta(\cdot)}e^{u(\cdot,\mathbf{y}^{\prime},q^{\prime})-\vartheta(\cdot)}\left\lvert(Dw)(\cdot,\mathbf{y}^{\prime},q^{\prime})\right\rvert:\mathcal{T}\to[0,\infty)$
is uniformly $\mu$-integrable for all $(q,q^{\prime})$ in some neighborhood of
$(p,p^{\prime})$.
Then a second application of the Leibniz Rule holds for the Mixed Logit choice
probabilities, which are also continuously differentiable on
$(\mathbf{0},\varsigma_{*}\mathbf{1})$.
This is proved in the same manner as Proposition 2.4.
We also observe the following.
###### Proposition 3.11.
If $P_{k}(\mathbf{p})=0$ then
$(D_{l}D_{k}\hat{\pi}_{f(k)})(\mathbf{p})=(D_{k}D_{l}\hat{\pi}_{f(l)})(\mathbf{p})=0$
for all $l\in\mathbb{N}(J)$.
The proof follows from the derivative formulae given above. Of course, if
$P_{k}(\mathbf{p})=0$ then $(D_{k}\hat{\pi}_{f(k)})(\mathbf{p})=0$ as well and
we have the following situation: (i) the Newton system is consistent for any
$s_{k}^{N}(\mathbf{p})\in\mathbb{R}$ and (ii) $s_{l}^{N}(\mathbf{p})$ does not
depend on $s_{k}^{N}(\mathbf{p})$ for all $l\in\mathbb{N}(J)\setminus\\{k\\}$.
Thus, in practice one can restrict attention to the Newton step defined by the
submatrix of $(D\tilde{\nabla}\hat{\pi})(\mathbf{p})$ formed by rows and
columns indexed by $\\{j:P_{j}(\mathbf{p})>\varepsilon_{P}\\}$.
The formulae above give the following expression of the profit Hessians.
###### Corollary 3.12.
Let $w$ be twice continuously differentiable in $p$ and suppose a second
application of the Leibniz Rule holds for the Mixed Logit choice
probabilities. Firm $f$’s profit Hessian is given by
$(D_{f}\nabla_{f}\hat{\pi}_{f})(\mathbf{p})=\boldsymbol{\Xi}_{f,f}(\mathbf{p})+2\boldsymbol{\Psi}_{f,f}(\mathbf{p})+\boldsymbol{\Xi}_{f,f}(\mathbf{p})^{\top}.$
#### 3.6.2. The $\boldsymbol{\eta}$ map.
For
$\mathbf{F}_{\eta}(\mathbf{p})=\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p})$,
we have
$(D\mathbf{F}_{\eta})(\mathbf{p})=\mathbf{I}-(D\boldsymbol{\eta})(\mathbf{p})$
where $(D\boldsymbol{\eta})(\mathbf{p})$ solves the linear matrix equation
$(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}(D\boldsymbol{\eta})(\mathbf{p})=-(\mathbf{A}(\mathbf{p})+(D\mathbf{P})(\mathbf{p})).$
Here
$(\mathbf{A}(\mathbf{p}))_{k,l}=\sum_{j\in\mathcal{J}_{f(k)}}(D_{l}D_{k}P_{j})(\mathbf{p})\eta_{j}(\mathbf{p})$.
This is easily derived from the defining formula
$(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}\boldsymbol{\eta}(\mathbf{p})=-\mathbf{P}(\mathbf{p})$.
#### 3.6.3. The $\boldsymbol{\zeta}$ map.
For
$\mathbf{F}_{\zeta}(\mathbf{p})=\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})$,
we have
$(D\mathbf{F}_{\zeta})(\mathbf{p})=\mathbf{I}-(D\boldsymbol{\zeta})(\mathbf{p})$
where $(D\boldsymbol{\zeta})(\mathbf{p})$ can be computed using the following
formula:
$\displaystyle(D_{l}\zeta_{k})$
$\displaystyle=\lambda_{k}^{-1}\Bigg{[}\delta_{k,l}\left[\int
P_{k}^{L}\big{(}(D^{2}w_{k})+(Dw_{k})^{2}\big{)}\left(\hat{\pi}_{f(k)}^{L}-\zeta_{k}\right)-\lambda_{k}\right]$
$\displaystyle\quad\quad\quad\quad\quad\quad+\zeta_{k}\phi_{k,l}+\gamma_{k,l}+\delta_{f(k),f(l)}\phi_{k,l}(p_{l}-c_{l})+\delta_{f(k),f(l)}\gamma_{l,k}-2\psi_{k,l}\Bigg{]}.$
## 4\. The GMRES-Newton Hookstep Method
In this section we provide some details regarding the GMRES-Newton Hookstep
method employed in Morrow and Skerlos (2010). For complete details, see
Morrow10b.
### 4.1. Inexact Newton Methods
A strong theory of “Inexact” Newton methods exists for the solution of systems
of nonlinear equations when there are “many” variables. Inexact Newton steps
are simply “inexact” solutions to the Newton system; that is, an inexact
Newton step $\mathbf{s}^{IN}$ is any vector that satisfies
(25)
$\lvert\lvert\mathbf{F}(\mathbf{x})+(D\mathbf{F})(\mathbf{x})\mathbf{s}^{IN}\rvert\rvert\leq\delta\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert$
for some fixed $\delta\in(0,1)$ (Dembo et al., 1982; Brown and Saad, 1990;
Eisenstat and Walker, 1994, 1996; Pernice and Walker, 1998). The name
“truncated” Newton method has also been used for the specific case when the
inexactness comes from the use of iterative linear system solvers like GMRES
(Saad and Schultz, 1986; Walker, 1988) or BiCGSTAB (van der Vorst, 1992;
Sleijpen and Fokkema, 1993). We focus on GMRES, a particularly simple yet
strong iterative method for general linear systems that has been consistently
used in the context of solving nonlinear systems (Brown and Saad, 1990).
By appropriately choosing a sequence of $\delta$’s, the local asymptotic
convergence rate of an inexact Newton’s method can be fully quadratic (Dembo
et al., 1982; Eisenstat and Walker, 1994). Of course, taking $\delta\to 0$ to
achieve the quadratic convergence rate will also require increasingly
burdensome computations of inexact Newton steps that satisfy increasingly
strict inexact Newton conditions. On the other hand, $\delta$ can be chosen to
be a constant if a linear locally asymptotic convergence rate is suitable
(Pernice and Walker, 1998).
Generally speaking there are three reasons to adopt the inexact perspective.
First, direct methods like QR factorization may not be the most effective
means to solve the Newton system when this system is large, because of
computational burden and accumulation of roundoff errors. Instead, iterative
solution methods are often used to solve linear systems with many variables;
see, e.g. Trefethen and Bau (1997). Second, iterative methods like GMRES
require only matrix-vector products $(D\mathbf{F})(\mathbf{p})\mathbf{s}$ that
can be approximated with finite directional derivatives (Brown and Saad, 1990;
Pernice and Walker, 1998). Thus inexact Newton’s methods can be “matrix-free”;
see Section 4.3.4 below. Third, Newton steps often point in inaccurate
directions when far from a solution (Pernice and Walker, 1998). Thus solving
for exact Newton steps may involve wasted effort, especially when there are
many variables.
matlab’s fsolve function implements a related approach using the
(preconditioned) Conjugate Gradient (CG) method applied to the normal equation
for the Newton system,
$(D\mathbf{F})(\mathbf{p})^{\top}(D\mathbf{F})(\mathbf{p})\mathbf{s}^{IN}=-(D\mathbf{F})(\mathbf{p})^{\top}\mathbf{F}(\mathbf{p})$.
Use of the normal equations is required because CG is applicable only to
symmetric systems (Trefethen and Bau, 1997). Note that this requires that the
Jacobian $(D\mathbf{F})$ is explicitly available. Although this holds for
price equilibrium problems under Mixed Logit models, it can be a significant
restriction for general problems. By requiring products
$(D\mathbf{F})(\mathbf{p})^{\top}\mathbf{h}$ in each step of the iterative
linear solver, this approach also increases the work by $\mathcal{O}(NJ^{2})$
flops where the solver takes $N$ steps. Finally, this approach can also be
less accurate: using the normal equation squares the linear problem’s
condition number, and thus risks serious degradation in solution quality
(Trefethen and Bau, 1997). Pernice and Walker (1998) describe a similar
approach using BiCGSTAB: the extension of CG to non-symmetric systems.
### 4.2. GMRES
The “Generalized Minimum Residuals” or GMRES method (Saad and Schultz, 1986)
solves a linear system $\mathbf{Ax}=\mathbf{b}$ by using the Arnoldi process
to compute an orthonormal basis of the successive Krylov subspaces
$\mathcal{K}^{(n)}$ and then takes approximate solutions from those subspaces
having least squares residuals. See Trefethen and Bau (1997) for a good
introduction to Krylov methods in general, including the Arnoldi process and
GMRES. In the $n^{\text{th}}$ stage, GMRES “factors” $\mathbf{A}$ as
$\mathbf{A}\mathbf{Q}^{(n)}=\mathbf{Q}^{(n+1)}\tilde{\mathbf{H}}^{(n)}$ where
$\mathbf{Q}^{(n)}\in\mathbb{R}^{N\times n}$ is an orthonormal basis for
$\mathcal{K}^{(n)}$, $\mathbf{Q}^{(n+1)}\in\mathbb{R}^{N\times(n+1)}$ is an
orthonormal basis for $\mathcal{K}^{(n+1)}\supset\mathcal{K}^{(n)}$, and
$\tilde{\mathbf{H}}^{(n)}\in\mathbb{R}^{(n+1)\times n}$ is upper-Hessenberg.
Any vector $\mathbf{x}\in\mathcal{K}^{(n)}\subset\mathbb{R}^{N}$ can be
written $\mathbf{x}=\mathbf{Q}^{(n)}\mathbf{y}$ for some
$\mathbf{y}\in\mathbb{R}^{n}$ and thus the least-squares residual problem
becomes
$\min_{\mathbf{x}\in\mathcal{K}^{(n)}}\lvert\lvert\mathbf{As}-\mathbf{b}\rvert\rvert_{2}=\min_{\mathbf{y}\in\mathbb{R}^{n}}\lvert\lvert\mathbf{A}\mathbf{Q}^{(n)}\mathbf{y}-\mathbf{b}\rvert\rvert_{2}=\min_{\mathbf{y}\in\mathbb{R}^{n}}\lvert\lvert\tilde{\mathbf{H}}^{(n)}\mathbf{y}-(\mathbf{Q}^{(n+1)})^{\top}\mathbf{b}\rvert\rvert_{2}.$
The orthonormal basis is typically chosen so that
$(\mathbf{Q}^{(n+1)})^{\top}\mathbf{b}=\beta\mathbf{e}_{1}$ for some
$\beta\in\mathbb{R}$, and hence the GMRES solution
$\mathbf{x}^{(n)}=\mathbf{Q}^{(n)}\mathbf{y}$ where $\mathbf{y}$ solves
$\min_{\mathbf{q}\in\mathbb{R}^{n}}\lvert\lvert\tilde{\mathbf{H}}^{(n)}\mathbf{y}-\beta\mathbf{e}_{1}\rvert\rvert_{2}$.
This least squares problem can be solved using the QR factorization of
$\tilde{\mathbf{H}}^{(n)}$. Furthermore this factorization can be efficiently
updated in each iteration, instead of computed from scratch. Moreover the
actual solution vector need not be formed until the residual is suitably
small.
#### 4.2.1. Householder GMRES
We have implemented a variant of GMRES based on Householder transformations
due to Walker (1988); this is also the version implemented in matlab’s gmres
code. We have verified that our implementation generates results matching
matlab’s implementation. In this version of the GMRES process applied to the
generic problem $\mathbf{Ax}=\mathbf{b}$, Householder reflectors
$\mathbf{P}^{(n)}\in\mathbb{R}^{N\times N}$ are used to generate the
orthonormal matrices
$\mathbf{Q}^{(n)}=\mathbf{P}^{(1)}\dotsb\mathbf{P}^{(n)}\begin{bmatrix}\mathbf{I}\\\
\mathbf{0}\end{bmatrix}\in\mathbb{R}^{N\times
n}\quad\quad(\mathbf{I}\in\mathbb{R}^{n\times
n},\;\mathbf{0}\in\mathbb{R}^{(N-n)\times n})$
satisfying
$\mathbf{A}\mathbf{Q}^{(n)}=\mathbf{P}^{(1)}\dotsb\mathbf{P}^{(n+1)}\mathbf{H}^{(n)}=\mathbf{Q}^{(n+1)}\tilde{\mathbf{H}}^{(n)}$
where $\mathbf{H}^{(n)}\in\mathbb{R}^{N\times n}$ is
$\mathbf{H}^{(n)}=\begin{bmatrix}\tilde{\mathbf{H}}^{(n)}\\\
\mathbf{0}\end{bmatrix}$
for upper Hessenberg $\tilde{\mathbf{H}}^{(n)}\in\mathbb{R}^{(n+1)\times n}$
and $\mathbf{0}\in\mathbb{R}^{(N-n-1)\times n}$. $\mathbf{P}^{(1)}$ is chosen
to satisfy $\mathbf{P}^{(1)}\mathbf{b}=-\beta\mathbf{e}_{1}$ where
$\beta=\mathrm{sign}(b_{1})\lvert\lvert\mathbf{b}\rvert\rvert_{2}$, and hence
$(\mathbf{Q}^{(n)})^{\top}\mathbf{b}=-\beta\mathbf{e}_{1}$. The
$n^{\text{th}}$ approximate solution $\mathbf{x}^{(n)}$ is taken to be
$\mathbf{x}^{(n)}=\mathbf{Q}^{(n)}\mathbf{y}^{(n)}$ where
$\mathbf{y}^{(n)}\in\mathbb{R}^{n}$ solves
$\min_{\mathbf{y}\in\mathbb{R}^{n}}\lvert\lvert\tilde{\mathbf{H}}^{(n)}\mathbf{y}-\beta\mathbf{e}_{1}\rvert\rvert_{2}.$
Again these problems can be solved cheaply by updating QR factorizations with
Givens rotations. Neither the solution vector nor the residual vector be
formed until GMRES converges. An efficient implementation requires
$\mathcal{O}(Jn)$ flops and a matrix multiply in the $n^{\text{th}}$
iteration, so that taking $N$ iterations requires $\mathcal{O}(JN^{2})$ of
“overhead” in addition to the $\mathcal{O}(NJ^{2})$ work required for the
matrix multiplications (using the actual Jacobians). So long as $N<J$, using
GMRES with the actual Jacobians is cheaper than solving for the actual
Jacobian with QR. With small $N$, as we achieve using $\boldsymbol{\eta}$ and
$\boldsymbol{\zeta}$, the savings is quite substantial.
We note the following formulae specific to the Newton system case. For
$\mathbf{A}=(D\mathbf{F})(\mathbf{x})$ and
$\mathbf{b}=-\mathbf{F}(\mathbf{x})$,
$\beta=-\mathrm{sign}(F_{1}(\mathbf{x}))\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}$
and
$-\beta\mathbf{e}_{1}=\mathbf{P}^{(1)}\mathbf{b}=-\mathbf{P}^{(1)}\mathbf{F}(\mathbf{x})$
so that
$\mathbf{P}^{(1)}\mathbf{F}(\mathbf{x})=\beta\mathbf{e}_{1}=-\mathrm{sign}(F_{1}(\mathbf{x}))\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}\mathbf{e}_{1}.$
Moreover, $\mathbf{P}^{(n)}\mathbf{e}_{1}=\mathbf{e}_{1}$ for all $n>1$ so
that
$(\mathbf{Q}^{(n)})^{\top}\mathbf{F}(\mathbf{x})=-\mathrm{sign}(F_{1}(\mathbf{x}))\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}\mathbf{e}_{1}.$
#### 4.2.2. Preconditioning
As is well known, preconditioning is key to the effectiveness of iterative
linear solvers; see Golub and Loan (1996). We have not found the linear
systems in $\boldsymbol{\eta}$-NM or $\boldsymbol{\zeta}$-NM to need
preconditioning. However we have found the preconditioned system
(26)
$\boldsymbol{\Lambda}(\mathbf{p})^{-1}(D\tilde{\nabla}\hat{\pi})(\mathbf{p})\mathbf{s}^{IN}=-\boldsymbol{\Lambda}(\mathbf{p})^{-1}(\tilde{\nabla}\hat{\pi})(\mathbf{p})=\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})-\mathbf{p}$
to be very necessary for rapid solution of the Newton system in CG-NM. This
preconditioner is motivated by the following relationship of the Jacobian of
$(\tilde{\nabla}\hat{\pi})$ to the Jacobian of $\boldsymbol{\zeta}$ in
equilibrium.
###### Lemma 4.1.
$\mathbf{I}-(D\boldsymbol{\zeta})(\mathbf{p})=\boldsymbol{\Lambda}(\mathbf{p})^{-1}(D\tilde{\nabla}\hat{\pi})(\mathbf{p})$
for any simultaneously stationary $\mathbf{p}$.
###### Proof.
This follows from differentiating
$(\tilde{\nabla}\hat{\pi})(\mathbf{p})=\boldsymbol{\Lambda}(\mathbf{p})(\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p}))$
via the product rule, recognizing that
$\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})=\mathbf{0}$ in
equilibrium and
$D[\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})]=\mathbf{I}-(D\boldsymbol{\zeta})(\mathbf{p})$.
∎
In other words, Newton’s methods applied to $\mathbf{F}_{\pi}(\mathbf{p})$
preconditioned as above ends up being essentially the same iteration as
$\mathbf{F}_{\zeta}(\mathbf{p})$, close enough to equilibrium.
GMRES, if used successfully on this preconditioned system Eqn. (26), will
ensure that
(27)
$\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}(\tilde{\nabla}\hat{\pi})(\mathbf{p})+\boldsymbol{\Lambda}(\mathbf{p})^{-1}(D\tilde{\nabla}\hat{\pi})(\mathbf{p})\mathbf{s}^{IN}\rvert\rvert\leq\delta^{\prime}\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}(\tilde{\nabla}\hat{\pi})(\mathbf{p})\rvert\rvert$
for some $\delta^{\prime}$. This is distinct from the inexact Newton condition
Eqn. (25). The following proposition gives modified tolerances for the
preconditioned system to ensure satisfaction of the original system.
###### Proposition 4.2.
Let $\delta>0$ be given. If Eqn. (27) is satisfied with
$\delta^{\prime}(\mathbf{p},\delta)\leq\delta$ given by
(28)
$\delta^{\prime}(\mathbf{p},\delta)=\left(\frac{\lvert\lvert(\tilde{\nabla}\hat{\pi})(\mathbf{p})\rvert\rvert_{2}}{\max_{j}\left\\{\left\lvert\lambda_{j}(\mathbf{p})\right\rvert\right\\}\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}(\tilde{\nabla}\hat{\pi})(\mathbf{p})\rvert\rvert_{2}}\right)\delta,$
then Eqn. (25) is satisfied.
This is a consequence of the following general result, which we state without
proof.
###### Lemma 4.3.
Let $\mathbf{b}\in\mathbb{R}^{N}$ and
$\mathbf{A},\mathbf{M}\in\mathbb{R}^{N\times N}$ be nonsingular. Then
(29)
$\frac{\lvert\lvert\mathbf{Ax}-\mathbf{b}\rvert\rvert}{\lvert\lvert\mathbf{b}\rvert\rvert}\leq\alpha\left(\frac{\lvert\lvert\mathbf{M}^{-1}\mathbf{Ax}-\mathbf{M}^{-1}\mathbf{b}\rvert\rvert}{\lvert\lvert\mathbf{M}^{-1}\mathbf{b}\rvert\rvert}\right)$
where $\alpha\in[1,\kappa(\mathbf{M})]$ is given by
$\alpha=\frac{\lvert\lvert\mathbf{M}\rvert\rvert\lvert\lvert\mathbf{M}^{-1}\mathbf{b}\rvert\rvert}{\lvert\lvert\mathbf{b}\rvert\rvert}=\lvert\lvert\mathbf{M}\rvert\rvert\left\lvert\left\lvert\mathbf{M}^{-1}\left(\frac{\mathbf{b}}{\lvert\lvert\mathbf{b}\rvert\rvert}\right)\right\rvert\right\rvert.$
This implies that
$\frac{\lvert\lvert\mathbf{Ax}-\mathbf{b}\rvert\rvert}{\lvert\lvert\mathbf{b}\rvert\rvert}\leq\delta\quad\quad\text{if}\quad\quad\frac{\lvert\lvert\mathbf{M}^{-1}\mathbf{Ax}-\mathbf{M}^{-1}\mathbf{b}\rvert\rvert}{\lvert\lvert\mathbf{M}^{-1}\mathbf{b}\rvert\rvert}\leq\frac{\delta}{\alpha}.$
Note that the preconditioned system must always be solved to a stricter
tolerance than is desired for the un-preconditioned system using this bound.
Additionally, computing $\alpha$ for a generic preconditioner $\mathbf{M}$
relies on the ability to compute $\lvert\lvert\mathbf{M}\rvert\rvert$.
Eqn. (28) simply adopts the 2-norm and applies the formula (Golub and Loan,
1996)
$\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})\rvert\rvert_{2}=\sqrt{\max_{j}\\{\left\lvert\lambda_{j}(\mathbf{p})\right\rvert^{2}\\}}=\max_{j}\\{\left\lvert\lambda_{j}(\mathbf{p})\right\rvert\\}$
Eqn. (29) also implies that if Eqn. (27) holds with $\delta^{\prime}>0$, then
$\frac{\lvert\lvert(\tilde{\nabla}\hat{\pi})(\mathbf{p})+(D\tilde{\nabla}\hat{\pi})(\mathbf{p})\mathbf{s}^{IN}\rvert\rvert_{2}}{\lvert\lvert(\tilde{\nabla}\hat{\pi})(\mathbf{p})\rvert\rvert_{2}}\leq\kappa_{2}(\boldsymbol{\Lambda}(\mathbf{p}))\delta^{\prime}$
where
$\kappa_{2}(\boldsymbol{\Lambda}(\mathbf{p}))=\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})\rvert\rvert_{2}\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\rvert\rvert_{2}$
is the (2-norm) condition number of $\boldsymbol{\Lambda}(\mathbf{p})$. This
equation, while the more compact representation, can also be overly
conservative as clearly illustrated in Fig. 1. It is unlikely that
$\kappa(\boldsymbol{\Lambda}(\mathbf{p}))$ is a tight upper bound on the
multiplier in Eqn. (28). In fact, the multiplier on $\delta$ depends only on
the norm of $\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{x}$ at a single
point on the surface of the unit sphere in $\mathbb{R}^{J}$ rather than
$\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\rvert\rvert_{2}$, the
maximum norm of $\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{x}$ over this
entire sphere. Our examples in Fig. 1 bear this out, having condition numbers
many orders of magnitude larger than the multiplier in Eqn. (28).
The power of the preconditioning is that the preconditioned system Eqn. (27)
appears to be solved to a relative error of
$\delta^{\prime}(\mathbf{p},\delta)$ much faster than the original system can
be solved to a relative error of $\delta$, even though
$\delta^{\prime}(\mathbf{p},\delta)\leq\delta$. As can be seen in Fig. 1,
solving the preconditioned system to $\delta^{\prime}(\mathbf{p},\delta)$ can
achieve a relative error in the original system below $\delta=10^{-10}$ in
roughly four orders of magnitude fewer iterations than solving the original
system to this same relative error for prices near equilibrium. Away from
equilibrium, GMRES may not be able to solve the original system to small
relative errors like $10^{-6}$ at all. Thus using the original system would
appear to slow, if not halt, an implementation of the inexact Newton’s method.
Figure 1. Relative error in computed solutions to the CG-NM Newton system and
its preconditioned form using GMRES in the vehicle example from Morrow and
Skerlos (2010) using the Berry et al. (1995) model. On the top, prices are
$\mathbf{p}=\mathbf{p}^{*}+100\boldsymbol{\nu}$ where $\mathbf{p}^{*}$ are
equilibrium prices and $\boldsymbol{\nu}\in[-\mathbf{1},\mathbf{1}]$ is a
sample from a uniformly distributed random vector. For this case
$\kappa(\boldsymbol{\Lambda}(\mathbf{p}))=1.56\times 10^{11}$ while the
multiplier in Eqn. (29) is only 106.41. On the bottom, prices are
$\mathbf{p}=20,000\boldsymbol{\nu}+5,000$ where $\boldsymbol{\nu}$ is a sample
from a random vector uniformly distributed on $[\mathbf{0},\mathbf{1}]$. For
this case $\kappa(\boldsymbol{\Lambda}(\mathbf{p}))=4.6\times 10^{4}$ while
the multiplier in Eqn. (29) is only 10.73. Abbreviations are as follows. REL:
relative error in the Newton System; PREL: relative error in the pre-
conditioned Newton System; OBREL: our bound, Eqn. (29), on the relative error
in the Newton System as determined from the relative error in the
preconditioned Newton system; CNBREL: condition number bound on the relative
error in the Newton System as determined from the relative error in the
preconditioned Newton system.
### 4.3. The GMRES Hookstep
Suitable modifications of each of the globalization strategies originally
developed for “exact” Newton methods can be applied in the inexact context.
Brown and Saad (1990) directly extend line search and a dogleg steps to GMRES-
Newton methods. Eisenstat and Walker (1996) and Pernice and Walker (1998)
apply a safeguarded backtracking line search to facilitate global convergence.
More recently, Pawlowski et al. (2006, 2008) have studied dogleg steps
suitable for GMRES-Newton methods in some detail. Finally Viswanath (2007) has
derived an elegant version of the hookstep method suitable for GMRES-Newton
methods. In contrast with the hookstep approach for the “exact” Newton method
with Jacobian $(D\mathbf{F})(\mathbf{p})$, Viswanath’s approach requires
computing the SVD only of a matrix whose size is determined by the number of
iterations taken by GMRES. For reasonable applications of GMRES, this can be
far less than the size of $(D\mathbf{F})(\mathbf{p})$ itself. For the examples
in Morrow and Skerlos (2010), the size difference is roughly two orders of
magnitude: the GMRES-Newton hookstep worked with roughly $10\times 10$ instead
of $1,000\times 1,000$ matrices. Thus, the GMRES-hookstep can accumulate a
tremendous savings over an exact-Newton implementation of the hookstep method.
Again, each of these approaches iterates until an acceptable step is found,
and can, in principle, involve many additional evaluations of $\mathbf{F}$ or
fail to find an acceptable step altogether.
Here we describe an implementation of the Levenberg-Marquardt method or
“hookstep” (Dennis and Schnabel, 1996) suitable for GMRES as first suggested
by Viswanath (2007). See also Viswanath (2009); Viswanath and Cvitanovic
(2009); Halcrow et al. (2009). First, we recall the basic structure of model
trust region methods; see (Dennis and Schnabel, 1996, Chapter 6, Section 4).
We then adopt this structure to the case of Krylov subspace methods,
particularly GMRES. Again, see Morrow10b for a more detailed discussion of
this method.
#### 4.3.1. Model Trust Region Methods.
Trust region methods assume that for steps $\mathbf{s}$ satisfying
$\lvert\lvert\mathbf{s}\rvert\rvert_{2}\leq\delta$, the function
$\hat{m}_{\mathbf{x}}(\mathbf{s})=\left(\frac{1}{2}\right)\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}^{2}+((D\mathbf{F})(\mathbf{x})^{\top}\mathbf{F}(\mathbf{x}))^{\top}\mathbf{s}+\left(\frac{1}{2}\right)\mathbf{s}^{\top}(D\mathbf{F})(\mathbf{x})^{\top}(D\mathbf{F})(\mathbf{x})\mathbf{s}$
is an accurate local model of
$f(\mathbf{x})=\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}^{2}/2$ for
suitably small steps. Note that $\hat{m}_{\mathbf{x}}$ is not the usual,
quadratic model of $f$ derived from a Taylor series because
$(D\mathbf{F})(\mathbf{x})^{\top}(D\mathbf{F})(\mathbf{x})\neq(D\nabla
f)(\mathbf{x})$ (Dennis and Schnabel, 1996, pg. 149). The idea is to solve
(30)
$\min_{\lvert\lvert\mathbf{s}\rvert\rvert_{2}\leq\delta}\hat{m}_{\mathbf{x}}(\mathbf{s}).$
The solution $\mathbf{s}_{*}$ is given as follows: take
$\mathbf{s}_{*}=\mathbf{s}^{N}=-(D\mathbf{F})(\mathbf{x})^{-1}\mathbf{F}(\mathbf{x})$
if $\lvert\lvert\mathbf{s}^{N}\rvert\rvert_{2}\leq\delta$; if
$\lvert\lvert\mathbf{s}^{N}\rvert\rvert_{2}>\delta$, take
$\mathbf{s}_{*}=\mathbf{s}(\mu_{*})$ where
$\mathbf{s}(\mu)=-\big{(}(D\mathbf{F})(\mathbf{x})^{\top}(D\mathbf{F})(\mathbf{x})+\mu\mathbf{I}\big{)}^{-1}(D\mathbf{F})(\mathbf{x})^{\top}\mathbf{F}(\mathbf{x})$
and $\mu_{*}>0$ is the unique $\mu>0$ such that
$\lvert\lvert\mathbf{s}(\mu)\rvert\rvert_{2}=\delta$. These follow from the
standard optimality conditions, or rather that the gradient
$(\nabla\hat{m}_{\mathbf{x}})(\mathbf{s})$ must lie in the negative normal
cone to
$\bar{\mathbb{B}}_{\delta}(\mathbf{0})=\\{\mathbf{y}\in\mathbb{R}^{N}:\lvert\lvert\mathbf{y}\rvert\rvert_{2}\leq\delta\\}$
at $\mathbf{x}$ (Clarke, 1975); see (Dennis and Schnabel, 1996, Lemma 6.4.1,
pg. 131).
Solving the problem above exactly generates the Levenberg-Marquardt method
(Levenberg, 1944; Marquardt, 1963) or “hookstep.” By computing the SVD of
$(D\mathbf{F})(\mathbf{x})$ we can easily solve for $\mathbf{s}(\mu)$ when
$\lvert\lvert\mathbf{s}^{N}\rvert\rvert_{2}>\delta$ (Dennis and Schnabel,
1996); see (Golub and Loan, 1996, Section 12.1, pgs. 580-583) for closely
related results. Let
$(D\mathbf{F})(\mathbf{x})=\mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^{\top}$. We
can then set $\mathbf{s}(\mu)=\mathbf{V}\mathbf{r}(\mu)$ where
$\mathbf{r}(\mu)=-(\boldsymbol{\Sigma}^{2}+\mu\mathbf{I})^{-1}\boldsymbol{\Sigma}\mathbf{U}^{\top}\mathbf{F}(\mathbf{x}).$
A simple single-dimensional iteration can then be used to solve for the unique
$\mu_{*}$ such that $\lvert\lvert\mathbf{s}(\mu_{*})\rvert\rvert_{2}=\delta$.
Morrow10b derives two globally convergent methods for this task using Newton’s
method and a nonlinear local model (Dennis and Schnabel, 1996). The difficulty
here is computing the SVD of $(D\mathbf{F})(\mathbf{x})$, requiring
$\mathcal{O}(J^{3})$ flops (Golub and Loan, 1996, Chapter 5, pg. 254).
The step $\mathbf{s}_{*}$ computed by either approach is acceptable if it
generates sufficient decrease in the squared 2-norm of $\mathbf{F}$.
Specifically, fix $\rho\in(0,1)$, $\alpha>1$, and $\beta_{2}\leq\beta_{1}<1$.
If
$\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}^{2}-\lvert\lvert\mathbf{F}(\mathbf{x}+\mathbf{s}_{*})\rvert\rvert_{2}^{2}\geq\rho(\lvert\lvert(D\mathbf{F})(\mathbf{x})\rvert\rvert_{2}^{2}-\lvert\lvert\mathbf{F}(\mathbf{x})+(D\mathbf{F})(\mathbf{x})\mathbf{s}_{*}\rvert\rvert_{2}^{2})$
then $\mathbf{p}\leftarrow\mathbf{p}+\mathbf{s}_{*}$ and a the step length
bound is expanded to $[\delta,\alpha\delta]$ for the next iteration.
Otherwise, $\delta$ is chosen from $[\beta_{1}\delta,\beta_{2}\delta]$ and the
corresponding $\mathbf{s}_{*}$ is computed. While this process of specifying
an acceptable $\mathbf{s}_{*}$ is iterative, much of the work required to
build a trial step does not need to be repeated. Specifically the SVD required
for the hookstep does not change (so long as it was computed in a previous
iteration) while in the doglep step the Newton and Cauchy steps remain the
same. However every time the step size bound is decreased $\mathbf{F}$ must be
re-evaluated at the new trial step, with a computational burden equivalent to
taking a fixed-point step.
#### 4.3.2. Model Trust Region Methods on a Subspace
A Krylov method for solving
$(D\mathbf{F})(\mathbf{x})\mathbf{s}^{N}=-\mathbf{F}(\mathbf{x})$ builds
approximate solutions in the successive Krylov subspaces $\mathcal{K}^{(n)}$.
This has the effect of further constraining the local model problem (30) to
(31)
$\min_{\mathbf{s}\in\mathcal{K}^{(n)},\;\lvert\lvert\mathbf{s}\rvert\rvert_{2}\leq\delta}\hat{m}_{\mathbf{x}}(\mathbf{s}).$
For any $\mathbf{Q}\in\mathbb{R}^{J\times n}$ with orthonormal columns
(generated by GMRES or not) we can set
$\hat{m}_{\mathbf{x},\mathbf{Q}}(\mathbf{y})=\hat{m}_{\mathbf{x}}(\mathbf{Q}\mathbf{y})$
and restrict attention to the trust region problem
$\min_{\lvert\lvert\mathbf{y}\rvert\rvert_{2}\leq\delta}\hat{m}_{\mathbf{x},\mathbf{Q}}(\mathbf{y})$.
See (Brown and Saad, 1990, pgs. 149-150). The first-order conditions for this
problem are equivalent to either
* (i)
$(\nabla\hat{m}_{\mathbf{x},\mathbf{Q}})(\mathbf{y})=\mathbf{0}$ and
$\lvert\lvert\mathbf{y}\rvert\rvert_{2}\leq\delta$
* (ii)
or
$(\nabla\hat{m}_{\mathbf{x},\mathbf{Q}})(\mathbf{y})+\mu\mathbf{y}=\mathbf{0}$
for $\lvert\lvert\mathbf{y}\rvert\rvert_{2}=\delta$ and some $\mu>0$.
By the definition of $\hat{m}_{\mathbf{x},\mathbf{Q}}$, (i) implies
$\mathbf{Q}^{\top}(D\mathbf{F})(\mathbf{x})^{\top}(D\mathbf{F})(\mathbf{x})\mathbf{Q}\mathbf{y}+\mathbf{Q}^{\top}(D\mathbf{F})(\mathbf{x})^{\top}\mathbf{F}(\mathbf{x})=\mathbf{0}$
and (ii) implies
$\left(\mathbf{Q}^{\top}(D\mathbf{F})(\mathbf{x})^{\top}(D\mathbf{F})(\mathbf{x})\mathbf{Q}+\mu\mathbf{I}\right)\mathbf{y}+\mathbf{Q}^{\top}(D\mathbf{F})(\mathbf{x})^{\top}\mathbf{F}(\mathbf{x})=\mathbf{0}.$
Note that these are square problems that can be solved exactly.
#### 4.3.3. The GMRES-Newton Hookstep
Using GMRES started at zero,
$(D\mathbf{F})(\mathbf{x})\mathbf{Q}^{(n)}=\mathbf{Q}^{(n+1)}\tilde{\mathbf{H}}^{(n)}$
and
$(\mathbf{Q}^{(n+1)})^{\top}\mathbf{F}(\mathbf{x})=-\mathrm{sign}(F_{1}(\mathbf{x}))\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}\mathbf{e}_{1}$.
Thus we consider the family of $n\times n$ linear systems
$\displaystyle(\mathbf{Q}^{(n)})^{\top}(D\mathbf{F})(\mathbf{x})^{\top}(D\mathbf{F})(\mathbf{x})\mathbf{Q}^{(n)}\mathbf{q}+\mu\mathbf{q}+(\mathbf{Q}^{(n)})^{\top}(D\mathbf{F})(\mathbf{x})^{\top}\mathbf{F}(\mathbf{x})$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad=\big{(}(\tilde{\mathbf{H}}^{(n)})^{\top}\tilde{\mathbf{H}}^{(n)}+\mu\mathbf{I}\big{)}\mathbf{q}-\mathrm{sign}(F_{1}(\mathbf{x}))\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}(\tilde{\mathbf{H}}^{(n)})^{\top}\mathbf{e}_{1}=\mathbf{0}$
defined for all $\mu\geq 0$.
By computing the (“thin”) Singular Value Decomposition of
$\tilde{\mathbf{H}}^{(n)}$,
$\tilde{\mathbf{H}}^{(n)}=\tilde{\mathbf{U}}\boldsymbol{\Sigma}\mathbf{V}^{\top}$
where $\tilde{\mathbf{U}}\in\mathbb{R}^{(n+1)\times n}$,
$\mathbf{V}\in\mathbb{R}^{n\times n}$, and
$\boldsymbol{\Sigma}\in\mathbb{R}^{n\times n}$, we can easily solve each such
problem. See (Golub and Loan, 1996, Section 12.1, pgs. 580-583) for closely
related results. Particularly,
$\displaystyle((\tilde{\mathbf{H}}^{(n)})^{\top}\tilde{\mathbf{H}}^{(n)}+\mu\mathbf{I})\mathbf{q}-\mathrm{sign}(F_{1}(\mathbf{x}))\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}(\tilde{\mathbf{H}}^{(n)})^{\top}\mathbf{e}_{1}=\mathbf{0}$
is solved by $\mathbf{q}(\mu)=\mathbf{V}\mathbf{r}(\mu)$ where
$\mathbf{r}(\mu)=\mathrm{sign}(F_{1}(\mathbf{x}))\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}(\boldsymbol{\Sigma}^{2}+\mu\mathbf{I})^{-1}\boldsymbol{\Sigma}\tilde{\mathbf{U}}^{\top}\mathbf{e}_{1}.$
Because the diagonal elements of $\boldsymbol{\Sigma}^{2}$ are positive,
$\mathbf{r}(\mu)$ is well defined for all $\mu\geq 0$. Note also that we only
need the first row of $\mathbf{U}$, but all of $\mathbf{V}$, to compute
$\mathbf{q}(\mu)$.
In particular,
$\mathbf{q}(0)=\mathrm{sign}(F_{1}(\mathbf{x}))\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}\mathbf{V}\boldsymbol{\Sigma}^{-1}\mathbf{U}^{\top}\mathbf{e}_{1}$.
Invoking the full SVD of $\tilde{\mathbf{H}}^{(n)}$,
$\displaystyle\tilde{\mathbf{H}}^{(n)}=\begin{bmatrix}\tilde{\mathbf{U}}&\mathbf{u}_{n+1}\end{bmatrix}\begin{bmatrix}\boldsymbol{\Sigma}\\\
\mathbf{0}^{\top}\end{bmatrix}\mathbf{V}^{\top}$
for some $\mathbf{u}_{n+1}\perp\mathrm{span}\\{\mathbf{u}_{i}\\}_{i=1}^{n}$,
we can write
$\displaystyle\lvert\lvert\tilde{\mathbf{H}}^{(n)}\mathbf{q}-\mathrm{sign}(F_{1}(\mathbf{x}))\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}\mathbf{e}_{1}\rvert\rvert_{2}$
$\displaystyle=\left\lvert\left\lvert\begin{bmatrix}\tilde{\boldsymbol{\Sigma}}\mathbf{V}^{\top}\mathbf{q}\\\
0\end{bmatrix}-\mathrm{sign}(F_{1}(\mathbf{x}))\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}\begin{bmatrix}\tilde{\mathbf{U}}^{\top}\mathbf{e}_{1}\\\
u_{1,n+1}\end{bmatrix}\right\rvert\right\rvert_{2}.$
We thus see that $\mathbf{q}(0)$ solves the $(n+1)\times n$ GMRES least
squares problem
$\min_{\mathbf{q}}\lvert\lvert\mathbf{H}^{(n+1,n)}\mathbf{q}-\mathrm{sign}(F_{1}(\mathbf{x}))\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}\mathbf{e}_{1}\rvert\rvert_{2}.$
with residual $\left\lvert
u_{1,n+1}\right\rvert\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}$.
$\left\lvert u_{1,n+1}\right\rvert$ is unique: First, note that
$\mathbf{u}_{n+1}$ is a unit vector in the span of a single vector, say
$\mathbf{v}$, that is orthogonal to the span of the columns of
$\tilde{\mathbf{U}}$. There are only two unit vectors in this span,
specifically $\pm\mathbf{v}/\lvert\lvert\mathbf{v}\rvert\rvert_{2}$, and thus
$\mathbf{u}_{n+1}\in\\{\pm\mathbf{v}/\lvert\lvert\mathbf{v}\rvert\rvert_{2}\\}$.
Thus $\left\lvert u_{1,n+1}\right\rvert\in\left\lvert\pm
v_{1}/\lvert\lvert\mathbf{v}\rvert\rvert_{2}\right\rvert=\left\lvert
v_{1}\right\rvert/\lvert\lvert\mathbf{v}\rvert\rvert_{2}$.
It is also easy to see that
$\displaystyle\mathbf{F}(\mathbf{x})^{\top}(D\mathbf{F})(\mathbf{x})\mathbf{s}^{(n)}(\mu)$
$\displaystyle=\mathbf{F}(\mathbf{x})^{\top}(D\mathbf{F})(\mathbf{x})\mathbf{Q}^{(n)}\mathbf{q}^{(n)}(\mu)$
$\displaystyle=\left(\big{(}\mathbf{Q}^{(n+1)}\big{)}^{\top}\mathbf{F}(\mathbf{x})\right)^{\top}\tilde{\mathbf{H}}^{(n)}\mathbf{q}^{(n)}(\mu)$
$\displaystyle=-\beta^{2}\left(\boldsymbol{\nu}_{1}^{\top}\mathbf{D}(\mu)\boldsymbol{\nu}_{1}\right)$
$\displaystyle=-\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}^{2}\left(\boldsymbol{\nu}_{1}^{\top}\mathbf{D}(\mu)\boldsymbol{\nu}_{1}\right)<0$
where $\boldsymbol{\nu}_{1}$ is the first row of $\tilde{\mathbf{U}}$ and
$\mathbf{D}(\mu)=\mathrm{diag}(d_{1}(\mu),\dotsc,d_{n}(\mu))$ for
$d_{i}(\mu)=\sigma_{i}^{2}/(\sigma_{i}^{2}+\mu)$. That is, the Householder
GMRES-Newton Hookstep always lies in a descent direction for the globalizing
objective
$f(\mathbf{x})=\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}^{2}/2$.
#### 4.3.4. Directional Finite Differences
Recall that one advantage to using an iterative solver like GMRES to solve the
Newton system is that only products of the type
$(D\mathbf{F})(\mathbf{p})\mathbf{s}$ will be required to solve the Newton
system for $\mathbf{F}$ at $\mathbf{p}$ (Brown and Saad, 1990; Pernice and
Walker, 1998). Such products can be approximated by a single additional
evaluation of $\mathbf{F}$ in a “directional” finite difference (Brown and
Saad, 1990; Pernice and Walker, 1998). For example, the first-order formula
$(D\mathbf{F})(\mathbf{x})\mathbf{s}\approx
h^{-1}\big{(}\mathbf{F}(\mathbf{x}+h\mathbf{s})-\mathbf{F}(\mathbf{x})\big{)},$
requires only a single additional evaluation of $\mathbf{F}$ per (approximate)
evaluation of $(D\mathbf{F})(\mathbf{x})\mathbf{s}$. Higher-order formulae
requiring 2 and 4 additional evaluations of $\mathbf{F}$ are easy to derive;
see Pernice and Walker (1998). In their implementation of the GMRES method in
the context of an inexact Newton method, Pernice and Walker (1998) only use
higher order finite-differencing formulas at restarts. Brown and Saad (1990)
provide a practical formula for computing an appropriate value of $h$.
Since directional finite derivatives must be repeated at each step of
iterative linear solvers, each step of an iterative Newton system solver using
directional finite differences could be at least as expensive as a
$\boldsymbol{\zeta}$-FPI step. That is, if an iterative solver should take 100
steps to compute an inexact Newton step having small enough residual to
satisfy the inexact Newton condition, then we could have equivalently taken
100, 200, and 400 $\boldsymbol{\zeta}$-FPI steps with the first, second, and
fourth order formulae available in Pernice and Walker (1998). In our examples,
using GMRES regularly solves the $\boldsymbol{\eta}$-NM and
$\boldsymbol{\zeta}$-NM Newton systems in approximately 10 steps. This implies
that each $\boldsymbol{\eta}$-NM and $\boldsymbol{\zeta}$-NM step is roughly
equivalent to $10$ $\boldsymbol{\zeta}$-FPI steps.
In the Newton context, whether efficiency is ultimately gained by using
directional finite differences instead of computing the Jacobian matrices and
using standard matrix-vector products depends on the number of steps taken by
the iterative linear solver. If GMRES takes $N\in\mathbb{N}$ iterations to
find an inexact Newton step for $\mathbf{F}$, computing and using the Jacobian
requires $\mathcal{O}((S+N)J^{2})$ flops while using directional finite
differences requires $\mathcal{O}(SN\sum_{f=1}^{F}J_{f}^{2})$ flops.
We have observed that for $\boldsymbol{\eta}$-NM and $\boldsymbol{\zeta}$-NM,
using the actual Jacobian takes roughly half the computation time than using
directional finite differences, even though GMRES converges in very few
iterations ($N\sim 10$). Fig. 2 plots the sample trials for the Boyd and
Mellman (1980) model provided in Morrow and Skerlos (2010) using both
analytical Jacobians and directional finite differences. First note that the
$\boldsymbol{\zeta}$-FPI regularly takes about 1 s per iteration. For
$\kappa=1$ USD, the single-step convergence of the GMRES-Newton Hookstep
method translates into about 10 $\boldsymbol{\zeta}$-FPI steps, or about $10$
s. Because GMRES itself requires some small overhead ($\mathcal{O}(Jn)$ in the
$n^{\text{th}}$ step), this is a somewhat reasonable estimate of the work
required. Two GMRES-Newton steps are required with $\kappa=10$ USD and we
would expect about $20$ s, a somewhat less sound estimate of the time
required. Three GMRES-Newton steps are required with $\kappa=100$ USD, leading
us to expect about $20$ s, a further less sound estimate of the time required.
(These observations can be matched with an asymptotic analysis of the work
required.) Note also that the $\boldsymbol{\eta}$-NM has the greatest increase
in time as a consequence of using the directional finite differences. This is
a consequence of having to repeat block QR factorizations when evaluating
$\boldsymbol{\eta}$ at different points, while evaluating
$(D\boldsymbol{\eta})$ requires only a single factorization.
Figure 2. Typical convergence curves for perturbation trials under the Boyd
and Mellman (1980) model using both analytical and directional finite
difference Jacobians. See also Fig. LABEL:FIG_BM80_BestCasePert. Convergence
curves for analytical Jacobian are drawn with solid lines, whereas convergence
curves for directional finite differences are drawn with dashed lines of the
same color.
Fig. 3 plots the sample trials for the Berry et al. (1995) model provided in
Morrow and Skerlos (2010) using both analytical Jacobians and directional
finite differences. Interestingly, in this case use of the directional finite
differences appears to generate a convergence rate improvement. Otherwise, the
story remains much the same as that discussed above for the Boyd and Mellman
(1980) model.
Figure 3. Typical convergence curves for perturbation trials under the Boyd
and Mellman (1980) model using both analytical and directional finite
difference Jacobians. See also Fig. LABEL:FIG_BLP95_BestCasePert. Convergence
curves for analytical Jacobian are drawn with solid lines, whereas convergence
curves for directional finite differences are drawn with dashed lines of the
same color.
## 5\. Other Methods
### 5.1. Variational Methods
Equilibrium problems are commonly formulated as variational inequalities or
complementarity problems (Harker and Pang, 1990; Ferris and Pang, 1997). To be
nontrivially distinct from nonlinear equations, such formulations require
restricting the variables to a proper, convex subset of $\mathbb{R}^{J}$. When
$\varsigma_{*}<\infty$ there is an appropriate variational formulation of the
equilibrium pricing problem:
(32) $\text{find}\quad\mathbf{p}\in[0,\varsigma_{*}]^{J}\quad\text{such
that}\quad(\tilde{\nabla}\hat{\pi})(\mathbf{p})^{\top}(\mathbf{p}-\mathbf{q})\geq
0\quad\text{for all}\quad\mathbf{q}\in[0,\varsigma_{*}]^{J}.$
#### 5.1.1. The VI formulation is poorly posed
Unfortunately, the Variational Inequality (32) is poorly posed when the
derivatives of profit vanish as prices approach $\varsigma_{*}<\infty$. There
are two specific issues with Eqn. (32) in this case. First,
$\varsigma_{*}\mathbf{1}\in\mathcal{P}^{J}$ is always a solution but never an
equilibrium when profits vanish as all prices approach $\varsigma_{*}$; see
Section LABEL:ECSUBSEC:Profits and Lemma 5.1. Second, Eqn. (32) can be solved
by any equilibrium of any differentiated product market model constructed with
a subset of the products offered (Prop. 5.2). Equilibria of such “sub-
problems” are not necessarily equilibria of the original problem, as
demonstrated in Example 8 below. This issue with Eqn. (32) is, in fact,
equivalent to the problem with CG-NM discussed in Section 3.2.
These issues imply that variational methods can compute many “spurious”
solutions. If an equilibrium problem and all its sub-problems have unique
equilibria with all prices less than $\varsigma_{*}$, Eqn. (32) has $2^{J}$
solutions that might be recovered by a global method such as PATH (Ralph,
1994; Dirkse and Ferris, 1995). However, only one of these solutions is an
equilibrium of the original problem, by assumption. A simple example
demonstrates this phenomenon.
###### Example 8.
Consider a monopoly with two products produced at the same unit cost $c$.
Demand is given by a simple Logit model with product-specific utility
functions $u_{j}(p_{j})=\alpha\log(\varsigma-p_{j})+v_{j}$ for
$j\in\\{1,2\\}$, where $\varsigma\in(c,\infty)$, $v_{1},v_{2}\in\mathbb{R}$,
and $\vartheta>-\infty$. The firm has unique profit-maximizing prices
$(p_{1}^{*},p_{2}^{*})$. Furthermore $p_{1}^{*},p_{2}^{*}<\varsigma$, and
$(p_{1}^{*},p_{2}^{*})$ is the unique fixed-point of the map
$\mathbf{c}+\boldsymbol{\zeta}(\cdot)$ on all of $\mathcal{P}^{2}$.
However the variational inequality formulation contains four distinct
solutions, only one of which is profit-maximizing. These four solutions are
$(p_{1}^{*},p_{2}^{*})$, $(\varsigma,\varsigma)$, $(q_{1}^{*},\varsigma)$, and
$(\varsigma,q_{2}^{*})$, where $q_{j}^{*}<\varsigma$ for $j\in\\{1,2\\}$ are
the unique profit-maximizing prices that exist should the firm offer only
product $1$ or $2$. Only the first solution, $(p_{1}^{*},p_{2}^{*})$, is
profit-maximizing.
###### Proof.
We complete the details of Example 8.
Consider a monopoly with two products produced at the same unit cost
($c=c_{1}=c_{2}>0$), $\vartheta>-\infty$, and simple Logit model with utility
$u_{1}(p_{1})=\alpha\log(\varsigma-p_{1})+v_{1}\quad\text{and}\quad
u_{2}(p_{2})=\alpha\log(\varsigma-p_{2})+v_{2}$
for some fixed $\varsigma\in(c,\infty)$, $\alpha>1$, and arbitrary
$v_{1},v_{2}\in\mathbb{R}$. Let $p_{2}\leq\varsigma$, and observe that
$\lim_{p_{1}\uparrow\varsigma}\Big{(}p_{1}-c-\zeta_{1}(p_{1},p_{2})\Big{)}=\varsigma-
c-
P_{2}(\varsigma,p_{2})(p_{2}-c)=(\varsigma-c)\left[1-P_{2}(\varsigma,p_{2})\left(\frac{p_{2}-c}{\varsigma-c}\right)\right].$
Since $p_{2}\leq\varsigma$ and $P_{2}(p_{1},p_{2})<1$ for all $p_{1},p_{2}$,
we have $\lim_{p_{1}\uparrow\varsigma}(p_{1}-c-\zeta_{1}(p_{1},p_{2}))>0$.
Thus $(D_{1}\hat{\pi})(p_{1},p_{2})<0$ for all $p_{1}$ sufficiently close to
$\varsigma$. A similar argument can be made for
$(D_{2}\hat{\pi})(p_{1},p_{2})$.
Note also that this proves that
$\varsigma+\epsilon>c+\zeta_{1}(\varsigma+\epsilon,p_{2})$ for any
$\epsilon\geq 0$ and $p_{2}$, where $\zeta_{1}$ is the extended map. A similar
result holds for $\zeta_{2}$, instead of $\zeta_{1}$. Thus no $(p_{1},p_{2})$
outside of $(0,\varsigma)$ is fixed for the extended map
$\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$.
We now prove that there exists a unique pair of profit-maximizing prices
$\mathbf{p}^{*}=(p_{1}^{*},p_{2}^{*})\in(0,\varsigma)^{2}$. Since
$\lim_{p_{j}\uparrow\varsigma}\Big{(}p_{j}-c-\zeta_{j}(p_{1},p_{2})\Big{)}<\infty$
for $j\in\\{1,2\\}$, $\boldsymbol{\zeta}=(\zeta_{1},\zeta_{2})$ is bounded and
continuous on $\mathcal{P}^{2}$. By Brower’s fixed-point theorem, there exists
a stationary point $\mathbf{p}^{*}=(p_{1}^{*},p_{2}^{*})$. Both prices must
both be less than $\varsigma$, since profits decrease for all prices
sufficiently close to $\varsigma$. We now show that these prices are also
unique, borrowing a technique from Morrow and Skerlos (2008).
The first step is to prove that $(D\nabla\hat{\pi})(\mathbf{p}^{*})$ is
negative definite at any stationary $\mathbf{p}^{*}$. Note that
$(D\nabla\hat{\pi})(\mathbf{p}^{*})=\boldsymbol{\Lambda}(\mathbf{p}^{*})(\mathbf{I}-(D\boldsymbol{\zeta})(\mathbf{p}^{*}))$;
this relationship is valid for Mixed Logit models with multiple firms as well.
Furthermore $\zeta_{j}(\mathbf{p})=\hat{\pi}(\mathbf{p})-(Dw_{k})(p_{k})^{-1}$
for any simple Logit model and any number of products. Hence
$(D_{k}\zeta_{j})(\mathbf{p})=(D_{k}\hat{\pi})(\mathbf{p})+\delta_{j,k}\left(\frac{(D^{2}w_{k})(p_{k})}{(Dw_{k})(p_{k})^{2}}\right),$
and $\mathbf{I}-(D\boldsymbol{\zeta})(\mathbf{p}^{*})$ is a diagonal matrix
with elements
$1-\frac{(D^{2}w_{k})(p_{k})}{(Dw_{k})(p_{k})^{2}}.$
In the case of this example,
$1-\frac{(D^{2}w_{1})(p_{1})}{(Dw_{1})(p_{1})^{2}}=1-\frac{(D^{2}w_{2})(p_{2})}{(Dw_{2})(p_{2})^{2}}=1+\frac{1}{\alpha}>0.$
Thus $(D\nabla\hat{\pi})(\mathbf{p}^{*})$ is negative definite at any
stationary point, and any stationary point maximizes profits.
The next step is to prove that the existence of only maximizers of profits
proves that there is a unique pair of profit-maximizing prices. Morrow and
Skerlos (2008) accomplish this with an application of the Poincare-Hopf
theorem (Milnor, 1965), as follows. Consider $-\hat{\pi}(\mathbf{p})$. This
function is minimized at any stationary $\mathbf{p}^{*}=(p_{1},p_{2})$, and
thus the gradient vector field $-(\nabla\hat{\pi})(\mathbf{p})$ has index $1$
at any stationary point $\mathbf{p}^{*}$ (Milnor, 1965). Note also that
$\displaystyle\mathrm{sign}\\{-(D_{j}\hat{\pi})(p_{1},p_{2})\\}$
$\displaystyle=\mathrm{sign}\left\\{p_{j}-c-\hat{\pi}(p_{1},p_{2})-\frac{\varsigma-
p_{j}}{\alpha}\right\\}$
for $j\in\\{1,2\\}$. This equation shows that the gradient vector field
$-(\nabla\hat{\pi})(\mathbf{p})$ points outward on the boundary of the
compact, convex set $[c,\varsigma]^{2}$, as can be checked. Thus the Poincare-
Hopf theorem states that the sum of the indices of the critical (stationary)
points equals one, the Euler characteristic of $[c,\varsigma]^{2}$. Since the
index of any critical (stationary) point of $-(\nabla\hat{\pi})(\mathbf{p})$
is one, there can only be one stationary point.
Using similar arguments, we see that the sub-problems formed by offering
product 1 or product 2 alone also have unique profit-maximizing prices
$q_{1}^{*}$ and $q_{2}^{*}$, respectively. Because $v_{1}$ and $v_{2}$ may be
distinct, these prices need not be the same.
We have claimed that variational formulation of this problem has four
solutions, only one of which is an equilibrium. Indeed, these four solutions
are $(p_{1}^{*},p_{2}^{*})$, $(q_{1}^{*},\varsigma)$, $(\varsigma,q_{2}^{*})$,
and $(\varsigma,\varsigma)$ but, as shown above, only $(p_{1}^{*},p_{2}^{*})$
is an equilibrium. While this follows from Props. 5.1 and 5.2 above, we prove
it directly here. Of course, $(p_{1}^{*},p_{2}^{*})$ is a solution since
$(\nabla\hat{\pi})(p_{1}^{*},p_{2}^{*})=(0,0)$. Since
$\lim_{p_{j}\uparrow\varsigma}\lambda_{j}(p_{1},p_{2})=\lim_{p_{j}\uparrow\varsigma}\left[\left(\frac{\varsigma-
p_{j}}{\alpha}\right)P_{j}(p_{1},p_{2})\right]=0$
for $j\in\\{1,2\\}$,
$\lim_{p_{j}\uparrow\varsigma}(D_{j}\hat{\pi})(p_{1},p_{2})=0$ (i.e.,
Assumption 3.1 holds). Thus $(\nabla\hat{\pi})(\varsigma,\varsigma)=(0,0)$,
and the variational inequality is satisfied at $(\varsigma,\varsigma)$.
Furthermore,
$(D_{1}\hat{\pi})(\varsigma,p_{2})(\varsigma-
q_{1})+(D_{2}\hat{\pi})(\varsigma,p_{2})(p_{2}-q_{2})=(D_{2}\hat{\pi})(\varsigma,p_{2})(p_{2}-q_{2})$
and thus $(\varsigma,q_{2}^{*})$ is also a solution to the variational
inequality. Similarly, $(q_{1}^{*},\varsigma)$ is also a solution. This
completes the proof. ∎
Example 8 is easily generalized to include $J>2$ products and a variational
inequality with $2^{J}$ solutions. One of these solutions is the unique vector
of profit-maximizing prices for the original problem, one is
$\varsigma\mathbf{1}\in\mathcal{P}^{J}$ and is not profit-maximizing for any
sub-problem, and the rest are profit-maximizing for some sub-problem but not
profit-maximizing for the original problem.
This property of variational formulations is especially problematic since
computations of equilibrium prices must often be performed using models with
$\varsigma_{*}<\infty$. Such models may be derived from simulation-based
approximations to Mixed Logit models with reservation prices that are finite
$\mu$-a.e., as in the Berry et al. (1995) model of Example 2.
Fortunately methods based on the $\boldsymbol{\zeta}$ map resolve only
equilibria of the original problem. In Section 5.1.3 we consider the important
class of simulation-based approximations to Mixed Logit models like those from
Example 2 and prove that fixed-points of
$\mathbf{c}+\boldsymbol{\zeta}(\cdot)$ cannot be equilibria of a sub-problem
that is not an equilibria of the original model. This is essentially a
consequence of Eqn. (21), which connects the sign of
$(D_{k}\hat{\pi}_{f})(\mathbf{p})$ directly to the sign of
$p_{k}-c_{k}-\zeta_{k}(\mathbf{p})$.
Similar results may apply to the markup equation. However because Eqn. (20)
involves $(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}$ instead of simply the
diagonal matrix $\boldsymbol{\Lambda}(\mathbf{p})$, the relationship between
the sign of $p_{k}-c_{k}-\eta_{k}(\mathbf{p})$ and the sign of
$(D_{k}\hat{\pi}_{f})(\mathbf{p})$ is not clear.
#### 5.1.2. General Results.
We now prove the results stated above concerning a variational formulation of
the price equilibrium problem when $\varsigma_{*}<\infty$.
###### Proposition 5.1.
Suppose $\varsigma_{*}<\infty$ and Assumptions 2.1-3.1 hold. Then the
variational inequality (32) always contains
$\varsigma_{*}\mathbf{1}\in\mathcal{P}^{J}$ as a solution.
###### Proof.
Since $(\tilde{\nabla}\hat{\pi})(\varsigma_{*}\mathbf{1})=\mathbf{0}$, Eqn.
(32) is trivially satisfied. ∎
The following proposition states that this variational formulation is poorly
posed in the sense that it contains solutions to all sub-problems.
###### Proposition 5.2.
Let $\varsigma_{*}<\infty$ and Assumptions 2.1-3.1 hold. Consider a proper
subset $\mathcal{J}^{\prime}\subset\mathbb{N}(J)$ of
$J^{\prime}=\left\lvert\mathcal{J}^{\prime}\right\rvert$ product indices, and
any solution
$\mathbf{p}_{\mathcal{J}^{\prime}}^{*}=\\{p_{j}^{*}:j\in\mathcal{J}^{\prime})$
to the sub-variational inequality
$\sum_{j\in\mathcal{J}^{\prime}}(D_{j}\hat{\pi}_{f(j)})(\mathbf{p}_{\mathcal{J}^{\prime}}^{*})(p_{j}^{*}-q_{j})\geq
0\quad\text{for
all}\quad\mathbf{q}_{\mathcal{J}^{\prime}}=\\{q_{j}:j\in\mathcal{J}^{\prime}\\}\subset[0,\varsigma_{*}]^{J^{\prime}}.$
If we define $\mathbf{p}\in[0,\varsigma_{*}]^{J}$ by $p_{j}=p_{j}^{*}$ for all
$j\in\mathcal{J}^{\prime}$ and $p_{k}=\varsigma_{*}$ for all
$k\notin\mathcal{J}^{\prime}$ then $\mathbf{p}$ solves the full variational
inequality (32).
###### Proof.
Because
$(D_{j}\hat{\pi}_{f(j)})(\mathbf{p})=\left\\{\begin{aligned}
&(D_{j}\hat{\pi}_{f(j)})(\mathbf{p}_{\mathcal{J}^{\prime}}^{*})&&\quad\text{if
}j\in\mathcal{J}^{\prime}\\\ &\quad\quad 0&&\quad\text{if
}j\notin\mathcal{J}^{\prime}\\\ \end{aligned}\right.$
we have
$\sum_{j=1}^{J}(D_{j}\hat{\pi}_{f(j)})(\mathbf{p})(p_{j}-q_{j})=\sum_{j\in\mathcal{J}^{\prime}}(D_{j}\hat{\pi}_{f(j)})(\mathbf{p}_{\mathcal{J}^{\prime}}^{*})(p_{j}^{*}-q_{j})\geq
0$
for all $\mathbf{q}\in[0,\varsigma_{*}]^{J}$. ∎
#### 5.1.3. The Resolution of Equilibria with $\boldsymbol{\zeta}$
We have shown that variational formulations of the equilibrium problem nest
equilibria of all sub-problems, which may not be equilibria of the original
problem as Example 8. In this section we show that methods based on the
$\boldsymbol{\zeta}$ map need not have this unfortunate shortcoming. This
result strongly distinguishes nonlinear system methods based on the
$\boldsymbol{\zeta}$ map from variational approaches.
We motivate this result with an example.
###### Example 9.
Consider a finite-sample approximation to the Berry et al. (1995) model of
Example 2. That is, choose $S\in\mathbb{N}$ and draw
$\\{\boldsymbol{\theta}_{s}\\}_{s=1}^{S}$ where
$\boldsymbol{\theta}_{s}=(\phi_{s},\boldsymbol{\beta}_{s},\beta_{0,s})$. These
samples could be drawn via standard sampling from $\mu$ or from another
technique like importance or quasi-random sampling. In any case, suppose that
the $\phi$’s drawn are distinct with probability one: $\phi_{s}\neq\phi_{r}$
for all $s,r\in\mathbb{N}(S)$ with probability one. Without loss of generality
we take $\phi_{1}<\phi_{2}<\dotsb<\phi_{S}$, and note that
$\varsigma_{*}=\phi_{S}<\infty$. If
$\mathbf{p}=\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$ and
$p_{k}>\varsigma_{*}$, then firm $f(k)$’s profits increase with the price of
the $k^{\text{th}}$ product in some neighborhood of $\varsigma_{*}$.
Thus if we compute some fixed-point
$\mathbf{p}=\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$ with
$p_{k}>\varsigma_{*}$, we know that excluding product $k$ is profit-optimal
for firm $f(k)$. As shown in Example 8, this is not the case with the VI
formulation.
###### Proof.
We will first define $\boldsymbol{\zeta}$ on all of $\mathcal{P}^{J}$, and
then consider fixed-points
$\mathbf{p}=\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$ with
$p_{k}\geq\phi_{S}=\varsigma_{*}$.
To extend $\boldsymbol{\zeta}$, we define
$\zeta_{k}(p_{1},\dotsc,p_{k},\dotsc,p_{J})=\zeta_{k}(p_{1},\dotsc,\varsigma_{*},\dotsc,p_{J})=\lim_{q\to\varsigma_{*}}\zeta_{k}(p_{1},\dotsc,q,\dotsc,p_{J}).$
when $p_{k}\geq\varsigma_{*}$. Note that for all $k$ and all
$\mathbf{p}\in(0,\varsigma_{*})^{J}$ we can write
$\zeta_{k}(\mathbf{p})=\sum_{s:\phi_{s}>p_{k}}\left(\sum_{j\in\mathcal{J}_{f(k)}}P_{j}^{L}(\boldsymbol{\theta}_{s},\mathbf{p})(p_{j}-c_{j})+\frac{\phi_{s}-p_{k}}{\alpha}\right)\left(\frac{P_{k}^{L}(\boldsymbol{\theta}_{s},\mathbf{p})/(\phi_{s}-p_{k})}{\sum_{r:\phi_{r}>p_{k}}P_{k}^{L}(\boldsymbol{\theta}_{r},\mathbf{p})/(\phi_{r}-p_{k})}\right)$
We first define $\lim_{p_{k}\uparrow\phi_{S}}\zeta_{k}(\mathbf{p})$, we first
note that for all $p_{k}\in(\phi_{S-1},\phi_{S})\neq\\{\emptyset\\}$, we have
$\zeta_{k}(\mathbf{p})=\sum_{j\in\mathcal{J}_{f(k)}}P_{j}^{L}(\boldsymbol{\theta}_{S},\mathbf{p})(p_{j}-c_{j})+\frac{\phi_{S}-p_{k}}{\alpha}$
since $p_{k}>\phi_{s}$ for all $s\in\\{1,\dotsc,S-1\\}$. Thus
$\lim_{p_{k}\uparrow\phi_{S}}\zeta_{k}(\mathbf{p})=\sum_{j\in\mathcal{J}_{f(k)}\setminus\\{k\\}}\left[\lim_{p_{k}\uparrow\phi_{S}}P_{j}^{L}(\boldsymbol{\theta}_{S},\mathbf{p})\right](p_{j}-c_{j}).$
In other words, as $p_{k}$ approaches $\phi_{S}=\varsigma_{*}$, $\zeta_{k}$
approaches the profits firm $f(k)$ accrues from selling all products other
than $p_{k}$ to the sampled individual with the highest income. This
establishes that the extended $\boldsymbol{\zeta}$ is well-defined and
continuous.
Now suppose $p_{k}=c_{k}+\zeta_{k}(\mathbf{p})$, where
$p_{k}>\phi_{S}=\varsigma_{*}$. Thus
$0=p_{k}-c_{k}-\zeta_{k}(\mathbf{p})>\phi_{S}-c_{k}-\zeta_{k}(\mathbf{p})=\lim_{q_{k}\uparrow\phi_{S}}\Big{(}q_{k}-c_{k}-\zeta_{k}(p_{1},\dotsc,q_{k},\dotsc,p_{J})\Big{)},$
and there must exist some $\delta>0$ such that
$q_{k}-c_{k}-\zeta_{k}(p_{1},\dotsc,q_{k},\dotsc,p_{J})<0$
for all $q_{k}\in(\varsigma_{*}-\delta,\varsigma_{*})$. Hence
$(D_{k}\hat{\pi}_{f(k)})(p_{1},\dotsc,q_{k},\dotsc,p_{J})>0$
$(D_{k}\hat{\pi}_{f(k)})(p_{1},\dotsc,q_{k},\dotsc,p_{J})=\lambda_{k}(p_{1},\dotsc,q_{k},\dotsc,p_{J})\big{(}q_{k}-c_{k}-\zeta_{k}(p_{1},\dotsc,q_{k},\dotsc,p_{J})\big{)}>0.$
In other words, if $\mathbf{p}=\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$ and
$p_{k}>\varsigma_{*}$, then firm $f(k)$’s profits increase with the price of
the $k^{\text{th}}$ product in some neighborhood of $\varsigma_{*}$. ∎
Fortunately this example is fairly general. In the following proposition we
prove that all finite-sample simulators generate $\boldsymbol{\zeta}$ maps
that do not have equilibria of sub-problems as fixed points unless they are,
in fact, equilibria of the original problem. Three assumptions are added:
utilities must be twice continuously differentiable in prices,
$\varsigma(\boldsymbol{\theta})$ is finite $\mu$-a.e. as in the Berry et al.
(1995) model, and the sampled values $\varsigma(\boldsymbol{\theta}_{s})$ must
be distinct with probability one.
###### Proposition 5.3.
Consider a Mixed Logit model satisfying Assumptions 2.1, 2.3, and 3.1 with
$w_{j}(\boldsymbol{\theta},\cdot):(0,\varsigma(\boldsymbol{\theta}))\to\mathbb{R}$
twice continuously differentiable in price and
$\varsigma:\mathcal{T}\to\mathcal{P}$ finite $\mu$-a.e..
Generate a finite-sample simulator to this Mixed Logit model with
$\\{\boldsymbol{\theta}_{s}\\}_{s=1}^{S}$ for some $S\in\mathbb{N}$. Let
$\varsigma_{s}=\varsigma(\boldsymbol{\theta}_{s})$, and assume that
$\varsigma_{s}\neq\varsigma_{r}$ with probability one for any $s\neq r$.
Subsequently, order the samples so that
$\varsigma_{1}<\dotsb<\varsigma_{S}=\varsigma_{*}$.
Suppose that $\mathbf{p}\in\mathcal{P}^{J}$ satisfies
$\mathbf{p}=\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$ where
$\boldsymbol{\zeta}$ is the extended map as in Example 9. If
$p_{k}\geq\varsigma_{S}$, then excluding product $k$ is profit-optimal for
firm $f=f(k)$; particularly, there exists $\delta>0$ such that
$(D_{k}\hat{\pi}_{f(k)})(p_{1},\dotsc,p_{k},\dotsc,p_{J})>0$
for all $p_{k}\in(\varsigma_{S}-\delta,\varsigma_{S})$.
###### Proof.
The case $p_{k}>\varsigma_{S}$ is handled exactly as in Example 9. We must
only consider the case where
$0=\varsigma_{S}-c_{k}-\lim_{p_{k}\uparrow\varsigma_{S}}\zeta_{k}(\mathbf{p})=\lim_{p_{k}\uparrow\varsigma_{S}}\Big{[}p_{k}-c_{k}-\zeta_{k}(\mathbf{p})\Big{]}.$
Our approach is to show that $D_{k}[p_{k}-c_{k}-\zeta_{k}(\mathbf{p})]>0$ for
all $p_{k}$ near enough to $\varsigma_{S}$, and thus
$\displaystyle p_{k}-c_{k}-\zeta_{k}(\mathbf{p})$
$\displaystyle=p_{k}-c_{k}-\zeta_{k}(\mathbf{p})-\Big{[}\varsigma_{S}-c_{k}-\lim_{p_{k}\uparrow\varsigma_{S}}\zeta_{k}(\mathbf{p})\Big{]}$
$\displaystyle\quad\quad\quad\quad=-\int_{p_{k}}^{\varsigma_{S}}D_{k}[p_{k}-c_{k}-\zeta_{k}(\mathbf{p})]dp_{k}<0$
(with a slight abuse of notation in the integral). More specifically, we prove
that
$\lim_{p_{k}\uparrow\varsigma_{S}}D_{k}[p_{k}-c_{k}-\zeta_{k}(\mathbf{p})]>0$,
which implies that $D_{k}[p_{k}-c_{k}-\zeta_{k}(\mathbf{p})]>0$ for all
$p_{k}$ near enough to $\varsigma_{S}$. Because
$p_{k}-c_{k}-\zeta_{k}(\mathbf{p})<0$ for $p_{k}$ near enough to
$\varsigma_{S}$,
$(D_{k}\hat{\pi}_{f(k)})(p_{1},\dotsc,q_{k},\dotsc,p_{J})=\lambda_{k}(\mathbf{p})\Big{(}p_{k}-c_{k}-\zeta_{k}(\mathbf{p})\Big{)}>0.$
As in Example 9, note that for all $p_{k}\in(\varsigma_{S-1},\varsigma_{S})$
we have
$\zeta_{k}(\mathbf{p})=\sum_{j\in\mathcal{J}_{f(k)}}P_{j}^{L}(\boldsymbol{\theta}_{S},\mathbf{p})(p_{j}-c_{j})-\frac{1}{(Dw_{k})(\boldsymbol{\theta}_{S},p_{k})}$
since $p_{k}>\varsigma_{s}$ for all $s\in\\{1,\dotsc,S-1\\}$. From this
equation we derive
$(D_{k}\zeta_{k})(\mathbf{p})=\sum_{j\in\mathcal{J}_{f(k)}}(D_{k}P_{j}^{L})(\boldsymbol{\theta}_{S},\mathbf{p})(p_{j}-c_{j})+P_{k}^{L}(\boldsymbol{\theta}_{S},\mathbf{p})+\frac{(D^{2}w_{k})(\boldsymbol{\theta}_{S},p_{k})}{(Dw_{k})(\boldsymbol{\theta}_{S},p_{k})^{2}}$
and thus
$\displaystyle D_{k}\Big{[}p_{k}-c_{k}-\zeta_{k}(\mathbf{p})\Big{]}$
$\displaystyle\quad\quad=1-(Dw_{k})(\boldsymbol{\theta}_{S},p_{k})P_{k}^{L}(\boldsymbol{\theta}_{S},\mathbf{p})\sum_{j\in\mathcal{J}_{f(k)}}P_{j}^{L}(\boldsymbol{\theta}_{S},\mathbf{p})(p_{j}-c_{j})-P_{k}^{L}(\boldsymbol{\theta}_{S},\mathbf{p})-\frac{(D^{2}w_{k})(\boldsymbol{\theta}_{S},p_{k})}{(Dw_{k})(\boldsymbol{\theta}_{S},p_{k})^{2}}$
Now
$\lim_{p_{k}\uparrow\varsigma_{S}}P_{k}^{L}(\boldsymbol{\theta}_{S},\mathbf{p})=0$,
we have assumed that
$\lim_{p_{k}\uparrow\varsigma_{S}}\big{[}(Dw_{k})(\boldsymbol{\theta}_{S},p_{k})P_{k}^{L}(\boldsymbol{\theta}_{S},\mathbf{p})\big{]}=0$
(Assumption 3.1), and
$\lim_{p_{k}\uparrow\varsigma_{S}}\left[\sum_{j\in\mathcal{J}_{f(k)}}P_{j}^{L}(\boldsymbol{\theta}_{S},\mathbf{p})(p_{j}-c_{j})\right]=\sum_{j\in\mathcal{J}_{f(k)}\setminus\\{k\\}}\lim_{p_{k}\uparrow\varsigma_{S}}\left[P_{j}^{L}(\boldsymbol{\theta}_{S},\mathbf{p})\right](p_{j}-c_{j})<\infty,$
we have
$\displaystyle\lim_{p_{k}\uparrow\varsigma_{S}}D_{k}\Big{[}p_{k}-c_{k}-\zeta_{k}(\mathbf{p})\Big{]}=1-\lim_{p_{k}\uparrow\varsigma_{S}}\left[\frac{(D^{2}w_{k})(\boldsymbol{\theta}_{S},p_{k})}{(Dw_{k})(\boldsymbol{\theta}_{S},p_{k})^{2}}\right]$
So long as
$\displaystyle\lim_{p_{k}\uparrow\varsigma_{S}}\left[\frac{(D^{2}w_{k})(\boldsymbol{\theta}_{S},p_{k})}{(Dw_{k})(\boldsymbol{\theta}_{S},p_{k})^{2}}\right]<1$
we have
$\lim_{p_{k}\uparrow\varsigma_{S}}D_{k}\Big{[}p_{k}-c_{k}-\zeta_{k}(\mathbf{p})\Big{]}>0$.
This must be true, as Claim 1 below demonstrates. This completes the proof. ∎
###### Claim 1.
Let $w:(0,\varsigma)\to\mathbb{R}$ be twice continuously differentiable, with
$(Dw)(p)<0$ for all $p\in(0,\varsigma)$ and $(Dw)(p)\downarrow-\infty$ as
$p\uparrow\varsigma$. Then
$\lim_{p\uparrow\varsigma}\left[\frac{(D^{2}w)(p)}{(Dw)(p)^{2}}\right]<1.$
###### Proof.
Proof We prove this by contradiction. Note that
$D\left[\frac{1}{\left\lvert(Dw)(p)\right\rvert}\right]=\frac{(D^{2}w)(p)}{(Dw)(p)^{2}}.$
Now if
$\displaystyle\lim_{p\uparrow\varsigma_{S}}\left[\frac{(D^{2}w)(p)}{(Dw)(p)^{2}}\right]\geq
1,$
there must exist some $\bar{p}\in(0,\varsigma)$ such that
$\displaystyle\frac{(D^{2}w)(p)}{(Dw)(p)^{2}}>0\quad\text{for all}\quad
p\in[\bar{p},\varsigma).$
But then
$0\leq\int_{p}^{\varsigma}\frac{(D^{2}w)(q)}{(Dw)(q)^{2}}dq=\int_{p}^{\varsigma}D\left[\frac{1}{\left\lvert(Dw)(q)\right\rvert}\right]dq=\lim_{q\uparrow\varsigma}\left[\frac{1}{\left\lvert(Dw)(q)\right\rvert}\right]-\frac{1}{\left\lvert(Dw)(p)\right\rvert}=-\frac{1}{\left\lvert(Dw)(p)\right\rvert}<0,$
a contradiction. ∎
### 5.2. Tatonnement
Some authors iterate best responses $-$ i.e. tatonnement $-$ to compute
equilibria. See, for example, Choi et al. (1990); CBO (2003); Michalek et al.
(2004); Austin and Dinan (2005); Bento et al. (2005); Hu and Ralph (2007). For
this process Newton’s method, or another algorithm of (unconstrained)
optimization, will be required. Tatonnement should be an efficient way to
compute “equilibrium” if all firm’s profit-maximizing prices are independent
of their competitor’s decisions, but wasteful if some firm’s optimal pricing
depends heavily on their competitors’ prices. Furthermore no convergence
guarantees exist for tatonnement while there are at least theoretical
guarantees that Newton’s method, properly constructed, will converge to
simultaneously stationary prices.
### 5.3. Least-Squares Minimization and the Gauss-Newton Method
In principle one could also use optimization methods to explicitly minimize
$f(\mathbf{p})=\lvert\lvert\mathbf{F}(\mathbf{p})\rvert\rvert_{2}^{2}/2$ for
any of our choices of $\mathbf{F}$. In fact, line search and trust-region
strategies for global convergence implicitly minimize this function (Dennis
and Schnabel, 1996). Computations of equilibrium prices benefit from leaving
this implicit, as explicit minimization via Newton’s method requires third-
order derivatives of $\mathbf{F}$, increasing both differentiability
requirements and computational burden. The Gauss-Newton method (Ortega and
Rheinboldt, 1970) is obtained by neglecting the influence of the third-order
derivatives of $\mathbf{F}$. This defines the Gauss-Newton step as a solution
to the (symmetric) normal equation
$(D\mathbf{F})(\mathbf{p})^{\top}(D\mathbf{F})(\mathbf{p})\mathbf{s}=-(D\mathbf{F})(\mathbf{p})^{\top}\mathbf{F}(\mathbf{p})$;
note that the same problem arises should one wish to use the Conjugate
Gradient method to solve the Newton system. So long as
$(D\mathbf{F})(\mathbf{p})$ is nonsingular the standard Newton steps will be
recovered from the Gauss-Newton method. However they are explicitly formulated
as solutions to linear systems that are more poorly conditioned (Golub and
Loan, 1996; Trefethen and Bau, 1997) and thus we should at least expect to
accumulate more error in the process of solving for the same steps. The burden
of computing these steps also increases because of the requirement to multiply
by the transpose of the Jacobian of $\mathbf{F}$.
## 6\. Acknowledgements
This research was supported by the National Science Foundation, the University
of Michigan Transportation Research Institute’s Doctoral Studies Program, and
a research fellowship at the Belfer Center for Science and International
Affairs at the Harvard Kennedy School. Both authors wish to thank Walter
McManus, Brock Palen, Divakar Viswanath, Erin MacDonald, the editors, and the
three anonymous reviewers for their contributions to this research. We would
also like to acknowledge helpful suggestions offered by Fred Feinberg, John
Hauser, Meredith Fowlie, Kenneth Judd, and the support of Kelly Sims-Gallagher
and Henry Lee at the Belfer Center. Three anonymous reviewers, an associate
editor, and Duncan Simester at Operations Research provided extremely helpful
suggestions.
## References
* Aguirregabiria and Vicentini (2006) Aguirregabiria, V. and G. Vicentini (2006, August). Dynamic spatial competition between multi-store firms. Working Paper, University of Toronto.
* Anderson and de Palma (1988) Anderson, S. P. and A. de Palma (1988, October). Spatial price discrimination with heterogeneous products. The Review of Economic Studies 55(4), 573–592.
* Austin and Dinan (2005) Austin, D. and T. Dinan (2005). Clearing the air: The costs and consequences of higher caf standards and increased gasoline taxes. Journal of Environmental Economics and Management 50, 562–582.
* Bartle (1966) Bartle, R. G. (1966). The Elements of Integration and Lebesgue Masure. Jon Wiley and Sons, Inc.
* Baye and Kovenock (2008) Baye, M. R. and D. Kovenock (2008). Bertrand competition. In S. N. Durlaf and L. E. Blume (Eds.), The New Pagrave Dictionary of Economics, Second Edition. Palgrave Macmillian.
* Bento et al. (2005) Bento, A. M., L. M. Goulder, E. Henry, M. R. Jacobsen, and R. H. von Haefen (2005, May). Distributional and efficiency impacts of gasoline taxes: An econometrically based multi-market study. The American Economic Review 95(2), 282–287.
* Beresteanu and Li (2008) Beresteanu, A. and S. Li (2008). Gasoline prices, government support, and the demand for hybrid vehicles in the u.s. Working Paper, Duke University.
* Berry et al. (1995) Berry, S., J. Levinsohn, and A. Pakes (1995, June). Automobile prices in market equilibrium. Econometrica 63(4), 841–890.
* Berry et al. (2004) Berry, S., J. Levinsohn, and A. Pakes (2004). Differentiated products demand systems from a combination of micro and macro data: The new car market. Journal of Political Economy 112(1), 68–105.
* Boyd and Mellman (1980) Boyd, J. H. and R. E. Mellman (1980). The effect of fuel economy standards on the u.s. automotive market: An hedonic demand analysis. Transportation Research A 14A, 367–378.
* Brown and Saad (1990) Brown, P. N. and Y. Saad (1990, May). Hybrid krylov methods for nonlinear systems of equations. SIAM Journal of Scientific and Statistical Computing 11(3), 450–481.
* Brownstone et al. (2000) Brownstone, D., D. S. Bunch, and K. Train (2000). Joint mixed logit models of stated and revealed preferences for alternative fuel vehicles. Transportation Research B 34, 315–338.
* Cai and Keyes (2002) Cai, X.-C. and D. E. Keyes (2002). Nonlinearly preconditioned inexact newton algorithms. SIAM Journal of Scientific and Statistical Computing 24(1), 183–200.
* Caplin and Nalebuff (1991) Caplin, A. and B. Nalebuff (1991). Aggregation and imperfect competition: On the existence of equilibrium. Econometrica 59(1), 25–59.
* CBO (2003) CBO (2003, December). The economic costs of fuel economy standards versus a gasoline tax. Technical report, Congressional Budget Office.
* Choi et al. (1990) Choi, S. C., W. S. Desarbo, and P. T. Harker (1990, February). Product positioning under price competition. Management Science 36(2), 175–199.
* Clarke (1975) Clarke, F. H. (1975, April). Generalized gradients and applications. Transactions of the American Mathematical Society 205, 247–262.
* Dembo et al. (1982) Dembo, R. S., S. C. Eisenstat, and T. Steihaug (1982, April). Inexact newton methods. SIAM Journal on Numerical Analysis 19(2), 400–408.
* Dennis and Schnabel (1996) Dennis, J. E. and R. B. Schnabel (1996). Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM.
* Dirkse and Ferris (1995) Dirkse, S. P. and M. C. Ferris (1995). The path solver: A non-monotone stabilization scheme for mixed complementarity problems. Optimization Methods and Software 5, 123–156.
* Doraszelski and Draganska (2006) Doraszelski, U. and M. Draganska (2006). Market segmentation strategies of multiproduct firms. Journal of Industrial Economics 54(1), 125–149.
* Draganska and Jain (2004) Draganska, M. and D. Jain (2004). A likelihood approach to estimating market equilibrium models. Management Science 50(5), 605–616.
* Dube et al. (2002) Dube, J.-P., P. Chintagunta, A. Petrin, B. Bronnenberg, R. Goettler, P. B. Seetharaman, K. Sudhir, R. Thomadsen, and Y. Zhao (2002). Structural applications of the discrete choice model. Marketing Letters 13(3), 207–220.
* Dube et al. (2008) Dube, J.-P., J. T. Fox, and C.-L. Su (2008, June). Improving the performance of blp static and dynamic discrete choice random coefficients demand estimation. Working paper, University of Chicago.
* Eisenstat and Walker (1994) Eisenstat, S. C. and H. F. Walker (1994, May). Globally convergent inexact newton methods. SIAM Journal on Optimization 4(2), 393–422.
* Eisenstat and Walker (1996) Eisenstat, S. C. and H. F. Walker (1996, January). Choosing the forcing terms in an inexact newton method. SIAM Journal of Scientific Computing 17(1), 16–32.
* Ferris and Pang (1997) Ferris, M. C. and J.-S. Pang (1997). Engineering and economic applications of complementarity problems. SIAM Review 39(4), 669–713.
* Goldberg (1995) Goldberg, P. K. (1995, June). Product differentiation and oligopoly in international markets: The case of the u.s. automobile industry. Econometrica 63(4), 891–951.
* Goldberg (1998) Goldberg, P. K. (1998). The effects of the corporate average fuel efficiency standards in the us. The Journal of Industrial Economics 46(1), 1–33.
* Golub and Loan (1996) Golub, G. H. and C. F. V. Loan (1996). Matrix Computations. The Johns Hopkins University Press.
* Goolsbee and Petrin (2004) Goolsbee, A. and A. Petrin (2004, March). The consumer gains from direct broadcast satellites and the competition with cable tv. Econometrica 72(2), 351–381.
* Halcrow et al. (2009) Halcrow, J., J. F. Gibson, P. Cvitanovic, and D. Viswanath (2009). Heteroclinic connections in plane couette flow. Journal of Fluid Mechanics 611, 365–376.
* Hanson and Martin (1996) Hanson, W. and K. Martin (1996, July). Optimizing multinomial logit profit functions. Management Science 42(7), 992–1003.
* Harker and Pang (1990) Harker, P. T. and J.-S. Pang (1990). Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms, and applications. Mathematical Programming 48, 161–220.
* Hu and Ralph (2007) Hu, X. and D. Ralph (2007, September-October). Using epecs to model bilevel games in restructured electricity markets with locational prices. Operations Research 55(5), 809–827.
* Jacobsen (2006) Jacobsen, M. R. (2006, November). Evaluating u. s. fuel economy standards in a model with producer and household heterogeneity. Working Paper, Stanford University.
* Judd (1998) Judd, K. L. (1998). Numerical Methods in Economics. MIT Press.
* Kelley (1995) Kelley, C. T. (1995). Iterative Methods for Linear and Nonlinear Equations, Volume 16 of Frontiers in Applied Mathematics. SIAM.
* Kelley (2003) Kelley, C. T. (2003). Solving Nonlinear Equations with Newton’s Method. Number Fundamentals of Algorithms. SIAM.
* Knittel and Metaxoglou (2008) Knittel, C. R. and K. Metaxoglou (2008, October). Estimation of random coefficient demand models: Challenges, difficulties, and warnings. Working Paper, University of California Davis.
* Levenberg (1944) Levenberg, K. (1944). A method for the solution of certain non-linear problems in least squares. The Quarterly Journal of Applied Mathematics 2, 164–168.
* Marquardt (1963) Marquardt, D. (1963). An algorithm for least-squares estimation of nonlinear parameters. SIAM Journal on Applied Mathematics 11, 431–441.
* McFadden (1989) McFadden, D. L. (1989). A method of simulated moments for estimation of discrete response models without numerical integration. Econometrica 57(5), 995–1026.
* Michalek et al. (2004) Michalek, J. J., P. Y. Papalambros, and S. J. Skerlos (2004). A study of fuel efficiency and emission policy impact on optimal vehicle design decisions. Journal of Mechanical Design 126(6), 1062–1070.
* Milgrom and Roberts (1990) Milgrom, P. and J. Roberts (1990, November). Rationalizability, learning, and equilibrium in games with strategic complementarities. Econometrica 58(6), 1255–1277.
* Milnor (1965) Milnor, J. W. (1965). Topology from the Differentiable Viewpoint. Princeton Landmarks in Mathematics. Princeton University Press.
* Morrow (2008) Morrow, W. R. (2008). A Fixed-Point Approach to Equilibrium Pricing in Differentiated Product Markets. Ph. D. thesis, Department of Mechanical Engineering, University of Michigan.
* Morrow and Skerlos (2008) Morrow, W. R. and S. J. Skerlos (2008). On the existence of bertrand-nash equilibrium prices under logit demand. Technical report, University of Michigan.
* Morrow and Skerlos (2010) Morrow, W. R. and S. J. Skerlos (2010). Fixed-Point Approaches to Computing Bertrand-Nash Equilibrium Prices Under Mixed-Logit Demand. Operations Research Forthcoming.
* Munkres (1991) Munkres, J. R. (1991). Analysis on Manifolds. Westview Press.
* Nevo (1997) Nevo, A. (1997, November). Mergers with differentiated products: The case of the ready-to-eat cereal industry. Technical report, University of California, Berkeley.
* Nevo (2000a) Nevo, A. (2000a). Mergers with differentiated products: The case of the ready-to-eat cereal industry. The RAND Journal of Economics 31(3), 395–421.
* Nevo (2000b) Nevo, A. (2000b). A practitioner’s guide to estimation of random-coefficients logit models of demand. Journal of Economics and Management Strategy 9(4), 513–548.
* Ortega and Rheinboldt (1970) Ortega, J. M. and W. C. Rheinboldt (1970). Iterative Solution of Nonlinear Equations in Several Variables. Society for Industrial and Applied Mathematics.
* Pawlowski et al. (2006) Pawlowski, R. P., J. N. Shadid, J. P. Simonis, and H. F. Walker (2006). Globalization techniques for newton-krylov methods and applications to the fully coupled solutions of the navier-stokes equations. SIAM Review 48(4), 700–721.
* Pawlowski et al. (2008) Pawlowski, R. P., J. P. Simonis, H. F. Walker, and J. N. Shadid (2008). Inexact newton dogleg methods. SIAM Journal of Numerical Analysis 46(4), 2112–2132.
* Pernice and Walker (1998) Pernice, M. and H. F. Walker (1998, January). Nitsol: A newton iterative solver for nonlinear systems. SIAM Journal of Scientific Computing 19(1), 302–318.
* Petrin (2002) Petrin, A. (2002). Quantifying the benefits of new products: The case of the minivan. Journal of Political Economy 110(4), 705–729.
* Ralph (1994) Ralph, D. (1994). Global convergence of damped newton’s method for nonsmooth equations, via the path search. Mathematics of Operations Research 19, 352–389.
* Saad and Schultz (1986) Saad, Y. and M. Schultz (1986, July). Gmres: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal of Scientific and Statistical Computing 7(3), 856–869.
* Sleijpen and Fokkema (1993) Sleijpen, G. L. G. and D. R. Fokkema (1993). Bicgstab(l) for linear equations involving unsymmetric matrices with complex spectrum. Electronic Transactions on Numerical Analysis 1, 11–32.
* Small and Rosen (1981) Small, K. and D. Rosen (1981). Applied welfare economics with discrete choice models. Econometrica 49(3), 105–130.
* Smith (2004) Smith, H. (2004). Supermarket choice and supermarket competition in market equilibrium. The Review of Economic Studies 71, 235–263.
* Su and Judd (2008) Su, C.-L. and K. L. Judd (2008, April). Constrained optimization approaches to estimation of structural models. Working paper, University of Chicago.
* Thomadsen (2005) Thomadsen, R. (2005, Winter). The effect of ownership structure on prices in geographically differentiated markets. The RAND Journal of Economics 36(4), 908–929.
* Train (2003) Train, K. (2003). Discrete Choice Methods with Simulation. Cambridge University Press.
* Trefethen and Bau (1997) Trefethen, L. N. and D. Bau (1997). Numerical Linear Algebra. SIAM.
* van der Vorst (1992) van der Vorst, H. A. (1992, March). Bi-cgstab: A fast and smoothly converging variant of bi-cg for the solution of nonsymmetric linear systems. SIAM Journal of Scientific and Statistical Computing 13(2), 631–644.
* Viswanath (2007) Viswanath, D. (2007). Recurrent motions within plane couette turbulence. Journal of Fluid Mechanics 580, 339–358.
* Viswanath (2009) Viswanath, D. (2009). The critical layer in pipe flow at high reynolds number. Philisophical Transactions of the Royal Society 367, 561–576.
* Viswanath and Cvitanovic (2009) Viswanath, D. and P. Cvitanovic (2009). Stable manifolds and the transition to turbulence in pipe flow. Journal of Fluid Mechanics 627, 215–233.
* Walker (1988) Walker, H. F. (1988). Implementation of the gmres method using householder transformations. SIAM Journal of Scientific and Statistical Computing 9(1), 152–163.
* Wards (2007) Wards (2004-2007). Wards Automotive Yearbooks. Wards Reports Inc.
|
arxiv-papers
| 2010-12-28T20:53:11 |
2024-09-04T02:49:16.026736
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "W. Ross Morrow and Steven J. Skerlos",
"submitter": "William Morrow",
"url": "https://arxiv.org/abs/1012.5836"
}
|
1012.5894
|
2010 Vol. XX No. XX, 000–000
11institutetext: Department of Astronomy, Peking University, Beijing 100871,
China; fanzuhui@pku.edu.cn
Received [year] [month] [day]; accepted [year] [month] [day]
# Comparison of Halo Detection from Noisy Weak Lensing Convergence Maps with
Gaussian Smoothing and MRLens Treatment
Y.-X. Jiao H.-Y. Shan and Z.-H. Fan
###### Abstract
Taking into account the noise from intrinsic ellipticities of source galaxies,
we study the efficiency and completeness of halo detections from weak lensing
convergence maps. Particularly, with numerical simulations, we compare the
Gaussian filter with the so called MRLens treatment based on the modification
of the Maximum Entropy Method. For a pure noise field without lensing signals,
a Gaussian smoothing results a residual noise field that is approximately
Gaussian in statistics if a large enough number of galaxies are included in
the smoothing window. On the other hand, the noise field after the MRLens
treatment is significantly non-Gaussian, resulting complications in
characterizing the noise effects. Considering weak-lensing cluster detections,
although the MRLens treatment effectively deletes false peaks arising from
noise, it removes the real peaks heavily due to its inability to distinguish
real signals with relatively low amplitudes from noise in its restoration
process. The higher the noise level is, the larger the removal effects are for
the real peaks. For a survey with a source density $n_{g}\sim 30\hbox{
arcmin}^{-2}$, the number of peaks found in an area of $3\times 3\hbox{
deg}^{2}$ after MRLens filtering is only $\sim 50$ for the detection threshold
$\kappa=0.02$, while the number of halos with $M>5\times 10^{13}\hbox{
M}_{\odot}$ and with redshift $z\leq 2$ in the same area is expected to be
$\sim 530$. For the Gaussian smoothing treatment, the number of detections is
$\sim 260$, much larger than that of the MRLens. The Gaussianity of the noise
statistics in the Gaussian smoothing case adds further advantages for this
method to circumvent the problem of the relatively low efficiency in weak-
lensing cluster detections. Therefore, in studies aiming to construct large
cluster samples from weak-lensing surveys, the Gaussian smoothing method
performs significantly better than the MRLens.
###### keywords:
cosmology: theory - gravitation - dark matter - gravitational lensing
## 1 Introduction
The weak gravitational lensing effect provides a unique tool in measuring the
matter distribution in the universe (e.g., Bartelmann & Schneider 2001;
Hoekstra et al. 2006; Massey et al. 2007). Its additional dependence on the
distances to the source, to the lens and between the source and lens makes it
an excellent probe in cosmological studies of dark energy (e.g., Albrecht et
al. 2006; Benjamin et al. 2007; Kilbinger et al. 2009; Li et al. 2009). On the
other hand, however, different observational and physical effects can affect
the weak lensing analyses significantly. Being extracted from shape distortion
of background galaxies, the weak lensing effect on individual source galaxies
is severely contaminated by their intrinsic ellipticities. Therefore
statistical analyses on a large number of galaxies are necessary in weak
lensing studies. Even so, intrinsic shape alignments of galaxies, including
intrinsic-intrinsic and shear-intrinsic correlations, can be an important
source of error in cosmic shear correlation analyses. For cluster detections
from weak lensing convergence maps reconstructed from shear measurements
(e.g., Kaiser & Squires 1993; Bartelmann 1995; Kaiser 1995; Schneider & Seitz
1995; Squires & Kaiser 1996; Bridle et al.1998; Marshall et al. 2002), even
randomly orientated intrinsic ellipticities can result false peaks by their
chance alignments, which can reduce the efficiency of cluster detections
significantly (e.g., Schneider 1996; van Waerbeke 2000; White et al. 2002;
Hamana et al. 2004; Fan 2007). Thus further treatments for a convergence map
are normally required to suppress the noise effects.
The noise from intrinsic ellipticities of source galaxies is essentially shot
noise, and thus by averaging over a relatively large number of source galaxies
in weak lensing analyses, the residual noise can be effectively reduced. This
leads to the normal smoothing treatment. It is clear that the residual noise
depends on the form of the window function and the smoothing scale. For a
Gaussian smoothing with a window function of the form
$W(\theta)\propto\exp(-\theta^{2}/\theta_{G}^{2})$, the residual noise can be
estimated by
$\sigma^{2}_{0}\approx{(\sigma^{2}_{\epsilon}/2)}{[1/(2\pi\theta_{G}^{2}n_{g})]}$,
where $\sigma_{\epsilon}$ is the rms of the intrinsic ellipticity of
individual source galaxies, $\theta_{G}$ is the smoothing scale, and $n_{g}$
is the surface number density of source galaxies. For $\sigma_{\epsilon}=0.3$,
$n_{g}=30\hbox{ arcmin}^{-2}$ and $\theta_{G}=1\hbox{ arcmin}$, we have
$\sigma_{0}\approx 0.015$.
Recently, Starck et al. (2006) proposed the MRLens filtering technique, which
is based on the Bayesian analyses with a multi-scale entropy prior applied.
The False Detection Rate (FDR) method is used to select significant/non-
significant wavelet coefficients (e.g., Starck et al. 2006; Pires et al.
2009). The MRLens method suppresses noise adaptively according to the strength
of the noise itself. A more detailed description of the method is given in §4.
In this paper, with numerical simulations, we compare Gaussian smoothing with
MRLens treatment, paying particular attention to the completeness and the
efficiency of weak lensing halo detections from convergence maps. The rest of
the paper is organized as follows. In §2, we describe briefly the weak-lensing
convergence reconstruction and the Gaussian smoothing. In §3, we present the
important aspects of the MRLens treatment. Results are shown in §4. Section 5
contains summaries and discussions.
## 2 Weak lensing convergence reconstruction
In the weak lensing regime, the convergence $\kappa(\vec{\theta})$ is
essentially related to the weighted projection of density fluctuations
$\delta$ along the unperturbed light path. Specifically, we have
$\kappa(\vec{\theta})={3H_{0}^{2}\Omega_{0}\over
2}\int_{0}^{w_{H}}dw\bar{W}(w)f_{K}(w){\delta[f_{K}(w)\vec{\theta},w]\over
a(w)}$ (1)
where $H_{0}$ is the present Hubble constant, $\Omega_{0}$ is the present
matter density of the universe in unit of the critical density, ${w}$ is the
radial coordinate, $a(w)$ is the scale factor of the universe, and, with $K$
being the spatial curvature of the universe,
$\displaystyle f_{K}(w)$ $\displaystyle=|K|^{-1/2}\sin(|K|^{1/2}w)\quad\quad\
\ \ (K>0)$ (2) $\displaystyle=w\qquad\qquad\qquad\qquad\qquad\quad(K=0)$
$\displaystyle=|K|^{-1/2}\sinh(|K|^{1/2}w)\qquad(K<0)\ .$
The factor $\bar{W}(w)$ is the weighting function that is related to the
source galaxy distribution $G(w)$ by
$\bar{W}(w)=\int_{w}^{w_{H}}dw^{\prime}G(w^{\prime}){f_{K}(w^{\prime}-w)\over
f_{K}(w^{\prime})}\ .$ (3)
The lensing potential $\phi$ is related to $\kappa$ by
$\kappa={\nabla^{2}\phi\over 2}\ ,$ (4)
and the shears $\gamma_{1}$ and $\gamma_{2}$ are
$\gamma_{1}={\partial_{11}\phi-\partial_{22}\phi\over 2}\
,\quad\gamma_{2}=\partial_{12}\phi.$ (5)
Since both $\kappa$ and $\gamma_{i}$ are determined by the lensing potential,
they are mutually dependent of each other. In the Fourier space, we have
(Kaiser & Squires 1993)
$\kappa(\vec{k})=c_{1}(k)\gamma_{1}(\vec{k})+c_{2}(k)\gamma_{2}(\vec{k}),$ (6)
where $[c_{1},c_{2}]=[\cos(2\phi),\sin(2\phi)]$ with
$\vec{k}=k(\cos\phi,\sin\phi)$.
Observationally, the shear $\gamma$ can be extracted from the shape
measurement of source galaxy images. Under the condition $\kappa<<1$, we have
$\vec{e}^{\rm obs}\approx\vec{\gamma}+\vec{e}^{S}\ ,$ (7)
where $\vec{e}^{\rm obs}$ and $\vec{e}^{S}$ are the observed ellipticity, and
the intrinsic ellipticity of a source galaxy, respectively. Reconstructed from
$\vec{e}^{\rm obs}$, the convergence $\kappa_{n}(\vec{k})$ then contains noise
from the intrinsic part, i.e.,
$\kappa_{n}({\vec{k}})=c_{\alpha}(k)e^{\rm
obs}_{\alpha}({\vec{k}})=\kappa({\vec{k}})+c_{\alpha}(k)e^{S}_{\alpha}({\vec{k}}).$
(8)
With the transformation back to the real 2-D space and applying a smoothing
with the window function $W(\vec{\theta})$, we can obtain the smoothed
quantities (e.g., van Waerbeke 2000)
$\Sigma^{\rm
obs}(\vec{\theta})=\Gamma(\vec{\theta})+\frac{1}{n_{g}}\Sigma^{N_{g}}_{i=1}W(\vec{\theta}-\vec{\theta}_{i})e^{\rm
S}(\vec{\theta}_{i})$ (9)
and
$K_{N}(\vec{\theta})=\int
d\vec{k}e^{-i\vec{k}\cdot\vec{\theta}}c_{\alpha}(k)\Sigma_{\alpha}^{\rm
obs}(\vec{k}),$ (10)
where $\Sigma^{\rm obs}$, $\Gamma$, and $K_{N}$ are the smoothed $e^{\rm
obs}$, $\gamma$ and $\kappa_{n}$, respectively, and $n_{g}$ and $N_{g}$ are
the surface number density and the total number of source galaxies in the
field. The noise part of $K_{N}$ due to the intrinsic ellipticities is then
$N(\vec{\theta})=\frac{1}{n_{g}}\Sigma^{N_{g}}_{i=1}\int
d\vec{k}W(\vec{k})e^{-i\vec{k}\cdot(\vec{\theta}-\vec{\theta}_{i})}c_{\alpha}(k)e^{S}_{\alpha}(\vec{\theta}_{i}),$
(11)
where $W(\vec{k})$ is the Fourier transformation of the window function with
the form
$W(\vec{k})=\frac{1}{(2\pi)^{2}}\int
d\vec{\theta}e^{i\vec{k}\cdot\vec{\theta}}W(\vec{\theta}).$ (12)
Without considering the intrinsic alignment of $e^{S}$, it is expected from
the central limit theorem that the smoothed noise field $N(\vec{\theta})$ is
approximately Gaussian in statistics if the effective number of galaxies
included in the smoothing window is larger than about $10$ (e.g., van Waerbeke
2000). In this case, smoothing leads to correlations in $N(\vec{\theta})$, and
its two-point correlation function is approximately
$<N(\vec{\theta})N(\vec{\theta}^{\prime})>=\frac{\sigma_{\epsilon^{2}}}{2n_{g}}(2\pi)^{2}\int
d\vec{k}e^{i\vec{k}\cdot(\vec{\theta}^{\prime}-\vec{\theta})}|W(\vec{k})|^{2},$
(13)
where $\sigma_{\epsilon}$ is the intrinsic dispersion of $e^{\rm obs}$.
The approximate Gaussianity of $N(\vec{\theta})$ allows us to quantify the
noise effects straightforwardly. The noise effects on cluster mass
reconstruction and the noise peak statistics are analyzed in van Waerbeke
(2000). Even with weak alignments of intrinsic ellipticities,
$N(\vec{\theta})$ can still be approximately described by a Gaussian random
field with a modified two-point correlation function including the effects of
intrinsic alignments. The enhancement of the noise peak abundance due to the
weakly intrinsic alignments are analyzed in Fan (2007). In Fan et al. (2010),
the effects of the presence of real dark matter halos on the noise peak
statistics around them as well as the effects of the noise on the peak height
of real halos are investigated in detail. They further present a model to
calculate the total peak abundance in a large-scale convergence map, including
the peaks corresponding to real halos and the noise peaks from the chance
alignment of the intrinsic ellipticities of source galaxies. Such a model
makes it possible for us to use directly the peaks from convergence maps as
cosmological probes without the need to differentiate real and false peaks.
Due to its simple operational procedure and the Gaussian statistics of the
residual noise field, the smoothing treatment has been widely applied in weak
lensing analyses. Different smoothing functions have been used in different
studies. In this paper, we consider the Gaussian smoothing function $W_{G}$,
which is one of the most commonly adopted window functions. Specifically, we
have
$W_{G}=\frac{1}{\pi\theta_{G}^{2}}\exp\left(-\frac{\theta^{2}}{\theta_{G}^{2}}\right)\
,$ (14)
where $\theta_{G}$ is the smoothing scale. Then from Eq. (13), the rms of the
noise $\sigma_{0}$ after smoothing is given by
$\sigma_{0}^{2}={\sigma_{\epsilon}^{2}\over 2}{1\over
2\pi\theta_{G}^{2}n_{g}}\ .$ (15)
In our analyses, we choose $\sigma_{\epsilon}=0.3$, the typical value for
lensing source galaxies, and $\theta_{G}=1\hbox{ arcmin}$, which is the
optimal smoothing scale considering cluster-sized halos. Then for a lensing
survey with $n_{g}=30\hbox{ arcmin}^{-2}$, $\sigma_{0}\approx 0.015$, which is
about $20$ times lower than $\sigma_{\epsilon}$.
## 3 MRLens method
Starck et al. (2006) introduce a new reconstruction and filtering method,
namely, Multi-scale Entropy Restoration (MRLens). It is developed from the
Maximum Entropy Method. The basic idea is to use only ‘signals’ selected by
the so called False Discovery Rate (FDR) (Benjamini & Hochberg 1995) to
reconstruct the convergence field through a Multi-scale Entropy prior. In the
following, we present specific steps of MRLens.
### 3.1 Wavelet decomposition
For an original convergence map $\kappa_{obs}$ with $N=n\times n$ pixels, the
first step of MRLens is to decompose the image map into different components
representing fine structures of different scales.
To do this, we first initialize $j=0$ and set $C_{0}(k,l)=\kappa_{obs}(k,l)$,
i.e., $j=0$ corresponds to the unprocessed map with detailed structures. Then
we progressively go to higher $j$ to obtain smoother maps through (Starck et
al. 2001)
$C_{j+1}(k,l)=\sum_{m}\sum_{n}h_{1D}(m)h_{1D}(n)C_{j}(k+2^{j}m,l+2^{j}n),$
(16)
where $h_{1D}(m)={[1/16,4/16,6/16,4/16,1/16]}$ for $m=-2,-1,0,1,2$,
respectively. Defining
$w_{j+1}(k,l)=C_{j}(k,l)-C_{j+1}(k,l),$ (17)
we finally obtain
$\kappa_{obs}(k,l)=C_{J}(k,l)+\sum_{j=1}^{J}w_{j}(k.l),$ (18)
where $J$ is a chosen number determined by specific considerations on how
smooth we want to go. Here we set $J=7$. In our following analyses, each map
is $3\times 3\hbox{ deg}^{2}$ discretized into $1024\times 1024$ pixels. Thus
$2^{J}=128$ pixels corresponding to $\sim 22\hbox{ arcmin}$. Because we do not
expect to see significant structures resulting purely from noise on such a
large scale, $J=7$ is an appropriate choice.
It can be seen from Eq. (18) that $C_{J}(k,l)$ is the most smoothed version of
the original map $\kappa_{obs}$, and the terms in the summation contain ever
smaller-scale information with smaller $j$.
### 3.2 Multiscale Entropy
With the multi-scale wavelet decomposition, one can then construct an entropy
with the obtained wavelet coefficients $w_{j}(k,l)$ at each grid $(k,l)$ with
$j=1,2,...,J$. It can generally be written as
$H(\kappa)=\sum_{k,l}h[C_{J}(k,l)]+\sum_{j=1}^{J}\sum_{k,l}h[w_{j}(k,l)].$
(19)
For $h$, there are different definitions (e.g., Starck et al. 2006).
Here we follow Starck et al. (2001) to choose the entropy of NOISE-MSE $h_{n}$
in our considerations. At each scale $j$, the noise entropy at each grid
$(k,l)$ is derived by weighting the entropy with a probability that
$w_{j}(k,l)$ is contributed by noise. Specifically, we have
$h_{n}[w_{j}(k,l)]=\int_{0}^{\mid w_{j}(k,l)\mid}P_{n}[\mid
w_{j}(k,l)\mid-u]\frac{\partial h(x)}{\partial x}|_{x=u}du,$ (20)
where $P_{n}[w_{j}(k,l)]$ is the probability that the coefficient $w_{j}(k,l)$
can be due to noise, and is given by
$P_{n}[w_{j}(k,l)]=\mathrm{Prob}[W>\mid w_{j}(k,l)\mid].$ (21)
Eq. (20) essentially regards the information contained in $w_{j}(k,l)$ to be
built up from the summation of $dh(u)$. For each newly added $dh(u)$,
depending on the difference $|w_{j}(k,l)|-u$, there is a probability that it
is due to noise.
For Gaussian noise with rms $\sigma_{j}$ at scale $j$, we have
$\displaystyle P_{n}[w_{j}(k,l)]$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{2\pi}\sigma_{j}}\int_{\mid
w_{j}(k,l)\mid}^{+\infty}\exp(-W^{2}/2\sigma^{2}_{j})dW$ (22) $\displaystyle=$
$\displaystyle\mbox{erfc}\bigg{(}\frac{\mid
w_{j,k,l}\mid}{\sqrt{2}\sigma_{j}}\bigg{)}$
and thus
$h_{n}[w_{j}(k,l)]=\frac{1}{\sigma_{j}^{2}}\int_{0}^{\mid
w_{j}(k,l)\mid}u\mbox{ erfc}\bigg{(}\frac{\mid
w_{j}(k,l)\mid-u}{\sqrt{2}\sigma_{j}}\bigg{)}du.$ (23)
### 3.3 Selecting significant wavelet coefficients using the False Discovery
Rate (FDR)
The Multiscale Entropy method applies regularizations on wavelet coefficients
to minimize noise contributions while keeping the signal information. Thus for
those coefficients which are clearly signals, they should be kept unchanged.
Then a new Multiscale Entropy is defined as (e.g., Starck et al. 2006)
$\tilde{h}_{n}[w_{j}(k,l)]=\bar{M}_{j}(k,l)h_{n}[w_{j}(k,l)],$ (24)
where
$\bar{M}_{j}(k,l)=1-M_{j}(k,l),$ (25)
and $M$ is the multi-resolution support defined as (Starck et al. 1995)
$M_{j}(k,l)=\\{\begin{array}[]{ll}1&\textrm{if $w_{j}(k,l)$ is significant}\\\
0&\textrm{if $w_{j}(k,l)$ is not significant}\end{array}.$ (26)
Therefore $\tilde{h}_{n}$ means that we only need to regularize those wavelet
coefficients which are ′not significant′, that is, they are likely due to
noise.
For judging the significance of a wavelet coefficient, a commonly used
criterion is a ‘$k\sigma$’ threshold. If a coefficient is above the threshold,
it is defined to be ′significant′. This is equivalent to set a threshold for
the ratio of ′significant′ detections over the total number of pixels being
analyzed. Considering a Gaussian noise, a $2\sigma$ criterion corresponds to a
probability of $0.05$ for a noise coefficient being mis-classified as
′significant′. If we have totally $N$ pixels to consider, the number of false
discoveries is then on average $0.05N$. If the number of pixels related to
real signals in the analyses is comparable to $0.05N$, the false discovery
rate with respect to the number of real signals can be much higher than
$0.05$. Increasing $k$ can lower the number of false detections at the
expense, however, of the power of real detections. To overcome such
difficulties, an alternative thresholding technique, the False Discovery Rate
(FDR), has been proposed (Benjamini & Hochberg 1995; Miller et al. 2001;
Hopkins et al. 2002; Starck et al. 2006).
This method can effectively control, in an adaptive manner, the fraction of
false discoveries over the total number of discoveries, rather than over the
total number of pixels analyzed.
Let $P_{1},\ldots,P_{N}$ denote the p-values ordered from low to high for the
N pixels, where p-value is defined as
$p_{value}=\frac{1}{\sqrt{2\pi}\sigma_{j}}\int_{w_{j}(k,l)}^{\infty}\exp[-(w-\bar{w}_{j})^{2}/2\sigma^{2}_{j}]dw,$
(27)
with $\bar{w}_{j}$ the average of $w_{j}(k,l)$ for the scale $j$ over all the
pixels. Define
$d_{j}=\max\left\\{k_{j}:\ P_{k_{j}}<\frac{k_{j}\alpha_{j}}{c_{N}N}\right\\},$
(28)
then all the $w_{j}(k,l)$ with their values larger than $d_{j}$ are classified
as ′significant′. Here $c_{N}=1$ if all the pixels are statistically
independent. The meaning of $\alpha_{j}$ is approximately the pre-defined
false discovery rate at scale $j$ with respect to the total number of
detections. The larger the $\alpha_{j}$ value, the larger the fraction of
$w_{j}(k,l)$ defined to be ′significant′. In our analyses, we adopt FDR to
find the values of $M$ in Eq. (26). In the MRLens program, $\alpha_{0}$ is an
adjustable parameter, and $\alpha_{j}=\alpha_{0}\times 2^{j}$ (Starck et al.
2006).
### 3.4 Multi-scale Entropy Filtering algorithm
Given the discussions in the previous subsections, the Multi-scale Entropy
restoration method reduces to find the reconstructed $\kappa_{f}$ that
minimizes $I(\kappa_{f})$ defined as
$I({\kappa_{f}})=\frac{\parallel{\kappa_{obs}-\kappa_{f}\parallel^{2}}}{2\sigma_{n}^{2}}+\beta\sum_{j=1}^{J}\sum_{k,l}\tilde{h}_{n}[({\cal
W}{\kappa_{f}})_{j}(k,l)],$ (29)
where $\sigma_{n}$ is the rms of noise in the original convergence map
$\kappa_{obs}$, $J$ is the number of wavelet scales, $\cal W$ is the wavelet
transform operator and $\tilde{h}_{n}[({\cal W}{\kappa_{f}})_{j}(k,l)]$ is the
multi-scale entropy defined only for non-significant coefficients selected by
the FDR method. The $\beta$ parameter is calculated under the restriction that
the residual should have a standard deviation equal to the rms of noise. The
best $\kappa_{f}$ is then obtained by iterative calculations. Full details of
the minimization algorithm can be found in Starck et al. (2001).
It can be seen that the two terms in the right of Eq. (29) are balancing each
other. While the first term tends to keep the information in $\kappa_{f}$ the
most, the second term has the effect to lower the noise as much as possible.
## 4 Results
In this section, we present the results of our analyses. For weak-lensing
effects from large-scale structures in the universe, we use the publicly
available ray-tracing weak-lensing maps provided kindly by White & Vale
(2004). The specific set of lensing maps we analyze are generated from large-
scale N-body simulations with cosmological parameters $\Omega_{M}=0.296$,
$\Omega_{\Lambda}=0.704$, $w=-1.0$, $h=0.7$, and $\sigma_{8}=0.93$. The box
size is $300\hbox{ Mpc}~{}h^{-1}$, the number of particles is $512^{3}$ with
$m\approx 1.7\times 10^{10}M_{\odot}~{}h^{-1}$ for each, and the softening
length is $\approx 20\hbox{ kpc}~{}h^{-1}$. There are totally $16$ convergence
maps and each has a size of $3\times 3\hbox{ deg}^{2}$ pixelized into
$1024\times 1024$ pixels. The redshift distribution of source galaxies follows
$p(z)\propto z^{2}\exp[-(z/z_{0})^{3/2}]$ with $z_{0}=2/3$.
For each map, we add in Gaussian noise due to the intrinsic ellipticities of
source galaxies with the variance given by (e.g., Hamana et al. 2004),
$\sigma_{\rm pix}^{2}=\frac{\sigma_{\epsilon}^{2}}{2}\frac{1}{n_{g}\theta_{\rm
pix}^{2}},$ (30)
where $\theta_{\rm pix}$ is the pixel size of the simulated
convergence-$\kappa$ map, and $\sigma_{\epsilon}$ is the rms of the intrinsic
ellipticites taken to be $\sigma_{\epsilon}=0.3$. The surface number density
$n_{g}$ depends on specific observations. Here we consider $n_{g}=30\hbox{
arcmin}^{-2}$ which is typical for ground-based observations, and
$n_{g}=100\hbox{ arcmin}^{-2}$ expected from space observations, respectively.
Figure 1 presents one set of convergence maps without (left) and with (right)
noise. It can be seen very clearly that the noise from intrinsic ellipticities
of source galaxies dominates the map, and certain post-processing procedures
are necessary in order to extract weak-lensing signals embedded under noise.
Here we compare two such methods, namely, the normal smoothing method with a
Gaussian smoothing function, and the MRLens treatment, paying particular
attention to their effects on weak-lensing peak statistics.
Figure 1: The convergence maps of $3\times 3\hbox{ deg}^{2}$. The left panel
is the noise-free map, and the right panel is the noisy convergence map with
$n_{g}=30\hbox{ arcmin}^{-2}$.
### 4.1 Statistical properties of residual noise
Post-processing procedures can reduce noise effectively. However, certain
levels of residual noise inevitably remain. It is thus important to understand
the statistical properties of the residual noise so that we can quantify their
effects on weak-lensing cosmological studies properly. For that, we first in
this subsection consider pure noise maps without including weak-lensing
signals. After applying Gaussian smoothing and MRLens, respectively, we
compare the residual noise-peak statistics in the two cases. This is highly
relevant to cosmological applications of weak-lensing cluster statistics, in
which, high peaks in convergence maps are thought to be related to clusters of
galaxies and their abundances contain important cosmological information. The
existence of residual noise can generate false peaks in convergence maps,
which in turn can contaminate the weak-lensing peak statistics significantly.
With Eq. (30), we generate a $3\times 3\hbox{ deg}^{2}$ noise map containing
$1024\times 1024$ pixels with $\theta_{pix}=0.176\hbox{ arcmin}$ and the
corresponding $\sigma_{pix}=0.22$ for $n_{g}=30\hbox{ arcmin}^{-2}$ and
$\sigma_{pix}=0.12$ for $n_{g}=100\hbox{ arcmin}^{-2}$. For Gaussian
smoothing, we take $\theta_{G}=1\hbox{ arcmin}$. For MRLens, we take
$\alpha_{0}=0.01$. In a smoothed map, a positive (maximum)/negative (minimum)
peak position is located if its value is above/below those of its eight
neighboring pixels (e.g., Jain & Van Waerbeke 2000; Miyazaki et al. 2002).
Figure 2 shows the probability distribution function (PDF) of peaks in the
residual noise field for the two cases, respectively, with the left for the
Gaussian smoothing and the right for the MRLens. In each panel, the solid and
dashed lines correspond to the results with $n_{g}=30\hbox{ arcmin}^{-2}$ and
$n_{g}=100\hbox{ arcmin}^{-2}$, respectively. The bin size is
$\Delta\kappa=0.005$. Both the positive and the negative peaks are counted in.
Two distinctly different distributions are seen. For the Gaussian smoothing
case, the peak number distribution has a double-peak behavior at
$\kappa/\sigma_{0}\sim\pm 1$, in good agreement with that expected for a
Gaussian random field (Bond & Efstathiou 1987; Van Waerbeke 2000). The rms of
the residual noise in this case is $\sigma_{0}\approx 0.016$ for
$n_{g}=30\hbox{ arcmin}^{-2}$ and $\sigma_{0}\approx 0.009$ for
$n_{g}=100\hbox{ arcmin}^{-2}$, in excellent agreement with the theoretical
value $0.015$ for $n_{g}=30\hbox{ arcmin}^{-2}$ and $\sigma_{0}\approx 0.008$
for $n_{g}=100\hbox{ arcmin}^{-2}$ calculated from Eq. (15). Considering
positive peaks that are relevant for weak-lensing analyses, the noise peaks
with $\kappa/\sigma_{0}\sim 1$, rather than with $\kappa/\sigma_{0}=0$, have
the highest occurrence probability. Such a property of noise can cause
statistically a positive shift for the peak height of a cluster measured from
noisy convergence maps. The shift depends on the density profile of the
cluster. This noise-induced shift can bias the cluster mass estimation from
weak-lensing observations. On the other hand, it can increase the weak-lensing
detectability of clusters, and thus affect the corresponding cosmological
studies significantly (Fan et al. 2010).
For the MRLens case, the residual noise after restoration treatment is low
with $\sigma_{0}\approx 0.0029$ for $n_{g}=30\hbox{ arcmin}^{-2}$ and
$\sigma_{0}\approx 0.0016$ for $n_{g}=100\hbox{ arcmin}^{-2}$, much less than
those of the Gaussian smoothing. However, the noise statistics is highly non-
Gaussian, which results significant complications in quantifying the noise
effect on weak-lensing signals. The number distribution of noise peaks is
narrowly concentrated around $\kappa=0$. Thus unlike the Gaussian smoothing,
it seems that we do not expect a systematic shift due to noise in weak-lensing
cluster peak measurement. It should be noted, however, in MRLens, the noise
filtering involves restoration procedures based on NOISE-MSE of Eq. (29). The
results depend on the noise properties [the second term in Eq. (29)] as well
as on the properties of signals we would like to detect [the first term in Eq.
(29)]. The higher the original noise is, the larger the fraction of the
wavelet coefficients that are suppressed. In such a treatment, the signals are
changed depending on the original noise level and their own properties.
Therefore considering the convergence peak for a cluster, the results after
MRLens restoration in the cases with and without noise are different. In this
sense, the existence of noise also induces a systematic bias for the peak
value of a cluster, though for a reason different from and much more
complicated than that of the Gaussian smoothing case. The quantitative
modeling of such a bias for MRLens needs to be further explored.
Figure 2: The probability distribution functions of peaks in pure noise maps.
The solid and dashed lines are for $n_{g}=30\hbox{ arcmin}^{-2}$ and
$n_{g}=100\hbox{ arcmin}^{-2}$, respectively. The left panel is for the
Gaussian smoothing with $\theta_{G}=1\hbox{ arcmin}$, and the right panel is
for the MRLens result with $\alpha_{0}=0.01$.
For MRLens, the $\alpha_{0}$ parameter in FDR affects the classification of
significant and non-significant wavelet coefficients. A smaller $\alpha_{0}$
results a more stringent criteria for the definition of a significant wavelet
coefficient, and thus stronger suppressions of noise. To test the
$\alpha_{0}$-dependence, we vary its value to obtain different restoration
results for pure noise maps. In Table 1, the rms of the residual noise for
different $\alpha_{0}$ and different $n_{g}$ are shown. With the increase of
$n_{g}$, the original noise level decreases with $(n_{g})^{-1/2}$. It is noted
that after MRLens treatment, the rms of the residual noise also approximately
follows $\sigma_{0}\propto(n_{g})^{-1/2}$. For the $\alpha_{0}$-dependence, as
expected, the residual noise decreases with the decrease of $\alpha_{0}$.
However, this dependence is rather weak. Changing $\alpha_{0}$ from $0.1$ to
$0.01$ only decreases $\sigma_{0}$ by $\sim 20\%$.
Table 1: Standard deviation of the reconstruction error with MRLens $\alpha_{0}$ | $\sigma_{0}$($n_{g}=15$) | $\sigma_{0}$($n_{g}=30$) | $\sigma_{0}$($n_{g}=50$) | $\sigma_{0}$($n_{g}=100$)
---|---|---|---|---
0.001 | 0.0038 | 0.0026 | 0.0021 | 0.0015
0.01 | 0.0041 | 0.0029 | 0.0023 | 0.0016
0.02 | 0.0041 | 0.0030 | 0.0023 | 0.0016
0.04 | 0.0044 | 0.0032 | 0.0024 | 0.0017
0.06 | 0.0045 | 0.0033 | 0.0026 | 0.0019
0.08 | 0.0047 | 0.0033 | 0.0027 | 0.0020
0.1 | 0.0050 | 0.0035 | 0.0027 | 0.0021
0.2 | 0.0051 | 0.0036 | 0.0029 | 0.0023
### 4.2 Peak statistics in noisy convergence maps
Now we consider peak statistics of noisy convergence maps. Figure 3 shows the
post-processed maps of the right panel of Figure 1 with Gaussian smoothing for
$\theta_{G}=1\hbox{ arcmin}$ (upper) and with MRLens for $\alpha_{0}=0.01$
(lower), respectively. The left panels are for $n_{g}=30\hbox{ arcmin}^{-2}$
and the right panels are for $n_{g}=100\hbox{ arcmin}^{-2}$.
Figure 3: Noisy convergence maps of $3\times 3\hbox{ deg}^{2}$. The upper and
lower panels are for the Gaussian smoothing with $\theta_{G}=1\hbox{ arcmin}$
and MRLens with $\alpha_{0}=0.01$, respectively. The left and right panels are
for $n_{g}=30\hbox{ arcmin}^{-2}$ and $n_{g}=100\hbox{ arcmin}^{-2}$,
respectively.
Comparing to the maps in Figure 1, we see that the post-processing procedures
can indeed filter out much of the noise so that the real structures in the
large-scale mass distribution can be detected. For $n_{g}=30\hbox{
arcmin}^{-2}$, the MRLens map looks very smooth with only very massive
structures left. On the other hand, in the Gaussian smoothing case, small
structures can also be seen. However, it contains many more noise peaks than
that of the MRLens case. For $n_{g}=100\hbox{ arcmin}^{-2}$, the map is
smoother for the Gaussian smoothing case than that with $n_{g}=30\hbox{
arcmin}^{-2}$. The MRLens map, however, appears lumpier for the lower noise
case. Such opposite trends seen in the Gaussian smoothing and in MRLens
reflect clearly the different underlying filtering mechanisms between the two
smoothing schemes. For the Gaussian smoothing, the filtering is mainly
performed through an averaging procedure. Given a smoothing scale, the peak
signals of real clusters are more or less similar regardless the noise level.
Meanwhile, the noise peaks with relatively high $\kappa$ values are
significantly reduced if the noise level is lowered. Thus the smoother
appearance of the upper right panel is mainly due to the less number of high
noise peaks than that of the upper left panel. For MRLens, it involves a
restoration procedure that depends on the original noise level. The smaller
the original noise is, the lower the fraction is for the wavelet coefficients
to be suppressed. It is important to note that the suppression leads to the
removal of both noise peaks and true peaks of relatively low amplitudes. Thus
the lumpier structures seen in the lower right panel is largely attributed to
the lower level of removal of real structures than that of the lower left
panel.
In Figure 4 and Figure 5, we show the probability distribution function of
peaks for Gaussian smoothing and for MRLens, respectively. The results for
each case are obtained by averaging over $16$ simulated maps with noise added.
Figure 4: The probability distribution function of convergence peaks for the
case of Gaussian smoothing with $\theta_{G}=1\hbox{ arcmin}$. The black solid
line is for the result of the noise-free convergence peaks, the red dashed and
red solid lines are for the pure noise peaks and noisy convergence peaks with
$n_{g}=30\hbox{ arcmin}^{-2}$, respectively, and the blue dashed and blue
solid lines are for the pure noise peaks and noisy convergence peaks with
$n_{g}=100\hbox{ arcmin}^{-2}$, respectively. The left upper panel includes
both maximum and minimum peaks, and the left lower panel shows the
distribution function for maximum peaks only. The right panel is the zoom-in
version of the left lower panel. Figure 5: Same as Figure 4 but for the case
of MRLens with $\alpha_{0}=0.01$.
For the Gaussian smoothing results in Figure 4, the black, red dashed, red
solid, blue dashed, and blue solid lines are for the results of noise free
peaks, pure noise peaks with $n_{g}=30\hbox{ arcmin}^{-2}$, noisy convergence
peaks with $n_{g}=30\hbox{ arcmin}^{-2}$, pure noise peaks with
$n_{g}=100\hbox{ arcmin}^{-2}$, noisy convergence peaks with $n_{g}=100\hbox{
arcmin}^{-2}$, respectively. We can see that in the Gaussian smoothing cases,
the noise peaks dominate over the real peaks at $\kappa<3\sigma_{0}$. At
larger $\kappa>3\sigma_{0}$, real peaks can be detected with high
efficiencies. Comparing the blue solid line with the red solid line, we see
that by reducing the noise level from $\sigma_{0}\sim 0.015$ to
$\sigma_{0}\sim 0.008$, we effectively reduce the number of noise peaks with
$\kappa>0.025$, and thus increase the real peak detection efficiencies
significantly.
In Figure 5 for the MRLens results, the line styles are the same as those in
Figure 4. Different from that in the Gaussian smoothing cases, here the noise
peaks (red and blue dashed lines) contribute little to the total number of
peaks with $\kappa>0.02$ in comparison with the real peaks (black solid line).
However, the suppression process in the MRLens treatment mistakenly removes a
large number of real peaks with $\kappa<0.1$. Thus we expect a high efficiency
but a low completeness in weak-lensing peak detections after MRLens filtering.
Reducing the original noise level by increasing $n_{g}$ form $30\hbox{
arcmin}^{-2}$ to $100\hbox{ arcmin}^{-2}$ leads to a less suppression effect.
Therefore more peaks with $\kappa<0.1$ are kept and the completeness of peak
detections increases considerably.
In the next subsection, we investigate and compare explicitly the efficiency
and completeness of weak-lensing cluster detections in the two smoothing
treatments.
### 4.3 Efficiency and completeness of weak-lensing cluster detection
The existence of noise from intrinsic ellipticities of source galaxies results
false peaks in convergence maps, and thus lowers considerably the efficiency
of weak-lensing cluster detections. Increasing the detection threshold can
increase the efficiency, however at the expense of completeness. In this
section, we compare the weak-lensing cluster detection with Gaussian smoothing
and with the MRLens, respectively. Following Hamana et al. (2004), we define
the efficiency $f_{e}$ and completeness $f_{c}$ of cluster detection with
respect to the number of clusters (dark matter halos) above a certain mass
threshold. Specifically, we have
$f_{e}=\frac{N_{iii}}{N_{i}},$ (31) $f_{c}=\frac{N_{iii}}{N_{ii}},$ (32)
where $N_{i}$ denotes the number of convergence peaks with their heights above
a detection threshold, $N_{ii}$ represents the number of dark matter halo with
mass above a certain mass threshold, and $N_{iii}$ is the number of peaks that
have correspondences with dark matter halos among $N_{ii}$. A peak is defined
to be associated with its nearest dark matter halo if the location of the peak
is within a radius of $12$ pixels (corresponds to $2.11\hbox{ arcmin}$) around
the halo. If there are two or more peaks associated with a same halo, the
highest peak is defined to have the correspondence with the halo.
Figure 6: The efficiency $f_{e}$ and completeness $f_{c}$ as functions of the
peak detection threshold $\kappa$. The left and right panels are for
$n_{g}=30\hbox{ arcmin}^{-2}$ and $n_{g}=100\hbox{ arcmin}^{-2}$,
respectively. The upper two panels are for the Gaussian smoothing and the
lower two panels are for MRLens. The lines with symbols are for the
completeness, and the lines without symbols are for the efficiency. The red,
green and blue lines are for the results of halos with $M>5\times
10^{13}M_{\odot}$, $M>1\times 10^{14}M_{\odot}$, and $M>2\times
10^{14}M_{\odot}$, respectively.
Figure 6 shows the results of $f_{e}$ and $f_{c}$ for Gaussian smoothing
(upper panels) and MRLens (lower panels). The left panels are for
$n_{g}=30\hbox{ arcmin}^{-2}$, and the right panels are for $n_{g}=100\hbox{
arcmin}^{-2}$. In each panel, the red, green and blue lines are for halos with
mass $M>5\times 10^{13}M_{\odot}$, $M>1\times 10^{14}M_{\odot}$, and
$M>2\times 10^{14}M_{\odot}$, respectively. The lines with and without symbols
are, respectively, for the results of completeness and efficiency. The
horizontal axis in each panel is the peak detection threshold $\kappa$.
We first analyze the Gaussian smoothing cases. As we discuss previously, such
a smoothing process reserves more or less all the real peaks with scales above
the smoothing scale. At mean time, the number of noise peaks is large at
$\kappa<3\sigma_{0}$. Thus a high completeness and a low efficiency are
expected when the peak detection threshold is low. For $n_{g}=30\hbox{
arcmin}^{-2}$ (upper left), we have $\sigma_{0}\sim 0.015$. At the detection
threshold $\kappa=0.02\sim 1.3\sigma_{0}$, we have the completeness $f_{c}\sim
50\%,65\%$ and $70\%$ for $M>5\times 10^{13}\hbox{ M}_{\odot}$, $1\times
10^{14}\hbox{ M}_{\odot}$ and $2\times 10^{14}\hbox{ M}_{\odot}$,
respectively. The corresponding efficiencies are $30\%,10\%$ and $2\%$. When
the detection threshold $\kappa>3\sigma_{0}$, the number of noise peaks drops
significantly, leading to a large increase in the detection efficiency. On the
other hand, a considerable fraction of halos are missed due to the high
detection threshold, resulting a decrease in the completeness. Specifically,
at $\kappa=0.045\sim 3\sigma_{0}$, the completeness $f_{c}\sim 20\%,35\%$ and
$60\%$, and the efficiency $f_{e}\sim 50\%,25\%$ and $10\%$, for $M>5\times
10^{13}\hbox{ M}_{\odot}$, $1\times 10^{14}\hbox{ M}_{\odot}$ and $2\times
10^{14}\hbox{ M}_{\odot}$, respectively. With the increase of $n_{g}$ to
$n_{g}=100\hbox{ arcmin}^{-2}$ (upper right), the noise level $\sigma_{0}$
decreases by a factor of $\sqrt{100/30}$ to $\sigma_{0}\sim 0.008$. Thus
$3\sigma_{0}$ corresponds to $\kappa\sim 0.025$. At this detection threshold,
the number of noise peaks is smaller and correspondingly the efficiency is
higher than those with $n_{g}=30\hbox{ arcmin}^{-2}$. On the other hand, the
number of real peaks does not change much as the noise level decreases, and
thus the completeness is similar to that of $n_{g}=30\hbox{ arcmin}^{-2}$.
Quantitatively, at the threshold $\kappa=0.025$, the efficiency $f_{e}\sim
50\%,20\%$ and $8\%$, in comparison with $f_{e}\sim 35\%,12\%$ and $3\%$ in
the case of $n_{g}=30\hbox{ arcmin}^{-2}$, for $M>5\times 10^{13}\hbox{
M}_{\odot}$, $1\times 10^{14}\hbox{ M}_{\odot}$ and $2\times 10^{14}\hbox{
M}_{\odot}$, respectively. For the completeness, we have $f_{c}\sim 30\%,50\%$
and $80\%$ for $n_{g}=100\hbox{ arcmin}^{-2}$. For $n_{g}=30\hbox{
arcmin}^{-2}$, $f_{c}\sim 45\%,60\%$ and $70\%$. While being similar, $f_{c}$
decreases somewhat for $M>5\times 10^{13}\hbox{ M}_{\odot}$ and $1\times
10^{14}\hbox{ M}_{\odot}$ with the decrease of noise level. This is in
accordance with the analyses of Fan et al. (2010) where they find that the
existence of noise generates a systematic shift for the real peaks toward
higher amplitudes. The shift depends on the density profile of dark matter
halos associated with the real peaks, and can be as high as $\sim 1\sigma_{0}$
for NFW halos with low concentrations. In terms of $\kappa$ values, the shift
is larger for larger $\sigma_{0}$. Thus, in the case of $n_{g}=30\hbox{
arcmin}^{-2}$, the relatively large $\sigma_{0}$ leads to a large shift of the
real peak heights and consequently a larger number of real peaks above the
detection threshold than that in the case of $n_{g}=100\hbox{ arcmin}^{-2}$.
For MRLens, with $n_{g}=30\hbox{ arcmin}^{-2}$ (lower left), the completeness
of the weak-lensing cluster detection is very low, and $f_{c}\sim 10\%,20\%$
and $40\%$ at the threshold $\kappa=0.02$, in comparison with $f_{c}\sim
50\%,65\%$ and $70\%$ in the corresponding Gaussian smoothing case. This is
because the suppression of the wavelet coefficients aiming to reduce noise
removes a large fraction of real peaks in the range of $\kappa<0.1$ as seen
from Figure 5. The total number of peaks in $3\times 3\hbox{ deg}^{2}$ with
$\kappa\geq 0.02$ is only $\sim 53$, while the total number of halos in the
area with $M>5\times 10^{13}\hbox{ M}_{\odot}$ is $\sim 530$. Thus although
the efficiency in MRLens here is rather high ($\sim 80\%$ for $M>5\times
10^{13}\hbox{ M}_{\odot}$), the very few number of detected halos makes the
MRLens method be disadvantageous in comparing with that of simple Gaussian
smoothing method. For $n_{g}=100\hbox{ arcmin}^{-2}$, the noise level is lower
and thus the removal effect is less significant than the case of
$n_{g}=30\hbox{ arcmin}^{-2}$. Consequently, the completeness increases
considerably with $f_{c}\sim 20\%,40\%$ and $65\%$. Meanwhile, the efficiency
decreases somewhat. In this low noise case, the differences between the MRLens
and Gaussian smoothing in terms of the completeness and efficiency are less
than those of high noise case. But still, the completeness is lower for
MRLens, especially considering relatively low mass halos with $M>5\times
10^{13}\hbox{ M}_{\odot}$.
To further demonstrate the differences between the Gaussian smoothing and the
MRLens, in Figure 7, we show the peak-halo correspondences explicitly in $z-M$
plane for one of our $3\times 3\hbox{ deg}^{2}$ simulation maps, where $M$ is
the halo mass in unit of $10^{13}M_{\odot}$ and $z$ is the halo redshift from
simulations. The halos are the ones located in the solid angle of $3\times
3\hbox{ deg}^{2}$ in the considered direction and their redshift and mass are
taken directly from the halo catalogs constructed by White and Vale (2004).
The upper and lower panels are for the Gaussian smoothing and the MRLens,
respectively. The left and right panels correspond to $n_{g}=30\hbox{
arcmin}^{-2}$ and $n_{g}=100\hbox{ arcmin}^{-2}$, respectively. In each panel,
the ‘+’ symbols denote the dark matter halos identified in simulations with
$M\geq 5\times 10^{13}M_{\odot}$ and in the redshift range of $0\leq z\leq 2$.
There are very few halos extending to redshift beyond $z=2$. The green squares
represent those halos that have corresponding convergence peaks with
$\kappa\geq 0.02$. The differences between the two filtering methods are
strikingly seen. For MRLens with $n_{g}=30\hbox{ arcmin}^{-2}$ (lower left), a
majority of halos with $M<10^{14}\hbox{ M}_{\odot}$ or with $z>0.8$ are missed
in weak-lensing detections, consistent with its extremely low completeness
shown in Figure 6. Lowering the noise level by increasing $n_{g}$ to
$n_{g}=100\hbox{ arcmin}^{-2}$ increases the number of halos with associated
peaks by nearly a factor of $2$ (lower right). But the number is still much
less than that in the Gaussian smoothing case. Therefore in studies aiming to
detect a large number of clusters from blind surveys and subsequent
cosmological applications, the Gaussian smoothing method is clearly much
better than the MRLens. In addition, the noise field after a Gaussian
smoothing with $\theta_{G}\sim 1\hbox{ arcmin}$ is approximately Gaussian in
statistics, and thus its effects on weak-lensing cluster detection can be
modeled much easier than the case of MRLens where the left-over noise is
statistically highly non-Gaussian (Fan et al. 2010).
Figure 7: Peak-halo correspondences in $z-M$ plane. The upper and lower panels
are for the Gaussian smoothing and MRLens, respectively. The left and right
panels are for $n_{g}=30\hbox{ arcmin}^{-2}$ and $n_{g}=100\hbox{
arcmin}^{-2}$, respectively. In each panel, the ‘+’ symbols show all the halos
with $M\geq 5\times 10^{13}M_{\odot}$ and the squares show the halos with
corresponding convergence peaks with $\kappa\geq 0.02$.
In MRLens, the $\alpha_{0}$ parameter plays a crucial role in classifying
significant and non-significant wavelet coefficients. A larger $\alpha_{0}$
leads to a larger fraction of significant coefficients, and thus a less
suppression effect in MRLens restoration. To see if the problem of low
completeness in MRLens cluster detection can be largely improved by increasing
$\alpha_{0}$, we analyze the $\alpha_{0}$ dependence for the completeness
$f_{c}$ as well as for the efficiency $f_{e}$. The results are shown in Figure
8. The upper and lower panels are for $n_{g}=30\hbox{ arcmin}^{-2}$ and
$n_{g}=100\hbox{ arcmin}^{-2}$, respectively. The red, green and blue lines
are for $M\geq 5\times 10^{13}M_{\odot}$, $1\times 10^{14}M_{\odot}$, and
$2\times 10^{14}M_{\odot}$, respectively. The peak detection threshold is set
to be $\kappa=0.02$. We can see that both $f_{c}$ and $f_{e}$ are not very
sensitive to $\alpha_{0}$. Increasing $\alpha_{0}$ from $0.01$ to $0.1$
improves the completeness only by $\Delta f_{c}\sim 10\%$ for both
$n_{g}=30\hbox{ arcmin}^{-2}$, and $n_{g}=100\hbox{ arcmin}^{-2}$. Meanwhile,
the efficiency decreases by $\Delta f_{e}\sim 10\%-20\%$. Therefore increasing
$\alpha_{0}$ cannot overcome the shortcoming of MRLens considerably. Comparing
to the Gaussian smoothing cases, the completeness is still low for MRLens
weak-lensing cluster detection even with $\alpha_{0}=0.1$.
We then conclude that in probing cosmologies with weak-lensing cluster
abundance analyses, in which a large sample of clusters is needed, the
Gaussian smoothing method performs much better than the MRLens method. To
overcome the relatively low efficiency for low peaks in the Gaussian smoothing
treatment, a detection threshold $\kappa>3\sigma_{0}$ is normally set. In Fan
et al. (2010), the noise effects on convergence peak statistics can be
accurately modeled for the Gaussian smoothing method. Therefore it is
potentially possible to even include peaks with $\kappa<3\sigma_{0}$ in the
abundance analyses, which can increase the number of detected clusters greatly
so that to strengthen the derived cosmological constraints on different
parameters. This will be explored further in our future studies.
Figure 8: The $\alpha_{0}$ dependence of the completeness $f_{c}$ and the
efficiency $f_{e}$. The upper and lower panels are for $n_{g}=30\hbox{
arcmin}^{-2}$ and $n_{g}=100\hbox{ arcmin}^{-2}$ respectively. The peak
detection threshold is set to be $\kappa=0.02$. The red, green and blue lines
are for the results of halos with $M>5\times 10^{13}M_{\odot}$, $M>1\times
10^{14}M_{\odot}$, and $M>2\times 10^{14}M_{\odot}$, respectively. The lines
with and without symbols are for the completeness $f_{c}$ and the efficiency
$f_{e}$, respectively.
## 5 Summary
Constructing cluster samples through their weak-lensing effects has been an
important aspect of weak-lensing studies. Their statistical abundance contains
valuable cosmological information. Observations have shown the feasibility in
detecting clusters with weak-lensing effects (e.g., Wittman et al. 2006;
Dietrich et al. 2007; Gavazzi & Soucail 2007; Schirmer et al. 2007; Hamana et
al. 2009). In conjunction with optical observations, the detailed analyses on
the completeness and efficiency of weak-lensing selected cluster samples also
become possible (e.g., Geller et al. 2010). It is noted, however, the
efficiency and completeness depend on the method applied to reconstruct the
convergence field from shear measurements. Different methods can result
residual noise with different statistical properties, and can also change the
weak-lensing signals differently. In order to extract cosmological information
from observations, it is therefore crucial to understand how a particular
reconstruction method affects the results in detail.
In this paper, we systematically compare the Gaussian smoothing method and the
MRLens treatment to suppress noise from intrinsic ellipticities in convergence
maps. We concentrate on convergence peak statistics. It is found that while
the MRLens method can remove noise very effectively, it mistakenly removes a
large fraction of real peaks associated with clusters of galaxies. For
$n_{g}=30\hbox{ arcmin}^{-2}$, the number of peaks with $\kappa\geq 0.02$
after MRLens filtering is only $\sim 50$ in an area of $3\times 3\hbox{
deg}^{2}$ in comparison with $\sim 530$ for the number of halos of $M>5\times
10^{13}\hbox{ M}_{\odot}$. On the other hand, for the Gaussian smoothing
treatment, the number of detected clusters is $\sim 260$. Even with the
detection threshold $\kappa=3\sigma_{0}\sim 0.045$, which is normally set in
the Gaussian smoothing treatment to reduce the number of noise peaks in the
peak catalog and thus to increase the cluster detection efficiency, the number
of detected clusters is $\sim 100$, twice as many as that in the MRLens
filtering with the threshold $\kappa=0.02$. As the accuracy of statistical
abundance analyses depends crucially on the number of detected clusters, the
Gaussian smoothing method is therefore strongly favored to detect clusters as
many as possible. Furthermore, the Gaussian smoothing leads to a noise field
which is approximately Gaussian in statistics, while the residual noise from
MRLens filtering is highly non-Gaussian. Therefore the noise effects can be
modeled more straightforwardly for the Gaussian smoothing case than that of
MRLens (e.g., van Waerbeke 2000; Fan 2007; Fan et al. 2010). The recent
studies of Fan et al. (2010) on the weak-lensing peak statistics with noise
included provide an analytical model for the efficiency of peak detections in
the Gaussian smoothing case. Thus it is possible for us to include peaks with
$\kappa<3\sigma_{0}$ in the analyses. Then the number of detected clusters can
increase considerably, which in turn can lead to a significant improvement in
the cosmological constraints derived from weak-lensing cluster statistics.
###### Acknowledgements.
This research is supported in part by the NSFC of China under grants 10373001,
10533010 and 10773001, and the 973 program No.2007CB815401. HuanYuan Shan is
very grateful for the hospitality of CPPM.
## References
* [Albrecht(2006)] Albrecht, A., Bernstein, G., Cahn, R., et al., 2006, astro-ph/0609591
* [Benjaminet al(2007)] Benjamin, J., Heymans, C., Semboloni, E., et al., 2007, MNRAS, 381, 702
* [Benjamini & Hochberg1995] Benjamini, Y. & Hochberg, Y., 1995, J.R.Stat.Soc.B, 57, 289
* [Bartelmann(1995)] Bartelmann, M., 1995, A&A, 303, 643
* [Bartelmannet al.(1996)] Bartelmann, M., Narayan, R., Seitz, S.,& Schneider, P., 1996,ApJ, 464, L115
* [Bartelmann & Schneider(2001)] Bartelmann, M.,& Schneider, P., 2001, Physics Reports, 340, 291
* [Bondet al(1987)] Bond, J. R., & Efstathiou, G., 1987, MNRAS, 226, 655
* [Bridleet al.(1998)] Bridle, S. L., Hobson, M. P., Lasenby, A. N.,& Saunders, R., 1998, MNRAS, 299, 895
* [Dietrichet al(2007)] Dietrich, J. P., Erben, T., Lamer, G., Schneider, P., Schwope, A., Hartlap, J., & Maturi, M., 2007, A&A, 470, 821
* [Fan(2007)] Fan, Z. H. 2007, ApJ, 669, 10
* [Fanet al(2010)] Fan, Z. H., Shan, H. Y., & Liu, J. Y., 2010, ApJ, 719, 1408
* [Gavazziet al(2007)] Gavazzi, R., & Soucail, G., 2007, A&A, 462, 459
* [Gelleret al(2010)] Geller, M. J.,Kurtz, M., J., Dell’Antonio, I. P., Ramella, M., & Fabricant, D. G., 2010, ApJ, 709, 832
* [Hamanaet al.(2004)] Hamana, T., Takada, M.,& Yoshida, N., 2004, MNRAS, 350, 893
* [Hamanaet al(2009)] Hamana, T.,Miyazaki, S., Kashikawa, N., Ellis, R. S., Massey, R. J., Refregier, A.,& Taylor, J. E., 2009, PASJ, 61, 833
* [Hoekstraet al.(2006)] Hoekstra, H., Mellier, Y., van Waerbeke, L., et al., 2006, ApJ, 647, 116
* [Hopkinset al.(2002)] Hopkins, A. M., Miller, C. J., Connolly, A. J., Genovese, C., Nichol, R. C., & Wasserman, L., 2002, AJ, 123, 1086
* [Jain & Van Waerbeke(2000)] Jain B.,& Van Waerbeke L., 2000, ApJ, 530, L1
* [Kaiser & Squires1993] Kaiser, N.,& Squires, G., 1993, ApJ, 404, 441
* [Kaiser(1995)] Kaiser, N., 1995,ApJ, 439, L1
* [Kilbingeret al(2009)] Kilbinger, M., Benabed, K., Guy, J., et al, 2009, A&A, 497, 677
* [Liet al.(2009)] Li, R., Mo, H. J., Fan, Z. H., Cacciato, M., van de Bosch, F., Yang, X. H., & More, S., 2009, MNRAS, 394, 1016
* [Marshallet al.(2002)] Marshall, P. J., Hobson, M. P., Gull, S. F.,& Bridle, S. L., 2002, MNRAS, 335, 1037
* [Masseyet al(2007)] Massey, R., Heymans, C.,& Berge, J., et al. , 2007, MNRAS, 376, 13
* [Milleret al.(2001)] Miller, C. J., Genovese, C., Nichol, R. C., et al., 2001,AJ, 122, 3492
* [Miyazakiet al.(2002)] Miyazaki, S., Hamana, T., Shimasaku, K., et al., 2002, ApJ, 580, 97
* [Pireset al.(2009)] Pires, S., Starck, J-L., Amara, A., Refregier, A.& Teyssier, R., 2009, A&A, 505, 969
* [Schirmeret al(2007)] Schirmer, M., Erben, T., Hetterscheidt, M., & Schneider, P., 2007, A&A, 462, 875
* [Schneider & Seitz(1995)] Schneider, P.& Seitz, C., 1995,A&A, 294, 411
* [Schneider(1996)] Schneider, P., 1996, MNRAS, 283, 837
* [Seitzet al.(1998)] Seitz, S., Schneider, P.,& Bartelmann, M., 1998,A&A, 337, 325
* [Squires & Kaiser(1996)] Squires, G.,& Kaiser, N., 1996, 473, 65
* [Starcket al.(1995)] Starck, J.-L., Murtagh, F., & Bijaoui, A., 1995,ASPC, 77, 279
* [Starcket al.(2001)] Starck, J.-L., Murtagh, F., Querre, P.,& Bonnarel, F., 2001,A&A, 368, 730
* [Starcket al.(2006)] Starck J.-L., Pires S. & Réfrégier A., 2006, A&A, 451, 1139
* [van Waerbeke(2000)] van Waerbeke L., 2000, MNRAS, 313, 524
* [Whiteet al(2002)] White, M., Van Waerbeke, L., & Jonathan, M., 2002, ApJ, 575, 640
* [White & Vale(2003)] White, M., & Vale, C., 2004, Astroparticle Physics, 22, 19
* [Wittmanet al(2006)] Wittman, D., Dell’Antonio, I. P., Hughes, J. P., Margoniner, V. E., Tyson, J. A., Cohen, J. G.,& Norman, D., 2006, ApJ, 643, 128
|
arxiv-papers
| 2010-12-29T09:23:43 |
2024-09-04T02:49:16.049113
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yangxiu Jiao, Huanyuan Shan, Zuhui Fan",
"submitter": "HuanYuan Shan",
"url": "https://arxiv.org/abs/1012.5894"
}
|
1101.0019
|
Current Address: ]Instituto de Física de São Carlos, Universidade de São
Paulo, São Carlos, 13560-970, Brazil
Primary Address: ]Department of Physics, Umeå University, SE-901 87, Umeå,
Sweden
Current Address: ]School of Physics, Monash University, 3800, Australia
# Superflow in a toroidal Bose-Einstein condensate:
an atom circuit with a tunable weak link
A. Ramanathan∗ Joint Quantum Institute, National Institute of Standards and
Technology and University of Maryland, Gaithersburg, MD, 20899, USA K. C.
Wright kcwright@nist.gov Joint Quantum Institute, National Institute of
Standards and Technology and University of Maryland, Gaithersburg, MD, 20899,
USA S. R. Muniz [ M. Zelan [ W. T. Hill III Joint Quantum Institute,
National Institute of Standards and Technology and University of Maryland,
Gaithersburg, MD, 20899, USA C. J. Lobb Joint Quantum Institute, National
Institute of Standards and Technology and University of Maryland,
Gaithersburg, MD, 20899, USA K. Helmerson [ W. D. Phillips Joint Quantum
Institute, National Institute of Standards and Technology and University of
Maryland, Gaithersburg, MD, 20899, USA G. K. Campbell Joint Quantum
Institute, National Institute of Standards and Technology and University of
Maryland, Gaithersburg, MD, 20899, USA
###### Abstract
We have created a long-lived ($\approx$ 40 s) persistent current in a toroidal
Bose-Einstein condensate held in an all-optical trap. A repulsive optical
barrier across one side of the torus creates a tunable weak link in the
condensate circuit, which can affect the current around the loop. Superflow
stops abruptly at a barrier strength such that the local flow velocity at the
barrier exceeds a critical velocity. The measured critical velocity is
consistent with dissipation due to the creation of vortex-antivortex pairs.
This system is the first realization of an elementary closed-loop atom
circuit.
###### pacs:
03.75.Lm, 03.75.Kk, 67.85.De
Quantum fluids can exhibit properties such as long-range coherence and
superfluidity that make them useful for constructing sensors and other
devices. For example, superconducting quantum interference devices (SQUIDs)
are sensitive magnetic field detectors Clarke and Braginski (2004), and
superfluid He circuits have been used to detect rotation Simmonds et al.
(2001); Hoskinson et al. (2001). Ultracold atomic-gas analogs of electronic
devices and circuits, or “atomtronics” have been proposed including diodes and
transistors atomtronics . Of particular interest is the realization of an
atomic-gas SQUID analog. SQUID circuits have been realized with either tunnel
or weak link junctions Clarke and Braginski (2004); Likharev (1979); Davis and
Packard (2002). In atomic Bose-Einstein condensates, Josephson junctions have
been demonstrated only between adjacent wells Albiez et al. (2005); Levy et
al. (2007). Here we present the first implementation of a non-trivial, closed-
loop atom circuit, and show that it is possible to control the current at the
single-quantum level by changing the strength of a weak link. This is an
essential step toward realizing an atomic SQUID analog.
Superfluids flow without dissipation if the flow velocity is below a threshold
determined by the lowest energy excitations allowed for the system Landau
(1941). In a homogeneous condensate the lowest energy excitations are phonons
end , and the Landau critical velocity is the speed of sound Pitaevskii and
Stringari (2003). Real systems are finite and therefore inhomogeneous;
consequently the lowest energy excitations can be vortex-like Barenghi et al.
(2001), and dissipation can occur at velocities well below the sound speed
Feynman (1955). Dissipation involving vortex-like excitations has been
previously observed in experiments with liquid He Avenel and Varoquaux (1985);
Amar et al. (1992), superconductors Huebener (2001), and in a simply-connected
condensate Neely et al. (2010).
Figure 1: Experimental configuration. (a) Schematic of the toroidal optical
dipole trap formed at the intersection of two red-detuned beams: a horizontal
“sheet” beam, and a vertical Laguerre-Gaussian beam (LG${}_{0}^{1}$) with a
ring-shaped intensity maximum. A pulsed pair of Raman beams (large downward
arrows) co-propagating with the LG trapping beam creates circulation in the
condensate. (b) Energy level diagram for the Raman transition:
$|F\\!=\\!1,m_{F}\\!=\\!-1\rangle\\!\rightarrow\\!|F\\!=\\!1,m_{F}\\!=\\!0\rangle$.
One Raman beam carries $\hbar$ orbital angular momentum per photon
(LG${}_{0}^{1}$), the other carries none (Gaussian); the condensate is
transferred to a quantized ($l$=1) circulating state. (c) False-color
absorption image showing the normalized column density of a condensate in the
trap, viewed from above. Arrows: Raman beam polarizations and magnetic bias.
The critical velocity in simply-connected condensates has been measured
previously by moving a defect created by a localized optical potential Onofrio
et al. (2000); Engels and Atherton (2007); Neely et al. (2010). When the
velocity of the defect was high enough, excitations and heating were observed.
In contrast to this earlier work, we create a quantized, persistent flow
around a multiply-connected (toroidal) condensate, and study the decay of that
flow in the presence of a stationary barrier, as a function of barrier height
and condensate atom number.
In previous experiments Ryu et al. (2007), we created persistent currents in a
harmonic magnetic potential pierced by a repulsive optical potential. Relative
drift between these potentials limited the flow lifetime to $\approx$ 10 s.
This motivated the construction of an all-optical trap which supports
persistent currents for up to 40 s, and allows us to carefully study the
stability of superflow.
To create a toroidal condensate,
$3^{2}S_{1/2}\,|F\\!=\\!1,m_{F}\\!=\\!-1\rangle\ $ 23Na atoms are cooled
almost to degeneracy in a magnetic trap and then transferred into an optical
dipole trap created by the intersection of red-detuned (1030 nm) “sheet” and
“ring” beams (Fig. 1a). The horizontal sheet beam has a vertical (horizontal)
$1/e^{2}$ half-width of $\approx$ 9 $\mu$m ($\approx$ 400 $\mu$m), and
provides vertical confinement. The vertical ring beam is Laguerre-Gaussian
(LG${}^{1}_{0}$), and confines the condensate to its $\approx$ 20 $\mu$m
radial intensity peak, generating a toroidal potential minimum. With the atoms
in the optical trap, the beam intensities are ramped down to force evaporative
cooling. At the end of the ramp, the trap depth is $\approx$ 700 nK, with trap
frequencies $\omega_{z}/2\pi=550$ Hz (vertical) and $\omega_{r}/2\pi=110$ Hz
(radial). This produces a toroidal condensate of up to 3$\times$105 atoms with
a chemical potential $\mu_{0}$ of up to $h\cdot 1200$ Hz, and temperature $<$
10 nK (no discernible non-condensed fraction). The azimuthal variation of the
potential minimum is less than $h\cdot 100$ Hz, as shown by the smooth
condensate density profile in Fig. 1c.
The condensate is initially nonrotating ywas . Superfluid circulation around
any closed path must be quantized, such that the wave function has a $2\pi l$
phase winding ($l\\!\in\\!\mathbb{Z}$). We create circulation by transferring
quantized angular momentum from optical fields during a Raman process Wright
et al. (2008). The co-propagating Raman beams, detuned 2.3 GHz below the D2
transitions, are in two-photon resonance with the
$|1,-1\rangle\\!\rightarrow\\!|1,\;0\rangle\ $ transition (Fig. 1b). They have
orthogonal linear polarizations, parallel and perpendicular to the horizontal
magnetic bias field (Fig. 1c). The nonlinear Zeeman shift from the 0.5 mT
field applied during the interaction prevents coupling to $|1,1\rangle$.
The angular momentum change of the condensate is determined by the spatial
mode of the Raman beams. With one beam Gaussian, and the other in an
LG${}^{1}_{0}$ spatial mode carrying $\hbar$ orbital angular momentum per
photon, the Raman process coherently transfers the condensate to the $l$=1
circulating state Andersen et al. (2006); Wright et al. (2008); Ryu et al.
(2007). With good mode matching and an optimized Raman $\pi$-pulse ($\approx$
100 $\mu$s), we achieve a minimum transfer efficiency of 90%, with only a few
percent atom loss due to spontaneous scattering. Residual atoms in
$|1,-1\rangle$ after the Raman pulse are quickly removed from the trap by
transferring them to $|2,-2\rangle$ with a microwave pulse, then ejecting them
from the trap with resonant imaging light (see below).
Circulation is detected by releasing the condensate from the trap and imaging
the density distribution after several milliseconds time-of-flight (TOF)
Madison et al. (2000). If the condensate is not rotating, the central hole
closes after a short time. When a rotating condensate is released, the angular
velocity of the flow prevents complete closure. The persistence of a central
hole after sufficiently long TOF is the signature of circulation in the ring
(see Fig 2(b) insets). The apparent size of the hole at a given time after
release is related to the the azimuthal flow velocity prior to release and the
velocity of the mean-field-driven inward expansion. For a rotating condensate
released directly from our narrow annular trap, the hole size is below our
imaging resolution for experimentally accessible TOFs ($<$ 15 ms). To make the
signature of circulation visible earlier, we first adiabatically reduce the
ring beam intensity by 90% over 100 ms, then release the condensate suddenly
($<$ 1 $\mu$s). We use this procedure, followed by 6 ms TOF, to detect
circulation.
Figure 2: (a) Flow survival as a function of chemical potential, $\mu_{0}$,
for two barrier heights: $\beta/h$ = 650 Hz (upper, blue), and $\beta/h$ = 780
Hz (lower, red). Presence or absence of flow for a single condensate is shown
by closed circles. Open circles are the average of data within the bins
(vertical lines), representing the flow survival probability
(P${}_{\textrm{flow}}$) of each bin. A critical chemical potential $\mu_{c}$
for stable flow is found from a sigmoidal fit (solid lines) to the data for
each $\beta$. Inset: In situ absorption image of a condensate near $\mu_{c}$
($\mu_{0}/h$ = 870 Hz, $\beta/h$ = 650 Hz). (b) Values of $\mu_{c}$ at
different $\beta$, determined by fits as described in (a). The open circles
correspond to the data in (a). The vertical error bars reflect the width of
the sigmoidal fit, $\pm 2\mu_{w}$. The horizontal error bars are the 20 Hz
uncertainty in calibrating $\beta$. The dotted line is a linear fit to the
data, with slope 1.6(2). Insets: typical TOF absorption images showing the
presence (top left) and absence (bottom right) of circulation.
The Raman beams used to create circulation also cause small-amplitude
oscillations in the radial density profile, due to small dipole forces and
atom loss. These oscillations have no observable impact on the stability of
the circulation, and damp out after $<0.5$ s. We add a wait time $\geq$ 3 s
after the Raman transfer to ensure complete damping. The circulation is
extremely robust, and continues until losses due to background collisions
reduce $\mu_{0}$ to the level of the nonuniformities in the trap. For a 30 s
vacuum-limited $1/e$ condensate lifetime, $\mu_{0}$ remains high enough for
flow to survive up to 40 s.
After the $\geq$ 3 s wait time, we insert a barrier into the path of the flow
and study the stability of the circulation. The repulsive barrier is created
with a blue-detuned (532 nm) laser beam focused to an elliptical spot. The
major axis (15 $\mu$m $1/e^{2}$ radius) is aligned in the radial direction of
the toroid, and the minor axis (4.3 $\mu$m $1/e^{2}$ radius) is parallel to
the flow, and exceeds the bulk condensate healing length
($\xi=\hbar/\sqrt{2m\mu}$ $<$ 1 $\mu$m). The barrier depletes the local
density of the condensate, $n$, as seen in the inset in Fig. 2(a). The
reduction in density increases the local flow velocity (roughly
$v\\!\propto\\!1/n$). Lowering the density also lowers the local interaction
energy, $\mu_{l}\\!\propto\\!n$, decreasing the local sound speed. To study
flow stability, we ramp up the barrier intensity over 100 ms to a chosen
barrier height $\beta$, hold for 2 s, then ramp back down in another 100 ms.
The presence or absence of flow is then detected in TOF as described above.
This procedure is repeated many times for the same $\beta$, varying the total
number of atoms (by varying the initial condensate number and/or the wait
time) until the range of atom number is well-sampled. We then change $\beta$
and repeat the procedure. If the barrier is not applied, the flow always
survives, so we can attribute the decay of the flow to the effect of the
barrier. Separate measurements indicate that the flow decays in $<$ 100 ms.
The analysis of flow stability depends on in situ observations of the
condensate density profile in the presence of the barrier, and from TOF images
after the barrier has been removed. From TOF images we determine whether the
flow survived [insets Fig. 2(b)], and measure the condensate atom number, $N$.
For an annular condensate with a Thomas-Fermi profile, the chemical potential
$\mu_{0}=\hbar\bar{\omega}\sqrt{\pi/2\cdot(Na_{s}/R)}$ where
$\bar{\omega}\equiv\sqrt{\omega_{z}\omega_{r}}$, $a_{s}$ is the s-wave
scattering length, and $R$ is the radius of the ring. This calculation does
not include small corrections ($\approx 6\%$) due to the azimuthal
nonuniformity of the potential minimum and displacement of atoms from the
barrier region, corrections which are less than the systematic uncertainty in
determining $\mu_{0}$ ($\approx 10\%$).
We calibrate $\beta$ by taking in situ images of the condensate and measuring
the reduction in column density at the location of the barrier (see Fig. 2a
inset). Due to the high optical depth (up to 10), we use a partial transfer
imaging technique Freilich et al. (2010); Ramanathan et al. , in which a
precise fraction (ranging from 15-40%) of the atoms is transferred to the
$|2,-1\rangle$ state using a microwave pulse, then resonantly imaged on the
$S_{1/2}\,F\\!=\\!2\rightarrow P_{3/2}\,F\\!=\\!3$ transition. The local
interaction energy $\mu_{l}$ can be found from the measured column density
$\tilde{n}$. For data where $\mu_{l}\\!<\\!\hbar\omega_{z}$, we assume the
axial density profile is that of the harmonic oscillator ground state, with
$\mu_{l}\\!=\\![8\pi\cdot(\hbar\omega_{z})(\hbar^{2}a_{s}^{2}\tilde{n}^{2}/m)]^{1/2}$.
For data where $\mu_{l}>\hbar\omega_{z}$, we assume a Thomas-Fermi profile,
with
$\mu_{l}\\!=\\![9\pi^{2}/2\cdot(\hbar\omega_{z})^{2}(\hbar^{2}a_{s}^{2}\tilde{n}^{2}/m)]^{1/3}$.
When measuring the column density at the barrier, we correct for loss of
contrast due to the imaging resolution, which reduces the apparent depth of
the density depletion by $\approx$ 15%. We take $\beta$ to be
$\mu_{0}-\mu_{l}$.
Figure 3: Critical flow velocity, $v_{c}$, above which the circulation becomes
unstable, for each of the barrier heights in Fig 2(b). The value of $v_{c}$ is
shown normalized to the effective sound speed at the barrier
($c_{\mathrm{eff}}$), and is plotted as a function of $c_{\mathrm{eff}}$. The
horizontal error bars are the estimated uncertainty in $c_{\mathrm{eff}}$. The
vertical error bars are the combined experimental uncertainty in $v_{c}$ and
$c_{\mathrm{eff}}$. The measured ratio $v_{c}/c_{\mathrm{eff}}\approx 0.6$,
and is independent of $c_{\mathrm{eff}}$ to within the experimental
uncertainty. Vortex-like excitations are expected to occur in this system at
and above a velocity $v_{F}$, where $v_{F}<c_{\mathrm{eff}}$ Feynman (1955).
The gray band indicates estimated upper/lower bounds (see text) on
$v_{F}/c_{\mathrm{eff}}$, using Feynman’s approximate expression for $v_{F}$.
The solid circles in Fig. 2(a) show the flow survival or decay for single
experimental runs, plotted against $\mu_{0}$ for that run. The open circles
are the average of the solid circles within the bins shown. The upper plot
(blue) is for $\beta/h$ = 650 Hz (upper); the lower (red) shows $\beta/h$ =
780 Hz. At low $\mu_{0}$, the flow is arrested by the barrier. At high
$\mu_{0}$ the flow survival probability becomes unity. In between, the
survival probability increases from zero to one over a narrow critical region.
We characterize this critical region for each $\beta$ by fitting a sigmoidal
function $P(\mu_{0})=1/(1+e^{(\mu_{c}-\mu_{0})/\mu_{w}})$ to each unbinned
data set, where the parameters $\mu_{c}$ and $\mu_{w}$ are the critical
chemical potential and the sigmoidal width respectively unc . The observed
width is consistent with observed shot-to-shot variations in the trapping
potential.
Figure 2(b) shows the values of $\mu_{c}$ extracted from the fits of the data
for seven different $\beta$. Over this range, $\mu_{c}$ increases
approximately linearly with $\beta$, with a slope greater than unity. The
functional dependence and slope are determined in a non-trivial way by trap
geometry and the condition of quantized circulation around the ring. The
experimental results are consistent with expectations for our geometry.
The physics behind Fig. 2 is more apparent when the data is recast in terms of
flow velocity and sound speed at the barrier. The barrier thickness is greater
than $\xi$, so we expect the flow to become unstable when the velocity in the
barrier region exceeds some local critical velocity $v_{c}$. The flow velocity
at the barrier cannot be determined just from $\mu_{0}$ and $\beta$. The
requirements of quantized circulation (global), and flow conservation (local),
make it necessary to self-consistently calculate the velocity distribution
around the entire ring. We do this by integrating the in situ column density
radially to make a 1D approximation of the density profile, then solving for
the velocity distribution of an $l=1$ circulation state.
The critical velocity is determined by the lowest energy excitations allowed
for the system Landau (1941). For phonon-like excitations in the ring, that
velocity should be approximately the local sound speed in the barrier region
Watanabe et al. (2009). We make an initial estimate for the critical velocity
from the local interaction energy at the peak of the barrier,
$c_{l}=\sqrt{\mu_{l}/m}$. However, the inhomogeneous (nearly parabolic) radial
density profile lowers the effective sound speed to
$c_{\mathrm{eff}}=c_{l}/\sqrt{2}$ for waves traveling azimuthally along the
annulus Stringari (1998). Figure 3 shows the observed critical velocity
normalized to $c_{\mathrm{eff}}$, as a function of $c_{\mathrm{eff}}$. As seen
in previous work with finite inhomogeneous atomic condensates Onofrio et al.
(2000); Engels and Atherton (2007); Neely et al. (2010), the observed critical
velocity is less than the sound speed. For all tested values of $\beta$,
$v_{c}/c_{\mathrm{eff}}\approx 0.6$ and is independent of $c_{\mathrm{eff}}$
to within the experimental uncertainty.
In this experiment, flow is confined to a narrow, flattened channel, raising
the possibility that vortex-like excitations are responsible for the observed
critical velocity. Numerical simulations Piazza et al. (2009) with a model
condensate similar to ours, but in an $l=8$ circulation state, showed vortices
traversing the barrier region when the barrier was raised above a critical
level. This suggests that for our $l=1$ circulation state, a similar decay
mechanism could be at work. For vortex-like excitations in our quasi-2D
geometry, the (Feynman) critical velocity $v_{F}$ can be estimated from
energetic arguments Feynman (1955) to be $v_{F}=(\hbar/md)\ln(d/a)$, where $d$
is the channel width, and $a$ is the vortex core size. We take $d$ to be the
Thomas-Fermi width, and $a$ the healing length, both calculated for the
barrier region. Both $d$ and $a$ depend on $c_{\mathrm{eff}}$ via the
interaction energy $\mu_{l}$. The grey band in Fig. 3 is an estimate of the
probable value of $v_{F}/c_{\mathrm{eff}}$ with $c_{l}\geq
c_{\mathrm{eff}}\geq c_{l}/\sqrt{2}$. While this calculation is in
surprisingly good agreement with our data, a more complete model including
geometric factors is needed to accurately calculate the energy of a vortex-
antivortex pair in the barrier region.
We have presented the first realization of a closed atomtronic circuit,
demonstrating precise control both in inducing and arresting superfluid flow.
We have clearly identified the critical velocity where flow stops, and our
observations are in agreement with theoretical predictions in which vortex-
antivortex excitations are the decay mechanism for the system. In future work,
we plan to investigate the role of barrier geometry, condensate temperature,
and dimensionality in determining the critical velocity and decay mode. In
addition, rotating a barrier around the ring (oscillating it azimuthally)
would be analogous to magnetically biasing (driving an AC current in) a SQUID.
The present work constitutes a significant step toward realizing such an
atomic SQUID analog.
The authors thank L. Mathey for helpful discussions, and R. B. Blakestad for
comments on the manuscript. This work was partially supported by ONR, the ARO
atomtronics MURI, and the NSF PFC at JQI.
## References
* Clarke and Braginski (2004) J. Clarke and A. I. Braginski, _The SQUID Handbook_ , vol. 1,2 (Wiley-VCH, Weinheim, 2004).
* Simmonds et al. (2001) R. W. Simmonds et al., Nature 412, 55 (2001).
* Hoskinson et al. (2001) E. Hoskinson, Y. Sato, R. Packard, Phys. Rev. B 74, 100509 (2006).
* (4) B. T. Seaman et al., Phys. Rev. A 75, 023615 (2007). R. A. Pepino et al., Phys. Rev. Lett. 103, 140405 (2009). A. Ruschhaupt, and J. G. Muga, Phys. Rev. A 70, 061604 (2004). J. J. Thorn et al., Phys. Rev. Lett. 100, 240407 (2008). J. A. Stickney, D. Z. Anderson, and A. A. Zozulya, Phys. Rev. A 75, 013608 (2007).
* Likharev (1979) K. K. Likharev, Rev. Mod. Phys. 51, 101 (1979).
* Davis and Packard (2002) J. C. Davis and, R. E. Packard, Rev. Mod. Phys. 74, 741 (2001).
* Albiez et al. (2005) M. Albiez et al., Phys. Rev. Lett. 95, 010402 (2005).
* Levy et al. (2007) S. Levy et al., Nature 449, 579 (2007).
* Landau (1941) L. D. Landau, J. Phys. (USSR) 5, 71 (1941).
* (10) This assumes that the spatial scale of any perturbing potential is much less than the healing length.
* Pitaevskii and Stringari (2003) L. P. Pitaevskii and S. Stringari, _Bose-Einstein Condensation_ (Clarendon, Oxford, 2003).
* Barenghi et al. (2001) C. F. Barenghi, R. J. Donnelly, , and W. F. Vinen, eds., _Quantized Vortex Dynamics and Superfluid Turbulence_ (Springer-Verlag, Berlin, 2001).
* Feynman (1955) R. Feynman, Prog. Low Temp. Phys. 1, 17 (1955).
* Avenel and Varoquaux (1985) O. Avenel and E. Varoquaux, Phys. Rev. Lett. 55, 2704 (1985).
* Amar et al. (1992) A. Amar et al., Phys. Rev. Lett. 68, 2624 (1992).
* Huebener (2001) R. P. Huebener, _Magnetic Flux Structures in Superconductors_ (Springer, 2001).
* Neely et al. (2010) T. W. Neely et al., Phys. Rev. Lett. 104, 160401 (2010).
* Onofrio et al. (2000) R. Onofrio et al., Phys. Rev. Lett. 85, 2228 (2000).
* Engels and Atherton (2007) P. Engels and C. Atherton, Phys. Rev. Lett. 99, 160405 (2007).
* Ryu et al. (2007) C. Ryu et al., Phys. Rev. Lett. 99, 260401 (2007).
* (21) We apply a strong barrier during evaporation, removing it adiabatically when well below the condensation temperature, several seconds before we create circulation.
* Wright et al. (2008) K. C. Wright, L. S. Leslie, and N. P. Bigelow, Phys. Rev. A 77, 041601 (2008).
* Andersen et al. (2006) M. F. Andersen et al., Phys. Rev. Lett. 97, 170406 (2006).
* Madison et al. (2000) K. W. Madison et al., Phys. Rev. Lett. 84, 806 (2000).
* Freilich et al. (2010) D. V. Freilich et al., Science 329, 1182 (2010).
* (26) A. Ramanathan et al., to be published.
* (27) Uncertainties herein are the uncorrelated combination of 1$\sigma$ statistical and systematic uncertainties unless stated otherwise.
* Watanabe et al. (2009) G. Watanabe et al., Phys. Rev. A 80, 053602 (2009).
* Stringari (1998) S. Stringari, Phys. Rev. A 58, 2385 (1998).
* Piazza et al. (2009) F. Piazza, L. A. Collins, and A. Smerzi, Phys. Rev. A 80, 021601 (2009).
|
arxiv-papers
| 2010-12-29T22:43:04 |
2024-09-04T02:49:16.064742
|
{
"license": "Public Domain",
"authors": "A. Ramanathan, K. C. Wright, S. R. Muniz, M. Zelan, W. T. Hill III, C.\n J. Lobb, K. Helmerson, W. D. Phillips, and G. K. Campbell",
"submitter": "Kevin Wright",
"url": "https://arxiv.org/abs/1101.0019"
}
|
1101.0082
|
#
Probabilistic Dynamic Logic of Phenomena and Cognition ††thanks: Evgenii
Vityaev is with the Department of Mathematical Logic, Sobolev Institute of
Mathematics of the Russian Academy of Sciences and with the Department of
Discrete mathematics and Informatics of the Novosibirsk State University,
630090, Novosibirsk, Russia, email: vityaev@math.nsc.ru ††thanks: Boris
Kovalerchuk is with the Department of Computer Science, Central Washington
University, Ellensburg, WA 98926-7520, e-mail: borisk@cwu.edu ††thanks: Leonid
Perlovsky is with the Harvard University and the Air Force Research
Laboratory, Sensors Directorate, Hanscom AFB, leonid@seas.harvard.edu
††thanks: Stanislav Smerdov, Novosibirsk State University, Sobolev Institute
of Mathematics of the Russian Academy of Sciences, 630090, Novosibirsk,
Russia, email: netid@ya.ru
Evgenii Vityaev, Boris Kovalerchuk, Leonid Perlovsky, Stanislav Smerdov
###### Abstract
The purpose of this paper is to develop further the main concepts of Phenomena
Dynamic Logic (P-DL) and Cognitive Dynamic Logic (C-DL), presented in the
previous paper. The specific character of these logics is in matching
vagueness or fuzziness of similarity measures to the uncertainty of models.
These logics are based on the following fundamental notions: _generality
relation, uncertainty relation, simplicity relation, similarity maximization
problem with empirical content and enhancement (learning) operator_. We
develop these notions in terms of logic and probability and developed a
Probabilistic Dynamic Logic of Phenomena and Cognition (P-DL-PC) that relates
to the scope of probabilistic models of brain. In our research the
effectiveness of suggested formalization is demonstrated by approximation of
the expert model of breast cancer diagnostic decisions. The P-DL-PC logic was
previously successfully applied to solving many practical tasks and also for
modelling of some cognitive processes.
## I Introduction
In the paper [1] there was introduced a Phenomena Dynamic Logic (P-DL) and
Cognitive Dynamic Logic (C-DL) as a generalization of the Dynamic Logic and
Neural Modelling Fields theory (NMF) introduced in the previous papers [2, 3].
Logics P-DL, C-DL provide the most general description of Dynamic Logic in the
following fundamental notions _generality relation, uncertainty relation,
simplicity relation, similarity maximization problem with empirical content
and enhancement (learning) operator_. This generalization provide
interpretation of P-DL, C-DL logics in the frame of other approaches.
In this paper we interpret logics P-DL, C-DL in terms of logic and
probability: uncertainty we interpret as probability, while the process of
learning as a semantic probabilistic inference [4, 9, 6, 5]. We also interpret
mentioned fundamental notions. The resulting Probabilistic Dynamic Logic of
Phenomena and Cognition (P-DL-PC) belong to the scope of the probabilistic
models of brain [19, 20]. Thus, through logics P-DL, C-DL we extend the
interpretation of Dynamic Logic and Neural Modelling Fields theory to
probabilistic models of brain. The P-DL-PC logic as probabilistic model of
brain was previously applied to modelling of some cognitive process [7, 8, 9,
21]. The effectiveness of P-DL-PC logic demonstrated in this paper by
approximation of the expert model of breast cancer diagnostic decisions.
## II Universal productions. Data for prediction
In our study learning models will be generated as sets of _universal
productions_ (_u-productions_), which are introduced in this section. Note
that every set of _universal formulas_ is logically equivalent to a certain
set of u-productions.
Consider a fixed first-order language $\mathfrak{L}$ in a countable signature.
Hereafter denote $\mathbf{A}_{\mathfrak{L}}$ the set of all atoms;
$\mathbf{L}_{\mathfrak{L}}$ – the set of all literals;
$\mathbf{S}_{\mathfrak{L}}^{0}$ – the set of ground sentences. The set of
ground atoms and the set of ground literals are denoted
$\mathbf{A}_{\mathfrak{L}}^{0}\rightleftharpoons\mathbf{A}_{\mathfrak{L}}\cap\mathbf{S}_{\mathfrak{L}}^{0}$
and
$\mathbf{L}_{\mathfrak{L}}^{0}\rightleftharpoons\mathbf{L}_{\mathfrak{L}}\cap\mathbf{S}_{\mathfrak{L}}^{0}$
correspondingly. Following examples of atoms and literals are given in the
section VIII for the task of approximation of the expert model of breast
cancer diagnostic decisions: ‘number of calcifications per $cm^{3}$ less than
20‘, ‘volume of calcifications in $cm^{3}$ not less or equal to 5‘, ‘total
number of calcifications more than 30 and etc.
Let $\Theta$ be the set of all substitutions and $\Theta^{0}\subseteq\Theta$
the set of ground substitutions, that are mappings variables to ground terms.
All necessary notions from model theory and logic programming are elementary
and can be easily found in books [12], [13, 14].
###### Definition.
A record of the type
$\mathrm{R}\leftrightharpoons\tilde{\forall}\left(\mathrm{A}_{1}\wedge\cdots\wedge\mathrm{A}_{m}\Leftarrow\mathrm{B}_{1}\wedge\cdots\wedge\mathrm{B}_{n}\right),$
where
$\mathrm{A}_{1},\cdots\mathrm{A}_{m},\mathrm{B}_{1},\cdots,\mathrm{B}_{n}$ are
literals, and $\tilde{\forall}$ stands for a bloc of quantifiers over all free
variables of the formulae in brackets (universal closure), is called a _u-
production_. _A variant of u-production $\mathrm{R}$_ is
$\mathrm{R}\theta\rightleftharpoons\tilde{\forall}\left(\mathrm{A}_{1}\theta\wedge\cdots\mathrm{A}_{m}\theta\Leftarrow\mathrm{B}_{1}\theta\wedge\cdots\wedge\mathrm{B}_{n}\theta\right),$
where $\theta$ is an arbitrary one-to-one correspondence over the set of
variables. Let ${\tt Prod}$ be the set of all u-productions.
For example in section X presented the following u-production that was
discovered by the learning model:
> IF TOTAL number of calcifications is more than 30, and VOLUME is more than 5
> $cm^{3}$, and DENSITY of calcifications is moderate,
> THEN Malignant.
Let $\mathtt{Fact}_{v}\subset\mathbf{A}_{\mathfrak{L}}$ be a set of atoms from
A that are valid for verification in algebraic system ${\mathfrak{B}}$
appearing in practice. Our aim is to investigate as much “extra” facts about
$\mathfrak{L}$ as possible, i.e., to predict or explain them. A natural
assumption is that we can verify (falsify) each element of
$\mathtt{Fact}_{\rm o}\leftrightharpoons\left\\{{\rm
A}\theta\mid\theta\in\Theta^{0},~{}{\rm A}\in\mathtt{Fact}_{v}\right\\}.$
Certainly we may postulate our ability to check any literal of
$\mathtt{Fact}_{v}^{\ast}\leftrightharpoons\mathtt{Fact}_{v}\cup\left\\{\neg{\rm
A}\mid{\rm A}\in\mathtt{Fact}_{v}\right\\}$. For the rest of the literals (and
their conjunctions) the machinery of _probabilistic prediction_ will be
defined later on. Note that ${\tt Fact}_{\rm o}^{\ast}={\tt Fact}_{\rm
o}\cup{\tt Fact}_{\rm o}^{\neg}$, where ${\tt Fact}_{\rm
o}^{\neg}\rightleftharpoons\left\\{\neg\mathrm{A}\mid\mathrm{A}\in{\tt
Fact}_{\rm o}\right\\}$ is _the complete set of alternatives_ allowing a real
test.
The _data_ are defined as a maximal (logically) consistent subset of the
complete set of alternatives, i.e., being given a mapping
$\zeta_{\mathfrak{B}}:{\tt Fact}_{\rm o}\mapsto\left\\{{\bot,\top}\right\\}$
(here _$\bot$ – “false”, $\top$ – “true”_) we conclude that
${\tt Data}\left[\mathfrak{B}\right]\rightleftharpoons\left\\{{{\rm A}\mid{\rm
A}\in{\tt Fact}_{\rm o}~{}\mbox{{and}}~{}\zeta_{\mathfrak{B}}\left({\rm
A}\right)=\top}\right\\}\cup$ $\left\\{{\neg{\rm A}\mid{\rm A}\in{\tt
Fact}_{\rm o}~{}\mbox{{and}}~{}\zeta_{\mathfrak{B}}\left({\rm
A}\right)=\bot}\right\\}.$
## III Generality relation between theories
The idea of a _generality relation_ between theories can be viewed, for
example, as a reduction of the set of properties predicted by the use of these
theories. A more general theory (potentially) predicts a greater number of
formal features. We start with a generality relation between one-element
specifications, i.e., between u-productions.
###### Definition.
For two productions
$\mathrm{R}_{1}\equiv\tilde{\forall}\left(\mathrm{A}_{1}\wedge\cdots\wedge\mathrm{A}_{m_{1}}\Leftarrow\mathrm{B}_{1}\wedge\cdots\wedge\mathrm{B}_{n_{1}}\right)$
and
$\mathrm{R}_{2}\equiv\tilde{\forall}\left(\mathrm{C}_{1}\wedge\cdots\wedge\mathrm{C}_{m_{2}}\Leftarrow\mathrm{D}_{1}\wedge\cdots\wedge\mathrm{D}_{n_{2}}\right)$
a relation $\mathrm{R}_{1}\succ\mathrm{R}_{2}$ (“more general than”) takes
place if and only if there exists $\theta\in\Theta$ such that
$\left\\{{\mathrm{B}_{1}\theta,\cdots,\mathrm{B}_{n_{1}}\theta}\right\\}\subseteq\left\\{{\mathrm{D}_{1},\cdots,\mathrm{D}_{n_{2}}}\right\\}$,
$\left\\{{\mathrm{A}_{1}\theta,\cdots,\mathrm{A}_{m_{1}}\theta}\right\\}\supseteq\left\\{{\mathrm{C}_{1},\cdots,\mathrm{C}_{m_{2}}}\right\\}$,
and $n_{1}\leqslant n_{2}$, $m_{1}\geqslant m_{2}$, $\not\vdash{\rm
R}_{1}\equiv{\rm R}_{2}$.
The inclusion of the sets of premises designates that the more general
u-production is, then the wider its field of application. The inverse
inclusion (for conclusions) says that $\mathrm{R}_{1}$ predicts a greater
number of properties using a smaller premise.
Let $S\subseteq{\tt Prod}$. Denote ${\tt Fact}\left[S;\mathfrak{B}\right]$ the
set of all $\mathrm{A}\in\mathbf{L}_{\mathfrak{L}}^{0}$ such that for some
${\rm R}\in S$ and $\theta\in\Theta^{0}$,
$\mathrm{R}\theta\equiv\left(\mathrm{A}_{1}\wedge\cdots\wedge\mathrm{A}_{m}\Leftarrow\mathrm{B}_{1}\wedge\cdots\wedge\mathrm{B}_{n}\right)$,
holds
$\left\\{\mathrm{B}_{1},\cdots,\mathrm{B}_{n}\right\\}\subseteq{\tt
Data}\left[\mathfrak{B}\right]$ and
$\mathrm{A}\in\left\\{\mathrm{A}_{1},\cdots,\mathrm{A}_{m}\right\\}$.
Thus, ${\tt Fact}\left[S;\mathfrak{B}\right]$ is the set of ground literals
predicted according to available data (about the model $\mathfrak{B}$)
together with u-productions in $S$.
In the sequel let $\succcurlyeq$ be a reflexive closure of $\succ$. One should
pay attention to the fact: $\mathrm{R}_{1}\succcurlyeq\mathrm{R}_{2}$ entails
that ${\tt Fact}\left[\left\\{\mathrm{R}_{1}\right\\};\mathfrak{B}\right]$
contains ${\tt
Fact}\left[\left\\{\mathrm{R}_{2}\right\\};\mathfrak{B}\right]$.
Thereafter it isn’t difficult to extend the domain of our generality relation
to subsets of ${\tt Prod}$.
###### Definition.
Let $S,S^{\prime}\subseteq{\tt Prod}$, and for any ${\rm R}^{\prime}\in
S^{\prime}$ we find ${\rm R}\in S$ such that
$\mathrm{R}\succcurlyeq\mathrm{R}^{\prime}$. In this case we say ‘$S$ is not
less general than $S^{\prime}$’ ($S\vartriangleright S^{\prime}$).
It’s straightforward to notice that ${\tt
Fact}\left[S^{\prime};\mathfrak{B}\right]\subseteq{\tt
Fact}\left[S;\mathfrak{B}\right]$ for $S$ and $S^{\prime}$ from the definition
above. Remark that $S$ may include u-productions apart from those, which are
generalizations of elements of $S^{\prime}$.
## IV Probability/degree of belief
The topic of distributing probability over formulas of propositional logic (as
well as over ground statements in a first order language) being widely
discussed in a literature and meets Kolmogorov’s understanding of probability
measure [11]. The following definition is given on the basis of analysis cited
in [10].
###### Definition.
A probability over $F\subseteq{\mathbf{S}}_{\mathfrak{L}}^{0}$ closed with
respect to $\wedge$, $\vee$ and $\neg$, is a function
$\mu:F\mapsto\left[{0,1}\right]$ satisfying the following conditions:
1. 1.
if $\vdash\Phi$ (“$\Phi$ is a tautology”), then $\mu\left(\Phi\right)=1$;
2. 2.
if $\vdash\neg\left({\Phi\wedge\Psi}\right)$, then
$\mu\left({\Phi\vee\Psi}\right)=\mu\left(\Phi\right)+\mu\left(\Psi\right)$.
For any ground instance of a u-production its probability is defined as
conditional, i.e.,
$\mu\left({\rm A}_{1}\wedge\cdots{\rm A}_{m}\Leftarrow{\rm
B}_{1}\wedge\cdots\wedge{\rm B}_{n}\right)=$
$=\mu\left({\rm A}_{1}\wedge\cdots{\rm A}_{m}\mid{\rm
B}_{1}\wedge\cdots\wedge{\rm B}_{n}\right)=\frac{\mu\left({{\rm
A}_{1}\wedge\cdots{\rm A}_{m}\wedge{\rm B}_{1}\wedge\cdots\wedge{\rm
B}_{n}}\right)}{\mu\left({{\rm B}_{1}\wedge\cdots\wedge{\rm B}_{n}}\right)}$
Let ${\rm R}\in{\tt Prod}$. Denote as ${\tt Sub}\left[{\rm R}\right]^{\mu}$
those substitutions $\theta\in\Theta^{0}$, for which the premise of
u-production ${\rm R}\theta$ has a non-zero probability.
${\tt Prod}^{\mu}\rightleftharpoons\left\\{{\rm R}\in{\tt Prod}\mid{\tt
Sub}\left[{\rm R}\right]^{\mu}\neq\varnothing\right\\}$;
$\underline{\mu}\left({\rm R}\right)\rightleftharpoons{\rm
inf}\left\\{\mu\left({{\rm R}\theta}\right)\mid\theta\in{\tt Sub}\left[{\rm
R}\right]^{\mu}\right\\}$, where ${\rm R}\in{\tt Prod}^{\mu}$.
A value of conditional probability serves to characterize our _degree of
belief_ (and responsible for an _uncertainty relation_) in reliability of
different causal connections included in temporary specification. Note that
two productions are not necessary comparable with respect to generality
relation $\succcurlyeq$; moreover, their premisses may not be contained in the
complete set of alternatives (and so these productions will be not valid for a
direct check in a real structure $\mathfrak{B}$).
## V Simplicity of probabilistic theories
Adding comparison of lower probabilistic estimations to the definition of
generality relation we obtain the following definition.
###### Definition.
Let $S,S^{\prime}\subseteq{\tt Prod}^{\mu}$. We say that $S$ is _more
$\mu$-general than_ $S^{\prime}$ iff for every $\mathrm{C}^{\prime}\in
S^{\prime}$ there exists $\mathrm{C}\in S$ such that
$\mathrm{C}\succcurlyeq\mathrm{C}^{\prime}$ and
$\underline{\mu}(\mathrm{C})\geqslant\underline{\mu}(\mathrm{C}^{\prime})$,
and in at least one of the cases the strong relation $\succ$ takes place.
Hence, $\mu$-generalization allows us to define a more general set $S$ in such
a way that the lower estimations of probabilities is not declined. When our
belief to the elements of $S$ is no less than that of $S^{\prime}$, then we
have a _simplicity relation_ – the set $S$ is simpler than $S^{\prime}$ in
order to describe/predict the properties.
## VI Similarity measure with the empirical content
By elaboration of u-productions we mean the gain of its conditional
probability.
###### Definition.
A relation ${\rm R}_{1}\sqsubset{\rm R}_{2}$ (‘probabilistic inference’) for
${\rm R}_{1},{\rm R}_{2}\in{\tt Prod}^{\mu}$ means that ${\rm R}_{1}\succ{\rm
R}_{2}$ and $\underline{\mu}\left({{\rm
R}_{1}}\right)<\underline{\mu}\left({{\rm R}_{2}}\right)$.
###### Definition.
Let $\pi$ be some requirements to be applied to elements of ${\tt
Prod}^{\mu}$, i.e. $\pi:{\tt Prod}^{\mu}\mapsto\left\\{\bot,\top\right\\}$
(value is equal to $\bot$, if u-production satisfies $\pi$, and $\top$ –
otherwise); $\mathsf{\Pi}\leftrightharpoons\left\\{{\rm R}\in{\tt
Prod}^{\mu}\mid\pi\left({\rm R}\right)=\top\right\\}$. We say that
$\mathrm{R}_{2}\in\mathsf{\Pi}$ is a _minimal follower of
$\mathrm{R}_{1}\in{\tt Prod}^{\mu}$ relative to $\sqsubset$ in $\mathsf{\Pi}$_
(denoted as $\mathrm{R}_{1}\sqsubset_{\pi}\mathrm{R}_{2}$), iff
$\mathrm{R}_{1}\sqsubset\mathrm{R}_{2}$ and there is no intermediate
u-production $\mathrm{R}_{3/2}\in\Pi$ such that
$\mathrm{R}_{1}\sqsubset\mathrm{R}_{3/2}\sqsubset\mathrm{R}_{2}$.
In the prediction of a literal ${\rm H}$ the _similarity measure_ for
u-productions, which are valid for verification and applicable to the goal
${\rm H}$, is equal to conditional probability
$\underline{\mu}\left(\cdot\right)$. Thus we deal with a uniform measure of
similarity.
## VII Learning operator
###### Definition.
A production
$\mathrm{R}\equiv\tilde{\forall}\left(\mathrm{A}_{1}\wedge\cdots\wedge\mathrm{A}_{m}\leftarrow\mathrm{B}_{1}\wedge\dots\wedge\mathrm{B}_{n}\right)$
is called a _maximal specific u-production (ums-production) for prediction of
a conjunction
$\mathrm{H}\equiv\left(\mathrm{H}_{1}\wedge\cdots\wedge\mathrm{H}_{k}\right)$,_
where
$\left\\{\mathrm{H}_{1},\cdots,\mathrm{H}_{k}\right\\}\subset\mathbf{L}_{\mathfrak{L}}$
and $m\leqslant k$, iff the following conditions are satisfied:
1. 1.
there is a substitution $\theta$ (not necessary ground) such that
$\left\\{\mathrm{A}_{1},\cdots,\mathrm{A}_{m}\right\\}\subseteq\left\\{\mathrm{H}_{1}\theta,\cdots,\mathrm{H}_{k}\theta\right\\}$,
$\left\\{{\mathrm{B}_{1},\dots,\mathrm{B}_{n}}\right\\}\subseteq\left\\{\mathrm{B}\theta\mid\mathrm{B}\in\mathtt{Fact}_{v}^{\ast}\right\\}$;
2. 2.
if $\mathrm{D}\in\left\\{\mathrm{A}_{1},\cdots,\mathrm{A}_{m}\right\\}$ and
$\theta_{\rm o}\in{\tt Sub}\left[{\rm R}\right]^{\mu}$, then
$\mu\left({\mathrm{A}_{1}\theta_{\rm
o}\wedge\cdots\wedge\mathrm{A}_{m}}\theta_{\rm o}\right)<\\\
\mu\left({\mathrm{A}_{1}\theta_{\rm
o}\wedge\cdots\wedge\mathrm{A}_{m}}\theta_{\rm
o}\mid{\mathrm{B}_{1}\theta_{\rm
o}\wedge\cdots\wedge\mathrm{B}_{n}}\theta_{\rm o}\right)$
and $\mu\left(\mathrm{D}\theta_{\rm o}\right)<\mu\left(\mathrm{D}\theta_{\rm
o}\mid{\mathrm{B}_{1}\theta_{\rm
o}\wedge\cdots\wedge\mathrm{B}_{n}}\theta_{\rm o}\right)$;
3. 3.
there is no ${\rm R^{\prime}}\in\mathtt{Prod}^{\mu}$, for which points (1–2)
are hold along with ${\rm R}\sqsubset{\rm R^{\prime}}$;
4. 4.
the u-production ${\rm R}$ can’t be generalized up to some ${\rm
R^{\prime}}\in\mathtt{Prod}^{\mu}$ satisfying all the previous points (1–3)
without decreasing its estimation $\underline{\mu}\left(\cdot\right)$.
The conditions above (for corresponding ums-productions) are denoted as
‘point.i’, $1\leqslant i\leqslant 4$.
###### Remark.
Though condition point.4 emphasizes the nature of definition, but it isn’t
necessary for indication. Indeed, if $\mathrm{R}$ may be generalized up to
$\mathrm{R}^{\prime}$ under preserving point.1–3, then
$\underline{\mu}\left(\mathrm{R}\right)\leqslant\underline{\mu}\left(\mathrm{R}^{\prime}\right)$
(otherwise we get ${\rm R^{\prime}}\sqsubset{\rm R}$ – that contradicts
point.3 for $\mathrm{R}$.
Let $\pi\left({\rm R}\right)=\top$ be fulfilled for ${\rm
R}\in\mathtt{Prod}^{\mu}$ iff conditions points.1–2 are satisfied for ${\rm
R}$ and ${\rm H}$ (the last one is fixed from this moment); denote
$\mathsf{\Pi}\leftrightharpoons\pi^{-1}\left(\top\right)$.
Define the _probabilistic fix-point operator_
$\mathrm{T}_{\pi}:2^{\mathtt{Prod}^{\mu}}\mapsto 2^{\mathtt{Prod}^{\mu}}$ as
follows: for a set $S\subseteq\mathtt{Prod}^{\mu}$ it produces
$S^{\prime}\leftrightharpoons\left\\{\mathrm{R}^{\prime}\mid\mathrm{R}\sqsubset_{\pi}\mathrm{R}^{\prime}\
\mbox{\emph{for some}}\ \mathrm{R}\in S\right\\}\cup$
$\cup\left\\{\mathrm{R}\mid\mathrm{R}\in S\cap\mathsf{\Pi}\ \mbox{\emph{and
there is no}}\ \mathrm{R}^{\prime}\ \mbox{\emph{such that}}\
\mathrm{R}\sqsubset_{\pi}\mathrm{R}^{\prime}\right\\}$.
Therefore the operator $\mathrm{T}_{\pi}:S\mapsto S^{\prime}$ possess
important properties:
1. 1.
the set $S^{\prime}$ is always more precise than $S$ (relative to
$\succcurlyeq$);
2. 2.
the conditional probabilities $\underline{\mu}\left(\cdot\right)$ increase
during the conversion to more particular cases (and so fuzziness decreases);
3. 3.
the similarity measure with the empirical content becomes greater for at least
one u-production (in $S$) when the operator converts $S$ to $S^{\prime}$ (if
not $S=S^{\prime}$, of course);
As a result the operator $\mathrm{T}_{\pi}$ is the _enhancement, or learning,
operator_ in the sense of [1].
###### Definition.
A fix-point (f.p., for short) $S$ of $\mathrm{T}_{\pi}$ is _optimal_ iff there
is no other f.p. $S^{\prime}$ of considered operator, which is more
$\mu$-general than $S$.
###### Statement.
A subset $S\subseteq\mathtt{Prod}^{\mu}$ is a fix-point of the operator
$\mathrm{T}_{\pi}$ iff every element of $S$ satisfies points.1–3 for
$\mathrm{H}$.
###### Corrolary.
A subset $S\subseteq\mathtt{Prod}^{\mu}$ is an optimal fix-point of the
operator $\mathrm{T}_{\pi}$ iff every element of $S$ is a ums-production for
prediction of $\mathrm{H}$.
Ums-productions may be viewed as a result of performing generalized scheme of
the _semantic probabilistic inference_ [4, 5], which is realized by the fix-
point operator described above. The program system ‘Discovery’ (see [16, 17,
9, 21]) was developed: it carries out the propositional version of the
probabilistic fix-point (learning) operator and was successfully applied to
solving many practical tasks [21].
## VIII Extraction of the expert model of breast cancer diagnostic decisions
We applied our method to approximation of the expert model of breast cancer
diagnostic decisions that was obtained from the radiologist J.Ruiz [17]. At
first we extract this model from the expert by the special procedure using
monotone boolean functions [17] and then apply the program system ‘Discovery’
[16] to approximate this model.
### VIII-A Hierarchical Approach
At first we ask an expert to describe particular cases using the binary
features. Then we ask a radiologist to evaluate a particular cases, when
features take on specific values. A typical query will have the following
format: ”If feature 1 has value $v_{1}$, feature 2 has value $v_{2}$, …,
feature n has value $v_{n}$, then is a case suspicious of cancer or not?”
Each set of values ($v_{1},v_{2},...,v_{n}$) represent a possible clinical
case. It is practically impossible to ask a radiologist to generate diagnosis
for thousands of possible cases. A hierarchical approach combined with the use
of the property of monotonicity makes the problem manageable. We construct a
hierarchy of medically interpretable features from a very generalized level to
a less generalized level. This hierarchy follows from the definition of the 11
medically oriented binary attributes. The medical expert indicate that the
original 11 binary attributes
$w_{1},w_{2},w_{3},y_{1},y_{2},y_{3},y_{4},y_{5},x_{3},x_{4},x_{5}$ could be
organized in terms of a hierarchy with development of two new generalized
attributes $x_{1}$, depending on attributes $w_{1},w_{2},w_{3}$, and $x_{2}$,
depending on attributes $y_{1},y_{2},y_{3},y_{4},y_{5}$.
A new generalized feature, $x_{1}$ – ‘Amount and volume of calcifications’
with grades (0 - ‘benign’ and 1 - ‘cancer’) was introduced based on features:
$w_{1}$ – number of calcifications/cm3, $w_{2}$ – volume of calcification, cm3
and $w_{3}$ – total number of calcifications. We view $x_{1}$ as a function
$g(w_{1},w_{2},w_{3})$ to be identified. Similarly a new feature $x_{2}$ –
‘Shape and density of calcification’ with grades: (1) for ‘cancer’ and
(0)-‘benign’ generalizes features: $y_{1}$ – ‘irregularity in shape of
individual calcifications’ $y_{2}$ – ‘variation in shape of calcifications’
$y_{3}$ – ‘variation in size of calcifications’ $y_{4}$ – ‘variation in
density of calcifications’ $y_{5}$ – ‘density of calcifications’. We view
$x_{2}$ as a function $x_{2}=h(y_{1},y_{2},y_{3},y_{4},y_{5})$ to be
identified for cancer diagnosis.
As result we have a decomposition of our task as follows:
$f\left(x_{1},x_{2},x_{3},x_{4},x_{5}\right)=$
$f\left(g\left(w_{1},w_{2},w_{3}\right),h\left(y_{1},y_{2},y_{3},y_{4},y_{5}\right),x_{3},x_{4},x_{5}\right).$
### VIII-B Monotonicity
Giving the above definitions we can represent clinical cases in terms of
binary vectors with five generalized features as:
$(x_{1},x_{2},x_{3},x_{4},x_{5})$. Let us consider two clinical cases that are
represented by the two binary sequences: (10110) and (10100). If radiologist
correctly diagnose (10100) as cancer, then, by utilizing the property of
monotonicity, we can also conclude that the clinical case (10110) should also
be cancer. Medical expert agreed with presupposition about monotonicity of the
functions $f\left(x_{1},x_{2},x_{3},x_{4},x_{5}\right)$ and
$h\left(y_{1},y_{2},y_{3},y_{4},y_{5}\right)$.
Let us describe the interview with an expert using minimal sequence of
questions to completely infer a diagnostic function using monotonicity. This
sequence is based on fundamental Hansel lemma [15]. We omit a detailed
description of the specific mathematical steps. They can be found in [18].
Table 1 illustrates this.
### VIII-C Expert model extraction
Columns 2 and 3 present values of above defined functions $f$ and $h$. We omit
a restoration of function $g\left(w_{1},w_{2},w_{3}\right)$ because few
questions are needed to restore this function. All 32 possible cases with five
binary features $\langle x_{1},x_{2},x_{3},x_{4},x_{5}\rangle$ are presented
in column 1 in table 1. They are grouped and the groups are called Hansel
chains [17]. The sequence of chains begins with the shortest chain 1 –
$(01100)<(11100)$ for five binary features. Then largest chain 10 consists of
6 ordered cases: $(00000)<(00001)<(00011)<(00111)<(01111)<(11111)$. The chains
are numbered there from 1 to 10 and each case has its number in the chain,
e.g., 1.2 means the second case in the first chain. Asterisks in columns 2 and
3 mark answers obtained from an expert, e.g., 1* for case (01100) in column 3
means that the expert answered ‘yes’. The answers for some other chains in
column 3 are automatically obtained using monotonicity. The value f(01100) = 1
for case 1.1 is extended for cases 1.2, 6.3. and 7.3 in this way. Similarly
values of the monotone Boolean functions h are computed using the table 1. The
attributes in the sequence (10010) are interpreted as
$y_{1},y_{2},y_{3},y_{4},y_{5}$ for the function h instead of
$x_{1},x_{2},x_{3},x_{4},x_{5}$. The Hansel chains are the same if the number
of attributes is the same five in this case.
Column 5 and 6 list cases for extending functions’ values without asking an
expert. Column 5 is for extending functions’ values from 1 to 1 and column 6
is for extending them from 0 to 0. If an expert gave an answer opposite
(f(01100) = 0) to that presented in table 1 for function $f$ in the case 1.1,
then this 0 value could be extended in column 2 for cases 7.1 (00100) and 8.1
(01000). These cases are listed in column 5 for case (01100). There is no need
to ask an expert about cases 7.1 (00100) and 8.1 (01000). Monotonicity
provides the answer. The negative answer f(01100) = 0 can not be extended for
f(11100). An expert should be queried regarding f(11100). If his/her answer is
negative f(11100) = 0 then this value can be extended for cases 5.1. and 3.1
listed in column 5 for case 1.2. Relying on monotonicity, the value of f for
them will also be 0.
The total number of cases with asterisk (*) in columns 2 and 3 are equal to 13
and 12. These numbers show that 13 questions are needed to restore the
function $f\left(x_{1},x_{2},x_{3},x_{4},x_{5}\right)$ and 12 questions are
needed to restore the function $h\left(y_{1},y_{2},y_{3},y_{4},y_{5}\right)$.
This is only 37.5% of 32 possible questions. The full number of questions for
the expert without monotonicity and hierarchy is $2^{11}=2048$.
## IX Approximation of the expert model by learning operator
For the Approximation of the expert model we used the program system
‘Discovery’ [16], that realizes the propositional case of the probabilistic
fix-point learning operator. We discovered several dozens diagnostic rules
that were statistically significant on the 0.01, 0.05 and 0.1 levels of
(F-criterion). Rules were extracted using 156 cases (73 malignant, 77 benign,
2 highly suspicious and 4 with mixed diagnosis). In the Round-Robin test our
rules diagnosed 134 cases and refused to diagnose 22 cases. The total accuracy
of diagnosis is 86%. Incorrect diagnoses were obtained in 19 cases (14% of
diagnosed cases). The false-negative rate was 5.2% (7 malignant cases were
diagnosed as benign) and the false-positive rate was 8.9% (12 benign cases
were diagnosed as malignant). Some of the rules are shown in table 2. This
table presents examples of discovered rules with their statistical
significance. In this table:
* •
‘NUM’ – number of calcifications per $cm^{3}$;
* •
‘VOL’ – volume in $cm^{3}$;
* •
‘TOT’ – total number of calcifications;
* •
‘DEN’ – density of calcifications;
* •
‘VAR’ – variation in shape of calcifications;
* •
‘SIZE’ – variation in size of calcifications;
* •
‘IRR’ – irregularity in shape of calcifications;
* •
‘SHAPE’ – shape of calcifications.
We studied three levels of similarity measure: 0.7, 0.85 and 0.95. A higher
level of conditional probability decreases the number of rules and diagnosed
patients, but increases accuracy of diagnosis.
Table 1. Dynamic sequence of questions to expert
---
1 | 2 | 3 | 4 | 5 | 6 | 7
Number | $f$ | $h$ | Monotonic extrapolation | Chain | Case
| Diagnose | Form and V | $1\mapsto 1$ | $0\mapsto 0$ | |
$(01100)$ | 1* | 1* | 1.2, 6.3, 7.3 | 7.1, 8.1 | Chain 1 | 1.1
$(11100)$ | 1 | 1 | 6.4, 7.4 | 5.1, 3.1 | | 1.2
$(01010)$ | 0* | 1* | 2.2, 6.3, 8.3 | 6.1, 8.1 | Chain 2 | 2.1
$(11010)$ | 1* | 1 | 6.4, 8.4 | 3.1, 6.1 | | 2.2
$(11000)$ | 1* | 1* | 3.2 | 8.1, 9.1 | Chain 3 | 3.1
$(11001)$ | 1 | 1 | 7.4, 8.4 | 8.2, 9.2 | | 3.2
$(10010)$ | 0* | 1* | 4.2, 9.3 | 6.1, 9.1 | Chain 4 | 4.1
$(10110)$ | 1* | 1 | 6.4, 9.4 | 6.2, 5.1 | | 4.2
$(10100)$ | 1* | 1* | 5.2 | 7.1, 9.1 | Chain 5 | 5.1
$(10101)$ | 1 | 1 | 7.4, 9.4 | 7.2, 9.2 | | 5.2
$(00010)$ | 0 | 0* | 6.2, 10.3 | 10.1 | Chain 6 | 6.1
$(00110)$ | 1* | 0* | 6.3, 10.4 | 7.1 | | 6.2
$(01110)$ | 1 | 1 | 6.4, 10.5 | | | 6.3
$(11110)$ | 1 | 1 | 10.6 | | | 6.4
$(00100)$ | 1* | 0* | 7.2, 10.4 | 10.1 | Chain 7 | 7.1
$(00101)$ | 1 | 0* | 7.3, 10.4 | 10.2 | | 7.2
$(01101)$ | 1 | 1* | 7.4, 10.5 | 8.2, 10.2 | | 7.3
$(11101)$ | 1 | 1 | 5.6 | | | 7.4
$(01000)$ | 0 | 1* | 8.2 | 10.1 | Chain 8 | 8.1
$(01001)$ | 1* | 1 | 8.3 | 10.2 | | 8.2
$(01011)$ | 1 | 1 | 8.4 | 10.3 | | 8.3
$(11011)$ | 1 | 1 | 10.6 | 9.3 | | 8.4
$(10000)$ | 0 | 1* | 9.2 | 10.1 | Chain 9 | 9.1
$(10001)$ | 1* | 1 | 9.3 | 10.2 | | 9.2
$(10011)$ | 1 | 1 | 9.4 | 10.3 | | 9.3
$(10111)$ | 1 | 1 | 10.6 | 10.4 | | 9.4
$(00000)$ | 0 | 0 | 10.2 | | Chain 10 | 10.1
$(00001)$ | 0* | 0 | 10.3 | | | 10.2
$(00011)$ | 1* | 0 | 10.4 | | | 10.3
$(00111)$ | 1 | 1* | 10.5 | | | 10.4
$(01111)$ | 1 | 1 | 10.6 | | | 10.5
$(11111)$ | 1 | 1 | | | | 10.6
Questions | 13 | 12 | | | |
Table 2. Examples of discovered diagnostic rules
---
Diagnosis | $f$-criteria | Value. $f$-criteria | Precision
rule | | 0.01 | 0.05 | 0.1 | on control
If $10<{\rm NUM}<20$ | NUM | 0.0029 | + | + | + | 93.3%
and ${\rm VOL}>5$ | VOL | 0.0040 | + | + | + |
then malignant | | | | | |
If ${\rm TOT}>30$ | TOT | 0.0229 | - | + | + | 100.0%
and ${\rm VOL}>5$ | VOL | 0.0124 | - | + | + |
and DEN is moderate | DEN | 0.0325 | - | + | + |
then malignant | | | | | |
If VAR is marked | VAR | 0.0044 | + | + | + | 100.0%
and $10<{\rm NUM}<20$ | NUM | 0.0039 | + | + | + |
and IRR is moderate | IRR | 0.0254 | - | + | + |
then malignant | | | | | |
If SIZE is moderate | SIZE | 0.0150 | - | + | + | 92.86%
and SHAPE is mild | SHAPE | 0.0114 | - | + | + |
and IRR is mild | IRR | 0.0878 | - | - | + |
then benign | | | | | |
Results for them are marked as Discovery1, Discovery2 and Discovery3. We
extracted 44 statistically significant diagnostic rules for 0.05 level of F
-criterion with a conditional probability no less than 0.75 (Discovery1).
There were 30 rules with a conditional probability no less than 0.85
(Discovery2) and 18 rules with a conditional probability no less than 0.95
(Discovery3). The most reliable 30 rules delivered a total accuracy of 90%,
and the 18 most reliable rules performed with 96.6% accuracy with only 3 false
positive cases (3.4%).
## X Decision rule (model) extracted from the expert through monotone Boolean
functions
We obtained Boolean expressions for function
$h\left(y_{1},y_{2},y_{3},y_{4},y_{5}\right)$ (‘shape and density of
calcification’) from the information depicted in table 1 with the following
steps:
1. -
Find all the maximal lower units for all chains as elementary conjunctions;
2. -
Take the disjunction of obtained conjunctions;
3. -
Exclude the redundant terms (conjunctions) from the end formula.
Using 1 and 3 columns we have
$x_{2}=h\left(y_{1},y_{2},y_{3},y_{4},y_{5}\right)=y_{2}y_{3}\vee
y_{2}y_{4}\vee y_{1}y_{2}\vee y_{1}y_{4}\vee y_{1}y_{3}\vee$ $\vee
y_{2}y_{3}y_{5}\vee y_{2}\vee y_{1}\vee y_{3}y_{4}y_{5}\equiv y_{2}\vee
y_{1}\vee y_{3}y_{4}y_{5}.$
Function $g\left(w_{1},w_{2},w_{3}\right)=w_{2}\vee w_{1}w_{3}$ we may obtain
by direct $2^{3}=8$ questions for the expert.
Using 1 and 2 columns we have
$f\left(\overline{x}\right)=x_{2}x_{3}\vee x_{1}x_{2}x_{4}\vee x_{1}x_{2}\vee
x_{1}x_{3}x_{4}\vee x_{1}x_{3}\vee x_{3}x_{4}\vee x_{3}$ $\vee x_{2}x_{5}\vee
x_{1}x_{5}\vee x_{4}x_{5}\equiv x_{1}x_{2}\vee
x_{3}\vee\left(x_{2}x_{1}x_{4}\right)x_{5}\equiv$ $\left(w_{2}\vee
w_{1}w_{3}\right)\left(y_{1}\vee y_{2}\vee y_{3}y_{4}y_{5}\right)\vee
x_{3}\vee$ $\vee\left(y_{1}\vee y_{2}\vee
y_{3}y_{4}y_{5}\right)\left(w_{2}\vee w_{1}w_{3}\right)x_{4}x_{5}.$
## XI Comparison of the expert model with its approximation by learning
operator
For compare rules discovered by the learning operator (Discovery system) with
the expert model we asked the expert to evaluate this rules. Below we present
some rules, discovered by Discovery system, and radiologists comments
regarding these rules as approximation of his model.
> IF TOTAL number of calcifications is more than 30, and VOLUME is more than 5
> $cm^{3}$, and DENSITY of calcifications is moderate,
> THEN Malignant.
$f$-criterion significant for $0.05$. Accuracy of diagnosis for test cases –
$100\%$. Radiologist’s comment - this rule might have promise, but I would
consider it risky.
> IF VARIATION in shape of calcifications is marked, and NUMBER of
> calcifications is between 10 and 20, and IRREGULARITY in shape of
> calcifications is moderate,
> THEN Malignant.
$f$-criterion significant for $0.05$. Accuracy of diagnosis for test cases –
$100\%$. Radiologist’s comment - I would trust this rule.
> IF variation in SIZE of calcifications is moderate, and variation in SHAPE
> of calcifications is mild, and IRREGULARITY in shape of calcifications is
> mild, THEN Benign.
$f$-criterion significant for $0.05$. Accuracy of diagnosis for test cases –
$92.86\%$. Radiologist’s comment - I would trust this rule.
## Acknowledgment
This work partially supported by the Russian Science Foundation grant
08-07-00272a and Integration projects of the Siberian Division of the Russian
Academy of science grants 47, 111, 119.
## References
* [1] Kovalerchuk B. Ya., Perlovsky L. I. _Dynamic logic of phenomena and cognition_. IJCNN, 2008, pp. 3530–3537.
* [2] Perlovsky L. I. _Toward physics of the mind: concepts, emotions, consciousness, and symbols_ // Physics of Life Reviews, 3, 2006, pp. 23–55.
* [3] Perlovsky L. I. _Neural Networks, Fuzzy Models and Dynamic Logic_ // R. Kohler and A. Mehler, eds., Aspects of Automatic Text Analysis (Festschrift in Honor of Burghard Rieger), Springer, Germany, 2007, pp. 363-386.
* [4] Vityaev E. E. The logic of prediction // Mathematical Logic in Asia 2005, Proceedings of the 9th Asian Logic Conference, eds. Goncharov S.S., Downey R. and Ono H., August 16–19, Novosibirsk, Russia, World Scientific Publisher, 2006, pp. 263–276.
* [5] Smerdov S. O., Vityaev E. E. Probability, logic & learning synthesis: formalizing prediction concept // Siberian Electronic Mathematical Reports, vol. 9, 2009, pp. 340–365., in russian, english abstract.
* [6] Vityaev E. E., Smerdov S. O. New definition of prediction without logical inference // Proceedings of the IASTED international conference on Computational Intelligence (CI 2009), ed. Kovalerchuk B. Ya., August 17 -19, Honolulu, Hawaii, USA, pp. 48–54.
* [7] Evgenii Vityaev, Principals of brain activity, supported the functional systems theory by P.K. Anokhin and emotional theory by P.V. Simonov, Neiroifformatics, v.3, N1, 2008, pp. 25-78
* [8] Akexander Demin, Evgenii Vityaev, Logical model of adaptive control system. Neiroifformatics, v.3, N1, 2008, pp. 79-107
* [9] Evgenii Vityaev. Knowledge discovery. Computational cognition. Cognitive processes modeling. Novosibirsk State University, Novosibirsk, 2006. pp.293.
* [10] Halpern J. Y. An analysis of first-order logics of probability. In: Artificial Intelligence, 46, 1990, pp. 311–350.
* [11] Shiryaev A. N. Probability. Springer, 1995.
* [12] Keisler H. J., Chang C. C. Model theory. Elsevier, 1990.
* [13] Maltsev A. I. Algebraic systems. Springer-Verlag, 1973.
* [14] Lloyd J.W. Foundations of logic programming. Springer-Verlag, 1987.
* [15] Hansel G. Sur le nombre des fonctions Boolenes monotones den variables // C. R. Acad. Sci. Paris, vol. 262, 20, 1966, pp. 1088–1090.
* [16] Kovalerchuk B. Ya., Vityaev E. E. Data mining in finance: advances in relational and hybrid methods. Kluwer Academic Publisher, 2000.
* [17] Kovalerchuk B. Ya., Vityaev E. E., Ruiz J. F. Consistent and complete data and “expert” mining in medicine // Medical data mining and knowledge discovery, Springer, 2001, pp. 238–280.
* [18] Kovalerchuk B, Talianski V. Comparison of empirical and computed fuzzy values of conjunction. Fuzzy Sets and Systems 46: 49-53, 1992.
* [19] The Probabilistic Mind. Prospects for Bayesian cognitive sciense // Eds. Nick Chater, Mike Oaksford, Oxfor University Press, 2008, pp.536
* [20] Probabilistic models of cognition // Special issue of the journal Trends in cognitive science, v.10, Issue 7, 2006, pp. 287-344
* [21] Scientific Discovery website.
http://math.nsc.ru/AP/ScientificDiscovery
|
arxiv-papers
| 2010-12-30T13:15:15 |
2024-09-04T02:49:16.074972
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Evgenii Vityaev, Boris Kovalerchuk, Leonid Perlovsky, Stanislav\n Smerdov",
"submitter": "Evgenii Vityaev",
"url": "https://arxiv.org/abs/1101.0082"
}
|
1101.0232
|
# Phase qubits fabricated with trilayer junctions
M. Weides1,2, R. C. Bialczak1, M. Lenander1, E. Lucero1, Matteo Mariantoni1,
M. Neeley1,3, A. D. O’Connell1, D. Sank1, H. Wang1,4, J. Wenner1, T.
Yamamoto1,5, Y. Yin1, A. N. Cleland1, and J. Martinis1 1Department of Physics,
University of California, Santa Barbara, CA 93106, USA 2Present address:
National Institute of Standards and Technology, Boulder, CO 80305, USA
3Present address: Lincoln Laboratory, Massachusetts Institute of Technology,
Lexington, MA 02420, USA 4Present address: Department of Physics, Zhejiang
University, Zhejiang 310027, China 5Present address: Green Innovation
Research Laboratories, NEC Corporation, Tsukuba, Ibaraki 305-8501, Japan
martin.weides@nist.gov martinis@physics.ucsb.edu ,
###### Abstract
We have developed a novel Josephson junction geometry with minimal volume of
lossy isolation dielectric, being suitable for higher quality trilayer
junctions implemented in qubits. The junctions are based on in-situ deposited
trilayers with thermal tunnel oxide, have micron-sized areas and a low subgap
current. In qubit spectroscopy only a few avoided level crossings are
observed, and the measured relaxation time of $T_{1}\approx 400\;\rm{nsec}$ is
in good agreement with the usual phase qubit decay time, indicating low loss
due to the additional isolation dielectric.
###### pacs:
74.50.+r, 85.25.Cp, 85.25.-j
## 1 Introduction
The energy relaxation time $T_{1}$ of superconducting qubits is affected by
dielectric loss, nonequilibrium quasiparticles [1], and charge or bias noise,
and varies between a few nano- to several microseconds, depending on qubit
type, material, and device layout. Superconducting qubits are commonly based
on $\mathrm{Al}$ thin films, and their central element, the non-linear
inductor given by a Josephson tunnel junction (JJ), is formed either by
overlap [2] or window-type geometries [3]. Qubit spectroscopy reveals coupling
to stochastically distributed two-level systems (TLSs) in the tunnel oxide [4,
5, 6, 7, 8] which provide a channel for qubit decoherence. While the physical
nature of TLSs is still under debate, their number was shown to decrease with
junction size and their density with higher atomic coordination number of the
tunnel oxide [3, 9]. The number of coherent oscillations in the qubit is
limited by, among other decoherence mechanisms such as nonequilibrium
quasiparticles, the _effective_ dielectric loss tangent
$\tan\delta_{\rm{eff}}$ [9]. The overlap geometry provides JJs with amorphous
barriers with no need for isolation dielectrics, being itself a source for
additional TLSs and dielectric losses. The window geometry is used for higher
quality, e.g. epitaxial, trilayer JJs with in-situ grown barriers. Besides
complex fabrication, they have the drawback of requiring additional isolation
dielectrics [10]. The importance of keeping the total dielectric volume in
qubits small to reduce the additional loss was shown in Ref. [9].
In this paper we give an overview of our standard technology for junction
fabrication, and present an alternative junction based on sputtered trilayer
stacks, which provide an intrinsically cleaner tunnel oxide and is well suited
for micron-sized trilayer qubit junctions. The so-called _side-wall passivated
JJs_ provide contact to the top electrode without adding too much lossy
dielectric to the circuitry, which would negatively affect the loss tangent.
The trilayer isolation is achieved via an electrolytic process. These novel
JJs were realized in a flux-biased phase qubit and characterized by i) current
transport measurements on reference junctions and ii) spectroscopy and time-
domain measurements of the qubit.
By systematically replacing only the Josephson junction, being central to any
superconducting qubit, we aim to analyze the loss contributions of this
specific element, and, ideally, develop low-loss Josephson junctions for
superconducting qubits and improve our qubit performance. We found performance
comparable to the current generation of overlap phase qubits.
## 2 Novel geometry
Figure 1: (Color online) Schematics of the a) overlap JJ and b) side-wall
passivated JJ, offering minimal volume of passivation region. Left (right)
part: before (after) the top-layer deposition. After the edge etch in the
trilayer stack, the side-wall oxide is grown by anodic oxidation. The trilayer
JJ has in-situ grown tunnel oxides to avoid sources of residual impurities.
Patterning of the top wiring and etching below the tunnel barrier yields the
tunnel junction.
Figure 1 depicts the patterning process for our standard overlap (a) and
trilayer junctions (b). Our standard process has an oxide layer grown on an
ion mill cleaned aluminum edge, which was previously chlorine etched. The top
wiring is then etched back below the oxide layer using argon with $\sim 10\%$
chlorine mixture. For the trilayer process, the in-situ sputtered
$\mathrm{Al}$-$\mathrm{Al}\mathrm{O}_{x}$-$\mathrm{Al}$ trilayer has a
thermally grown tunnel oxide barrier, formed for 10 min at $140\>\rm{mTorr}$
at room temperature. After deposition of the trilayer stack an edge is etched.
The bottom electrode of the trilayer stack is isolated from the top electrode
wiring by a self-aligned nanometer thin dielectric layer, grown for
$\mathrm{Al}$ (or other suitable electrode metals such as $\mathrm{Nb}$) by
anodic oxidation [11]. The metallic aluminium serves as partly submerged anode
in a liquid electrolytic mixture of $156\;\rm{g}$ ammonium pentaborate,
$1120\;\rm{ml}$ ethylene glycol and $760\;\rm{ml}$ $\mathrm{H}_{2}\mathrm{O}$
at room temperature. A gold-covered metal served as cathode and the electric
contact was made outside the electrolyte to the anode. By protecting parts of
the aluminum electrode with photoresist only a well-defined area was oxidized
by passing a constant current through the Al film and converting the metallic
surface to its oxide form. The oxide thickness can be controlled by the
voltage drop across the electrolyte. After a light ion clean and top wiring
deposition the resist is patterned to define the junction area. Finally, the
trilayer is etched below the tunnel barrier, yielding Josephson junctions with
planar tunnel barrier and isolation dielectric on just one side of the tunnel
area. For $\mathrm{Nb}$ junctions a similar patterning process, without
minimizing the dielectric loss contribution, was developed using anodic
$\mathrm{Nb}$ oxide and covered by $\mathrm{Si}\mathrm{O}_{2}$ [12]. The in-
situ grown tunnel oxide avoids sources of residual impurities such as
hydrogen, hydroxide or carbon at the interface vicinity, which may remain even
after ion-milling in our standard process. These trilayer junctions are fully
compatible with our standard process using overlap patterning and no junction
side-wall.
### 2.1 Transport
Transport measurements on a $\sim 3\>\rm{\mu m^{2}}$ reference junction at
$100\;\rm{mK}$ are shown in Fig. 2. The critical current $I_{c}$ is
$1.80\>\rm{\mu A}$, with normal resistance $R_{n}=150\rm{\Omega}$ yielding
$I_{c}R_{n}=270\rm{\mu V}$, close to the calculated Ambegaokar-Baratoff value
of $I_{c}R_{n}=298\rm{\mu V}$ for the measured superconducting gap of
$190\;\rm{\mu V}$. The back bending of the voltage close to the gap voltage is
attributed to self-heating inside the junction. The retrapping current of
$\approx 0.01\cdot I_{c}$ indicates a very small subgap current. The current
transport is consistent with tunneling, and we can exclude transport via
metallic pinholes, located in the $\sim 5\;\rm{nm}$ thin side-wall dielectric.
As a further check, the $I_{c}(T)$ dependence is as expected, see inset in
Fig. 2, with a critical temperature $T_{c}$ of $1.2\;\rm{K}$.
Figure 2: Current-voltage-characteristic at $100\;\rm{mK}$ and $I_{c}(T)$
dependence (lower inset) of a $3\;\rm{\mu m^{2}}$ side-wall passivated
trilayer junction. Top inset: dielectric circuit elements of the junction. The
tunnel oxide capacitance $C_{\rm{t}}$ is connected in parallel with the
capacitor formed by the side-wall oxide $C_{\rm{sw}}$.
## 3 Measurement
The qubit is a flux-biased phase qubit that is coupled via a tunable mutual
inductance to the readout-SQUID [13]. The total qubit capacitance $C_{total}$,
see upper inset of Fig. 3, is given by the tunnel oxide $C_{\rm{t}}$, the
anodic side-wall oxide $C_{\rm{sw}}$ and shunt capacitor $C_{\rm{s}}\approx
1250\;\rm{fF}$ dielectric, provided by a parallel plate capacitor with
relative permittivity $\epsilon^{\prime}\simeq 11.8$ made from hydrogenated
amorphous silicon (a-Si:H). The measurement process follows the standard phase
qubit characterization [14].
### 3.1 Spectroscopy
When operated as a qubit, spectroscopy over a range of more than
$2.5\;\rm{GHz}$ revealed clean qubit resonance spectra with just two avoided
level crossings of $40$–$50\;\rm{MHz}$ coupling strength (at $6.96$ and
$7.32\>\rm{GHz}$, as shown in Fig. 3). The excitation pulse length is
$1\;\rm{\mu sec}$, and the qubit linewidth is about $3\>\rm{MHz}$ in the weak
power limit. The qubit visibility, measured in a separate experiment, is about
$86\%$, which is in the range we found for our standard phase qubits.
Qualitatively, the TLS number and coupling strength per qubit is lower than in
other trilayer systems [3], that have larger tunnel areas. The TLS density per
qubit has roughly the same order of magnitude as in conventional overlap
qubits with similar tunnel area dimensions [2].
### 3.2 Relaxation
Figure 3: (Color online) 2D spectroscopy of side-wall passivated trilayer
qubit at $25\;\rm{mK}$. Two avoided level crossings due to qubit-TLS coupling
are observed at $6.96$ and $7.32\>\rm{GHz}$ (arrows). Top inset: Dielectric
circuit schematics of the qubit. Bottom inset: Qubit relaxation measurement.
Qubit relaxation measurements via $\pi$ pulse excitation and time-varied delay
before readout pulse were obtained when operated outside the avoided level
structures. We measured a relaxation time $T_{1}$ of about $400\;\rm{nsec}$,
as shown in the lower inset of Fig. 3. This relaxation time is similar to that
observed in the overlap qubits, which consistently have
$300\textrm{--}500\>\rm{nsec}$ for $\sim 2$-$4\;\rm{\mu m^{2}}$ JJ size. Apart
from the change to trilayer junctions, no modification from the previous
design was made.
## 4 Loss estimation
dielectric elements | | capacitance | loss $\tan\delta_{i}$ | $\frac{C_{i}}{C_{total}}\tan\delta_{i}$ |
---|---|---|---|---|---
| | $[\rm{fF}$] | | |
shunt capacitor a-Si:H | $C_{\rm{s}}$ | 1250 | $2\cdot 10^{-5}$ | $1.83\cdot 10^{-5}$ | [15]
anodic side-wall oxide | $C_{\rm{sw}}$ | 3.2 | $<1.6\cdot 10^{-3}$ | $<3.7\cdot 10^{-6}$ | [9]
tunnel barrier | $C_{\rm{t}}$ | 116 | $<1.6\cdot 10^{-3}$ | $<1.36\cdot 10^{-4}$ | [9]
_measured_ $\tan\delta_{m}$ | | | | $6.6\cdot 10^{-5}$ |
Table 1: Dielectric parameters for anodic oxide, shunt capacitance, and tunnel
barrier. $\tan\delta$ is given for low temperature and low power at microwave
frequencies. The capacitance for the tunnel oxide $\mathrm{Al}\mathrm{O}_{x}$
is taken and corrected for $\mathrm{Al}$ electrodes from Ref. [16] for the
dimensions given in the text. For qubits the loss tangent is calculated away
from TLS resonances, as the losses in small size anodic side-wall oxide and
tunnel barrier are smaller than the bulk value considered for the specific
loss contribution $\frac{C_{i}}{C_{total}}\tan\delta_{i}$. The measured loss
$\tan\delta_{m}$ is a factor 2-3 smaller than $\tan\delta_{\rm{eff}}$, the
weighted sum of all specific loss contributions.
We estimate the additional dielectric loss due to the sidewall oxide. The
_effective_ loss tangent of a parallel combination of capacitors is given by
$\tan{\delta_{\rm{eff}}}=\frac{\epsilon^{\prime\prime}_{\rm{eff}}}{\epsilon^{\prime}_{\rm{eff}}}=\frac{\sum\limits_{i}\epsilon^{\prime\prime}_{i}\frac{A_{i}}{d_{i}}}{\sum\limits_{i}\epsilon^{\prime}_{i}\frac{A_{i}}{d_{i}}}=\frac{\sum\limits_{i}C_{i}\tan\delta_{i}}{\sum\limits_{i}C_{i}}$
with $\epsilon^{\prime}_{i}$ and $\epsilon^{\prime\prime}_{i}$ being the real
and imaginary part of the individual permittivity for capacitor $i$ with area
$A_{i}$ and dielectric thickness $d_{i}$.
Now, we discuss the individual loss contributions for all dielectrics. We
design the qubit so that the dominant capacitance comes from the shunt
capacitor made from a-Si:H, which has a relatively low loss tangent of $2\cdot
10^{-5}$. Including the non-negligible capacitance of the tunnel junction,
this gives an effective loss tangent to the qubit of $1.83\cdot 10^{-5}$.
Because the junction capacitance is about 10% of the shunting capacitance, the
effective junction loss tangent is 10 times less than the loss tangent of the
junction oxide. We statistically avoid the effects of two-level systems by
purposely choosing to bias the devices away from the deleterious resonances.
The loss tangent of the junction is smaller than the value for bulk aluminum
oxide, approximately $1.6\cdot 10^{-3}$, and probably smaller than $5\cdot
10^{-5}$ since long energy decay times ($500\;\rm{nsec}$) have been observed
for an unshunted junction when operated away from resonances [9].
The anodic side-wall oxide contributes a small capacitance of about 3.2 fF,
which can be calculated assuming a parallel plate geometry. Here, we use the
dielectric constant $\epsilon^{\prime}=9$ for aluminum oxide, assume an area
given by $2\,\mu$m, the width of the overlap, multiplied by $0.1\mu$m the
thickness of the base layer, and estimate the thickness of the oxide $\simeq
5\,$nm as determined by the anodic process [11]. The anodic oxide is assumed
to have a bulk loss tangent of $1.6\cdot 10^{-3}$ [15], which gives a net
qubit loss contribution of $3.7\cdot 10^{-6}$, about 5 times lower than for
the a-Si:H capacitor. Note that we expect the loss from this capacitance to be
even lower because of statistical avoidance of the TLS loss [9]. The small
volume of the capacitor, equivalent to a $\sim 0.5\,\mu\textrm{m}^{2}$ volume
tunnel junction, implies that most biases do not put the qubit on resonance
with two-level systems in the anodic oxide.
## 5 Qubit lifetime and effective loss tangent
From the measured energy decay time $T_{1}=400\;\rm{nsec}$ , we determine the
loss tangent of the qubit to be $\delta_{m}=(T_{1}\;\omega_{10})^{-1}\approx
6.6\cdot 10^{-5}$, using a qubit frequency $\omega_{10}/2\pi=6\;\rm{GHz}$.
This is 3-5 times larger than our estimation of our dielectric losses, as
shown in Table 1. We believe the qubit dissipation mechanism comes from some
other energy loss sources as well, such as non-equilibrium quasiparticles [1].
## 6 Conclusion
In conclusion, we have shown that the use of a anodic oxide, self-aligned to
the junction edge, does not degrade the coherence of present phase qubits
[14]. We found performance comparable to the current generation of overlap
phase qubits.
The new junction geometry may provide a method to integrate submicron sized,
superior quality junctions (lower TLS densities) grown, for example, by MBE
epitaxy to eliminate the need for shunt dielectrics. Also, our nanometer-thin,
three dimensional-conform anodic passivation layer can be replaced by a self-
aligned isolation dielectric at the side-wall, which could be used for all
types of trilayer stacks.
Devices were made at the UCSB Nanofabrication Facility, a part of the NSF-
funded National Nanotechnology Infrastructure Network.
The authors would like to thank D. Pappas for stimulating discussions. This
work was supported by IARPA under grant W911NF-04-1-0204. M.W. acknowledges
support from AvH foundation and M.M. from an Elings Postdoctoral Fellowship.
## References
* [1] John M. Martinis, M. Ansmann, and J. Aumentado. Energy decay in superconducting josephson-junction qubits from nonequilibrium quasiparticle excitations. Phys. Rev. Lett., 103(9):097002, Aug 2009.
* [2] Matthias Steffen, M. Ansmann, R. McDermott, N. Katz, Radoslaw C. Bialczak, Erik Lucero, Matthew Neeley, E. M. Weig, A. N. Cleland, and John M. Martinis. State tomography of capacitively shunted phase qubits with high fidelity. Phys. Rev. Lett., 97(5):050502, Aug 2006.
* [3] Jeffrey S Kline, Haohua Wang, Seongshik Oh, John M Martinis, and David P Pappas. Josephson phase qubit circuit for the evaluation of advanced tunnel barrier materials. Supercond. Sci. Technol., 22(1):015004, 2009.
* [4] R. W. Simmonds, K. M. Lang, D. A. Hite, S. Nam, D. P. Pappas, and John M. Martinis. Decoherence in josephson phase qubits from junction resonators. Phys. Rev. Lett., 93(7):077003, Aug 2004.
* [5] A. Lupaşcu, P. Bertet, E. F. C. Driessen, C. J. P. M. Harmans, and J. E. Mooij. One- and two-photon spectroscopy of a flux qubit coupled to a microscopic defect. Phys. Rev. B, 80(17):172506, Nov 2009.
* [6] Jürgen Lisenfeld, Clemens Müller, Jared H. Cole, Pavel Bushev, Alexander Lukashenko, Alexander Shnirman, and Alexey V. Ustinov. Rabi spectroscopy of a qubit-fluctuator system. Phys. Rev. B, 81(10):100511, Mar 2010.
* [7] P. Bushev, C. Müller, J. Lisenfeld, J. H. Cole, A. Lukashenko, A. Shnirman, and A. V. Ustinov. Multiphoton spectroscopy of a hybrid quantum system. Phys. Rev. B, 82(13):134530, Oct 2010.
* [8] F. Deppe, M. Mariantoni, E. P. Menzel, S. Saito, K. Kakuyanagi, H. Tanaka, T. Meno, K. Semba, H. Takayanagi, and R. Gross. Phase coherent dynamics of a superconducting flux qubit with capacitive bias readout. Phys. Rev. B, 76(21):214503, Dec 2007.
* [9] John M. Martinis, K. B. Cooper, R. McDermott, Matthias Steffen, Markus Ansmann, K. D. Osborn, K. Cicak, Seongshik Oh, D. P. Pappas, R. W. Simmonds, and Clare C. Yu. Decoherence in josephson qubits from dielectric loss. Phys. Rev. Lett., 95(21):210503, Nov 2005.
* [10] A. Barone and G. Paterno. Physics and Applications of the Josephson Effect. John Wiley & Sons, 1982.
* [11] H. Kroger, L. N. Smith, and D. W. Jillie. Selective niobium anodization process for fabricating josephson tunnel-junctions. Appl. Phys. Lett., 39:280, 1981.
* [12] F. Müller, H. Schulze, R. Behr, J. Kohlmann, and J. Niemeyer. The Nb-Al technology at PTB: a common base for different types of Josephson voltage standards. Physica C, 354:66, 2001.
* [13] Matthew Neeley, M. Ansmann, Radoslaw C. Bialczak, M. Hofheinz, N. Katz, Erik Lucero, A. O’Connell, H. Wang, A. N. Cleland, and John M. Martinis. Transformed dissipation in superconducting quantum circuits. Phys. Rev. B, 77(18):180508, May 2008.
* [14] John M. Martinis. Superconducting phase qubits. Quantum Inf. Process., 8:81, 2009.
* [15] Aaron D. O’Connell, M. Ansmann, R. C. Bialczak, M. Hofheinz, N. Katz, Erik Lucero, C. McKenney, M. Neeley, H. Wang, E. M. Weig, A. N. Cleland, and J. M. Martinis. Microwave dielectric loss at single photon energies and millikelvin temperatures. Appl. Phys. Lett., 92(11):112903, 2008.
* [16] H. S. J. van der Zant, R. A. M. Receveur, T. P. Orlando, and A. W. Kleinsasser. One-dimensional parallel josephson-junction arrays as a tool for diagnostics. Appl. Phys. Lett., 65(16):2102–2104, 1994.
|
arxiv-papers
| 2010-12-31T11:54:34 |
2024-09-04T02:49:16.087858
|
{
"license": "Public Domain",
"authors": "M. Weides, R. C. Bialczak, M. Lenander, E. Lucero, Matteo Mariantoni,\n M. Neeley, A. D. O'Connell, D. Sank, H. Wang, J. Wenner, T. Yamamoto, Y. Yin,\n A. N. Cleland, and J. Martinis",
"submitter": "Martin Weides",
"url": "https://arxiv.org/abs/1101.0232"
}
|
1101.0470
|
SNSN-323-63
$B\rightarrow K^{(*)}\ell^{+}\ell^{-}$ from B-factories and Tevatron
Gerald Eigen
representing the BABAR collaboration 111Work supported by the Norwegian
Research Council.
Department of Physics and Technology
University of Bergen, 5007 Bergen, NORWAY
> BABAR and Belle measurements of branching fractions, rate asymmetries and
> angular observables in the decay modes $B\rightarrow
> K^{(*)}\ell^{+}\ell^{-}$ are reviewed and new results from CDF on
> $B\rightarrow K^{(*)}\mu^{+}\mu^{-}$ branching fractions and angular
> observables are discussed. A first search for $B^{+}\rightarrow
> K^{+}\tau^{+}\tau^{-}$ is presented.
> PRESENTED AT
>
>
>
>
> CKM workshop 2010
> Warwick, UK, September 06–10, 2010
## 1 Introduction
The decays $b\rightarrow s\ell^{+}\ell^{-}$, where $\ell^{+}\ell^{-}$ is an
$e^{+}e^{-},\mu^{+}\mu^{-}$ or $\tau^{+}\tau^{-}$ pair, are flavor-changing
neutral current (FCNC) processes, which are forbidden in the Standard Model
(SM) at tree level but are allowed to proceed via electroweak loops and weak
box diagrams. An effective Hamiltonian is used to calculate decay amplitudes
[1], which depend on three effective Wilson coefficients, $C_{7}^{eff}$,
$C_{9}^{eff}$, and $C_{10}^{eff}$. The first is extracted from the
$B\rightarrow X_{s}\gamma$ branching fraction, the latter two respectively
represent the vector and axial vector part of the weak penguin and box
diagrams. New Physics effects involve new loops that interfere with the SM
processes modifying the measured values of $C_{7}^{eff}$, $C_{9}^{eff}$, and
$C_{10}^{eff}$ with respect to the SM predictions [2]. In addition, scalar and
pseudoscalar processes may contribute that introduce new Wilson coefficients
$C_{s}$ and $C_{p}$ that are forbidden in the SM. Thus, it is important to
measure many observables in order to overconstrain the complex Wilson
coefficients [3]. These electroweak penguin modes contribute in probing New
Physics at a scale of a few TeV [4]. In this review, we focus on exclusive
decays presenting results from BABAR , Belle and CDF. The data samples are
based on luminosities of $349~{}\rm fb^{-1}$, $605~{}\rm fb^{-1}$ and
$4.4~{}\rm fb^{-1}$ corresponding to 384 million $B\overline{B}$ events, 656
Million $B\overline{B}$ events and $2\times 10^{10}~{}b\overline{b}$ events,
respectively.
## 2 Selection of $B\rightarrow K^{(*)}e^{+}e^{-}$ and $B\rightarrow
K^{(*)}\mu^{+}\mu^{-}$ Events
BABAR and Belle fully reconstruct ten $B\rightarrow K^{(*)}e^{+}e^{-}$ and
$B\rightarrow K^{(*)}\mu^{+}\mu^{-}$ final states, in which a
$K^{+},K^{0}_{S},K^{+}\pi^{-},K^{+}\pi^{0}$ or $K^{0}_{S}\pi^{+}$ recoils
against the lepton pair***Charge conjugation is implied unless otherwise
stated., while CDF reconstructs $K^{+}\mu^{+}\mu^{-}$ and
$K^{+}\pi^{-}\mu^{+}\mu^{-}$ final states. BABAR (Belle) selects lepton
candidates with momenta $p_{e}>0.3(0.4)~{}\rm GeV/c$ and
$p_{\mu}>0.7(0.7)~{}\rm GeV/c$. BABAR and Belle require good particle
identification (PID) for $e,\mu,K$, and $\pi$, and select $K^{0}_{S}$ in the
$\pi^{+}\pi^{-}$ channel. CDF requires muons with $p_{T}(\mu)>0.4~{}\rm
GeV/c$, kaons and pions with $p_{T}(K,\pi)>\rm 1~{}GeV/c$ and $B$-mesons with
$p_{T}(B)>6~{}\rm GeV/c$. Both, muons and hadrons must have good PID and the
muon pair must originate from a secondary vertex. All three experiments
suppress combinatorial $B\overline{B}$ and $q\overline{q}$ continuum
backgrounds ($q=u,d,s,c)$. Here, the leptons dominantly originate from
semileptonic $b$ and $c$ decays. BABAR trains neural networks (NN) using event
shape variables, vertex information, missing energy, and lepton separation
near the interaction region (IR) optimized in each mode and each $q^{2}$
bin†††This is the squared momentum transfer into the dilepton system.. Belle
trains a Fisher discriminant using event shape variables, missing mass, B
flavor tagging, and lepton separation in z near the IR. CDF trains NNs using
vertex information, the angle between the signed vertex displacement with
respect to the B momentum, and the $\mu$ separation. BABAR and Belle select
signal candidates using the beam-energy substituted mass
$m_{ES}=\sqrt{E^{*2}_{beam}-p^{*2}_{B}}$ and the energy difference $\Delta
E=E^{*}_{B}-E^{*}_{beam}$, where $E^{*}_{beam},E^{*}_{B}$ and $p^{*}_{B}$ are
the beam energy, B-meson energy and B-meson momentum in the $\mathchar
28935\relax(4S)$ center-of-mass frame, respectively. BABAR extracts the signal
yield from a one-dimensional unbinned extended maximum log-likelihood fit in
$m_{ES}$, while Belle performs a one (two) dimensional unbinned extended
maximum log-likelihood fit in $m_{ES}$ (and $m_{K\pi}$) for
$K^{(*)}\ell^{+}\ell^{-}$ modes. CDF selects signal candidates from an
unbinned maximum log-likelihood fit in the B invariant-mass distribution. All
experiments reject events in the $J/\psi$ and $\psi(2S)$ mass regions and
require that $K\mu$ and $K\pi\mu$ masses are not consistent with a $D$ mass to
reject background from $B\rightarrow DX$ decays. The rejected charmonium
events are used as control samples for various cross checks.
## 3 Results for $B\rightarrow K^{(*)}e^{+}e^{-}$ and $B\rightarrow
K^{(*)}\mu^{+}\mu^{-}$ Modes
Figure 1 (left) shows total branching fractions for $B\rightarrow
K^{(*)}\ell^{+}\ell^{-}$ ($e^{+}e^{-}$ and $\mu^{+}\mu^{-}$ modes combined)
[6, 7, 8] and $B\rightarrow X_{s}\ell^{+}\ell^{-}$[9, 7] in comparison to the
SM predictions [5]. The individual exclusive measurements are summarized in
Table 1. The Belle inclusive measurement is a recent update based on a
luminosity of $\rm 605~{}fb^{-1}$, yielding ${\cal B}(B\rightarrow
X_{s}\ell^{+}\ell^{-})=3.33\pm 0.8^{+0.19}_{-0.24})\times 10^{-6}$ [10]. The
partial branching fractions measured in the three experiments are also
consistent with the SM predictions.
Experiment | Mode | ${\cal B}~{}[10^{-6)}]$ | ${\cal A}_{CP}$ | ${\cal R}_{K^{(*)}}$
---|---|---|---|---
BABAR [6] | $K\ell^{+}\ell^{-}$ | $0.394^{+0.073}_{-0.069}\pm 0.02$ | $-0.18^{+0.18}_{-0.18}\pm 0.01$ | $0.96^{+0.44}_{-0.34}\pm 0.05$
BABAR [6] | $K^{*}\ell^{+}\ell^{-}$ | $1.11^{+0.19}_{-0.18}\pm 0.07$ | $-0.01^{+0.16}_{-0.15}\pm 0.01$ | $1.10^{+0.42}_{-0.32}\pm 0.07$
Belle [7] | $K\ell^{+}\ell^{-}$ | $0.48^{+0.05}_{-0.04}\pm 0.03$ | $0.04\pm 0.1\pm 0.02$ | $1.03\pm 0.19\pm 0.06$
Belle [7] | $K^{*}\ell^{+}\ell^{-}$ | $1.07^{+0.11}_{-0.10}\pm 0.09$ | $-0.10\pm 0.1\pm 0.01$ | $0.83\pm 0.17\pm 0.8$
CDF [8] | $K\mu^{+}\mu^{-}$ | $0.38^{+0.05}_{-0.05}\pm 0.03$ | |
CDF [8] | $K^{*}\mu^{+}\mu^{-}$ | $1.06^{+0.14}_{-0.14}\pm 0.09$ | |
Table 1: Branching fractions, $C\\!P$ asymmetries and lepton flavor ratios for
$B\rightarrow K^{(*)}\ell^{+}\ell^{-}$ modes in the entire $q^{2}$ region from
BABAR, Belle, and CDF. Uncertainties are statistical and systematic,
respectively.
Figure 1: (Left) Total branching fractions measurements of $B\rightarrow
K^{(*)}\ell^{+}\ell^{-}$ and $B\rightarrow X_{s}\ell^{+}\ell^{-}$ modes from
BABAR (red dots), Belle (blue triangles) and CDF (magenta squares) in
comparison to the SM prediction (grey-shaded region). For BABAR and Belle,
$\ell^{+}\ell^{-}$ is a combination of $e^{+}e^{-}$ and $\mu^{+}\mu^{-}$
modes, for CDF it is $\mu^{+}\mu^{-}$. (Right) Isospin asymmetry measurements
for $B\rightarrow K^{(*)}\ell^{+}\ell^{-}$ versus $q^{2}$ from BABAR (black
squares, blue dots) and Belle (red triangles, green triangles).
Rate asymmetries are more precisely measured than branching fractions, since
many uncertainties cancel [11]. The isospin asymmetry [12]
${\cal A}_{I}(q^{2})=\frac{d{\cal B}(B^{0}\rightarrow
K^{(*)0}\ell^{+}\ell^{-})/dq^{2}-(\tau_{B^{0}}/\tau_{B^{+}})d{\cal
B}(B^{+}\rightarrow K^{(*)+}\ell^{+}\ell^{-})/dq^{2}}{d{\cal
B}(B^{0}\rightarrow
K^{(*)0}\ell^{+}\ell^{-})/dq^{2}+(\tau_{B^{0}}/\tau_{B^{+}})d{\cal
B}(B^{+}\rightarrow K^{(*)+}\ell^{+}\ell^{-})/dq^{2}},$ (1)
corrected for the different $B^{0}$ and $B^{+}$ lifetimes
($\tau_{B^{0}}/\tau_{B^{+}}$), is expected to be small in the SM ($d{\cal
A}_{I}(q^{2})/dq^{2}$ is $-0.01$ for $q^{2}=2.7-6~{}\rm GeV^{2}/c^{4}$ after
dropping from $\simeq 0.075$ at $q^{2}=0.1~{}\rm GeV^{2}/c^{4}$ and crossing
zero near $q^{2}=1.7~{}\rm GeV^{2}/c^{4}$) [12]. Figure 1 (right) shows the
BABAR and Belle ${\cal A}_{I}$ measurements for different $q^{2}$ regions. The
$q^{2}$ integrated isospin asymmetry and ${\cal A}_{I}$ for $q^{2}$ values
above the $J/\psi$ are consistent with the SM prediction. Below the $J/\psi$,
however, BABAR observes a negative ${\cal A}_{I}$ that deviates significantly
from the SM prediction ($3.9\sigma$ from ${\cal A}_{I}=0$) . For models in
which the sign in $C^{eff}_{7}$ is flipped with respect to the value in the
SM, a small negative ${\cal A}_{I}$ is expected [12, 13], but it is too small
to explain the BABAR measurement. For low $q^{2}$, the Belle results are
consistent with both BABAR and the SM.
In the SM, the direct $C\\!P$ asymmetry
${\cal A}_{CP}=\frac{{\cal B}(\overline{B}\rightarrow
K^{(*)}\ell^{+}\ell^{-})-{\cal B}(B\rightarrow K^{(*)}\ell^{+}\ell^{-})}{{\cal
B}(\overline{B}\rightarrow K^{(*)}\ell^{+}\ell^{-})+{\cal B}(B\rightarrow
K^{(*)+}\ell^{+}\ell^{-})}.$ (2)
is expected to be ${\cal O}(10^{-3})$, and new physics at the electroweak
scale may provide significant enhancements [14]. BABAR performs a simultaneous
fit to $B^{+}\rightarrow K^{+}\ell^{+}\ell^{-}$ and $B\rightarrow
K^{*}\ell^{+}\ell^{-}$ modes. The results summarized in Table 1 together with
Belle’s measurements are consistent with the SM expectations.
In the SM, the lepton flavor ratios ${\cal R}_{K}={\cal B}(B\rightarrow
K\mu^{+}\mu^{-})/{\cal B}(B\rightarrow Ke^{+}e^{-})$ and ${\cal
R}_{K^{*}}={\cal B}(B\rightarrow K^{*}\mu^{+}\mu^{-})/{\cal B}(B\rightarrow
K^{*}e^{+}e^{-})$ integrated over all $q^{2}$ are predicted to be one and
0.75, respectively. The theoretical uncertainties are just a few percent. For
example, in two-Higgs-doublet models the presence of a SUSY Higgs might give
$\sim 10\%$ corrections to ${\cal R}_{K^{(*)}}$ for large $\tan\beta$ [13].The
BABAR and Belle measurements summarized in Table 1 are consistent with the SM
expectations.
The $B\rightarrow K^{*}\ell^{+}\ell^{-}$ angular distribution depends on three
angles: $\theta_{K}$, the angle between the K momentum and the B momentum in
the $K^{*}$ rest frame, $\theta_{\ell}$, the angle between the
$\ell^{+}(\ell^{-})$ momentum and the $B(\overline{B})$ momentum in the
$\ell^{+}\ell^{-}$ rest frame, and $\phi$, the angle between the two decay
planes. The angular distribution involves 12 $q^{2}$-dependent coefficients
$J_{i}$ [15, 16] that can be extracted from a full angular fit in individual
bins of $q^{2}$. Since large data samples are necessary for this study, BABAR
, Belle and CDF have analyzed only the one-dimensional angular distributions
$\displaystyle W(\cos\theta_{K})=\frac{3}{2}{\cal
F}_{L}\cos^{2}\theta_{K}+\frac{3}{4}(1-{\cal F}_{L})\sin^{2}\theta_{K},\ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (3) $\displaystyle
W(\cos\theta_{\ell})=\frac{3}{4}{\cal
F}_{L}\sin^{2}\theta_{\ell}+\frac{3}{8}(1-{\cal
F}_{L})(1+\cos^{2}\theta_{\ell})+{\cal A}_{FB}\cos\theta_{\ell},$ (4)
where ${\cal F}_{L}$ is the $K^{*}$ longitudinal polarization and ${\cal
A}_{FB}$ is the lepton forward-backward asymmetry. While Belle and CDF measure
${\cal F}_{L}$ and ${\cal A}_{FB}$ in six $q^{2}$ bins, BABAR measured ${\cal
F}_{L}$ and ${\cal A}_{FB}$ in two $q^{2}$ bins due to the limited data
sample. An update with the full BABAR data set in six $q^{2}$ bins is in
progress. The measured $m_{ES}$ and angular distributions are fitted with
signal, combinatorial background and peaking background components. After
determining the signal yield from the $m_{ES}$ spectrum, ${\cal F}_{L}$ is
extracted from a fit to the $\cos\theta_{K}$ distribution for fixed signal
yield. Finally, ${\cal A}_{FB}$ is extracted from the $\cos\theta_{\ell}$
distribution for fixed signal yield and fixed ${\cal F}_{L}$.
Figure 2 shows the BABAR, Belle, and CDF results for ${\cal F}_{L}$ (left) and
${\cal A}_{FB}$ (right) in comparison to the SM prediction (lower red curve)
[18] and for flipped-sign $C^{eff}_{7}$ models (upper blue curve) [20, 23]. In
the SM, ${\cal A}_{FB}$ is negative for small $q^{2}$, crosses zero at
$q^{2}_{0}=(4.2\pm 0.6)~{}\rm GeV^{2}/c^{4}$ and is positive for large
$q^{2}$, while for flipped-sign $C^{eff}_{7}$ models ${\cal A}_{FB}$ is
positive for all $q^{2}$. Table 2 summarized the ${\cal F}_{L}$ and ${\cal
A}_{FB}$ measurements from $B\rightarrow K^{*}\ell^{+}\ell^{-}$ in the low
$q^{2}$ region in comparison to the SM prediction. For ${\cal F}_{L}$, the
three measurements are consistent with each other and the SM prediction. For
${\cal A}_{FB}$, the three measurements are in good agreement. Though they are
in better agreement with the flipped-sign $C^{eff}_{7}$ model, they are
consistent with the SM prediction. For $B\rightarrow K\ell^{+}\ell^{-}$,
${\cal A}_{FB}$ is consistent with zero as expected in the SM.
Experiment | $q^{2}$ bin $\rm[GeV^{2}/c^{4}]$ | ${\cal F}_{L}$ | ${\cal A}_{FB}$
---|---|---|---
BABAR [17] | 0.1-6.25 | $0.35\pm 0.16\pm 0.04$ | $0.24^{+0.18}_{-0.23}\pm 0.05$
Belle [7] | 1-6 | $0.67\pm 0.23\pm 0.04$ | $0.26^{+0.27}_{-0.30}\pm 0.07$
CDF [8] | 1-6 | $0.5^{+0.27}_{-0.30}\pm 0.04$ | $0.43^{+0.36}_{-0.37}\pm 0.06$
SM [24] | 1-6 | $0.73^{+0.13}_{-0.23}$ | $-0.05^{+0.03}_{-0.04}$
Table 2: BABAR, Belle, and CDF measurements of ${\cal F}_{L}$ and ${\cal
A}_{FB}$ from $B\rightarrow K^{*}\ell^{+}\ell^{-}$ modes in the low $q^{2}$
region.
Figure 2: (left) Measurements of ${\cal F}_{L}$ and (right) Measurements of
${\cal A}_{FB}$ in $B\rightarrow K^{(*)}\ell^{+}\ell^{-}$ modes by BABAR
(black squares), Belle ( brown dots) and CDF (green triangles). The SM
prediction (flipped-sign $C^{eff}_{7}$ model) is shown by the upper red (lower
blue) curve for ${\cal F}_{L}$ and the lower red (upper blue) curve for ${\cal
A}_{FB}$.
## 4 Search for $B^{+}\rightarrow K^{+}\tau^{+}\tau^{-}$
In the SM, the $q^{2}$ dependence of the $B\rightarrow X_{s}\tau^{+}\tau^{-}$
decay rate has a shape similar to that of $B\rightarrow X_{s}\mu^{+}\mu^{-}$
in the high $q^{2}$ region. The $B^{+}\rightarrow K^{+}\tau^{+}\tau^{-}$
branching fraction is predicted to be $\sim 2\times 10^{-7}$ in the SM, which
is $50-60\%$ of the total inclusive branching fraction [21]. Enhancements are
predicted in models beyond the SM. In the next-to-minimal supersymmetric
models (NMSSM), for example, the rate may be enhanced by the squared tau-to-
muon mass ratio $(m_{\tau}/m_{\mu})^{2}\sim 280$. Since signal final states
contain 2-4$\nu$, a different analysis strategy is needed here to control
backgrounds.
BABAR has performed the first search for $B^{+}\rightarrow
K^{+}\tau^{+}\tau^{-}$ using an integrated luminosity of $423~{}\rm fb^{-1}$
which corresponds to 465 $B\overline{B}$ events. The recoiling (”tag”) $B$ is
reconstructed in many hadronic final states, $B^{-}\rightarrow D^{(*)0,+}X$,
where $X$ represents up to six hadrons
($\pi^{\pm},\pi^{0},K^{\pm},K^{0}_{S}$). Using $m_{ES}$ and $\Delta E$ the tag
is selected with an efficiency of $\sim 0.2\%$. The single-prong $\tau$ decays
$\tau\rightarrow e\nu\overline{\nu},\tau\rightarrow\mu\nu\overline{\nu}$ and
$\tau\rightarrow\pi\nu$ are selected as signal modes. Thus, signal candidates
are required to have only three charged particles of which one is an
identified kaon with charge opposite to the tag $B$ and $0.44<p_{K}<1.4~{}\rm
GeV/c$ in the center-of-mass frame. The two remaining particles must have
opposite charge, be consistent with the signal $\tau$ decays, have
$p<1.59~{}\rm GeV/c$ and a mass $M_{pair}<2.89~{}\rm GeV/c^{2}$. Further
requirements are $q^{2}=(\vec{p}_{\mathchar
28935\relax(4S)}-\vec{p}_{tag}-\vec{p}_{K})^{2}/c^{2}>14.23~{}\rm
GeV^{2}/c^{4}$, a missing energy ($i.e.$ the energy carried off by neutrinos
estimated as the difference between $\mathchar 28935\relax(4S)$ energy and
that of all observed particles) of $1.39<E_{miss}<3.38~{}\rm GeV$, and neutral
energy deposited in the electromagnetic calorimeter $E_{extra}<0.74~{}\rm
GeV$. Continuum background is suppressed by $|\cos\theta_{T}|<0.8$, where
$\theta_{T}$ is the opening angle between the thrust axis of the tag and that
of the rest of the event. The largest remaining background originates from
$B^{+}\rightarrow D^{0}X^{+}$, which is suppressed by combining the signal
$K^{+}$ with the $\tau$ daughter of opposite charge assigned the $\pi$ mass
hypothesis and requiring a mass $M_{K\pi}>1.96~{}\rm GeV/c^{2}$.
BABAR observes 47 events with an expected background of $64.7\pm 7.3$ events.
Including systematic uncertainties a branching fraction upper limit of ${\cal
B}(B\rightarrow K^{+}\tau^{+}\tau^{-})<3.3\times 10^{-3}$ is set at $90\%$
confidence level (CL).
## 5 Conclusion
BABAR and Belle have measured branching fractions, rate asymmetries and
angular observables in $B\rightarrow K^{(*)}\ell^{+}\ell^{-}$ final states.
Recently, CDF contributed new measurements on branching fractions and angular
observables in $B\rightarrow K^{(*)}\mu^{+}\mu^{-}$. Except for the isospin
asymmetry at low values of $q^{2}$ all other measurements are consistent with
the SM, though ${\cal F}_{L}$ and ${\cal A}_{FB}$ agree also with the flipped-
sign $C^{eff}_{7}$ model. BABAR has performed the first search for
$B^{+}\rightarrow K^{+}\tau^{+}\tau^{-}$ setting a branching fraction upper
limit of ${\cal B}(B^{+}\rightarrow K^{+}\tau^{+}\tau^{-})<3.3\times 10^{-3}$
at $90\%~{}CL$. Although all experiments are expected to update results with
the final data sets, significant improvement in precision will come from LHCb
and the Super B-factories. In these new experiments, sufficiently large data
samples will be collected to measure the full angular distribution from which
the 12 observables $J_{i}$ [15] can be measured with high precision in
different bins of $q^{2}$. In turn, the Wilson coefficients can be determined
with high precision to reveal small discrepancies with respect to the SM
predictions [3, 23].
ACKNOWLEDGEMENTS
I would like to thank my BABAR colleague K. Flood for useful discussions. This
work has been supported by the Norwegian Research Council.
## References
* [1] G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996); C. Bobeth, M. Misiak and J. Urban, Nucl. Phys. B574, 291 (2000); H.H Asatryan $et~{}al.$, Phys. Rev. D65, 034009 (2002); Phys. Lett. B507, 162, (2001); G. Hiller and F.Krüger, Phys.Rev. D69, 074020 (2004); M. Beneke, Th. Feldmann, and D. Seidel; Nucl. Phys.B612, 25 (2001);M. Beneke, Th. Feldmann, and D. Seidel; Eur.Phys.J. C41, 173 (2005).
* [2] G. Burdman, Phys. Rev. D52, 6400 (1995); J. L. Hewett and J. D. Wells, Phys. Rev. D55, 5549 (1997); W. J. Li, Y. B. Dai and C. S. Huang, Eur. Phys. J. C40, 565 (2005); Y. G. Xu, R. M. Wang and Y. D. Yang, Phys. Rev. D74, 114019 (2006); P. Colangelo et al., Phys. Rev. D73, 115006 (2006); C.-H. Chen and C.Q. Geng, Phys. Rev. D 66 094018 (2002);C. Bobeth et al, Phys. Rev. D64 074014 (2001).
* [3] K.S.M. Lee et al., Phys. Rev. D75, 034016 (2007).
* [4] G. Isidori, Y. Nir, G. Prerez, arXiv:1002.0900 (2010).
* [5] A. Ali, E. Lunghi, C. Greub and G. Hiller, Phys. Rev. D 66, 034002 (2002).
* [6] B. Aubert et al. (BABAR collaboration), Phys. Rev. Lett.102, 091803 (2009).
* [7] J.T. Wei et al. (Belle collaboration), Phys. Rev. Lett.103, 171801 (2009).
* [8] T. Aaltonen et al. (CDF collaboration), CDF note 10047 (2010).
* [9] B. Aubert et al. (BABAR collaboration), Phys. Rev. Lett.93, 081862 (2004).
* [10] C.C.Chiang (Belle collaboration), talk at ICHEP10 (2010).
* [11] F. Krüger, L. M. Sehgal, N. Sinha and R. Sinha, Phys. Rev. D61, 114028 (2000), [Erratum-ibid. D63, 019901 (2001)].
* [12] T. Feldmann and J. Matias, JHEP 0301, 074 (2003).
* [13] Q. S. Yan, C. S. Huang, W. Liao and S. H. Zhu, Phys. Rev. D 62, 094023 (2000).
* [14] C. Bobeth, G. Hiller and G. Piranishvili, JHEP 0807, 106 (2008).
* [15] F. Krüger et al., Phys. Rev. D61, 114028 (2000); Erratum-ibid D63, 019901 (2001).
* [16] C.S. Kim et al., Phys. Rev. D 62, 034013 (2000).
* [17] B. Aubert et al. (BABAR collaboration), Phys. Rev. D79, 031102 (2009).
* [18] G. Buchalla $et~{}al.$, Phys. Rev. D63, 014015 (2001).
* [19] G. Buchalla $et~{}al.$, Phys. Rev. D63, 014015 (2001).
* [20] A. Hovhannisyan, W. S. Hou and N. Mahajan, Phys. Rev. D 77, 014016 (2008).
* [21] J.L. Hewett, Phys. Rev. D53, 4964 (1995).
* [22] K. Flood, talk at the Int. Conf. on HEP, Paris July 22-28 (2010).
* [23] F. Krüger and J. Matias, Phys. Rev. D71, 094009 (2005).
* [24] C. Bobeth, G. Hiller and D. van Dyk, JHEP 1007, 098 (2010).
|
arxiv-papers
| 2011-01-03T08:44:20 |
2024-09-04T02:49:16.098438
|
{
"license": "Public Domain",
"authors": "Gerald Eigen",
"submitter": "Gerald Eigen",
"url": "https://arxiv.org/abs/1101.0470"
}
|
1101.0649
|
aainstitutetext: Department of Physics, Kobe University,
1-1 Rokkodai-Cho, Nada-Ku, Kobe 657-8501, JAPAN
# Higgs production and decay processes via loop diagrams in various 6D
Universal Extra Dimension Models at LHC
Kenji Nishiwaki nishiwaki@stu.kobe-u.ac.jp
###### Abstract
We calculate loop-induced Higgs production and decay processes which are
relevant for the LHC in various six-dimensional Universal Extra Dimension
models. More concretely, we focus on the Higgs production through gluon fusion
and the Higgs decay into two photons induced by loop diagrams. They are one-
loop leading processes and the contribution of Kaluza-Klein particles is
considered to be significant. These processes are divergent in six dimensions.
Therefore, we employ a momentum cutoff, whose size is fixed from the validity
of perturbative calculation through naive dimensional analysis. In these six-
dimensional Universal Extra Dimension models, the Higgs production cross
section through gluon fusion is highly enhanced and the Higgs decay width into
two photons is suppressed. In particular in the case of the compactification
on Projective Sphere, these effects are remarkable. The deviation of the
$h^{(0)}\rightarrow 2\gamma$ signal from the prediction of the Standard model
is much greater than that in the case of the five-dimensional minimal UED
model. We also consider threshold corrections in the two processes and these
effect are noteworthy even when we take a higher cutoff and/or a heavy KK
scale. Comparing our calculation to the recent LHC results which were
published at the Lepton-Photon 2011 and at the December of 2011 is performed
briefly.
###### Keywords:
Universal Extra Dimension model, Collider Physics
††arxiv: 1101.0649 [hep-ph]††dedication: KOBE-TH-10-04
## 1 Introduction
After a long shutdown, the LHC (Large Hadron Collider) restarted and new era
of particle physics comes. Stimulated by the advent of two renowned works
ArkaniHamed:1998rs ; Randall:1999ee , phenomenology in extra dimension has
been well studied. Universal Extra Dimension (UED) is one of the interesting
possibility along this direction and has been studied very well. In this
model, all the fields describing particles of the Standard Model (SM)
propagate in the bulk space.111 This possibility is first considered within
string theory context Antoniadis:1990ew . The minimal UED (mUED) model is
constructed with one extra spacial dimension of orbifold $S^{1}/Z_{2}$
Appelquist:2000nn . This orbifold imposes the identification between the extra
spacial coordinate $y$ and $-y$ and there are two fixed points at $y=0,\pi R$,
where $R$ is the radius of $S^{1}$. Due to this identification four-
dimensional (4D) chiral fermions describing the SM fermions appear. One of the
interesting points of UED model is that the constraints from the current
experiments are very loose. The Kaluza-Klein (KK) mass scale
${M_{\text{KK}}}$, which is defined by the inverse of the compactification
radius $R$, is constrained Appelquist:2000nn ; Agashe:2001ra ; Agashe:2001xt ;
Appelquist:2001jz ; Appelquist:2002wb ; Oliver:2002up ; Chakraverty:2002qk ;
Buras:2002ej ; Colangelo:2006vm ; Gogoladze:2006br in the mUED case. In UED
model, the zero mode profile takes constant value and the overlap integral
between zero modes and KK modes does not generate large deviation from the SM
result. Therefore we can take the lower KK mass scale than in the other types
of extra dimensional models. In addition, the existence of dark matter
candidate is naturally explained by the KK parity, which is the remnant of the
translational invariance along the extra spacial direction. The particle
cosmology in the five-dimensional (5D) UED models has been studied strenuously
Cheng:2002ej ; Servant:2002aq ; Kakizaki:2005en ; Matsumoto:2005uh ;
Burnell:2005hm ; Kakizaki:2006dz ; Kong:2005hn ; Matsumoto:2007dp ;
Kakizaki:2005uy ; Belanger:2010yx ; Hisano:2010yh . The collider signature of
the 5D UED models is similar to the one of the supersymmetric theory with
neutralino dark matter Cheng:2002ab . The discrimination between these models
is also well studied Datta:2005zs ; Matsumoto:2009tb .
And another thing, UED models with two spacial dimensions have been studied
energetically. Six-dimensional (6D) UED models have remarkable theoretical
properties, for example, prediction of the number of matter generations
imposed by the condition of (global) anomaly cancellation Dobrescu:2001ae ,
ensuring proton stability Appelquist:2001mj , generating electroweak symmetry
breaking ArkaniHamed:2000hv ; Hashimoto:2003ve ; Hashimoto:2004xz . These
topics drive us into considering such a class of models. In phenomenological
point of view, there are also interesting aspects in 6D UED model. In 6D case,
the KK mass spectrum is not equally-spaced, up to radiative corrections
Cheng:2002iz ; Ponton:2005kx . And a 4D new scalar particle named “spinless
adjoint” emerges in the model corresponding to a 6D gauge boson. These are un-
eaten physical scalars associated to the 4D vector components of the 6D gauge
bosons. These two points exert considerable influence on collider physics and
particle cosmology Burdman:2006gy ; Dobrescu:2007xf ; Dobrescu:2007ec ;
Freitas:2007rh ; Freitas:2008vh ; Ghosh:2008dp ; Bertone:2009cb ;
Blennow:2009ag . These studies are executed on the 6D UED model based on two-
torus $T^{2}$ Dobrescu:2004zi ; Burdman:2005sr ; Cacciapaglia:2009pa . It is
noted that recently the UED models based on two-sphere $S^{2}$ are proposed
and these models have interesting properties Maru:2009wu ; Dohi:2010vc . In
the $S^{2}$-based models, the KK mass spectrum is totally different from that
of the ordinary $T^{2}$-based models and we consider that this difference
would have an impact on collider and cosmological phenomenology.222In 5D case,
there are also many approaches of considering the non-minimal UED models
Flacke:2008ne ; Park:2009cs ; Csaki:2010az ; Haba:2009uu ; Haba:2009pb ;
Haba:2009wa ; Haba:2010xz ; Nishiwaki:2010te .
In this paper, we focus on the Higgs boson production and decay sequences
through one-loop leading processes expected to occur at the LHC. In one-loop
leading processes, the contribution of KK particles is considered to be
significant. More concretely, we consider the Higgs production by gluon fusion
and the Higgs decay to two photons. The former process is very important
because it is the dominant Higgs production process at the LHC. The latter
process becomes important in the case when Higgs boson mass is about from
$120$ GeV to $150$ GeV. Actually, the ATLAS and CMS experiments at the CERN
LHC have presented their latest results for the $\simeq 2\,\text{fb}^{-1}$ of
data at the center of mass energy 7 TeV at the Lepton-Photon 2011, Mumbai,
India, 22-27 August 2011 and the Higgs decay to two photons process plays a
significant role at this range ATLAS-CONF-2011-135 ; CMS_PAS_HIG-11-022 .333
During revising this paper, both the ATLAS and CMS have published the new
results, which claim that there is a peak around 125 GeV ATLAS:2012ae ;
Chatrchyan:2012tx . In the SM, the branching ratio of the decay into two
photons is too small, but the signal of this process is very clear at the LHC
experiments. Using the result of the above two processes, we can perform a
crude estimation of the difference of the number of the decay events to two
photons from the SM expectation value. By naive power counting argument, the
production and the decay processes are known to be divergent logarithmically.
We adopt the regularization scheme by use of KK momentum cutoff, which is
determined by naive dimensional analysis. We also consider threshold
corrections in the two processes and these effect are noteworthy, especially
when we choose low cutoff scale in 6D UED.
As the end of the introduction, we show the organization of this paper
briefly. In Section 2, we give a brief review of 6D UED model on
$T^{2}/Z_{4}$, which is one of the $T^{2}$-based 6D UED model and has been
studied well. In Section 3, we calculate the rate of the Higgs production
process through gluon fusion and the Higgs decay process to two photons via
loop diagrams in the 6D UED model on $T^{2}/Z_{4}$. These results can be
applied for the $S^{2}$-based 6D UED cases with some modifications. In Section
4, we get an overview of gauge theory on $S^{2}$ and give a brief review of
the two types of $S^{2}$-based 6D UED models. In Section 5, we estimate the
maximal cutoff scale, where the validity of perturbation will break down. In
Section 6, we estimate the deviation of the rate of the Higgs production and
decay processes and evaluate the difference of the event number from the SM
results with/without threshold corrections. Section 7 is devoted to summary
and discussions.
## 2 Universal Extra Dimension on $T^{2}/Z_{4}$
We give a brief review of UED model on $T^{2}/Z_{4}$. A detailed construction
of the minimal 5D UED based on $S^{1}/Z_{2}$ is studied in Appelquist:2000nn .
We consider a gauge theory on six-dimensional spacetime $M^{4}\times
T^{2}/Z_{4}$, which is a direct product of the four-dimensional Minkowski
spacetime $M^{4}$ and two-torus $T^{2}$ with $Z_{4}$ orbifolding. We use the
coordinate of six-dimensional spacetime defined by $x^{M}=(x^{\mu},y,z)$ and
the mostly-minus metric convention
$\eta_{MN}=\text{diag}(1,-1,-1,-1,-1,-1)$.444Latin indices ($M,N$) run for
$0,1,2,3,y,z$ and Greek indices ($\mu,\nu$) run for $0,1,2,3$. The
representation of Clifford algebra which we adopt is
$\Gamma^{\mu}=\gamma^{\mu}\otimes I_{2}=\begin{bmatrix}\gamma^{\mu}&0\\\
0&\gamma^{\mu}\end{bmatrix}\ ,\ \Gamma^{y}=\gamma^{5}\otimes
i\sigma_{1}=\begin{bmatrix}0&i\gamma^{5}\\\ i\gamma^{5}&0\end{bmatrix}\ ,\
\Gamma^{z}=\gamma^{5}\otimes i\sigma_{2}=\begin{bmatrix}0&\gamma^{5}\\\
-\gamma^{5}&0\end{bmatrix},$ (1)
where $\gamma^{5}$ is 4D chirality operator and $\sigma_{i}\ (i=1,2,3)$ are
Pauli matrices. To obtain 4D Weyl fermion from 6D Weyl fermion, we choose the
type of orbifold as $Z_{4}$, not as $Z_{2}$ in 5D case Dobrescu:2004zi ;
Burdman:2005sr . $Z_{4}$ symmetry is realized as the rotation on the $y-z$
plane by an angle $\frac{\pi}{2}$ on $T^{2}$. This means a bulk scalar field
$\Phi(x;y,z)$ obeys the following relation:
$\Phi_{t}(x,-z,y)=t\Phi_{t}(x,y,z).$ (2)
$t$ is $Z_{4}$ parity which takes the possible values $t=\pm 1,\pm i$ and all
the fields are classified according to their parity. Following the general
prescription Georgi:2000ks , mode functions of $T^{2}/Z_{4}$
$f^{(m,n)}_{t}(y,z)$ are obtained as follows:555For simplicity, we drop the
overall $-i$ factor for $t=\pm i$ cases.
$\displaystyle
f^{(m,n)}_{t}(y,z)=\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{2\pi
R}\frac{1}{\sqrt{1+3\delta_{m,0}\delta_{n,0}}}\Big{[}\cos\left(\frac{my+nz}{R}\right)+\cos\left(\frac{ny-
mz}{R}\right)\Big{]}&\text{for}\ t=1,\\\ \displaystyle\frac{1}{2\pi
R}\Big{[}\cos\left(\frac{my+nz}{R}\right)-\cos\left(\frac{ny-
mz}{R}\right)\Big{]}&\text{for}\ t=-1,\\\ \displaystyle\frac{1}{2\pi
R}\Big{[}\sin\left(\frac{my+nz}{R}\right)-i\sin\left(\frac{ny-
mz}{R}\right)\Big{]}&\text{for}\ t=i,\\\ \displaystyle\frac{1}{2\pi
R}\Big{[}\sin\left(\frac{my+nz}{R}\right)+i\sin\left(\frac{ny-
mz}{R}\right)\Big{]}&\text{for}\ t=-i,\end{array}\right.$ (3)
where $m$ and $n$ are $y$ and $z$ directional KK numbers, respectively and
take the values $m\geq 1,n\geq 0$ or $m=n=0$ (only for $t=1$).666The complex
factor $i$ in $f_{t=\pm i}^{(m,n)}$ generates CP violating interactions after
KK expansion in KK sector Lim:2009pj .
And realizing cancellation of 6D gravitational and SU(2)L global anomalies
requires the choice of 6D chiralities, for example, as follows Dobrescu:2001ae
:
$(\mathcal{Q}_{-},\mathcal{U}_{+},\mathcal{D}_{+},\mathcal{L}_{-},\mathcal{E}_{+},\mathcal{N}_{+}),$
(4)
whose zero modes form single generation of the standard model;
$\mathcal{Q}_{-}^{(0)}=(u,d)_{L},\ \mathcal{U}_{+}^{(0)}=u_{R},\
\mathcal{D}_{+}^{(0)}=d_{R},\ \mathcal{L}_{-}^{(0)}=(\nu,l)_{L},\
\mathcal{E}_{+}^{(0)}=l_{R},\ \mathcal{N}_{+}^{(0)}=\nu_{R}$. The $\pm$
suffixes represent 6D chirality of each field and 6D chirality operator is
defined as
$\Gamma_{7}=\gamma^{5}\otimes\sigma_{3}.$ (5)
Using 6D chiral projective operator
$\Gamma_{\pm}\equiv\frac{1}{2}(1\pm\Gamma_{7})$, 6D Weyl fermions $\Psi_{\pm}$
are described as follows;
$\Psi_{+}=\begin{pmatrix}\psi_{+R}\\\
\psi_{+L}\end{pmatrix},\quad\Psi_{-}=\begin{pmatrix}\psi_{-L}\\\
\psi_{-R}\end{pmatrix},$ (6)
where $\psi_{L(R)}$ is a left(right)- handed 4D Weyl fermion. We can take the
boundary condition of 6D fermion $\Psi_{6}=(\psi,\Psi)^{\mathrm{T}}$
($\mathrm{T}:$ transpose) as in Scrucca:2003ut :
$\displaystyle\Psi_{6}(x,-z,y)$
$\displaystyle=(i)^{\frac{1}{2}+r}\left(\frac{1+\Gamma^{y}\Gamma^{z}}{\sqrt{2}}\right)P\Psi_{6}(x,y,z)$
$\displaystyle{\Longleftrightarrow}\begin{pmatrix}\psi\\\
\Psi\end{pmatrix}(x,-z,y)$ $\displaystyle=\begin{pmatrix}i^{r}&0\\\
0&i^{r+1}\end{pmatrix}P\begin{pmatrix}\psi\\\ \Psi\end{pmatrix}(x,y,z).$ (7)
$r$ is $Z_{4}$ twist factor which can takes the values ($r=0,1,2,3$) and $P$
is group twist matrix for fundamental representation with the $Z_{4}$
identification $(P^{4}=1)$, which we discuss soon later. When we choose the
values of $r$ as $0$ or $3$, zero mode sectors of $\Psi_{\pm}$ become 4D
chiral.
Next, we go on to the gauge sector. The boundary condition of this part is as
in Scrucca:2003ut :
$A_{\mu}(x,i\omega)=PA_{\mu}(x,\omega)P^{-1},\quad
A_{\omega}(x,i\omega)=(-i)PA_{\omega}(x,\omega)P^{-1}.$ (8)
Here we define a complexified coordinate and a vector field component for
clarity as
$\omega\equiv\frac{y+iz}{\sqrt{2}},\quad
A_{\omega}\equiv\frac{A_{y}-iA_{z}}{\sqrt{2}}.$ (9)
In UED model, we do not break the gauge symmetry by boundary condition. Then
the matrix $P$ is selected as $P=\mathbf{1}$. This means that none of the
fields belonging to $A_{\omega}$ (or $A_{\bar{\omega}}$) takes zero mode,
which is an would-be exotic SM particle. Finally we discuss the 6D scalar
$\Phi$. The boundary condition for this field is very simple:
$\Phi(x,i\omega)=P\Phi(x,\omega).$ (10)
Choosing $P=\mathbf{1}$, $\Phi$’s zero mode remains and can be identified as
the SM Higgs field. From above discussion, we can form the zero mode sector
just as the SM one.
We write down the part of the 6D UED Lagrangian which is requisite for our
calculation. The 6D action takes the form as follows:
$\displaystyle S$ $\displaystyle=\int_{0}^{2\pi R}dy{\int_{0}^{2\pi R}}dz\int
d^{4}x\Bigg{\\{}-\frac{1}{2}\sum_{i=1}^{3}\mathrm{Tr}\big{[}F_{MN}^{(i)}F^{(i)MN}\big{]}$
$\displaystyle\qquad{+(D^{M}H)^{\dagger}(D_{M}H)+\bigg{[}\mu^{2}|H|^{2}-\frac{\lambda_{6}^{(H)}}{4}|H|^{4}\bigg{]}}$
$\displaystyle\qquad+i\bar{\mathcal{Q}}_{3-}\Gamma^{M}D_{M}\mathcal{Q}_{3-}+i\bar{\mathcal{U}}_{3+}\Gamma^{M}D_{M}\mathcal{U}_{3+}-\bigg{[}\lambda_{6}^{(t)}\bar{\mathcal{Q}}_{3-}(i\sigma_{2}H^{\ast})\mathcal{U}_{3+}+\mathrm{h.c.}\bigg{]}\Bigg{\\}}.$
(11)
$F^{(i)}_{MN}$ are the field strengths of gauge fields, where
$F^{(i)}_{MN}=\partial_{M}A_{N}^{(i)}-\partial_{N}A_{M}^{(i)}-ig_{6}^{(i)}[A_{M}^{(i)},A_{N}^{(i)}]$,
and the gauge groups are those for $U(1)_{Y}\ (i=1),SU(2)_{L}\ (i=2)$ and
$SU(3)_{C}\ (i=3)$ in the SM. The covariant derivatives $D_{M}$ are expressed
in our convention as
$D_{M}=\partial_{M}-i\sum_{i=1}^{3}g_{6}^{(i)}T^{(i)a}A^{(i)a}_{M},$ (12)
where $g_{6}^{(i)}$ are the six-dimensional gauge couplings and $T^{(i)a}$ are
the group generators of each corresponding gauge group. $H$ is the Higgs
doublet, and $\mu$, $\lambda_{6}^{(H)}$ and $\lambda_{6}^{(t)}$ are the usual
Higgs mass, Higgs self coupling and Yukawa coupling of the top quark in 6D
theory, respectively.777All the six-dimensional couplings are dimensionful.
After the KK expansion, corresponding 4D couplings become dimensionless as
they should be so. $\mathcal{Q}_{3-}$ is the quark doublet in third generation
and $\mathcal{U}_{3+}$ is the top quark singlet.
We are ready to derive the four-dimensional effective action by expanding all
the 6D fields by use of Eq. (3). The concrete forms of KK expansion are as
follows:
$\displaystyle A^{(i)}_{\mu}(x;y,z)$ $\displaystyle=\frac{1}{2\pi
R}\bigg{\\{}A^{(i)(0)}_{\mu}(x)$ $\displaystyle\qquad{+{\sum_{m\geq 1,n\geq
0}}A_{\mu}^{(i)(m,n)}(x)\Big{[}\cos\left(\frac{my+nz}{R}\right)+\cos\left(\frac{ny-
mz}{R}\right)\Big{]}\bigg{\\}}},$ (13) $\displaystyle A^{(i)}_{\omega}(x;y,z)$
$\displaystyle=\frac{1}{2\pi R}\bigg{\\{}{\sum_{m\geq 1,n\geq
0}}A_{\omega}^{(i)(m,n)}(x)\Big{[}\sin\left(\frac{my+nz}{R}\right)+i\sin\left(\frac{ny-
mz}{R}\right)\Big{]}\bigg{\\}},$ (14) $\displaystyle H(x;y,z)$
$\displaystyle=\frac{1}{2\pi R}\bigg{\\{}H^{(0)}(x)+{\sum_{m\geq 1,n\geq
0}}H^{(m,n)}(x)\Big{[}\cos\left(\frac{my+nz}{R}\right)+\cos\left(\frac{ny-
mz}{R}\right)\Big{]}\bigg{\\}},$ (15) $\displaystyle\mathcal{Q}_{3-}(x;y,z)$
$\displaystyle=\frac{1}{2\pi R}\begin{pmatrix}\displaystyle
Q_{3L}^{(0)}(x)+{\sum_{m\geq 1,n\geq
0}}Q_{3L}^{(m,n)}\Big{[}\cos\left(\frac{my+nz}{R}\right)+\cos\left(\frac{ny-
mz}{R}\right)\Big{]}\\\ \qquad\displaystyle{\sum_{m\geq 1,n\geq
0}}Q_{3R}^{(m,n)}\Big{[}\sin\left(\frac{my+nz}{R}\right)-i\sin\left(\frac{ny-
mz}{R}\right)\Big{]}\end{pmatrix},$ (16)
$\displaystyle\mathcal{U}_{3+}(x;y,z)$ $\displaystyle=\frac{1}{2\pi
R}\begin{pmatrix}\displaystyle t_{R}^{(0)}(x)+{\sum_{m\geq 1,n\geq
0}}t_{R}^{(m,n)}\Big{[}\cos\left(\frac{my+nz}{R}\right)+\cos\left(\frac{ny-
mz}{R}\right)\Big{]}\\\ \qquad\displaystyle{\sum_{m\geq 1,n\geq
0}}t_{L}^{(m,n)}\Big{[}\sin\left(\frac{my+nz}{R}\right)-i\sin\left(\frac{ny-
mz}{R}\right)\Big{]}\end{pmatrix}.$ (17)
In the fermionic part, we choose all the twist factors as $r=0$. Now we can
find the SM fields
$A^{(0)(i)}_{\mu},H^{(0)},Q^{(0)}_{3L}\big{(}=\big{(}t^{(0)}_{L},b^{(0)}_{L}\big{)}^{\mathrm{T}}\big{)}$
and $t^{(0)}_{R}$ in the zero mode sectors. Here we focus on the 5D Higgs
doublet in terms of 4D component fields:
$H^{(0)}=\begin{pmatrix}\phi^{+(0)}\\\
\frac{1}{\sqrt{2}}\big{(}v+h^{(0)}+i\chi^{(0)}\big{)}\end{pmatrix},\quad
H^{(m,n)}=\begin{pmatrix}\phi^{+(m,n)}\\\
\frac{1}{\sqrt{2}}\big{(}h^{(m,n)}+i\chi^{(m,n)}\big{)}\end{pmatrix}.$ (18)
At the zero mode part, $v$ and $h^{(0)}$ are the ordinary four-dimensional
Higgs Vacuum Expectation Value (VEV) and the usual SM physical Higgs field.
$\phi^{+(0)}$ is the would-be Nambu-Goldstone boson of $W_{\mu}^{+(0)}$ and
generate the longitudinal d.o.f. for $W^{+(0)}_{\mu}$ and $\chi^{(0)}$ is for
$Z^{(0)}_{\mu}$. Subsequently, we take notice of the (4D) scalar KK excitation
modes. In addition to the Higgs KK excitation modes
${\\{}h^{(m,n)},\phi^{+(m,n)},\chi^{(m,n)}{\\}}$, there are other excitation
modes closely related to the (zero mode) massive gauge bosons, which are $y$
and $z$ components of 6D gauge fields.
Throughout this paper, we use information about W boson zero mode and its KK
particles and their associative particles, which are zero and KK modes of
$\phi^{{+}}$, $W^{{+y}}$ and $W^{{+z}}$. In what follows, we discuss only the
free Lagrangian with respect to the non-zero KK modes of W boson and their
associative particles since the zero mode part is the same with the SM one.
From Eq. (11), we can read off the free Lagrangian part
$S^{W}|_{{\text{free}}}$ as
$\displaystyle S^{W}|_{{\text{free}}}=\int d^{4}x{\sum_{m\geq 1,n\geq
0}}\Bigg{\\{}-\frac{1}{2}\Big{[}F_{\mu\nu}^{W(m,n)}F^{W(m,n)\mu\nu}\Big{]}_{{\text{quad}}}$
$\displaystyle\
+\frac{1}{2}\Big{[}(\partial_{\mu}\phi^{+(m,n)})(\partial^{\mu}\phi^{-(m,n)})+(\partial_{\mu}W^{+(m,n)y})(\partial^{\mu}W^{-(m,n)y})+(\partial_{\mu}W^{+(m,n)z})(\partial^{\mu}W^{-(m,n)z})\Big{]}$
$\displaystyle\
+\big{(}m_{W}^{2}+m_{(m,n)}^{2}\big{)}W_{\mu}^{+(m,n)}W^{\mu-(m,n)}-m_{(n)}^{2}W^{+(m,n)y}W^{-(m,n)y}$
$\displaystyle\
-m_{(m)}^{2}W^{+(m,n)z}W^{-(m,n)z}+m_{(m)}m_{(n)}\Big{[}W^{+(m,n)y}W^{-(m,n)z}+W^{-(m,n)y}W^{+(m,n)z}\Big{]}$
$\displaystyle\
-m_{(m,n)}^{2}\phi^{+(m,n)}\phi^{-(m,n)}-im_{W}\phi^{-(m,n)}\Big{[}m_{(m)}W^{+(m,n)y}+m_{(n)}W^{+(m,n)z}\Big{]}$
$\displaystyle\
+im_{W}\phi^{+(m,n)}\Big{[}m_{(m)}W^{-(m,n)y}+m_{(n)}W^{-(m,n)z}\Big{]}$
$\displaystyle\
-m_{W}^{2}\Big{[}W^{+(m,n)y}W^{-(m,n)y}+W^{+(m,n)z}W^{-(m,n)z}\Big{]}$
$\displaystyle\
-im_{W}\Big{[}(\partial^{\mu}\phi^{-(m,n)})W_{\mu}^{+(m,n)}-(\partial^{\mu}\phi^{+(m,n)})W_{\mu}^{-(m,n)}\Big{]}$
$\displaystyle\
-\Big{[}m_{(m)}(\partial^{\mu}W^{+(m,n)y})+m_{(n)}(\partial^{\mu}W^{+(m,n)z})\Big{]}W^{-(m,n)}_{\mu}$
$\displaystyle\
-\Big{[}m_{(m)}(\partial^{\mu}W^{-(m,n)y})+m_{(n)}(\partial^{\mu}W^{-(m,n)z})\Big{]}W^{+(m,n)}_{\mu}\Bigg{\\}},$
(19)
where
$\Big{[}F_{\mu\nu}^{W(m,n)}F^{W(m,n)\mu\nu}\Big{]}_{{\text{quad}}}=\big{(}\partial^{\mu}W^{+(m,n)\nu}-\partial^{\nu}W^{+(m,n)\mu}\big{)}\big{(}\partial_{\mu}W^{-(m,n)}_{\nu}-\partial_{\nu}W^{-(m,n)}_{\mu}\big{)}$
is the KK W-boson’s kinetic term, $m_{W}$ is the W-boson mass;
$m_{(m)}=\frac{m}{R}$ and $m_{(m,n)}^{2}=m_{(m)}^{2}+m_{(n)}^{2}$ are
describing the KK masses.
Here we adopt the following type of gauge-fixing term about W boson to
eliminate cross terms in Eq. (19) as
$\displaystyle S^{W}_{{\text{gf}}}$
$\displaystyle=-\frac{1}{\xi}\int_{0}^{2\pi R}dy{\int_{0}^{2\pi R}}dz\int
d^{4}x\Big{[}\partial_{\mu}W^{+\mu}+\xi\big{(}\partial_{y}W^{+y}+\partial_{z}W^{+z}-im_{W}\phi^{+}\big{)}\Big{]}$
$\displaystyle\qquad\times\Big{[}\partial_{\mu}W^{-\mu}+\xi\big{(}\partial_{y}W^{-y}+\partial_{z}W^{-z}+im_{W}\phi^{-}\big{)}\Big{]}.$
(20)
From Eq. (19), the mass of the field $W^{+(m,n)}_{\mu}$ is determined as
$m_{W,(m,n)}^{2}=m_{W}^{2}+m_{(m,n)}^{2}$. Meanwhile, we have to diagonalize
the scalar mass terms about $\phi^{+(m,n)}$, $W^{+(m,n)y}$ and $W^{+(m,n)z}$
to execute perturbative calculations. When we focus on this part
$S^{W}_{\text{scalar-mass}}$ out of $S^{W}+S^{W}_{{\text{gf}}}$,
$S^{W}_{\text{scalar-mass}}=-\int d^{4}x{\sum_{m\geq 1,n\geq
0}}\Big{(}W^{+(m,n)y},W^{+(m,n)z},\phi^{+(m,n)}\Big{)}\mathcal{M}_{(m,n)}\begin{pmatrix}W^{-(m,n)y}\\\
W^{-(m,n)z}\\\ \phi^{-(m,n)}\end{pmatrix},$ (21)
$\mathcal{M}_{(m,n)}=\begin{bmatrix}m_{W}^{2}+\xi
m_{(m)}^{2}+m_{(n)}^{2}&&-(1-\xi)m_{(m)}m_{(n)}&&-i(1+\xi)m_{W}m_{(m)}\\\
-(1-\xi)m_{(m)}m_{(n)}&&m_{W}^{2}+m_{(m)}^{2}+\xi
m_{(n)}^{2}&&-i(1+\xi)m_{W}m_{(n)}\\\
+i(1+\xi)m_{W}m_{(m)}&&+i(1+\xi)m_{W}m_{(n)}&&\xi
m_{W}^{2}+m_{(m)}^{2}+m_{(n)}^{2}\end{bmatrix}.$ (22)
By using those mass eigenstates ${\\{}G^{+(m,n)},a^{+(m,n)},H^{+(m,n)}{\\}}$,
we can diagonalize the matrix $\mathcal{M}_{(m,n)}$ to the following form:
$\begin{pmatrix}G^{\pm(m,n)}\\\ a^{\pm(m,n)}\\\
H^{\pm(m,n)}\end{pmatrix}=N_{(m,n)}^{\pm}\begin{pmatrix}W^{\pm(m,n)y}\\\
W^{\pm(m,n)z}\\\ \phi^{\pm(m,n)}\end{pmatrix},$ (23)
$N_{(m,n)}^{\pm}=\frac{1}{m_{W,(m,n)}m_{(m,n)}}\begin{bmatrix}m_{(m)}m_{(m,n)}&&m_{(n)}m_{(m,n)}&&\mp
im_{W}m_{(m,n)}\\\ \mp im_{W}m_{(m)}&&\mp im_{W}m_{(n)}&&m_{(m,n)}^{2}\\\
-m_{(n)}m_{W,(m,n)}&&+m_{(m)}m_{W,(m,n)}&&0\end{bmatrix},$ (24)
$N^{-}_{(m,n)}\mathcal{M}_{(m,n)}\big{(}N^{-}_{(m,n)}\big{)}^{\dagger}=\text{diag}\big{(}\xi
m_{W,(m,n)}^{2}\ ,\ m_{W,(m,n)}^{2}\ ,\ m_{W,(m,n)}^{2}\big{)}.$ (25)
This result means that $G^{+(m,n)}$ is the would-be Nambu-Goldstone boson of
$W^{+(m,n)}_{\mu}$ and the others $a^{+(m,n)},H^{+(m,n)}$ are physical 4D
scalars. It is noted that ${H}^{+(m,n)}$ is called “spinless adjoint” because
${H}^{+(m,n)}$ is constructed only by extra spacial components of the 6D gauge
boson $W^{+(m,n)}_{\mu}$. They contribute to $h^{(0)}\rightarrow 2\gamma$
Higgs decay process via loop diagrams.
Next, we derive the mass eigenstates of fermions. Just like the case mentioned
above, we again consider the KK part only. The kinetic terms are diagonal, and
therefore there is no need to discuss the part. The mass term of $(m,n)$-th KK
mode fermions arising from Eq. (11) is
$\Big{(}\bar{t}^{(m,n)}_{R}\ ,\
\bar{Q}^{(m,n)}_{tR}\Big{)}\begin{pmatrix}-m_{(m)}+im_{(n)}&m_{t}\\\
m_{t}&m_{(m)}+im_{(n)}\end{pmatrix}\begin{pmatrix}t_{L}^{(m,n)}\\\
Q_{tL}^{(m,n)}\end{pmatrix}+\text{h.c.},$ (26)
where $m_{t}$ is the zero mode top quark mass and $Q_{t}^{(m,n)}$ is the upper
component of the SU(2) doublet $Q_{3}^{(m,n)}$. By using the following unitary
transformation including chiral rotation, we can derive the ordinary
diagonalized Dirac mass term as follows:
$\begin{pmatrix}t^{(m,n)}\\\
Q_{t}^{(m,n)}\end{pmatrix}=\begin{pmatrix}e^{\frac{i}{2}\gamma^{5}\varphi_{(m,n)}}&0\\\
0&e^{-\frac{i}{2}\gamma^{5}\varphi_{(m,n)}}\end{pmatrix}\begin{pmatrix}-\cos{\alpha_{(m,n)}}\gamma^{5}&\sin{\alpha_{(m,n)}}\\\
\sin{\alpha_{(m,n)}}\gamma^{5}&\cos{\alpha_{(m,n)}}\end{pmatrix}\begin{pmatrix}t^{{}^{\prime}(m,n)}\\\
Q_{t}^{{}^{\prime}(m,n)}\end{pmatrix},$ (27)
where $t^{{}^{\prime}(m,n)}$ and $Q_{t}^{{}^{\prime}(m,n)}$ are mass
eigenstates of their corresponding fields with degenerate $(m,n)$-th level
masses; $m_{t,(m,n)}^{2}=m_{t}^{2}+m_{(m,n)}^{2}$. The mixing angles
$\varphi_{(m,n)}$ and $\alpha_{(m,n)}$ are determined as
$\tan{\varphi_{(m,n)}}=-\frac{m_{(n)}}{m_{(m)}},\quad\cos{{2}\alpha_{(m,n)}}=\frac{m_{(m,n)}}{m_{t,(m,n)}},$
(28)
from the condition to obtain the ordinary diagonalized Dirac mass matrix. Now
we are ready to calculate the rates of Higgs processes at the LHC. Some
requisite interactions in this paper are discussed at the next section.
## 3 Calculation of one loop Higgs production and decay processes
We calculate some virtual effects of KK particle via loop diagrams in the
Higgs production process through gluon ($g$) fusion $2g\rightarrow h^{(0)}$
and the Higgs decay process to two photon ($\gamma$) $h^{(0)}\rightarrow
2\gamma$. Those processes are 1-loop leading and it is expected that the
effects of massive KK particles are significant. In addition, there is another
1-loop leading Higgs decay process to photon and Z-boson ($Z$)
$h^{(0)}\rightarrow\gamma Z$, which we do not discuss in this paper. Before
the concrete discussion about interactions, we have to understand the general
structure of interactions which is needed for our study. In the scope of this
paper, all external particles are SM particles, which are described by zero
modes. This means the effective couplings which we use are obtained by the
following type of integrals concerning mode functions $f_{t}^{(m,n)}$,
$\displaystyle\int_{0}^{2\pi R}dy{\int_{0}^{2\pi
R}}dz\left\\{f_{t_{i}}^{(0,0)}f_{t_{j}}^{(m,n)}f_{t_{k}}^{(m^{\prime},n^{\prime})}\right\\}\
[\text{3-point}],$ (29) $\displaystyle\int_{0}^{2\pi R}dy{\int_{0}^{2\pi
R}}dz\left\\{f_{t_{l}}^{(0,0)}f_{t_{i}}^{(0,0)}f_{t_{j}}^{(m,n)}f_{t_{k}}^{(m^{\prime},n^{\prime})}\right\\}\
[\text{4-point}],$ (30)
where the Latin indices $i,j,k,l$ indicate types of the particles. The
$Z_{4}$-parities $t_{l},t_{i}$ are determined as $t_{l}=t_{i}=1$ and the
condition $(t_{j})^{\ast}=t_{k}$ is required from $Z_{4}$ invariance of the
action in Eq. (11). Because of orthonormality of mode functions, we know that
the integrals are non-vanishing only when $(m,n)=(m^{\prime},n^{\prime})$ and
the integrals can be reduced to the ones for the zero modes alone. In other
words, the value of the vertex containing KK modes, which are described by the
above integrals, is exactly the same with the value of the corresponding
vertex for the zero mode alone in the basis of gauge eigenstates.
We give a comment on the Higgs mass $m_{h}$ and the lowest KK mass
${M_{\text{KK}}}$, which is defined as $1/R$ on the geometry of $T^{2}$. In
UED model, those two parameters are free, which means they are not determined
by the theory, but there are some constraints on these parameters. From the
result of LEP2 experiment, $m_{h}$ is bounded from below as $m_{h}>114\
\text{GeV}$. And recently another bound is announced from the LHC experiments
ATLAS-CONF-2011-135 ; CMS_PAS_HIG-11-022 ; ATLAS:2012ae ; Chatrchyan:2012tx .
We discuss this point in Section 6.
We ignore the graviton contributions. In any 6D UED model, 6D Planck scale
$M_{\ast}$ is related to 4D Planck scale $M_{\text{pl}}$ through a KK mass
scale ${M_{\text{KK}}}$ as follows:
$M_{\ast}^{2}\sim{M_{\text{KK}}}M_{\text{pl}}.$ (31)
$M_{\text{pl}}$ is approximately $10^{18}$ GeV and we are interested in the
case ${M_{\text{KK}}}\sim\mathcal{O}(1)$ TeV. Then the magnitude of $M_{\ast}$
is estimated easily as $\sim 10^{10}$ GeV and gravitons are still weekly
coupled to other fields.
### 3.1 $2g\rightarrow h^{(0)}$ process
This gluon fusion process gets contribution only from the fermion triangle
loops at 1-loop level. The SM contribution is calculated in Eq. Georgi:1977gs
; Rizzo:1979mf . In UED model, the intermediate fermions are not only SM ones
(zero modes) but also their KK excitations. Studies of the production process
for the case of 5D minimal UED Petriello:2002uu and 6D $S^{2}/Z_{2}$ UED
Maru:2009cu are made. We consider only contributions from the top quark and
its KK states. The reason why we ignore other types of quarks and its KK modes
is that the coupling of fermions to the Higgs is proportional to each zero
mode quark mass, and thereby those effects are negligible in our analysis. In
terms of the fermion mass eigenstates, the interactions of KK quarks are
$\displaystyle S^{t}_{{\text{int}}}$ $\displaystyle=\int d^{4}x{\sum_{m\geq
1,n\geq 0}}$ $\displaystyle\times\Bigg{\\{}\Big{(}\bar{t}^{{}^{\prime}(m,n)}\
,\ \bar{Q}^{{}^{\prime}(m,n)}_{t}\Big{)}\Bigg{[}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}g^{(3)}\gamma^{\mu}g_{\mu}$
$\displaystyle\qquad-\frac{m_{t}}{v}h^{(0)}\begin{pmatrix}\sin{2\alpha_{(m,n)}}&\cos{2\alpha_{(m,n)}}\gamma^{5}\\\
-\cos{2\alpha_{(m,n)}}\gamma^{5}&\sin{2\alpha_{(m,n)}}\end{pmatrix}\Bigg{]}\begin{pmatrix}t^{{}^{\prime}(m,n)}\\\
Q_{t}^{{}^{\prime}(m,n)}\end{pmatrix}\Bigg{\\}}.$ (32)
The production cross section of $2g\rightarrow h^{(0)}$ process is given as
follows:
${\sigma_{2g\rightarrow h^{(0)}}=\frac{\sqrt{2}\pi
G_{F}}{64}\left(\frac{\alpha_{s}}{\pi}\right)^{2}|F_{\text{{gluonfusion}}}|^{2},}$
(33)
where $G_{F}$ is the Fermi constant and $\alpha_{s}$ is the QCD coupling.
$F_{\text{{gluonfusion}}}$ is the loop function, which consists of the SM top
quark effect $F_{t}^{{\text{SM}}}$, the KK top quark effect
$F_{t}^{{\text{KK}}}$ and the threshold correction
$F_{\text{gluonfusion}}^{\text{TC}}$. Then we can write
$F_{\text{{gluonfusion}}}=F_{t}^{{\text{SM}}}+F_{t}^{{\text{KK}}}{+F_{\text{gluonfusion}}^{\text{TC}}}$
and $F_{t}^{{\text{SM}}}$ is given in Ref. Rizzo:1979mf in our notation as
$F_{t}^{\text{{SM}}}={-2}\lambda(m_{t}^{2}){+}\lambda(m_{t}^{2})(1-4\lambda(m_{t}^{2}))J\left(\lambda(m_{t}^{2})\right),$
(34)
where $\lambda(m^{2})$ and the loop function $J(\lambda)$ are defined as
$\displaystyle{\lambda(m^{2})}$ $\displaystyle{=m^{2}/m_{h}^{2},}$ (35)
$\displaystyle{J(\lambda)}$ $\displaystyle{=\int_{0}^{1}{dx\over
x}\ln\left[{x(x-1)\over\lambda}+1-i\epsilon\right]}$
$\displaystyle{=\begin{cases}\displaystyle-2\left[\arcsin{1\over\sqrt{4\lambda}}\right]^{2}&\text{(for
$\lambda\geq{1\over 4}$)},\\\ \displaystyle{1\over
2}\left[\ln{1+\sqrt{1-4\lambda}\over
1-\sqrt{1-4\lambda}}-i\pi\right]^{2}&\text{(for $\lambda<{1\over
4}$)},\end{cases}}$ (36)
respectively.888Our loop function $J$ is related to the three-point scalar
Passarino-Veltman function $C_{0}$ in Refs. Passarino:1978jh ; Denner:1991kt
as $J=m_{h}^{2}C_{0}$. The KK top quark coupling to the gluon and the zero
mode Higgs is shown in Eq. (32). After some calculation, we can get the form
of $F_{t}^{{\text{KK}}}$ in 6D UED model, where the concrete form is
$\displaystyle F_{t}^{{\text{KK}}}$ $\displaystyle=2\sum_{m\geq 1,n\geq
0}\left(\frac{m_{t}}{m_{t,(m,n)}}\right)^{2}$
$\displaystyle\quad\times\left\\{{-2}\lambda(m_{t,(m,n)}^{2}){+}\lambda(m_{t,(m,n)}^{2})(1-4\lambda(m_{t,(m,n)}^{2}))J\left(\lambda(m_{t,(m,n)}^{2})\right)\right\\}.$
(37)
It is noted that $F_{t}^{\text{SM}}$ and $F_{t}^{\text{KK}}$ contain the
$(-1)$ factor due to fermionic loop. Our result is directly related to the
minimal 5D case in Ref. Petriello:2002uu . The reason is that the only
difference between 5D and 6D case is the KK top quark mass spectrum and the
structure of the Feynman diagrams itself describing this process is completely
the same. The $m^{2}$ in ${\lambda(={m^{2}}/{m_{h}^{2}})}$ indicates the
intermediate mass scale propagating in the loops and we consider the situation
that KK scale $m_{(m,n)}$ is much greater than the Higgs scale $m_{h}$. It is
noted that we only focus on the light Higgs possibility;
$120\,\text{GeV}\lesssim m_{h}\lesssim 150\,\text{GeV}$ and thereby we have to
only consider the ${\lambda}\geq 1/4$ case. Finally the contribution from
threshold correction is obtained as
$F_{\text{gluonfusion}}^{\text{TC}}=\left[\left(\frac{\alpha_{s}}{\pi}\right)\frac{1}{v}\right]^{-1}C^{\prime}_{hgg},$
(38)
where $C^{\prime}_{hgg}$ is a dimensionful coefficient describing the
threshold correction and is related to the dimensionless constant in a part of
the Lagrangian $C_{hgg}$ with the UED cutoff scale ${\Lambda_{\text{UED}}}$ as
$C^{\prime}_{hgg}=\frac{C_{hgg}\left(v\over\sqrt{2}\right)}{{\Lambda_{\text{UED}}}^{2}}.$
(39)
We see the details in Appendix B.
From naive power counting, this result is logarithmically divergent. The
reason is the following. Higgs decay through gluon fusion is described with
dimension-six operator in four-dimensional picture after KK reduction. In UED
model, there is no shift symmetry alleviating divergence, then this process
obeys the above simple estimation.999In 5D UED, we can calculate this process
without cutoff dependence. Therefore, we introduce a cutoff scale
${\Lambda_{\text{UED}}}$ to regularize the $F_{t}^{\text{KK}}$ in Eq. (37). We
estimate an upper bound of ${\Lambda_{\text{UED}}}$ by use of Naive
Dimensional Analysis (NDA) technique Appelquist:2000nn in Section 5.
### 3.2 $h^{(0)}\rightarrow 2\gamma$ process
Now we turn to the Higgs decay process $h^{(0)}\rightarrow 2\gamma$, which is
the experimentally favorable at the LHC with Higgs mass region
$120\,\text{GeV}\lesssim m_{h}\lesssim 150\,\text{GeV}$. The Feynman diagrams
describing $h^{(0)}\rightarrow 2\gamma$ process due to the contribution of W
boson and its associated particles are shown in Fig. 1, 2, 3, and 4.
$\omega_{W}^{(m,n)}$ and $\bar{\omega}_{W}^{(m,n)}$ indicate $(m,n)$-th ghost
and anti-ghost modes originated from $W^{(m,n)}_{\mu}$ boson, respectively. We
also need to consider a flipped $(\mu\leftrightarrow\nu)$ one for each diagram
if it exists. It is noted that there are another triangle loop diagrams
contributing to this process, whose intermediate particles are the top quark
and its KK states. But we can take these effects into account by use of the
previous result in Eq. (37) with some modifications. The decay width can be
written as
${\Gamma_{h^{(0)}\rightarrow
2\gamma}=\frac{\sqrt{2}G_{F}}{16\pi}\left(\frac{\alpha_{\text{EM}}}{\pi}\right)^{2}m_{h}^{3}|F_{\text{decay}}|^{2},}$
(40)
where $\alpha_{\text{EM}}$ is the electromagnetic coupling strength. In this
process, the function describing loop effects $F_{\text{decay}}$ is written by
$F_{\text{decay}}=F_{W}+3Q_{t}^{2}\left({F_{t}^{\text{SM}}+F_{t}^{\text{KK}}}\right){+F^{\text{TC}}_{\text{decay}}},$
(41)
where the first term represents the effect of W boson and its associated
particles, and the second term represents that of the top quark and its KK
states.1010103 is color factor and $Q_{t}$ is the electromagnetic charge of
the top quark $(=\frac{2}{3})$. The third term describes the threshold
correction. The SM result for $F_{t}^{{\text{SM}}}$ is previously obtained in
Eq. (34) and the concrete form of $F_{W}^{{\text{SM}}}$ is derived in
Ellis:1975ap as
${F_{W}^{{\text{SM}}}=\frac{1}{2}+3\lambda{(m_{W}^{2})}-3\lambda{(m_{W}^{2})}(1-2\lambda{(m_{W}^{2})})J\left(\lambda(m_{W}^{2})\right)},$
(42)
where $J$ is given in Eq. (36). We set
$F_{W}=F_{W}^{{\text{SM}}}+F^{{\text{KK}}}_{W}$, where $F_{t}^{{\text{KK}}}$
has been already discussed and $F_{W}^{{\text{KK}}}$ represents the
contribution of KK W boson and its associated KK particles. And we decompose
$F_{W}^{{\text{KK}}}$ into four pieces as
$F_{W}^{{\text{KK}}}=F^{{\text{KK}}}_{\text{gauge}}+F^{{\text{KK}}}_{\text{NG}}+F^{{\text{KK}}}_{\text{scalar1}}+F^{{\text{KK}}}_{\text{scalar2}},$
(43)
where each term $F_{W}^{{\text{KK}}}$ indicates the loop effects coming from
gauge, would-be NG boson, scalar particles, respectively and corresponding
Feynman diagrams are found in Fig. 1, 2, 3, and 4, respectively. The four sets
of diagrams are $U(1)_{EM}$ gauge invariant and we can check this fact by use
of Ward identity.
Figure 1: Feynman diagrams of 4D gauge sector describing $h^{(0)}\rightarrow
2\gamma$ process. Figure 2: Feynman diagrams of 4D would-be NG boson sector
describing $h^{(0)}\rightarrow 2\gamma$ process. Figure 3: Feynman diagrams
of 4D scalar sector describing $h^{(0)}\rightarrow 2\gamma$ process. Figure
4: Feynman diagrams of 4D scalar (“spinless adjoint”) sector describing
$h^{(0)}\rightarrow 2\gamma$ process.
After some tedious but straightforward calculation, we can get the result as
follows:111111In Appendix A, we write down some Feynman rules to calculate
this process.
$\displaystyle F^{{\text{KK}}}_{\text{gauge}}$ $\displaystyle=\sum_{m\geq
1,n\geq
0}\left\\{3\lambda(m_{W}^{2})+2\lambda(m_{W}^{2})\big{(}3\lambda(m_{W,(m,n)}^{2})-2\big{)}J\left(\lambda(m_{W,(m,n)}^{2})\right)\right\\},$
(44) $\displaystyle F^{{\text{KK}}}_{\text{NG}}$ $\displaystyle=\sum_{m\geq
1,n\geq
0}\left(\frac{1}{2}\frac{m_{h}^{2}}{m_{W,(m,n)}^{2}}\right)\lambda(m_{W}^{2})\left\\{1+2\lambda(m_{W,(m,n)}^{2})J\left(\lambda(m_{W,(m,n)}^{2})\right)\right\\},$
(45) $\displaystyle F^{{\text{KK}}}_{\text{scalar1}}$
$\displaystyle=\sum_{m\geq 1,n\geq
0}\left(\frac{1}{2}\frac{1}{m_{W,(m,n)}^{2}}\right)\left[\frac{m_{h}^{2}}{m_{W}^{2}}m_{(m,n)}^{2}+{2}m_{W,(m,n)}^{2}\right]$
$\displaystyle\quad\times\lambda(m_{W}^{2})\left\\{1+2\lambda(m_{W,(m,n)}^{2})J\left(\lambda(m_{W,(m,n)}^{2})\right)\right\\},$
(46) $\displaystyle F^{{\text{KK}}}_{\text{scalar2}}$
$\displaystyle=\sum_{m\geq 1,n\geq
0}\lambda(m_{W}^{2})\left\\{1+2\lambda(m_{W,(m,n)}^{2})J\left(\lambda(m_{W,(m,n)}^{2})\right)\right\\},$
(47)
By adding up Eqs. (44)-(47), the concrete form of $F_{W}^{{\text{KK}}}$ is
given as
$\displaystyle F_{W}^{{\text{KK}}}$ $\displaystyle=\sum_{m\geq 1,n\geq
0}\Big{\\{}\frac{1}{2}+5\lambda(m_{W}^{2})-\Big{[}\lambda(m_{W}^{2})(4-10\lambda(m_{W,(m,n)}^{2}))$
$\displaystyle\qquad\qquad-\lambda(m_{W,(m,n)}^{2})\Big{]}J\left(\lambda(m_{W,(m,n)}^{2})\right)\Big{\\}},$
(48)
where we use the relation $m_{W,{(m,n)}}^{2}=m_{W}^{2}+m_{(m,n)}^{2}$. This
loop-induced process is also described by dimension-six operator in 4D point
of view and we have to introduce the cutoff scale ${\Lambda_{\text{UED}}}$ to
regularize the summations. The concrete form of the third term in Eq. (41),
which originates form threshold correction, is as follows:
$F^{\text{TC}}_{\text{decay}}=\left[\left(\frac{\alpha_{\text{EM}}}{\pi}\right)\frac{2}{v}\right]^{-1}C^{\prime}_{h\gamma\gamma},$
(49)
where $C^{\prime}_{h\gamma\gamma}$ is a dimensionful coefficient describing
the threshold correction and is related to the dimensionless constant in a
part of the Lagrangian $C_{h\gamma\gamma}$ with the UED cutoff scale
${\Lambda_{\text{UED}}}$ as
$C^{\prime}_{h\gamma\gamma}=\frac{C_{h\gamma\gamma}\left(v\over\sqrt{2}\right)}{{\Lambda_{\text{UED}}}^{2}}.$
(50)
We also see the details in Appendix B.
## 4 Universal Extra Dimension Models based on $S^{2}$
Recently, Universal Extra Dimension Models based on $S^{2}$ are proposed in
Refs. Maru:2009wu ; Dohi:2010vc . After an overview of gauge theory on
$S^{2}$, we give a brief review of these models.
### 4.1 Gauge Theory on $S^{2}$
We consider a gauge theory on six-dimensional spacetime $M^{4}\times S^{2}$,
which is a direct product of the four-dimensional Minkowski spacetime $M^{4}$
and two-sphere $S^{2}$. We use the coordinate of six-dimensional spacetime
defined by $x^{M}=(x^{\mu},\theta,\phi)$. $\theta\ (\phi)$ is zenith
(azimuthal) angle of $S^{2}$, respectively and we use the same coordinate
conventions as in Section 2. The metric ansatz of $M^{4}\times S^{2}$ is
$g_{MN}=\text{diag}(1,-1,-1,-1,-R^{2},-R^{2}\sin^{2}{\theta})$ (51)
and we also need to introduce the vielbein $e_{M}^{\ \
\underline{N}}=\text{diag}(1,1,1,1,R,R\sin{\theta})$ to describe tangent space
which fermions live in. In this tangent space, the coordinate is expressed
with barred letters and we choose the same representation of Clifford algebra
as in Eq. (1). $S^{2}$ has a positive curvature and then a radius of $S^{2}$
described by $R$ only can take an infinite value by the consistency with the
6D Einstein equation. To stabilize the system, we introduce a $U(1)_{X}$ gauge
field which has a monopole-like configuration in classical level $X^{c}_{M}$
RandjbarDaemi:1982hi . This configuration is defined as follows:
$[X^{c}_{\phi}(x^{\mu},\theta,\phi)]^{{N}\atop{S}}={n\over
2g^{(X)}_{6}}(\cos{\theta}\mp 1),\quad(\text{other components})=0,$ (52)
where $g^{(X)}_{6}$ is a $U(1)_{X}$ gauge coupling and $n$ is a monopole
index. The superscript $N\atop S$ indicates that the field is given in north
(involving the $\theta=0$ point) and south (involving the $\theta=\pi$ point)
patches, respectively and we use this notation throughout the rest of this
paper. The gauge transformation from the north to the south patch is given by
$[X_{M}(x^{\mu},\theta,\phi)]^{S}=[X_{M}(x^{\mu},\theta,\phi)]^{N}+\frac{1}{g_{6}^{(X)}}\partial_{M}\alpha(x^{\mu},\theta,\phi){,}$
(53)
where the function $\alpha(x^{\mu},\theta,\phi)=n\phi$. Because of the
monopole-like configuration, the radius of $S^{2}$ is stabilized spontaneously
as
$R^{2}=\left({n\over 2g_{6}^{(X)}M_{\ast}^{2}}\right)^{2}.$ (54)
Every 6D field $\Phi$ on $S^{2}$ is KK expanded by use of the spin-weighted
spherical harmonics ${}_{s}Y_{jm}(\theta,\phi)$ as follows:121212Newman-
Penrose edth formalism Newman:1966ub is useful for description of spin
weighted spherical harmonics.
$\Phi(x,\theta,\phi)^{N\atop
S}=\sum_{j=|s|}^{\infty}\sum_{m=-j}^{j}\varphi^{(j,m)}(x)f_{\Phi}^{(j,m)}(\theta,\phi)^{N\atop
S},\quad f_{\Phi}^{(j,m)}(\theta,\phi)^{N\atop
S}:={{}_{s}Y_{jm}(\theta,\phi)e^{\pm is\phi}\over R},$ (55)
where $s$ is the spin weight of the field $\Phi$. The spin-weighted spherical
harmonics ${}_{s}Y_{jm}(\theta,\phi)$ matches the orthonormal condition as
${\int_{0}^{2\pi}d\phi\int_{-1}^{1}d\cos{\theta}\overline{{}_{s}Y_{jm}(\theta,\phi)}{}_{s}Y_{j^{\prime}m^{\prime}}(\theta,\phi)=\delta_{jj^{\prime}}\delta_{mm^{\prime}}.}$
(56)
A spin weight of a fermion is closely related to its $U(1)_{X}$ charge. When
we assign $U(1)_{X}$ charges of 6D Weyl fermions $\Psi_{\pm}$ as
$q_{\Psi_{\pm}}$, the corresponding spin weights of 4D Weyl fermions
$\\{\psi_{+{R\atop L}},\psi_{-{R\atop L}}\\}$ are given as follows in our
convention:
$s_{+{R\atop L}}=-\left({nq_{\Psi_{+}}\mp 1\over 2}\right),\quad s_{-{R\atop
L}}=-\left({nq_{\Psi_{-}}\pm 1\over 2}\right).$ (57)
We can find the fact that if a 6D Weyl fermion takes the $s=0$ spin weight,
one zero mode $(j=0)$ appears as a 4D Weyl fermion with no KK mass. This means
we can get the SM fermions without orbifolding in the case of $S^{2}$. When we
take the values as $(s_{+R},s_{+L},s_{-R},s_{-L})=(0,-1,-1,0)$, we can create
the same situation as in $T^{2}/Z_{4}$ which we discussed before. A spin
weight of a 4D vector component of a 6D gauge boson is $s=0$ and then there is
a zero mode which we can assign as a SM gauge boson. However, extra
dimensional components of 6D gauge boson are expanded by the $|s|=1$ spin-
weighted spherical harmonics. This is because these parts are closely related
to $S^{2}$ structure. Concretely speaking, the combinations of components
$A_{\pm}=\frac{1}{\sqrt{2}}(A_{\underline{\theta}}\pm iA_{\underline{\phi}})$
are KK expanded with $s=\pm 1$ spin weighted spherical harmonics,
respectively, where $A_{\underline{M}}$ is a gauge field on tangent space
defined as $A_{\underline{M}}=e_{\underline{M}}^{\ \ N}A_{N}$. Then there is
no zero mode in these parts. After the introduction of gauge fixing term
concerning a gauge field $A_{M}$, whose concrete form is
$-\frac{1}{\xi}\text{tr}\left(\eta^{\mu\nu}\partial_{\mu}A_{\nu}-\frac{\xi}{R^{2}\sin{\theta}}\partial_{\theta}\sin{\theta}A_{\theta}-\frac{\xi}{R^{2}\sin^{2}{\theta}}A_{\phi}\right)^{2}\quad(\xi:\text{gauge
fixing parameter}),$ (58)
the mass eigenstates are obtained as follows:
$\begin{pmatrix}A_{\underline{\theta}}\\\
A_{\underline{\phi}}\end{pmatrix}=\begin{pmatrix}\partial_{\theta}&-\csc{\theta}\partial_{\phi}\\\
\partial_{\theta}&+\csc{\theta}\partial_{\phi}\end{pmatrix}\begin{pmatrix}\phi_{1}^{(A)}\\\
\phi_{2}^{(A)}\end{pmatrix}.$ (59)
$\phi_{1}^{(A)}$ and $\phi_{2}^{(A)}$ are 4D physical scalar field and
unphysical would-be Nambu-Goldstone mode, respectively. A 6D scalar field can
take a nonzero spin weight through the interaction with the $U(1)_{X}$ gauge
boson. But we would like to regard the zero mode of a 6D scalar field as the
SM Higgs, then the value of the spin weight must be $s=0$.
In our configuration, any $(j,m)$-th KK mode has the KK mass,
$m_{(j,m)}^{2}=\frac{j(j+1)}{R^{2}}.$ (60)
An important point is that the form of the above KK mass is independent of the
index of $m$. This means there are $2j+1$ degenerated modes for each $j$. It
is noted that each KK mode summation over $j$ begins from one. In contrast to
the $T^{2}$ case, the value of the first KK mass is represented as
$M_{\text{KK}}=\sqrt{2}/R$.
We can construct an Universal Extra Dimension model on $S^{2}$ along the
direction which we have discussed. But there are two problems in this model.
One is absence of KK parity. In usual UED models based on orbifold, there are
fixed points of orbifold discrete symmetry and KK parity is realized as a
remnant of extra spatial symmetry, which is an invariance of system in
exchange of fixed points. It ensures the existence of dark matter candidate in
these models. But the geometry of $S^{2}$ do not have fixed point and thereby
the UED on $S^{2}$ cannot possess KK parity. The other is more serious. As we
discussed before, a 4D vector component of a 6D gauge boson has zero mode in
$S^{2}$. In case of the $U(1)_{X}$ gauge boson, which has the monopole-like
configuration, this is true. We should notice that the gauge coupling of an
extra massless gauge boson is severely constrained to be $g^{(X)}\lesssim
10^{-23}$ by a torsion balance experiment Smith:1999cr . $g^{(X)}$ is the 4D
effective coupling of the 6D $U(1)_{X}$ gauge coupling $g^{(X)}_{6}$ and is
described as $g^{(X)}={{g^{(X)}_{6}}/{\sqrt{4\pi R^{2}}}}$. By use of (54), we
can estimate the value of $g^{(X)}$ in the UED model on $S^{2}$ as
$g^{(X)}{\simeq}\frac{n{M_{\text{KK}}}}{M_{\text{pl}}}{{.}}$ (61)
In the viewpoint of our phenomenological motivation, ${M_{\text{KK}}}$ must be
$\sim\mathcal{O}(1)$ TeV. In such a situation, $g^{(X)}$ becomes $\sim
10^{-15}\cdot n$ and its value is far from the experimental bound. Since
monopole charge $n$ only can take integer value, we cannot resolve this
pathology by tuning of the parameter $n$.
Fortunately, we can solve these problems by some modifications in the $S^{2}$
geometry. In the rest of this section, we follow some essential points of
these ideas.
### 4.2 UED on $S^{2}/Z_{2}$
Following Ref. Maru:2009wu , we take a $Z_{2}$ orbifold on the geometry of
$S^{2}$. On this orbifold, the point ${(\theta,\phi)}$ is identified with
$(\pi-\theta,-\phi)$. The 6D action is as follows:
$\displaystyle S$ $\displaystyle=\int_{0}^{\pi}d\theta\int_{0}^{2\pi}d\phi\int
d^{4}x\sqrt{-g}\Bigg{\\{}-\frac{1}{2}\sum_{i=1}^{3}\mathrm{g}^{MN}g^{KL}\text{Tr}\big{[}F_{MK}^{(i)}F^{(i)}_{NL}\big{]}-\frac{1}{4}g^{MN}g^{KL}\big{[}F_{MK}^{(X)}F^{(X)}_{NL}\big{]}$
$\displaystyle\qquad+g^{MN}(D_{M}H)^{\dagger}(D_{N}H)+\bigg{[}\mu^{2}|H|^{2}-\frac{\lambda_{6}^{(H)}}{4}|H|^{4}\bigg{]}$
$\displaystyle\qquad+i\bar{\mathcal{Q}}_{3-}\Gamma^{M}D_{M}\mathcal{Q}_{3-}+i\bar{\mathcal{U}}_{3+}\Gamma^{M}D_{M}\mathcal{U}_{3+}-\bigg{[}\lambda_{6}^{(t)}\bar{\mathcal{Q}}_{3-}(i\sigma_{2}H^{\ast})\mathcal{U}_{3+}+\mathrm{h.c.}\bigg{]}\Bigg{\\}}$
(62)
where $\sqrt{-g}=R^{2}\sin{\theta}$. In this model, the form of 6D action and
matter content are almost the same with these of the $T^{2}/Z_{4}$ except the
existence of the $U(1)_{X}$ gauge field and $F_{MN}^{(X)}$ has the classical
part arising from the monopole-like configuration as
$F_{\theta\phi}=-\frac{n}{2g_{6}^{(X)}}\sin{\theta},\quad(\text{other
components})=0.$ (63)
The covariant derivative of the Higgs is given in an ordinary form as
$D_{M}=\partial_{M}-i\sum_{i=1}^{3}g_{6}^{(i)}T^{(i)a}A^{(i)a}_{M},$ (64)
and the covariant derivatives of fermions are obtained as follows:
$D_{M}=\partial_{M}-i\sum_{i=1}^{3}g_{6}^{(i)}T^{(i)a}A^{(i)a}_{M}-ig_{6}^{(X)}q_{\Psi}(X^{c}_{M}+X_{M})+\Omega_{M}.$
(65)
$q_{\Psi}$ is a $U(1)_{X}$ charge of a fermion and $X^{c}_{M}$ is the
monopole-like classical configuration in Eq. (52). The other additional term
$\Omega_{M}$ is the spin connection in $S^{2}$, whose concrete form is
$(\Omega_{\phi})^{N\atop S}=\frac{i}{2}(\cos{\theta}\mp
1)\begin{pmatrix}1_{4}&0\\\ 0&-1_{4}\end{pmatrix},\quad(\text{other
components})=0,$ (66)
where $1_{4}$ is a four-by-four unit matrix. We can easily construct mode
functions of $S^{2}/Z_{2}$ $f_{s,t}^{(j,m)}(\theta,\phi)$ with spin weight $s$
in both north and south patches following the general prescription in Ref.
Georgi:2000ks as follows:
$f_{s,t}^{(j,m)}(\theta,\phi)^{N\atop
S}=\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{2R}\left[{}_{s}Y_{jm}(\theta,\phi)+(-1)^{j-s}{}_{s}Y_{j-m}(\theta,\phi)\right]e^{\pm
is\phi}&\text{for}\ t=+1\\\
\displaystyle\frac{1}{2R}\left[{}_{s}Y_{jm}(\theta,\phi)-(-1)^{j-s}{}_{s}Y_{j-m}(\theta,\phi)\right]e^{\pm
is\phi}&\text{for}\ t=-1\end{array}\right.,$ (67)
where $t=\pm 1$ is the $Z_{2}$ parity. These mode functions have the property
that $f_{s,t=\pm 1}^{(j,m)}(\pi-\theta,-\phi)^{N\atop S}=\pm f_{s,t=\pm
1}^{(j,m)}(\theta,\phi)^{S\atop N}$. To realize the $Z_{2}$ symmetry, we
identify a field at $(\theta,\phi)$ in the north patch with the same field at
$(\pi-\theta,-\phi)$ in the south patch. The conditions are as follows:
$\displaystyle H(x,\pi-\theta,-\phi)^{N\atop S}$
$\displaystyle=+H(x,\theta,\phi)^{S\atop N},$ (68)
$\displaystyle\\{A^{(i)}_{\mu},X_{\mu}\\}(x,\pi-\theta,-\phi)^{N\atop S}$
$\displaystyle=+\\{A^{(i)}_{\mu},X_{\mu}\\}(x,\theta,\phi)^{S\atop N},$ (69)
$\displaystyle\\{A^{(i)}_{\theta,\phi},X_{\theta,\phi}\\}(x,\pi-\theta,-\phi)^{N\atop
S}$
$\displaystyle=-\\{A^{(i)}_{\theta,\phi},X_{\theta,\phi}\\}(x,\theta,\phi)^{S\atop
N},$ (70)
$\displaystyle\\{\mathcal{Q}_{3-},\mathcal{U}_{3+}\\}(x,\pi-\theta,-\phi)^{N\atop
S}$
$\displaystyle=+i\Gamma^{\underline{y}}\Gamma^{\underline{z}}\\{\mathcal{Q}_{3-},\mathcal{U}_{3+}\\}(x,\theta,\phi)^{S\atop
N},$ (71)
where we take the choice that all gauge twist matrices are trivial ($P={\bf
1}$). And we define the transformation of 6D Weyl fermion $\Psi_{\pm}$ from
the north to the south patch as
$\Psi^{S}_{\pm}(x,\theta,\phi)=\exp(iq_{\Psi_{\pm}}\alpha+2\phi\Sigma^{\underline{y}\underline{z}})\Psi^{N}_{\pm}(x,\theta,\phi){,}$
(72)
where $\alpha$ is the $U(1)_{X}$ gauge transformation function in Eq. (53) and
$\Sigma^{\underline{y}\underline{z}}$ is the $(\underline{y},\underline{z})$
component of the local Lorentz generator of a 6D Weyl fermion.131313In our
notation,
$\Sigma^{\underline{y}\underline{z}}=\frac{-i}{2}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}$. The Higgs does not transform along the patches because the
Higgs does not have spin and interaction with the $U(1)_{X}$ gauge field. By
use of the above facts and some specific information of this model,141414We
can find the details in Ref. Dohi:2010vc . we can show that the action in Eq.
(62) is equal at both the north and the south patches. Combining this result
with Eqs. (68)-(71), it is clear that the $Z_{2}$ symmetry is entailed on the
action in Eq. (62).
The specific forms of each KK expansion are as follows:
$\displaystyle\\{A^{(i)}_{\mu},X_{\mu}\\}(x,\theta,\phi)^{N\atop S}$
$\displaystyle=\frac{1}{\sqrt{4\pi}R}\\{A^{(i)(0)}_{\mu},X^{(0)}_{\mu}\\}(x)$
$\displaystyle\quad+\sum_{j=1}^{\infty}\sum_{m=0}^{j}\\{A^{(i)(j,m)}_{\mu},X^{(j,m)}_{\mu}\\}(x)\cdot(\sqrt{2}(i)^{j+m})f_{s=0,t=+1}^{(j,m)}(\theta,\phi)^{N\atop
S},$ (73) $\displaystyle\\{A^{(i)}_{\pm},X_{\pm}\\}(x,\theta,\phi)^{N\atop S}$
$\displaystyle=\sum_{j=1}^{\infty}\sum_{m=0}^{j}\\{A^{(i)(j,m)}_{\pm},X^{(j,m)}_{\pm}\\}(x)\cdot(\sqrt{2}(i)^{j+m+1})f_{s=\pm
1,t=-1}^{(j,m)}(\theta,\phi)^{N\atop S},$ (74) $\displaystyle
H(x,\theta,\phi)^{N\atop S}$
$\displaystyle=\frac{1}{\sqrt{4\pi}R}H^{(0)}(x)+\sum_{j=1}^{\infty}\sum_{m=0}^{j}H^{(j,m)}(x)\cdot\sqrt{2}f_{s=0,t=+1}^{(j,m)}(\theta,\phi)^{N\atop
S},$ (75) $\displaystyle\mathcal{Q}_{3-}(x,\theta,\phi)^{N\atop S}$
$\displaystyle=\begin{pmatrix}\displaystyle\frac{1}{\sqrt{4\pi}R}Q_{3L}^{(0)}(x)+\sum_{j=1}^{\infty}\sum_{m=0}^{j}Q_{3L}^{(j,m)}(x)\cdot\sqrt{2}f_{s=0,t=+1}^{(j,m)}(\theta,\phi)^{N\atop
S}\\\
\displaystyle\sum_{j=1}^{\infty}\sum_{m=0}^{j}Q_{3R}^{(j,m)}(x)\cdot\sqrt{2}f_{s=-1,t=-1}^{(j,m)}(\theta,\phi)^{N\atop
S}\end{pmatrix},$ (76) $\displaystyle\mathcal{U}_{3+}(x,\theta,\phi)^{N\atop
S}$
$\displaystyle=\begin{pmatrix}\displaystyle\frac{1}{\sqrt{4\pi}R}t_{R}^{(0)}(x)+\sum_{j=1}^{\infty}\sum_{m=0}^{j}t_{R}^{(j,m)}(x)\cdot\sqrt{2}f_{s=0,t=+1}^{(j,m)}(\theta,\phi)^{N\atop
S}\\\
\displaystyle\sum_{j=1}^{\infty}\sum_{m=0}^{j}t_{L}^{(j,m)}(x)\cdot\sqrt{2}f_{s=-1,t=-1}^{(j,m)}(\theta,\phi)^{N\atop
S}\end{pmatrix}.$ (77)
Here we introduce suitable normalization factor $(\sqrt{2})$ in each KK modes
and some phase factors ($(i)^{j+m},(i)^{j+m+1}$) in Eqs. (73,74) to ensure the
reality of these fields. The range of the summation over $m$ shrinks from
$[-j,j]$ to $[0,j]$ after the $Z_{2}$ identification. This system has two
fixed points of the $Z_{2}$ symmetry at
$(\theta,\phi)=(\frac{\pi}{2},0),(\frac{\pi}{2},\pi)$ and under the
transformation of $(\theta,\phi)\rightarrow(\theta,\phi+\pi)$, mode functions
behave as
$\displaystyle f_{s=0,t=+1}^{(j,m)}(\theta,\phi+\pi)^{N\atop S}$
$\displaystyle=(-1)^{m}f_{s=0,t=+1}^{(j,m)}(\theta,\phi)^{N\atop S},$
$\displaystyle f_{s=\pm 1,t=-1}^{(j,m)}(\theta,\phi+\pi)^{N\atop S}$
$\displaystyle=-(-1)^{m}f_{s=\pm 1,t=-1}^{(j,m)}(\theta,\phi)^{N\atop S}.$
(78)
Thereby after the fields redefinition as
$\displaystyle\\{A^{(i)(j,m)}_{\pm},X^{(j,m)}_{\pm},Q^{(j,m)}_{3R},t^{(j,m)}_{L}\\}\rightarrow(-1)\\{A^{(i)(j,m)}_{\pm},X^{(j,m)}_{\pm},Q^{(j,m)}_{3R},t^{(j,m)}_{L}\\},$
(79)
we can find that each KK field has KK parity $(-1)^{m}$, whose origin is
considered to be a remnant of KK angular momentum conservation.
We focus on the $m=0$ modes of each $j$ level. When we see the concrete forms
of mode functions in $m=0$, which are
$\displaystyle f_{s=0,t=+1}^{(j,m=0)}(\theta,\phi)^{N\atop S}$
$\displaystyle=\frac{1}{2R}(1+(-1)^{j})\cdot{}_{0}Y_{j0}(\theta,\phi),$ (80)
$\displaystyle f_{s=+1,t=-1}^{(j,m=0)}(\theta,\phi)^{N\atop S}$
$\displaystyle=\frac{1}{2R}(1+(-1)^{j})\cdot{}_{1}Y_{j0}(\theta,\phi)e^{\pm
i\phi},$ (81) $\displaystyle f_{s=-1,t=-1}^{(j,m=0)}(\theta,\phi)^{N\atop S}$
$\displaystyle=\frac{1}{2R}(1+(-1)^{j})\cdot{}_{-1}Y_{j0}(\theta,\phi)e^{\mp
i\phi},$ (82)
we find that $m=0$ modes appear only in the case of even $j$. Then the
degeneracy of KK masses is
$\begin{array}[]{cl}j+1&\text{for}\quad j=\text{even},\\\ j&\text{for}\quad
j=\text{odd},\end{array}$ (83)
since $m$ runs from $0$ to $j$. These results play an essential role at the
Higgs production and decay processes via loop diagrams.
After the $Z_{2}$ identification, the massless zero mode of $U(1)_{X}$ gauge
boson survives. In this model, it is assumed that the $U(1)_{X}$ symmetry is
anomalous and it is broken at the quantum level Scrucca:2003ra . Therefore
gauge bosons should be heavy and decoupled from the low energy physics.
### 4.3 UED on Projective Sphere
We can also construct a UED model based on a non-orbifolding idea in Ref.
Dohi:2010vc .151515 In Dohi:2010vc the terminology “real projective plane” is
employed for the compactified space, the sphere with its antipodal points
being identified. The projective sphere $(PS)$ is a sphere $S^{2}$ with its
antipodal points identified by $(\theta,\phi)\sim(\pi-\theta,\phi+\pi)$. In
the UED model based on ${PS}$, the 6D action takes a different form from that
of ordinary 6D UED model. It is written as follows:
$\displaystyle S$ $\displaystyle=\int_{0}^{\pi}d\theta\int_{0}^{2\pi}d\phi\int
d^{4}x\sqrt{-g}\Bigg{\\{}-\frac{1}{2}\sum_{i=1}^{3}\mathrm{g}^{MN}g^{KL}\text{Tr}\big{[}F_{MK}^{(i)}F^{(i)}_{NL}\big{]}-\frac{1}{4}g^{MN}g^{KL}\big{[}F_{MK}^{(X)}F^{(X)}_{NL}\big{]}$
$\displaystyle\qquad+g^{MN}(D_{M}H)^{\dagger}(D_{N}H)+\bigg{[}\mu^{2}|H|^{2}-\frac{\lambda_{6}^{(H)}}{4}|H|^{4}\bigg{]}$
$\displaystyle\qquad+\frac{1}{2}\bigg{[}i\bar{\mathcal{Q}}_{3-}\Gamma^{M}D_{M}\mathcal{Q}_{3-}+i\bar{\mathcal{Q}}_{3+}\Gamma^{M}D_{M}\mathcal{Q}_{3+}\bigg{]}+\frac{1}{2}\bigg{[}i\bar{\mathcal{U}}_{3+}\Gamma^{M}D_{M}\mathcal{U}_{3+}+i\bar{\mathcal{U}}_{3-}\Gamma^{M}D_{M}\mathcal{U}_{3-}\bigg{]}$
$\displaystyle\qquad-\frac{1}{2}\bigg{[}\lambda_{6}^{(t)}\Big{(}\bar{\mathcal{Q}}_{3-}(i\sigma_{2}H^{\ast})\mathcal{U}_{3+}+\bar{\mathcal{Q}}_{3+}(i\sigma_{2}H^{\ast})^{\text{T}}\mathcal{U}_{3-}\Big{)}+\mathrm{h.c.}\bigg{]}\Bigg{\\}}.$
(84)
Here the $``1/2"$ factors are introduced for a later convenience. Like the
$S^{2}/Z_{2}$ case, $F_{MN}^{(X)}$ has the classical part. A new feature of
this model is that we introduce “mirror” 6D Weyl fermions
$\\{\mathcal{Q}_{3+},\mathcal{U}_{3-}\\}$, which have opposite 6D chirality
and opposite SM and $U(1)_{X}$ charges when compared with the fields
$\\{\mathcal{Q}_{3-},\mathcal{U}_{3+}\\}$, respectively. And the covariant
derivatives in this model are given as
$\displaystyle D_{M}$
$\displaystyle=\partial_{M}-i\sum_{i=1}^{3}g_{6}^{(i)}T^{(i)a}A^{(i)a}_{M},$
$\displaystyle\text{for}\quad H,$ (85) $\displaystyle D_{M}$
$\displaystyle=\partial_{M}-i\sum_{i=1}^{3}g_{6}^{(i)}T^{(i)a}A^{(i)a}_{M}-ig_{6}^{(X)}q_{\Psi}(X^{c}_{M}+X_{M})+\Omega_{M},$
$\displaystyle\text{for}\quad\mathcal{Q}_{3-},\mathcal{U}_{3+},$ (86)
$\displaystyle D_{M}$
$\displaystyle=\partial_{M}-i\sum_{i=1}^{3}g_{6}^{(i)}\big{[}-T^{(i)a}\big{]}^{\text{T}}A^{(i)a}_{M}-ig_{6}^{(X)}q_{\Psi}(X^{c}_{M}+X_{M})+\Omega_{M},$
$\displaystyle\text{for}\quad\mathcal{Q}_{3+},\mathcal{U}_{3-}.$ (87)
The covariant derivative of the Higgs is the same with that in the
$S^{2}/Z_{2}$ case, but there is a difference between fermions and these
“mirror” fermions. We discuss these points shortly below.
${PS}$ is a non-orientable manifold and has no fixed point. Therefore, we
cannot perform identification like the $S^{2}/Z_{2}$ case. We focus on the 6D
$P$ and $CP$ transformations, which are defined as
$[\text{6D}\
P]=\left\\{\begin{array}[]{lcr}A_{\mu}(x,\theta,\phi)&\rightarrow&A_{\mu}(x,\pi-\theta,\phi+\pi),\\\
A_{\theta}(x,\theta,\phi)&\rightarrow&-A_{\theta}(x,\pi-\theta,\phi+\pi),\\\
A_{\phi}(x,\theta,\phi)&\rightarrow&A_{\phi}(x,\pi-\theta,\phi+\pi),\\\
\Psi(x,\theta,\phi)&\rightarrow&P\Psi(x,\pi-\theta,\phi+\pi),\\\
H(x,\theta,\phi)&\rightarrow&H(x,\pi-\theta,\phi+\pi),\end{array}\right.$ (88)
$[\text{6D}\
CP]=\left\\{\begin{array}[]{lcr}A_{\mu}(x,\theta,\phi)&\rightarrow&A_{\mu}^{C}(x,\pi-\theta,\phi+\pi),\\\
A_{\theta}(x,\theta,\phi)&\rightarrow&-A_{\theta}^{C}(x,\pi-\theta,\phi+\pi),\\\
A_{\phi}(x,\theta,\phi)&\rightarrow&A_{\phi}^{C}(x,\pi-\theta,\phi+\pi),\\\
\Psi(x,\theta,\phi)&\rightarrow&P\Psi^{C}(x,\pi-\theta,\phi+\pi),\\\
H(x,\theta,\phi)&\rightarrow&H^{\ast}(x,\pi-\theta,\phi+\pi).\end{array}\right.$
(89)
Like before, we consider $\Psi$ is a 6D fermion and the concrete shapes of 6D
$C$ and $P$ transformations are
$A_{M}^{C}=-A_{M}^{\text{T}}=-A_{M}^{\ast},\quad\Psi^{C}=\Gamma^{\underline{2}}\Gamma^{\underline{y}}\Psi^{\ast},\quad
P=\Gamma^{\underline{y}}.$ (90)
It must be noted that the monopole-like configuration of the $U(1)_{X}$ gauge
boson in Eq. (52) behaves under the antipodal identification as
$\\{X^{c}_{\phi}\\}^{N\atop
S}(x,\pi-\theta,\phi+\pi)=-\\{X^{c}_{\phi}\\}(x,\theta,\phi)^{S\atop
N}=\\{(X^{c}_{\phi})^{C}\\}(x,\theta,\phi)^{S\atop N}.$ (91)
We use a property of $U(1)$ gauge field $(X_{M}^{\text{T}}=X_{M})$. We can
notice that the monopole-like configuration is invariant under the 6D $CP$
transformation and transition between patches. Then we consider the
identification of the $U(1)_{X}$ gauge field as 161616We pay attention the
fact that identification conditions of classical field $(X_{\phi}^{c})$ and
quantum field $(X_{\phi})$ must be the same.
$\left\\{\begin{array}[]{lcc}X_{\mu}(x,\pi-\theta,\phi+\pi)^{N\atop
S}&=&X_{\mu}^{C}(x,\theta,\phi)^{S\atop N},\\\
X_{\theta}(x,\pi-\theta,\phi+\pi)^{N\atop
S}&=&-X_{\theta}^{C}(x,\theta,\phi)^{S\atop N},\\\
\\{X_{\phi}^{c},X_{\phi}\\}(x,\pi-\theta,\phi+\pi)^{N\atop
S}&=&\\{(X_{\phi}^{c})^{C},X_{\phi}^{C}\\}(x,\theta,\phi)^{S\atop
N}.\end{array}\right.$ (92)
These conditions ensure the monopole-like configuration after the antipodal
identification and projected out the non-desirable $U(1)_{X}$ zero mode. It is
clearly understood by the additional minus factor coming from the 6D CP
transformation of gauge field in Eq. (90). In contrast, since we want the zero
modes which describe the SM gauge bosons in UED model construction,
identification of $A^{(i)}_{M}$ should be done by another condition. We adopt
the 6D $P$ transformation and those identifications are written as
$\left\\{\begin{array}[]{lcc}A^{(i)}_{\mu}(x,\pi-\theta,\phi+\pi)^{N\atop
S}&=&A^{(i)}_{\mu}(x,\theta,\phi)^{S\atop N},\\\
A^{(i)}_{\theta}(x,\pi-\theta,\phi+\pi)^{N\atop
S}&=&-A^{(i)}_{\theta}(x,\theta,\phi)^{S\atop N},\\\
A^{(i)}_{\phi}(x,\pi-\theta,\phi+\pi)^{N\atop
S}&=&A^{(i)}_{\phi}(x,\theta,\phi)^{S\atop N},\end{array}\right.$ (93)
where it is evident that $A_{\mu}^{(i)}$’s zero mode survives. We also
identify the Higgs with the 6D $P$ transformation to obtain its zero mode as
$H(x,\pi-\theta,\phi+\pi)^{N\atop S}=H(x,\theta,\phi)^{S\atop N}.$ (94)
Finally, we discuss the identification of 6D Weyl fermions. Since 6D Weyl
fermions have $U(1)_{X}$ charge and interact with the $U(1)_{X}$ gauge boson,
they should be identified by the 6D $CP$ transformation. But if we do not
consider the “mirror” fermions, a fundamental problem arises. The 6D $P$
transformation of fermion changes the 6D chirality like the ordinary 4D
transformation. However, the 6D $C$ transformation of fermion does not change
the 6D chirality unlike the ordinary 4D case. This means 6D chirality flips
under the 6D $CP$ transformation and we should introduce the “mirror” fermions
with opposite 6D chirality and opposite SM and $U(1)_{X}$ charges to perform
identification. The specific forms are as follows:
$\\{\mathcal{Q}_{3+},\mathcal{U}_{3-}\\}(x,\pi-\theta,\phi+\pi)^{N\atop
S}=P\\{\mathcal{Q}_{3-}^{C},\mathcal{U}_{3+}^{C}\\}(x,\theta,\phi)^{S\atop
N}=\Gamma^{\underline{2}}\\{\mathcal{Q}_{3-}^{\ast},\mathcal{U}_{3+}^{\ast}\\}(x,\theta,\phi)^{S\atop
N}.$ (95)
And we determine the forms of the covariant derivatives in Eqs. (86,87) on the
criterion of invariance of the action under the 6D $CP$ transformation in
advance. Using the identification conditions in Eqs. (92)-(95), we can see
that the “mirror” fermions vanish from the action in Eq. (84) after the
identifications as
$\displaystyle S$
$\displaystyle\longrightarrow\int_{0}^{\pi}d\theta\int_{0}^{2\pi}d\phi\int
d^{4}x\sqrt{-g}\Bigg{\\{}-\frac{1}{2}\sum_{i=1}^{3}\mathrm{g}^{MN}g^{KL}\text{Tr}\big{[}F_{MK}^{(i)}F^{(i)}_{NL}\big{]}-\frac{1}{4}g^{MN}g^{KL}\big{[}F_{MK}^{(X)}F^{(X)}_{NL}\big{]}$
$\displaystyle\qquad+g^{MN}(D_{M}H)^{\dagger}(D_{N}H)+\bigg{[}\mu^{2}|H|^{2}-\frac{\lambda_{6}^{(H)}}{4}|H|^{4}\bigg{]}$
$\displaystyle\qquad+\bigg{[}i\bar{\mathcal{Q}}_{3-}\Gamma^{M}D_{M}\mathcal{Q}_{3-}\bigg{]}+\bigg{[}i\bar{\mathcal{U}}_{3+}\Gamma^{M}D_{M}\mathcal{U}_{3+}\bigg{]}-\bigg{[}\lambda_{6}^{(t)}\Big{(}\bar{\mathcal{Q}}_{3-}(i\sigma_{2}H^{\ast})\mathcal{U}_{3+}\Big{)}+\mathrm{h.c.}\bigg{]}\Bigg{\\}},$
(96)
and we obtain a usual type of UED model action.
Next we discuss the mass spectrum of the UED model on ${PS}$. Roughly
speaking, about a half of modes are projected out. First, we focus on the
$U(1)_{X}$ gauge boson. By use of properties of spin weighted spherical
harmonics, we can conclude that its identification conditions in terms of 4D
KK fields are as follows:
$\displaystyle X_{\mu}^{(j,m)}(x)$
$\displaystyle=(-1)^{j}(X^{(j,m)}_{\mu})^{\text{c}}(x)=(-1)^{j+1}(X^{(j,m)}_{\mu})(x),$
(97) $\displaystyle X_{\pm}^{(j,m)}(x)$
$\displaystyle=(-1)^{j+1}(X^{(j,m)}_{\mp})^{\text{c}}(x),$ (98)
$\displaystyle\phi^{(X)(j,m)}_{1}(x)$
$\displaystyle=(-1)^{j+1}(\phi^{(X)(j,m)}_{1})^{\text{c}}(x)=(-1)^{j}(\phi^{(X)(j,m)}_{1})(x),$
(99) $\displaystyle\phi^{(X)(j,m)}_{2}(x)$
$\displaystyle=(-1)^{j}(\phi^{(X)(j,m)}_{2})^{\text{c}}(x)=(-1)^{j+1}(\phi^{(X)(j,m)}_{2})(x),$
(100)
where the superscript ${}^{\text{c}}$ means 4D charge conjugation and has the
property that $(X_{M}^{(j,m)})^{\text{c}}(x)=-(X_{M}^{(j,m)})^{\text{T}}$.
$\phi_{1,2}^{(X)}$ are a 4D physical scalar field and an unphysical would-be
Nambu-Goldstone mode of $U(1)_{X}$ gauge field, respectively in Eq. (59). In
Eq. (97), it is clear that its unwanted zero mode is projected out correctly.
In ${PS}$ case, the range of the summation over $m$ does not shrink under the
identification and is still $[-j,j]$. This means that degeneracy of KK masses
is $2j+1$ in this model. But from Eqs. (97)-(100), we can find that the even
$j$ modes of both $X_{\mu}^{(j,m)}$ and $\phi^{(X)(j,m)}_{2}$ and the odd $j$
modes of $\phi^{(X)(j,m)}_{1}$ are projected out. The structure of these mass
spectrums is one of the most characteristic feature in the UED model on ${PS}$
and influences the rates of the Higgs production and decay processes via loop
diagrams.
Next, we go on to the gauge bosons $A_{M}^{(i)}$ and the Higgs $H$. These
field are identified by the 6D $P$ transformation and its identification
conditions in terms of 4D KK fields are as follows:
$\displaystyle A_{\mu}^{(i)(j,m)}(x)$
$\displaystyle=(-1)^{j}(A^{(i)(j,m)}_{\mu})(x),$ (101) $\displaystyle
A_{\pm}^{(i)(j,m)}(x)$ $\displaystyle=(-1)^{j+1}({A}^{(i)(j,m)}_{\mp})(x),$
(102) $\displaystyle\phi^{(i)(j,m)}_{1}(x)$
$\displaystyle=(-1)^{j+1}(\phi^{(i)(j,m)}_{1})(x),$ (103)
$\displaystyle\phi^{(i)(j,m)}_{2}(x)$
$\displaystyle=(-1)^{j}(\phi^{(i)(j,m)}_{2})(x).$ (104)
$H^{(j,m)}(x)=(-1)^{j}H^{(j,m)}(x).$ (105)
From Eqs. (101)-(105), it is obvious that the even $j$ modes of
$\phi^{(i)(j,m)}_{1}$ and the odd $j$ modes of
$A_{\mu}^{(i)(j,m)},\phi^{(i)(j,m)}_{2}$ and $H^{(j,m)}$ are projected out. As
a previous argument, the zero modes of $A_{\mu}^{(i)(j,m)}$ do not vanish.
Finally, 6D Weyl fermion must be discussed. It is important that the “mirror”
fermions are completely projected out from the action in Eq. (96) after the
antipodal identification. This is interpreted that all modes of the “mirror”
fermions $\\{\mathcal{Q}_{3+},\mathcal{U}_{3-}\\}$ are erased and no mode of
$\\{\mathcal{Q}_{3-},\mathcal{U}_{3+}\\}$ is projected out.
We comment on the dark matter candidate briefly. In this model, there is no
KK-parity because of lack of fixed points. But alternatively, the conservation
of KK angular momentum exists and it implies that the lightest KK particle is
stable.
## 5 Naive Dimensional Analysis
In 6D UED models, since the gluon fusion Higgs production and Higgs decay to
two photons processes are logarithmically divergent, we must consider upper
limit of the summations of KK number in such models. We review Naive
Dimensional Analysis (NDA) in these 6D models briefly. Following the concept
of NDA, a loop expansion parameter $\epsilon$ in D-dimensional SU(N) gauge
theory at a scale $\mu$ is obtained as
$\displaystyle\epsilon{(\mu)}$
$\displaystyle=\frac{1}{2}\frac{2\pi^{D/2}}{(2\pi)^{D}\Gamma(D/2)}N_{g}\,g_{Di}^{2}(\mu)\,{\Lambda_{\text{UED}}}^{D-4},$
(106)
where $N_{g}$ is a group index, $g_{Di}$ is a gauge coupling in $D$-dimensions
and ${\Lambda_{\text{UED}}}$ is a cutoff scale. The index $i$ is introduced to
express the type of gauge interaction and the remaining part is originated
from D-dimensional momentum loop integral. We should mention that
$g_{Di}({\mu})$, which is the effective running coupling, has energy
dependency and obeys power-of-two law scaling. When we consider a 6D theory
$(D=6)$ with two compact spacial directions, an effective 4D gauge coupling
$g_{i}$ emerges after KK decomposition and it is described with the volume of
two extra dimensions $V_{2}$ as $g_{i}={g_{6i}}/{\sqrt{V_{2}}}$. The cutoff
scale $\mu$ is the scale where the perturbation breaks down
$\epsilon({\Lambda_{\text{UED}}})\sim 1$. It is obvious that the upper bound
of ${\Lambda_{\text{UED}}}$ depends on the value of $V_{2}$, whose value is
$(2\pi R)^{2}$ in $T^{2}$ and $4\pi R^{2}$ in $S^{2}$, where $R$ is the radius
of $T^{2}$ or $S^{2}$.
Next we would like to focus on the behavior of running of the 4D effective
gauge coupling strength $\alpha_{i}({\Lambda_{\text{UED}}})$ along the energy.
We consider the following renormalization group equation:
$\displaystyle{\alpha_{4i}^{{-1}}(\mu)=\alpha_{4i}^{{-1}}(m_{Z})-\frac{\textsf{b}_{i}^{\text{SM}}}{2\pi}\ln{\mu\over
m_{Z}}+2C\,{\textsf{b}_{i}^{\text{6D}}\over 2\pi}\ln{\mu\over
M_{\text{KK}}}-C\,\frac{\textsf{b}_{i}^{\text{6D}}}{2\pi}\left[\left(\frac{\mu}{M_{\text{KK}}}\right)^{2}-1\right],}$
(107)
where $C$ represents $\pi/2\,(1)$ in the case of $T^{2}$ ($S^{2}$) geometry.
We note that the coefficient of the quadratic term for $T^{2}$ coincides with
that in Refs. Dienes:1998vh ; Dienes:1998vg obtained from a different
regularization scheme. The value of $C$ differs due to the structure of the
background geometry.171717Readers who are interested in the details see
Appendix in Ref. Nishiwaki:2011gm . In Eq. (107), we take a scheme of
approximation; masses of particles are almost degenerated in each KK level
regardless of type of the fields, the effect of KK particles appears after the
reference energy $\mu$ exceeds the value of $M_{\text{KK}}$.181818 When we
consider PS model with non-orientable manifold, there are differences in KK
spectrum of gauge and Higgs fields compared to that of the other “ordinary”
UED models as we discussed before. We ignore the effect coming from this in
our analysis. The coefficients are summarized in Table. 1.191919 Note that we
do not employ the GUT normalization for the $U(1)_{Y}$ coupling and the beta
function.
gauge group | SM contribution ($\textsf{b}^{\text{SM}}_{i}$) | KK contribution ($\textsf{b}^{\text{6D}}_{i}$)
---|---|---
$SU(3)_{C}$ | $\displaystyle-7$ | $-2$
$SU(2)_{W}$ | $\displaystyle-{19}/{6}$ | $\displaystyle{3}/{2}$
$U(1)_{Y}$ | $\displaystyle{41/6}$ | $\displaystyle{27}/{2}$
Table 1: Coefficients of renormalization group equation in Eq. (107).
Considering only the quadratic term, Eq. (107) reads
$\displaystyle\alpha_{4i}^{-1}(\Lambda_{\text{UED}})\sim\alpha_{4i}^{-1}({m_{Z}})-\frac{C\textsf{b}^{\text{6D}}_{i}}{{2}\pi}\frac{\Lambda_{\text{UED}}^{2}}{M_{\text{KK}}^{2}}.$
(108)
From Eq. (108) and $\epsilon({\Lambda_{\text{UED}}})\sim 1$, we get
$\displaystyle\Lambda_{\text{UED}}^{2}\sim{{4\pi M_{\text{KK}}^{2}\over
C\left(N_{g}+2\textsf{b}_{i}^{\text{6D}}\right)\alpha_{4i}(m_{Z})},}$ (109)
In the above analysis, we take values of $N_{g}$ as $3$, $2$ and $1$ in each
case of $SU(3)_{C}$, $SU(2)_{W}$ and $U(1)_{Y}$, respectively, and adopt some
latest data announced by Particle Data Group (PDG) as
$\left\\{\begin{array}[]{rcl}\alpha_{U(1)_{Y}}(m_{Z})^{-1}\big{|}_{\text{MS}}&=&97.99,\\\
\alpha_{SU(2)_{W}}(m_{Z})^{-1}\big{|}_{\text{MS}}&=&29.46,\\\
\alpha_{SU(3)_{C}}(m_{Z})^{-1}\big{|}_{\text{MS}}&=&8.445,\\\
m_{Z}&=&91.18\,\text{[GeV]}.\end{array}\right.$ (110)
We do not consider “TeV-scale gauge coupling unification” in this paper.
In the both $T^{2}$ and $S^{2}$ cases, the most stringent bounds come from the
$U(1)_{Y}$ cutoff scales, which restrict the effective range of the
perturbation the most severely. Therefore we can conclude that the “cutoff”
scales are as follows:
$\displaystyle{\Lambda_{\text{UED}}}$
$\displaystyle\lesssim{{5.3}\,M_{\text{KK}}},$ $\displaystyle\text{for
$T^{2}$-case}\ (V_{2}=(2\pi R)^{2},\,M_{\text{KK}}=1/R),$ (111)
$\displaystyle{\Lambda_{\text{UED}}}$
$\displaystyle\lesssim{{6.6}\,M_{\text{KK}}},$ $\displaystyle\text{for
$S^{2}$-case}\ (V_{2}=4\pi R^{2},\,M_{\text{KK}}=\sqrt{2}/R).$ (112)
We truncate the KK mode summations up to these upper bounds in each case to
regularize the process. Before going on to the concrete calculation, we have
to declare our choice of the UED cutoff scales. We choose three patterns in
$T^{2}$ and $S^{2}$ cases separately and the concrete forms are summarized in
Table 2. We also list up the value of the QCD and electromagnetic coupling
strengths ${\\{}\alpha_{s},\alpha_{\text{EM}}{\\}}$ at the cutoff scales by
use of Eq. (108) in Table 3. It is noted that the values derived form Eq.
(108) do not depend on the value of the KK mass scale $M_{\text{KK}}$ up to
our approximation in Eq. (108). The electromagnetic coupling strength is
defined by using $\alpha_{SU(2)_{W}}$ and $\alpha_{U(1)_{Y}}$ as
$\displaystyle\alpha_{\text{EM}}(\mu)^{-1}=\alpha_{SU(2)_{W}}(\mu)^{-1}+\alpha_{U(1)_{Y}}(\mu)^{-1}.$
(113)
| $T^{2}$-based | $S^{2}$-based
---|---|---
| high | low | high | low
KK index | $m^{2}+n^{2}\leq 30$ | $m^{2}+n^{2}\leq 10$ | $j(j+1)\leq 100$ | $j(j+1)\leq 30$
UV cutoff | $\Lambda_{\text{UED}}\sim 5{M_{\text{KK}}}$ | $\Lambda_{\text{UED}}\sim 3{M_{\text{KK}}}$ | $\Lambda_{\text{UED}}\sim 7{M_{\text{KK}}}$ | $\Lambda_{\text{UED}}\sim 4{M_{\text{KK}}}$
Table 2: Two choices of high and low upper bounds for KK indices and for the corresponding UV cutoff scale. | $T^{2}$-based | $S^{2}$-based
---|---|---
| high | low | high | low
$\alpha_{s}(\Lambda_{\text{UED}})^{{-1}}$ | $20.9$ | $12.9$ | $24.0$ | $13.5$
$\alpha_{\text{EM}}(\Lambda_{\text{UED}})^{{-1}}$ | $33.7$ | $93.7$ | $10.5$ | $89.3$
Table 3: The value of the QCD and electromagnetic coupling strengths at the
cutoff scales.
## 6 The deviation of the rates of Higgs production and its decay from the
standard model predictions
### 6.1 Formulation of calculation
From the discussions which we have done, we evaluate the ratio (fractional
deviation) of the Higgs production cross section through gluon fusion and the
Higgs decay width into two photons to the SM ones in the three types of 6D UED
models, which are denoted by $\mathcal{R}_{2g\rightarrow h^{(0)}}$ and
$\mathcal{R}_{h^{(0)}\rightarrow 2\gamma}$, respectively. These ratio are
represented as
$\mathcal{R}_{2g\rightarrow h^{(0)}}\equiv\frac{\sigma(2g\rightarrow h^{(0)};\
\text{UED})}{\sigma(2g\rightarrow h^{(0)};\
\text{SM})}=\left(1+{F_{t}^{{\text{KK}}}{+F_{\text{gluonfusion}}^{\text{TC}}}\over
F_{t}^{{\text{SM}}}}\right)^{2},$ (114) $\mathcal{R}_{h^{(0)}\rightarrow
2\gamma}\equiv\frac{\Gamma(h^{(0)}\rightarrow 2\gamma;\
\text{UED})}{\Gamma(h^{(0)}\rightarrow 2\gamma;\
\text{SM})}=\left(1+{F_{W}^{{\text{KK}}}+3Q_{t}^{2}F_{t}^{{\text{KK}}}{+F_{\text{decay}}^{\text{TC}}}\over
F_{W}^{{\text{SM}}}+3Q_{t}^{2}F_{t}^{{\text{SM}}}}\right)^{2}.$ (115)
We have obtained
$F_{W}^{{\text{KK}}},F_{t}^{{\text{KK}}},{F_{\text{gluonfusion}}^{\text{TC}}}$
and ${F_{\text{decay}}^{\text{TC}}}$ in Section 3 in the case of $T^{2}/Z_{4}$
by 1-loop calculation and we can apply these results for the $S^{2}/Z_{2}$ and
the ${PS}$ cases with some modifications. It is important that the $U(1)_{X}$
gauge boson does not contribute to either the production process and the decay
process at the 1-loop level. Therefore no new type of diagram appears and only
difference appears in the KK mass spectrum and the multiplicity of each KK
mode. Once the $Z_{2}$ orbifolding or the antipodal identification is
understood, the structure of KK mass spectrum itself is the same as the case
of $S^{2}$ up to degeneracy. We summarize the information which is needed for
the estimation in Table 4.
type of field | $S^{2}/Z_{2}$ case | ${PS}$ case
---|---|---
fermion | $\begin{array}[]{cl}j+1&\text{for}\ j=\text{even}\\\ j&\text{for}\ j=\text{odd}\end{array}$ | $\begin{array}[]{cl}2j+1&\text{for}\ j=\text{even}\\\ 2j+1&\text{for}\ j=\text{odd}\end{array}$
“mirror” fermion | N/A | $\begin{array}[]{cl}0&\text{for}\ j=\text{even}\\\ 0&\text{for}\ j=\text{odd}\end{array}$
| gauge boson & would-be NG boson
---
& scalar(Higgs)
$\begin{array}[]{cl}j+1&\text{for}\ j=\text{even}\\\ j&\text{for}\ j=\text{odd}\end{array}$ | $\begin{array}[]{cl}2j+1&\text{for}\ j=\text{even}\\\ 0&\text{for}\ j=\text{odd}\end{array}$
scalar(“spinless adjoint”) | $\begin{array}[]{cl}j+1&\text{for}\ j=\text{even}\\\ j&\text{for}\ j=\text{odd}\end{array}$ | $\begin{array}[]{cl}0&\text{for}\ j=\text{even}\\\ 2j+1&\text{for}\ j=\text{odd}\end{array}$
Table 4: Multiplicities of fields at j level in $S^{2}$-based UED models.
In the $S^{2}/Z_{2}$ case, the KK state multiplicity is the same irrespective
of the type of field. With the modification
${\sum_{m\geq 1,n\geq 0}\rightarrow\sum_{j\geq 1}}\ n_{{S^{2}/Z_{2}}}(j),\quad
m_{(m,n)}\rightarrow m_{(j,m)},$ (116)
where $n_{S^{2}/Z_{2}}(j)$ shows the multiplicity of each level of KK modes
($j+1$ for $j=\text{even}$ or $j$ for $j=\text{odd}$) and $m_{(j,m)}$ is the
KK mass on $S^{2}$ in Eq. (60), we can obtain the results as follows:
$\displaystyle F_{t}^{{\text{KK}}}$ $\displaystyle=2\sum_{j\geq
1}n_{S^{2}/Z_{2}}(j)\left(\frac{m_{t}}{m_{t,(j,m)}}\right)^{2}$
$\displaystyle\quad\times\left\\{{-2}\lambda(m_{t,(j,m)}^{2}){+}\lambda(m_{t,(j,m)}^{2})(1-4\lambda(m_{t,(j,m)}^{2}))J\left(\lambda(m_{t,(j,m)}^{2})\right)\right\\},$
(117) $\displaystyle F_{W}^{{\text{KK}}}$ $\displaystyle=\sum_{j\geq
1}n_{S^{2}/Z_{2}}(j)\Big{\\{}\frac{1}{2}+5\lambda(m_{W}^{2})-\Big{[}\lambda(m_{W}^{2})(4-10\lambda(m_{W,(j,m)}^{2}))$
$\displaystyle\qquad\qquad-\lambda(m_{W,(j,m)}^{2})\Big{]}J\left(\lambda(m_{W,(j,m)}^{2})\right)\Big{\\}}{,}$
(118)
where we use $J(m^{2})$ in Eq. (36). In ${PS}$ case, we should pay attention
to the KK state multiplicity of each type of field. There is no contribution
from the “mirror” fermions. The concrete forms are as follows:
$\displaystyle{F_{t}^{\text{KK}}}$ $\displaystyle=2\sum_{j\geq
1}(2j+1)\left(\frac{m_{t}}{m_{t,(j,m)}}\right)^{2}$
$\displaystyle\quad\times\left\\{{-2}\lambda(m_{t,(j,m)}^{2}){+}\lambda(m_{t,(j,m)}^{2})(1-4\lambda(m_{t,(j,m)}^{2}))J\left(\lambda(m_{t,(j,m)}^{2})\right)\right\\},$
(119) $\displaystyle F^{{\text{KK}}}_{\text{gauge}}$
$\displaystyle={\sum_{j\geq
1}n_{PS\text{even}}(j)\left\\{3\lambda(m_{W}^{2})+2\lambda(m_{W}^{2})\big{(}3\lambda(m_{W,(j,m)}^{2})-2\big{)}J\left(\lambda(m_{W,(j,m)}^{2})\right)\right\\}},$
(120) $\displaystyle F^{{\text{KK}}}_{\text{NG}}$ $\displaystyle={\sum_{j\geq
1}n_{PS\text{even}}(j)\left(\frac{1}{2}\frac{m_{h}^{2}}{m_{W,(j,m)}^{2}}\right)\lambda(m_{W}^{2})\left\\{1+2\lambda(m_{W,(j,m)}^{2})J\left(\lambda(m_{W,(j,m)}^{2})\right)\right\\}},$
(121) $\displaystyle F^{{\text{KK}}}_{\text{scalar1}}$
$\displaystyle=\sum_{j\geq
1}n_{PS\text{even}}(j)\Big{(}\frac{1}{2}\frac{1}{m_{W,(j,m)}^{2}}\Big{)}\Big{[}\frac{m_{h}^{2}}{m_{W}^{2}}m_{(j,m)}^{2}+{2}m_{W,(j,m)}^{2}\Big{]}$
$\displaystyle\quad\times\lambda(m_{W}^{2})\left\\{1+2\lambda(m_{W,(j,m)}^{2})J\left(\lambda(m_{W,(j,m)}^{2})\right)\right\\},$
(122) $\displaystyle F^{{\text{KK}}}_{\text{scalar2}}$
$\displaystyle={\sum_{j\geq
1}n_{PS\text{odd}}(j)\lambda(m_{W}^{2})\left\\{1+2\lambda(m_{W,(j,m)}^{2})J\left(\lambda(m_{W,(j,m)}^{2})\right)\right\\}},$
(123)
where $n_{PS\text{even}}(j)$: $2j+1$ for $j=$even, $0$ for $j=$odd and
$n_{PS\text{odd}}(j)$: $0$ for $j=$even, $2j+1$ for $j=$odd. We have already
discussed the cutoff scale in both the $T^{2}$ and $S^{2}$ cases concretely in
Section 5 and we are ready to estimate the ratio in the various 6D UED models.
### 6.2 Results without threshold corrections
The numerical results of the ratios of the production cross section via gluon
fusion to the standard model prediction $\mathcal{R}_{2g\rightarrow h^{(0)}}$
are given as functions of the first KK mass scale $(M_{\text{KK}})$ in a unit
of GeV in Fig. 5. In this paper, we consider two possibilities of Higgs mass;
$m_{h}=120\,\text{GeV}$ and $m_{h}=145\,\text{GeV}$ and take the KK mass range
between $600\,\text{GeV}$ and $2000\,\text{GeV}$. We use the values of the W
boson mass $m_{W}$ and the top quark mass $m_{t}$, which are
$m_{W}=80.3\,\text{GeV},m_{t}=173\,\text{GeV}$.
From top to bottom, the green, blue, red curves represent the results of
${PS}$, $S^{2}/Z_{2}$, $T^{2}/Z_{4}$ with $m_{h}=120\,\,\text{GeV}$, providing
no threshold correction, respectively. Each black dashed line near the lines
for $m_{h}=120\,\text{GeV}$ corresponds to that with $m_{h}=145\,\text{GeV}$.
The left, right are these with the high, low cutoff choices, respectively. It
is noted that the $\mathcal{R}_{2g\rightarrow h^{(0)}}=1$ shows the SM
predictions and there are few differences between $m_{h}=120\,\text{GeV}$ case
and $m_{h}=145\,\text{GeV}$ case in all the models. Contrast to the case of
little Higgs Han:2003gf ; Dib:2003zj ; Chen:2006cs or gauge-Higgs unification
Falkowski:2007hz ; Maru:2007xn ; Maru:2008cu , the contribution from KK
fermions is constructive and the results of UED cases are enhanced compared to
the SM prediction. These results are naturally understood because the number
of the intermediate particles are much greater than these of the
SM.202020These results are consistent with the results in Ref. Maru:2009cu .
It is expected that 6D UED models predict a significant collider signature in
the Higgs production at the LHC, especially in the ${PS}$ case. The origin of
the remarkable enhancement in the ${PS}$ case is that numerous fermions
contribute to the production process in each KK level. Besides, we can find
the fact that when we choose the higher cutoff, the larger number of KK top
modes propagate in the triangle loop and therefore the deviation from the SM
gets significant. This tendency do not depend on the type of the 6D UED
models.
Figure 5: These plots represent the ratios of the Higgs boson production
cross sections via gluon fusion to the SM prediction
$\mathcal{R}_{2g\rightarrow h^{(0)}}$ in 6D UED on ${PS}$(green),
$S^{2}/Z_{2}$(blue), $T^{2}/Z_{4}$(red) with $m_{h}=120\,\text{GeV}$ providing
no threshold correction from top to bottom. Each black dashed line near the
lines for $m_{h}=120\,\text{GeV}$ corresponds to that with
$m_{h}=145\,\text{GeV}$. The left, right are these with the high, low cutoff
choices, respectively.
The numerical results of the ratios of the rate of Higgs decay into two
photons $\mathcal{R}_{h^{(0)}\rightarrow 2\gamma}$ are also given as functions
of the first KK mass scale ($M_{\text{KK}}$) in a unit of GeV in Fig. 6. From
bottom to top, the green, blue, red curves represent the results of ${PS}$,
$S^{2}/Z_{2}$, $T^{2}/Z_{4}$ with $m_{h}=120\,\,\text{GeV}$, providing no
threshold correction, respevtively. Each black dashed line located above the
lines for $m_{h}=120\,\text{GeV}$ corresponds to that with
$m_{h}=145\,\text{GeV}$. The left, right are these with the high, low cutoff
choices, respectively. Differently from the production, the ratios are
suppressed because the contributions from quarks and gauge bosons are
destructive each other. The reason of the large reduction in the ${PS}$ case
is understood as the results of the enormous effects of KK top quarks, which
we discussed before. In any type of 6D UED models, this ratio takes the lower
value than 5D mUED one. We can find some differences between
$m_{h}=120\,\text{GeV}$ and $m_{h}=145\,\text{GeV}$ in each case of 6D UED
model, which are sizable in particular at the KK mass range between
$600\,\text{GeV}$ and $1200\,\text{GeV}$.
Figure 6: These plots represent the ratios of the Higgs boson decay width to
two photons to the SM prediction $\mathcal{R}_{h^{(0)}\rightarrow 2\gamma}$ in
6D UED on ${PS}$(green), $S^{2}/Z_{2}$(blue), $T^{2}/Z_{4}$(red) with
$m_{h}=120\,\text{GeV}$ providing no threshold correction from bottom to top.
Each black dashed line located above the lines for $m_{h}=120\,\text{GeV}$
corresponds to that with $m_{h}=145\,\text{GeV}$. The left, right are these
with the high, low cutoff choices, respectively.
We now define a value defined as
$\displaystyle\Delta\equiv\mathcal{R}_{2g\rightarrow
h^{(0)}}\times\mathcal{R}_{h^{(0)}\rightarrow 2\gamma},$ (124)
which shows the “total ratio” of the deviation of the $h^{(0)}\rightarrow
2\gamma$ signals coming from the $2g\rightarrow h^{(0)}$ Higgs production. At
the LHC, the Higgs production process through gluon fusion is dominant and the
value of $\Delta$ is considered to be an appropriate approximation of the
$h^{(0)}\rightarrow 2\gamma$ signal deviation coming from all the Higgs
production processes. Of course the numerical results of $\Delta$ are given as
functions of the first KK mass scale ($M_{\text{KK}}$) in a unit of GeV and
are shown in Fig. 7. It should be noted that in 6D UED models, the collider
signal deviations from the SM in the $h^{(0)}\rightarrow 2\gamma$ process take
the greater values than in the 5D mUED case. When we take the reference value
as ${M_{\text{KK}}}=800$ GeV in the $m_{h}=120$ GeV and each high cutoff case,
approximately $40\%(T^{2}/Z_{4})$, $60\%(S^{2}/Z_{2})$, $110\%({PS})$
enhancements from the SM expectation value can be seen. It should be mentioned
that the shapes of $\Delta$ in each model do not have large dependence on the
value of the UED cutoff $\Lambda_{\text{UED}}$. This reason can be considered
that the behavior of the ratios of the gluon fusion Higgs production and the
Higgs decay to two photons is opposite when we change the value of the cutoff
and a large part of the distinctions due to the value of the cutoff are
cancelled out. This property is accidental but do not depend on the type of
the background geometry and thereby we consider that this is one of the
interesting aspects of 6D UED model. The difference between the above results
and the SM expectation value is significant and we hope that this could be
tested at the LHC experiments in the near future.
Finally, we comment on the up-to-date collider experimental results at the
LHC. The ATLAS group announced their results, which conclude the upper limit
of the cross section of the $h^{(0)}\rightarrow 2\gamma$ process in the form
of the ratio to the SM result $(\sigma/\sigma_{\text{SM}})$ based on the
$1.7\,\text{fb}^{-1}$ data within the $95\%$ confidence level in the August of
2011. According to ATLAS-CONF-2011-135 , the value of the upper bound of
$(\sigma/\sigma_{\text{SM}})$ is about $3.5$ $(5.0)$ at the point of
$m_{h}=120\,\text{GeV}$ ($m_{h}=145\,\text{GeV}$). The CMS group also
announced their results, which says that the value of the upper bound of
$(\sigma/\sigma_{\text{SM}})$ is about $3.5$ $(4.0)$ at the point of
$m_{h}=120\,\text{GeV}$ ($m_{h}=145\,\text{GeV}$) CMS_PAS_HIG-11-022 . And at
the December of 2011, the new results have been published by both the ALTAS
and CMS. The ATLAS claims that there is an excess of events close to 126 GeV
with a 3.6 $\sigma$ confidence ATLAS:2012ae . On the other hand, the excess
also have been observed by the CMS, but the location of the peak is 124 GeV
with a 3.1 $\sigma$ confidence Chatrchyan:2012tx . It is noted that both
results are these before taking looking-elsewhere effect. The allowed region
of the SM Higgs becomes highly constrained as
$115.5\,\,\text{GeV}<m_{h}<127\,\,\text{GeV}$ except the unexplored high mass
region $m_{h}>600\,\,\text{GeV}$.
We do not execute detailed analysis in this paper on this topics but we can
conclude from our result in Fig. 7 that the $T^{2}/Z_{4}$, $S^{2}/Z_{2}$, and
$PS$ 6D UED with $m_{h}=120\,\,\text{GeV}$ still survive only in the KK mass
region above ${M_{\text{KK}}=600,750,1150\,\text{GeV}}$, respectively, when we
consider the high cutoffs, judging from the constraint on the value of
$\sigma/\sigma_{\text{SM}}$ in the December’s ALTAS and CMS results, whose
maximum value is roughly 1.6.212121 We note that the newer CMS diphoton data
set includes vector boson fusion (VBF) events that occurs at the tree level in
the SM. The VBF Higgs production process is not significantly enhanced by the
UED loop effects.
It is obvious that the possibility of $m_{h}=145\,\,\text{GeV}$ is discarded
in all the 6D UED models since the signals are expected to be greater than
these in the SM.
Figure 7: These plots represent the values of $\Delta$ (total ratio) in 6D
UED on ${PS}$(green), $S^{2}/Z_{2}$(blue), $T^{2}/Z_{4}$(red) with
$m_{h}=120\,\text{GeV}$ providing each high cutoff case and no threshold
correction from top to bottom. Each black dashed line located above the lines
for $m_{h}=120\,\text{GeV}$ corresponds to that with $m_{h}=145\,\text{GeV}$.
The left, right are these with the high, low cutoff choices, respectively.
### 6.3 Results with threshold corrections
When we switch on the threshold corrections accompanying the processes of
$2g\rightarrow h^{(0)}$ and $h^{(0)}\rightarrow 2\gamma$, the shapes of the
ratios ${\\{}\mathcal{R}_{2g\rightarrow
h^{(0)}},\mathcal{R}_{h^{(0)}\rightarrow 2\gamma}{\\}}$ and the total ratios
$\Delta$ are modified forcefully. There are two dimensionless new parameters
in Eqs. (39,50) $C_{hgg},C_{h\gamma\gamma}$, which describe the threshold
correction in the process of $2g\rightarrow h^{(0)}$ or $h^{(0)}\rightarrow
2\gamma$, respectively. In this paper, we only consider some extremal choices
of $C_{hgg},C_{h\gamma\gamma}$ as
$\displaystyle C_{hgg}=0,\pm 1,\quad C_{h\gamma\gamma}=0,\pm 1.$ (125)
We mention that the plus/minus sign of $C_{hgg},C_{h\gamma\gamma}$ determines
the direction of interference effects to the UED part. We show the results of
our numerical calculations in Figs. 8-16. We write down our convention about
the color/shape of curves in Figs. 14,15,16 (total ratios) in Table 5. In the
range of our approximation, the values of $\alpha_{s}^{-1}(m_{h})$ and
$\alpha_{\text{EM}}^{-1}(m_{h})$ only appears in the terms describing the
threshold corrections in Eqs. (39,50) and we adopt these values at the Z boson
mass pole as $\alpha_{s}^{-1}(m_{Z})=8.44,\alpha_{\text{EM}}^{-1}(m_{Z})=127$
with ignoring the renormalization group effects between $m_{Z}$ and $m_{h}$
$(=120\ \text{or}\ 145\,\,\text{GeV})$. We make several comments in order.
* •
In the gluon fusion process in each model, due to the $(-1)$ factor which
originates from Fermi statistics in $F_{t}^{\text{SM}}$, the interference term
between the threshold correction and the UED effect is destructive
(constructive) in case of $C_{hgg}=+1\ (-1)$, respectively. It is considered
that the degree of a threshold correction is inversely proportional to the
value of a cutoff. When we look at the PS case with its high cutoff choice in
Fig. 10, we notice that the threshold correction works a little compared to
the cases of the $T^{2}/Z_{4}$ or $S^{2}/Z_{2}$. We can find some differences
between the cases of $PS$ and $S^{2}/Z_{2}$ with a same cutoff value, which
stem from the differences in the corresponding UED contributions. We mention
that the threshold correction is still observable in the many cases with
$M_{\text{KK}}=2\,\text{TeV}$. By contraries, in all the low cutoff cases in
Figs. 8,9,10, the threshold correction plays a very important role. Here we
mention that the cases with $m_{h}=145\,\text{GeV}$ are almost identical with
these with $m_{h}=120\,\text{GeV}$.
* •
In the Higgs decay to two photons in each model, unlike with the previous
gluon fusion case, the interference term between the threshold correction and
the UED effect is constructive (destructive) in case of $C_{h\gamma\gamma}=+1\
(-1)$, respectively. In this case, the degree of the effect which only comes
from the threshold correction is also smaller than the others. It is an
interesting point that in some cases with $C_{g\gamma\gamma}=+1$, the value of
the ratio $(\mathcal{R}_{h^{(0)}\rightarrow 2\gamma})$ exceeds one, which we
never find in the no-threshold-correction cases in the range of the parameter
region of $M_{\text{KK}}$ which we consider. Another remarkable point compared
to the gluon fusion, the cases with $m_{h}=145\,\text{GeV}$ are not identical
with these with $m_{h}=120\,\text{GeV}$ but this difference is still not so
significant since the other effects (cutoff scale, threshold correction and so
on) are more effective. Of course, in all the low cutoff cases in Figs.
11,12,13, the threshold correction works very well. We mention that we can
find the $10\sim 20\,\%$ deviations from the no-threshold-correction cases
even with $M_{\text{KK}}=2\,\text{TeV}$ in all the 6D UED models.
* •
After combining the above two results, we can estimate the total ratio
$\Delta$ in each 6D UED model with the extremal threshold corrections. In this
analysis, we only consider the $m_{h}=120\,\text{GeV}$ cases. There are nine
curves in each graph and our convention about the color/shape of curves is
summarized in Table 5. Here we would like to only focus on a few important
topics. Firstly, we can find the tendency that every orange line
$(C_{hgg}=-1,C_{h\gamma\gamma}=+1)$ is located at the top of each graph and
any cyan one $(C_{hgg}=+1,C_{h\gamma\gamma}=-1)$ is located at the bottom of
each graph. The reason is that the two threshold corrections function toward
maximally enhancing (suppressing) the process in the former (latter) case.
Secondly, the deviation from the no-threshold-correction case (black dot-
dashed curve) is noteworthy, in particular, in the $M_{\text{KK}}$ range below
$1\,\text{TeV}$. All the results tend to converge with the no-threshold-
correction curve proportional to the value of $M_{\text{KK}}$ and it is
notable even around $M_{\text{KK}}=2\,\text{TeV}$ in many choices of $C_{hgg}$
and $C_{h\gamma\gamma}$ because the tens of percents of the deviations still
remain. Thirdly, in the low cutoff cases, the interference effects dominate
the whole process and the predictions about two photon signals via the gluon
fusion Higgs production possibly become extraordinary. Finally we comment on
the constraint from the LHC results briefly. We also do not execute detailed
analysis in this case but we can conclude that some cases which predict too
great value of the total ratio are already excluded. On the other hand, the
total ratio can be suppressed in some choices of the parameters describing the
threshold corrections, and in this case the possibility with
$m_{h}=145\,\,\text{GeV}$ is not totally rejected.
At the end of this section, we give comments for more precise analysis. We
need to take into account the correction from QCD (parton distribution
function and K-factor) Rai:2005vy . However the KK contributions would receive
almost the same QCD corrections as in the case of the SM and this deviation
from the SM result would not be large. Actually, to get the ratios of event
rates in 6D UED models to that in the SM, the partial decay width in $\Delta$
should be replaced by corresponding branching ratios. We, however, expect that
the effects of heavy KK particles to the leading decay processes at the tree
level are small because of decoupling. Thus $\Delta$ is expected to be enough
for crude estimation. But there are two other one-loop leading decay processes
of $h^{(0)}\rightarrow 2g$ and $h^{(0)}\rightarrow\gamma Z$, which may
possibly give considerable contribution to the Branching Ratio. These points
are beyond the scope of this paper and are left for future work
Nishiwaki:2011gk .
## 7 Summary
In this paper, we have discussed the main Higgs production process through
gluon fusion and the important one-loop leading decay channel to two photons
at the LHC in various 6D UED models. The Higgs production cross sections in 6D
UED models are much enhanced than the prediction of the SM or the 5D mUED. In
contrast, the decay width in 6D UED models are decreased because of the
destructive contribution between quarks and gauge bosons. In both cases, the
results of ${PS}$ model are significant. This is because numerous fermions
contribute to the process in each level of KK modes. We also have discussed
the threshold corrections in the processes and their effects become
significant even when we take the higher cutoff and/or a heavy KK scale. Some
parameter region are already excluded by the current LHC experimental results
obviously. By use of the data announced by the ALTAS and CMS experiments in
the December of 2011, we can estimate the lower bounds of the KK scale as
$M_{\text{KK}}=600\,(T^{2}/Z_{4}),750\,(S^{2}/Z_{2}),1150\,(PS)\,\text{GeV}$
when we consider the high cutoffs with no threshold correction. These results
are modified by the threshold corrections substantially. The SM with the
$145\,\,\text{GeV}$ Higgs boson is rejected but 6D UED with the Higgs mass
parameter is still survived in some ranges of the parameters describing the
threshold corrections.
Our results are affected by ultraviolet physics because the calculation has
logarithmic cutoff scale dependence. There seem to be some ambiguities coming
from this fact. We expect that our prediction would be verified by forthcoming
LHC experimental results. Detailed analysis of the final states in the single
Higgs production at the LHC is important for discriminating UED from the other
models.
The collider physics and particle cosmology of $S^{2}$-based 6D UED models are
unexplored and we would like to pursue these topics in future work
Nishiwaki:2011gm ; Nishiwaki:2011gk .
Acknowledgments
We are most grateful to C. S. Lim and Kin-ya Oda for valuable comments and
discussions. In particular C. S. Lim red the manuscript carefully and gave us
very useful comments. And we also thank Nobuhito Maru, Naoya Okuda, Makoto
Sakamoto, Joe Sato, Takashi Shimomura, Ryoutaro Watanabe and Masato Yamanaka
for fruitful discussions. Yasuhiro Okada and Hideo Ito suggest the recent
Tevatron experimental results to us. We express our appreciation to them very
much. We again appreciate C. S. Lim and Kin-ya Oda for advising me in the
revision. Finally we appreciate the referee for giving a lot of useful
comments.
Figure 8: These plots represent the values of the ratios $\mathcal{R}_{2g\rightarrow h^{(0)}}$ in 6D UED on $T^{2}/Z_{4}$ with/without threshold correction. The red, blue, green curves show these with $C_{hgg}=0,+1,-1$, respectively. Each black dashed line near the lines for $m_{h}=120\,\text{GeV}$ corresponds to that with $m_{h}=145\,\text{GeV}$. In the left and right plots, which correspond to the high and low cutoff cases, respectively, we take the value of the QCD coupling strength as that at the Z boson mass scale. Figure 9: These plots represent the values of the ratios $\mathcal{R}_{2g\rightarrow h^{(0)}}$ in 6D UED on $S^{2}/Z_{2}$ with/without threshold correction. The red, blue, green curves show these with $C_{hgg}=0,+1,-1$, respectively. Each black dashed line near the lines for $m_{h}=120\,\text{GeV}$ corresponds to that with $m_{h}=145\,\text{GeV}$. In the left and right plots, which correspond to the high and low cutoff cases, respectively, we take the value of the QCD coupling strength as that at the Z boson mass scale. Figure 10: These plots represent the values of the ratios $\mathcal{R}_{2g\rightarrow h^{(0)}}$ in 6D UED on $PS$ with/without threshold correction. The red, blue, green curves show these with $C_{hgg}=0,+1,-1$, respectively. Each black dashed line near the lines for $m_{h}=120\,\text{GeV}$ corresponds to that with $m_{h}=145\,\text{GeV}$. In the left and right plots, which correspond to the high and low cutoff cases, respectively, we take the value of the QCD coupling strength as that at the Z boson mass scale. Figure 11: These plots represent the values of the ratios $\mathcal{R}_{h^{(0)}\rightarrow 2\gamma}$ in 6D UED on $T^{2}/Z_{4}$ with/without threshold correction. The red, blue, green curves show these with $C_{h\gamma\gamma}=0,+1,-1$, respectively. Each black dashed line near the lines for $m_{h}=120\,\text{GeV}$ corresponds to that with $m_{h}=145\,\text{GeV}$. In the left and right plots, which correspond to the high and low cutoff cases, respectively, we take the value of the electromagnetic coupling strength as that at the Z boson mass scale. Figure 12: These plots represent the values of the ratios $\mathcal{R}_{h^{(0)}\rightarrow 2\gamma}$ in 6D UED on $S^{2}/Z_{2}$ with/without threshold correction. The red, blue, green curves show these with $C_{h\gamma\gamma}=0,+1,-1$, respectively. Each black dashed line near the lines for $m_{h}=120\,\text{GeV}$ corresponds to that with $m_{h}=145\,\text{GeV}$. In the left and right plots, which correspond to the high and low cutoff cases, respectively, we take the value of the electromagnetic coupling strength as that at the Z boson mass scale. Figure 13: These plots represent the values of the ratios $\mathcal{R}_{h^{(0)}\rightarrow 2\gamma}$ in 6D UED on $PS$ with/without threshold correction. The red, blue, green curves show these with $C_{h\gamma\gamma}=0,+1,-1$, respectively. Each black dashed line near the lines for $m_{h}=120\,\text{GeV}$ corresponds to that with $m_{h}=145\,\text{GeV}$. In the left and right plots, which correspond to the high and low cutoff cases, respectively, we take the value of the electromagnetic coupling strength as that at the Z boson mass scale. value of $C_{hgg}$ | value of $C_{h\gamma\gamma}$ | color/shape of curve
---|---|---
$C_{hgg}=0$ | $C_{h\gamma\gamma}=0$ | black, dot-dashed
$C_{hgg}=+1$ | $C_{h\gamma\gamma}=0$ | red
$C_{hgg}=0$ | $C_{h\gamma\gamma}=+1$ | blue
$C_{hgg}=-1$ | $C_{h\gamma\gamma}=0$ | green
$C_{hgg}=0$ | $C_{h\gamma\gamma}=-1$ | magenta
$C_{hgg}=+1$ | $C_{h\gamma\gamma}=+1$ | yellow, dotted
$C_{hgg}=-1$ | $C_{h\gamma\gamma}=+1$ | orange, dotted
$C_{hgg}=+1$ | $C_{h\gamma\gamma}=-1$ | cyan, dotted
$C_{hgg}=-1$ | $C_{h\gamma\gamma}=-1$ | brown, dotted
Table 5: Our convention about the color/shape of curves in Figs. 14,15,16
(total ratios). Figure 14: These plots represent the values of the total
ratios $\Delta$ in 6D UED on $T^{2}/Z_{4}$ with/without threshold corrections
with $m_{h}=120\,\text{GeV}$. The color/shape convention is summarized in
Table 5. In the left and right plots, which correspond to the high and low
cutoff cases, respectively, we take the values of the QCD and electromagnetic
coupling strengths as that at the Z boson mass scale. Figure 15: These plots
represent the values of the total ratios $\Delta$ in 6D UED on $S^{2}/Z_{2}$
with/without threshold corrections with $m_{h}=120\,\text{GeV}$. The
color/shape convention is summarized in Table 5. In the left and right plots,
which correspond to the high and low cutoff cases, respectively, we take the
values of the QCD and electromagnetic coupling strengths as that at the Z
boson mass scale. Figure 16: These plots represent the values of the total
ratios $\Delta$ in 6D UED on $PS$ with/without threshold corrections with
$m_{h}=120\,\text{GeV}$. The color/shape convention is summarized in Table 5.
In the left and right plots, which correspond to the high and low cutoff
cases, respectively, we take the values of the QCD and electromagnetic
coupling strengths as that at the Z boson mass scale.
## Appendix A Feynman Rules containing scalar particle
In this appendix, we list the Feynman rules containing scalar particle in the
’t Hooft-Feynman gauge. We omit the rules containing no scalar particle, which
are the same with the corresponding rules of the SM for the zero modes alone.
In the vertices all momenta $(k_{1},k_{2})$ and directions of propagation are
considered as incoming. $g^{(2)}$ and $e$ are $SU(2)_{L}$ the 4D gauge
coupling and the 4D elementary electric charge, respectively.
$\displaystyle=-i(k_{1}-k_{2})_{\mu}\mathcal{F}$,
$\begin{array}[]{|c|c|c||c|}\hline\cr A_{\mu}&B&C&\mathcal{F}\\\ \hline\cr
W^{+(m,n)}_{\mu}&G^{-(m,n)}&h^{(0)}&\frac{g^{(2)}}{2}\frac{-im_{W}}{m_{W,(m,n)}}\\\
\hline\cr
W^{+(m,n)}_{\mu}&a^{-(m,n)}&h^{(0)}&\frac{g^{(2)}}{2}\frac{m_{(m,n)}}{m_{W,(m,n)}}\\\
\hline\cr
W^{-(m,n)}_{\mu}&G^{+(m,n)}&h^{(0)}&\frac{g^{(2)}}{2}\frac{-im_{W}}{m_{W,(m,n)}}\\\
\hline\cr
W^{-(m,n)}_{\mu}&a^{+(m,n)}&h^{(0)}&\frac{g^{(2)}}{2}\frac{-m_{(m,n)}}{m_{W,(m,n)}}\\\
\hline\cr A^{(0)}_{\mu}&G^{+(m,n)}&G^{-(m,n)}&e\\\ \hline\cr
A^{(0)}_{\mu}&a^{+(m,n)}&a^{-(m,n)}&e\\\ \hline\cr
A^{(0)}_{\mu}&H^{+(m,n)}&H^{-(m,n)}&e\\\ \hline\cr\end{array}$
$\displaystyle=i\eta_{\mu\nu}\mathcal{F}$,
$\begin{array}[]{|c|c|c||c|}\hline\cr A&B_{\mu}&C_{\nu}&\mathcal{F}\\\
\hline\cr h^{(0)}&W^{+(m,n)}_{\mu}&W^{-(m,n)}_{\nu}&m_{W}g^{(2)}\\\ \hline\cr
G^{+(m,n)}&W^{-(m,n)}_{\mu}&A^{(0)}_{\nu}&-iem_{W,(m,n)}\\\ \hline\cr
G^{-(m,n)}&W^{+(m,n)}_{\mu}&A^{(0)}_{\nu}&iem_{W,(m,n)}\\\
\hline\cr\end{array}$
$\displaystyle=-im_{W}g^{(2)}\mathcal{F}$,
$\begin{array}[]{|c|c|c||c|}\hline\cr A&B&C&\mathcal{F}\\\ \hline\cr
h^{(0)}&G^{+(m,n)}&G^{-(m,n)}&\frac{m_{h}^{2}}{2m_{W,(m,n)}^{2}}\\\ \hline\cr
h^{(0)}&G^{+(m,n)}&a^{-(m,n)}&i\frac{m_{(m,n)}}{2m_{W,(m,n)}^{2}}\left(\frac{m_{h}^{2}}{m_{W}}-\frac{m_{W,(m,n)}^{2}}{m_{W}}\right)\\\
\hline\cr
h^{(0)}&a^{+(m,n)}&G^{-(m,n)}&i\frac{m_{(m,n)}}{2m_{W,(m,n)}^{2}}\left(-\frac{m_{h}^{2}}{m_{W}}+\frac{m_{W,(m,n)}^{2}}{m_{W}}\right)\\\
\hline\cr
h^{(0)}&a^{+(m,n)}&a^{-(m,n)}&\frac{1}{2m_{W,(m,n)}^{2}}\left(\frac{m_{h}^{2}}{m_{W}^{2}}m_{(m,n)}^{2}+2{m_{W,(m,n)}^{2}}\right)\\\
\hline\cr h^{(0)}&H^{+(m,n)}&H^{-(m,n)}&1\\\ \hline\cr\end{array}$
$\displaystyle=i\eta_{\mu\nu}\mathcal{F}$,
$\begin{array}[]{|c|c|c|c||c|}\hline\cr A_{\mu}&B_{\nu}&C&D&\mathcal{F}\\\
\hline\cr
W_{\mu}^{+(m,n)}&A_{\nu}^{(0)}&G^{-(m,n)}&h^{(0)}&\frac{eg^{(2)}}{2}\left(\frac{im_{W}}{m_{W,(m,n)}}\right)\\\
\hline\cr
W_{\mu}^{-(m,n)}&A_{\nu}^{(0)}&G^{+(m,n)}&h^{(0)}&\frac{eg^{(2)}}{2}\left(\frac{-im_{W}}{m_{W,(m,n)}}\right)\\\
\hline\cr
W_{\mu}^{+(m,n)}&A_{\nu}^{(0)}&a^{-(m,n)}&h^{(0)}&\frac{eg^{(2)}}{2}\left(\frac{-m_{(m,n)}}{m_{W,(m,n)}}\right)\\\
\hline\cr
W_{\mu}^{-(m,n)}&A_{\nu}^{(0)}&a^{+(m,n)}&h^{(0)}&\frac{eg^{(2)}}{2}\left(\frac{-m_{(m,n)}}{m_{W,(m,n)}}\right)\\\
\hline\cr A_{\mu}^{(0)}&A_{\nu}^{(0)}&G^{+(m,n)}&G^{-(m,n)}&2e^{2}\\\
\hline\cr A_{\mu}^{(0)}&A_{\nu}^{(0)}&a^{+(m,n)}&a^{-(m,n)}&2e^{2}\\\
\hline\cr A_{\mu}^{(0)}&A_{\nu}^{(0)}&H^{+(m,n)}&H^{-(m,n)}&2e^{2}\\\
\hline\cr\end{array}$
## Appendix B Detail on threshold correction
In this Appendix, we explain the concrete forms of threshold corrections in
the gluon fusion $(2g\rightarrow h^{(0)})$ and the Higgs decay to two photons
$(h^{(0)}\rightarrow 2\gamma)$. The parts of the Lagrangian describing the
former ($\mathcal{L}_{hgg}$) and the latter ($\mathcal{L}_{h\gamma\gamma}$)
processes are defined as
$\displaystyle\mathcal{L}_{hgg}$
$\displaystyle=-\frac{1}{4}\frac{C_{hgg}}{{\Lambda_{\text{UED}}}^{2}}V_{2}F_{MN}^{[\text{QCD}]}F^{[\text{QCD}]MN}H^{\dagger}H,$
(126) $\displaystyle\mathcal{L}_{h\gamma\gamma}$
$\displaystyle=-\frac{1}{4}\frac{C_{h\gamma\gamma}}{{\Lambda_{\text{UED}}}^{2}}V_{2}F_{MN}^{[\text{QED}]}F^{[\text{QED}]MN}H^{\dagger}H,$
(127)
where $C_{hgg}$ and $C_{h\gamma\gamma}$ are dimensionless coefficients
characterizing the processes, $\Lambda_{\text{UED}}$ is 6D UED cutoff, $V_{2}$
is the volume of the two extra dimensions, $H$ is the 6D Higgs doublet, and
$F_{MN}^{[\text{QCD}]}$ ($F_{MN}^{[\text{QED}]}$) is the 6D field strength of
gluon (photon), respectively. It is an important thing that the Higgs doublet
should be introduced in these effective operators in a bilinear form of
$H^{\dagger}H$ because the electroweak symmetry breaking (EWSB) is realized by
the usual Higgs mechanism in 6D UED models. After EWSB and KK reduction, the
Higgs doublet can acquire the VEV as $\langle
H\rangle=\left(0,v\right)^{\text{T}}/\sqrt{2V_{2}}$, where $v\simeq
246\,\text{GeV}$, and we would like to focus on the parts, which are
$\displaystyle\mathcal{L}_{hgg}$
$\displaystyle\supset-\frac{v/\sqrt{2}}{4}\frac{C_{hgg}}{{\Lambda_{\text{UED}}}^{2}}F_{\mu\nu}^{(0)[\text{QCD}]}F^{(0)[\text{QCD}]\mu\nu}h^{(0)},$
(128) $\displaystyle\mathcal{L}_{h\gamma\gamma}$
$\displaystyle\supset-\frac{v/\sqrt{2}}{4}\frac{C_{h\gamma\gamma}}{{\Lambda_{\text{UED}}}^{2}}F_{\mu\nu}^{(0)[\text{QED}]}F^{(0)[\text{QED}]\mu\nu}h^{(0)}.$
(129)
The superscript “$(0)$” means that the fields are zero modes and we take
integration toward the two extra spacial directions in Eqs. (128) and (129).
The two operators are understood as dimension-six operators in 4D point of
view. Finally, we write down the concrete forms of Feynman rules as follows:
$\displaystyle\quad=-i\frac{C_{hgg}v/\sqrt{2}}{{\Lambda_{\text{UED}}}^{2}}\left(k_{2\mu}k_{1\nu}-(k_{1}\cdot
k_{2})\eta_{\mu\nu}\right)\delta^{ab}$, (130)
$\displaystyle\quad=-i\frac{C_{h\gamma\gamma}v/\sqrt{2}}{{\Lambda_{\text{UED}}}^{2}}\left(k_{2\mu}k_{1\nu}-(k_{1}\cdot
k_{2})\eta_{\mu\nu}\right)$, (131)
where $a,b$ are gluon indices.
## References
* (1) N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, The hierarchy problem and new dimensions at a millimeter, Phys. Lett. B429 (1998) 263–272, [hep-ph/9803315].
* (2) L. Randall and R. Sundrum, A large mass hierarchy from a small extra dimension, Phys. Rev. Lett. 83 (1999) 3370–3373, [hep-ph/9905221].
* (3) I. Antoniadis, A Possible new dimension at a few TeV, Phys. Lett. B246 (1990) 377–384.
* (4) T. Appelquist, H.-C. Cheng, and B. A. Dobrescu, Bounds on universal extra dimensions, Phys. Rev. D64 (2001) 035002, [hep-ph/0012100].
* (5) K. Agashe, N. G. Deshpande, and G. H. Wu, Can extra dimensions accessible to the SM explain the recent measurement of anomalous magnetic moment of the muon?, Phys. Lett. B511 (2001) 85–91, [hep-ph/0103235].
* (6) K. Agashe, N. G. Deshpande, and G. H. Wu, Universal extra dimensions and b $\rightarrow$ s gamma, Phys. Lett. B514 (2001) 309–314, [hep-ph/0105084].
* (7) T. Appelquist and B. A. Dobrescu, Universal extra dimensions and the muon magnetic moment, Phys. Lett. B516 (2001) 85–91, [hep-ph/0106140].
* (8) T. Appelquist and H.-U. Yee, Universal extra dimensions and the Higgs boson mass, Phys. Rev. D67 (2003) 055002, [hep-ph/0211023].
* (9) J. F. Oliver, J. Papavassiliou, and A. Santamaria, Universal extra dimensions and Z $\rightarrow$ b anti-b, Phys. Rev. D67 (2003) 056002, [hep-ph/0212391].
* (10) D. Chakraverty, K. Huitu, and A. Kundu, Effects of Universal Extra Dimensions on $B^{0}-{\bar{B}^{0}}$ Mixing, Phys. Lett. B558 (2003) 173–181, [hep-ph/0212047].
* (11) A. J. Buras, M. Spranger, and A. Weiler, The Impact of Universal Extra Dimensions on the Unitarity Triangle and Rare K and B Decays, Nucl. Phys. B660 (2003) 225–268, [hep-ph/0212143].
* (12) P. Colangelo, F. De Fazio, R. Ferrandes, and T. N. Pham, Exclusive $B\rightarrow K^{(\ast)}l_{+}l_{-}$, $B\rightarrow K^{(\ast)}\nu\bar{\nu}$ and $B\rightarrow K^{\ast}\gamma$ transitions in a scenario with a single universal extra dimension, Phys. Rev. D73 (2006) 115006, [hep-ph/0604029].
* (13) I. Gogoladze and C. Macesanu, Precision electroweak constraints on Universal Extra Dimensions revisited, Phys. Rev. D74 (2006) 093012, [hep-ph/0605207].
* (14) H.-C. Cheng, J. L. Feng, and K. T. Matchev, Kaluza-Klein dark matter, Phys. Rev. Lett. 89 (2002) 211301, [hep-ph/0207125].
* (15) G. Servant and T. M. P. Tait, Is the lightest Kaluza-Klein particle a viable dark matter candidate?, Nucl. Phys. B650 (2003) 391–419, [hep-ph/0206071].
* (16) M. Kakizaki, S. Matsumoto, Y. Sato, and M. Senami, Significant effects of second KK particles on LKP dark matter physics, Phys. Rev. D71 (2005) 123522, [hep-ph/0502059].
* (17) S. Matsumoto and M. Senami, Efficient coannihilation process through strong Higgs self-coupling in LKP dark matter annihilation, Phys. Lett. B633 (2006) 671–674, [hep-ph/0512003].
* (18) F. Burnell and G. D. Kribs, The abundance of Kaluza-Klein dark matter with coannihilation, Phys. Rev. D73 (2006) 015001, [hep-ph/0509118].
* (19) M. Kakizaki, S. Matsumoto, and M. Senami, Relic abundance of dark matter in the minimal universal extra dimension model, Phys. Rev. D74 (2006) 023504, [hep-ph/0605280].
* (20) K. Kong and K. T. Matchev, Precise calculation of the relic density of Kaluza-Klein dark matter in universal extra dimensions, JHEP 01 (2006) 038, [hep-ph/0509119].
* (21) S. Matsumoto, J. Sato, M. Senami, and M. Yamanaka, Relic abundance of dark matter in universal extra dimension models with right-handed neutrinos, Phys. Rev. D76 (2007) 043528, [arXiv:0705.0934].
* (22) M. Kakizaki, S. Matsumoto, Y. Sato, and M. Senami, Relic abundance of LKP dark matter in UED model including effects of second KK resonances, Nucl. Phys. B735 (2006) 84–95, [hep-ph/0508283].
* (23) G. Belanger, M. Kakizaki, and A. Pukhov, Dark matter in UED: The Role of the second KK level, JCAP 1102 (2011) 009, [arXiv:1012.2577].
* (24) J. Hisano, K. Ishiwata, N. Nagata, and M. Yamanaka, Direct Detection of Vector Dark Matter, Prog.Theor.Phys. 126 (2011) 435–456, [arXiv:1012.5455].
* (25) H.-C. Cheng, K. T. Matchev, and M. Schmaltz, Bosonic supersymmetry? Getting fooled at the CERN LHC, Phys. Rev. D66 (2002) 056006, [hep-ph/0205314].
* (26) A. Datta, K. Kong, and K. T. Matchev, Discrimination of supersymmetry and universal extra dimensions at hadron colliders, Phys. Rev. D72 (2005) 096006, [hep-ph/0509246].
* (27) S. Matsumoto, J. Sato, M. Senami, and M. Yamanaka, Productions of second Kaluza-Klein gauge bosons in the minimal universal extra dimension model at LHC, Phys. Rev. D80 (2009) 056006, [arXiv:0903.3255].
* (28) B. A. Dobrescu and E. Poppitz, Number of fermion generations derived from anomaly cancellation, Phys. Rev. Lett. 87 (2001) 031801, [hep-ph/0102010].
* (29) T. Appelquist, B. A. Dobrescu, E. Ponton, and H.-U. Yee, Proton stability in six dimensions, Phys. Rev. Lett. 87 (2001) 181802, [hep-ph/0107056].
* (30) N. Arkani-Hamed, H.-C. Cheng, B. A. Dobrescu, and L. J. Hall, Self-breaking of the standard model gauge symmetry, Phys. Rev. D62 (2000) 096006, [hep-ph/0006238].
* (31) M. Hashimoto, M. Tanabashi, and K. Yamawaki, Topped MAC with extra dimensions?, Phys. Rev. D69 (2004) 076004, [hep-ph/0311165].
* (32) M. Hashimoto and D. K. Hong, Topcolor breaking through boundary conditions, Phys. Rev. D71 (2005) 056004, [hep-ph/0409223].
* (33) H.-C. Cheng, K. T. Matchev, and M. Schmaltz, Radiative corrections to Kaluza-Klein masses, Phys. Rev. D66 (2002) 036005, [hep-ph/0204342].
* (34) E. Ponton and L. Wang, Radiative effects on the chiral square, JHEP 11 (2006) 018, [hep-ph/0512304].
* (35) G. Burdman, B. A. Dobrescu, and E. Ponton, Resonances from two universal extra dimensions, Phys. Rev. D74 (2006) 075008, [hep-ph/0601186].
* (36) B. A. Dobrescu, K. Kong, and R. Mahbubani, Leptons and photons at the LHC: Cascades through spinless adjoints, JHEP 07 (2007) 006, [hep-ph/0703231].
* (37) B. A. Dobrescu, D. Hooper, K. Kong, and R. Mahbubani, Spinless photon dark matter from two universal extra dimensions, JCAP 0710 (2007) 012, [arXiv:0706.3409].
* (38) A. Freitas and K. Kong, Two universal extra dimensions and spinless photons at the ILC, JHEP 02 (2008) 068, [arXiv:0711.4124].
* (39) A. Freitas and U. Haisch, Anti-B $\rightarrow$ X(s) gamma in two universal extra dimensions, Phys. Rev. D77 (2008) 093008, [arXiv:0801.4346].
* (40) K. Ghosh and A. Datta, Probing two Universal Extra Dimensions at International Linear Collider, Phys. Lett. B665 (2008) 369–373, [arXiv:0802.2162].
* (41) G. Bertone, C. B. Jackson, G. Shaughnessy, T. M. P. Tait, and A. Vallinotto, The WIMP Forest: Indirect Detection of a Chiral Square, Phys. Rev. D80 (2009) 023512, [arXiv:0904.1442].
* (42) M. Blennow, H. Melbeus, and T. Ohlsson, Neutrinos from Kaluza-Klein dark matter in the Sun, JCAP 1001 (2010) 018, [arXiv:0910.1588].
* (43) B. A. Dobrescu and E. Ponton, Chiral compactification on a square, JHEP 03 (2004) 071, [hep-th/0401032].
* (44) G. Burdman, B. A. Dobrescu, and E. Ponton, Six-dimensional gauge theory on the chiral square, JHEP 02 (2006) 033, [hep-ph/0506334].
* (45) G. Cacciapaglia, A. Deandrea, and J. Llodra-Perez, A Dark Matter candidate from Lorentz Invariance in 6 Dimensions, JHEP 03 (2010) 083, [arXiv:0907.4993].
* (46) N. Maru, T. Nomura, J. Sato, and M. Yamanaka, The Universal Extra Dimensional Model with $S^{2}/Z_{2}$ extra-space, Nucl. Phys. B830 (2010) 414–433, [arXiv:0904.1909].
* (47) H. Dohi and K.-y. Oda, Universal Extra Dimensions on Real Projective Plane, Phys. Lett. B692 (2010) 114–120, [arXiv:1004.3722].
* (48) T. Flacke, A. Menon, and D. J. Phalen, Non-minimal universal extra dimensions, Phys. Rev. D79 (2009) 056009, [arXiv:0811.1598].
* (49) S. C. Park and J. Shu, Split-UED and Dark Matter, Phys. Rev. D79 (2009) 091702, [arXiv:0901.0720].
* (50) C. Csaki, J. Heinonen, J. Hubisz, S. C. Park, and J. Shu, 5D UED: Flat and Flavorless, JHEP 1101 (2011) 089, [arXiv:1007.0025].
* (51) N. Haba, K.-y. Oda, and R. Takahashi, Top Yukawa Deviation in Extra Dimension, Nucl. Phys. B821 (2009) 74–128, [arXiv:0904.3813].
* (52) N. Haba, K.-y. Oda, and R. Takahashi, Dirichlet Higgs in extra-dimension, consistent with electroweak data, Acta Phys.Polon. B42 (2011) 33–44, [arXiv:0910.3356].
* (53) N. Haba, K.-y. Oda, and R. Takahashi, Diagonal Kaluza-Klein expansion under brane localized potential, Acta Phys. Polon. B41 (2010) 1291–1316, [arXiv:0910.4528].
* (54) N. Haba, K.-y. Oda, and R. Takahashi, Phenomenological Aspects of Dirichlet Higgs Model from Extra-Dimension, JHEP 07 (2010) 079, [arXiv:1005.2306].
* (55) K. Nishiwaki and K.-y. Oda, Unitarity in Dirichlet Higgs Model, Eur.Phys.J. C71 (2011) 1786, [arXiv:1011.0405].
* (56) The ATLAS collaboration Collaboration, Update of the Combination of Higgs Boson Searches in 1.0 to 2.3 $\text{fb}^{-1}$ of $pp$ Collisions Data Taken at $\sqrt{s}=7$ TeV with the ATLAS Experiment at the LHC, . ATLAS NOTE, ATLAS-CONF-2011-135.
* (57) The CMS collaboration Collaboration, Search for standard model Higgs boson in pp collisions at $\sqrt{s}=7$ TeV and integrated luminosity up to 1.7 $\text{fb}^{-1}$, . CMS PAS HIG-11-022.
* (58) ATLAS Collaboration Collaboration, G. Aad et. al., Combined search for the Standard Model Higgs boson using up to 4.9 fb-1 of $pp$ collision data at $\sqrt{s}=7$ TeV with the ATLAS detector at the LHC, Phys.Lett. B710 (2012) 49–66, [arXiv:1202.1408].
* (59) CMS Collaboration Collaboration, S. Chatrchyan et. al., Combined results of searches for the standard model Higgs boson in $pp$ collisions at $\sqrt{s}=7$ TeV, Phys.Lett. B710 (2012) 26–48, [arXiv:1202.1488].
* (60) H. Georgi, A. K. Grant, and G. Hailu, Brane couplings from bulk loops, Phys. Lett. B506 (2001) 207–214, [hep-ph/0012379].
* (61) C. S. Lim, N. Maru, and K. Nishiwaki, CP Violation due to Compactification, Phys.Rev. D81 (2010) 076006, [arXiv:0910.2314].
* (62) C. A. Scrucca, M. Serone, L. Silvestrini, and A. Wulzer, Gauge-Higgs Unification in Orbifold Models, JHEP 02 (2004) 049, [hep-th/0312267].
* (63) H. M. Georgi, S. L. Glashow, M. E. Machacek, and D. V. Nanopoulos, Higgs Bosons from Two Gluon Annihilation in Proton Proton Collisions, Phys. Rev. Lett. 40 (1978) 692.
* (64) T. G. Rizzo, GLUON FINAL STATES IN HIGGS BOSON DECAY, Phys. Rev. D22 (1980) 178.
* (65) F. J. Petriello, Kaluza-Klein effects on Higgs physics in universal extra dimensions, JHEP 05 (2002) 003, [hep-ph/0204067].
* (66) N. Maru, T. Nomura, J. Sato, and M. Yamanaka, Higgs Production via Gluon Fusion in a Six Dimensional Universal Extra Dimension Model on $S^{2}/Z_{2}$, Eur. Phys. J. C66 (2010) 283–287, [arXiv:0905.4554].
* (67) G. Passarino and M. J. G. Veltman, One Loop Corrections for e+ e- Annihilation Into mu+ mu- in the Weinberg Model, Nucl. Phys. B160 (1979) 151.
* (68) A. Denner, Techniques for calculation of electroweak radiative corrections at the one loop level and results for W physics at LEP-200, Fortschr. Phys. 41 (1993) 307–420, [arXiv:0709.1075].
* (69) J. R. Ellis, M. K. Gaillard, and D. V. Nanopoulos, A Phenomenological Profile of the Higgs Boson, Nucl. Phys. B106 (1976) 292.
* (70) S. Randjbar-Daemi, A. Salam, and J. A. Strathdee, Spontaneous Compactification in Six-Dimensional Einstein- Maxwell Theory, Nucl. Phys. B214 (1983) 491–512.
* (71) E. T. Newman and R. Penrose, Note on the Bondi-Metzner-Sachs group, J. Math. Phys. 7 (1966) 863–870.
* (72) G. L. Smith et. al., Short range tests of the equivalence principle, Phys. Rev. D61 (2000) 022001.
* (73) C. A. Scrucca, M. Serone, and L. Silvestrini, Electroweak symmetry breaking and fermion masses from extra dimensions, Nucl. Phys. B669 (2003) 128–158, [hep-ph/0304220].
* (74) K. R. Dienes, E. Dudas, and T. Gherghetta, Extra spacetime dimensions and unification, Phys. Lett. B436 (1998) 55–65, [hep-ph/9803466].
* (75) K. R. Dienes, E. Dudas, and T. Gherghetta, Grand unification at intermediate mass scales through extra dimensions, Nucl. Phys. B537 (1999) 47–108, [hep-ph/9806292].
* (76) K. Nishiwaki, K.-y. Oda, N. Okuda, and R. Watanabe, Heavy Higgs at Tevatron and LHC in Universal Extra Dimension Models, Phys.Rev. D85 (2012) 035026, [arXiv:1108.1765].
* (77) T. Han, H. E. Logan, B. McElrath, and L.-T. Wang, Loop induced decays of the little Higgs: H $\rightarrow$ g g, gamma gamma, Phys. Lett. B563 (2003) 191–202, [hep-ph/0302188].
* (78) C. Dib, R. Rosenfeld, and A. Zerwekh, Higgs production and decay in the little Higgs model, hep-ph/0302068.
* (79) C.-R. Chen, K. Tobe, and C. P. Yuan, Higgs boson production and decay in little Higgs models with T-parity, Phys. Lett. B640 (2006) 263–271, [hep-ph/0602211].
* (80) A. Falkowski, Pseudo-Goldstone Higgs Production via Gluon Fusion, Phys. Rev. D77 (2008) 055018, [arXiv:0711.0828].
* (81) N. Maru and N. Okada, Gauge-Higgs Unification at LHC, Phys. Rev. D77 (2008) 055010, [arXiv:0711.2589].
* (82) N. Maru, Finite Gluon Fusion Amplitude in the Gauge-Higgs Unification, Mod. Phys. Lett. A23 (2008) 2737–2750, [arXiv:0803.0380].
* (83) S. K. Rai, UED effects on Higgs signals at LHC, Int. J. Mod. Phys. A23 (2008) 823–834, [hep-ph/0510339].
* (84) K. Nishiwaki, K.-y. Oda, N. Okuda, and R. Watanabe, A Bound on Universal Extra Dimension Models from up to $\mathrm{2fb}^{-1}$ of LHC Data at 7TeV, Phys.Lett. B707 (2012) 506–511, [arXiv:1108.1764].
|
arxiv-papers
| 2011-01-04T04:02:16 |
2024-09-04T02:49:16.109296
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kenji Nishiwaki",
"submitter": "Kenji Nishiwaki",
"url": "https://arxiv.org/abs/1101.0649"
}
|
1101.0711
|
# Optical Intraday Variability Studies of Ten Low Energy Peaked Blazars
Bindu Rani1, Alok C. Gupta1, U. C. Joshi2, S. Ganesh2, Paul J. Wiita3
1Aryabhatta Research Institute of Observational Sciences (ARIES), Manora Peak,
Nainital – 263129, India
2Physical Research Laboratory, Navrangpura, Ahmedabad-380 009, India
3Department of Physics, The College of New Jersey, P.O. Box 7718, Ewing, NJ
08628, USA E-mail: bindu@aries.res.in
(Accepted ……. Received ……; in original form ……)
###### Abstract
We have carried out optical (R band) intraday variability (IDV) monitoring of
a sample of ten bright low energy peaked blazars (LBLs). Forty photometric
observations, of an average of $\sim 4$ hours each, were made between 2008
September and 2009 June using two telescopes in India. Measurements with good
signal to noise ratios were typically obtained within 1–3 minutes, allowing
the detection of weak, fast variations using N-star differential photometry.
We employed both structure function and discrete correlation function analysis
methods to estimate any dominant timescales of variability and found that in
most of the cases any such timescales were longer than the duration of the
observation. The calculated duty cycle of IDV in LBLs during our observing run
is $\sim$52$\%$, which is low compared to many earlier studies; however, the
relatively short periods for which each source was observed can probably
explain this difference. We briefly discuss possible emission mechanisms for
the observed variability.
###### keywords:
galaxies: active — galaxies: BL Lacs — galaxies: photometry
††pagerange: Optical Intraday Variability Studies of Ten Low Energy Peaked
Blazars–9††pubyear: 2010
## 1 Introduction
Blazars are a subclass of radio-loud AGNs characterized by strong and rapid
flux variability across the entire EM spectrum and strong polarization from
radio to optical wavelengths. Microvariability, or intraday variability (IDV)
is commonly observed across much of the electromagnetic (EM) spectrum of AGNs
but is particularly common in blazars. A change of flux of $\sim$1–15% within
a few minutes to hours reflect extreme physical conditions embedded in small
sub-parsec scales. According to the usually accepted orientation based unified
model of radio-loud AGNs, blazar jets usually make an angle $\leq 10^{\circ}$
to our line-of-sight (Urry & Padovani, 1995). The Doppler boosting of the jet
emission means that most of what we see from blazars arises in those jets.
These jet dominated AGNs provide a natural laboratory to study the mechanisms
of energy extraction from the vicinity of central supermassive black holes,
the physical properties of jets and perhaps also accretion disks.
The radiation of blazars across the whole EM spectrum is predominantly non-
thermal. At lower frequencies (through the UV or X-ray bands) the mechanism is
almost certainly synchrotron emission while at higher frequencies it is very
probably dominated by Inverse-Compton (IC) emission (Sikora & Madejski, 2001;
Krawczynski, 2004). The spectral energy distributions (SEDs) of blazars have a
double-peaked structure (Fossati et al., 1998; Ghisellini et al., 1997). Based
on the location of the peak of their SEDs, blazars are often sub-classified
into the low energy peaked blazars (LBLs) and high energy peaked blazars
(HBLs); the first component peaks in the near-infrared (NIR)/optical in case
of LBLs and in the UV/X-rays in HBLs, while the second component usually peaks
at GeV energies in LBLs and at TeV energies in HBLs.
The study of variability is one of the most powerful tools for revealing the
nature of blazars and probing the various processes occurring in them. Based
on their different timescales, the variability of blazars can be broadly
divided into three classes, IDV, short-term variability (STV), and long-term
variability (LTV). Variations in the flux of source up to a few tenths of
magnitude over a time scale of a day or less is known as IDV (Wagner & Witzel,
1995) or microvariability, or intra-night optical variability. Variations of
days to a few months are often considered to be STV, while those from several
months to many years are usually called LTV (e.g., Gupta et al., 2004); both
of these classes of variations for blazars typically exceed $\sim$ 1 magnitude
and can exceed even 5 magnitudes. Over the last two decades, the optical
variability of blazars has been extensively studied on diverse timescales
(e.g., Heidt & Wagner, 1996; Sillanpää et al., 1996a, b; Gupta et al., 2004,
2008a, 2008b, 2008c, 2009; Ciprini et al., 2003, 2007, and references
therein).
There are several theoretical models that might be able to explain the
observed variability over wide time-scales for all bands, with the leading
contenders all variants of models based upon shocks propagating down
relativistic jets (Marscher & Gear, 1985; Qian et al., 1991; Hughes et al.,
1991; Marscher et al., 1992; Wagner & Witzel, 1995). Some of the variability
may arise from helical structures, precession or other geometrical effects
occurring within the jets (e.g., Camenzind & Krockenberger, 1992; Gopal-
Krishna & Wiita, 1992) and some of the radio variability is due to extrinsic
propagative effects (Rickett et al., 2001). Hot spots or other disturbances in
or above accretion disks surrounding the black holes at the centres of AGNs
(e.g., Mangalam & Wiita, 1993; Chakrabarti & Wiita, 1993) are likely to play a
key role in the variability of non-blazar AGNs and might provide seed
fluctuations that could be advected into a rotating blazar jet and then be
Doppler amplified.
Despite the large amount of information we have about blazars that is very
briefly summarized above, we still lack sufficient understanding of basic
parameters of the emission regions, such as jet composition, a quantitative
assessment of beaming parameters, or the processes leading to the origin of
shocks in the jets. These physical quantities are obviously important in
understanding the physics of jets and their emission regions and additional
IDV studies leading to statistically valid pictures of many blazars can help
constrain them.
In this paper we present the results of an extensive IDV studies of a sample
of ten LBLs including six BL Lacs and four FSRQs. The work presented here is
focused on intraday variations in the R passband magnitudes of these sources,
which were the brightest blazars visible from ARIES, Nainital, India and PRL,
Mount Abu India. We also compare our observational results to some of those
presented in the literature.
The paper is structured as follows. In Section 2, we present the observations
and data reduction procedure. Section 3 provides our analysis and results. We
present our discussion and conclusions in Section 4.
## 2 Observations and data reduction
We have carried out optical R band photometric observations of ten LBLs from
September 2008 to June 2009 using two telescopes in India. The details of the
telescopes and instruments used are given in Table 1 and the observation
details are in Table 2.
For doing image processing, or data pre-processing, we generated a master bias
frame for each observing night by taking the median of all bias frames; the
master bias frame was subtracted from all flat and source image frames taken
on that night. Then the master flat in each passband was generated by median
combine of flat frames in that passband. Finally, the normalized master flat
in each passband was generated. As usual, each source image frame was divided
by the normalized master flat in the respective passband to remove pixel-to-
pixel inhomogeneities (flat fielding). Finally cosmic ray removal was done
from all source image frames. This pre-processing of the data was done by
using standard routines in the Image Reduction and Analysis Facility111IRAF is
distributed by the National Optical Astronomy Observatories, which are
operated by the Association of Universities for Research in Astronomy, Inc.,
under cooperative agreement with the National Science Foundation. (IRAF)
software.
Our data analysis, or processing of the data, utilizes Dominion Astronomical
Observatory Photometry (DAOPHOT II) software to perform aperture photometry
(Stetson, 1987, 1992). We carried out aperture photometry with four different
aperture radii, 1$\times$FWHM, 2$\times$FWHM, 3$\times$FWHM and 4$\times$FWHM.
On comparing the results, we observed that aperture radii = 3$\times$FWHM
provided the best S/N ratio, and we have adopted that in this work. For all of
the ten blazars, we observed more than three local standard stars. The
magnitudes of the standard stars we used in the fields of our sources are
given in Table 3. The multiple comparison stars were used to check that the
usual standard stars were non-variable. We have used two non-varying standard
stars from each blazar field and plotted their differential instrumental
magnitudes in Figs. 1$-$9\. Finally, for the calibration of blazar data, we
have used the one standard star that has a colour close to the blazar from
those two standards stars. The calibrated light curves of the blazars are
plotted in the same panel with the differential instrumental magnitudes of two
standard stars.
## 3 Analysis and Results
### 3.1 Variability Parameters
We checked for the presence of microvariability both by using the $F$-test (de
Diego, 2010) and the variability detection parameter, $C,$ for the sake of
comparison with earlier papers, which nearly always used that approach. For
two sample variances, say s${}^{2}_{Q}$, for the quasar differential light
curves and s${}^{2}_{stS}$, for that of the standard star
$F=\frac{s^{2}_{Q}}{s^{2}_{stS}}.$ (1)
The $F$-statistic is compared with a critical value corresponding to the
significance level set for the test. We have used the inbuilt F test code
available in R222R: A language and environment for statistical computing. R
Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL
http://www.R-project.org.. A $p$-value of $\leq$ 0.01 ($\geq$99$\%$
significance level) is adopted for our variability detection criterion.
The variability detection parameter is defined as Romero et al. (e.g., 1999))
the average of $C1$and $C2$ where
$\displaystyle C1=\frac{\sigma(BL-starA)}{\sigma(starA-starB)}\hskip
14.22636pt\rm{and}\hskip 14.22636ptC2=\frac{\sigma(BL-starB)}{\sigma(starA-
starB)}.$ (2)
Here (BL $-$ starA) and (BL $-$ starB) are the differential instrumental
magnitudes of the blazar and standard star A and the blazar and standard star
B, respectively, while $\sigma$(BL$-$starA) and $\sigma$(BL$-$starB) are the
observational scatters of the differential instrumental magnitude of the
blazar and star A and the blazar and star B, respectively. If $C\geq 2.57$, a
conservative confidence level of a variability detection is $>99$%, and we
consider this to be a positive detection of a variation using this criterion.
As noted by de Diego (2010) this C-statistic does not behave as a proper
statistic should, but as it has been used in most of the IDV studies in
literature, we also employed this test.
The percentage variation in the intraday light curves of LBLs is calculated by
using the variability amplitude parameter, $A$, introduced by Heidt & Wagner
(1996), defined as
$\displaystyle A=100\times\sqrt{{(A_{max}-A_{min}})^{2}-2\sigma^{2}}(\%),$ (3)
where $A_{max}$ and $A_{min}$ are the maximum and minimum magnitudes in the
calibrated light curves of the blazar and $\sigma$ is the average measurement
error of the blazar light curve. The calculated $F$-statistics,
$C$-“statistics” and variability amplitude parameter, $A$, values are listed
in Table 4.
### 3.2 Structure Function
The first order structure function (SF) is a very useful tool to search for
periodicities and timescales of variability in time series data trains
(Simonetti et al., 1985). Here we give only a very brief introduction to the
method; for details refer to Rani et al. (2009). The first order SF for a data
train, $a$, is defined as
$\displaystyle
D^{1}_{a}(k)={\frac{1}{N^{1}_{a}(k)}}{\sum_{i=1}^{N}}w(i)w(i+k){[a(i+k)-a(i)]}^{2},$
(4)
where $k$ is the time lag, ${N^{1}_{a}(k)}=\sum w(i)w(i+k)$, and the weighting
factor, $w(i)$ is 1 if a measurement exists for the $i^{th}$ interval, and 0
otherwise.
The behaviour of the first order SF can be simply summarized. The SF curves
for AGN usually at first rise with time lag. Following this rising portion,
the SF will then fall into one of the following classes: (i) if no plateau
exists, any time scale of variability exceeds the length of the data train;
(ii) if there are one or more plateaus, each one indicates a possible time
scale of variability; and (iii) if that plateau is followed by a dip in the
SF, the lag corresponding to the minimum of that dip indicates a possible
periodic cycle (unless such a dip is seen at a lag close to the maximum length
of the data train, when it is probably an artifact). However, (iv)
uncorrelated data produce a white noise behaviour, characterized by a constant
slope (Ciprini et al., 2003).
We have carried out the SF analysis of all of those LCs which satisfy the
variability detection criteria. Recently, some weaknesses of the SF method,
including spurious indications of timescales and periodicities have been
discussed by Emmanoulopoulos et al. (2010), so we cross check the SF results
by using the discrete correlation function (DCF) method. The timescales of
variability calculated using the SF analysis are listed in Table 4.
### 3.3 Discrete Correlation Functions
The Discrete Correlation function (DCF) method was first introduced by Edelson
& Krolik (1988) and it was later generalized to provide better error estimates
(Hufnagel & Bregman, 1992). Here we give only a brief introduction to the
method; for details refer to Hovatta et al. (2007), Rani et al. (2009), and
references therein.
The first step is to calculate the unbinned correlation (UDCF) using the given
time series through (Hovatta et al., 2007)
$UDCF_{ij}={\frac{(a(i)-\bar{a})(b(j)-\bar{b})}{\sqrt{\sigma_{a}^{2}\sigma_{b}^{2}}}}.$
(5)
Here $a(i)$ and $b(j)$ are the individual points in two time series $a$ and
$b$, respectively, $\bar{a}$ and $\bar{b}$ are respectively the means of the
time series, and $\sigma_{a}^{2}$ and $\sigma_{b}^{2}$ are their variances.
The correlation function is binned after calculation of the UDCF. The DCF
method does not automatically define a bin size, so several values need to be
tried. If the bin size is too big, useful information is lost, but if the bin
size is too small, a spurious correlation can be found. Taking $\tau$ as the
centre of a time bin and $n$ as the number of points in each bin, the DCF is
found from the UDCF via
$DCF(\tau)={\frac{1}{n}}\sum~{}UDCF_{ij}(\tau).$ (6)
The error for each bin can be calculated using
$\sigma_{\mathrm{d}ef}(\tau)={\frac{1}{n-1}}\Bigl{\\{}\sum~{}\bigl{[}UDCF_{ij}-DCF(\tau)\bigr{]}^{2}\Bigr{\\}}^{0.5}.$
(7)
A DCF analysis is frequently used for finding the correlation and possible
lags between multi-frequency AGN data where different data trains are used in
the calculation (e.g., Villata et al., 2004; Raiteri et al., 2003; Hovatta et
al., 2007, and references therein). When the same data train is used, so
$a$=$b$, there is obviously a peak at zero DCF, indicating that there is no
time lag between the two, but any other strong peaks in the DCF give
indications of variability timescales. The calculated $t_{v}$ values using DCF
analyses are listed in Table 4.
### 3.4 Intraday Variability (IDV) of Individual Blazars
The R filter light curves (LCs) of the blazars in which significant
variability has been detected, along with their corresponding SF and DCF
analysis curves, are displayed in Figures 1$-$7; the remaining non-variable
LCs of the blazars are plotted along with the corresponding curves of the
standard stars used for comparison in Figures 8 and 9. The complete observing
log for the blazars in given in Table 3. The values of $A$, $C$, $F$-test
along with any estimated timescales of variability found using SF and DCF
analysis methods on the individual blazar LCs are listed in Table 4. A
detailed multiband optical short term variability (STV) study of the fluxes
and colours of all of these blazars over the same time period of observation
is reported in Rani et al. (2010b). There we showed that the colour versus
brightness correlations seen in these sources support the hypothesis that BL
Lacs tends to be bluer with increase in brightness while FSRQs shows the
opposite trend. We now report some key individual results for each of our
sources, placed in the context of earlier work.
3C 66A: This is a low energy peaked blazar (LBL) at redshift $z=0.444$
(Lanzetta et al., 1993) and belongs to the class of BL Lac objects. Since its
optical identification (Wills & Wills, 1974), the source has been regularly
monitored at many observable frequencies, although less regularly at radio
frequencies (Aller et al., 1992; Takalo et al., 1996). Fan & Lin (1999, 2000)
have studied the long-term optical and IR variability of the source and
reported a variation of $\leq$1.5 mag at time scales of $\sim$1 week to
several years at those two frequencies. Böttcher et al. (2005) reported a
large microvariability of $\sim$0.2 mag within 6 hours; they also reported
several major outbursts in the source separated by $\sim$50 days and argued
that the outbursts seem to have a quasi-periodic behaviour. At the end of 2007
the source was found to be in an optically active phase, which triggered a new
Optical-IR-Radio Whole Earth Blazar Telescope (WEBT) campaign on the source
(Böttcher et al., 2009).
Our optical IDV observations of the source 3C 66A comprise a total of seven
LCs, spanning a time period between October 2008 and January 2009. During this
period a change of $\sim$1 magnitude in brightness of the source is seen (Rani
et al., 2010b). The source showed significant microvariability only on October
22 and 26, 2008 (Fig. 1). There is a continuous fading trend of $\sim$0.08 and
$\sim$0.06 magnitude, respectively on those two nights. The SF and DCF
analysis of the LCs revealed that any timescale of variability in this source
at those epochs is greater than the lengths of our observations.
AO 0235+164: The blazar AO 0235+164 at $z=0.94$ (Nilsson et al., 1996) was
classified as a BL Lac object by Spinrad & Smith (1975). Over the past few
decades this blazar has been seen to be highly variable over all timescales
and at all frequencies (Ghosh & Soundararajaperumal, 1995; Heidt & Wagner,
1996; Raiteri et al., 2001) and a very high fractional polarization of
$\sim$40$\%$ has been reported in the source both at visible and IR
frequencies (Impey et al., 1982). By analysing 25 years (1975$-$2000) of
optical and radio data, Raiteri et al. (2001) argued that the source seemed to
have an optical outburst period of $\sim$5.70 years but the expected outburst
in 2004 was not detected by a 2003–2005 multiwavelength WEBT observing
campaign (Raiteri et al., 2006a). A more detailed long term optical data
analysis suggested a possible outburst period of $\sim$8 years in this source
(Raiteri et al., 2006b) and this period was supported by subsequent
observations Gupta et al. (2008c). Strong IDV flux variations of 9.5$\%$ and
13.7$\%$ during two nights were observed by Gupta et al. (2008c). Recently,
Rani et al. (2009) reported the possible presence of nearly periodic
fluctuations, with a timescale of $\sim$17 days, in a 12.7 year long X-ray
light curve of AO 0235$+$164 obtained by the All Sky Monitor (ASM) instrument
on the Rossi X-ray Timing Explorer (RXTE) satellite.
We observed three IDV LCs of the source AO 0235+164 between October 2008 and
January 2009. The brightness of the source decayed by $\sim$2.2 magnitude
during this period (Rani et al., 2010b). We found a significant flux variation
in two out of three LCs. A continuous fading trend of $\sim$0.13 magnitude and
both brightening and decaying of $\sim$0.04 magnitude were observed on October
20 (Fig. 1) and October 23, 2008 (Fig. 2), respectively. A possible timescale
of variability is $\sim$5.2 hr (from the SF analysis) for the LC observed on
20 October, while any such timescale exceeds the length of our observation on
23 October. However this putative timescale is not supported by the DCF
analysis.
PKS 0420$-$014: The blazar PKS 0420$-$014 is classified as a FSRQ and has a
redshift of 0.915. It has been observed in optical bands since 1969. Several
papers have reported multiple optically active and bright phases of the source
and perhaps regular major flaring cycles (e.g., Villata et al., 1997; Webb et
al., 1998; Raiteri et al., 1998, and references therein). Webb et al. (1998)
reported that there were increases of $\sim$ 2$-$3 magnitudes during the
active phases of this blazar during their observations that stretched from
December 1969 to January 1986. Clements et al. (1995) have reported variations
of $\Delta$mag $\cong$ 2.8 mag with a time scale of $\sim$22 years.
We found a variation of $>$0.1 magnitude within 2 hrs in the brief optical LCs
of the source on both of the days of observation in October and December 2008.
During this period the source brightened by a factor of $\sim$0.7 magnitude
(Rani et al., 2010b). The nominal timescale of variability (from the peak of
the SF) is 0.12 hrs for the LC observed on December 26 and multiple dips might
indicate a possible periodicity around 0.18 hrs, which is weakly supported by
the DCF curve. The other LC is irregular and shows no hint of a timescale
(Fig. 2).
S5 0716$+$714: The blazar S5 0716$+$714 is classified as a BL Lac object.
Nilsson et al. (2008) made a recent claim of redshift determination of the
source to be $z=0.31\pm 0.08$. This source has been extensively studied at all
observable wavelengths from radio to $\gamma$-rays on diverse time scales
(Wagner et al., 1990; Heidt & Wagner, 1996; Villata et al., 2000; Raiteri et
al., 2003; Montagni et al., 2006; Foschini et al., 2006; Ostorero et al.,
2006; Gupta et al., 2008a, c). This source is one of the brightest BL Lacs in
optical bands with an IDV duty cycle of nearly 1. Unsurprisingly, it has been
the subject of several optical monitoring campaigns on IDV timescales (Heidt &
Wagner, 1996; Montagni et al., 2006; Gupta et al., 2008c). This source has
shown five major optical outbursts (Gupta et al., 2008c) at intervals of
$\sim$3.0$\pm$0.3 years. High optical polarizations of $\sim$ 20$\%$ \- 29$\%$
has also been observed in the source (Takalo et al., 1994; Fan et al., 1997).
Gupta et al. (2009) reported good evidence for nearly periodic oscillations in
a few of the intraday optical light curves of the source observed by Montagni
et al. (2006). Good evidence of presence of a $\sim$15 minute periodic
oscillation at optical frequencies has been reported by Rani et al. (2010a).
Our optical IDV observations of S5 0716$+$714 spans a time period from October
2008 to January 2009. The source brightened by a factor of $\sim$2 magnitude
during this period (Rani et al., 2010b). We found significant microvariability
of $\sim$0.1 magnitude in three out of four LCs of the source. The LCs
observed on December 24, 2008 and January 03, 2009 show continuous fading
trends trend of the order of $\sim$0.1 magnitude, though the former is abrupt
while the latter is gradual. Both fading and brightening and fading trends of
$\sim$0.05 magnitude were observed over just a few minutes on December 23,
2008. The calculated possible variability timescales are listed in Table 4,
but the lack of agreement between the SF and DCF possibilities leads us to
discount their reality.
PKS 0735$+$178: The blazar PKS 0735$+$718 has been classified as a BL Lac
object (Carswell et al., 1974). Papers concerning its redshift determination
(Carswell et al., 1974; Falomo & Ulrich, 2000) had set a lower limit of
$z>0.4$ and $z=0.424$ was reported for the source using a HST snapshot image
(Sbarufatti et al., 2005). Since its optical identification, the source has
been extensively observed across the whole electromagnetic spectrum
(Teräsranta et al., 2004; Gu et al., 2006; Gupta et al., 2008c; Ciprini et
al., 2007). A periodicity of $\sim$14 years has been suggested to be present
in the source using a century long, but still sparse, optical light curve (Fan
et al., 1997). Optical variability on IDV and STV timescales has been observed
for 0735$+$178 (Xie et al., 1992; Massaro et al., 1995; Fan et al., 1997;
Zhang et al., 2004; Ciprini et al., 2007; Gupta et al., 2008c). A significant
amount of polarization ($\sim$ 1$\%$ to 30$\%$) has been observed in the
source both at optical and IR bands (Mead et al., 1990; Takalo, 1991; Takalo
et al., 1992b; Valtaoja et al., 1991a, 1993; Tommasi et al., 2001).
Our IDV observations of the source PKS 0735$+$718 comprise four LCs taken
between December 2008 and January 2009\. The brightness of the source changes
by $\sim$0.6 magnitude during this period (Rani et al., 2010b). We found
significant microvariations of an order of $\sim$0.1 magnitude in three out of
four observed LCs of the source (Fig. 4). The only conceivable hints of
timescales for variability from the SFs are $\sim$2.3 hrs for the LC observed
on December 28, 2008 and $\sim$0.58 hrs for January 04, 2009. However, since
both of these peaks are close to the total lengths of the observations they
are not likely to be real, nor supported by DCF results.
OJ 287: The blazar OJ 287 at $z=0.306$ is one of the most extensively observed
and best studied BL Lac objects. It is also among the very few AGN’s for which
more than a century of optical observations are available (Sillanpää et al.,
1996a, b; Fan et al., 1998; Abraham, 2000; Gupta et al., 2008c; Fan et al.,
2009). Using the binary black hole model (Sillanpää et al., 1988) for the
long-term optical light curve of the source, an outburst with a predicted
$\sim$12 year period was detected in the source by the OJ-94 programme
(Sillanpää et al., 1996a; Valtonen et al., 2008). A very high optical
polarization and variability in the degree and angle of polarization has been
also reported for OJ 287 (Efimov et al., 2002). The observational properties
of the source from radio to X-ray energy bands have been reviewed by Takalo et
al. (1994). Recently, Fan et al. (2009), reported large variations in the
source of $\Delta$V = 1.96 mag, $\Delta$R = 2.36 mag, and $\Delta$I = 1.95 mag
during their observations spanning 2002 to 2007.
Our observations of OJ 287 span a period from December 2008 to January 2009,
and during this time the brightness of the source changed significantly by
$\sim$1 magnitude (Rani et al., 2010b). The source showed significant
microvariations only in one out of three observed nightly LCs during which the
brightness of the source continuously faded by $\sim$0.08 magnitude within 2
hrs (Fig. 5). The SF and DCF analysis showed that any timescale of variability
is longer than the timescale of observation.
3C 273: The FSRQ 3C 273 was the first quasar discovered and has a redshift of
0.158 (Schmidt, 1963). It is categorized as a LBL (Nieppola et al., 2006) and
its spectral energy distribution, correlations among different energy band
flares and the approaching jet orientation have been extensively studied at
all wavelengths. There are many papers covering the observational properties
of the source in the optical band (Angione, 1971; Sitko et al., 1982; Corso et
al., 1985, 1986; Moles et al., 1986; Hamuy & Maza, 1987; Sillanpää et al.,
1991; Valtaoja et al., 1991b; Takalo et al., 1992a, b; Elvis et al., 1994;
Lichti et al., 1995; Ghosh et al., 2000). An analysis of the optical light
curve of 3C 273 spanning over 100 years can be interpreted to suggest a LTV
timescale of $\sim$13.5 years (Fan et al., 2001). Recently, the short-term
optical variability and colour index properties of the source have been
studied by Dai et al. (2009).
Our IDV observations of the source 3C 273 span the period from December 2008
to April 2009 during which a total of eight LCs were obtained. There is no
change in overall flux of the source during this period (Rani et al., 2010b).
This source showed microvariations (of $\sim$0.05 magnitude) in only two out
of eight observed LCs (Fig. 5). On the night of 19 April 2009 there is a hint
of a timescale of variability of $\sim$6 hrs from both the SF and DCF
approaches.
PKS 1510$-$089: The blazar PKS 1510$-$089 is classified as a FSRQ and has
$z=0.361$. It also belongs to the category of highly polarized quasars.
Significant optical flux variations in the source were first reported by Lu
(1972) over a time span of $\sim$5 years. The historical light curve of the
source shows a large variation of $\Delta$B = 5.4 mag during an outburst in
1948 after which it faded by $\sim$2.2 mag within 9 days (Liller & Liller,
1975). Strong variations on IDV time scales also have been reported for PKS
1510$-$089; e.g., $\Delta$R = 0.65 mag within 13 min. (Xie et al., 2001),
$\Delta$R = 2.0 mag in 42 min. (Dai et al., 2001), $\Delta$V = 1.68 mag in 60
min. (Xie et al., 2002a). In the optical light curves of this source, a few
deep minima have been observed on different days (Xie et al., 2001; Dai et
al., 2001; Xie et al., 2002b), that nominally correspond to a time scale of
$\sim$42 min, though no more than 3 such dips were ever seen in a single
night. Nonetheless, an eclipsing binary black hole model was actually proposed
to explain the occurrences of these minima (Wu et al., 2005). Other
observations by this group have yielded a claim of another possible time scale
between minima of $\sim$89 min (Xie et al., 2004).
We carried out IDV observations of the source from April to June 2009. There
is a large change in the brightness of the source during this period , with
$\Delta$Rmag = 1.5 (Rani et al., 2010b). We found significant microvariations
of $\sim$0.05–0.08 magnitude in three out of four LCs (Fig. 6). We found that
any timescale of variability is larger than the timescale of observations,
except perhaps for the LC observed on April 19, 2009 for which it is formally
$\sim$0.6 hrs from the SF curve and $\sim$00.5 hr from the DCF; however, this
LC has too few points to allow the production of a crisp SF or DCF, so this
evidence is very weak.
BL Lac: The object BL Lac at $z=0.069$ (Miller et al., 1978) is the archetype
of its class. Observations over the past few decades have showed that its
optical and radio emissions are highly variable and polarized and the
polarization at those widely separated frequencies is found to be strongly
correlated (Sitko et al., 1985). It is among the very few sources for which
more than 100 years of optical data is available in the literature (Shen,
1970; Webb et al., 1988; Fan et al., 1998). An optical variation of $\Delta$B
= 5.3 mag and a possible periodicity of $\sim$14 years has been reported for
BL Lac by Fan et al. (1998). Very recently, Nieppola et al. (2009) have
studied the long term variability of the source at radio frequencies and
generalized the shock model that can explain it.
Our IDV observations of the source BL Lac were made between September 2008 and
June 2009. The brightness of the source faded by $\sim$1.6 magnitude during
this period (Rani et al., 2010b). BL Lac faded by $\sim$0.05 magnitude within
1 hr of observation on September 04, 2008 which had a nominal variability
timescale of $\sim$2.5 hrs according to both the SF and DCF plots (Fig. 7). A
continuous rising trend of $\sim$0.1 magnitude in the LC of the source was
observed on June 21, 2009, and the calculated timescale of variability for the
LC that night exceeds the time period of the observations.
3C 454.3: A FSRQ at a redshift of 0.859, 3C 454.3 is among the most intense
and variable sources. The source has been detected in the flaring state in
July 2007 and July 2008 at $\gamma$-ray frequencies and those flares have been
found to be well correlated with optical and longer wavelength flares
(Ghisellini et al., 2007; Raiteri et al., 2008; Villata et al., 2007). The
long term observational properties of the source at optical and radio
frequencies have been well studied through multiwavelength campaigns (e.g.,
Villata et al., 2006, 2007). The IDV of the source was recently studied by
Gupta et al. (2008c). They have reported that the amplitude varied by $\sim$5
- 17$\%$ during their observing span. An extraordinary flaring activity above
100 MeV has been reported in the source in December 2009 (Striani et al.,
2010).
We observed two IDV LCs of the source on October 24 and 28, 2008. During the
period the brightness of the source increases by $\sim$0.4 magnitude (Rani et
al., 2010b). The source showed significant microvariations of $\sim$0.13
magnitude on October 24, 2008 (Fig. 7) while no significant variations has
been detected for the LC observed on October 28, 2008. The SF and DCF analysis
revealed that any timescale of variability exceeds the length of the
observation on the first of those nights.
## 4 Discussion and Conclusions
We have carried out optical R band IDV observations of ten LBLs spanning over
a time period of 2008 September to 2009 June. The sources PKS 0420$-$014 and
PKS 1510$-$089 were in faint states; AO 0235$+$164, BL Lac and 3C 454.3 were
possibly in post-outburst states; S5 0716$+$714 and 3C 66A were in pre-
outburst states, while PKS 0735$+$178 and OJ 287 were in some intermediate
states during the observing run (Rani et al., 2010b). In our search of
microvariations in ten LBLs we found significant IDV in 21 out of 40 observed
LCs; so the calculated duty cycle of IDV in LBLs during our observing run is
$\sim$52$\%$. We performed the SF and DCF analysis to calculate the nominal
timescales of variability; however we found that in most of the cases this
timescale of variability is longer than the length of observations and in a
substantial majority of the cases where the SF indicated a possible timescale
the DCF did not support it.
The blazar emission mechanism in the outburst state is quantitatively
understood by relativistic shocks propagating through a relativistic jet of
plasma. In general, blazar emission in the outburst state is non-thermal
Doppler-boosted emission from jets enhanced by that arising from shocks in the
flows. (Blandford & Rees, 1978; Marscher & Gear, 1985; Marscher et al., 1992).
The other models of AGNs that can explain the IDV in any type of AGN are
optical flares, disturbances or hot spots on the accretion disk surrounding
the black hole of the AGN (e.g., Mangalam & Wiita, 1993, and references
therein). Models based on the instabilities on the accretion disk could
convinancibly yield blazar IDV only when the blazar is in the very low state.
When a blazar is in the low state, any contribution from the jets, if at all
present, is very weak. So, we consider that the observed IDV in the sources AO
0235$+$164, BL Lac, 3C 454.3, S5 0716$+$714, 3C 66A, 0735$+$178 and OJ 287 is
almost certainly related to a shock propagating through a relativistic jet
(Blandford & Konigl, 1979; Marscher & Gear, 1985). Turbulence behind a shock
propagating down such a jet is a very feasible way to explain the observed IDV
(Marscher, 1996). Since the blazars PKS 0420$-$014 and PKS 1510$-$089 were
observed in relatively faint states, there is a chance that the observed
optical IDV in these source may be because of hot spots or any other enhanced
emission on the accretion disk (Mangalam & Wiita, 1993).
In one source, 3C 273, we are unable to classify the state of the source since
this blazar has been in an essentially steady state for last six years (Dai et
al., 2009; Rani et al., 2010b). This source showed significant microvariations
in two out of eight observed LCs and on both of days of observation the
brightness of the source follows both rising and fading trends of $\sim$0.05
magnitude. Whatever may be the mechanism responsible for the origin of of
microvariability in this source, it does not seem to be strong enough to
introduce day-to-day variations in the flux of the source.
It is worth noting that an extrinsic mechanism can also be responsible for
some of the observed IDV in blazars. Extrinsic IDV could be caused by
refractive interstellar scintillation, which is only relevant in low-frequency
radio observations, or microlensing, which is achromatic. We note that the
blazar AO 0235$+$164 has two foreground galaxies at $z=0.524$ and $z=0.851$
(Cohen et al., 1987; Nilsson et al., 1996; Webb et al., 2000). The flux of
this source is contaminated and absorbed by foreground galaxies, the stars of
which can act as gravitational microlenses. Thus the observed optical IDV in
AO 0235+164 could arise, at least partially, from gravitational microlensing.
Using two separated telescopes to simultaneously observe 0716$+$714 Pollock,
Webb & Azarnia (2007) showed that instrumental and atmospheric effects cannot
account for the microvariations they measured for that blazar.
We found significant IDV in 21 out of 40 observed LCs of ten LBLs; so the
calculated duty cycle of IDV in LBLs during our observing run is $\sim$52$\%$.
The average duration of our observation was 3.7 hrs per LC. The SF and DCF
curves revealed that in $\sim$60$\%$ cases any timescale of variability is
longer than the timescale of observations. In quite a few cases there were
hints of timescales in the data from the SF plots, but only a few of those
hints were supported by the DCF plots.
Extensive IDV studies of different subclasses of AGNs revealed that the
occurrence of IDV in blazars observed on a timescale of $<$6 hrs is
$\sim$60$-$65$\%$ and if the blazar is observed more than 6 hrs then the
possibility of IDV detection is 80$-$85$\%$ (Carini, 1990; Gupta & Joshi,
2005, and references therein). If we consider observations over days to
months, i.e., at STV timescales then the observed duty cycle of variations is
$>$92$\%$ (e.g. Rani et al., 2010b) and at LTV timescales it is almost
100$\%$, which confirms that the probability of detection of variability in
blazars rises with the duration of the observations. Although the duty cycle
from our current observations is less than that reported by Gupta & Joshi
(2005), since the average duration of our observations is $<$4 hrs, this is
unsurprising.
## Acknowledgments
Work at PRL is supported by the Department of Space, Govt. of India.
## References
* Abraham (2000) Abraham, Z. 2000, , 355, 915
* Aller et al. (1992) Aller, M. F., Aller, H. D., & Hughes, P. A. 1992, , 399, 16
* Angione (1971) Angione, R. J. 1971, , 76, 412
* Blandford & Konigl (1979) Blandford, R. D., & Konigl, A. 1979, , 232, 34
* Blandford & Rees (1978) Blandford, R. D., & Rees, M. J. 1978, , 17, 265
* Böttcher et al. (2009) Böttcher, M., et al. 2009, , 694, 174
* Böttcher et al. (2005) Böttcher, M., et al. 2005, , 631, 169
* Camenzind & Krockenberger (1992) Camenzind, M., & Krockenberger, M. 1992, , 255, 59
* Carini (1990) Carini, M. T. 1990, Ph.D. thesis, Georgia State Univ., Atlanta.
* Carswell et al. (1974) Carswell, R. F., Strittmatter, P. A., Williams, R. E., Kinman, T. D., & Serkowski, K. 1974, , 190, L101
* Chakrabarti & Wiita (1993) Chakrabarti, S. K., & Wiita, P. J. 1993, , 411, 602
* Ciprini et al. (2007) Ciprini, S., et al. 2007, , 467, 465
* Ciprini et al. (2003) Ciprini, S., Tosti, G., Raiteri, C. M., Villata, M., Ibrahimov, M. A., Nucciarelli, G., & Lanteri, L. 2003, , 400, 487
* Clements et al. (1995) Clements, S. D., Smith, A. G., Aller, H. D., & Aller, M. F. 1995, , 110, 529
* Cohen et al. (1987) Cohen, R. D., Smith, H. E., Junkkarinen, V. T., & Burbidge, E. M. 1987, , 318, 577
* Corso et al. (1986) Corso, G. J., Schultz, J., & Dey, A. 1986, , 98, 1287
* Corso et al. (1985) Corso, G. J., Schultz, J., Purcell, B., Garino, G., & Dey, A. 1985, , 97, 118
* Dai et al. (2009) Dai, B. Z., et al. 2009, , 392, 1181
* Dai et al. (2001) Dai, B. Z., Xie, G. Z., Li, K. H., Zhou, S. B., Liu, W. W., & Jiang, Z. J. 2001, , 122, 2901
* de Diego (2010) de Diego, J. A. 2010, , 139, 1269
* Edelson & Krolik (1988) Edelson, R. A., & Krolik, J. H. 1988, , 333, 646
* Efimov et al. (2002) Efimov, Y. S., Shakhovskoy, N. M., Takalo, L. O., & Sillanpää, A. 2002, , 381, 408
* Elvis et al. (1994) Elvis, M., et al. 1994, , 95, 1
* Emmanoulopoulos et al. (2010) Emmanoulopoulos, D., McHardy, I. M., & Uttley, P. 2010, , 404, 931
* Falomo & Ulrich (2000) Falomo, R., & Ulrich, M. 2000, , 357, 91
* Fan et al. (1997) Fan, J. H., Cheng, K. S., Zhang, L., & Liu, C. H. 1997, , 327, 947
* Fan & Lin (1999) Fan, J. H., & Lin, R. G. 1999, , 121, 131
* Fan & Lin (2000) Fan, J. H., & Lin, R. G. 2000, , 537, 101
* Fan et al. (2001) Fan, J. H., Qian, B. C., & Tao, J. 2001, , 369, 758
* Fan et al. (1998) Fan, J. H., Xie, G. Z., Pecontal, E., Pecontal, A., & Copin, Y. 1998, , 507, 173
* Fan et al. (2009) Fan, J. H., Zhang, Y. W., Qian, B. C., Tao, J., Liu, Y., & Hua, T. X. 2009, , 181, 466
* Fiorucci & Tosti (1996) Fiorucci, M., & Tosti, G. 1996, , 116, 403
* Foschini et al. (2006) Foschini, L., et al. 2006, , 455, 871
* Fossati et al. (1998) Fossati, G., Maraschi, L., Celotti, A., Comastri, A., & Ghisellini, G. 1998, , 299, 433
* Ghisellini et al. (2007) Ghisellini, G., Foschini, L., Tavecchio, F., & Pian, E. 2007, , 382, L82
* Ghisellini et al. (1997) Ghisellini, G., et al. 1997, , 327, 61
* Ghosh et al. (2000) Ghosh, K. K., Ramsey, B. D., Sadun, A. C., & Soundararajaperumal, S. 2000, , 127, 11
* Ghosh & Soundararajaperumal (1995) Ghosh, K. K., & Soundararajaperumal, S. 1995, , 100, 37
* Gopal-Krishna & Wiita (1992) Gopal-Krishna, & Wiita, P. J. 1992, , 259, 109
* Gu et al. (2006) Gu, M. F., Lee, C., Pak, S., Yim, H. S., & Fletcher, A. B. 2006, , 450, 39
* Gupta et al. (2004) Gupta, A. C., Banerjee, D. P. K., Ashok, N. M., & Joshi, U. C. 2004, , 422, 505
* Gupta & Joshi (2005) Gupta, A. C., & Joshi, U. C. 2005, , 440, 855
* Gupta et al. (2008a) Gupta, A. C., et al. 2008a, , 136, 2359
* Gupta et al. (2008b) Gupta, A. C., Deng, W. G., Joshi, U. C., Bai, J. M., & Lee, M. G. 2008b, , 13, 375
* Gupta et al. (2008c) Gupta, A. C., Fan, J. H., Bai, J. M., & Wagner, S. J. 2008c, , 135, 1384
* Gupta et al. (2009) Gupta, A. C., Srivastava, A. K., & Wiita, P. J. 2009, , 690, 216
* Hamuy & Maza (1987) Hamuy, M., & Maza, J. 1987, , 68, 383
* Heidt & Wagner (1996) Heidt, J., & Wagner, S. J. 1996, , 305, 42
* Hovatta et al. (2007) Hovatta, T., Tornikoski, M., Lainela, M., Lehto, H. J., Valtaoja, E., Torniainen, I., Aller, M. F., & Aller, H. D. 2007, , 469, 899
* Hufnagel & Bregman (1992) Hufnagel, B. R., & Bregman, J. N. 1992, , 386, 473
* Hughes et al. (1991) Hughes, P. A., Aller, H. D., & Aller, M. F. 1991, , 374, 57
* Impey et al. (1982) Impey, C. D., Brand, P. W. J. L., & Tapia, S. 1982, , 198, 1
* Krawczynski (2004) Krawczynski, H. 2004, , 48, 367
* Lanzetta et al. (1993) Lanzetta, K. M., Turnshek, D. A., & Sandoval, J. 1993, , 84, 109
* Lichti et al. (1995) Lichti, G. G., et al. 1995, , 298, 711
* Liller & Liller (1975) Liller, M. H., & Liller, W. 1975, , 199, L133
* Lu (1972) Lu, P. K. 1972, , 77, 829
* Mangalam & Wiita (1993) Mangalam, A. V., & Wiita, P. J. 1993, , 406, 420
* Marscher (1996) Marscher, A. P. 1996, in Astronomical Society of the Pacific Conference Series, Vol. 110, Blazar Continuum Variability, ed. H. R. Miller, J. R. Webb, & J. C. Noble, 248
* Marscher & Gear (1985) Marscher, A. P., & Gear, W. K. 1985, , 298, 114
* Marscher et al. (1992) Marscher, A. P., Gear, W. K., & Travis, J. P. 1992, in Variability of Blazars, 85
* Massaro et al. (1995) Massaro, E., Nesci, R., Perola, G. C., Lorenzetti, D., & Spinoglio, L. 1995, , 299, 339
* Mead et al. (1990) Mead, A. R. G., Ballard, K. R., Brand, P. W. J. L., Hough, J. H., Brindle, C., & Bailey, J. A. 1990, , 83, 183
* Miller et al. (1978) Miller, J. S., French, H. B., & Hawley, S. A. 1978, , 219, L85
* Moles et al. (1986) Moles, M., Garcia-Pelayo, J. M., Masegosa, J., & Garrido, R. 1986, , 92, 1030
* Montagni et al. (2006) Montagni, F., Maselli, A., Massaro, E., Nesci, R., Sclavi, S., & Maesano, M. 2006, , 451, 435
* Nieppola et al. (2009) Nieppola, E., Hovatta, T., Tornikoski, M., Valtaoja, E., Aller, M. F., & Aller, H. D. 2009, , 137, 5022
* Nieppola et al. (2006) Nieppola, E., Tornikoski, M., & Valtaoja, E. 2006, , 445, 441
* Nilsson et al. (1996) Nilsson, K., Charles, P. A., Pursimo, T., Takalo, L. O., Sillanpaeae, A., & Teerikorpi, P. 1996, , 314, 754
* Nilsson et al. (2008) Nilsson, K., Pursimo, T., Sillanpää, A., Takalo, L. O., & Lindfors, E. 2008, , 487, L29
* Ostorero et al. (2006) Ostorero, L., et al. 2006, , 451, 797
* Pollock, Webb & Azarnia (2007) Pollock J. T., Webb J. R., Azarnia G., 2007, , 133, 487
* Qian et al. (1991) Qian, S. J., Quirrenbach, A., Witzel, A., Krichbaum, T. P., Hummel, C. A., & Zensus, J. A. 1991, , 241, 15
* Raiteri et al. (1998) Raiteri, C. M., Ghisellini, G., Villata, M., de Francesco, G., Lanteri, L., Chiaberge, M., Peila, A., & Antico, G. 1998, , 127, 445
* Raiteri et al. (2001) Raiteri, C. M., et al. 2001, , 377, 396
* Raiteri et al. (2006a) Raiteri, C. M., et al. 2006a, , 459, 731
* Raiteri et al. (2006b) Raiteri, C. M., Villata, M., Kadler, M., Krichbaum, T. P., Böttcher, M., Fuhrmann, L., & Orio, M. 2006b, , 452, 845
* Raiteri et al. (2008) Raiteri, C. M., et al. 2008, , 491, 755
* Raiteri et al. (2003) Raiteri, C. M., et al. 2003, , 402, 151
* Rani et al. (2010a) Rani, B., Gupta, A. C., Joshi, U. C., Ganesh, S., & Wiita, P. J. 2010a, , 719, L153
* Rani et al. (2010b) Rani, B., et al. 2010b, , 404, 1992
* Rani et al. (2009) Rani, B., Wiita, P. J., & Gupta, A. C. 2009, , 696, 2170
* Rickett et al. (2001) Rickett, B. J., Witzel, A., Kraus, A., Krichbaum, T. P., & Qian, S. J. 2001, , 550, L11
* Romero et al. (1999) Romero, G. E., Cellone, S. A., & Combi, J. A. 1999, , 135, 477
* Sbarufatti et al. (2005) Sbarufatti, B., Treves, A., & Falomo, R. 2005, , 635, 173
* Schmidt (1963) Schmidt, M. 1963, , 197, 1040
* Shen (1970) Shen, B. S. P. 1970, , 228, 1070
* Sikora & Madejski (2001) Sikora, M., & Madejski, G. 2001, in American Institute of Physics Conference Series, Vol. 558, American Institute of Physics Conference Series, ed. F. A. Aharonian & H. J. Völk, 275
* Sillanpää et al. (1988) Sillanpää, A., Haarala, S., Valtonen, M. J., Sundelius, B., & Byrd, G. G. 1988, , 325, 628
* Sillanpää et al. (1991) Sillanpää, A., Mikkola, S., & Valtaoja, L. 1991, , 88, 225
* Sillanpää et al. (1996a) Sillanpää, A., et al. 1996a, , 305, L17
* Sillanpää et al. (1996b) Sillanpää, A., et al. 1996b, , 315, L13
* Simonetti et al. (1985) Simonetti, J. H., Cordes, J. M., & Heeschen, D. S. 1985, , 296, 46
* Sitko et al. (1985) Sitko, M. L., Schmidt, G. D., & Stein, W. A. 1985, , 59, 323
* Sitko et al. (1982) Sitko, M. L., Stein, W. A., Zhang, Y., & Wisniewski, W. Z. 1982, , 259, 486
* Smith & Balonek (1998) Smith, P. S., & Balonek, T. J. 1998, , 110, 1164
* Smith et al. (1985) Smith, P. S., Balonek, T. J., Heckert, P. A., Elston, R., & Schmidt, G. D. 1985, , 90, 1184
* Spinrad & Smith (1975) Spinrad, H., & Smith, H. E. 1975, , 201, 275
* Stetson (1987) Stetson, P. B. 1987, , 99, 191
* Stetson (1992) Stetson, P. B. 1992, , 86, 71
* Striani et al. (2010) Striani, E., et al. 2010, , 718, 455
* Takalo (1991) Takalo, L. O. 1991, , 90, 161
* Takalo et al. (1992a) Takalo, L. O., Kidger, M. R., de Diego, J. A., & Sillanpää, A. 1992a, , 261, 415
* Takalo et al. (1992b) Takalo, L. O., Sillanpää, A., Nilsson, K., Kidger, M., de Diego, J. A., & Piirola, V. 1992b, , 94, 37
* Takalo et al. (1994) Takalo, L. O., Sillanpaeae, A., & Nilsson, K. 1994, , 107, 497
* Takalo et al. (1996) Takalo, L. O., et al. 1996, , 120, 313
* Teräsranta et al. (2004) Teräsranta, H., et al. 2004, , 427, 769
* Tommasi et al. (2001) Tommasi, L., et al. 2001, , 376, 51
* Urry & Padovani (1995) Urry, C. M., & Padovani, P. 1995, , 107, 803
* Valtaoja et al. (1993) Valtaoja, L., Karttunen, H., Efimov, Y., & Shakhovskoy, N. M. 1993, , 278, 371
* Valtaoja et al. (1991a) Valtaoja, L., Sillanpää, A., Valtaoja, E., Shakhovskoi, N. M., & Efimov, I. S. 1991a, , 101, 78
* Valtaoja et al. (1991b) Valtaoja, L., et al. 1991b, , 102, 1946
* Valtonen et al. (2008) Valtonen, M., Kidger, M., Lehto, H., & Poyner, G. 2008, , 477, 407
* Villata et al. (1997) Villata, M., et al. 1997, , 121, 119
* Villata et al. (1998) Villata, M., Raiteri, C. M., Lanteri, L., Sobrito, G., & Cavallone, M. 1998, , 130, 305
* Villata et al. (2000) Villata, M., et al. 2000, , 363, 108
* Villata et al. (2004) Villata, M., et al. 2004, , 424, 497
* Villata et al. (2006) Villata, M., et al. 2006, , 453, 817
* Villata et al. (2007) Villata, M., et al. 2007, , 464, L5
* Wagner et al. (1990) Wagner, S., Sanchez-Pons, F., Quirrenbach, A., & Witzel, A. 1990, , 235, L1
* Wagner & Witzel (1995) Wagner, S. J., & Witzel, A. 1995, , 33, 163
* Webb et al. (1988) Webb, J. R., Smith, A. G., Leacock, R. J., Fitzgibbons, G. L., Gombola, P. P., & Shepherd, D. W. 1988, , 95, 374
* Webb et al. (1998) Webb, J. R., et al. 1998, , 115, 2244
* Webb et al. (2000) Webb J. R., Howard E., Benítez E., Balonek T., McGrath E., Shrader C., Robson I., Jenkins P., 2000, AJ, 120, 41
* Wills & Wills (1974) Wills, B. J., & Wills, D. 1974, , 190, L97
* Wu et al. (2005) Wu, J., Zhou, X., Peng, B., Ma, J., Jiang, Z., & Chen, J. 2005, , 361, 155
* Xie et al. (2001) Xie, G. Z., Li, K. H., Bai, J. M., Dai, B. Z., Liu, W. W., Zhang, X., & Xing, S. Y. 2001, , 548, 200
* Xie et al. (1992) Xie, G. Z., Li, K. H., Liu, F. K., Lu, R. W., Wu, J. X., Fan, J. H., Zhu, Y. Y., & Cheng, F. Z. 1992, , 80, 683
* Xie et al. (2002a) Xie, G. Z., Liang, E. W., Zhou, S. B., Li, K. H., Dai, B. Z., & Ma, L. 2002a, , 334, 459
* Xie et al. (2002b) Xie, G. Z., Zhou, S. B., Dai, B. Z., Liang, E. W., Li, K. H., Bai, J. M., Xing, S. Y., & Liu, W. W. 2002b, , 329, 689
* Xie et al. (2004) Xie, G. Z., Zhou, S. B., Li, K. H., Dai, H., Chen, L. E., & Ma, L. 2004, , 348, 831
* Zhang et al. (2004) Zhang, X., Zhang, L., Zhao, G., Xie, Z., Wu, L., & Zheng, Y. 2004, , 128, 1929
Table 1: Properties of Telescopes and Instruments Site: | ARIES Nainital | PRL Mount Abu
---|---|---
Telescope: | 1.04-m RC Cassegrain | 1.20 m Cassegrain
CCD model: | Wright 2K CCD | Andor EMCCD
Chip size: | $2048\times 2048$ pixels | $2048\times 2048$ pixels
Pixel size: | $24\times 24$ $\mu$m | $13\times 13$ $\mu$m
Scale: | 0.37″/pixel | 0.17″/pixel
Field: | $13\arcmin\times 13\arcmin$ | $3\arcmin\times 3\arcmin$
Gain: | 10 $e^{-}$/ADU | 5 $e^{-}$/ADU
Read Out Noise: | 5.3 $e^{-}$ rms | 4.9 $e^{-}$ rms
Binning used: | $2\times 2$ | $2\times 2$
Typical seeing : | 1″to 2.8″ | 1″to 2.6″
Table 2: Observation Log
Blazar Name | Date of Observation | Telescope | Filter | Data Points | Duration (hrs)
---|---|---|---|---|---
3C 66A | 2008 Oct 22 | A | R | 68 | 3.86
| 2008 Oct 26 | A | R | 72 | 3.50
| 2008 Dec 23 | B | R | 90 | 1.45
| 2008 Dec 24 | B | R | 113 | 1.85
| 2008 Dec 27 | B | R | 417 | 3.46
| 2008 Dec 28 | B | R | 203 | 2.23
| 2009 Jan 03 | B | R | 320 | 3.54
AO 0235$+$164 | 2008 Oct 20 | A | R | 98 | 6.30
| 2008 Oct 23 | A | R | 45 | 2.60
| 2008 Dec 26 | B | R | 248 | 1.23
PKS 0420$-$014 | 2008 Oct 23 | A | R | 18 | 1.94
| 2008 Dec 26 | B | R | 29 | 0.73
S5 0716+714 | 2008 Oct 24 | A | R | 25 | 1.35
| 2008 Dec 23 | B | R | 114 | 0.40
| 2008 Dec 24 | B | R | 292 | 1.62
| 2009 Jan 03 | B | R | 2685 | 3.73
PKS 0735$+$178 | 2008 Dec 23 | B | R | 90 | 0.63
| 2008 Dec 28 | B | R | 300 | 3.20
| 2009 Jan 04 | B | R | 50 | 1.02
| 2009 Jan 20 | A | R | 67 | 3.98
OJ 287 | 2008 Dec 26 | B | R | 70 | 0.57
| 2009 Jan 20 | A | R | 60 | 3.87
| 2009 Jan 22 | A | R | 70 | 3.35
3C 273 | 2008 Dec 23 | B | R | 130 | 0.72
| 2009 Jan 22 | A | R | 90 | 3.88
| 2009 Feb 25 | A | R | 101 | 3.61
| 2009 Mar 24 | A | R | 36 | 1.38
| 2009 Apr 01 | A | R | 110 | 5.60
| 2009 Apr 18 | A | R | 91 | 3.54
| 2009 Apr 19 | A | R | 195 | 7.42
| 2009 Apr 27 | A | R | 66 | 3.31
PKS 1510$-$089 | 2009 Apr 17 | A | R | 77 | 3.82
| 2009 Apr 19 | A | R | 22 | 1.09
| 2009 Apr 27 | A | R | 64 | 4.47
| 2009 June 21 | A | R | 68 | 4.41
BL Lac | 2008 Sep 04 | A | R | 83 | 5.10
| 2008 Oct 26 | A | R | 73 | 4.25
| 2009 June 21 | A | R | 62 | 2.98
3C 454.3 | 2008 Oct 24 | A | R | 55 | 3.01
| 2008 Oct 28 | A | R | 65 | 4.25
A : 1.04 m Sampuranand Telescope, ARIES, Nainital, India
B : 1.20 m Telescope, PRL, Mount Abu, India
Table 3: Standard Stars in the Blazar Fields
Source | Standard | R magnitude | Refrencesa
---|---|---|---
Name | star | (error) |
3C 66A | 1 | 13.36(0.01) | 5
| 2 | 14.28(0.04) | 5
| 3 | 15.46(0.12) | 5
| 4 | 12.70(0.04) | 6
| 5 | 13.62(0.05) | 6
AO 0235+164 | 1 | 12.69(0.02) | 1
| 2 | 12.23(0.02) | 1
| 3 | 12.48(0.03) | 1
| 6 | 13.64(0.04) | 6
| 8 | 15.79(0.10) | 1
| C1 | 14.23(0.05) | 6
PKS 0420$-$014 | 1 | 12.09(0.03) | 4
| 2 | 12.80(0.02) | 5
| 3 | 12.89(0.01) | 5
| 4 | 14.47(0.01) | 5
| 5 | 14.37(0.03) | 4
| 6 | 14.70(0.03) | 4
| 7 | 14.91(0.03) | 4
| 8 | 15.46(0.03) | 4
| 9 | 15.58(0.04) | 4
S5 0716+714 | 1 | 10.63(0.01) | 2
| 2 | 11.12(0.01) | 2, 3
| 3 | 12.06(0.01) | 2, 3
| 4 | 12.89(0.01) | 2
| 5 | 13.18(0.01) | 2, 3
| 6 | 13.26(0.01) | 2, 3
| 7 | 13.32(0.01) | 2
| 8 | 13.79(0.02) | 2
PKS 0735+178 | A | 13.14(0.05) | 1
| C | 13.87(0.06) | 1
| D | 15.45(0.06) | 1
OJ 287 | 2 | 12.46(0.05) | 1
| 4 | 13.72(0.06) | 1
| 10 | 14.26(0.06) | 1
| 11 | 14.67(0.07) | 1
3C 273 | C | 11.30(0.04) | 1
| D | 12.31(0.04) | 1
| E | 12.27(0.05) | 1
| G | 13.16(0.05) | 1
PKS 1510$-$089 | 1 | 11.23(0.03) | 5
| 2 | 12.95(0.03) | 5
| 3 | 13.98(0.09) | 5
| 4 | 14.34(0.05) | 5
| 5 | 14.35(0.05) | 4
| 6 | 14.61(0.02) | 4
BL Lac | B | 11.93(0.05) | 1
| C | 13.69(0.03) | 1
| H | 13.60(0.03) | 1
| K | 14.88(0.05) | 1
3C 454.3 | A | 15.32(0.09) | 6, 9
| B | 14.73(0.05) | 6, 9
| C | 13.98(0.02) | 9, 10
| D | 13.22(0.01) | 9, 10
| E | 14.92(0.08) | 6, 9
| F | 14.83(0.03) | 4, 9
| G | 14.83(0.02) | 4, 9
| H | 13.10(0.04) | 6, 9
| C1 | 15.27(0.06) | 6
a1\. (Smith et al., 1985); 2. (Villata et al., 1998); 3. (Ghisellini et al.,
1997); 4. (Raiteri et al., 1998); 5. (Smith & Balonek, 1998); 6\. (Fiorucci &
Tosti, 1996); 7. Craine E.R.:Handbook of Quasistellar and BL Lacertae Objects,
(Angione, 1971); 8\. http://www.lsw.uni-
heidelberg.de/projects/extragalactic/charts/2251+158.html
Table 4: IDV Results Blazar Name | Date of | C - Test | F - Test | V | A ($\%$) | tv
---|---|---|---|---|---|---
| Observation | value | F-value | p-value | | | SF (hrs) | DCF (hrs)
3C 66A | 2008 Oct 22 | 3.59 | 3.65 | 3.1e-7 | V | 8.1 | $>$3.86 | $>$3.86
| 2008 Oct 26 | 2.65 | 7.03 | 1.9e-14 | V | 5.7 | $>$3.50 | $>$3.50
| 2008 Dec 23 | 0.78 | 0.41 | 0.06 | NV | | |
| 2008 Dec 24 | 3.31 | 0.61 | 0.02 | NV | | |
| 2008 Dec 27 | 0.88 | 0.58 | 0.26 | NV | | |
| 2008 Dec 28 | 0.75 | 0.37 | 0.3 | NV | | |
| 2009 Jan 03 | 0.83 | 0.63 | 0.35 | NV | | |
AO 0235$+$164 | 2008 Oct 20 | 8.32 | 76.84 | 2.2e-16 | V | 13.4 | 5.20? | $>$6.30
| 2008 Oct 23 | 3.23 | 10.53 | 1.4e-14 | V | 4.2 | $>$2.60 | $>$2.60
| 2008 Dec 26 | 1.09 | 1.43 | 0.04 | NV | | |
PKS 0420$-$014 | 2008 Oct 23 | 5.44 | 31.44 | 6.6e-15 | V | 14.4 | |
| 2008 Dec 26 | 5.42 | 31.24 | 6.1e-15 | V | 12.3 | 0.12,0.18 | 0.18
S5 0716$+$714 | 2008 Oct 24 | 0.92 | 1.57 | 0.27 | NV | | |
| 2008 Dec 23 | 3.47 | 7.77 | 2.2e-16 | V | 9.1 | 0.0014 | 0.0028
| 2008 Dec 24 | 4.51 | 3.07 | 2.2e-16 | V | 14.6 | $>$1.62 | $>$1.62
| 2009 Jan 03 | 3.71 | 1.26 | 7.7e-7 | V | 31.6 | 3.28 | $>$3.73
PKS 0735$+$178 | 2008 Dec 23 | 3.14 | 9.49 | 2.2e-16 | V | 11.1 | |
| 2008 Dec 28 | 4.22 | 18.36 | 2.2e-16 | V | 19.3 | 2.90 | 2.30
| 2009 Jan 04 | 2.31 | 4.64 | 3.1e-7 | V | 9.8 | 0.58 | 0.60
| 2009 Jan 20 | 1.30 | 1.99 | 0.055 | NV | | |
OJ 287 | 2008 Dec 26 | 1.11 | 1.48 | 0.10 | NV | | |
| 2009 Jan 20 | 0.88 | 1.20 | 0.47 | NV | | |
| 2009 Jan 22 | 3.21 | 11.13 | 2.2e-16 | V | 7.9 | $>$3.35 | $>$3.35
3C 273 | 2008 Dec 23 | 0.91 | 0.73 | 0.08 | NV | | |
| 2009 Jan 22 | 0.81 | 0.65 | 0.05 | NV | | |
| 2009 Feb 25 | 0.75 | 0.49 | 0.06 | NV | | |
| 2009 Mar 24 | 0.76 | 0.50 | 0.05 | NV | | |
| 2009 Apr 01 | 2.59 | 2.58 | 1.2e-6 | V | 5.2 | $>$5.2 | $>$5.60
| 2009 Apr 18 | 1.59 | 0.25 | 0.11 | NV | | |
| 2009 Apr 19 | 3.74 | 2.22 | 4.7e-8 | V | 4.5 | 6.0 | 6.0
| 2009 Apr 27 | 0.86 | 0.84 | 0.49 | NV | | |
PKS 1510$-$089 | 2009 Apr 17 | 2.67 | 2.31 | 0.0003 | V | 9.3 | $>$3.82 | $>$3.82
| 2009 Apr 19 | 2.72 | 7.11 | 3.3e-5 | V | 7.6 | 0.60 | 0.50
| 2009 Apr 27 | 1.27 | 1.54 | 0.09 | NV | | |
| 2009 June 21 | 4.46 | 2.65 | 0.0001 | V | 8.3 | $>$4.41 | $>$4.41
BL Lac | 2008 Sep 04 | 2.73 | 5.71 | 1.0e-13 | V | 8.7 | 2.5 | 2.5
| 2008 Oct 26 | 1.05 | 1.56 | 0.06 | NV | | |
| 2009 June 21 | 2.87 | 7.34 | 4.9e-13 | V | 12.9 | $>$2.98 | $>$2.98
3C 454.3 | 2008 Oct 24 | 5.14 | 22.95 | 2.2e-16 | V | 16.3 | $>$3.01 | $>$3.01
| 2008 Oct 28 | 1.42 | 2.09 | 0.01 | NV | | |
Figure 1: R band optical IDV LCs of the blazars 3C 66A and AO 0235$+$164 and
their respective SFs and DCFs. Figure 2: R band optical IDV LCs of the
blazars AO 0235$+$164 and PKS 0420$-$014 and their respective SFs and DCFs.
Figure 3: R band optical IDV LCs of the blazar S5 0716$+$714 and their
respective SFs and DCFs. Figure 4: R band optical IDV LCs of the blazar PKS
0735$+$178 and their respective SFs and DCFs. Figure 5: R band optical IDV LCs
of the blazars 3C 273 and OJ 287 and their respective SFs and DCFs. Figure 6:
R band optical IDV LCs of blazar PKS 1510$-$089 and their respective SFs and
DCFs. Figure 7: R band optical IDV LC of the blazars BL Lac and 3C 454.3 and
their respective SFs and DCFs. Figure 8: The R filter LCs of several blazars
during which no significant variability has been detected, plotted along with
the differential magnitudes of standard stars. Figure 9: The R filter LCs of
several blazars during which no significant variability has been detected,
plotted along with the differential magnitudes of standard stars.
|
arxiv-papers
| 2011-01-04T11:57:23 |
2024-09-04T02:49:16.125128
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bindu Rani (1), Alok C. Gupta (1), U. C. Joshi (2), S. Ganesh (2),\n Paul J. Wiita (3), ((1) Aryabhatta Research Institute of Observational\n Sciences (ARIES), India, (2) Physical Research Laboratory, Navrangpura,\n India, (3) Department of Physics, The College of New Jersey)",
"submitter": "Bindu Rani Ms.",
"url": "https://arxiv.org/abs/1101.0711"
}
|
1101.0811
|
# Measuring the Spins of Accreting Black Holes
Jeffrey E. McClintock11affiliation: Harvard-Smithsonian Center for
Astrophysics, 60 Garden Street, Cambridge, MA 02138 , Ramesh
Narayan11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden
Street, Cambridge, MA 02138 , Shane W. Davis22affiliation: Canadian Institute
for Theoretical Astrophysics, Toronto, ON M5S3H4, Canada , Lijun
Gou11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden
Street, Cambridge, MA 02138 , Akshay Kulkarni11affiliation: Harvard-
Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 ,
Jerome A. Orosz33affiliation: Department of Astronomy, San Diego State
University, 5500 Companile Drive, San Diego, CA 92182 , Robert F.
Penna11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden
Street, Cambridge, MA 02138 , Ronald A. Remillard44affiliation: MIT Kavli
Institute for Astrophysics and Space Research, MIT, 70 Vassar Street,
Cambridge, MA 02139 , James F. Steiner11affiliation: Harvard-Smithsonian
Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138
###### Abstract
A typical galaxy is thought to contain tens of millions of stellar-mass black
holes, the collapsed remnants of once massive stars, and a single nuclear
supermassive black hole. Both classes of black holes accrete gas from their
environments. The accreting gas forms a flattened orbiting structure known as
an accretion disk. During the past several years, it has become possible to
obtain measurements of the spins of the two classes of black holes by modeling
the X-ray emission from their accretion disks. Two methods are employed, both
of which depend upon identifying the inner radius of the accretion disk with
the innermost stable circular orbit (ISCO), whose radius depends only on the
mass and spin of the black hole. In the Fe K$\alpha$ method, which applies to
both classes of black holes, one models the profile of the relativistically-
broadened iron line with a special focus on the gravitationally redshifted red
wing of the line. In the continuum-fitting method, which has so far only been
applied to stellar-mass black holes, one models the thermal X-ray continuum
spectrum of the accretion disk. We discuss both methods, with a strong
emphasis on the continuum-fitting method and its application to stellar-mass
black holes. Spin results for eight stellar-mass black holes are summarized.
These data are used to argue that the high spins of at least some of these
black holes are natal, and that the presence or absence of relativistic jets
in accreting black holes is not entirely determined by the spin of the black
hole.
## 1 Introduction
Our focus throughout is on the two principal methods for measuring the spins
of accreting black holes: modeling the thermal continuum X-ray spectrum and
modeling the profile of the relativistically-broadened Fe K$\alpha$ line. The
continuum-fitting (CF) method, has thus far only been applied to stellar-mass
black holes in X-ray binaries, whereas the Fe K$\alpha$ method has been
applied to both stellar-mass ($M\sim 10$ $M_{\odot}$) and supermassive ($M\sim
10^{6}-10^{10}$ $M_{\odot}$) black holes1111 $M_{\odot}$ = 1 solar mass =
$2.0\times 10^{33}$ g.. This paper is chiefly focused on measuring the spins
of stellar-mass black holes via the CF method because we have been deeply
engaged in this work during the past several years. In Section 6, we
secondarily discuss the Fe K$\alpha$ method, which is very important because
it is the primary approach to measuring the spins of supermassive black holes.
We note that spin estimates have also been obtained by modeling the high-
frequency X-ray oscillations (100–450 Hz) observed for several stellar-mass
black holes (Wagoner et al. 2001; Török et al. 2005). At present, this method
is not providing dependable results because the correct model of these
oscillations is not known. X-ray polarimetry is another potential avenue for
measuring spin (Dovčiak et al. 2008; Li et al. 2009; Schmoll et al. 2009),
which may be realized soon with the 2014 launch of the Gravity and Extreme
Magnetism Small Explorer (Swank et al. 2008). Meanwhile, several other methods
of measuring spin have been proposed or applied (e.g., Takahashi 2004; Barai
et al. 2004; Huang et al. 2007; Suleimanov et al. 2008; Shcherbakov & Huang
2011).
## 2 Stellar-Mass Black Holes in X-ray Binaries
Observations in 1972 of the X-ray binary Cygnus X-1 provided the first strong
evidence that black holes exist. Today, a total of 23 such X-ray binary
systems are known that contain a compact object too massive to be a neutron
star or a degenerate star of any kind (i.e., $M>3$ $M_{\odot}$; Özel et al.
2010). These compact objects, which have typical masses of $10$$M_{\odot}$,
are referred to as black holes. Their host systems are mass-exchange binaries
containing a nondegenerate star that supplies gas to the black hole via a
stellar wind or via Roche-lobe overflow in a stream that emanates from the
inner Lagrangian point. The mass-donor star in the Roche-lobe overflow systems
is typically a low mass ($M\sim 1$ $M_{\odot}$) sun-like star, and the X-ray
source is transient, alternating between yearlong outbursts ($L_{\rm max}\sim
L_{\rm Edd}=1.3{\times}10^{39}{M/10M_{\odot}}$ erg s-1) and years or decades
of quiescence ($L\sim 10^{-7}L_{\rm Edd}$)222$L_{\rm Edd}$ is the critical
Eddington luminosity above which radiation pressure exceeds gravity.. The
wind-fed X-ray sources, on the other hand, are fueled by massive hot stars
($M\gtrsim 10M_{\odot}$) and are persistently luminous. A schematic sketch to
scale of 21 of these systems is shown in Figure 1: The four at top are
persistent systems and the 17 at bottom are the transients. For a review of
the properties of black hole binaries, see Remillard & McClintock (2006).
## 3 The Continuum-Fitting Method
A definite prediction of relativity theory is the existence of an innermost
stable circular orbit (ISCO) for a particle orbiting a black hole. This
inherently relativistic effect has a major impact on the structure of an
accretion disk. At radii $R\geq R_{\rm ISCO}$ (the radius of the ISCO),
accreting gas moves on nearly circular orbits and slowly spirals in toward the
black hole. At the ISCO, however, the dynamics change suddenly and the gas,
finding no more stable circular orbits, plunges into the hole. In the
continuum-fitting (CF) method, one identifies the inner edge of the accretion
disk with the ISCO (see Secs. 4 and 5 for supporting evidence) and estimates
$R_{\rm ISCO}$ by fitting the X-ray continuum spectrum. Since the
dimensionless radius $r_{\rm ISCO}\equiv R_{\rm ISCO}/(GM/c^{2})$ is solely a
monotonic function of the black hole spin parameter $a_{*}$ (Fig. 2)333We
express black hole spin in the customary way as the dimensionless quantity
$a_{*}\equiv cJ/GM^{2}$ with $|a_{*}|\leq 1$, where $M$ and $J$ are
respectively the black hole mass and angular momentum., knowing its value
allows one immediately to infer the value of $a_{*}$. We note that the
truncation of the disk at the ISCO is also a crucial assumption of the Fe
K$\alpha$ method of measuring spin (Reynolds & Fabian 2008).
Before describing the CF methodology, we stress that for this technique to
succeed it is essential to have accurate measurements of the distance $D$ to
the source, the inclination $i$ of the accretion disk, and the mass $M$ of the
black hole (for reasons discussed below). The methodologies for measuring $D$,
$i$ and $M$ are firmly established. Therefore, rather than digressing to
discuss how these measurements are made, we refer the interested reader to
some recent papers on the subject (Orosz et al. 2007, 2009, 2011; Cantrell et
al. 2010).
The gaseous matter flowing from the companion star to the black hole has
appreciable angular momentum as a consequence of the binary orbital motion. As
the gas flows, viscous forces cause it to spread out into an orbiting
structure known as an accretion disk. The gas flowing into the outer disk
spirals slowly inward on Keplerian orbits on a time scale of weeks, reaching a
typical temperature near the ISCO of $kT\sim 1$ keV. Because the accretion
disk is of fundamental importance to the measurement of black hole spin, we
now describe in some detail the thin-disk model we employ.
The model we use is that described in Novikov & Thorne (1973, hereafter NT),
which is a relativisitic generalization of a Newtonian model developed by
Shakura & Sunyaev (1973). The NT model describes an axisymmetric radiatively-
efficient accretion flow which, for a given black hole mass $M$, mass
accretion rate $\dot{M}$ and black hole spin parameter $a_{*}$, gives a
precise prediction for the local radiative flux $F(R)$ emitted at each radius
$R$ of the disk. Moreover, the accreting gas is optically thick and the
emission is thermal and blackbody-like, making it straightforward to compute
the spectrum of the emission. Most importantly, the inner edge of the disk is
located at the ISCO. Therefore, from the measurement of $R_{\rm ISCO}$, and if
we know the mass $M$ of the black hole, we can immediately obtain $a_{*}$
(Fig. 2). This is the principle behind the CF method of estimating black hole
spin, which was first described by Zhang et al. (1997).
Before discussing how to measure $R_{\rm ISCO}$ of a disk, we remind the
reader how one measures the radius $R_{*}$ of a star. Given the distance $D$
to the star, the radiation flux $F_{\rm obs}$ received from the star, and the
temperature $T$ of its continuum radiation, the luminosity of the star is
given by
$L_{*}=4\pi D^{2}F_{\rm obs}=4\pi R_{*}^{2}\sigma T^{4},$ (1)
where $\sigma$ is the Stefan-Boltzmann constant. From $F_{\rm obs}$ and $T$,
we immediately obtain the quantity $\pi(R_{*}/D)^{2}$, which is the solid
angle subtended by the star. From this and the distance $D$, we immediately
obtain $R_{*}$. For accurate results we must allow for limb darkening and
other non-blackbody effects in the stellar emission by computing a stellar
atmosphere model, but this is a detail.
The same principle applies to an accretion disk, but with some differences.
First, since the flux $F(R)$ emitted locally by the disk varies with radius
$R$, the radiation temperature $T(R)$ also varies with $R$. But the precise
variation is known (if we assume the NT disk model), so it is easily
incorporated into the model. Second, since the bulk of the emission is from
the very inner regions of the disk, the effective area of the radiating
surface is directly proportional to the square of the disk inner radius,
$A_{\rm eff}=CR_{\rm ISCO}^{2}$, where the constant $C$ is known. Third, the
observed flux $F_{\rm obs}$ depends not only on the luminosity and the
distance, but also on the inclination $i$ of the disk to the line of sight.
Allowing for these differences, one can write a relation for the disk problem
similar in spirit to eq. (1), i.e., given $F_{\rm obs}$ and a characteristic
$T$ (from X-ray observations), one obtains the solid angle subtended by the
ISCO: $\pi\cos i\,(R_{\rm ISCO}/D)^{2}$. If we further know $i$ and $D$, we
obtain $R_{\rm ISCO}$; and if we also know $M$, we obtain $a_{*}$ (Fig. 2).
This is the basic idea of the CF method.
There are three main issues that must be dealt with before applying the
method: (1) One must carefully trace rays from the disk to the observer in the
Kerr metric of the rotating black hole in order to compute accurately the
observed flux and spectrum. To this end, we have developed an accretion disk
model called kerrbb444This model name and those that follow designate
publicly-available programs that comprise a suite of X-ray data analysis
software known as XSPEC (Arnaud 1996;
http://heasarc.gsfc.nasa.gov/docs/xanadu/xspec/index.html). (Li et al. 2005)
for fitting X-ray data. kerrbb assumes the NT model of the disk and carries
out all the necessary ray-tracing to relate disk properties to observables.
(2) One must have an accurate model for computing the spectral hardening
factor $f=T/T_{\rm eff}$, where $T$ is the temperature of the radiation at a
given radius and $T_{\rm eff}$ is the effective temperature at the same radius
defined by $F(R)=\sigma T_{\rm eff}^{4}(R)$. This correction for non-blackbody
effects is important at the high temperatures typically found in black hole
disks. To carry out this correction we use the advanced disk atmosphere models
of Davis et al. (2005) and Davis & Hubeny (2006). (3) Most importantly, the
inner accretion disk must be well described by the standard geometrically-
thin, optically-thick NT disk model that we employ. To ensure this, we
restrict our attention strictly to observations with a strong thermal
component (Steiner et al. 2009a) and with disk luminosities below 30% of the
Eddington limit (McClintock et al. 2006). In the two sections that follow, we
present observational and theoretical evidence that at these luminosities the
NT model is quite accurate.
For a full description of the mechanics of the CF method we refer the reader
to Section 4 in McClintock et al. (2006). In brief, we fit the broadband X-ray
continuum spectrum in conjunction with other components as needed, principally
a Compton tail component that in more recent work is described using an
empirical model of Comptonization called simpl (Steiner et al. 2009b). The
accretion-disk component, which is key for the CF method, is modeled using
kerrbb (Li et al. 2005), which includes all relativistic effects within the
context of the NT model, and also incorporates the advanced treatment of
spectral hardening mentioned above. It furthermore includes self-irradiation
of the disk (“returning radiation”) and the effects of limb darkening. The key
fit parameters returned are the black hole spin parameter $a_{*}$ and the mass
accretion rate $\dot{M}$.
## 4 Truncation of the Accretion Disk at the ISCO: Observational Evidence
A crucial assumption that underlies both the CF and Fe K$\alpha$ approaches to
measuring spin is that the accretion disk is quite sharply truncated at the
ISCO. This assumption, which is fundamental to the NT model, is clearly valid
if one considers only geodesic forces in the midplane. However, there are
strong magnetohydrodynamic (MHD) forces acting in black hole accretion disks,
and it is therefore unclear a priori that the disk terminates sharply at the
ISCO. In this section, we present the observational evidence that there exists
a fixed inner-disk radius in black hole binaries and, in the following
section, we discuss the theoretical evidence for identifying this radius with
$r_{\rm ISCO}$.
There is a long history of evidence suggesting that fitting the X-ray
continuum is a promising approach to measuring black hole spin. This history
begins in the mid-1980s with the application of a simple nonrelativistic
multicolor disk model (Mitsuda et al. 1984; Makishima et al. 1986), now known
as diskbb, which returns the color temperature $T_{\rm in}$ at the inner-disk
radius $R_{\rm in}$. In their review paper on black hole binaries, Tanaka &
Lewin (1995) summarize examples of the steady decay (by factors of 10–100) of
the thermal flux of transient sources during which $R_{\rm in}$ remains quite
constant (see their Fig. 3.14). They remark that the constancy of $R_{\rm in}$
suggests that this fit parameter is related to the radius of the ISCO. Zhang
et al. (1997) then outlined how, using a relativistic disk model and
corrections for the effects of radiative transfer, the fixed inner disk radius
provides an observational basis to infer black hole spin. More recently, the
evidence for a constant inner radius in the thermal state has been presented
for a number of sources via plots showing that the bolometric luminosity of
the thermal component is approximately proportional to $T_{\rm in}^{4}$ (e.g.,
Kubota et al. 2001; Kubota & Makishima 2004; Gierliński & Done 2004; Abe et
al. 2005; McClintock et al. 2009; Dunn et al. 2010).
A recent study of the persistent source LMC X-3 presents the most compelling
evidence to date for a constant inner-disk radius (Steiner et al. 2010a). We
analyzed many spectra collected during eight X-ray missions that span 26
years. As shown in Figure 3, for a selected sample of hundreds of spectra
obtained using the Rossi X-ray Timing Explorer (RXTE), we find that to within
$\approx 2$ percent the inner radius of the accretion disk is constant over
time and unaffected by the gross variability of the source (top panel).
Meanwhile, even considering an ensemble of eight X-ray missions, we find
consistent values of the radius to within $\approx 4$ percent. These results
provide compelling evidence for the existence of a fixed inner-disk radius and
establish a firm foundation for the measurement of black hole spin. The only
reasonable inference is that this radius is closely associated with the radius
of the ISCO, as we show to be the case in the following section.
## 5 Truncation of the Accretion Disk at the ISCO: Theoretical Evidence
The relativistic NT model on which the CF method is currently built assumes,
as does its predecessor the Newtonian model of Shakura & Sunyaev (1973), that
the accretion disk under consideration is geometrically thin. That is, the
model assumes that the vertical thickness $H$ at any radius $R$ satisfies
$H\ll R$. Assuming in addition that the disk is axisymmetric and in steady
state, the model derives a number of relations which follow directly from
basic conservation laws. One of the powerful results of this analysis is a
formula for the disk flux profile $F(R)$ which depends only on the mass $M$
and spin $a_{*}$ of the black hole and the mass accretion rate $\dot{M}$, but
is independent of messy details such as the viscosity of the accreting gas. It
is the existence of this robust result for $F(R)$ that enables the CF method
to work so well.
There is, however, one unproven assumption in the NT model which is
incorporated via a boundary condition: The model assumes that the shear stress
(which drives the accretion at larger radii) vanishes at the ISCO. This “zero-
torque” assumption is intuitively reasonable (since the gas switches to a
rapidly plunging state once it crosses the ISCO, why should there be a stress
at the transition radius?), but as NT themselves realized, it is ultimately an
assumption. Paczyński (2000) and Afshordi & Paczyński (2003) argued that
deviations from the NT model decrease monotonically with decreasing disk
thickness and that thin disks with $H/R\ll 1$ should be very well described by
the model. Their argument, which was based on a hydrodynamical description of
the disk, was confirmed by detailed calculations carried out by Shafee et al.
(2008b). However, this leaves open the question of whether magnetized disks
might deviate substantially from the NT model. In their paper, NT explicitly
mention that magnetized disks could very well violate the zero-torque boundary
condition.
Arguments have been advanced to suggest that a magnetized accreting gas will
indeed have a non-zero shear stress at the ISCO (Krolik 1999; Gammie 1999),
and that furthermore this stress could be so large that it may completely
invalidate the NT model even in very thin disks. This is clearly an important
question that strikes at the heart of the CF method. A number of recent
studies of magnetized disks using three-dimensional general relativistic
magnetohydrodynamic (GRMHD) simulations have explored this question (Shafee et
al. 2008a; Reynolds & Fabian 2008; Noble & Krolik 2009; Noble et al. 2010;
Penna et al. 2010). The conclusion of these authors is that the shear stress
and the luminosity of the simulated disks do differ from the NT model, but
perhaps not by a large amount.
Figure 4, taken from Kulkarni et al. (2010), shows the disk luminosity
distribution $dL/d\ln R=4\pi R^{2}F(R)$ derived from a set of four GRMHD thin-
disk models simulated by Penna et al. (2010). These models have thicknesses
$H/R\sim 0.045-0.08$ (see Penna et al. and Kulkarni et al. for a precise
definition of $H$, which varies directly with luminosity). As is clear from
the figure, the numerical models follow the NT model reasonably well, although
they do deviate from it. Two kinds of deviation are seen. First, the numerical
models produce some radiation inside the ISCO, whereas the NT model predicts
no radiation there. Second, the peak of the emission in the simulated disks is
shifted inward relative to the peak in the NT model. Both of these effects
cause the disk to appear to have a smaller ISCO radius. This in turn means
that, if one were to infer the black hole spin by fitting this luminosity
distribution (or the corresponding spectrum) using the NT model, one would
infer an erroneously large value for the spin. This systematic error arises
because the NT model is not a perfect description of the simulated disk. To
the extent that the simulated disk is a closer match to a real accretion disk
than is the NT model, this allows us to estimate the corresponding systematic
error in our measurements of spin.
How serious is this systematic error? We answer this in three parts (see
Kulkarni et al. 2010, for details).
1\. Each of the models shown in Figure 4 causes an error in the spin estimate
that is smallest for a low disk inclination (face-on disk; see Fig. 1) and
largest for a high disk inclination ($i=75^{\circ}$). For the latter (most
unfavorable) case, the four models, which correspond to true spin values of
$a_{*}=0$, 0.7, 0.9 and 0.98, give via the CF method spin values of 0.17,
0.83, 0.936 and 0.991, respectively.
2\. A similar exercise can be carried out for disks with other thicknesses. It
is found that the error in the spin estimate is larger for thicker disks and
smaller for thinner disks. Very roughly, the error appears to scale as $H/R$.
Thus, the conclusion of Paczyński (2000), Afshordi & Paczyński (2003) and
Shafee et al. (2008b), that deviations from the NT model vanish in the limit
of vanishingly small disk thickness, appears to be valid also for MHD disks.
3\. The particular models shown in Figure 4 correspond to disk luminosities in
the range $L/L_{\rm Edd}\sim 0.4-0.8$, based on their $H/R$. (It is difficult
to be very quantitative since the mapping between luminosity and $H/R$ is not
known precisely.) Since the CF method is applied only to observations at
$L/L_{\rm Edd}<0.3$, the systematic error due to inaccuracies in the
theoretical model could be up to a factor of 2 smaller than the errors quoted
in point 1 above.
Although the above results are based on numerical simulations that do not
necessarily mimic real disks perfectly, we believe they still provide an
estimate of the likely magnitude of the systematic error. The key point is
that the level of systematic error we find is not serious at the current time.
The observational errors considered in the following section are significantly
larger in all cases. Stating this differently, while magnetized disks do
behave as if their inner edges are shifted inward relative to the position of
the ISCO (Fig. 4), the effect is quantitatively not serious for the disk
luminosities (or disk thicknesses) at which the CF method is applied.
## 6 Results of Continuum Fitting
The spin results obtained to date for eight stellar-mass black holes are
summarized in Table 1. The spins we find cover the full allowable range of
prograde spins (Fig. 2) from $a_{*}\approx 0$ (Schwarzschild) to $a_{*}\approx
1$ (extreme Kerr). Interestingly, the spin values that have been obtained
(Table 1), while spanning the full range of prograde spins, are all in the
canonical physical range, namely $|a_{*}|\leq 1$. This is an important result
and not at all a foregone conclusion: Given the hard external constraints on
the dynamical model parameters ($D$, $i$ and $M$; Section 3), it is entirely
possible that a black hole will be found that implies a spin beyond the reach
of our current model, i.e., $|a_{*}|>1$. Such a result could simply be caused
by large systematic errors in $D$, $i$ and $M$. Or, more interestingly, it
could falsify our model, or even possibly point to new physics. This is why we
are excited about improving our measurements of $D$, $i$ and $M$ for the near-
extreme Kerr hole GRS 1915+105 (see Sec. 6.2).
Table 1: Spin Results to Date for Eight Black HolesaaErrors are quoted at the 68% level of confidence. | Source | Spin $a_{*}$ | Reference
---|---|---|---
1 | GRS 1915+105 | $>0.98$ | McClintock et al. 2006
2 | LMC X–1 | $0.92_{-0.07}^{+0.05}$ | Gou et al. 2009
4 | M33 X–7 | $0.84\pm 0.05$ | Liu et al. 2008, 2010
3 | 4U 1543–47 | $0.80\pm 0.05$ | Shafee et al. 2006
5 | GRO J1655–40 | $0.70\pm 0.05$ | Shafee et al. 2006
6 | XTE J1550–564 | $0.34_{-0.28}^{+0.20}$ | Steiner et al. 2010b
7 | LMC X–3 | $<0.3$bbProvisional result pending improved measurements of $M$ and $i$. | Davis et al. 2006
8 | A0620–00 | $0.12\pm 0.18$ | Gou et al. 2010
Error estimates are primitive for the first four spin results published in
2006 (Table 1). In recent work on the other four black holes, the principal
sources of observational error, as well as the uncertainties in the key model
parameters (e.g., the viscosity parameter), have been treated in detail. In
particular, in our most recent paper on XTE J1550–564, we exhaustively
explored many different sources of error (see Table 3 and Appendix A in
Steiner et al. 2010b). The upshot of the work to date is that in every case
the uncertainty in $a_{*}$ is completely dominated by the errors in the three
key dynamical parameters that we input when fitting the X-ray spectral data.
As discussed in Section 3, these parameters are the distance $D$, the black
hole mass $M$, and the inclination of the inner disk $i$ (which we assume is
aligned with the orbital angular momentum vector of the binary; Li et al.
2009). In order to determine the error in $a_{*}$ due to the combined
uncertainties in $D$, $M$ and $i$, we perform Monte Carlo simulations assuming
that these parameters are normally and independently distributed (e.g., see
Gou et al. 2009). Note that the errors introduced by our use of the NT model
(Sec. 5), which are not considered here, are significantly smaller than the
observational errors (see Table 7 and Sec. 4 in Kulkarni et al. 2010).
We now discuss the results for four black holes in some detail. We first
consider the persistent source M33 X–7. We then turn to GRS 1915+105, the
prototype microquasar (Mirabel & Rodríguez 1994), which hosts a near-extreme
Kerr hole. Finally, we consider the microquasars A0620–00 and XTE J1550–564,
contrasting their behavior with that of GRS 1915+105.
### 6.1 M33 X-7: The First Eclipsing Black Hole
A long observation of the galaxy M33 with the Chandra X-ray Observatory led to
the discovery of a black hole that is eclipsed by its companion star (Pietsch
et al. 2006). We made a detailed follow-up dynamical study of the optical
counterpart of M33 X-7, the first such study of a black hole binary beyond the
environs of the Milky Way. We determined a precise mass for the black hole,
$M=15.65\pm 1.45$ $M_{\odot}$ (Orosz et al. 2007), as well as the mass of its
exceptional companion star ($\approx 70$ $M_{\odot}$). As we discuss in Orosz
et al., it is difficult to understand the origin of this system – a massive
black hole in a 3.5-day orbit, separated by only 42 solar radii from its
supergiant companion (see Fig. 1). Recently, a consistent evolutionary model
has been proposed that accounts for all the key properties of the system
(Valsecchi et al. 2010). It assumes that M33 X-7 started as a primary of $\sim
95$ $M_{\odot}$ and a secondary of $\sim 30$ $M_{\odot}$, with an orbital
period that is close to its present 3.5-day value.
Using as input our precise values for the black hole mass and orbital
inclination angle, and the well-established distance of M33, we fitted 15
Chandra and XMM-Newton X-ray spectra and obtained a precise value for the spin
of the black hole primary, $a_{*}=0.84\pm 0.05$ (Liu et al. 2008, 2010).
Remarkably, given that an (uncharged) astrophysical black hole is described by
just its mass and spin, this result yields a complete description of an
asteroid-size object at a distance of 2.74 million light-years (840 kpc5551
parsec (pc) = 3.26 light-years.).
What is the origin of the spin of M33 X-7? Was the black hole born with its
present spin, or was it torqued up gradually via the accretion flow supplied
by its companion? In order to achieve a spin of $a_{*}=0.84$ via disk
accretion, an initially non-spinning black hole must accrete $5.7M_{\odot}$
from its donor star (King & Kolb 1999) in becoming the $M=15.65M_{\odot}$
black hole that we observe today. However, to transfer this much mass even in
the case of Eddington-limited accretion requires $>17$ million
years666$\dot{M}_{\rm Edd}\equiv L_{\rm Edd}/{\eta}c^{2}$, where $L_{\rm
Edd}=1.3{\times}10^{39}{M/10M_{\odot}}$ erg s-1 and the efficiency $\eta$
increases from 5.7% to 13.3% as $a_{*}$ increases from 0 to 0.84 (Shapiro &
Teukolsky 1983)., whereas the massive companion star, and hence its host
system, can not be older than about 2–3 million years (Orosz et al. 2007).
Thus, it appears that the spin of M33 X-7 must be chiefly natal – i.e., the
event horizon trapped much of the angular momentum of the collapsing stellar
core – a conclusion that has been reached for two other stellar-mass black
holes (McClintock et al. 2006; Shafee et al. 2006). (However, see Moreno
Méndez et al. 2008 on hypercritical accretion).
### 6.2 GRS 1915+105: A Near-Extreme Kerr Black Hole
GRS 1915+105 has unique and striking properties that sharply distinguish it
from the 50 or so known binaries that are believed to contain a stellar-mass
black hole (McClintock et al. 2006; Özel et al. 2010). It is the most reliable
source of relativistic radio jets in the Milky Way and is the prototype of the
microquasars (Mirabel & Rodríguez 1994). It frequently displays extraordinary
X-ray variability that is not mimicked by any other black hole system. The
properties of its high-frequency X-ray oscillations are equally extraordinary
(Remillard & McClintock 2006). Among the 17 transient black hole systems, GRS
1915+105 is unique in having remained active for more than a decade since its
discovery in 1992. The system has an orbital period of 30.8 days, and it is
the largest of the black hole binary systems (Fig. 1).
The pc-scale radio jets of GRS 1915+105, with apparent velocities greater than
the speed of light (superluminal motion), are the analogue of the kpc-scale
jets that have long been observed for quasars. The source episodically ejects
material at relativistic speeds, which can easily be tracked for weeks at
centimeter wavelengths as clouds of plasma moving outward on the plane of the
sky (Mirabel & Rodríguez 1994; Fender et al. 1999). Based on a kinematic
model, the jet velocity for a plausible distance of $D\sim 11$ kpc is $v_{\rm
J}/c>0.9$, and the inclination of the jet from our line of sight is $i_{\rm
J}\approx 65^{\circ}$.
Based on the analysis of X-ray spectral data for reasonable estimates of $D$,
$M$ and $i$, we discovered that GRS 1915+105 contains a near-extreme Kerr hole
with $a_{*}>0.98$ (McClintock et al. 2006). However, the current estimates of
both $D$ and $M$ are poor: The distance is uncertain by a factor of $\approx
2$ (Fig. 5$b$), and the mass is uncertain by $\approx 30$% ($M=14.4\pm 4.4$
$M_{\odot}$; Greiner et al. 2001; Harlaftis & Greiner 2004).
Remarkably, the extraordinarily high spin of GRS 1915+105 and other properties
of the source allow one – for this source only – to place tight constraints on
the allowable values of $M$ and $D$, which must lie within the triangular
region shown in Figure 5$a$. Our spin model constrains the black hole’s mass
and distance to lie to the right of the slanted line (99% confidence level)
because the model implies values of $a_{*}>1$ to the left of the line. (Our
model is only valid for $a_{*}<0.999$). Distances $>12$ kpc are ruled out by
the kinematic model of the radio jets (Fender et al. 1999). The lower bound on
$M$ is an estimate and is based on work in progress: We (Danny Steeghs et al.)
are in the act of obtaining near-infrared spectroscopic data at ESO’s VLT
Observatory that we fully expect will improve the measurement of $M$ by at
least a factor of two. When this result is in hand, we will have $M$ and $D$
constrained to lie within the small shaded triangle, thereby constraining the
distance to lie within the range 9.5–12 kpc.
As noted above, the distance is also highly uncertain. Furthermore, all
measurements to date are model-dependent estimates with large, systematic
uncertainties that are difficult to assess (Fig. 5$b$). With Mark Reid, we are
in the act of obtaining a model-independent trigonometric distance with an
uncertainty of 10% via a parallax measurement (the gold-standard method in
astronomy) using the Very Long Baseline Array (VLBA), a worldwide array of
radio telescopes. We have made successful observations at four epochs in
2008–2010 and anticipate that observations at several additional epochs will
be required to reach our goal. Two hypothetical and possible outcomes of these
VLBA observations are indicated in Figure 5$a$: VLBA2 would confirm our spin
model and VLBA1 would rule it out.
### 6.3 A0620–00 and XTE J1550–564: Two Schwarzschild-Like Black Holes
The host systems of these two black holes are quite small (Fig. 1). The
optical companion in A0620–00, which has an orbital period of only 0.3 days,
is a star somewhat cooler than the Sun with about half its size and mass.
During its yearlong X-ray outburst in 1975–1976, this nearby transient
($D\approx 1$ kpc) became the brightest celestial X-ray source ever observed
(apart from the Sun). For several days, the flux at Earth from this source was
greater than the combined flux of all of the hundreds of other X-ray binaries
in our galaxy. During this period, A0620–00 was also a bright transient radio
source, which was observed with the early radio telescopes of the day. A
reanalysis of these data by Kuulkers et al. (1999) indicates that multiple jet
ejections occurred. The authors find that, like GRS 1915+105, the radio source
was extended on parsec scales, and they infer a relativistic expansion
velocity.
The cool companion star in XTE J1550–564 also has a mass about half that of
the Sun, although its radius is about twice as great (Fig. 1); the orbital
period of the system is 1.5 days (Orosz et al. 2011). During its principal
1998–1999 outburst cycle, this transient source produced one of the most
remarkable X-ray flare events ever observed for a black hole binary. For
$\approx 1$ day, the source intensity rose fourfold and the flux in the
dominant nonthermal component of emission rose by the same factor. Then, just
as quickly, the source intensity declined to its pre-outburst level (Sobczak
et al. 2000). Four days later, small-scale superluminal radio jets were
observed (Hannikainen et al. 2009); their separation angle and relative
velocity link them to the impulsive X-ray flare. The subsequent detection of
pc-scale radio jets in 2000 led to the discovery of relativistic X-ray jets
(Corbel et al. 2002). All of the available evidence strongly indicates that
these pc-scale X-ray and radio jets are associated with the powerful X-ray
flare.
Using our recently-determined estimates of $D$, $M$ and $i$ for A0620–00 and
XTE J1550–564 (Cantrell et al. 2010; Orosz et al. 2011), we fitted the X-ray
spectra of these black holes and determined their spins (Table 1). Figure 6
shows a pair of fitted spectra for the latter source. The spectrum in the left
panel is completely dominated by the thermal component and is therefore ideal
for the determination of spin. Now, however, using our improved methodologies
(Steiner et al. 2009a, 2010b), we are able to obtain useful and consistent
values of spin as well for spectra that have a strong Compton component of
emission, like the one shown in the right panel of Figure 6. The origin of
this component is widely attributed to Compton upscattering of the soft disk
photons by coronal electrons (see Sec. 7).
The CF spins of both A0620–00 and XTE J1550-564 are quite low: $a_{*}\approx
0.1$ and $a_{*}\approx 0.3$, respectively. The corresponding nominal radii of
their ISCOs are $5.7M$ and $5.0M$, which differ only modestly from the
Schwarzschild value of $6M$. The low spins of these two microquasars challenge
the long-standing and widely-held belief that there is a strong connection
between black hole spin and relativistic jets (Blandford & Znajek 1977,
hereafter BZ). If relativistic jets are powered by black hole spin, then
theory predicts that jet power will increase dramatically with increasing
$a_{*}$ (Tchekhovskoy et al. 2010). For low spins, the black hole contributes
very little power; in fact, for $a_{*}<0.4$, the accretion disk apparently
provides more power than the black hole (McKinney 2005).
Given the low spins of XTE J1550–564 and A0620–00, it would appear that their
episodic jets are driven largely by their accretion disks. One well-known
candidate mechanism is the centrifugally driven outflow of matter from a disk
described by Blandford & Payne (1982, hereafter BP). A useful comparison of
the operational regimes of BP and BZ is given by Garofalo et al. (2010). They
show that BP is always viable, but that BZ is a more likely mechanism for the
most rapidly rotating sources, such as the extreme-Kerr black hole GRS
1915+105 (see Sec. 6.2). In closing, we note that a statistical study by
Fender et al. (2010), which is based on data of uneven quality, found no
evidence that black hole spin powers jets.
## 7 The Fe K$\alpha$ Reflection Method
In the Fe K$\alpha$ method, one determines $r_{\rm ISCO}$ by modeling the
profile of the broad and skewed iron line, which is formed in the inner disk
by Doppler effects, light bending, relativistic beaming, and gravitational
redshift (Fabian et al. 2000; Reynolds & Nowak 2003; Miller 2007). Of central
importance is the effect of the gravitational redshift on the red wing of the
line. This wing extends to very low energies for a rapidly rotating black hole
($a_{*}\sim 1$) because in this case gas can orbit near the event horizon,
deep in the potential well of the black hole. Relative to the CF method,
measuring the extent of the red wing in order to infer $a_{*}$ is hindered by
the relative faintness of the signal. However, the Fe K$\alpha$ method has the
virtues that it is independent of $M$ and $D$, while the blue wing of the line
even allows an estimate if $i$. As noted in Section 1, this method, while
applicable to both classes of black holes, is presently the only viable
approach to measuring the spins of supermassive black holes.
For stellar-mass black holes, in addition to the thermal disk component of
emission (which is central to the CF method), a higher-energy power-law
component of emission is always observed (e.g., see Fig. 6). For supermassive
black holes in active galactic nuclei (AGN), this power-law component is
dominant, and it is thought to be produced by inverse Compton scattering of
soft thermal photons in a hot ($kT\sim 100$ keV) corona (Reynolds & Nowak
2003; Done et al. 2007). Meanwhile, the disks of both stellar-mass and
supermassive black holes are too cool ($kT\sim 1$ and 0.01 keV, respectively)
to produce the observed Fe K$\alpha$ emission line. Rather, this dominant
line, and a host of other lines, are generated via X-ray fluorescence as a
result of irradiation of the disk by the hard, coronal power-law component.
The complex spectrum so generated is referred to as a “reflection spectrum.”
In order to determine the spin using the Fe-line method, one must model the
reflection spectrum in detail (Ross & Fabian 2005, 2007).
The spins of several stellar-mass black holes have been measured using the Fe
K$\alpha$ method. An early suggestion of high spins for two black holes was
made by Miller et al. (2002, 2004) and preliminary results for a total of
eight stellar-mass black holes are given in Miller et al. (2009). Other
important papers on the spins of stellar-mass black holes include Reis et al.
(2008, 2009). For a review, see Miller (2007). Very recently, we teamed up
with Fe K$\alpha$ experts to measure the spin of XTE J1550–564 (Steiner et al.
2010b). The spin estimate obtained using the Fe K$\alpha$ method is
$a_{*}=0.55_{-0.22}^{+0.15}$, which is quite consistent with the CF value (see
Table 1).
The spins of several supermassive black holes have been reported, which range
from $a_{*}\approx 0.6$ to $>0.98$ (Brenneman & Reynolds 2006; Fabian et al.
2009; Schmoll et al. 2009; Miniutti et al. 2009). By far, the most well
studied of these is the Seyfert 1 galaxy MCG–6–30–15 (for background, see
Reynolds & Nowak 2003). The 6.4 keV Fe K$\alpha$ line of this AGN is extremely
broad and skewed. Brenneman & Reynolds (2006) and Miniutti et al. (2009) show
that the red wing extends downward to below 4 keV and conclude that
$a_{*}>0.98$.
## 8 Conclusion
We have discussed the only two established classes of black holes, stellar-
mass and supermassive777There is evidence for a class of intermediate-mass
($100-10^{5}$) black holes (Miller & Colbert 2004). However, to date there
existence remains uncertain because no direct and confirming measurement of
mass has been made., and the two principal approaches to measuring their
spins, the continuum-fitting and Fe-K$\alpha$ methods. Spin measurements for
eight stellar-mass black holes are presented, and these data are used to argue
that the high spin of M33 X-7 is natal, and that at least some relativistic
black-hole jets are powered by their accretion disks, not the spin of the
black hole.
Two aspects of this work excite us greatly.
First, by measuring a black hole’s spin, after earlier measuring its mass, we
are able to completely characterize the intrinsic properties of each of the
black holes we study. The No Hair Theorem states that a macroscopic black
hole, regardless of how massive it may be, is described by just two
parameters: $M$ and $a_{*}$888In principle it could also have an electric
charge, but astrophysical black holes are unlikely to have enough charge to be
dynamically important.. But is the No Hair Theorem really true? The only way
we will answer this question is by first measuring $M$ and $a_{*}$ for a good
sample of black holes, and then testing whether the Kerr metric corresponding
to these values of $M$ and $a_{*}$ is completely consistent with all
observables that are sensitive to the space-time near the black hole. In a
sense, we have attempted the first test of the No Hair Theorem by measuring
the spin of XTE J1550-564 by two independent methods, the continuum fitting
method and the Fe K$\alpha$ method (Sec. 7), and finding agreement. However,
the errors in the two measurements are currently rather large, and we do not
yet understand all systematic sources of error, so it would be premature to
claim success. But this example provides a taste of how astrophysics can
contribute to deep questions in physics.
The other aspect that excites us is all the areas of astrophysics that our
work ties to, e.g., the connections that are beginning to be made between
measurements of spin and the phenomenology and theory of relativistic jets
(Sec. 6.3), and the processes that lead to black hole formation (Sec. 6.1). We
hope to see spin data used to help constrain models of gamma-ray bursts, black
hole formation, black-hole binary evolution, high- and low-frequency X-ray
oscillations, black hole X-ray states and state transitions, models of X-ray
coronae, etc.
These two strong motivations stimulate us to continue firming up the
measurements of black hole spin, with the goal of amassing a good sample of a
total of 12–15 measurements during the next several years.
We conclude by noting that it is reasonable to expect LIGO, LISA and other
gravitational-wave observatories to provide us with intimate knowledge
concerning black holes. However, these breakthroughs are still some years in
the future whereas astrophysical techniques are providing information on black
holes today. Also, gravitational-wave facilities are unlikely to help us
understand MHD accretion flows in strong fields, or the origin of relativistic
jets, or the formation of relativistically-broadened Fe lines and high-
frequency quasi-periodic oscillations, etc., phenomena that are now routinely
observed for black holes. In short, observations of accreting black holes show
us uniquely how a black hole interacts with its environment.
There is no straight path to unlocking the mysteries of black holes, probing
the extreme physical conditions they generate, and understanding their
importance to astrophysics and cosmology. It behooves us to explore widely
because it is the synergistic exploration of all paths that will enlighten us.
Therefore, it is important to maintain balance between gravitational-wave and
electromagnetic studies of black holes.
J.E.M. acknowledges support from NASA grant NNX08AJ55G and the Smithsonian
Endowment Funds. R.N. was supported in part by NSF grant AST-0805832 and NASA
grant NNX08AH32G. We thank Laura Brenneman and Jon Miller for helpful comments
on the Fe-line method.
## References
* Abe et al. (2005) Abe, Y., Fukazawa, Y., Kubota, A., Kasama, D., & Makishima, K. 2005, PASJ, 57, 629
* Afshordi & Paczyński (2003) Afshordi, N. & Paczyński, B. 2003, ApJ, 592, 354
* Arnaud (1996) Arnaud, K. A. 1996, in Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, ed. G. H. Jacoby & J. Barnes, 17
* Barai et al. (2004) Barai, P., Das, T. K., & Wiita, P. J. 2004, ApJ, 613, L49
* Blandford & Payne (1982) Blandford, R. D. & Payne, D. G. 1982, MNRAS, 199, 883
* Blandford & Znajek (1977) Blandford, R. D. & Znajek, R. L. 1977, MNRAS, 179, 433
* Brenneman & Reynolds (2006) Brenneman, L. W. & Reynolds, C. S. 2006, ApJ, 652, 1028
* Cantrell et al. (2010) Cantrell, A. G. et al. 2010, ApJ, 710, 1127
* Corbel et al. (2002) Corbel, S., Fender, R. P., Tzioumis, A. K., Tomsick, J. A., Orosz, J. A., Miller, J. M., Wijnands, R., & Kaaret, P. 2002, Science, 298, 196
* Davis et al. (2005) Davis, S. W., Blaes, O. M., Hubeny, I., & Turner, N. J. 2005, ApJ, 621, 372
* Davis et al. (2006) Davis, S. W., Done, C., & Blaes, O. M. 2006, ApJ, 647, 525
* Davis & Hubeny (2006) Davis, S. W. & Hubeny, I. 2006, ApJS, 164, 530
* Done et al. (2007) Done, C., Gierliński, M., & Kubota, A. 2007, A&A Rev., 15, 1
* Dovčiak et al. (2008) Dovčiak, M., Muleri, F., Goosmann, R. W., Karas, V., & Matt, G. 2008, MNRAS, 391, 32
* Dunn et al. (2010) Dunn, R. J. H., Fender, R. P., Körding, E. G., Belloni, T., & Cabanac, C. 2010, MNRAS, 403, 61
* Fabian et al. (2000) Fabian, A. C., Iwasawa, K., Reynolds, C. S., & Young, A. J. 2000, PASP, 112, 1145
* Fabian et al. (2009) Fabian, A. C. et al. 2009, Nature, 459, 540
* Fender et al. (2010) Fender, R. P., Gallo, E., & Russell, D. 2010, MNRAS, 406, 1425
* Fender et al. (1999) Fender, R. P., Garrington, S. T., McKay, D. J., Muxlow, T. W. B., Pooley, G. G., Spencer, R. E., Stirling, A. M., & Waltman, E. B. 1999, MNRAS, 304, 865
* Gammie (1999) Gammie, C. F. 1999, ApJ, 522, L57
* Garofalo et al. (2010) Garofalo, D., Evans, D. A., & Sambruna, R. M. 2010, MNRAS, 406, 975
* Gierliński & Done (2004) Gierliński, M. & Done, C. 2004, MNRAS, 347, 885
* Gou et al. (2009) Gou, L. et al. 2009, ApJ, 701, 1076
* Gou et al. (2010) Gou, L., McClintock, J. E., Steiner, J. F., Narayan, R., Cantrell, A. G., Bailyn, C. D., & Orosz, J. A. 2010, ApJ, 718, L122
* Greiner et al. (2001) Greiner, J., Cuby, J. G., & McCaughrean, M. J. 2001, Nature, 414, 522
* Hannikainen et al. (2009) Hannikainen, D. C. et al. 2009, MNRAS, 397, 569
* Harlaftis & Greiner (2004) Harlaftis, E. T. & Greiner, J. 2004, A&A, 414, L13
* Huang et al. (2007) Huang, L., Cai, M., Shen, Z., & Yuan, F. 2007, MNRAS, 379, 833
* King & Kolb (1999) King, A. R. & Kolb, U. 1999, MNRAS, 305, 654
* Krolik (1999) Krolik, J. H. 1999, ApJ, 515, L73
* Kubota & Makishima (2004) Kubota, A. & Makishima, K. 2004, ApJ, 601, 428
* Kubota et al. (2001) Kubota, A., Makishima, K., & Ebisawa, K. 2001, ApJ, 560, L147
* Kulkarni et al. (2010) Kulkarni, A. K. et al. 2010, MNRAS submitted
* Kuulkers et al. (1999) Kuulkers, E., Fender, R. P., Spencer, R. E., Davis, R. J., & Morison, I. 1999, MNRAS, 306, 919
* Li et al. (2009) Li, L., Narayan, R., & McClintock, J. E. 2009, ApJ, 691, 847
* Li et al. (2005) Li, L., Zimmerman, E. R., Narayan, R., & McClintock, J. E. 2005, ApJS, 157, 335
* Liu et al. (2008) Liu, J., McClintock, J. E., Narayan, R., Davis, S. W., & Orosz, J. A. 2008, ApJ, 679, L37
* Liu et al. (2010) — 2010, ApJ, 719, L109
* Makishima et al. (1986) Makishima, K., Maejima, Y., Mitsuda, K., Bradt, H. V., Remillard, R. A., Tuohy, I. R., Hoshi, R., & Nakagawa, M. 1986, ApJ, 308, 635
* McClintock et al. (2009) McClintock, J. E., Remillard, R. A., Rupen, M. P., Torres, M. A. P., Steeghs, D., Levine, A. M., & Orosz, J. A. 2009, ApJ, 698, 1398
* McClintock et al. (2006) McClintock, J. E., Shafee, R., Narayan, R., Remillard, R. A., Davis, S. W., & Li, L. 2006, ApJ, 652, 518
* McKinney (2005) McKinney, J. C. 2005, ApJ, 630, L5
* Miller (2007) Miller, J. M. 2007, ARA&A, 45, 441
* Miller et al. (2004) Miller, J. M. et al. 2004, ApJ, 606, L131
* Miller et al. (2002) — 2002, ApJ, 570, L69
* Miller et al. (2009) Miller, J. M., Reynolds, C. S., Fabian, A. C., Miniutti, G., & Gallo, L. C. 2009, ApJ, 697, 900
* Miller & Colbert (2004) Miller, M. C. & Colbert, E. J. M. 2004, International Journal of Modern Physics D, 13, 1
* Miniutti et al. (2009) Miniutti, G., Panessa, F., de Rosa, A., Fabian, A. C., Malizia, A., Molina, M., Miller, J. M., & Vaughan, S. 2009, MNRAS, 398, 255
* Mirabel & Rodríguez (1994) Mirabel, I. F. & Rodríguez, L. F. 1994, Nature, 371, 46
* Mitsuda et al. (1984) Mitsuda, K. et al. 1984, PASJ, 36, 741
* Moreno Méndez et al. (2008) Moreno Méndez, E., Brown, G. E., Lee, C., & Park, I. H. 2008, ApJ, 689, L9
* Noble & Krolik (2009) Noble, S. C. & Krolik, J. H. 2009, ApJ, 703, 964
* Noble et al. (2010) Noble, S. C., Krolik, J. H., & Hawley, J. F. 2010, ApJ, 711, 959
* Novikov & Thorne (1973) Novikov, I. D. & Thorne, K. S. 1973, in Black Holes (Les Astres Occlus), ed. C. Dewitt & B. S. Dewitt, (New York: Gordon & Breach), 343–450
* Orosz et al. (2007) Orosz, J. A. et al. 2007, Nature, 449, 872
* Orosz et al. (2009) — 2009, ApJ, 697, 573
* Orosz et al. (2011) Orosz, J. A., Steiner, J., McClintock, J. E., Torres, M. A. P., Remillard, R. A., & Bailyn, C. D. 2011, ApJ in press, arXiv:1101.2499v1 [astro-ph.SR]
* Özel et al. (2010) Özel, F., Psaltis, D., Narayan, R., & McClintock, J. E. 2010, ApJ, 725, 1918
* Paczyński (2000) Paczyński, B. 2000, arXiv:astro-ph/0004129v1
* Penna et al. (2010) Penna, R. F., McKinney, J. C., Narayan, R., Tchekhovskoy, A., Shafee, R., & McClintock, J. E. 2010, MNRAS, 408, 752
* Pietsch et al. (2006) Pietsch, W., Haberl, F., Sasaki, M., Gaetz, T. J., Plucinsky, P. P., Ghavamian, P., Long, K. S., & Pannuti, T. G. 2006, ApJ, 646, 420
* Reis et al. (2009) Reis, R. C., Fabian, A. C., Ross, R. R., & Miller, J. M. 2009, MNRAS, 395, 1257
* Reis et al. (2008) Reis, R. C., Fabian, A. C., Ross, R. R., Miniutti, G., Miller, J. M., & Reynolds, C. 2008, MNRAS, 387, 1489
* Remillard & McClintock (2006) Remillard, R. A. & McClintock, J. E. 2006, ARA&A, 44, 49
* Reynolds & Fabian (2008) Reynolds, C. S. & Fabian, A. C. 2008, ApJ, 675, 1048
* Reynolds & Nowak (2003) Reynolds, C. S. & Nowak, M. A. 2003, Phys. Rep., 377, 389
* Ross & Fabian (2005) Ross, R. R. & Fabian, A. C. 2005, MNRAS, 358, 211
* Ross & Fabian (2007) — 2007, MNRAS, 381, 1697
* Schmoll et al. (2009) Schmoll, S. et al. 2009, ApJ, 703, 2171
* Shafee et al. (2006) Shafee, R., McClintock, J. E., Narayan, R., Davis, S. W., Li, L., & Remillard, R. A. 2006, ApJ, 636, L113
* Shafee et al. (2008a) Shafee, R., McKinney, J. C., Narayan, R., Tchekhovskoy, A., Gammie, C. F., & McClintock, J. E. 2008a, ApJ, 687, L25
* Shafee et al. (2008b) Shafee, R., Narayan, R., & McClintock, J. E. 2008b, ApJ, 676, 549
* Shakura & Sunyaev (1973) Shakura, N. I. & Sunyaev, R. A. 1973, A&A, 24, 337
* Shapiro & Teukolsky (1983) Shapiro, S. L. & Teukolsky, S. A. 1983, Black holes, white dwarfs, and neutron stars: The physics of compact objects, ed. Shapiro, S. L. & Teukolsky, S. A. (New York: Wiley)
* Shcherbakov & Huang (2011) Shcherbakov, R. V. & Huang, L. 2011, MNRAS, 410, 1052
* Sobczak et al. (2000) Sobczak, G. J., McClintock, J. E., Remillard, R. A., Cui, W., Levine, A. M., Morgan, E. H., Orosz, J. A., & Bailyn, C. D. 2000, ApJ, 544, 993
* Steiner et al. (2010a) Steiner, J. F., McClintock, J. E., Remillard, R. A., Gou, L., Yamada, S., & Narayan, R. 2010a, ApJ, 718, L117
* Steiner et al. (2009a) Steiner, J. F., McClintock, J. E., Remillard, R. A., Narayan, R., & Gou, L. 2009a, ApJ, 701, L83
* Steiner et al. (2009b) Steiner, J. F., Narayan, R., McClintock, J. E., & Ebisawa, K. 2009b, PASP, 121, 1279
* Steiner et al. (2010b) Steiner, J. F. et al. 2010b, MNRAS submitted, arXiv:1010.1013v2 [astro-ph.HE]
* Suleimanov et al. (2008) Suleimanov, V. F., Lipunova, G. V., & Shakura, N. I. 2008, A&A, 491, 267
* Swank et al. (2008) Swank, J., Kallman, T., & Jahoda, K. 2008, in COSPAR, Plenary Meeting, Vol. 37, 37th COSPAR Scientific Assembly, 3102
* Takahashi (2004) Takahashi, R. 2004, ApJ, 611, 996
* Tanaka & Lewin (1995) Tanaka, Y. & Lewin, W. H. G. 1995, in X-ray Binaries, ed. W. H. G. Lewin, J. van Paradijs, & E. P. J. van den Heuvel, (Cambridge: Cambridge Univ. Press), 126–174
* Tchekhovskoy et al. (2010) Tchekhovskoy, A., Narayan, R., & McKinney, J. C. 2010, ApJ, 711, 50
* Török et al. (2005) Török, G., Abramowicz, M. A., Kluźniak, W., & Stuchlík, Z. 2005, A&A, 436, 1
* Valsecchi et al. (2010) Valsecchi, F., Glebbeek, E., Farr, W. M., Fragos, T., Willems, B., Orosz, J. A., Liu, J., & Kalogera, V. 2010, Nature, 468, 77
* Wagoner et al. (2001) Wagoner, R. V., Silbergleit, A. S., & Ortega-Rodríguez, M. 2001, ApJ, 559, L25
* Zhang et al. (1997) Zhang, S. N., Cui, W., & Chen, W. 1997, ApJ, 482, L155
Figure 1: Scale drawings of 21 black hole binaries. The size of the Sun and
the Sun-Mercury distance (0.4 AU) are indicated at the top. The systems range
in size from the giant GRS 1915+105 with an orbital period of 30.8 days to
tiny XTE J1118+480 with an orbital period of 0.2 days. The shapes of the
tidally distorted stars are accurately rendered, and the black hole is located
at the center of the accretion disk (see key in inset). The inclination of the
binary to our line of sight is indicated by the tilt of the accretion disk; an
inclination angle of $i=0^{\circ}$ corresponds to a system whose accretion
disk lies in the plane of the sky and is viewed face on (e.g., $i=21^{\circ}$
for 4U 1543–47 and $i=75^{\circ}$ for SAX J1819.3-2525).
Figure 2: Radius of the ISCO in units of $GM/c^{2}$ versus the black hole spin
parameter. Negative values of $a_{*}$ correspond to retrograde motion, with
the black hole spinning in the opposite sense of the disk. Stellar black holes
are expected to have prograde spins ($a_{*}>0$) as a consequence of their
formation in a binary system, whereas the spins of supermassive black holes,
which are conditioned by galaxy merger events, may be either prograde or
retrograde (e.g., Garofalo et al. 2010).
Figure 3: $(top)$ Accretion-disk luminosity in Eddington-scaled units (for
$M=10$ $M_{\odot}$) versus time for all the 766 spectra considered in the
study of LMC X-3 by Steiner et al. (2010a). The downward arrows show RXTE data
which are off scale. Data in the unshaded region satisfy our thin-disk
selection criterion $L/L_{\rm Edd}<0.3$ (Sec. 3). The dotted line indicates
the lower luminosity threshold (5% $L/L_{\rm Edd}$) set to avoid confusion
with strongly Comptonized data. $(bottom)$ Fitted values of the inner disk
radius $r_{\rm in}\equiv R_{\rm in}/(GM/c^{2})$ are shown for thin-disk data
in the top panel that meet the selection criteria of the study (a total of 411
spectra). Despite large variations in luminosity, $r_{\rm in}$ remains
constant to within a few percent over time. The median value for just the 391
selected RXTE spectra is shown as a red dashed line.
Figure 4: Luminosity profiles from GRMHD simulations (solid lines) compared
with those from the Novikov & Thorne model (dashed lines) for $a_{*}=0$,
$0.7$, $0.9$ and $0.98$ (bottom to top). The ISCO is located at the radius
where the NT disk luminosity goes to zero.
Figure 5: ($a$) Allowed values of black hole mass and distance for GRS
1915+105 fall within the shaded triangular region (see text). ($b$) Six
estimates of the distance to GRS 1915+105 are shown. They range from below 7
to above 12 kpc. We are working toward a 10% trigonometric distance. Two
hypothetical and possible outcomes of our VLBA observations, labeled VLBA1 and
VLBA2, are indicated at the top of panel $a$. For references on distance
estimates, see Figure 18 in McClintock et al. (2006).
Figure 6: Model fits for a pair of spectra of XTE J1550–564. $(left)$ A
spectrum with a strongly dominant thermal component; shown are the data, the
fit to the data and the fitted thermal component. $(right)$ A strongly
Comptonized spectrum. Note the intensity of the power-law component relative
to its intensity in the left panel. For details, see Figure 4 and text in
Steiner et al. (2010b).
|
arxiv-papers
| 2011-01-04T21:00:02 |
2024-09-04T02:49:16.137963
|
{
"license": "Public Domain",
"authors": "Jeffrey E. McClintock, Ramesh Narayan, Shane W. Davis, Lijun Gou,\n Akshay Kulkarni, Jerome A. Orosz, Robert F. Penna, Ronald A. Remillard, James\n F. Steiner",
"submitter": "Jeffrey McClintock",
"url": "https://arxiv.org/abs/1101.0811"
}
|
1101.0827
|
# O Algoritmo Usado No Programa de Criptografia PASME
Péricles Lopes Machado Laboratório de Análises Numéricas em Eletromagnetismo
(LANE), Universidade Federal do Pará, caixa postal 8619, CEP 66073-900,
Brasil; e-mail: pericles.machado@itec.ufpa.br
###### Abstract
Neste trabalho será apresentado o principal algoritmo de criptografia da
ferramenta PASME, a qual permite encriptação e ocultamento de informações em
diversos tipos de arquivos. O algoritmo utiliza o fato da fatoração de números
grandes ser um problema difícil do ponto de vista computacional, efetuando
assim, os principais passos da encriptação.
###### Index Terms:
Criptografia, Teoria dos números, Teoria da informação
This work will present the main encryption algorithm of the PASME tool, PASME
allows encrypt and hide information in various types of files. The algorithm
uses the fact that factoring large numbers is a difficult issue in terms of
computational performing to make the main steps of the encryption.
## I Introdução
A ideia fundamental de qualquer algoritmo de criptografia é modificar a
representação de uma informação para garantir proteção contra acesso
indevidos.
No decorrer dos anos, muitos algoritmos de criptografia foram desenvolvidos.
Um dos mais antigos realiza uma permutação no alfabeto que contém todos os
símbolos da mensagem que será encriptada. Contudo, este algoritmo apresenta
grande vulnerabilidade a uma análise da frequência de ocorrência de
determinados símbolos, principalmente quando aplicados à textos escritos.
Outro método clássico, usado em mensagens binárias, consiste em inverter
certos bits e armazenar a posição dos bits que foram invertidos em outra
palavra, a folha-chave, com o mesmo tamanho da mensagem que foi encriptada. Um
problema desse método é que o tamanho da folha-chave pode ser muito grande,
inviabilizando o processo de encriptação.
Muitos algoritmos modernos utilizam estratégias envolvendo teoria dos números
através da utilização de problemas que atualmente são intratáveis do ponto de
vista computacional. Um exemplo clássico desta classe de algoritmo é o RSA [2]
[3].
A ideia do presente trabalho é utilizar a intratabilidade da fatoração de
inteiros grandes para realizar os passos-chave de sua encriptação. Nas
próximas seções, serão descritos os passos realizados pelo algoritmo de
encriptação PASME, além de serem comentados alguns detalhes de sua
implementação [1].
## II Algumas funções fundamentais
### II-A A função $\mp$ (inflar)
A ideia fundamental do algoritmo PASME é a mudança na base de representação de
um número. Mudar a base de representação de um número inteiro
$n=a_{0}a_{1}a_{2}...a_{k}$ para a base $b$ consiste em realizar a operação
descrita na equação 1
$T(n,b)=a_{0}b^{k}+a_{1}b^{k-1}+...+a_{k}b^{0}$ (1)
A função $\mp$ descrita em 2 é uma mudança de base onde a cada digito é
adicionado um ”lixo”.
$\mp(n,b,v)=(a_{0}+c_{0})b^{1}+(a_{1}+c_{1})b^{2}+...+(a_{k}+c_{k})b^{k+1}$
(2)
Onde $c_{i}$ é descrito na equação 5.
$\displaystyle c_{i}=\left\\{\begin{array}[]{clcr}\triangleright(v)&,se&i=0\\\
\triangleright(c_{i-1})&,se&i>0\end{array}\right.$ (5)
Nas equações 2 e 5, $\triangleright(x)$ é o próximo primo depois de $x$, $v$ é
um inteiro qualquer, $n$ é a informação representada na forma de um inteiro,
$a_{k}$ é um digito de $n$ na base original e $b$ é a base alvo.
### II-B A função $\pm$ (sujar)
$\pm$ é semelhante a função $\mp$, só que o ”lixo” $v$ usado é o mesmo em
todos dígitos, conforme pode ser visto na equação 6.
$\pm(n,b,v)=(a_{0}+v)b^{0}+(a_{1}+v)b^{1}+...+(a_{k}+v)b^{k}$ (6)
## III O algoritmo de encriptação PASME
A seguir, serão descritos os procedimentos para encriptar ou desencriptar uma
mensagem usando o algoritmo PASME. O algoritmo $PASME(n,key)$ encripta uma
mensagem $n$ usando a frase-chave $key$.
### III-A Encriptando uma mensagem
O processo de encriptação inicia com a geração de 7 números aleatórios (de
preferência, grandes) $r_{i},i=1...7$. Em seguida, são definidos 4 números
$K_{i}=\triangleright(r_{i})$, para $i=1...5$ e $i\neq 3$,
$K_{3}=\triangleright(K_{5}+d_{max}+r_{3}+1)$, $d_{max}$ é o maior digito da
base em que a informação originalmente está representada.
Para continuar o processo de encriptação, uma frase-chave $key$ tem de ser
fornecida. Usando-se a frase-chave, são gerados os números
$W=\mp(key,K_{3},K_{2})+K_{1}$, $Q=\triangleright(\pm(n,K_{3},K_{5})+r_{7})$,
$P=WQ+K_{4}$, e $X=\pm(n,K_{3},K_{5})$ xor $Q$. $X$ é a mensagem $n$
encriptada.
As informações divulgadas são os números $K_{i}(i=1...5)$, $P$ e $X$.
### III-B Desencriptando uma mensagem
Para desencriptar uma mensagem, é preciso que sejam fornecidos os números
$K_{i}(i=1...5)$, $P$ e $X$, além da frase-chave $key$.
O primeiro passo da desencriptação é a validação da chave, para realizar essa
operação, gera-se o número $W^{\prime}=\mp(key,K_{3},K_{2})+K_{1}$ e é
verificado se $P$ mod $W^{\prime}=K_{4}$. Efetuada a validação, pode-se
recuperar $Q=(P-K_{4})/W^{\prime}$ .
Com $Q$ recuperado, a mensagem $n$ ocultada em $X$ poderá ser revelada. Para
revelar a mensagem $n$, gera-se o número $Y=X$ xor $Q$ e o procedimento
descrito a seguir tem de ser efetuado.
1. 1.
$X^{\prime}=\emptyset$, $X^{\prime}$ é uma palavra vazia
2. 2.
Enquanto Y $\neq$ 0:
1. (a)
$a\leftarrow Y$ mod $K_{3}$
2. (b)
$Y\leftarrow Y-a$
3. (c)
$Y\leftarrow Y/K_{3}$, efetua-se a divisão inteira de $Y$ por $K_{3}$.
4. (d)
$a\leftarrow a-K_{5}$
5. (e)
$X^{\prime}\leftarrow X^{\prime}\oplus a$, $\oplus$ é a operação de
concatenação, ou seja, a união de duas palavras (por exemplo,$33\oplus
5=335$).
3. 3.
$X^{\prime}$ é a mensagem desencriptada
## IV Comentários sobre a implementação de PASME disponível em [1]
Em [1] está disponível uma implementação do algoritmo de criptografia descrito
na seção III. Essa implementação utiliza a biblioteca GMP [4] para realizar as
operações envolvendo inteiros presentes no algoritmo PASME. Como a ferramenta
[1] permite encriptar arquivos com tamanho variáveis, usar o algoritmo PASME
nem sempre é uma boa escolha, já que dependendo do tamanho da mensagem o tempo
de execução pode ser alto. Então, por questões de eficiência, a implementação
[1] utiliza o processo de encriptação de dois passos descrito a seguir para
encriptar uma mensagem $n$.
1. 1.
Gera-se uma folha-chave $fc$ com um tamanho de $L(fc)$ bytes.
2. 2.
Cria-se aleatoriamente uma frase-chave $key$ com $L(key)$ bytes de tamanho.
3. 3.
Utiliza-se o algoritmo descrito em III para encriptar a folha-chave $fc$.
4. 4.
Quebra-se a mensagem $n$ em $L(n)$ bytes,
5. 5.
$i\leftarrow 0$
6. 6.
$k\leftarrow 0$
7. 7.
$X\leftarrow\emptyset$
8. 8.
Enquanto $i\leq L(n)$:
1. (a)
$X\leftarrow X\oplus(n_{i}$ xor $fc_{k})$, $n_{i}$ é i-ésimo byte da mensagem
$n$ e $fc_{k}$ é o k-ésimo byte da folha-chave $fc$.
2. (b)
$i\leftarrow i+1$
3. (c)
$k\leftarrow(k+1)$ mod $L(fc)$
Para desencriptar, o passo (1) do algoritmo anterior não é executado, no passo
(2) é fornecido a frase-chave que ”abre” a mensagem e no passo (3) é chamado o
algoritmo de desencriptação descrito em III.
Na implementação [1], cada simbolo (digito num número) tem 1 byte (8 bits) de
comprimento.
A implementação [1] armazena em um arquivo-alvo as informações públicas
geradas pelo algoritmo III e a mensagem $X$ gerada pelo procedimento anterior.
Para ocultar informações em arquivos, [1] primeiramente verifica o tamanho, em
bytes, da informação que será ocultada. Após isso, a informação é concatenada
ao arquivo e, por fim, concatena-se o tamanho da informação (em [1], um
inteiro com 4 bytes de comprimento). O procedimento para recuperar a
informação é semelhante, só que, primeiramente, recupera-se o tamanho $L$ (em
[1], os 4 últimos bytes do arquivo) da informação que está oculta, depois
recua-se $L-4$ bytes a partir do fim do arquivo, no caso de [1], e armazena-se
os $L$ bytes seguintes em um arquivo-alvo.
A interface gráfica da implementação [1] foi criada utilizando-se o framework
QT4 [5].
## V Conclusões
Este trabalho apresentou um algoritmo de encriptação que usa o fato da mesma
informação ter significados distintos dependendo da base em que está
representada e de, atualmente, certos problemas em teoria dos números serem
intratáveis. Tal algoritmo faz parte da ferramenta PASME que permite a
encriptação e ocultamento da informação em arquivos nos mais diversos
formatos.
## VI Agradecimentos
O autor agradece a Diego Aranha por apontar uma falha no algoritmo inicial, a
João Augusto Palmitesta Neto por sugestões e testes na implemtentação [1] do
algoritmo e Fabio Lobato por revisar o artigo.
## References
* [1] “Projeto pasme,” _http://sourceforge.net/projects/pasme/_.
* [2] T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, _Algoritmos_. Editora Campus, 2002\.
* [3] L. Lovász, J. Pelikán, and K. Vesztergombi, _Matemática Discreta_. Sociedade Brasileira de Matemática, 2003.
* [4] “The gnu multiple precision arithmetic library,” _http://gmplib.org/_.
* [5] “qt - cross platform and ui framework,” _http://qt.nokia.com/_.
|
arxiv-papers
| 2011-01-04T21:35:59 |
2024-09-04T02:49:16.147871
|
{
"license": "Public Domain",
"authors": "P\\'ericles Lopes Machado",
"submitter": "P\\'ericles Lopes Machado Machado",
"url": "https://arxiv.org/abs/1101.0827"
}
|
1101.0869
|
# A New Variation of Hat Guessing Games
Tengyu Ma Xiaoming Sun Huacheng Yu
Institute for Theoretical Computer Science
Tsinghua University, Beijing, China
Email: matengyu1989@gmail.comEmail: xiaomings@tsinghua.edu.cnEmail:
yuhch123@gmail.com
###### Abstract
Several variations of hat guessing games have been popularly discussed in
recreational mathematics. In a typical hat guessing game, after initially
coordinating a strategy, each of $n$ players is assigned a hat from a given
color set. Simultaneously, each player tries to guess the color of his/her own
hat by looking at colors of hats worn by other players. In this paper, we
consider a new variation of this game, in which we require at least $k$
correct guesses and no wrong guess for the players to win the game, but they
can choose to “pass”.
A strategy is called perfect if it can achieve the simple upper bound
$\frac{n}{n+k}$ of the winning probability. We present sufficient and
necessary condition on the parameters $n$ and $k$ for the existence of perfect
strategy in the hat guessing games. In fact for any fixed parameter $k$, the
existence of perfect strategy can be determined for every sufficiently large
$n$.
In our construction we introduce a new notion: $(d_{1},d_{2})$-regular
partition of the boolean hypercube, which is worth to study in its own right.
For example, it is related to the $k$-dominating set of the hypercube. It also
might be interesting in coding theory. The existence of
$(d_{1},d_{2})$-regular partition is explored in the paper and the existence
of perfect $k$-dominating set follows as a corollary.
Keywords: Hat guessing game; perfect strategy; hypercube; k-dominating set;
perfect code
## 1 Introduction
Several different hat guessing games have been studied in recent years [1, 2,
3, 4, 5, 6, 7]. In this paper we investigate a variation where players can
either give a guess or pass. It was first proposed by Todd Ebert in [3]. In a
standard setting there are $n$ players sitting around a table, who are allowed
to coordinate a strategy before the game begins. Each player is assigned a hat
whose color is chosen randomly and independently with probability $1/2$ from
two possible colors, red and blue. Each player is allowed to see all the hats
but his own. Simultaneously, each player guesses its own hat color or passes,
according to their pre-coordinated strategy. If at least one player guesses
correctly and no player guesses wrong, the players win the game. Their goal is
to design a strategy to maximum their winning probability.
By a simple counting argument there is an upper bound of the maximum winning
probability, $n/(n+1)$. It is known that this upper bound can be achieved if
and only if $n$ has the form $2^{t}-1$ [3]. It turns out that the existence of
such perfect strategy that achieves the upper bound corresponds to the
existence of perfect 1-bit error-correcting code in $\\{0,1\\}^{n}$.
In this paper, we present a natural generalization of Ebert’s hat guessing
problem: The setting is the same as in the original problem, every player can
see all other hats except his own, and is allowed to guess or pass. However,
the requirement for them to win the game is generalized to be that at least
$k$ players from them should guess correctly, and no player guesses wrong
($1\leq k\leq n$). Note that when $k=1$, it is exactly the original problem.
We denote by $P_{n,k}$ the maximum winning probability of players. Similarly
to the $k=1$ case, $P_{n,k}$ has a simple upper bound
$P_{n,k}\leq\frac{n}{n+k}$. We call a pair $(n,k)$ perfect if this upper bound
can be achieved, i.e. $P_{n,k}=\frac{n}{n+k}$. There is a simple necessary
condition for a pair $(n,k)$ to be perfect, and our main result states that
this condition is almost sufficient:
###### Theorem 1.
For any $d,k,s\in\mathbb{N}$ with $s\geq 2\lceil\lg k\rceil$,
$(d(2^{s}-k),dk)$ is perfect, in particular, $(2^{s}-k,k)$ is perfect.
There exists pair $(n,k)$ with the necessary condition but not perfect, see
the remark in Section 4.
Here is the outline of the proof: first we give a general characterization of
the winner probability $P_{n,k}$ by using the size of the minimum
$k$-dominating set of the hypercube. Then we convert the condition of $(n,k)$
perfect to some kind of regular partition of the hypercube (see the definition
in Section 2). Our main contribution is that we present a strong sufficient
condition for the existence of such partition, which nearly matches the
necessary condition. Then we can transform it into a perfect hat guessing
strategy.
As a corollary of Theorem 1, we also give asymptotic characterization of the
value $P_{n,k}$. For example, we show that for any fixed $k$, the maximum
winning probability approaches 1 as $n$ tends to the infinity.
Related work:
Feige [4] considered some variations including the discarded hat version and
the everywhere balanced version. Lenstra and Seroussi [6] studied the case
that $n$ is not of form $2^{m}-1$, they also considered the case with multiple
colors. In [2], Butler, Hajiaghayi, Kleinberg and Leighton considered the
worst case of hat placement with sight graph $G$, in which they need to
minimize the maximum wrong guesses over all hat placements. In [5] Feige
studied the case that each player can see only some of other players’ hats
with respect to the sight graph $G$. In [7], Peterson and Stinson investigated
the case that each player can see hats in front of him and they guess one by
one. Very recently, Buhler, Butler, Graham and Tressler [1] studied the case
that every player needs to guess and the players win the game if either
exactly $k_{1}$ or $k_{2}$ players guess correctly, they showed that the
simple necessary condition is also sufficient in this game.
The rest of the paper is organized as follows: Section 2 describes the
definitions, notations and models used in the paper. Then, Section 3 presents
the result of the existence of $(d_{1},d_{2})$-regular partition of hypercube
while Section 4 shows the main result of the hat guessing game. Finally, we
conclude the paper in Section 5 with some open problems.
## 2 Preliminaries
We use $Q_{n}$ to denote the the $n$ dimension boolean hypercube
$\\{0,1\\}^{n}$. Two nodes are adjacent on $Q_{n}$ if they differ by only one
bit. We encode the blue and red color by 0 and 1. Thus the placement of hats
on the $n$ players’ heads can be represented as a node of $Q_{n}$. For any
$x\in Q_{n}$, $x^{(i)}$ is used to indicate the string obtained by flipping
the $i^{th}$ bit of $x$. Throughout the paper, all the operations are over
$\mathbb{F}_{2}$. We will clarify explicitly if ambiguity appears.
Here is the model of the hat guessing game we consider in this paper: The
number of players is denoted by $n$ and players are denoted by
$p_{1},\ldots,p_{n}$. The colors of players’ hats will be denoted to be
$h_{1},\dots,h_{n}$, which are randomly assigned from $\\{0,1\\}$ with equal
probability. $h=(h_{1},\ldots,h_{n})$. Let $h_{-i}\in Q_{n-1}$ denote the
tuple of colors $(h_{1},\dots,h_{i-1},h_{i+1},\dots,h_{n})$ that player
$p_{i}$ sees on the others’ heads. The strategy of player $p_{i}$ is a
function $s_{i}:Q_{n-1}\rightarrow\\{0,1,\bot\\}$, which maps the tuple of
colors $h_{-i}$ to $p_{i}$’s answer, where $\bot$ represents $p_{i}$ answers
“pass” (if some player answers pass, his answer is neither correct nor wrong).
A strategy $\mathcal{S}$ is a collection of $n$ functions
$(s_{1},\ldots,s_{n})$. The players win the game if at least $k$ of them guess
correctly and no one guesses wrong. We use $P_{n,k}$ to denote the maximum
winning probability of the players. The following two definitions are very
useful in characterization $P_{n,k}$:
###### Definition 2.
A subset $D\subseteq V$ is called a $k$-dominating set of graph $G=(V,E)$ if
for every vertex $v\in V\setminus D$, $v$ has at least $k$ neighbors in $D$.
###### Definition 3.
A partition $(V_{1},V_{2})$ of hypercube $Q_{n}$ is called a
$(d_{1},d_{2})$-regular partition if each node in $V_{1}$ has exactly $d_{1}$
neighbors in $V_{2}$, and each node in $V_{2}$ has exactly $d_{2}$ neighbors
in $V_{1}$.
For example, consider the following partition $(V_{1},V_{2})$ of $Q_{3}$:
$V_{1}=\\{000,111\\}$, and $V_{2}=Q_{3}\setminus V_{1}$. For each vertex in
$V_{1}$, there are $3$ neighbors in $V_{2}$, and for each vertex in $V_{2}$,
there is exactly one neighbor in $V_{1}$. Thus $(V_{1},V_{2})$ forms a
$(3,1)$-regular partition of $Q_{3}$.
## 3 ($d_{1},d_{2}$)-Regular Partition of $Q_{n}$
In this section we study the existence of $(d_{1},d_{2})$-regular partition of
$Q_{n}$.
###### Proposition 4.
Suppose $d_{1},d_{2}\leq n$, if there exists a $(d_{1},d_{2})$-regular
partition of hypercube $Q_{n}$, then the parameters $d_{1},d_{2},n$ should
satisfy $d_{1}+d_{2}=\gcd(d_{1},d_{2})2^{s}$ for some $s\leq n$.
###### Proof.
Suppose the partition is $(V_{1},V_{2})$, we count the total number of
vertices
$\left|V_{1}\right|+\left|V_{2}\right|=2^{n},$
and the number of edges between two parts
$d_{1}\left|V_{1}\right|=d_{2}\left|V_{2}\right|.$
By solving the equations, we obtain
$\left|V_{1}\right|=\frac{d_{2}}{d_{1}+d_{2}}2^{n},\ \
\left|V_{2}\right|=\frac{d_{1}}{d_{1}+d_{2}}2^{n}.$
Both $\left|V_{1}\right|$ and $\left|V_{2}\right|$ should be integers,
therefore $d_{1}+d_{2}=\gcd(d_{1},d_{2})2^{s}$ holds, since
$\gcd(d_{1},d_{1}+d_{2})=\gcd(d_{2},d_{1}+d_{2})=\gcd(d_{1},d_{2})$. ∎
###### Proposition 5.
If there exists a $(d_{1},d_{2})$-regular partition of hypercube $Q_{n}$, then
there exists a $(d_{1},d_{2})$-regular partition of $Q_{m}$ for every $m\geq
n$.
###### Proof.
It suffices to show that the statement holds when $m=n+1$, since the desired
result follows by induction. $Q_{n+1}$ can be treated as the union of two
copies of $Q_{n}$ (for example partition according to the last bit), i.e.
$Q_{n+1}=Q_{n}^{(1)}\cup Q_{n}^{(2)}$. Suppose $(V_{1},V_{2})$ is a
$(d_{1},d_{2})$-regular partition of $Q_{n}^{(1)}$. We can duplicate the
partition $(V_{1},V_{2})$ to get another partition
$(V_{1}^{\prime},V_{2}^{\prime})$ of $Q_{n}^{(2)}$. Then we can see that
$(V_{1}\cup V_{1}^{\prime},V_{2}\cup V_{2}^{\prime})$ forms a partition of
$Q_{n+1}$, in which each node has an edge to its duplicate through the last
dimension. Observe that each node in $V_{1}$ ($V_{1}^{\prime}$) still has
$d_{1}$ neighbors in $V_{2}$ ($V_{2}^{\prime}$) and same for $V_{2}$
($V_{2}^{\prime}$), and the new edges introduced by the new dimension are
among $V_{1}$ and $V_{1}^{\prime}$, or $V_{2}$ and $V_{2}^{\prime}$, which
does not contribute to the edges between two parts of the partition. Therefore
we constructed a $(d_{1},d_{2})$-regular partition of $Q_{n+1}$. ∎
###### Proposition 6.
If there exists a $(d_{1},d_{2})$-regular partition of $Q_{n}$, then there
exists $(td_{1},td_{2})$-regular partition of $Q_{tn}$, for any positive
integer $t$.
###### Proof.
Suppose $(V_{1},V_{2})$ is a $(d_{1},d_{2})$-regular partition of $Q_{n}$. Let
$x=x_{1}x_{2}\cdots x_{nt}$ be a node in $Q_{nt}$. We can divide $x$ into $n$
sections of length $t$, and denote the sum of $i^{th}$ section by $w_{i}$,
i.e.
$w_{i}(x)=\sum_{j=ti-t+1}^{ti}x_{j},\ \ (1\leq i\leq n).$
Let $R(x)=w_{1}(x)w_{2}(x)\ldots w_{n}(x)\in Q_{n}$. Define
$V_{i}^{\prime}=\\{x\in Q_{nt}|R(x)\in V_{i}\\},\ (i=1,2).$
We claim that $(V_{1}^{\prime},V_{2}^{\prime})$ is a $(td_{1},td_{2})$-regular
partition of $Q_{nt}$. This is because for any vertex $x$ in $V_{1}^{\prime}$,
$R(x)$ is in $V_{1}$. So $R(x)$ has $d_{1}$ neighbors in $V_{2}$, and each of
which corresponds $t$ neighbors of $x$ in $V_{2}^{\prime}$, thus in total
$td_{1}$ neighbors in $V_{2}^{\prime}$. It is the same for vertices in
$V_{2}^{\prime}$. ∎
By Proposition 4-6 we only need to consider the existence of
$(d_{1},d_{2})$-regular partition of $Q_{n}$ where $\gcd(d_{1},d_{2})=1$ and
$d_{1}+d_{2}=2^{s}$ (where $s\leq n$), or equivalently, the existence of
$(d,2^{s}-d)$-regular partition of $Q_{n}$, where $s\leq n$ and $d$ is odd.
The following Lemma from [1] showed that when $n=2^{s}-1$ such regular
partition always exists.
###### Lemma 7.
[1] There exists a $(t,2^{s}-t)$-regular partition of $Q_{2^{s}-1}$, for any
integer $s,t$ with $0<t<2^{s}$.
The following theorem shows how to construct the $(t,2^{s}-t)$-regular
partition for $n=2^{s}-r$ (where $r\leq t$).
###### Theorem 8.
Suppose there exists a $(t,2^{s}-t)$-regular partition for $Q_{2^{s}-r}$ and
$t>r$, then there exists a $(t,2^{s}-t)$-regular partition for
$Q_{2^{s+1}-\min\\{t,2r\\}}$.
###### Proof.
For convenience, let $m=2^{s}-r$, and $l=2r-\min\\{t,2r\\}(\geq 0)$. Observe
that $2^{s+1}-\min\\{t,2r\\}=2m+l$, and if $t\geq 2r$ then $l=0$. Suppose that
$(V_{1},V_{2})$ is a $(t,2^{s}-t)$-regular partition for $Q_{m}$. We want to
construct a $(t,2^{s+1}-t)$-regular partition for $Q_{2m+l}$. The basic idea
of the construction is as follows:
We start from set $V_{2}$. We construct a collection of linear equation
systems, each of which corresponds to a node in $V_{2}$. The variables of the
linear systems are the $(2m+l)$ bits of node $x\in Q_{2m+l}$. Let
$V_{2}^{\prime}$ be the union of solutions of these linear equation systems,
and $V_{1}^{\prime}$ be the complement of $V_{2}^{\prime}$. Then
$(V_{1}^{\prime},V_{2}^{\prime})$ is the $(t,2^{s+1}-t)$-regular partition as
we desired.
Here is the construction. Since $(V_{1},V_{2})$ is a $(t,2^{s}-t)$ regular
partition for $Q_{m}$, the subgraph induced by $V_{2}$ of $Q_{m}$ is a
$(t-r)$-regular graph, i.e. for every node $p\in V_{2}$, there are $(t-r)$
neighbors of $p$ in $V_{2}$. For each $p\in V_{2}$, arbitrarily choose a
subset $N(p)\subseteq V_{2}$ of neighbors of node $p$ with size
$\left|N(p)\right|=r-l$. (here $r-l=r-(2r-\min\\{t,2r\\})=\min\\{t,2r\\}-r$,
so $r-l\leq t-r$, and $r-l>0$ since $t>r$)
Now for each node $p=(p_{1},\dots,p_{m})\in V_{2}$, we construct a linear
equation system as follows:
$\begin{cases}x_{1}+x_{2}=p_{1},\\\ x_{3}+x_{4}=p_{2},\\\ \ \ldots\ \ \ldots\
\ \ldots,\\\ x_{2m-1}+x_{2m}=p_{m},\\\ \sum_{j=1}^{m}x_{2j-1}+\sum_{j\in
N(p)}x_{2j}+\sum_{1\leq j\leq l}x_{2m+j}=0.\\\ \end{cases}$ (1)
Note that in the last equation the last term $\sum_{1\leq j\leq l}x_{2m+j}$
vanishes if $l=0$. Denote by $S(p)\subseteq Q_{2m+l}$ the solutions of this
linear system. For convenience, let $f:Q_{2m+l}\rightarrow Q_{m}$ be the
operator such that
$f(x_{1},\ldots,x_{2m+l})=(x_{1}+x_{2},x_{3}+x_{4},\dots,x_{2m-1}+x_{2m}).$
Then in the linear system (1) the first $m$ equations is nothing but $f(x)=p$.
Let $V_{2}^{\prime}=\cup_{p\in V_{2}}S(p)$, and
$V_{1}^{\prime}=Q_{2m+l}\setminus V_{2}^{\prime}$ be its complement. We claim
that $(V_{1}^{\prime},V_{2}^{\prime})$ is a $(t,2^{s+1}-t)$-regular partition
of $Q_{2m+l}$.
To begin with, observe the following two facts.
Observation 1 For every $x\in V_{2}^{\prime}$, we have $f(x)\in V_{2}$. It can
be seen clearly from the first $m$ equations in each equation system.
Observation 2 If $k\leq 2m$, then $f(x^{(2k)})=f(x^{(2k-1)})=(f(x))^{(k)}$. If
$k>2m$, $f(x^{(k)})=f(x)$. Recall the $x^{(i)}$ is the node obtained by
flipping the $i^{th}$ bit of $x$. The observation can be seen from the
definition of $f(x)$.
For any node $x\in V_{1}^{\prime}$, we show that there are $t$ different ways
of flipping a bit of $x$ so that we can get a node in $V_{2}^{\prime}$. There
are two possible cases:
Case 1: $f(x)\not\in V_{2}$. In this case if we flip the $i^{th}$ bit of $x$
for some $i>2m$, then from Observation 2, $f(x^{(i)})=f(x)$, so $f(x^{(i)})$
will remain not in $V_{2}$, and therefore $x^{(i)}$ will not be in
$V_{2}^{\prime}$, by Observation 1. So we can only flip the bit in
$(x_{1},\ldots,x_{2m})$.
Suppose by flipping the $i^{th}$ bit of $x$ we get $x^{(i)}\in V_{2}^{\prime}$
($i\in[2m]$), from the definition of $V_{2}^{\prime}$ we have : $f(x^{(i)})\in
V_{2}$, and $x^{(i)}$ satisfies the last equation in the equation systems
corresponding to $f(x^{(i)})$:
$\sum_{j=1}^{m}x_{2j-1}+\sum_{j\in N(f(x^{(i)}))}x_{2j}+\sum_{1\leq j\leq
l}x_{2m+j}=0.$ (2)
Since $f(x)\notin V_{2}$ and $(V_{1},V_{2})$ is a $(t,2^{s}-t)$-regular
partition of $Q_{m}$, so there are exactly $t$ neighbors of $f(x)$ in $V_{2}$,
which implies there are $t$ bits of $f(x)$ by flipping which we can get a
neighbor of $f(x)$ in $V_{2}$. Let $\\{j_{1},\ldots,j_{t}\\}\subseteq[m]$ be
these bits, i.e. $f(x)^{(j_{1})},\ldots,f(x)^{(j_{t})}\in V_{2}$, by
Observation 2,
$f(x^{(2j_{k}-1)})=f(x^{(2j_{k})})=f(x)^{(j_{k})}\in V_{2},\ \
(k=1,\ldots,t).$
But exactly one of $\\{x^{(2j_{k}-1)},x^{(2j_{k})}\\}$ satisfies the equation
(2) (here we use the fact $f(x)\notin V_{2}$, note that $j_{k}\not\in
N(f(x)^{(j_{k})})$. Thus totally, there are $t$ possible $i$ such that
$x^{(i)}\in V_{2}^{\prime}$.
Case 2: $f(x)\in V_{2}$. Since $x\notin V_{2}^{\prime}$, the last linear
equation must be violated, i.e.
$\sum_{j=1}^{m}x_{2j-1}+\sum_{j\in N(f(x^{(i)}))}x_{2j}+\sum_{1\leq j\leq
l}x_{2m+j}=1.$ (3)
We further consider three cases here: flip a bit in
$\\{x_{1},\ldots,x_{2m}\\}\setminus\\{x_{2j},x_{2j-1}:j\in N(f(x))\\}$; flip a
bit in $\\{x_{2j},x_{2j-1}:j\in N(f(x))\\}$; flip a bit in
$\\{x_{2m+1},\ldots,x_{2m+l}\\}$:
a) if $i\in[m]$ , $i\notin N(f(x))$, and $f(x)^{(i)}\in V_{2}$. Since $f(x)\in
V_{2}$, $(V_{1},V_{2})$ is a $(t,2^{s}-t)$-regular partition of $Q_{m}$, there
are $m-(2^{s}-t)-\left|N(f(x))\right|=(2^{s}-r)-(2^{s}-t)-(r-l)=t-2r+l$ such
index $i$, and $x^{(2i-1)}$ is the exactly the one in
$\\{x^{(2i-1)},x^{(2i)}\\}$ which is in $V_{2}^{\prime}$. (determined by
Equation 3). Thus in this case there are $(t-2r+l)$ neighbors of $x$ in
$V_{2}^{\prime}$.
b) if $i\in N(f(x))$, then both of $x^{(2i-1)},x^{(2i)}$ are in
$V_{2}^{\prime}$, there are $2\cdot\left|N(f(x))\right|=2(r-l)$ such
neighbors.
c) if $i>2m$, then every $x^{(i)}$ is in $V_{2}^{\prime}$,
($i=2m+1,\ldots,2m+l$), there are $l$ such neighbors.
Hence, totally $x$ has $(t-2r+l)+2(r-l)+l=t$ neighbors in $V_{2}^{\prime}$.
The rest thing is to show that every node $x\in V_{2}^{\prime}$ has
$(2^{s+1}-t)$ neighbors in $V_{1}^{\prime}$. The proof is similar to the proof
of Case 2 above, we consider three cases:
a) If $i\in[m]$, $i\not\in N(f(x))$, and $f(x)^{(i)}\in V_{2}$. Then exactly
one of $x^{(2k-1)},x^{(2k)}$ in $V_{2}^{\prime}$, thus there are
$m-(2^{s}-t)-\left|N(f(x))\right|=2^{s}-r-(2^{s}-t)-(r-l)=t-2r+l$ such
neighbors of $x$ in $V_{2}^{\prime}$.
b) If $i\in N(f(x))$ both $x^{(2i-1)},x^{(2i)}$ are not in $V_{2}^{\prime}$.
c) If $i>2m$, then every $x^{(i)}$ is not in $V_{2}^{\prime}$.
Hence totally, $x$ has $(t-2r+l)$ neighbors in $V_{2}^{\prime}$, and therefore
$(2m+l)-(t-2r+l)=2^{s+1}-t$ neighbors in $V_{1}^{\prime}$.
Hence we prove that $(V_{1}^{\prime},V_{2}^{\prime})$ is indeed a
$(t,2^{s+1}-t)$-regular partition of $Q_{2^{s+1}-\min\\{t,2r\\}}$. ∎
###### Theorem 9.
For any odd number $t$ and any $c\leq t$, when $s\geq\lceil\lg
t\rceil+\lceil\lg c\rceil$, there exists a $(t,2^{s}-t)$-regular partition of
$Q_{2^{s}-c}$.
###### Proof.
Let $s_{0}=\lceil\lg t\rceil$. By Lemma 7, there exists a
$(t,2^{s_{0}}-t)$-regular partition of $Q_{2^{s_{0}}-1}$. By repeatedly using
Theorem 8, we obtain that there exists $(t,2^{s_{0}+1}-t)$-regular partition
of $Q_{2^{s_{0}+1}-2}$, $(t,2^{s_{0}+2}-t)$-regular partition of
$Q_{2^{s_{0}+2}-2^{2}}$, etc., $(t,2^{s_{0}+\lceil\lg{c}\rceil-1}-t)$-regular
partition of $Q_{2^{s_{0}+\lceil\lg{c}\rceil-1}-2^{\lceil\lg{c}\rceil-1}}$,
and $(t,2^{s_{0}+\lceil\lg{c}\rceil}-t)$-regular partition of
$Q_{2^{s_{0}+\lceil\lg{c}\rceil}-c}$. By using Proposition 5, we get that
there exists a $(t,2^{s}-t)$-regular partition of $Q_{2^{s}-c}$, for any
$s\geq s_{0}+\lceil\lg{c}\rceil=\lceil\lg t\rceil+\lceil\lg{c}\rceil$. ∎
Combining Proposition 6 and Theorem 9, we have the following corollary.
###### Corollary 10.
Suppose $d_{1}=dt,d_{2}=d(2^{s}-t),n=d(2^{s}-c)$, where $d,t,s$ are positive
integers with $0<t<2^{s}$, $c\leq t$ and $s\geq\lceil\lg c\rceil+\lceil\lg
t\rceil$, then there exists a $(d_{1},d_{2})$-regular partition for $Q_{n}$.
## 4 The Maximum Winning Probability $P_{n,k}$
The following lemma characterizes the relationship between the maximum winner
probability $P_{n,k}$ and the minimum $k$-dominating set of $Q_{n}$. The same
result was showed in [5] for $k=1$.
###### Lemma 11.
Suppose $D$ is a $k$-dominating set of $Q_{n}$ with minimum number of
vertices. Then
$P_{n,k}=1-\frac{|D|}{2^{n}}.$
###### Proof.
Given a $k$-dominating set $D$ of $Q_{n}$, the following strategy will have
winning probability at least $1-\frac{\left|D\right|}{2^{n}}$: For any certain
placement of hats, each player can see all hats but his own, so player $p_{i}$
knows that current placement $h$ is one of two adjacent nodes
$\\{x,x^{(i)}\\}$ of $Q_{n}$. If $x\in D$ (or $x^{(i)}\in D$), he guesses that
the current placement is $x^{(i)}$ (or $x$), otherwise he passes. We claim
that by using this strategy, players win the game when the placement is a node
which is not in $D$. Observe that since $D$ is a $k$-dominating set, for any
node $y\notin D$, $y$ has $l$ neighbors
$y^{(i_{1})},y^{(i_{2})},\ldots,y^{(i_{l})}$ that are in $D$, where $l\geq k$.
According to the strategy desribed, players $p_{i_{1}},\ldots,p_{i_{l}}$ would
guess correctly and all other players will pass. This shows the winning
probability is at least $1-\frac{\left|D\right|}{2^{n}}$.
Next we show that $P_{n,k}\leq 1-\frac{\left|D\right|}{2^{n}}$. Suppose we
have a strategy with winning probability $P_{n,k}$. We prove that there exists
a $k$-dominating set $D_{0}$, such that $|D_{0}|=2^{n}(1-P_{n,k})$. The
construction is straightforward: Let $D_{0}=\\{h\in Q_{n}:h\textrm{ is not a
winning placement}\\}$. Thus $|D_{0}|=N(1-P_{n,k})$. For every winning
placement $h\notin D_{0}$, suppose players $p_{i_{1}},\ldots,p_{i_{l}}$ will
guess correctly ($l\geq k$), consider the placement $h^{(i_{1})}$, which
differs from $h$ only at player $p_{i_{1}}$’s hat, so player $p_{i_{1}}$ will
guess incorrectly in this case, thus $h^{(i_{1})}\in D_{0}$. Similarly
$h^{(i_{2})},\ldots,h^{(i_{l})}\in D_{0}$, therefore $D_{0}$ is a
$k$-dominating set. We have
$|D|\leq|D_{0}|=2^{n}(1-P_{n,k}),$
which implies
$P_{n,k}\leq 1-\frac{\left|D\right|}{2^{n}}.$
Combining these two results, we have $P_{n,k}=1-\frac{\left|D\right|}{2^{n}}$
as desired. ∎
###### Proposition 12.
The following properties hold:
1. (a)
If $n_{1}<n_{2}$ then $P_{n_{1},k}\leq P_{n_{2},k}$.
2. (b)
$(n,k)$ is perfect iff there exists a $(k,n)$-regular partition of $Q_{n}$.
3. (c)
For any $t\in\mathbb{N}$, $P_{nt,kt}\geq P_{n,k}$. As a consequence, if
$(n,k)$ is perfect, $(nt,kt)$ is perfect.
###### Proof.
For part (a), suppose that $D$ is a minimum $k$-dominating set of $Q_{n_{1}}$.
We make $2^{n_{2}-n_{1}}$ copies of $Q_{n_{1}}$, and by combining them we get
a $Q_{n_{2}}$, which has dominating set of size $2^{n_{2}-n_{1}}|D|$. By Lemma
11, $P_{n_{2},k}\geq 1-\frac{2^{n_{2}-n_{1}}|D|}{2^{n_{2}}}=P_{n_{1},k}$.
For part (b), suppose $(U,V)$ is a $(k,n)$-regular partition of $Q_{n}$, note
that $V$ is a $k$-dominating set of $Q_{n}$ and $|V|=\frac{k}{n+k}\cdot
2^{n}$, thus $V$ is a minimum $k$-dominating set of $Q_{n}$. We have that
$P_{n,k}=1-\frac{|V|}{2^{n}}=\frac{n}{n+k}$, which implies that $(n,k)$ is
perfect.
On the other hand, if $(n,k)$ is perfect, suppose $D$ is the minimum
$k$-dominating set, we have $\left|D\right|=\frac{k}{n+k}\cdot 2^{n}$. It can
be observed that $\left(Q_{n}\setminus D,D\right)$ is a $(k,n)$-regular
partition of $Q_{n}$.
For part (c), since $\frac{n}{n+k}=\frac{nt}{nt+kt}$, once $P_{nt,kt}\geq
P_{n,k}$ holds, it’s an immediate consequence that the perfectness of $(n,k)$
implies the perfectness of $(nk,nt)$.
Suppose for $n$ players, we have a strategy $\mathcal{S}$ with probability of
winning $P_{n,k}$. For $nt$ players, we divide them into $n$ groups, each of
which has $t$ players. Each placement $h=(h_{1},h_{2},\ldots,h_{nt})$ of $nt$
players can be mapped to a placement $P(h)$ of $n$ players in the following
way: for Group $i$, suppose the sum of colors in the group is $w_{i}$, i.e.
$w_{i}(h)=\sum_{j=ti-t+1}^{ti}h_{j},\ \ (1\leq i\leq n).$
Let $P(h)=(w_{1}(h),w_{2}(h),\ldots,w_{n}(h))$ be a placement of $n$ players.
Each player in Group $i$ knows the color of all players in $P(h)$ other than
Player $i$, thus he uses Player $i$’s strategy $s_{i}$ in $\mathcal{S}$ to
guess the sum of colors in Group $i$ or passes. Moreover once he knows the
sum, his color can be uniquely determined.
Note that the players in Group $i$ would guess correctly or incorrectly or
pass, if and only if Player $i$ in the $n$-player-game would do. Since the hat
placement is uniformly at random, the probability of winning using this
strategy is at least $P_{n,k}$, thus $P_{nt,kt}\geq P_{n,k}$. ∎
Now we can prove our main theorem:
###### Theorem 1.
For any $d,k,s\in\mathbb{N}$ with $s\geq 2\lceil\lg k\rceil$,
$(d(2^{s}-k),dk)$ is perfect, in particular, $(2^{s}-k,k)$ is perfect.
###### Proof.
It’s an immediate corollary of part (b) of Proposition 12 and Theorem 9. ∎
Remark: By Proposition 4 and Proposition 12(b) there is a simple necessary
condition for $(n,k)$ to be perfect, $n+k=\gcd(n,k)2^{t}$. Theorem 1 indicates
that when $n+k=\gcd(n,k)2^{t}$ and $n$ is sufficiently large, $(n,k)$ is
perfect. The necessary condition and sufficient condition nearly match in the
sense that for each $k$, there’s only a few $n$ that we don’t know whether
$(n,k)$ is perfect. Moreover, the following proposition shows that the simple
necessary condition can’t be sufficient. The first counterexample is $(5,3)$,
it is not perfect while it satisfies the simple necessary condition. But
$(13,3)$ is perfect by Theorem 1 and more generally for all $s\geq 4$,
$(2^{s}-3,3)$ is perfect. We verified by computer program that
$(2^{4}-5,5)=(11,5)$ is not perfect, while by our main theorem
$(2^{6}-5,5)=(59,5)$ is perfect. But we still don’t know whether the case
between them, $(2^{5}-5,5)=(27,5)$, is perfect.
###### Proposition 13.
$(n,k)$ is not perfect unless $2k+1\leq n$ when $n\geq 2$ and $k<n$.
###### Proof.
Suppose $(n,k)$ is perfect. According to part (b) of Proposition 12, we can
find $(U,V)$, a $(k,n)$-regular partition of $Q_{n}$. Suppose $x$ is some node
in $U$, and $y$ is some neighbor of $x$ which is also in $U$, $y$ has $k$
neighbors in $V$. They all differ from $x$ at exactly $2$ bits and one of them
is what $y$ differs from $x$ at, i.e. each of them “dominates” $2$ neighbors
of $x$, one of them is $y$. So $x$ has totally $k+1$ neighbors “dominated” by
$k$ of nodes in $V$. Since all nodes in $V$ are pairwise nonadjacent, these
$k+1$ nodes must be in $U$. Now we have $k+1$ neighbors of $x$ are in $U$ and
$k$ neighbors are in $V$, it has totally $n$ neighbors. We must have $2k+1\leq
n$. ∎
For each odd number $k$, let $s(k)$ be the smallest number such that
$(2^{s(k)}-k,k)$ is perfect. We know that
$s(k)\in[\lceil\lg{k}\rceil,2\lceil\lg{k}\rceil]$. The following proposition
indicates that all $s\geq s(k)$, $(2^{s}-k,k)$ is also perfect.
###### Proposition 14.
If $(2^{s}-k,k)$ is perfect, $(2^{s+1}-k,k)$ is perfect.
###### Proof.
If $(2^{s}-k,k)$ is perfect, by Proposition 12(b) there is a
$(k,2^{s}-k)$-regular partition of $Q_{2^{s}-k}$. Thus by Proposition 5, we
have a $(k,2^{s}-k)$-regular partition of $Q_{2^{s}-k+1}$. Combine this
partition and Theorem 8, we get a $(k,2^{s+1}-k)$-regular partition of
$Q_{2^{s+1}-k}$. Therefore $(2^{s+1}-k,k)$ is perfect. ∎
Using Theorem 1 we can give a general lower bound for the winning probability
$P_{n,k}$. Recall that there’s upper bound $P_{n,k}\leq 1-\frac{k}{n+k}$.
###### Lemma 15.
$P_{n,k}>1-\frac{2k}{n+k}$, when $n\geq 2^{2\lceil\lg k\rceil}-k$.
###### Proof.
Let $n^{\prime}$ be the largest integer of form $2^{t}-k$ which is no more
than $n$. By Theorem 1, $(n^{\prime},k)$ is perfect, i.e.
$P_{n^{\prime},k}=1-\frac{k}{n^{\prime}+k}$. By part (a) of Proposition 12,
$P_{n,k}\geq P_{n^{\prime},k}$. On the other hand we have $n+k<2^{t+1}$, so we
have
$P_{n,k}\geq 1-\frac{k}{n^{\prime}+k}=1-\frac{2k}{2^{t+1}}>1-\frac{2k}{n+k}.$
∎
###### Corollary 16.
For any integer $k>0$, $\lim_{n\rightarrow\infty}P_{n,k}=1$.
## 5 Conclusion
In this paper we investigated the existence of regular partition for boolean
hypercube, and its applications in finding perfect strategies of a new hat
guessing games. We showed a sufficient condition for $(n,k)$ to be perfect,
which nearly matches the necessary condition. Several problems remain open:
for example, determine the minimum value of $s(k)$ such that $(2^{s(k)}-k,k)$
is perfect, and determine the exact value of $P_{n,k}$. It is also very
interesting to consider the case when there are more than two colors in the
game.
## References
* [1] Joe Buhler, Steve Butler, Ron Graham, and Eric Tressler. Hypercube orientations with only two in-degrees. http://arxiv.org/abs/1007.2311, 2010.
* [2] Steve Butler, Mohammad T. Hajiaghayi, Robert D. Kleinberg, and Tom Leighton. Hat guessing games. SIAM J. Discrete Math., 22(2):592–605, 2008.
* [3] Todd T. Ebert. Applications of recursive operators to randomness and complexity. PhD thesis, University of California at Santa Barbara, 1998.
* [4] Uriel Feige. You can leave your hat on (if you guess its color). Technical Report MCS04-03, Computer Science and Applied Mathematics, The Weizmann Institute of Science, 2004.
* [5] Uriel Feige. On optimal strategies for a hat game on graphs. SIAM Journal of Discrete Mathematics, 2010.
* [6] Hendrik W. Lenstra and Gadiel Seroussi. On hats and other covers. http://arxiv.org/abs/cs/0509045, 2005.
* [7] Maura B . Peterson and Douglas R. Stinson. Yet another hatt game. the electronic journal of combinatorics, 17(1), 2010.
|
arxiv-papers
| 2011-01-05T02:13:50 |
2024-09-04T02:49:16.154026
|
{
"license": "Public Domain",
"authors": "Tengyu Ma and Xiaoming Sun and Huacheng Yu",
"submitter": "Xiaoming Sun",
"url": "https://arxiv.org/abs/1101.0869"
}
|
1101.1136
|
Marginal Likelihood Estimation via Arrogance Sampling
By Benedict Escoto
###### Abstract
This paper describes a method for estimating the marginal likelihood or Bayes
factors of Bayesian models using non-parametric importance sampling
(“arrogance sampling”). This method can also be used to compute the
normalizing constant of probability distributions. Because the required inputs
are samples from the distribution to be normalized and the scaled density at
those samples, this method may be a convenient replacement for the harmonic
mean estimator. The method has been implemented in the open source R package
margLikArrogance.
## 1 Introduction
When a Bayesian evaluates two competing models or theories, $T_{1}$ and
$T_{2}$, having observed a vector of observations $\boldsymbol{x}$, Bayes’
Theorem determines the posterior ratio of the models’ probabilities:
$\frac{p(T_{1}|\boldsymbol{x})}{p(T_{2}|\boldsymbol{x})}=\frac{p(\boldsymbol{x}|T_{1})}{p(\boldsymbol{x}|T_{2})}\frac{p(T_{1})}{p(T_{2})}.$
(1)
The quantity $\frac{p(\boldsymbol{x}|T_{1})}{p(\boldsymbol{x}|T_{2})}$ is
called a _Bayes factor_ and the quantities $p(\boldsymbol{x}|T_{1})$ and
$p(\boldsymbol{x}|T_{2})$ are called the theories’ _marginal likelihoods_.
The types of Bayesian models considered in this paper have a fixed finite
number of parameters, each with their own probability function. If
$\boldsymbol{\theta}$ are parameters for a model $T$, then
$p(\boldsymbol{x}|T)=\int
p(\boldsymbol{x}|\boldsymbol{\theta},T)p(\boldsymbol{\theta}|T)\,d\boldsymbol{\theta}=\int
p(\boldsymbol{x}\wedge\boldsymbol{\theta}|T)\,d\boldsymbol{\theta}$ (2)
Unfortunately, this integral is difficult to compute in practice. The purpose
of this paper is to describe one method for estimating it.
Evaluating integral (2) is sometimes called the problem of computing
normalizing constants. The following formula shows how $p(\boldsymbol{x}|T)$
is a normalizing constant.
$p(\boldsymbol{\theta}|\boldsymbol{x},T)=\frac{p(\boldsymbol{\theta}\wedge\boldsymbol{x}|T)}{p(\boldsymbol{x}|T)}$
(3)
Thus the marginal likelihood $p(\boldsymbol{x}|T)$ is also the normalizing
constant of the posterior parameter distribution
$p(\boldsymbol{\theta}|\boldsymbol{x},T)$ assuming we are given the density
$p(\boldsymbol{\theta}\wedge\boldsymbol{x}|T)$ which is often easy to compute
in Bayesian models. Furthermore, Bayesian statisticians typically produce
samples from the posterior parameter distribution
$p(\boldsymbol{\theta}|\boldsymbol{x},T)$ even when not concerned with theory
choice. In these case, computing the marginal likelihood is equivalent to
computing the normalizing constant of a distribution from which samples and
the scaled density at these samples are available. The method described in
this paper takes this approach.
## 2 Review of Literature
Given how basic (1) is, it is perhaps surprising that there is no easy and
definitive way of applying it, even for simple models. Furthermore, as the
dimensionality and complexity of probability distributions increase, the
difficulty of approximation also increases. The following three techniques for
computing bayes factors or marginal likelihoods are important but will not be
mentioned further here.
1. 1.
Analytic asymptotic approximations such as Laplace’s method, see for instance
Kass and Raftery (1995),
2. 2.
Bridge sampling/path sampling/thermodynamic integration (Gelman and Meng,
1998), and
3. 3.
Chib’s MCMC approximation (Chib, 1995; Chib and Jeliazkov, 2005).
Kass and Raftery (1995) is a popular overview of the earlier literature on
Bayes factor computation. All these methods can be very successful in the
right circumstances, and can often handle problems too complex for the method
described here. However, the method of this paper may still be useful due to
its convenience.
The rest of section 2 describes three approaches that are relevant to this
paper.
### 2.1 Importance Sampling
Importance sampling is a technique for reducing the variance of monte carlo
integration. This section will note some general facts; see Owen and Zhou
(1998) for more information.
Suppose we are trying to compute the (possibly multidimensional) integral $I$
of a well-behaved function $f(\boldsymbol{\theta})$. Then
$I=\int
f(\boldsymbol{\theta})\,d\boldsymbol{\theta}=\int\frac{f(\boldsymbol{\theta})}{g(\boldsymbol{\theta})}g(\boldsymbol{\theta})\,d(\boldsymbol{\theta})$
so if $g(\boldsymbol{\theta})$ is a probability density function and
$\boldsymbol{\theta}_{i}$ are independent samples from it, then
$I=\mbox{E}_{g}[f(\boldsymbol{\theta})/g(\boldsymbol{\theta})]\approx\frac{1}{n}\sum_{i=1}^{n}\frac{f(\boldsymbol{\theta}_{i})}{g(\boldsymbol{\theta}_{i})}=I_{n}.$
(4)
$I_{n}$ is an unbiased approximation to $I$ and by the central limit theorem
will tend to a normal distribution. It has variance
$\mbox{Var}[I_{n}]=\frac{1}{n}\int\left(\frac{f(\boldsymbol{\theta})}{g(\boldsymbol{\theta})}-I\right)^{2}g(\boldsymbol{\theta})\,d\boldsymbol{\theta}=\frac{1}{n}\int\frac{(f(\boldsymbol{\theta})-Ig(\boldsymbol{\theta}))^{2}}{g(\boldsymbol{\theta})}\,d\boldsymbol{\theta}$
(5)
Sometimes $f$ is called the _target_ and $g$ is called the _proposal_
distribution.
Assuming that $f$ is non-negative, then minimum variance (of $0!$) is achieved
when $g=f/I$—in other words when $g$ is just the normalized version of $f$.
This cannot be done in practice because normalizing $f$ requires knowing the
quantity $I$ that we wanted to approximate; however (5) is still important
because it means that the more similar the proposal is to the target, the
better our estimator $I_{n}$ becomes. In particular, $f$ must go to 0 faster
than $g$ or the estimator will have infinite variance.
To summarize this section:
1. 1.
Importance sampling is a monte carlo integration technique which evaluates the
target using samples from a proposal distribution.
2. 2.
The estimator is unbiased, normally distributed, and its variance (if not 0 or
infinity) decreases as $O(n^{-1})$ (using big-$O$ notation).
3. 3.
The closer the proposal is to the target, the better the estimator. The
proposal also needs to have longer tails than the target.
### 2.2 Nonparametric Importance Sampling
A difficulty with importance sampling is that it is often difficult to choose
a proposal distribution $g$. Not enough is known about $f$ to choose an
optimal distribution, and if a bad distribution is chosen the result can have
large or even infinite variance. One approach to the selection of proposal $g$
is to use non-parametric techniques to build $g$ from samples of $f$. I call
this class of techniques self-importance sampling, or arrogance sampling for
short, because they attempt to sample $f$ from itself without using any
external information. (And also isn’t it a bit arrogant to try to evaluate a
complex, multidimensional integral using only the values at a few points?) The
method of this paper falls into this class and particularly deserves the name
because the target and proposal (when they are both non-zero) have exactly the
same values up to a multiplicative constant.
Two papers which apply nonparametric importance sampling to the problem of
marginal likelihood computation (or computation of normalizing constants) are
Zhang (1996) and Neddermeyer (2009). Although both authors apply their methods
to more general situations, here I will use the framework suggested by (3) and
assume that we can compute $p(\boldsymbol{\theta}\wedge\boldsymbol{x}|T)$ for
arbitrary $\boldsymbol{\theta}$ and also that we can sample from the posterior
parameter distribution $p(\boldsymbol{\theta}|\boldsymbol{x},T)$. The goal is
to estimate the normalizing constant, the marginal likelihood
$p(\boldsymbol{x}|T)$.
Zhang’s approach is to build the proposal $g$ using traditional kernel density
estimation. $m$ samples are first drawn from
$p(\boldsymbol{\theta}|\boldsymbol{x},T)$ and used to construct $g$. Then $n$
samples are drawn from $g$ and used to evaluate $p(\boldsymbol{x}|T)$ as in
traditional importance sampling. This approach is quite intuitive because
kernel estimation is a popular way of approximating an unknown function. Zhang
proves that the variance of his estimator decreases as
$O(m^{\frac{-4}{4+d}}n^{-1})$ where $d$ is the dimensionality of
$\boldsymbol{\theta}$, compared to $O(n^{-1})$ for standard (parametric)
importance sampling.
There were, however, a few issues with Zhang’s method:
1. 1.
A kernel density estimate is equal to 0 at points far from the points the
kernel estimator was built on. This is a problem because importance sampling
requires the proposal to have longer tails than the target. This fact forces
Zhang to make the restrictive assumption that
$p(\boldsymbol{\theta}|\boldsymbol{x},T)$ has compact support.
2. 2.
It is hard to compute the optimal kernel bandwidth. Zhang recommends using a
plug-in estimator because the function
$p(\boldsymbol{\theta}\wedge\boldsymbol{x}|T)$ is available, which is unusual
for kernel estimation problems. Still, bandwidth selection appears to require
significant additional analysis.
3. 3.
Finally, although the variance may decrease as $O(m^{\frac{-4}{4+d}}n^{-1})$
as $m$ increases, the difficulty of computing $g(\boldsymbol{\theta})$ also
increases with $m$, because it requires searching through the $m$ basis points
to find all the points close to $\boldsymbol{\theta}$. In multiple dimensions,
this problem is not trivial and may outweigh the $O(m^{\frac{-4}{4+d}})$
speedup (in the worst case, practical evaluation of $g(\boldsymbol{\theta})$
at a single point may be $O(m)$). See Zlochin and Baram (2002) for some
discussion of these issues.
Neddermeyer (2009) uses a similar approach to Zhang and also achieves a
variance of $O(m^{\frac{-4}{4+d}}n^{-1})$. It improves on Zhang’s approach in
two ways relevant to this paper:
1. 1.
The support of $p(\boldsymbol{\theta}|\boldsymbol{x},T)$ is not required to be
compact.
2. 2.
Instead of using kernel density estimators, linear blend frequency polynomials
(LBFPs) are used instead. LBFPs are basically histograms whose density is
interpolated between adjacent bins. As a result, the computation of
$g(\boldsymbol{\theta})$ requires only finding which bin $\boldsymbol{\theta}$
is in, and looking up the histogram value at that and adjacent bins ($2^{d}$
bins in total).
As we will see in section 3, the arrogance sampling described in this paper is
similar to the methods of Zhang and Neddermeyer.
### 2.3 Harmonic Mean Estimator
The harmonic mean estimator is a simple and notorious method for calculating
marginal likelihoods. It is a kind of importance sampling, except the proposal
$g$ is actually the distribution
$p(\boldsymbol{\theta}|\boldsymbol{x},T)=p(\boldsymbol{\theta}\wedge\boldsymbol{x}|T)/p(\boldsymbol{x}|T)$
to be normalized and the target $f$ is the known distribution
$p(\boldsymbol{\theta}|T)$. Then if $\boldsymbol{\theta}_{i}$ are samples from
$p(\boldsymbol{\theta}|x,T)$, we apparently have
$1\approx\frac{1}{n}\sum_{i=1}^{n}\frac{p(\boldsymbol{\theta}_{i}|T)}{p(\boldsymbol{\theta}_{i}|\boldsymbol{x},T)}=\frac{1}{n}\sum_{i=1}^{n}\frac{p(\boldsymbol{\theta}_{i}|T)}{p(\boldsymbol{x}|\boldsymbol{\theta}_{i},T)p(\boldsymbol{\theta}_{i}|T)/p(\boldsymbol{x}|T)}=\frac{1}{n}\sum_{i=1}^{n}\frac{1}{p(\boldsymbol{x}|\boldsymbol{\theta}_{i},T)/p(\boldsymbol{x}|T)}$
hence
$p(\boldsymbol{x}|T)\stackrel{{\scriptstyle?}}{{\approx}}\left(\frac{1}{n}\sum_{i=1}^{n}\frac{1}{p(\boldsymbol{x}|\boldsymbol{\theta}_{i},T)}\right)^{-1}$
(6)
Two advantages of the harmonic mean estimator are that it is simple to compute
and only depends on samples from $p(\boldsymbol{\theta}|x,T)$ and the
likelihood $p(\boldsymbol{x}|\boldsymbol{\theta},T)$ at those samples. The
main drawback of the harmonic mean estimator is that it doesn’t work—as
mentioned earlier the importance sampling proposal distribution needs to have
longer tails than the target. In this case the target
$p(\boldsymbol{\theta}|T)$ typically has longer tails than the proposal
$p(\boldsymbol{\theta}|\boldsymbol{x},T)$ and thus (6) has infinite variance.
Despite not working, the harmonic mean estimator continues to be popular
(Neal, 2008).
## 3 Description of Technique
This paper’s arrogance sampling technique is a simple method that applies the
nonparametric importance techniques of Zhang and Neddermeyer in an attempt to
develop a method almost as convenient as the harmonic mean estimator.
The only required inputs are samples $\boldsymbol{\theta}_{i}$ from
$p(\boldsymbol{\theta}|\boldsymbol{x},T)$ and the values
$p(\boldsymbol{\theta}_{i}\wedge\boldsymbol{x}|T)=p(x|\boldsymbol{\theta}_{i},T)p(\boldsymbol{\theta}_{i}|T)$.
This is similar to the harmonic mean estimator, but perhaps slightly less
convenient because $p(\boldsymbol{\theta}_{i}\wedge\boldsymbol{x}|T)$ is
required instead of $p(\boldsymbol{x}|\boldsymbol{\theta}_{i},T)$.
There are two basic steps:
1. 1.
Take $m$ samples from $p(\boldsymbol{\theta}|\boldsymbol{x},T)$ and using
modified histogram density estimation, construct probability density function
$f(\boldsymbol{\theta})$.
2. 2.
With $n$ more samples from $p(\boldsymbol{\theta}|\boldsymbol{x},T)$, estimate
$1/p(\boldsymbol{x}|T)$ via importance sampling with target $f$ and proposal
$p(\boldsymbol{\theta}|\boldsymbol{x},T)$.
These steps are described in more detail below.
### 3.1 Construction of the Histogram
Of the $N$ total samples $\boldsymbol{\theta}_{i}$ from
$p(\boldsymbol{\theta}|\boldsymbol{x},T)$, the first $m$ will be used to make
a histogram. The optimal choice of $m$ will be discussed below, but in
practice this seems difficult to determine. An arbitrary rule of
$\mbox{min}(0.2N,2\sqrt{N})$ can be used in practice.
With a traditional histogram, the only available information is the location
of the sampled points. In this case we also know the (scaled) heights
$p(\boldsymbol{\theta}\wedge\boldsymbol{x}|T)$ at each sampled point. We can
use this extra information to improve the fit.
Our “arrogant” histogram $f$ is constructed the same as a regular histogram,
except the bin heights are not determined by the number of points in each bin,
but rather by the minimum density over all points in the bin. If a bin
contains no sampled points, then $f(\boldsymbol{\theta})=0$ for
$\boldsymbol{\theta}$ in that bin. Then $f$ is normalized so that $\int
f(\boldsymbol{\theta})\,d\boldsymbol{\theta}=1$.
To determine our bin width, we can simply and somewhat arbitrarily set our bin
width $h$ so that the histogram is positive for 50% of the sampled points from
the distribution $p(\boldsymbol{\theta}|\boldsymbol{x},T)$. To approximate
$h$, we can use a small number of samples (say, 40) from
$p(\boldsymbol{\theta}|\boldsymbol{x},T)$ and set $h$ so that
$f(\boldsymbol{\theta})>0$ for exactly half of these samples.
Figure 1 compares the traditional and new histograms for a one dimensional
normal distribution based on 50 samples. The green rug lines indicate the $50$
sampled points which are the same for all. The arrogant histogram’s bin width
is chosen as above. The traditional histogram’s optimal bin width was
determined by Scott’s rule to minimize mean squared error. As the figure
shows, the modified histogram is much smoother for a given bin width, so a
smaller bin width can be used. On the other hand, $f$ will either equal 0 or
have about twice the original density at each point, while the traditional
histogram’s density is numerically close to the original density.
Figure 1: Histogram Comparison
### 3.2 Importance Sampling
The remaining $n=N-m-40$ sampled points can be used for importance sampling.
Using equation (4) with histogram $f$ as our target and
$p(\boldsymbol{\theta}|\boldsymbol{x},T)$ as the proposal, we have
$1\approx
I_{n}=\frac{1}{n}\sum_{i=1}^{n}\frac{f(\boldsymbol{\theta}_{i})}{p(\boldsymbol{\theta}_{i}|\boldsymbol{x},T)}=\frac{1}{n}\sum_{i=1}^{n}\frac{f(\boldsymbol{\theta}_{i})}{p(\boldsymbol{\theta}_{i}\wedge\boldsymbol{x}|T)/p(\boldsymbol{x}|T)}$
hence
$p(\boldsymbol{x}|T)\approx
p(\boldsymbol{x}|T)/I_{n}=\left(\frac{1}{n}\sum_{i=1}^{n}\frac{f(\boldsymbol{\theta}_{i})}{p(\boldsymbol{\theta}_{i}\wedge\boldsymbol{x}|T)}\right)^{-1}=A_{n}$
(7)
To underscore the self-important/arrogant nature of this approximation
$A_{n}$, we can rewrite (7) as
$p(\boldsymbol{x}|T)\approx
H\left(\frac{1}{n}\sum_{i=1}^{n}\frac{\mbox{min}\\{p(\boldsymbol{\theta}_{j}\wedge\boldsymbol{x}|T):\boldsymbol{\theta}_{j}\mbox{
and }\boldsymbol{\theta}_{j}\mbox{ are in the same
bin}\\}}{p(\boldsymbol{\theta}_{i}\wedge\boldsymbol{x}|T)}\right)^{-1}$
where $H$ is the histogram normalizing constant. This equation shows that all
the values in the numerator and the denominator of our importance sampling are
from the same distribution $p(\boldsymbol{\theta}\wedge\boldsymbol{x}|T)$.
Note that the histogram $f$ is the target of the importance sampling and
$p(\boldsymbol{\theta}\wedge\boldsymbol{x}|T)$ is the proposal. This is
backwards from the usual scheme where the unknown distribution is the target
and the known distribution is the proposal. Instead here the unknown
distribution is the proposal, as in the harmonic mean estimator (see Robert
and Wraith (2009) for another example of this.)
As in section 2.1, our approximation of $p(\boldsymbol{x}|T)^{-1}$ tends to a
normal distribution as $n\to\infty$ by the central limit theorem. This fact
can be used to estimate a confidence interval around $p(\boldsymbol{x}|T)$.
## 4 Validity of Method
This section will investigate the performance of the method. First, note that
this method is just an implementation of importance sampling, so $A_{n}^{-1}$
should converge to $p(\boldsymbol{x}|T)^{-1}$ with finite variance as long as
the proposal density $p(\boldsymbol{\theta}|\boldsymbol{x},T)$ exists and is
finite and positive on the compact region where the target histogram density
is positive.
To calculate the speed of convergence we will use equation (5) where $f$ is
the histogram,
$g(\boldsymbol{\theta})=p(\boldsymbol{\theta}|\boldsymbol{x},T)$, and $I=1$
because the histogram has been normalized. Unless otherwise noted, we will
assume below that $g:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is finite, twice
differentiable and positive, and that $\int\frac{\lVert\nabla\cdot
g(\boldsymbol{\theta})\rVert^{2}}{g(\boldsymbol{\theta})}d\boldsymbol{\theta}$
is finite.
### 4.1 Histogram Bin Width
One important issue will be how quickly the $d$-dimensional histogram’s
selected bin width $h$ goes to 0 as the number of samples
$m\rightarrow\infty$. This section will only offer an intuitive argument. For
any $m$, the histogram will enclose about the same probability ($\frac{1}{2}$)
and will have about the same average density in a fixed region. Each bin has
volume $h^{d}$, so if $l$ is the number of bins then $lh^{d}=O(1)$ and
$h\propto l^{-d}$.
Furthermore, the distribution of the sampled points converges to the actual
distribution $g(\boldsymbol{\theta})$. If $m>O(l)$, an unbounded number of
sampled points would end up in each bin. If $m<O(l)$, then some bins would
have no points in them. Neither of these is possible because exactly one
sampled point is necessary to establish each bin. Thus $m\propto l$ and
$h\propto m^{-d}$.
### 4.2 Conditional Variance
Before estimating the convergence rate of $A_{n}$ we will prove something
about the conditional variance of importance sampling. Let
$A=\\{\boldsymbol{\theta}:f(\boldsymbol{\theta})>0\\}$, $\mathbf{1}_{A}$ be
the characteristic function of $A$, and
$q=\int_{A}g(\boldsymbol{\theta})\,d\boldsymbol{\theta}$. Define
$g_{A}(\boldsymbol{\theta})=\left\\{\begin{array}[]{cl}g(\boldsymbol{\theta})/q&\mbox{
if }\boldsymbol{\theta}\in A\\\ 0&\mbox{ otherwise }\end{array}\right.$
Then $g_{A}$ is the density of $g$ conditional on $f>0$. Define
$\mbox{Var}_{A}$ and $\mbox{E}_{A}$ to mean the variance and expectation
conditional on $f(\boldsymbol{\theta})>0$. Thus
$\displaystyle\mbox{Var}(f(\boldsymbol{\theta})/g(\boldsymbol{\theta}))$
$\displaystyle=$
$\displaystyle\mbox{Var}(\mbox{E}(f(\boldsymbol{\theta})/g(\boldsymbol{\theta})|\mathbf{1}_{A}))+\mbox{E}(\mbox{Var}(f(\boldsymbol{\theta})/g(\boldsymbol{\theta})|\mathbf{1}_{A}))$
$\displaystyle=$
$\displaystyle\mbox{Var}\left(\begin{array}[]{cl}\mbox{E}_{A}(f(\boldsymbol{\theta})/g(\boldsymbol{\theta}))&\mbox{
if }\boldsymbol{\theta}\in A\\\ 0&\mbox{ otherwise}\\\ \end{array}\right)$
$\displaystyle+\,\mbox{E}\left(\begin{array}[]{cl}\mbox{Var}_{A}(f(\boldsymbol{\theta})/g(\boldsymbol{\theta}))&\mbox{
if }\boldsymbol{\theta}\in A\\\ 0&\mbox{ otherwise}\\\ \end{array}\right)$
$\displaystyle=$ $\displaystyle\mbox{Var}\left(\begin{array}[]{cl}1/q&\mbox{
if }\boldsymbol{\theta}\in A\\\ 0&\mbox{ otherwise}\\\
\end{array}\right)+q\mbox{Var}_{A}(f(\boldsymbol{\theta})/g(\boldsymbol{\theta}))$
$\displaystyle=$
$\displaystyle(1/q)^{2}q(1-q)+\frac{1}{q}\mbox{Var}_{A}(f(\boldsymbol{\theta})/qg(\boldsymbol{\theta}))$
$\displaystyle=$
$\displaystyle\frac{1-q}{q}+\frac{1}{q}\mbox{Var}_{A}(f(\boldsymbol{\theta})/g_{A}(\boldsymbol{\theta}))$
We will assume below that $q=\frac{1}{2}$, so that
$\mbox{Var}(f(\boldsymbol{\theta})/g(\boldsymbol{\theta}))=1+2\mbox{Var}_{A}(f(\boldsymbol{\theta})/g_{A}(\boldsymbol{\theta}))$
(11)
### 4.3 Importance Sampling Convergence
With $f$, $g$, and $A$ as defined above, $f$ and $g_{A}$ have the same domain.
Assuming errors in estimating $q$ and normalization errors are of a lesser
order of magnitude, we can treat the histogram heights as being sampled from
$g_{A}$. Suppose the histogram has $l$ bins $\\{B_{j}\\}$, each with width $h$
and based around the points $g_{A}(\boldsymbol{\theta}_{j})$. Then by equation
(5),
$\displaystyle\mbox{Var}_{A}(f(\boldsymbol{\theta})/g_{A}(\boldsymbol{\theta}))$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{l}\int_{B_{j}}\frac{(f(\boldsymbol{\theta})-g_{A}(\boldsymbol{\theta}))^{2}}{g_{A}(\boldsymbol{\theta})}d\boldsymbol{\theta}$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{l}\int_{B_{j}}\frac{(g_{A}(\boldsymbol{\theta})+\nabla
g_{A}(\boldsymbol{\theta})\cdot(\boldsymbol{\theta}_{j}-\boldsymbol{\theta})+O((\boldsymbol{\theta}_{j}-\boldsymbol{\theta})^{2})-g_{A}(\boldsymbol{\theta}))^{2}}{g_{A}(\boldsymbol{\theta})}d\boldsymbol{\theta}$
$\displaystyle=$ $\displaystyle\sum_{j=1}^{l}\int_{B_{j}}\frac{(\nabla
g_{A}(\boldsymbol{\theta})\cdot(\boldsymbol{\theta}_{j}-\boldsymbol{\theta}))^{2}+O((\boldsymbol{\theta}_{j}-\boldsymbol{\theta})^{3})}{g_{A}(\boldsymbol{\theta})}d\boldsymbol{\theta}$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{l}\int_{B_{j}}\frac{\lVert\nabla\cdot
g_{A}(\boldsymbol{\theta})\rVert^{2}h^{2}}{g_{A}(\boldsymbol{\theta})}d\boldsymbol{\theta}$
$\displaystyle=$ $\displaystyle h^{2}\int\frac{\lVert\nabla\cdot
g_{A}(\boldsymbol{\theta})\rVert^{2}}{g_{A}(\boldsymbol{\theta})}d\boldsymbol{\theta}$
Because $h\propto m^{-d}$ where $d$ is the number of dimensions, and $m$ is
the number of samples used to make the histogram,
$\mbox{Var}_{A}(f(\boldsymbol{\theta})/g_{A}(\boldsymbol{\theta}))\leq
Cm^{-2/d}\\\ $
where $C\propto\int\frac{\lVert\nabla\cdot
g_{A}(\boldsymbol{\theta})\rVert^{2}}{g_{A}(\boldsymbol{\theta})}d\boldsymbol{\theta}$.
Putting this together with (11), we get
$\mbox{Var}(I_{n})=\mbox{Var}(p(\boldsymbol{x}|T)/A_{n})=n^{-1}(1+O(Cm^{-2/d}))$
(12)
## 5 Implementation Issues
### 5.1 Speed of Convergence
The variance of $n^{-1}(1+O(Cm^{-2/d}))$ given by (12) is asymptotically equal
to $n^{-1}$, which is the typical importance sampling rate. In practice
however, the asymptotic results cannot distinguish useful from impractical
estimators. If $Cm^{-2/d}$ is small and
$\mbox{Var}(p(\boldsymbol{x}|T)/A_{n})\approx n^{-1}$, then
$p(\boldsymbol{x}|T)$ can be approximated in only 1000 samples to about
$6\%=\frac{1.96}{\sqrt{1000}}$ with 95% confidence. For many theory choice
purposes, this is quite sufficient. Thus in typical problem cases the factor
of $Cm^{-2/d}$ will be very significant. If $Cm^{-2/d}\gg 1$, then the
convergence rate may in practice be similar to $n^{-1}m^{-2/d}$. Compare this
to the rate of $n^{-1}m^{-4/(4+d)}$ for the methods proposed by Zhang and
Neddermeyer.
This method also uses simple histograms, instead of a more sophisticated
density estimation method (Zhang uses kernel estimation, Neddermeyer uses
linear blend frequency polynomials). Although simple histograms converge
slower for large $d$ as shown above, they are much faster to compute for large
$d$.
Neddermeyer’s LBFP algorithm is quite efficient compared to Zhang’s, but its
running time is $O(2^{d}d^{2}n^{\frac{d+5}{d+4}})$. $d$ is a constant for any
fixed problem, but if, say, $d=10$, then the dimensionality constant
multiplies the running time by $2^{10}10^{2}\approx 10^{5}$.
By contrast, this paper’s method takes only $O(dm\mbox{log}(m))$ time to
construct the initial histogram, and an additional $O(dn\mbox{log}(m))$ time
to do the importance sampling. The main reason for the difference is that
querying a simple histogram can be done in $\mbox{log}(m)$ time by computing
the bin coordinates and looking up the bin’s height in a tree structure.
However, querying a LBFP requires blending all nearby bins and is thus
exponential in $d$.
### 5.2 When $g=0$
Our discussion assumed that
$g(\boldsymbol{\theta})=p(\boldsymbol{\theta}|\boldsymbol{x},T)$ was always
positive. If $g$ goes to 0 where the histogram is positive, the variance of
$A_{n}^{-1}$ will be infinite. However, this paper’s method can still be used
if $g(\boldsymbol{\theta})$ is 0 over some well-defined area.
For instance, suppose one dimension $\theta_{k}$ of $p(\boldsymbol{\theta}|T)$
is defined by a gamma distribution, so that $p(\theta_{k}|T)=0$ if and only if
$\theta_{k}\leq 0$. Then we can ensure the variance is not infinite by
checking that the histogram is only defined where $\theta_{k}>\epsilon>0$ for
some fixed $\epsilon$.
The margLikArrogance package contains a simple mechanism to do this. The user
may specify a range along each dimension of $\boldsymbol{\theta}$ where it is
known that $g>0$. If the histogram is non-zero outside of this range, the
method aborts with an error.
Note that the variance of the estimator increases with
$\int\frac{\lVert\nabla\cdot
g_{A}(\boldsymbol{\theta})\rVert^{2}}{g_{A}(\boldsymbol{\theta})}d\boldsymbol{\theta}$.
In practice the estimator will work well only when $g$ doesn’t go to 0 too
quickly where the histogram is positive. In these cases the histogram will be
defined well away from any region where $g=0$ and infinite variance won’t be
an issue even if $g=0$ somewhere.
### 5.3 Bin Shape
Cubic histogram bins were used above—their widths were fixed at $h$ in each
dimension. Although the asymptotic results aren’t affected by the shape of
each bin, for usable convergence rates the bins’ dimensions need to compatible
with the shape of the high probability region of
$p(\boldsymbol{\theta}|\boldsymbol{x},T)$. Unfortunately, it is difficult to
determine the best bin shapes.
The margLikArrogance package contains a simple workaround: by default the
distribution is first scaled so that the sampled standard deviation along each
dimension is constant. This is equivalent to setting each bin’s width by
dimension in proportion to that dimension’s standard deviation. If this simple
rule of thumb is insufficient, the user can scale the sampled values of
$p(\boldsymbol{\theta}|\boldsymbol{x},T)$ manually (and make the corresponding
adjustment to the estimate $A_{n}$).
## 6 Conclusion
This paper has described an “arrogance sampling” technique for computing the
marginal likelihood or Bayes factor of a Bayesian model. It involves using
samples from the model’s posterior parameter distribution along with the
scaled values of the distribution’s density at those points. These samples are
divided into two main groups: $m$ samples are used to build a histogram; $n$
are used to importance sample the histogram using the posterior parameter
distribution as the proposal.
This method is simple to implement and runs quickly in
$O(d(m+n)\mbox{log}(m))$ time. Its asymptotic convergence rate,
$n^{-1}(1+O(Cm^{-2/d}))$, is not remarkable, but in practice convergence is
fast for many problems. Because the required inputs are similar to those of
the harmonic mean estimator, it may be a convenient replacement for it.
## 7 References
1. 1.
S. Chib. “Marginal Likelihood from the Gibbs Output” _Journal of the American
Statistical Association_. Vol 90, No 432. (1995)
2. 2.
S. Chib and I. Jeliazkov. “Accept-reject Metropolis-Hastings sampling and
marginal likelihood estimation” _Statistica Neerlandica_. Vol 59, No 1. (2005)
3. 3.
A. Gelman and X. Meng. “Simulating Normalizing Constants: From Importance
Sampling to Bridge Sampling to Path Sampling” _Statistical Science_. Vol 13,
No 2. (1998)
4. 4.
R. Kass and A. Raftery. “Bayes Factors” _Journal of the American Statistical
Association_. Vol 90, No 430. (1995)
5. 5.
R. Neal. “The Harmonic Mean of the Likelihood: Worst Monte Carlo Method Ever”.
Blog post, http://radfordneal.wordpress.com/2008/08/17/the-harmonic-mean-of-
the-likelihood-worst-monte-carlo-method-ever/. (2008)
6. 6.
J. Neddermeyer. “Computationally Efficient Nonparametric Importance Sampling”
_Journal of the American Statistical Association_. Vol 104, No 486. (2009)
arXiv:0805.3591v2
7. 7.
A. Owen and Y. Zhou. “Safe and effective importance sampling” _Journal of the
American Statistical Association_. Vol 95, No 449. (2000)
8. 8.
C. Robert and D. Wraith. “Computational methods for Bayesian model choice”
arXiv:0907.5123v1
9. 9.
P. Zhang. “Nonparametric Importance Sampling” _Journal of the American
Statistical Association_. Vol 91, No 435. (1996)
10. 10.
M. Zlochin and Y. Baram. “Efficient Nonparametric Importance Sampling for
Bayesian Inference” _Proceedings of the 2002 International Joint Conference on
Neural Networks_ 2498–2502. (2002)
|
arxiv-papers
| 2011-01-06T04:30:27 |
2024-09-04T02:49:16.168435
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Benedict Escoto",
"submitter": "Benedict Escoto",
"url": "https://arxiv.org/abs/1101.1136"
}
|
1101.1193
|
Dedicated to
Alexei Sisakian
co-author $\&$ friend
On the connection between quantum and classical descriptions
J.Manjavidze
Joint Institute for Nuclear Research
(VBLHEP & LNP, Dubna, Russia)
$\&$
Andronikashvili Institute of Physics
(LThP, Tbilisi State University, Georgia),
Tel: (09621) 6 35 17, Fax: (09621) 6 66 66, E-mail: joseph@jinr.ru
###### Abstract
The review paper presents generalization of d’Alembert’s variational
principle: the dynamics of a quantum system for an external observer is
defined by the exact equilibrium of all acting in the system forces, including
the random quantum force $\hbar j$, $\forall\hbar$. Spatial attention is
dedicated to the systems with (hidden) symmetries. It is shown how the
symmetry reduces the number of quantum degrees of freedom down to the
independent ones. The sin-Gordon model is considered as an example of such
field theory with symmetry. It is shown why the particles $S$-matrix is
trivial in that model.
###### Contents
1. 1 Introduction
2. 2 Simplest examples
1. 2.1 Introduction
2. 2.2 The generalized stationary-phase method
3. 2.3 Complex trajectories
4. 2.4 Conclusions
3. 3 Path integrals on Dirac measure
1. 3.1 Introduction
2. 3.2 Canonical transformation
3. 3.3 Selection rule
4. 3.4 Coordinate transformation
5. 3.5 Conclusions
4. 4 Reduction of quantum degrees of freedom
1. 4.1 Introduction
2. 4.2 Unitary definition of the path-integral measure
3. 4.3 Perturbation theory on the cotangent manifold
4. 4.4 Cancelation of angular perturbations
5. 4.5 Conclusions
5. 5 Example: H-atom
1. 5.1 Introduction
2. 5.2 Mapping
3. 5.3 Reduction
4. 5.4 Perturbations
5. 5.5 Conclusions
6. 6 Example: sin-Gordon model
1. 6.1 Introduction
2. 6.2 Reduction procedure
3. 6.3 Quantum corrections
7. 7 Summary
8. 8 Conclusion
Preface
Present paper is the review of the works which was done after my first paper
[1]. I returned from time to time to the idea [1] that it seems interesting to
embed the total probability conservation condition into the quantum field
theory formalism and discuss it with Alexei Sissakian during our team-work. It
seems that this suggestion is unnecessary noting that the $S$-matrix is the
unitary operator and it is not evident why this attempt can give something
new. But it turns out that exist the correspondence among quantum theory and
classics which is independent from the value of quantum corrections. Besides
this new quantum field theory is free from divergences and the value of
quantum corrections ingenuously depend on the topology of classical field. All
that is new from the point of view of ordinary theory and at last Alexei
Sissakian propose to write on paper all result in details. Present
introductory paper devoted to simplest examples and more interesting field
theory models will be published later.
## 1 Introduction
The basis of the method of calculations is the following [1]. The $S$-matrix
unitarity condition, $S^{\dagger}S=1$, in terms of amplitudes, $S=1+iA$, looks
as follows:
$iA^{\dagger}A=(A-A^{\dagger}).$ (1\. 1)
The nonlinearity of this equality points on existence of the cancelations
mechanism (of the real part of amplitude) which reduces quadratic form down
the linear one. Our purpose is to show how this reduction removes the
”unwanted” contributions111This means that the theory must be formulated
directly in terms of probability. But notice that it is the frequently used
method of particle physics. For example, one must integrate over unobserved
final state in the inclusive approach to the multiple production phenomena.
Another example: describing the very high multiplicity (VHM) processes the
number of produced particles $n$ must be considered as the dynamical
parameter. In the frame of $S$-matrix thermodynamics, where the ”rough”
description of final state is used, one must also integrate over final
particles momenta. In all cases one must consider quantities $\sim|A|^{2}$
directly, where $A$ is the corresponding amplitude..
One may consider the simplest vacuum-into-vacuum transition probability,
$|Z|^{2}$, as the main quantity, where $Z$ is the functional integral over
fields,
$Z=\int D\varphi~{}e^{iS(\varphi)},~{}D\varphi=\prod_{x}d\varphi(x).$ (1\. 2)
One may include into the action, $S$, also the linear over field $\varphi$
term,
$\int dxJ(x)\varphi(x)$ (1\. 3)
to describe production of particles. We will assume on the early stages that
$J=0$. Then the vacuum-into-vacuum transition probability
$|Z|^{2}=\int
D\varphi_{+}D\varphi^{*}_{-}~{}e^{iS(\varphi_{+})-iS^{*}(\varphi_{-})},$ (1\.
4)
where $\varphi_{+}$ and $\varphi_{-}$ are completely independent fields.
It can be shown that Eq. (1\. 1) means that a reduced form must also exist
[1]:
$|Z|^{2}=\lim_{j=e=0}e^{i\hat{\mathbb{K}}(j,e)}\int DMe^{iU(\varphi,e)},$ (1\.
5)
where $\hat{\mathbb{K}}=\hat{\mathbb{K}}(j,e)$ is a definite differential
operator over $j(x)$ and $e(x)$, see the examples (2\. 42), (6\. 7). The
expansion of $\exp\\{i\hat{\mathbb{K}}\\}$ generates perturbation series. The
functional $U(\varphi,e)$ introduces interaction among quantum degrees of
freedom and the integral measure is $\delta$-functional:
$DM=\prod_{x}d\varphi(x)\delta\left(\frac{\delta
S(\varphi)}{\delta\varphi(x)}+\hbar j(x)\right).$ (1\. 6)
Sometimes the $\delta$-like measure [2] is called in mathematical literature
as the ”Dirac measure”. It follows from (1\. 6) that
— the quantum system for an externa observer looks like classical which is
excited by the external random force $\hbar j,~{}\forall\hbar$.
The established generalized correspondence principle222This formulation of the
principle was offered by A.Sisakian. is the main consequence of Eq. (1\. 1).
Therefore the complete set of acceptable field states for external
observer333Since the ”probability” is considered is known having (1\. 6).
It is important that the restricted problem is considered. We will calculate
the imaginary part of amplitude believing that it will be sufficient for us.
In this case the unmeasurable phase of amplitude stay undefined444 Therewith
why must the calculations of unnecessary, i.e. unmeasurable, phase be
performed? Just in this sense the unitarity condition (1\. 1) is a necessary
one. It says that the real part is the ”unwanted” part of the amplitude.. A
main mathematical problem in the searching representation (1\. 5) is to find
the way how to find the imaginary part from the modulo square of amplitude. To
be more precise, we will find the imaginary part as a result of cancelation of
”unwanted” contribution in the modulo square of amplitude.
The $\delta$-function (1\. 6) solves the problem of definition of
contributions into the path integral but can not solve the problem completely
since the action of operator $\hat{\mathbb{K}}$ remains unknown. It must be
noted that $\exp\\{i\hat{\mathbb{K}}\\}$ generates the asymptotic series
ordinary in quantum theories [3] and it seems that $\delta$-like measure gives
nothing new555Looking at the approach from the point of view of the stationary
phase methods. In other words, one can think that the present approach gives
nothing new to the Bohr’s correspondence principle.. But this is not entirely
so. I would like draw attention to the appearance of source of quantum
excitations $\hbar j$ in the r.h.s. of classical Lagrange equation, i.e. the
changes of l.h.s. in equation of motion leads to the change of $j$. It is
crucially important that (1\. 6) is rightful independently from the value of
$\hbar$.
The theory defined on the Dirac measure (1\. 6) for this reason has quite
unexpected properties, e.g. allows to perform transformation of the path
integral variables. So, it will be shown that in theories with symmetry the
reduced form of representation (1\. 5) exists:
$|Z|^{2}=\lim_{j=e=0}e^{i\hat{\mathbb{K}}(j,e)}\int
DM(j)e^{iU(\varphi_{c},e)},$ (1\. 7)
where $\hat{\mathbb{K}}$ is again the perturbations generating operator and
$U$ introduces interactions. Note that $\hat{\mathbb{K}}$ and $U$ in (1\. 7)
depends on the sets $\\{j_{\xi_{k}},j_{\eta_{k}}\\}$,
$\\{e_{\xi_{k}},e_{\xi_{k}}\\}$ of new variables. One must take this auxiliary
variables equal to zero at the very end of calculations. At the same time the
transformed measure $DM$ is again $\delta$-like:
$DM=\prod_{k}\prod_{t}d\xi_{k}(t)d\eta_{k}(t)\times$
$\times\delta\left(\dot{\xi}_{k}(t)-\frac{\delta
h}{\delta\eta_{k}(t)}-j_{\xi_{k}}(t)\right)\delta\left(\dot{\eta}_{k}(t)+\frac{\delta
h}{\delta\xi_{k}(t)}+j_{\eta_{k}}(t)\right),$ (1\. 8)
where $t$ is the time variable and $h=h(\eta)$ is the transformed Hamiltonian:
$h(\eta)=H(\varphi_{c}),$ (1\. 9)
where $\varphi_{c}=\varphi_{c}({\mathbf{x}};\xi,\eta)$ is given solution of
Lagrange equation at $j=0$.
The formulae (1\. 8) is the main result. Therefore, as it follows from it the
problem of the quantum field theory with symmetry is reduced down to quantum
mechanical one, with potential defined by $\varphi_{c}$.
(A) The Dirac measure (1\. 6) prescribes that $|Z|^{2}$ is defined by the
$sum$ of $strict$ solutions of equation of motion:
$\frac{\delta S(\varphi)}{\delta\varphi(x)}=\hbar j(x),$ (1\. 10)
in vicinity of $j=0$, i.e. by definition Eq. (1\. 10) must be solved expanding
the solution over $j$ 666It should be noted that it may be that the limit
$j=0$ is absent. For example it may happen if the system is unstable against
symmetry breaking. This important possibility will not be considered in
present paper.. Following the ordinary rule we obviously leave the
contribution which ensures the minimal vacuum energy. On the other hand,
having theory on Dirac measure, which calls for the complete set of
contributions, we have offered another selection rule in our dynamic theory of
$S$-matrix. Namely, we simply propose777This selection rule is used widely in
classical mechanics, see e.g. formulation of Kolmogorov-Arnold-Mozer (KAM)
theorem [4]. that
— the largest terms in the sum over solutions of (1\. 10) are significant from
the physics point of view.
To be more precise, this selection rule means that if $G$ is the symmetry of
action and $TG^{*}$ is the symmetry of the extremum of the action, then in the
situation of a general position only the trajectories with the highest
dimension factor group, $(G/TG^{*})$, are sufficient.
It will be seen that this kind of definition of the ”ground state” extracts
the maximally ”feeling” symmetry contributions since other ones will be
realized on a zero measure, or, more precisely, only maximal symmetry breaking
field configurations, $\varphi_{c}$, are most probable. We will call such
solution of the problem the field theory with symmetry. It is the main formal
distinction of present approach.
It is important here that the zero width of $\delta$-function excludes the
interference among contributions from various trajectories. Therefore the
formalism naturally takes into account the orthogonality of Hilbert spaces
built on various trajectories. This is achieved through the special boundary
conditions in the frame of which the total action of the product
$Z\cdot Z^{*}=<in|out>\cdot<out|in>$
always describes a closed path, i.e. the necessary for d’Alembert variational
principle time reversible motion. It points to the necessity to be careful
with boundary conditions in a considered formalism888 The necessity to count
all possible boundary conditions of a given problem was mention to author by
L.Lipatov..
(B) The Dowker’s theorem [5] insists that the semiclassical approximation to
be exact for path integrals on the simple Lie group manifolds. For this reason
one can expect that the quantum-mechanical problems, as well as the field-
theoretical ones, may be at least transparent to the symmetry manifolds.
However we know how to construct correctly the path integral formalism only in
the restricted case of canonical variables [6]. At first sight the path
integrals in terms of generalized coordinates can be defined through the
corresponding transformation. But there is an opinion that it is impossible to
perform the transformation of path-integral variables: the naive
transformation of coordinates give wrong results because of their stochastic
nature in quantum theories999One can find corresponding examples in [6, 7].
The mostly popular method of transformation of the path-integral variables is
a ”time-sliced” method [8], which induces corrections to the interaction
Lagrangian proportional at least to $\hbar^{2}$ [9], i.e. the problem of
transformation is of a quantum nature. For this reason the usage of the ”time-
sliced” method in general case is cumbersome, see also [10].. That is why such
general principle as the conservation of total probability (1\. 1) should play
important role. Indeed, it is evident that $\delta$-like Dirac measure (1\. 6)
allows to perform arbitrary transformation [1] just as in the classical
mechanics.
Therefore, the theory on Dirac measure straight away leads to the new for
quantum field theory selection rule and latter one gives the theory with
symmetry. All this is attained by transition to the appropriate variables,
$(\xi,\eta)\in W$ in our notations. The last circumstance means that we go
away from ordinary spectral analysis of quantum fluctuations to the
description of the classical trajectories topology conserving deformations,
since $\varphi_{c}=\varphi_{c}({\mathbf{x}};\xi,\eta)$ is given, of symmetry
manifold, $W$ 101010It will be seen from our selection rule that the measure
on which particles mechanics is realized is equal to zero in the field
theories with symmetry.. It must be underlined that our method of
transformations is rightful for arbitrary case, i.e. not only for simple Lie
group manifolds, where the semiclassical approximation is exact.
Next, the dimensions of initial phase space of field and of the transformed
space of independent degrees of freedom, i.e. of the symmetry manifold, will
not coincide. That means that the mapping to the independent degrees of
freedom, $(\xi,\eta)$, will be singular. For this reason the transformation
$\varphi_{c}:\varphi\rightarrow(\xi,\eta)$
will be irreversible and the notion of particle should be considered as the
wrong idea of quantum field theory with symmetry 111111Considering gluon
production in the frame of Yang-Mills field theory with symmetry the
conclusion that gluons can not be created should be confirmed by direct
calculations, taking into account also the quark fields. That was mentioned to
the author by P.Culish and will be shown in later publications. It is
noticeable that the mapping in quantum mechanics is not singular and for this
reason both representations before and after transformation have the equal
status..
(C) It will be shown that the result of action of the operator
$\exp\\{i\hat{\mathbb{K}}\\}$ for transformed theories may be expressed as the
sum of contributions on all boundaries $\partial W$:
$|Z|^{2}=|Z|^{2}_{sc}+\sum_{k}\int
d\xi_{k}(0)\frac{\partial}{\partial\xi_{k}(0)}C_{\xi}+\sum_{k}\int
d\eta_{k}(0)\frac{\partial}{\partial\eta_{k}(0)}C_{\eta}$ (1\. 11)
where the first term presents a semiclassical contribution and $C_{\xi}$,
$C_{\eta}$ contains quantum corrections. This result shows that the quantum
corrections greatly depend on the topology of classical trajectory.
This important observation solves a number of problems. For instance, it is
known that the Coulomb trajectory is closed because of Bargman-Fock symmetry,
independent from the initial conditions. For this reason the corrections on
$\partial W$ of Coulomb problem are canceled and the H-atom problem is pure
semiclassical. We will find the same for sine-Gordon model [11] as the
consequence of mapping on the Arnold’s hypertorus [12].
It is extremely important to keep in mind that the symmetry constraints can
not be taken into account perturbatively over the interaction constant, $g$.
Indeed, we will see below that the expansion in $polinomial$ theories with
symmetry is performed in terms of the inverse interaction constant, $1/g$. It
points to the absence of the weak-coupling limit in such theories.
In the end our present aim is
— to find representation (1\. 5);
— to investigate the main properties of theory defined on the Dirac measure
(1\. 6);
— to investigate the structure of perturbation theory generated by operator
$\hat{\mathbb{K}}$ on the measure (1\. 8);
— to find particles production probabilities for theories with symmetry.
I understand that the perturbations scheme in terms of new variables,
especially in theories with symmetry, is outside the habitual one121212See
[13, 14, 15] and for this reason the approach will be describe in more
details, giving step-by-step the properties of a new quantization scheme by
appropriate examples. I think that such non-formal scheme of the description
is much more transparent, although the text may contain reiterations with the
used method of description far from completeness.
## 2 Simplest examples
### 2.1 Introduction
It it has mentioned above a technical aspect of our idea is the suggestion to
calculate the probability without the intermediate step of calculations of the
amplitudes. In present Section we restrict ourselves to the simplest problem -
to the motion of a particle in a potential $V(x)$.
Let the amplitude $A(x_{2},T;x_{1},0)$ describes the motion of the particle
from the point $x_{1}$ to the point $x_{2}$ during the time $T$. Using the
spectral representation:
$A(x_{2},T;x_{1},0)=\sum_{n}\psi_{n}(x_{2})\psi_{n}^{*}(x_{1})e^{iE_{n}T},$
(2\. 1)
we have for probability:
$W(x_{2},T;x_{1},0)=\sum_{n_{1},n_{2}}\psi_{n_{1}}(x_{2})\psi_{n_{1}}^{*}(x_{1})\psi_{n_{2}}^{*}(x_{2})\psi_{n_{2}}(x_{1})e^{i(E_{n_{1}}-E_{n_{2}})T}.$
(2\. 2)
Taking into account the ortho-normalizability condition:
$\int dx\psi_{n}(x)\psi_{m}^{*}(x)=\delta_{n,m},$ (2\. 3)
the total probability:
$\int dx_{2}dx_{1}W(x_{2},T;x_{1},0)=\sum_{n}\delta_{n,n}=\Omega$ (2\. 4)
is the time independent quantity which coincides with the number of existing
physics states. Therefore, the amplitude (2\. 1) is time dependent, but the
total probability (2\. 4) is not. This means that the time is the unwanted
parameter from the point of view of experiment described by the probability
(2\. 4). Notice also the role of boundary condition (2\. 3).
The quantity (2\. 4) is of no interest to experiment. Much more interesting
the probability $\rho(E)$, where $E$ is the energy experimentally measured.
The Fourier transform of $A(x_{2},T;x_{1},0)$ with respect to $T$
$a(x_{2},x_{1};E)=\sum_{n}\frac{\psi_{n}(x_{2})\psi_{n}^{*}(x_{1})}{E-(E_{n}+i\varepsilon)}$
(2\. 5)
leads to the probability
$\omega(x_{2},x_{1};E)=|a(x_{2},x_{1};E)|^{2}=\sum_{n_{1},n_{2}}\frac{\psi_{n_{1}}(x_{2})\psi_{n_{1}}^{*}(x_{1})}{E-(E_{n_{1}}+i\varepsilon)}\frac{\psi_{n_{2}}^{*}(x_{2})\psi_{n_{2}}(x_{1})}{E-(E_{n_{2}}-i\varepsilon)}$
(2\. 6)
and the total probability:
$\rho(E)=\int
dx_{1}dx_{2}\omega(x_{2},x_{1};E)=\sum_{n}\left|\frac{1}{E-E_{n}-i\varepsilon}\right|^{2}=$
$=\frac{1}{\varepsilon}\sum_{n}{\rm
Im}\frac{1}{E-E_{n}-i\varepsilon}=\frac{\pi}{\varepsilon}\sum_{n}\delta(E-E_{n}).$
(2\. 7)
The total probability $\rho(E)$ again coincides with number of existing states
but for all that it is seen that the unphysical, i.e. needless, states from
the point of view of measurement with $E\neq E_{n}$ was canseled131313Such
contributions enter into the real part of $a(x_{2},x_{1};E)$..
Let us use now the proper-time representation:
$a(x_{1},x_{2};E)=\sum_{n}\Psi_{n}(x_{1})\Psi^{*}_{n}(x_{2})i\int^{\infty}_{0}dTe^{i(E-E_{n}+i\varepsilon)T}$
(2\. 8)
to see the integral form of cancelation of unwanted contributions and insert
it into definition of total probability ($\varepsilon\rightarrow+0$):
$\rho(E)=\int
dx_{1}dx_{2}|a(x_{1},x_{2};E)|^{2}=\sum_{n}\int^{\infty}_{0}dT_{+}dT_{-}e^{-(T_{+}+T_{-})\varepsilon}e^{i(E-E_{n})(T_{+}-T_{-})}.$
(2\. 9)
We will introduce new time variables instead of $T_{\pm}$:
$T_{\pm}=T\pm\tau,$ (2\. 10)
where, as it follows from Jacobian of transformation, $|\tau|\leq T,~{}0\leq
T\leq\infty$. But we can put $|\tau|\leq\infty$ since $T\sim
1/\varepsilon\rightarrow\infty$ is essential in integral over $T$. As a
result,
$\rho(E)=4\pi\sum_{n}\int^{\infty}_{0}dTe^{-2\varepsilon
T}\int^{+\infty}_{-\infty}\frac{d\tau}{\pi}e^{2i(E-E_{n})\tau}=\frac{\pi}{\varepsilon}\sum_{n}\delta(E-E_{n}).$
(2\. 11)
In the last integral all contributions with $E\neq E_{n}$ has been canceled
and only the acceptable from physics point of view contributions with
$E=E_{n}$ has survived. This peculiarity of considered interference phenomena
which is the consequence of unitarity condition, i.e. its ability to extract
only physics states, would have the significant applications.
Note also that the product of amplitudes $a\cdot a^{*}$ was ”linearized” after
introduction of ”virtual” time $\tau=(T_{+}-T_{-})/2$, i.e. after
transformation (2\. 10) we start calculation of the imaginary part. The
meaning of such variables will be discussed also in Sec.2.2.
### 2.2 The generalized stationary-phase method
1. 0-dimensional model
Let us practise considering the ”$0$-dimensional” integral:
$A=\int^{+\infty}_{-\infty}\frac{dx}{(2\pi)^{1/2}}e^{i(\frac{1}{2}ax^{2}+\frac{1}{3}bx^{3})},$
(2\. 12)
with ${\rm Im}a\rightarrow+0$ and $b>0$. This example is useful since it
allows to illustrate practically all technical tricks of the approach.
We want to compute the ”probability”
$R=|A|^{2}=\int^{+\infty}_{-\infty}\frac{dx_{+}dx_{-}}{2\pi}e^{i(\frac{1}{2}ax_{+}^{2}+\frac{1}{3}bx_{+}^{3})-i(\frac{1}{2}a^{*}x_{-}^{2}+\frac{1}{3}bx_{-}^{3})}.$
(2\. 13)
New variables:
$x_{\pm}=x\pm e$ (2\. 14)
will be introduced to find out the cancelation phenomenon. In result:
$R=\int^{+\infty}_{-\infty}\frac{dxde}{\pi}e^{-2(x^{2}+e^{2}){\rm
Im}a}e^{2i({\rm Re}a\;x+2bx^{2})e}e^{2i\frac{b}{3}e^{3}},$ (2\. 15)
where the prescription that ${\rm Im}a\rightarrow+0$ has been used. Note that
integrations are performed along the real axis.
We will compute the integral over $e$ perturbatively. For this purpose the
transformation:
$F(e)=\lim_{j=e^{\prime}=0}e^{\frac{1}{2i}\hat{j}\hat{e}^{\prime}}e^{2ije}F(e^{\prime}),$
(2\. 16)
which is valid for any differentiable function, will be used. In (2\. 16) two
auxiliary variables $j$ and $e^{\prime}$ has been introduced and the ”hat”
symbol means the differential over corresponding quantity:
$\hat{j}=\frac{\partial}{\partial
j},\;\;\;\hat{e^{\prime}}=\frac{\partial}{\partial e^{\prime}}.$ (2\. 17)
The auxiliary variables must be taken equal to zero at the very end of
calculations.
Choosing
$\ln F(e)=-2e^{2}{\rm Im}a+2i\frac{b}{3}e^{3}$ (2\. 18)
we will find:
$R=\lim_{j=e=0}e^{\frac{1}{2i}\hat{j}\hat{e}}\int^{+\infty}_{-\infty}dxe^{-2(x^{2}+e^{2}){\rm
Im}a}e^{2i\frac{b}{3}e^{3}}\delta({\rm Re}a~{}x+bx^{2}+j).$ (2\. 19)
Therefore, the destructive interference among two exponents in the product
$a\cdot a^{*}$ unambiguously determines both integrals, over $x$ and over $e$.
The integral over difference $e=(x_{+}-x_{-})/2$ gives $\delta$-function and
then this $\delta$-function defines the contributions in the last integral
over $x=(x_{+}+x_{-})/2$. Following the definition of $\delta$-function only a
strict solutions of equation
${\rm Re}a\;x+bx^{2}+j=0$ (2\. 20)
gives the contribution into $R$.
But one can note that this is not the complete solution of the problem: the
expansion of operator exponent $\exp\\{\frac{1}{2i}\hat{j}\hat{e}\\}$
generates the asymptotic series. Note also that it is impossible to remove the
source, $j$, dependence (only harmonic case, $b=0$, is free from $j$).
The equation (2\. 20) at $j=0$ has the solutions, at $x_{1}=0$ and at
$x_{2}=-a/b$. Performing trivial transformation $e\rightarrow ie$,
$\hat{e}\rightarrow-i\hat{e}$ of auxiliary variable we find at the limit ${\rm
Im}a=0$ that the contribution from $x_{1}$ extremum (minimum) has the
expression141414The contribution of $x_{2}$ leads to divergent series.:
$R=\frac{1}{a}e^{-\frac{1}{2}\hat{j}\hat{e}}(1-4bj/a^{2})^{-1/2}e^{2\frac{b}{3}e^{3}}$
(2\. 21)
and the expansion of an operator exponent gives the asymptotic series:
$R=\frac{1}{a}\sum^{\infty}_{n=0}(-1)^{n}\frac{(6n-1)!!}{n!}\left(\frac{2b^{4}}{3a^{6}}\right)^{n},\;\;\;\;(-1)!!=0!!=1.$
(2\. 22)
This series is convergent in Borel’s sense. Therefore the described
destructive interference has not an action upon the value of perturbation
series convergence radii.
Let us calculate now $R$ using stationary phase method. The contribution from
the minimum $x_{1}$ gives $({\rm Im}a=0)$:
$A=e^{-i\hat{j}\hat{x}}e^{-\frac{i}{2a}j^{2}}e^{i\frac{b}{3}x^{3}}({i}/{a})^{1/2}.$
(2\. 23)
The corresponding “probability” is
$R=\frac{1}{a}e^{-i(\hat{j}_{+}\hat{x}_{+}-\hat{j}_{-}\hat{x}_{-})}e^{-\frac{i}{2a}(j_{+}^{2}-j_{-}^{2})}e^{i\frac{b}{3}(x_{+}^{3}-x_{-}^{3})}.$
(2\. 24)
Introducing new auxiliary variables:
$j_{\pm}=j\pm j_{1},\;\;\;\;x_{\pm}=x\pm e$ (2\. 25)
and, correspondingly,
$\hat{j}_{\pm}=(\hat{j}\pm\hat{j}_{1})/2,\;\;\;\;\hat{x}_{\pm}=(\hat{x}\pm\hat{e})/2$
(2\. 26)
we find from (2\. 24):
$R=\frac{1}{a}e^{-\frac{1}{2}\hat{j}\hat{e}}e^{2\frac{b}{3}e^{3}}e^{\frac{2b}{a^{2}}ej^{2}}$
(2\. 27)
This expression does not coincide with (2\. 21) but it leads to the same
asymptotic series (2\. 22). We may conclude that both considered methods of
calculation of $R$ are equivalent since the Borel’s regularization scheme of
asymptotic series gives a unique result.
The difference between this two methods of calculation is in different
organization of perturbations. So, if $F(e)$, instead of (2\. 18), is chosen
in the form:
$\ln F(e)=-2e^{2}{\rm Im}a+2i\frac{b}{3}e^{3}+2ibx^{2}e,$ (2\. 28)
we may find (2\. 27) straightforwardly.
Therefore, our method has the freedom in choice of (quantum) source
$j$151515This freedom was mentioned firstly by A.Ushveridze.. Indeed, the
transition from perturbation theory with Eq.(2\. 18) to the theory with
Eq.(2\. 28) formally looks like following transformation of the argument of
$\delta$-function:
$\delta(ax+bx^{2}+j)=\lim_{e^{\prime}=j^{\prime}=0}e^{-i\hat{j}^{\prime}\hat{e}^{\prime}}e^{i(bx^{2}+j)e^{\prime}}\delta(ax+j^{\prime}).$
(2\. 29)
Here the transformation (2\. 16) of the Fourier image of $\delta$-function was
used. Inserting Eq.(2\. 29) into (2\. 19) we easily find (2\. 27).
During analytic calculations it will be useful to have a corresponding quantum
sources of the new dynamical variables. Formally this will be done using
transformation (2\. 29). Note that this transformation will not lead to
changing of the Borel’s regularization procedure.
2\. 1-dimensional model
Let us calculate now the probability using the path-integral definition of
amplitudes [1]. Calculating the quantity
$|A|^{2}=\rm<in|out><in|out>^{*}=<in|out><out|in>,$ (2\. 30)
the converging and diverging waves in the product $A\cdot A^{*}$ interfere in
such a way that the continuum of contributions cancel each other. Indeed, the
amplitude
$A(x_{2},T;x_{1},0)=\int^{x(T)=x_{2}}_{x(0)=x_{1}}\frac{Dx}{C_{T}}e^{-iS_{T}(x)},~{}Dx=\prod_{t=0}^{T}\frac{dx(t)}{(2\pi)^{1/2}},$
(2\. 31)
where the action $S_{T}$ is given by the expression:
$S_{T}(x)=\int^{T}_{0}dt\left(\frac{1}{2}~{}\dot{x}^{2}-v(x)\right)$ (2\. 32)
and $C_{T}$ is the standard normalization coefficient:
$C_{T}=\int^{x(T)=x_{2}}_{x(0)=x_{1}}Dxe^{\frac{i}{2}\int^{T}_{0}dt~{}\dot{x}^{2}}$
(2\. 33)
Let us calculate the quantity
$R(x_{2},T;x_{1},0)=\int^{x_{\pm}(T)=x_{2}}_{x_{\pm}(0)=x_{1}}\frac{Dx_{+}}{C_{T}}\frac{Dx_{-}}{C_{T}^{*}}e^{-iS_{T}(x_{+})+iS_{T}(x_{-})}$
(2\. 34)
We assume for simplicity that the integration in (2\. 31) is performed over
real trajectories. Later a general case of complex trajectories will be
considered.
The convergence of functional integral at that is not important. One may
restrict the range of integration for better confidence, or introduce into the
Lagrangian $i\varepsilon$ term, and later remove the restriction in the
expression (2\. 40). It is interesting that the interference phenomena
naturally regularize divergent integrals of (2\. 31) type, accumulating
divergence into $\delta$-function.
In order to take into account explicitly the interference between
contributions of the trajectories $x_{+}(t)$ and $x_{-}(t)$ we shall go over
from the integration over two independent trajectories $x_{+}$ and $x_{-}$ to
the pair $(x,e)$:
$x_{\pm}(t)=x(t)\pm e(t).$ (2\. 35)
It must be stressed that the transformation (2\. 35) is linear and for this
reason may be done in the path integral. Substituting (2\. 35) into (2\. 34)
the argument of the exponent takes the form
$S_{T}(x+e)-S_{T}(x-e)=2\int_{0}^{T}dte(\ddot{x}+v^{\prime}(x))-U_{T}(x,e),$
(2\. 36)
where $U_{T}(x,e)$ is the remainder of the expansion in powers of $e(t)$
($U_{T}=O(e^{3})$). Note that in (2\. 36) we have discarded the ”surface” term
$\int_{0}^{T}dt\partial_{t}(e\dot{x})=e(T)\dot{x}(T)-e(0)\dot{x}(0)=0,$ (2\.
37)
since the boundary points of the trajectories $x_{+}(0)=x_{-}(0)=x_{1}$ and
$x_{+}(T)=x_{-}(T)=x_{2}$ are not varied, i.e.
$e(0)=e(T)=0.$ (2\. 38)
Next,
$Dx_{+}Dx_{-}=JDxDe=2\pi J\prod_{t=0}^{T}dx(t)\prod_{t\neq
0,T}\frac{de(t)}{2\pi},$ (2\. 39)
where $J$ is an unimportant Jacobian of the transformation.
As a result of the replacement (2\. 35) we have
$R(x_{2},T;x_{1},0)=2\pi
J\int^{x(T)=x_{2}}_{x(0)=x_{1}}\frac{Dx}{|C_{T}|^{2}}\int^{e(T)=0}_{e(0)=0}De~{}e^{2i\int_{0}^{T}dte(\ddot{x}+v^{\prime}(x))+U_{T}(x,e)}.$
(2\. 40)
One can make use of the formulae
$e^{iU_{T}(x,e)}=e^{\hat{\mathbb{K}}(e^{\prime},j)}e^{iU_{T}(x,e^{\prime})}e^{-2i\int_{0}^{T}e(t)j(t)dt},$
(2\. 41)
where we have introduced the operator
$\hat{\mathbb{K}}(e,j)=\lim_{e=j=0}\exp\left\\{-\frac{1}{2i}\int_{0}^{T}\frac{\delta}{\delta
j(t)}\frac{\delta}{\delta e(t)}\right\\},$ (2\. 42)
after which from (2\. 40) we have found that
$R(x_{2},T;x_{1},0)=2\pi
Je^{\hat{\mathbb{K}}(e^{\prime},j)}\int^{x(T)=x_{2}}_{x(0)=x_{1}}\frac{Dx}{|C_{T}|^{2}}e^{iU_{T}(x,e^{\prime})}\times$
$\times\int^{e(T)=0}_{e(0)=0}De~{}\exp\left\\{2i\int_{0}^{T}dt(\ddot{x}+v^{\prime}(x)-j)e\right\\}=$
$=2\pi
Je^{\hat{\mathbb{K}}(e,j)}\int^{x(T)=x_{2}}_{x(0)=x_{1}}\frac{Dx}{|C_{T}|^{2}}e^{iU_{T}(x,e)}\prod_{t\neq
0,T}\delta(\ddot{x}+v^{\prime}(x)-j),$ (2\. 43)
where the functional $\delta$-function
$\prod_{t\neq
0,T}\delta(\ddot{x}+v^{\prime}(x)-j)=\int^{e(T)=0}_{e(0)=0}De~{}\exp\left\\{2i\int_{0}^{T}dt(\ddot{x}+v^{\prime}(x)-j)e\right\\}$
(2\. 44)
has arisen as a result of total reduction of unnecessary contributions from
the point of view of equation of motion
$\ddot{x}(t)+V^{\prime}(x)=j(t).$ (2\. 45)
The operator (2\. 42) is Gaussian so that the system is perturbed by the
random force $j(t)$.
If $x(t)$ is the ”true” trajectory and the virtual deviation is $e(t)$ then
the quantity $e(\ddot{x}+v^{\prime}(x)-j)$ coincides with the virtual work. It
must be equal to zero in classical mechanics since only the time reversible
motion is considered. In result we came to equation of motion since $e$ is
arbitrary in classics.
The difference $S_{T}(x_{+})-S_{T}(x_{-})$ in (2\. 34) with boundary
conditions (2\. 38) coincides with the action of reversible motion. Upon the
substitution (2\. 35) we have identified the mean trajectory, $x(t)$, and the
deviation from it, $e(t)$. One must integrate over $e(t)$ in quantum case, in
contrast to classical one. In result the measure of the remaining path
integral over mean trajectory $x(t)$ takes the Dirac $\delta$-function form
which unambiguously chooses the ”true” trajectory.
In other words, the proposed definition of the measure of the path integral is
generalization of classical d’Alambert’s principle on the quantum case. The
theory in the frame of this principle can take into account any external
perturbations, $j(t)$ in our case, if the time reversibility of motion is
conserved. In quantum case the reversibility is established through the
boundary conditions (2\. 38). Next, one may generalize the approach adding
also the probe force which can lead to dynamical symmetry breaking
[16]161616It is important that if the expectation value of the probe force is
not equal to zero then the symmetry is broken. This important possibility will
not be considered in present work..
In the semiclassical approximation $\hat{\mathbb{K}}(e,j)=1$ and taking the
limit $e=j=0$ we find that
$R(x_{2},T;x_{1},0)=2\pi
J\int^{x(T)=x_{2}}_{x(0)=x_{1}}\frac{Dx}{|C_{T}|^{2}}\prod_{t\neq
0,T}\delta(\ddot{x}+v^{\prime}(x)),$ (2\. 46)
Let the solution of the homogeneous equation
$\ddot{x}+v^{\prime}(x)=0$ (2\. 47)
be $x_{c}(t)$, with $x_{c}(0)=x_{1}$ and $x_{c}(T)=x_{2}$. Then
$R(x_{2},T;x_{1},0)=2\pi
J\int^{x(T)=x_{2}}_{x(0)=x_{1}}\frac{Dx}{|C_{T}|^{2}}\prod_{t\neq
0,T}\delta(\ddot{x}+v^{\prime\prime}(x_{c})x),$ (2\. 48)
The remaining integral is calculated by the standard methods171717Here it is
more convenient to represent (2\. 48) as a production of two Gauss integrals;
later on more effective method of calculation of the functional determinant
will be offered.. As a result we find
$R(x_{2},T;x_{1},0)=\frac{1}{2\pi}\left|\frac{\partial^{2}S_{T}(x_{c})}{\partial
x_{c}(0)\partial x_{c}(T)}\right|_{x_{c}(0)=x_{1},x_{c}(T)=x_{2}}.$ (2\. 49)
Next, let us recall that the full derivative of the classical action is
$dS=p_{2}dx_{2}-p_{1}dx_{1},$ (2\. 50)
where $p_{2}$ and $p_{1}$ are, respectively, the final and initial momentum.
Noting this definition,
$\left|\frac{\partial^{2}S_{T}}{\partial x_{1}\partial
x_{2}}\right|dx_{2}=dp_{1},$ (2\. 51)
and in result we find that
$\int
dx_{1}dx_{2}R(x_{2},T;x_{1},0)=\int\frac{dx_{1}dp_{1}}{2\pi}=\Omega^{2},$ (2\.
52)
which coincides with (2\. 4), i.e. it agree with conservation of total
probability since (2\. 52) again coincides with the total number of physical
states.
Deriving (2\. 52) we somewhat simplify the problem considering a unique
solution of Eq.(2\. 47). A more complicate and important examples will be
considered in the next Sections.
### 2.3 Complex trajectories
Let us consider the one dimensional motion with fixed energy $E$ on the
complex trajectory181818The necessity to extend the formalism on the case of
complex trajectories was mention to the author by A.Slavnov.. The
corresponding amplitude has the form:
$A(x_{1},x_{2};E)=i\int^{\infty}_{0}dTe^{iET}\int_{x_{1}=x(0)}^{x_{2}=x(T)}D_{C_{+}}xe^{iS_{C_{+}}(x)},$
(2\. 53)
where the action
$S_{C_{+}}(x)=\int_{C_{+}}dt(\frac{1}{2}\dot{x}^{2}-v(x))$ (2\. 54)
and the measure
$D_{C_{+}}x=\prod_{t\in C_{+}}\frac{dx(t)}{(2\pi)^{1/2}}$ (2\. 55)
are defined on the shifted in the upper half time plane Mills’ contour
$C_{+}=C_{+}(T)$ [17]:
$t\rightarrow t+i\varepsilon,\;\;\;\varepsilon\rightarrow+0,\;\;\;0\leq t\leq
T.$ (2\. 56)
Therefore, we will consider integration over real functions of complex
variables:
$x^{*}(t)=x(t^{*}).$ (2\. 57)
It must be underlined also that the boundary conditions in (2\. 53) have the
classical meaning, i.e. they do not vary, and $x_{1}$, $x_{2}$ are the real
quantities.
The probability looks as follows:
$R(E)=\int^{\infty}_{0}e^{iE(T_{+}-T_{-})}\int^{x_{\pm}(T_{\pm})=x_{2}}_{x_{\pm}(0)=x_{1}}D_{C_{+}}x_{+}D_{C_{-}}x_{-}\times$
$\times e^{iS_{C_{+}(T_{+})}(x_{+})-iS_{C_{-}(T_{-})}(x_{-})},$ (2\. 58)
where $C_{-}(T)=C^{*}_{+}(T)$ is the time contour in the lower half of complex
time plane.
New time variables
$T_{\pm}=T\pm\tau$ (2\. 59)
will be used. Considering ${\rm Im}E\rightarrow+0$ we can consider $T$ and
$\tau$ as the independent variables:
$0\leq T\leq\infty,\;\;\;-\infty\leq\tau\leq\infty.$ (2\. 60)
We will apply the boundary conditions, see (2\. 58):
$x_{1}=x_{+}(0)=x_{-}(0),~{}~{}x_{2}=x_{+}(T_{+})=x_{-}(T_{-}).$ (2\. 61)
Inserting (2\. 59) one can find in zero order over $\tau$ from (2\. 61) that
$x_{+}(0)=x_{-}(0),~{}~{}x_{+}(T)=x_{-}(T),$ (2\. 62)
Now we will introduce also the mean trajectory $x(t)=(x_{+}(t)+x_{-}(t))/2$
and the deviation $e(t)$ from $x(t)$:
$x_{\pm}(t)=x(t)\pm e(t).$ (2\. 63)
We have consider $e(t)$ and $\tau$ as the virtual quantities. The integrals
over $e$ and $\tau$ will be calculated perturbatively. In zero order over $e$
and $\tau$, i.e. in the semiclassical approximation, $x$ is the classical path
and $T$ is the total time of classical motion. Note that one can do surely the
linear transformations (2\. 63) in the path integrals.
The higher terms over $\tau$ put a unphysical constrains on the trajectory
$x(t)$:
$\frac{d^{(2n+1)}x(T)}{dT^{(2n+1)}}=0,~{}n=0,1,2,...,$
since $e(t)$ must be arbitrary. Therefore, to avoid this constraints and since
the boundaries have classical unvaried meaning we will use the minimal
boundary conditions:
$e(0)=e(T)=0,$ (2\. 64)
which ensures the time reversibility. Note that it is sufficient to have (2\.
64) if the integrals over $e(t)$ are calculated perturbatively. At the same
time
$x(0)=x_{1},~{}x(T)=x_{2}.$ (2\. 65)
Let us extract now the linear over $e$ and $\tau$ terms from the closed-path
action:
$S_{C_{+}(T_{+})}(x_{+})-S_{C_{-}(T_{-})}(x_{-})=$ $=-2\tau
H_{T}(x)-\int_{C^{(+)}(T)}dte(\ddot{x}+v^{\prime}(x))-\tilde{H}_{T}(x;\tau)-U_{T}(x,e),$
(2\. 66)
where
$C^{(+)}(T)=C_{+}(T)+C_{-}(T)$ (2\. 67)
is the total-time path, $H_{T}$ is the Hamiltonian:
$2H_{T}(x)=-\frac{\partial}{\partial T}(S_{C_{+}(T)}(x)+S_{C_{-}(T)}(x)),$
(2\. 68)
and
$-\tilde{H}_{T}(x;\tau)=S_{C_{+}(T+\tau)}(x)-S_{C_{-}(T-\tau)}(x)+2\tau
H_{T}(x),$ (2\. 69)
$-U_{T}(x,e)=S_{C_{+}(T)}(x+e)-S_{C_{-}(T)}(x-e)+\int_{C^{(+)}}dte(\ddot{x}+v^{\prime}(x))$
(2\. 70)
are the remainder terms, where $v^{\prime}(x)=\partial v(x)/\partial x$.
Deriving the decomposition (2\. 66) the definition
$C_{-}(T)=C^{*}_{+}(T)$ (2\. 71)
and the boundary conditions (2\. 64) was used.
One can find the compact form of expansion of
$e^{-i\tilde{H}_{T}(x;\tau)-iU_{T}(x,e)}$
over $\tau$ and $e$ using formulae (2\. 16):
$\exp\\{-i\tilde{H}_{T}(x;\tau)-iU_{T}(x,e)\\}=\exp\left\\{\frac{1}{2i}\hat{\omega}\hat{\tau}^{\prime}-i\int_{C^{(+)}(T)}dt\hat{j}(t)\hat{e}^{\prime}(t)\right\\}\times$
$\times\exp\left\\{2i\omega\tau+i\int_{C^{(+)}(T)}dtj(t)e(t)\right\\}\exp\\{-i\tilde{H}_{T}(x;\tau^{\prime})-iU_{T}(x,e^{\prime})\\}.$
(2\. 72)
At the end of calculations the auxiliary variables
$(\omega,\tau^{\prime},j,e^{\prime})$ should be taken equal to zero.
Using (2\. 66) and (2\. 72) we find from (2\. 58) that
$R(E)=2\pi\int^{\infty}_{0}dT\exp\left\\{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dt\hat{j}(t)\hat{e}(t)\right\\}\times$
$\times\int Dx\exp\\{-i\tilde{H}_{T}(x;\tau)-iU_{T}(x,e)\\}\delta(E+\omega-
H_{T}(x))\prod_{C^{(+)}}\delta(\ddot{x}+v^{\prime}(x)-j).$ (2\. 73)
The expansion over the differential operators:
$\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dt\hat{j}(t)\hat{e}(t)=\frac{1}{2i}\left(\frac{\partial}{\partial\omega}\frac{\partial}{\partial\tau}+{\rm
Re}\int_{C+}dt\frac{\delta}{\delta j(t)}\frac{\delta}{\delta e(t)}\right)$
(2\. 74)
will generate the perturbation series. We propose that it is summable in Borel
sense.
The first $\delta$-function in (5\. 33) fixes the conservation of energy:
$E+\omega=H_{T}(x)$ (2\. 75)
where $E$ is the observed energy, $H_{T}(x)$ is the energy at the mean
trajectory at the time moment $T$ and $\omega$ is the energy of quantum
fluctuations. The second $\delta$-function191919Following shorthand entry of
$\delta$-function of the complex argument:
$\prod_{C^{(+)}}\delta(f(t))=\prod_{C_{+}}\delta(f(t))\prod_{C_{-}}\delta(f(t))=\prod_{C_{+}}\delta({\rm
Re}f(t)+i{\rm Im}f(t))\delta({\rm Re}f(t)-i{\rm
Im}f(t))=\prod_{C_{+}}\delta({\rm Re}f(t))\cdot\\\ \delta({\rm Im}f(t))$ will
be useful during calculations. The condition (2\. 57) is important here. The
inessential constant can be canceled by normalization. So, in the result of
analytical continuation of $C_{\pm}$ on the real axis the product of two
$\delta$-functions reduces to single one since $\delta^{2}({\rm
Re}f(x))=\delta(0)\delta({\rm Re}f(x))=\delta(0)\delta(f(x))$ and $\delta(0)$
must be canceled by normalization. Offered abbreviated notation will allow to
consider $\delta$-function on the complex time contour as the ordinary one.
$\prod_{t\in
C^{(+)}}\delta(\ddot{x}+v^{\prime}(x)-j)=(2\pi)^{2}\int\prod_{t\in
C^{(+)}}\frac{de(t)}{\pi}\delta(e(0))\delta(e(T))\times$ $\times e^{-2i{\rm
Re}\int_{C_{+}}dte(\ddot{x}+v^{\prime}(x)-j)}=\prod_{t\in C_{+}(T)}\delta({\rm
Re}(\ddot{x}+v^{\prime}(x)-j))\delta({\rm Im}(\ddot{x}+v^{\prime}(x)-j))$ (2\.
76)
fixes the function $x(t)$ of complex argument on $C^{(+)}$ completely by the
equation
$\ddot{x}+v^{\prime}(x)=j.$ (2\. 77)
The physics meaning of $\delta$-function (2\. 76) was discussed in Sec.2.3
noting that the unitarity condition of quantum theories plays the same role as
the d’Alambert’s variational principle in classical mechanics.
In (2\. 77) $j(t)$ describes the external quantum force. The solution
$x_{j}(t)$ of this equation will be found expanding it over $j(t)$:
$x_{j}(t)=x_{c}(t)+\int dt_{1}G(t,t_{1})j(t_{1})+...$ (2\. 78)
This is sufficient since $j(t)$ is the auxiliary variable202020See also
footnote 15.. In this decomposition $x_{c}(t)$ is the strict solution of
unperturbed equation:
$\ddot{x}+v^{\prime}(x)=0$ (2\. 79)
Note that the functional $\delta$-function in (2\. 76) does not contain the
end-point values of $x(t)$, at $t=0$ and $t=T$. This means that if we
integrate over $x_{1}$ and $x_{2}$ then the initial conditions to the Eq.(2\.
79) are not fixed and the integration over them must be performed.
Inserting (2\. 78) into (2\. 77) we find the equation for Green function:
$(\partial^{2}+v^{\prime\prime}(x_{c}))_{t}G(t,t^{\prime};x_{c})=\delta(t-t^{\prime}).$
(2\. 80)
It is too hard to find the exact solution of this equation if $x_{c}(t)$ is
the nontrivial function of $t$. We will see that the canonical transformation
to the (action-angle)-type variables can help to avoid this problem, see
following Section.
### 2.4 Conclusions
1\. The path integral must be defined on the Mills time contour. This
condition will be important in the field theories with high space-time
symmetries (such as the Yang-Mills type theory) since it seems that for such
theories with symmetry one can not perform surely the analytic continuation
over time variable212121The fact that a theory must satisfy certain conditions
upon analytic continuation over time variable is clear from [18]..
2\. The quantization can be performed without transition to the canonical
formalism, using only the Lagrange one which is more natural for relativistic
field theories.
3\. Only the exact solutions of the equation of motion must be taken into
account defining the contributions into the functional integral.
## 3 Path integrals on Dirac measure
### 3.1 Introduction
In present Section we will offer two methods which may simplify calculation of
path integrals on Dirac measure. They are based on the possibility to perform
transformation of the path-integral variables.
We will consider two examples. In the first example the transformation to the
(action,angle)-type variables will be considered. This example shows how much
the calculations of path integrals may be simplified.
In the second part of present Section the coordinate transformation will be
described. For the sake of definiteness the transformation to cylindrical
coordinates will be considered.
### 3.2 Canonical transformation
Let us introduce the first-order formalism. We will insert in (2\. 73)
$1=\int Dp\prod_{t}\delta(p-\dot{x}).$ (3\. 1)
As a result,
$R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}(\hat{\omega}\hat{\tau}+{\rm
Re}\int_{C_{+}(T)}dt\hat{j}(t)\hat{e}(t))}\int
DxDpe^{-i\tilde{H}_{T}(x;\tau)-iU_{T}(x,e)}\times$ $\times\delta(E+\omega-
H_{T}(x))\prod_{t}\delta\left(\dot{x}-\frac{\partial H_{j}}{\partial
p}\right)\delta\left(\dot{p}+\frac{\partial H_{j}}{\partial x}\right),$ (3\.
2)
where
$H_{j}=\frac{1}{2}p^{2}+v(x)-jx$ (3\. 3)
may be considered as the total Hamiltonian which is time dependent through
$j(t)$. Notice that in present simplest case $x$ and $p$ are independent
parameters and therefore (3\. 3) define the Hamiltonian.
Instead of pare $(x(t),p(t))$ we introduce new pare $(\theta(t),h(t))$
inserting in (3\. 2)
$1=\int\prod_{t}d\theta
dh\delta\left(h-\frac{1}{2}p^{2}-v(x)\right)\delta\left(\theta-\int^{x}dx(2(h-v(x)))^{-1/2}\right).$
(3\. 4)
Note that the integral measures in (3\. 2) and (3\. 4) are both $\delta$-like,
i.e. have the equal power. It allows to change the order of integration and
firstly integrate over $(x,p)$. We find that
$R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}(\hat{\omega}\hat{\tau}+{\rm
Re}\int_{C_{+}(T)}dt\hat{j}(t)\hat{e}(t))}\int D\theta
Dhe^{-i\tilde{H}_{T}(x_{c};\tau)-iU_{T}(x_{c},e)}\times$
$\times\delta(E+\omega-h(T))\prod_{t}\delta\left(\dot{\theta}-\frac{\partial
H_{c}}{\partial h}\right)\delta\left(\dot{h}+\frac{\partial
H_{c}}{\partial\theta}\right),$ (3\. 5)
where
$H_{c}=h-jx_{c}(h,\theta)$ (3\. 6)
is the transformed Hamiltonian and $x_{c}(\theta,h)$ is the given solution of
algebraic equation:
$\theta=\int^{x}dx(2(h-v(x)))^{-1/2},$ (3\. 7)
i.e. $x_{c}$ is the classical trajectory parametrized in terms of $h(t)$ and
$\theta(t)$.
As it follows from (3\. 5) new variables, $h(t)$ and $\theta(t)$, are
subjected to the action of quantum force $j(t)$ and the topology of classical
trajectory $x_{c}$ remains unchanged.
So, instead of Eq.(2\. 77) we must solve the equations:
$\dot{h}=j\frac{\partial
x_{c}}{\partial\theta},\;\;\;\;\;\dot{\theta}=1-j\frac{\partial
x_{c}}{\partial h},$ (3\. 8)
which have a simpler structure. Expanding the solutions over $j$ we will find
the infinite set of recursive equations. This is the important peculiarity of
used quantization scheme.
Note now that $j\partial x_{c}/\partial\theta$ and $j\partial x_{c}/\partial
h$ in the r.h.s. can be considered as the new sources. We will use this
property of Eqs.(3\. 8) and introduce in the perturbation theory new
”renormalized” sources:
$j_{h}=j\frac{\partial
x_{c}}{\partial\theta},\;\;\;\;\;j_{\theta}=j\frac{\partial x_{c}}{\partial
h},$ (3\. 9)
i.e. $j_{\xi}$ and $j_{\eta}$ are the forces on the cotangent bundle. We will
use transformation (2\. 29):
$\prod_{t}\delta(\dot{h}-j\frac{\partial
x_{c}}{\partial\theta})=e^{\frac{1}{2i}{\rm
Re}\int_{C_{+}}dt\hat{j}_{h}(t)\hat{e}_{h}(t)}e^{2i{\rm
Re}\int_{C_{+}}e_{h}j\frac{\partial
x_{c}}{\partial\theta}}\prod_{t}\delta(\dot{h}-j_{h})$ (3\. 10)
and
$\prod_{t}\delta(\dot{\theta}-1+j\frac{\partial x_{c}}{\partial
h})=e^{\frac{1}{2i}{\rm
Re}\int_{C_{+}}dt\hat{j}_{\theta}(t)\hat{e}_{\theta}(t)}e^{2i{\rm
Re}\int_{C_{+}}e_{\theta}j\frac{\partial x_{c}}{\partial
h}}\prod_{t}\delta(\dot{\theta}-1-j_{\theta})$ (3\. 11)
to introduce them. The re-scaling of source $j$ lead to the re-scaling of
auxiliary field $e$. In the new perturbation theory we will have two sources
$j_{h}$, $j_{\theta}$ and two auxiliary fields $e_{h}$, $e_{\theta}$. Notice
that the momentum $p$ never arose.
Inserting (3\. 10), (3\. 11) into (3\. 5) we find:
$R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}(\hat{\omega}\hat{\tau}-i\int_{C^{(+)}}dt(\hat{j}_{h}(t)\hat{e}_{h}(t)+\hat{j}_{\theta}(t)\hat{e}_{\theta}(t)))}\times$
$\times\int DhD\theta
e^{-i\tilde{H}_{T}(x_{c};\tau)-iU_{T}(x_{c},e_{c})}\times$
$\times\delta(E+\omega-h(T))\prod_{t}\delta(\dot{\theta}-1-j_{\theta})\delta(\dot{h}-j_{h}),$
(3\. 12)
where
$e_{c}=e_{h}\frac{\partial x_{c}}{\partial\theta}-e_{\theta}\frac{\partial
x_{c}}{\partial h}$ (3\. 13)
carry the simplectic structure of Hamilton equations of motion and the ”hat”
symbol means differential operator over corresponding quantity. At the very
end one should take all auxiliary variables,
$(e_{h},j_{h},e_{\theta},j_{\theta})$, equal to zero.
Hiding the $x_{c}(t)$ dependence into $e_{c}$ we solve the problem of the
functional determinants, see (3\. 12), and simplify the Hamilton equations of
motion as much as possible:
$\dot{h}(t)=j_{h}(t),\;\;\;\;\;\dot{\theta}(t)=1+j_{\theta}(t)$ (3\. 14)
We will use the boundary conditions
$h(0)=h_{0},\;\;\;\;\theta(0)=\theta_{0},$ (3\. 15)
as the extension of boundary conditions in (2\. 58). This lead to the
following Green function of transformed perturbation theory:
$g(t-t^{\prime})=\Theta(t-t^{\prime}),$ (3\. 16)
with the properties of projection operator:
$\displaystyle\int dtdt^{\prime}g^{2}(t-t^{\prime})=\int
dtdt^{\prime}g(t-t^{\prime}),$ $\displaystyle\int
dtdt^{\prime}g(t-t^{\prime})g(t^{\prime}-t)=0$ (3\. 17)
and, at the same time, we will assume that
$g(0)=1.$ (3\. 18)
It is important to note that ${\rm Im}g(t)$ is regular on the real time axis.
This is the very simplification of the perturbation theory since it eliminates
the doubling of degrees of freedom. One may use here the analytical
continuation to the real time axis.
In result, shifting $C_{+}$ and $C_{-}$ contours on the real time axis we
find:
$R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}(\hat{\omega}\hat{\tau}+\int^{\infty}_{0}dt_{1}dt_{2}\Theta(t_{1}-t_{2})(\hat{e}_{h}(t_{1})\hat{h}(t_{2})+\hat{e}_{\theta}(t_{1})\hat{\theta}(t_{2})))}\times$
$\times\int
dh_{0}d\theta_{0}e^{-i\tilde{H}_{T}(x_{c};\tau)-iU_{T}(x_{c},e_{c})}\delta(E+\omega-
h_{0}+h(T)),$ (3\. 19)
where the solutions of eqs.(3\. 14) was used. In this expression
$x_{c}(t)=x_{c}(h_{0}-h(t),t+\theta_{0}-\theta(t))$ and
$(h(t),e_{h}(t),\theta(t),e_{\theta}(t))$ are the auxiliary fields. At the
very end one must take them equal to zero.
### 3.3 Selection rule
Let us consider the theory with Lagrangian
$L(x)=\frac{1}{2}\dot{x}^{2}-\frac{1}{2}\omega^{2}x^{2}-\frac{g}{4}x^{4}.$
(3\. 20)
The Dirac measure gives the equation (of motion):
$\ddot{x}+\omega^{2}x+gx^{3}=j.$ (3\. 21)
It has two solutions:
$x_{1}(t)=x_{c}(t)+O(j),~{}~{}x_{2}(t)=O(j).$ (3\. 22)
For this reason
$R(E)=R_{1}(E;x_{1})+R_{2}(E;x_{2})$ (3\. 23)
and which one defines $R(E)$ is a question. Following to our selection rule
just $R_{1}$. This will be shown.
Let us return now to the example with Lagrangian (3\. 20). In the
semiclassical approximation
$R_{1}(E;x_{1})=\int_{0}^{\infty}dT\int_{0}^{\infty}dh_{0}\int_{-\infty}^{+\infty}d\theta_{0}e^{-iU_{T}(x_{c},0)}\delta(E-h_{0}).$
(3\. 24)
Therefore,
$R_{1}(E;x_{1})\sim\int_{-\infty}^{+\infty}d\theta_{0}\equiv\Omega,$ (3\. 25)
i.e. it is proportional to the volume of group of time translations.
At the same time
$R_{2}(E;x_{2})=O(1)$ (3\. 26)
in the semiclassical approximation. Therefore,
$R=R_{1}(1+O(1/\Omega)).$ (3\. 27)
This result explains the source of chosen selection rule.
### 3.4 Coordinate transformation
In this section the coordinate transformation of two dimensional quantum
mechanical model with potential
$v=v((x^{2}_{1}+x^{2}_{2})^{1/2})$ (3\. 28)
will be considered. Repeating calculations of previous sections,
$R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dt\hat{\vec{j}}(t)\hat{\vec{e}}(t)}\int
D^{(2)}M(x)e^{-i\tilde{H}_{T}(x;\tau)-iU_{T}(x,e)},$ (3\. 29)
where the $\delta$-like Dirac measure
$D^{(2)}M(x)=\delta(E+\omega-
H_{T}(x))\prod_{t}d^{2}x(t)\delta^{(2)}(\ddot{x}+v^{\prime}(x)-j).$ (3\. 30)
In the classical mechanics the problem with potential (3\. 28) is solved in
the cylindrical coordinates:
$x_{1}=r\cos\phi,\;\;\;\;\;x_{2}=r\sin\phi.$ (3\. 31)
We insert into (3\. 29)
$1=\int
DrD\phi\prod_{t}\delta(r-(x^{2}_{1}+x^{2}_{2})^{1/2})\delta(\phi-\arctan\frac{x_{2}}{x_{1}}).$
(3\. 32)
to perform the transformation. Note that the transformation (3\. 31) is not
canonical. In result we will find a new measure:
$D^{(2)}M(r,\phi)=\delta(E+\omega-H_{T}(x))\prod_{t}drd\phi J(r,\phi),$ (3\.
33)
where the Jacobian of transformation
$J(r,\phi)=\int\prod
d^{2}x\delta^{(2)}(\ddot{x}+v^{\prime}(x)-j)\delta(\phi-\arctan\frac{x_{2}}{x_{1}})\delta(r-(x^{2}_{1}+x^{2}_{2})^{1/2})$
(3\. 34)
is the product of two $\delta$-functions:
$J(r,\phi)=\prod_{t}r^{2}(t)\delta(\ddot{r}-\dot{\phi}^{2}r+v^{\prime}(r)-j_{r})\delta(\partial_{t}(\dot{\phi}r^{2})-rj_{\phi}),$
(3\. 35)
where $v^{\prime}(r)=\partial v(r)/\partial r$ and
$j_{r}=j_{1}\cos\phi+j_{2}\sin\phi,\;\;\;\;j_{\phi}=-j_{1}\sin\phi+j_{2}\cos\phi$
(3\. 36)
are the components of $\vec{j}$ in the cylindrical coordinates.
It is useful to organize the perturbation theory in terms of $j_{r}$ and
$j_{\phi}$. For this purpose following transformation of arguments of
$\delta$-functions will be used:
$\prod_{t}\delta(\ddot{r}-\dot{\phi}^{2}r+v^{\prime}(r)-j_{r})=e^{-i\int_{C^{(+)}}dt\hat{j}^{\prime}_{r}\hat{e}_{r}}e^{i\int_{C^{(+)}}dtj_{r}e_{r}}\prod_{t}\delta(\ddot{r}-\dot{\phi}^{2}r+v^{\prime}(r)-j^{\prime}_{r})$
(3\. 37)
and
$\prod_{t}\delta(\partial_{t}(\dot{\phi}r^{2})-rj_{\phi})=e^{-i\int_{C^{(+)}}dt\hat{j}^{\prime}_{\phi}\hat{e}_{\phi}}e^{i\int_{C^{(+)}}dtj_{\phi}re_{\phi}}\prod_{t}r(t)\delta(\partial_{t}(\dot{\phi}r^{2})-j^{\prime}_{\phi}).$
(3\. 38)
Here $j_{r}$ and $j_{\phi}$ was defined in (3\. 36). In result, we get to the
path integral formalism written in terms of cylindrical coordinates. This is a
very simplification which will help to solve a lot of mechanical problems. One
can note that in result of mapping our problem reduced to the description of
quantum fluctuations of the surface of cylinder:
$R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dt(\hat{j}_{r}(t)\hat{e}_{r}(t)+\hat{j}_{\phi}(t)\hat{e}_{\phi}(t))}\times$
$\times\int D^{(2)}M(r,\phi)e^{-i\tilde{H}_{T}(x;\tau)-iU_{T}(x,e_{C})},$ (3\.
39)
where
$\displaystyle D^{(2)}M(r,\phi)=\delta(E+\omega-
H_{T}(r,\phi))\prod_{t}r^{2}(t)dr(t)d\phi(t)\times$
$\displaystyle\times\delta(\ddot{r}-\dot{\phi}^{2}r+v^{\prime}(r)-j_{r})\delta(\partial_{t}(\dot{\phi}r^{2})-j_{\phi})$
(3\. 40)
and
$e_{C,1}=e_{r}\cos\phi-
re_{\phi}\sin\phi,\;\;\;\;e_{C,2}=e_{r}\sin\phi+re_{\phi}\cos\phi.$ (3\. 41)
This is the final result. The transformation looks quite classically but (3\.
39) can not be deduced from naive coordinate transformation of initial path
integral for amplitude.
Inserting
$1=\int DpDl\prod_{t}\delta(p-\dot{r})\delta(l-\dot{\phi}r^{2})$ (3\. 42)
into (3\. 39) we can introduce the motion in the phase space with Hamiltonian
$H_{j}=\frac{1}{2}p^{2}+\frac{l^{2}}{2r^{2}}+v(r)-j_{r}r-j_{\phi}\phi.$ (3\.
43)
The Dirac’s measure becomes four dimensional:
$D^{(4)}M(r,\phi,p,l)=\delta(E+\omega-
H_{T}(r,\phi,p,l))\prod_{t}dr(t)d\phi(t)dp(t)dl(t)\times$
$\times\delta\left(\dot{r}-\frac{\partial H_{j}}{\partial
p}\right)\delta\left(\dot{\phi}-\frac{\partial H_{j}}{\partial
l}\right)\delta\left(\dot{p}+\frac{\partial H_{j}}{\partial
r}\right)\delta\left(\dot{l}+\frac{\partial H_{j}}{\partial\phi}\right)$ (3\.
44)
Note absence of the coefficient $r^{2}$ in this expression. This is the result
of special choice of transformation (3\. 38).
Since the Hamilton’s group manifolds are more rich then Lagrange ones the
measure (3\. 44) can be considered as the starting point of farther
transformations. One must to note that the $(action,angle)$ variables are
mostly useful [12]. Note also that to avoid the technical problems with
equations of motion and with functional determinants it is useful to linearize
the argument of $\delta$-functions in (3\. 44) hiding nonlinear terms in the
corresponding auxiliary variables $e_{c}$.
### 3.5 Conclusions
1\. Our perturbation theory describes the quantum fluctuations of the
parameters $(h,\theta)$ of classical trajectory $x_{c}$. It is more
complicated than canonical one, over an interaction constant [19], since
demands investigation of analytic properties of $4N$-dimensional integrals,
where $2N$ is the phase space dimension. Indeed, in the considered case with
$N=1$ the perturbations generating operator, $\hat{\mathbb{K}}$, see (3\. 12),
contain derivatives over four auxiliary parameters,
$(j_{h},e_{h},j_{\theta},e_{\theta})$.
Our transformed theory describes the ”direct” deformations of classical
trajectory $x_{c}=x_{c}(h,\theta)$, i.e. just $h$ and $\theta$ are the objects
of quantization in the considered example. In another words, the quantum
deformations of the invariant hypersurface, $(h,\theta)$, is described in the
new quantum theory. This possibility is the consequence of $\delta$-likeness
of measure, i.e. it based on the conservation of total probability.
Dirac measure allows to perform classical transformations of the measure and
to use high resources of classical mechanics. For example, the interesting
possibility may arise in connection with Kolmogorov-Arnold-Mozer (KAM) theorem
[4]: the system which is not strictly integrable can show the stable motion
peculiar to integrable systems. This is the argument in favor of the idea that
there may be another, non-topological, mechanism of suppression of the quantum
excitations.
2\. One can note that the transformed perturbation theory describes only the
retarded quantum fluctuations, see definition of Green function (3\. 16). This
feature of the theory can lead to the imaginary time irreversibility of
quantum processes and it must be explained.
The starting expression (2\. 58) describes the reversible in time motion since
total action $S_{C_{+}(T_{+})}(x_{+})-S_{C_{-}(T_{-})}(x_{-})$ is time
reversible. But the unitarity condition forced us to consider the interference
picture between expanding and converging waves. This is fixed by the boundary
conditions $e(0)=e(T)=0$. The quantum theory remain time reversible up to
canonical transformation to the invariant hypersurface of the constant energy.
The causal Green function $G(t,t^{\prime})$ , see (2\. 80), is able to
describe both advanced and retarded perturbations and the theory contains the
doubling of degrees of freedom. It means that the theory ”keeps in mind” the
time reversibility. But after the canonical transformation, using above
mentioned boundary conditions, and continuing the theory to the real time, the
quantum perturbations were transferred on the inner degrees of freedom of
classical trajectory. In result the memory of doubling of the degrees of
freedom was disappeared and the theory becomes ”time irreversible”.
The key step in this calculations was an extraction of the classical
trajectory $x_{c}$ which can not be defined without definition of boundary
conditions. Just $x_{c}$ introduces the direction of motion and the order of
quantum perturbations of trajectories inner degrees of freedom play no role,
i.e. the mechanical motion is time reversible while the corrections to energy
of trajectory, $h$, and to the phase, $\theta$, can not be time reversible.
Therefore, the considered irreversibility of the quantum mechanics in terms of
$(h,\theta)$ seems to be imaginary.
## 4 Reduction of quantum degrees of freedom
### 4.1 Introduction
It will be shown in this Section that the quantum fluctuations of angular
variables may be removed if the classical motion is periodic. This cancelation
mechanism can be used for path-integral explanation of integrability of the
quantum-mechanical problems, for example of H-atom problem where the classical
trajectories is closed independently from the initial conditions222222The
approach may be extended on the case of rigid rotator problem [20]. Last one
is isomorphic to the Pocshle-Teller problem [21]. The main result of present
Section is based on the statement that the topology properties of classical
trajectory takes special significance232323Since the action of perturbations
generating operator of transformedf theory, $\hat{\mathbb{K}}$, maps quantum
corrections on the boundaries of cotangent foliation, $\partial W$, see (4\.
41)..
Our technical problem consist in necessity to extract the quantum angular
degrees of freedom. For this purpose we will define path integral in the phase
space of action-angle variables. For simplicity the effect of cancelations we
will demonstrate on the one-dimensional $\lambda x^{4}$ model. In the
following subsection the brief description of unitary definition of the path-
integral measure will be given. The perturbation theory in terms of action-
angle variables will be contracted in Sec.4.3 (the scheme of transformed
perturbation theory was given firstly in [1]). In Sec.4.4 the cancelation
mechanism will be demonstrated.
### 4.2 Unitary definition of the path-integral measure
We will calculate the probability
$R(E)=\int dx_{1}dx_{2}|A(x_{1},x_{2};E)|^{2},$ (4\. 1)
to introduce the unitary definition of path-integral measure [1]. Here
$A(x_{1},x_{2};E)=i\int^{\infty}_{0}dTe^{iET}\int_{x(0)=x_{1}}^{x(T)=x_{2}}Dxe^{iS_{C_{+}(T)}(x)}$
(4\. 2)
is the amplitude of the particle with energy $E$ moving from $x_{1}$ to
$x_{2}$. The action
$S_{C_{+}(T)}(x)=\int_{C_{+}(T)}dt(\frac{1}{2}\dot{x}^{2}-\frac{\omega_{0}^{2}}{2}x^{2}-\frac{\lambda}{4}x^{4})$
(4\. 3)
is defined on the Mills’ contour [17]:
$C_{\pm}(T):t\rightarrow t\pm
i\epsilon,\;\;\;\epsilon\rightarrow+0,\;\;\;0\leq t\leq T.$ (4\. 4)
So, we will omit the calculation of the amplitude.
Inserting (4\. 2) into (4\. 1) we find, see previous Section, that
$R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dt\hat{j}(t)\hat{e}(t)}\int
Dxe^{-i\tilde{H}(x;\tau)-iU_{T}(x,e)}\times$ $\times\delta(E+\omega-
H_{T}(x))\prod_{t}\delta(\ddot{x}+\omega_{0}^{2}x+\lambda x^{3}-j).$ (4\. 5)
The ”hat” symbol means differentiation over corresponding auxiliary quantity.
For instance,
$\hat{\omega}\equiv\frac{\partial}{\partial\omega},~{}~{}~{}\hat{j}(t)=\frac{\delta}{\delta
j(t)}.$ (4\. 6)
It will be assumed that
$\hat{j}(t\in C_{\pm})j(t^{\prime}\in C_{\pm})=\delta(t-t^{\prime}),$
$\hat{j}(t\in C_{\pm})j(t^{\prime}\in C_{\mp})=0.$ (4\. 7)
The time integral over contour $C^{(\pm)}(T)$ means that
$\int_{C^{(\pm)}(T)}=\int_{C_{+}(T)}\pm\int_{C_{-}(T)}.$ (4\. 8)
At the end of calculations the limit $(\omega,\tau,j,e)=0$ must be calculated.
The explicit form of $\tilde{H}(x;\tau)$ $U_{T}(x,e)$ will be given later;
$H_{T}(x)$ is the Hamiltonian at the time moment $t=T$.
The functional $\delta$-function unambiguously determines the contributions in
the path integral. For this purpose we must find the strict solution
$x_{j}(t)$ of the equation of motion:
$\ddot{x}+\omega_{0}^{2}x+\lambda x^{3}-j=0,$ (4\. 9)
expanding it over $j$. In zero order over $j$ we have the classical trajectory
$x_{c}$ which is defined by the equation of motion:
$\ddot{x}+\omega_{0}^{2}x+\lambda x^{3}=0.$ (4\. 10)
This equation is equivalent to the following one:
$t+\theta_{0}=\int^{x}dx\\{2(h_{0}-\omega_{0}^{2}x^{2}-\lambda
x^{4})\\}^{-1/2}.$ (4\. 11)
The solution of this equation is the periodic elliptic function.
Here $(h_{0},\theta_{0})$ are the constants of integration of Eq.(4\. 10),
i.e. $(h_{0},\theta_{0})$ are the coordinates of point on the surface defined
by elliptic function. The integration over $(h_{0},\theta_{0})$ is assumed
since the integration over all trajectories in (4\. 2) must be performed, i.e.
$(h_{0},\theta_{0})$ takes on all values available by elliptic function. Let
$W$ be the corresponding manyfold. One can say therefore that classical
trajectory belongs $W$ completely.
The mapping of our problem on the action-angle phase space will be performed
using representation (4\. 5) [22]. Using the obvious definition of the action:
$I=\frac{1}{2\pi}\oint\\{2(h-\omega_{0}^{2}x^{2}-\lambda x^{4})\\}^{1/2},$
(4\. 12)
and of the angle
$\phi=\frac{\partial h}{\partial
I}\int^{x_{c}}\\{2(h-\omega_{0}^{2}x^{2}-\lambda x^{4})\\}^{-1/2}$ (4\. 13)
variables [12] we easily find from (4\. 5) that
$R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dt\hat{j}(t)\hat{e}(t)}\int
DID\phi e^{-i\tilde{H}(x_{c};\tau)-iU_{T}(x_{c},e)}\times$
$\times\delta(E+\omega-h_{T}(I))\prod_{t}\delta(\dot{I}-j\frac{\partial
x_{c}}{\partial\phi})\delta(\dot{\phi}-\Omega(I)+j\frac{\partial
x_{c}}{\partial I}),$ (4\. 14)
where $x_{c}=x_{c}(I,\phi)$ is the solution of Eq.(4\. 13) with $h=h(I)$ as
the solution of Eq.(4\. 12) and the frequency
$\Omega(I)=\frac{\partial h}{\partial I}.$ (4\. 15)
Representation (4\. 14) is not the full solution of our problem: the action
and angle variables are still interdependent since they both are exited by the
same source $j(t)$. This reflects the Lagrange nature of the path-integral
description of phase-space motion. The true Hamilton’s description must
contain independent quantum sources of action and angle variables.
### 4.3 Perturbation theory on the cotangent manifold
The structure of source terms, $j\partial x_{c}/\partial\phi$ and $j\partial
x_{c}/\partial I$, show that the source of quantum fluctuations is the
classical trajectories perturbation and $j$ is the auxiliary variable. It
allows to regroup the perturbation series in a following manner. Let us
consider the action of the perturbation-generating operators on
$\delta$-functions:
$e^{-i\int_{C^{(+)}(T)}dt\hat{j}(t)\hat{e}(t)}e^{-iU_{T}(x,e)}\prod_{t}\delta\left(\dot{I}+j\frac{\partial
x_{c}}{\partial\phi}\right)\delta\left(\dot{\phi}-\Omega(I)-j\frac{\partial
x_{c}}{\partial I}\right)=$ $=\int
D_{C^{(+)}}e_{I}D_{C^{(+)}}e_{\phi}e^{i\int_{C^{(+)}}dt(e_{I}\dot{I}+e_{\phi}(\dot{\phi}-\Omega(I)))}e^{-iU_{T}(x,e_{c})},$
(4\. 16)
where
$e_{c}(e_{I},e_{\phi})=e_{I}\frac{\partial
x_{c}}{\partial\phi}-e_{\phi}\frac{\partial x_{c}}{\partial I}.$ (4\. 17)
The integrals over $(e_{I},e_{\phi})$ will be calculated perturbatively:
$e^{-iU_{T}(x,e_{c})}=\sum^{\infty}_{n_{I},n_{\phi}=0}\frac{1}{n_{I}!n_{\phi}!}\int\prod^{n_{I}}_{k=1}(dt_{k}e_{I}(t_{k}))\prod^{n_{\phi}}_{k=1}(dt^{\prime}_{k}e_{\phi}(t^{\prime}_{k}))\times$
$\times
P_{n_{I},n_{\phi}}(x_{c},t_{1},...,t_{n_{I}},t^{\prime}_{1},...,t_{n_{\phi}}),$
(4\. 18)
where
$P_{n_{I},n_{\phi}}(x_{c},t_{1},...,t_{n_{I}},t^{\prime}_{1},...,t_{n_{\phi}})=\prod^{n_{I}}_{k=1}\hat{e}^{\prime}_{I}(t_{k})\prod^{n_{\phi}}_{k=1}\hat{e}^{\prime}_{\phi}(t^{\prime}_{k})e^{-iU_{T}(x,e^{\prime}_{c})},$
(4\. 19)
where $e^{\prime}_{c}\equiv e_{c}(e^{\prime}_{I},e^{\prime}_{\phi})$ and the
derivatives in (4\. 19) are calculated at $e^{\prime}_{I}=0$,
$e^{\prime}_{\phi}=0$. At the same time,
$\prod^{n_{I}}_{k=1}e_{I}(t_{k})\prod^{n_{\phi}}_{k=1}e_{\phi}(t^{\prime}_{k})=\prod^{n_{I}}_{k=1}(i\hat{j}_{I}(t_{k}))\prod^{n_{\phi}}_{k=1}(i\hat{j}_{\phi}(t^{\prime}_{k}))e^{-i\int_{C^{(+)}}dt(j_{I}(t)e_{I}(t)+j_{\phi}(t)e_{\phi}(t))}.$
(4\. 20)
The limit $(j_{I},j_{\phi})=0$ is assumed. Inserting (4\. 19), (4\. 20) into
(4\. 16) we will find new representation for $R(E)$:
$R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dt(\hat{j}_{I}(t)\hat{e}_{I}(t)+\hat{j}_{\phi}(t)\hat{e}_{\phi}(t))}\times$
$\times\int DID\phi e^{-i\tilde{H}(x_{c};\tau)-iU_{T}(x_{c},e_{c})}\times$
$\times\delta(E+\omega-
h_{T}(I))\prod_{t}\delta(\dot{I}-j_{I})\delta(\dot{\phi}-\Omega(I)-j_{\phi}),$
(4\. 21)
in which the action and the angle are the decoupled degrees of freedom.
Solving the canonical equations of motion:
$\dot{I}=j_{I},\;\;\;\dot{\phi}=\Omega(I)+j_{\phi}$ (4\. 22)
the boundary conditions:
$I_{j}(0)=I_{0},\;\;\;\phi_{j}(0)=\phi_{0}$ (4\. 23)
will be used. This will lead to the following Green function:
$g(t-t^{\prime})=\Theta(t-t^{\prime}),$ (4\. 24)
with boundary condition: $\Theta(0)=1$. The solutions of eqs.(4\. 22) have the
form:
$\displaystyle I_{j}(t)=I_{0}+\int
dt^{\prime}g(t-t^{\prime})j_{I}(t^{\prime})\equiv I_{0}+I^{\prime}(t),$
$\displaystyle\phi_{j}(t)=\phi_{0}+\tilde{\Omega}(I_{j})t+\int
dt^{\prime}g(t-t^{\prime})j_{\phi}(t^{\prime})\equiv\phi_{0}+\tilde{\Omega}(I_{0}+I^{\prime})t+\phi^{\prime}(t),$
(4\. 25)
where
$\tilde{\Omega}(I_{j})=\frac{1}{t}\int
dt^{\prime}g(t-t^{\prime})\Omega(I_{0}+I^{\prime}(t^{\prime})).$ (4\. 26)
Inserting (4\. 25) into (4\. 21) we find:
$R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dt(\hat{j}{{}_{I}}(t)\hat{e}_{I}(t)+\hat{j}_{\phi}(t)\hat{e}_{\phi}(t))}\times$
$\times\int^{\infty}_{0}dI_{0}\int^{2\pi}_{0}d\phi_{0}e^{-i\tilde{H}(x_{c};\tau)-iU_{T}(x_{c},e_{c})}\delta(E+\omega-
h_{T}(I_{j})),$ (4\. 27)
where
$x_{c}=x_{c}(I_{j},\phi_{j})=x_{c}(I_{0}+I(t),\phi_{0}+\tilde{\Omega}(I_{0}+I)t+\phi(t))$
(4\. 28)
and $e_{c}$ was defined in (4\. 17). Note that the measure of the integrals
over $(I_{0},\phi_{0})$ was defined without of the Faddeev-Popov’s ansatz and
there is not any “hosts” since the Jacobian of transformation is equal to one.
We can extract the Green function into the perturbation-generating operator
using the equalities:
$\hat{j}_{I}(t)=\int dt^{\prime}g(t-t^{\prime})\hat{I}(t),\hat{j}_{\phi}=\int
dt^{\prime}g(t-t^{\prime})\hat{\phi}(t),$ (4\. 29)
which evidently follows from (4\. 25). In result,
$R(E)=2\pi\int^{\infty}_{0}dTe^{\\{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dtdt^{\prime}g(t^{\prime}-t)(\hat{I}(t)\hat{e}_{I}(t^{\prime})+\hat{\phi}(t)\hat{e}_{\phi}(t^{\prime}))\\}}\times$
$\times\int^{\infty}_{0}dI_{0}\int^{2\pi}_{0}d\phi_{0}e^{-i\tilde{H}(x_{c};\tau)-iU_{T}(x_{c},e_{c})}\delta(E+\omega-
h_{T}(I_{0}+I)),$ (4\. 30)
where $x_{c}$ was defined in (4\. 28).
We can define the formalism without doubling of the degrees of freedom. One
can use fact that the action of perturbation-generating operators and the
analytical continuation to the real times are commuting operations. This can
be seen easily using the definition (4\. 7). In result the expression:
$R(E)=2\pi\int^{\infty}_{0}dTe^{\\{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{0}^{T}dtdt^{\prime}\Theta(t^{\prime}-t)(\hat{I}(t)\hat{e}_{I}(t^{\prime})+\hat{\phi}(t)\hat{e}_{\phi}(t^{\prime}))\\}}\times$
$\times\int^{\infty}_{0}dI_{0}\int^{2\pi}_{0}d\phi_{0}e^{-i\tilde{H}(x_{c};\tau)-iU_{T}(x_{c},e_{c})}\delta(E+\omega-
h_{T}(I_{0}+I(T)),$ (4\. 31)
where
$\tilde{H}_{T}(x_{c};\tau)=2\sum^{\infty}_{n=1}\frac{\tau^{2n+1}}{(2n+1)!}\frac{d^{2n}}{dT^{2n}}h(I_{0}+I(T))$
(4\. 32)
and
$-U_{T}(x_{c},e_{c})=S(x_{c}+e_{c})-S(x_{c}-e_{c})-2\int_{0}^{T}dte_{c}\frac{\delta
S(x_{c})}{\delta x_{c}}$ (4\. 33)
defines quantum theory on the cotangent manifold $W$.
Now we can use the last $\delta$-function:
$R(E)=2\pi\int^{\infty}_{0}dTe^{\\{\frac{1}{2i}(\hat{\omega}\hat{\tau}+\int_{0}^{T}dtdt^{\prime}\Theta(t^{\prime}-t)(\hat{I}(t)\hat{e}_{I}(t^{\prime})+\hat{\phi}(t)\hat{e}_{\phi}(t^{\prime}))\\}}\times$
$\times\int^{\infty}_{0}dI_{0}\int^{2\pi}_{0}\frac{d\phi_{0}}{\Omega(E+\omega)}e^{-i\tilde{H}(x_{c};\tau)-iU_{T}(x_{c},e_{c})}.$
(4\. 34)
Here
$x_{c}(t)=x_{c}(I_{0}(E+\omega)+I(t)-I(T),\phi_{0}+\tilde{\Omega}t+\phi(t)).$
(4\. 35)
Eq.(4\. 34) contains unnecessary contributions: the action of the operator
$\int^{T}_{0}dtdt^{\prime}\Theta(t-t^{\prime})\hat{e}_{I}(t)\hat{I}(t^{\prime})$
(4\. 36)
on $\tilde{H}_{T}$, defined in (4\. 32), leads to the time integrals with zero
integration range:
$\int^{T}_{0}dt\Theta(T-t)\Theta(t-T)=0.$
Using this fact,
$\displaystyle
R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}\int_{0}^{T}dtdt^{\prime}\Theta(t^{\prime}-t)(\hat{I}(t)\hat{e}_{I}(t^{\prime})+\hat{\phi}(t)\hat{e}_{\phi}(t^{\prime}))}\times$
$\displaystyle\times\int^{\infty}_{0}dI_{0}\int^{2\pi}_{0}\frac{d\phi_{0}}{\Omega(E)}e^{-iU_{T}(x_{c},e_{c})},$
(4\. 37)
where
$x_{c}(t)=x_{c}(I_{0}(E)+I(t)-I(T),\phi_{0}+\tilde{\Omega}t+\phi(t)).$ (4\.
38)
is the periodic function:
$x_{c}(I_{0}(E)+I(t)-I(T),(\phi_{0}+2\pi)+\tilde{\Omega}t+\phi(t))=$
$=x_{c}(I_{0}(E)+I(t)-I(T),\phi_{0}+\tilde{\Omega}t+\phi(t)).$ (4\. 39)
Now we can consider the cancelation of angular perturbations.
### 4.4 Cancelation of angular perturbations
1\. Simples example
Introducing the perturbation-generating operator into the integral over
$\phi_{0}$:
$\displaystyle
R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}\int_{0}^{T}dtdt^{\prime}\Theta(t^{\prime}-t)\hat{I}(t)\hat{e}_{I}(t^{\prime})}\times$
$\displaystyle\times\int^{\infty}_{0}dI_{0}\int^{2\pi}_{0}\frac{d\phi_{0}}{\Omega(E)}e^{\frac{1}{2i}\int_{0}^{T}dtdt^{\prime}\Theta(t^{\prime}-t)\hat{\phi}(t)\hat{e}_{\phi}(t^{\prime})}e^{-iU_{T}(x_{c},e_{c})},$
(4\. 40)
the mechanism of cancelations of the angular perturbations becomes evident.
One can formulate the statement:
(i) if
$e^{\frac{1}{2i}\int_{0}^{T}dtdt^{\prime}\Theta(t^{\prime}-t)\hat{\phi}(t)\hat{e}_{\phi}(t^{\prime})}e^{-iU_{T}(x_{c},e_{c})}=e^{-iU_{T}(x_{c},e_{c})}|_{e_{\phi}=\phi=0}+dF(\phi_{0})/d\phi_{0},$
(4\. 41)
and
(ii) if
$F(\phi_{0}+2\pi)=F(\phi_{0}),$ (4\. 42)
then:
$R(E)=2\pi\int^{2\pi}_{0}\frac{d\phi_{0}}{\Omega(E)}\int^{\infty}_{0}dTdI_{0}e^{\frac{1}{2i}\int_{0}^{T}dtdt^{\prime}\Theta(t^{\prime}-t)(\hat{I}(t)\hat{e}_{I}(t^{\prime})}$
$\times e^{S(x_{c}+e\partial x_{c}/\partial\phi_{0})-S(x_{c}-e\partial
x_{c}/\partial\phi_{0})},$ (4\. 43)
i.e. we find the expression in which the angular corrections was canceled. In
this case the problem becomes semiclassical over the angular degrees of
freedom.
For the $(\lambda x^{4})_{1}$-model
$S(x_{c}+e\partial x_{c}/\partial\phi_{0})-S(x_{c}-e\partial
x_{c}/\partial\phi_{0})=S_{0}(x_{c})-2\lambda\int^{T}_{0}dtx_{c}(t)\\{e\partial
x_{c}/\partial\phi_{0}\\}^{3},$ (4\. 44)
where [1]
$S_{0}(x_{c})=\oint_{T}dt\left(\frac{1}{2}\dot{x}_{c}^{2}-\frac{\omega_{0}^{2}}{2}x_{c}^{2}-\frac{\lambda}{4}x_{c}^{4}\right)$
(4\. 45)
is the closed time-path action and
$x_{c}(t)=x_{c}(I_{0}(E)+I(t)-I(T),\phi_{0}+\tilde{\Omega}t).$ (4\. 46)
Here $I(t)$ and $I(T)$ are the auxiliary variables.
The condition (4\. 42) requires that the classical trajectory $x_{c}$ with all
derivatives over $I_{0}$, $\phi_{0}$ is the periodic function. In the
considered case of $(\lambda x^{4})_{1}$-model $x_{c}$ is periodic function
with period $1/\Omega$, see (4\. 39). Therefore, we can concentrate the
attention on the condition (4\. 41) only.
Expanding $F(\phi_{0})$ over $\lambda$:
$F(\phi_{0})=\lambda F_{1}(\phi_{0})+\lambda^{2}F_{2}(\phi_{0})+...$ (4\. 47)
we find that
$\frac{d}{d\phi_{0}}F_{1}(\phi_{0})=$
$=\int^{T}_{0}\prod^{3}_{k=1}dt^{\prime}_{k}\hat{\phi}(t^{\prime}_{k})\left(\left(-\frac{6}{(2i)^{3}}\right)\int^{T}_{0}dt\prod^{3}_{k=1}\Theta(t-t^{\prime}_{k})x_{c}(t)(\partial
x_{c}/\partial I_{0})^{3}e^{iS_{0}(x_{c})}_{k}\right)=$
$=\int^{T}_{0}dt^{\prime}\hat{\phi}(t^{\prime})B_{1}(\phi),$ (4\. 48)
where
$B_{1}(\phi)=\left\\{-\frac{6}{(2i)^{3}}\int^{T}_{0}dt\Theta(t-t^{\prime})\right.\times$
$\times\left.\prod^{2}_{k=1}(\Theta(t-t^{\prime}_{k})\hat{\phi}(t^{\prime}_{k}))x_{c}(t)(\partial
x_{c}/\partial I_{0})^{3}e^{iS_{0}(x_{c})}\right\\}$ (4\. 49)
This example shows that the sum over all powers of $\lambda$ can be written in
the form:
$\frac{d}{d\phi_{0}}F(\phi_{0})=\int^{T}_{0}dt^{\prime}\hat{\phi}(t^{\prime})B(\phi),$
(4\. 50)
where, using the definition (4\. 35),
$B(\phi)=\int^{T}_{0}dt\tilde{B}(\phi_{0}+\phi(t)).$ (4\. 51)
Therefore,
$\hat{\phi}(t^{\prime})B(\phi)=\frac{d}{d\phi_{0}}\int^{T}_{0}dt\delta(t-t^{\prime})\tilde{B}(\phi_{0}+\phi(t))$
(4\. 52)
coincides with the total derivative over initial phase $\phi_{0}$, and
$F(\phi_{0})=\tilde{B}(\phi_{0}+\phi(t))|_{\phi=0}.$ (4\. 53)
This result ends the prove of (4\. 41).
2\. General case
Now we will offer following important statement:
— each order of perturbation theory in the invariant subspace can be
represented as the sum of total derivative over the subspace coordinate.
This statement directly follows from structure of perturbations generating
operator $\hat{\mathbb{K}}$ and the assumption (3\. 18). It explains the
statement, offered in Preface.
Let us remind that integration with last $\delta$-function gives the result of
action of operator $\hat{\mathbb{K}}$ written in the form:
$R(E)=2\pi\int_{0}^{\infty}dT\int_{0}^{2\pi}\frac{d\varphi_{0}}{\Omega(E)}:e^{-iU_{(}x_{c},\hat{e}/2i)}:,$
(4\. 54)
where the colons mean normal product,
$\hat{e}=\hat{j}_{\varphi}\frac{\partial x_{c}}{\partial
I}-\hat{j}_{I}\frac{\partial x_{c}}{\partial\varphi},$ (4\. 55)
and by definition $U_{T}$ is the odd over $\hat{e}_{c}$ functional:
$U_{T}(x_{c},e_{c})=2\int_{0}^{T}\sum_{n=1}(\hat{e}_{c}(t)/2i)^{2n+1}u_{n}(x_{c}),$
(4\. 56)
where $u_{n}$ is the function of only $x_{c}$ at the time $t$. Inserting (4\.
55) one can write:
$:e^{-iU_{(}x_{c},\hat{e}/2i)}:=\prod_{n=1}^{\infty}\prod_{k=0}^{2n+1}:e^{-iU_{k,n}(j,x_{c})}:,$
(4\. 57)
where
$U_{k,n}(j,x_{c})=\int_{0}^{T}dt(\hat{j}_{\varphi}(t))^{2n-k+1}(\hat{j}_{I}(t))^{k}b_{k,n}(x_{c}(t))$
(4\. 58)
and the explicit form of $b_{k,n}(x_{c})$ is not important.
Using the evident definition:
$\hat{j}_{X}=\int_{0}^{T}dt^{\prime}\Theta(t-t^{\prime})\hat{X}(t^{\prime}),~{}~{}X=\varphi,I,$
it is easy to find that
$j_{X}(t_{1})b_{k,n}(x_{c}(t_{2}))=\Theta(t_{1}-t_{2})\partial
b_{k,n}(x_{c}(t_{2}))/\partial X_{0},$
since $x_{c}=x_{c}(X+X_{0})$, or shortly:
$j_{1}b_{2}=\Theta_{12}\partial_{X_{0}}b_{2}=\partial_{X_{0}}(\Theta_{12}b_{2})$
(4\. 59)
since the indexes $(k,n)$ are not important.
Let us start consideration from the first term with $k=0$. In this case we
describe only the angular fluctuations. Noting that $\partial_{X_{0}}$ and
$\hat{j}$ commute we can consider the lowest order over $\hat{j}$. The typical
term looks as follows (omitting the index $X_{0}$):
$\hat{j}_{1}\hat{j}_{2}\cdots\hat{j}_{m}b_{1}b_{2}\cdots b_{m}.$
It is sufficient to show that this expression is the total derivative over
$X_{0}$.
Case $m=1$. In this approximation we have, see (4\. 59):
$\hat{j}_{1}b_{1}=\Theta_{11}\partial_{0}b_{1}\neq 0.$ (4\. 60)
Here (3\. 18) was used.
Case $m=2$. This order is less trivial:
$\hat{j}_{1}\hat{j}_{2}b_{1}b_{2}=\Theta_{21}b_{1}^{2}b_{2}+b_{1}^{1}b_{2}^{1}+\Theta_{12}b_{1}b_{2}^{2},$
(4\. 61)
where
$b_{i}^{n}\equiv\partial^{n}b_{i}.$ (4\. 62)
At first glance (4\. 61) is not the total derivative. But inserting
$1=\Theta_{12}+\Theta_{21}$
we can symmetrize it:
$\hat{j}_{1}\hat{j}_{2}b_{1}b_{2}=\Theta_{21}(b_{1}^{2}b_{2}+b_{1}^{1}b_{2}^{1})+\Theta_{12}(b_{1}b_{2}^{2}+b_{1}^{1}b_{2}^{1})=$
$=\partial_{0}(\Theta_{21}b_{1}^{1}b_{2}+\Theta_{12}b_{1}b_{2}^{1})\equiv$
$\equiv\partial_{0}(b_{1}^{1}\rightarrow b_{2}+b_{2}^{1}\rightarrow b_{1})$
(4\. 63)
since the explicit form of the function is not important. Therefore, the
second order term can be also reduced to the total derivative. Notice that
(4\. 63) shows time reversibility.
Case $m=3$. In this order one can find that
$\hat{j}_{1}\hat{j}_{2}\hat{j}_{3}b_{1}b_{2}b_{3}=\partial_{0}\left\\{\sum_{i\neq
j\neq k=1}^{3}(i^{2}\rightarrow j\rightarrow k+i^{1}\rightarrow
j^{1}\rightarrow k)\right\\}$ (4\. 64)
The $m$-th order contribution is also total derivative:
$\hat{j}_{1}\hat{j}_{2}\cdots\hat{j}_{m}b_{1}b_{2}\cdots
b_{m}=\partial_{0}\\{\sum_{i_{1}\neq i_{2}\neq i_{3}\neq\cdots\neq
i_{m}=1}^{m}(i_{1}^{m}\rightarrow i_{2}\rightarrow
i_{3}\rightarrow\cdots\rightarrow i_{m}+$ $+i_{1}^{m-1}\rightarrow
i_{2}^{1}\rightarrow i_{3}\rightarrow\cdots\rightarrow
i_{m}+i_{1}^{m-2}\rightarrow i_{2}^{1}\rightarrow
i_{3}^{1}\rightarrow\cdots\rightarrow i_{m}+\cdots$
$\cdots+i_{1}^{1}\rightarrow i_{2}^{1}\rightarrow
i_{3}^{1}\rightarrow\cdots\rightarrow i_{m-1}^{1}\rightarrow i_{m})\\}$ (4\.
65)
Let us consider now the case with $k\neq 0$. The typical term looks as
follows:
$\hat{j}_{1}^{1}\hat{j}_{2}^{1}\cdots\hat{j}_{l}^{1}\hat{j}_{l+1}^{2}\hat{j}_{l+2}^{2}\cdots\hat{j}_{m}^{2}b_{1}b_{2}\cdots
b_{m},~{}0<l<m,$ (4\. 66)
where, for instance
$\hat{j}_{k}^{1}\equiv\hat{j}_{I}(t_{k}),~{}~{}\hat{j}_{k}^{2}\equiv\hat{j}_{\varphi}(t_{k})$
(4\. 67)
and
$\hat{j}_{1}^{i}b_{2}=\Theta_{12}\partial_{0}^{i}b_{2}.$ (4\. 68)
Case $m=2,l=1$.In this case:
$\hat{j}_{1}^{1}\hat{j}_{2}^{2}b_{1}b_{2}=\Theta_{21}(b_{2}\partial_{0}^{1}\partial_{0}^{2}b_{1}+(\partial_{0}^{2}b_{2})(\partial_{0}^{1}\partial_{0}^{2}b_{1}))+\Theta_{12}(b_{1}\partial_{0}^{1}\partial_{0}^{2}b_{2}+(\partial_{0}^{2}b_{2})(\partial_{0}^{1}\partial_{0}^{2}b_{1}))=$
$=\partial_{0}^{1}(\Theta_{21}b_{2}\partial_{0}^{2}b_{1}+\Theta_{12}b_{1}\partial_{0}^{2}b_{2})+\partial_{0}^{2}(\Theta_{21}b_{2}\partial_{0}^{1}b_{1}+\Theta_{12}b_{1}\partial_{0}^{1}b_{2}).$
(4\. 69)
Therefore we have the total-derivative structure yet. This property is
conserved in arbitrary order over $m$ and $l$ since the time-ordered structure
does not depends from upper index of $\hat{j}$, see (4\. 68).
One can conclude that the contribution are defined by topology properties of
classical trajectory $x_{c}$. We will see that this important property of
perturbation theory remains unchanged also for field theories with symmetry.
### 4.5 Conclusions
1\. It was shown that the real-time quantum problem can be semiclassical over
the part of the degrees of freedom and quantum over another ones. Following to
the result of this Section one may introduce the (probably naive)
interpretation of the quantum systems integrability (we suppose that the
classical system is integrable and can be mapped on the compact hypersurface
in the phase space [12]): the quantum system is strictly integrable in result
of cancelation of all quantum degrees of freedom. The mechanism of cancelation
of the quantum corrections is varied from case to case.
For some problems (as the rigid rotator, or the Pocshle-Teller) the
cancelation of angular degrees of the freedom is enough since they carry only
the angular ones. In an another case (as in the Coulomb problem, or in the
one-dimensional models) the problem may be partly integrable since the quantum
fluctuations of action degrees of freedom just survive. Theirs absence in the
Coulomb problem needs special discussion (one must take into account the
dynamical (hidden) symmetry of Coulomb problem [23]).
The transformation to the action-angle variables maps the $N$-dimensional
Lagrange problem on the $2N$-dimensional phase-space torus. If the winding
number on this hypertorus is a constant (i.e. the topological charge is
conserved) one can expect the same cancelations. This is important for the
field-theoretical problems (for instance, for sin-Gordon model [24]).
2\. In the classical mechanics following approximated method of calculations
is used [12]. The canonical equations of motion:
$\dot{I}=a(I,\phi),~{}~{}\dot{\phi}=b(I,\phi)$ (4\. 70)
are changed on the averaged equations:
$\dot{J}=\frac{1}{2\pi}\int^{2\pi}_{0}d\phi
a(J,\phi),~{}~{}\dot{\phi}=b(J,\phi),$ (4\. 71)
It is possible if the oscillations can be extracted from the systematic
evolution of the degrees of freedoms.
In our case
$a(I,\phi)=j\partial x_{c}/\partial\phi,~{}~{}b(I,\phi)=\Omega(I)-j\partial
x_{c}/\partial I.$ (4\. 72)
Inserting this definitions into (4\. 71) we find evidently wrong result since
in this approximation the problem looks like pure semiclassical for the case
of periodic motion:
$\dot{J}=0,~{}~{}\dot{\phi}=\Omega(J).$ (4\. 73)
The result of this Section was used here. This shows that the procedure of
extraction of the oscillations from the systematic evolution is not trivial
and this method should be used carefully in the quantum theories. (This
approximation of dynamics is ”good” on the time intervals $\sim 1/|a|$ [12].)
## 5 Example: H-atom
### 5.1 Introduction
The mapping
$J:T\rightarrow W,$ (5\. 1)
where $T$ is the $2N$-dimensional phase space and $W$ is a linear space solves
the mechanical problem iff
$J=\otimes^{N}_{1}J_{i},$ (5\. 2)
where $J_{i}$ are the first integrals in involution, see e.g. [12]242424The
formalism of reduction (5\. 1) in classical mechanics is described also in
[25]. The aim of this Section is to adopt this procedure for H-atom.
The mapping (5\. 1) introduces integral $manifold$ $J_{\omega}=J^{-1}(\omega)$
in such a way that the $classical$ phase space flaw belongs to $J_{\omega}$
$completely$. We wish quantize the $J_{\omega}$ manifold instead of flow in
$T$ noting that the quantum trajectory also should belong to $J_{\omega}$
completely. This important conclusion was demonstrated in previous Section by
transformation of the path-integral measure to the canonical variables
$(\xi,\eta)$. New perturbation theory is extremely simple since $W$ is the
linear space.
The ”direct” mapping (5\. 1) used in [26] assumes that $J$ is known. But it
seems inconvenient having in mind the general problem of nonlinear waves
quantization, when the number of degrees of freedom $N=\infty$, or if the
transformation is not canonical. We will consider by this reason the ”inverse”
approach assuming that just the classical flow is known. Then, since the flow
belongs to $J_{\omega}$ completely [26], we would be able to find the quantum
motion in $W$. It is the main technical result illustrated in this Section.
The manifold $J_{\omega}$ is invariant relatively to some subgroup
$G_{\omega}$ [27] in accordance to topological class of classical flaw. This
introduces the $J_{\omega}$ classification and summation over all (homotopic)
classes should be performed. Note, the classes are separated by the boundary
bifurcation lines in $W$ [27]. If the quantum perturbations switched on
adiabatically then the homotopic group should stay unbroken. It is the
ordinary statement for quantum mechanics, but, generally speaking, this is not
true for field theories.
We will calculate the bound state energies in the Coulomb potential252525 We
will restrict ourselves by the plane problem. Corresponding phase space
$T=(p,l,r,\varphi)$ is 4-dimensional.. This popular problem was considered by
many authors, using various methods, see e.g. [23]. The path-integral solution
of this problem was offered firstly in [28].
The classical flaw of this problem can be parameterized by the angular
momentum $l$, corresponding angle $\varphi$ and by the normalized on total
Hamiltonian Runge-Lentz vector length $n$. So, we will consider the mapping
($p$ is the conjugate to $r$ radial momentum in the cylindrical coordinates):
$J_{l,n}:(p,l,r,\varphi)\rightarrow(l,n,\varphi)$ (5\. 3)
to construct the perturbation theory in the $W=(l,n,\varphi)$ space. I.e. $W$
is not considered as the cotangent foliation on $T$.
The mapping (5\. 3) assumes additional reduction of the four-dimensional
incident phase space up to three-dimensional linear subspace262626$W$ would
not have the simplectic structure. Actually in considered case $W=R+TW$, where
$R$ is the zero-modes space and $TW$ is the simplectic subspace.. Just this
reduction phenomena leads to corresponding stability of $n$ concerning quantum
perturbations and will allow to solve our H-atom problem completely272727 In
other words, we would demonstrate that the hidden Bargman-Fock [23] $O(4)$
symmetry is stay unbroken concerning quantum perturbations..
In Subsec. 5.2 we will show how the mapping (5\. 3) can be performed for path-
integral differential measure. In Subsec. 5.3 the consequence of reduction
will be derived and in Subsec. 5.4 the perturbation theory in the $W$ space
will be analyzed. The calculations are based on the formalism offered in
previous Sections.
### 5.2 Mapping
We will calculate the integral [26]:
$\rho(E)=\int^{\infty}_{0}dTe^{-i{\hat{\mathbb{K}}}(j,e)}\int
DM(p,l,r,\varphi)e^{-iU(r,e)},$ (5\. 4)
where $\rho(E)$ is the $probability$ to find a particle with energy $E$, i.e.
we should find [22] that normalized on the zero-modes volume
$\rho(E)=\pi\sum_{n}\delta(E-E_{n}),$ (5\. 5)
where $E_{n}$ are the bound states energies. For $H$-atom problem $E_{n}\leq
0$. This condition will define considered homotopy class.
Expansion over operator
${\hat{\mathbb{K}}}(j,e)=\frac{1}{2}\int^{T}_{0}dt(\hat{j}_{r}\hat{e}_{r}+\hat{j}_{\varphi}\hat{e}_{\varphi}),~{}~{}~{}\hat{X}(t)\equiv\delta/\delta
X(t),$ (5\. 6)
generates the perturbation series. It will be seen that in our case we may
omit the question of perturbation theories convergence.
The differential measure
$DM(p,l,r,\varphi)=\delta(E-H_{0})\prod_{t}dr(t)dp(t)dl(t)d\varphi(t)\times$
$\delta\left(\dot{r}-\frac{\partial H_{j}}{\partial
p}\right)\delta\left(\dot{p}+\frac{\partial H_{j}}{\partial
r}\right)\delta\left(\dot{\varphi}-\frac{\partial H_{j}}{\partial
l}\right)\delta\left(\dot{l}+\frac{\partial H_{j}}{\partial\varphi}\right),$
(5\. 7)
with total Hamiltonian ($H_{0}=H_{j}|_{j=0}$)
$H_{j}=\frac{1}{2}p^{2}-\frac{l^{2}}{2r^{2}}-\frac{1}{r}-j_{r}r-j_{\varphi}\varphi$
(5\. 8)
allows perform arbitrary transformation of variables because of its
$\delta$-likeness. Notice that $H_{j}$ contains only the ”Lagrange forces”
$j_{r}$ and $j_{\varphi}$.
The functional
$U(r,e)=-s_{0}(r)+$
$+\int^{T}_{0}dt\left[\frac{1}{((r+e_{r})^{2}+r^{2}e_{\varphi}^{2})^{1/2}}-\frac{1}{((r-e_{r})^{2}+r^{2}e_{\varphi}^{2})^{1/2}}+2\frac{e_{r}}{r}\right]$
(5\. 9)
describes the interaction between various quantum modes and $s_{0}(r)$ defines
the non-integrable phase factor [22]. The quantization of this factor
determines the bound state energy. Such factor will appear if the phase of
amplitude can not be fixed 282828As, for instance, in the Aharonov-Bohm case..
Note that the Hamiltonian (5\. 8) contains the energy of radial $j_{r}r$ and
angular $j_{\varphi}\varphi$ excitation independently.
Let us introduce the functional
$\Delta=\int\prod_{t}d^{2}\xi d^{2}\eta\times$
$\times\delta(r(t)-r_{c}(\xi,\eta))\delta(p(t)-p_{c}(\xi,\eta))\delta(l(t)-l_{c}(\xi,\eta))\delta(\varphi(t)-\varphi_{c}(\xi,\eta))$
(5\. 10)
which is defined by given functions
$(r_{c},p_{c},\varphi_{c},l_{c})(\xi,\eta)$. If given functions $(\xi,\eta)$
zeroes argument of $\delta$-functions in (5\. 10) then it is assumed that the
functional determinant
$\displaystyle\Delta_{c}=\int\prod_{t}d^{2}\bar{\xi}d^{2}\bar{\eta}\delta\left(\frac{\partial
r_{c}}{\partial\xi}\cdot\bar{\xi}+\frac{\partial
r_{c}}{\partial\eta}\cdot\bar{\eta}\right)\delta\left(\frac{\partial
p_{c}}{\partial\xi}\cdot\bar{\xi}+\frac{\partial
p_{c}}{\partial\eta}\cdot\bar{\eta}\right)\times$
$\displaystyle\times\delta\left(\frac{\partial\varphi_{c}}{\partial\xi}\cdot\bar{\xi}+\frac{\partial\varphi_{c}}{\partial\eta}\cdot\bar{\eta}\right)\delta\left(\frac{\partial
l_{c}}{\partial\xi}\cdot\bar{\xi}+\frac{\partial
l_{c}}{\partial\eta}\cdot\bar{\eta}\right)\neq 0.$ (5\. 11)
Note that this is the condition only for
$(r_{c},p_{c},\varphi_{c},l_{c})(\xi,\eta)$.
To perform the mapping we will insert
$1=\Delta/\Delta_{c}$ (5\. 12)
into (5\. 4) and integrate over $r(t)$, $p(t)$, $\varphi(t)$ and $l(t)$. In
result we find the measure:
$\displaystyle
DM(\xi,\eta)=\frac{1}{\Delta_{c}}\delta(E-H_{0})\prod_{t}d^{2}\xi
d^{2}\eta\delta\left(\dot{r_{c}}-\frac{\partial H_{j}}{\partial
p_{c}}\right)\times$
$\displaystyle\times\delta\left(\dot{p_{c}}+\frac{\partial H_{j}}{\partial
r_{c}}\right)\delta\left(\dot{\varphi_{c}}-\frac{\partial H_{j}}{\partial
l_{c}}\right)\delta\left(\dot{l_{c}}+\frac{\partial
H_{j}}{\partial\varphi_{c}}\right),$ (5\. 13)
Note that the functions $(r_{c},p_{c},\varphi_{c},l_{c})(\xi,\eta)$ must obey
only one condition (5\. 11).
A simple algebra gives:
$\displaystyle
DM(\xi,\eta)=\frac{\delta(E-H_{0})}{\Delta_{c}}\prod_{t}d^{2}\xi
d^{2}\eta\int\prod_{t}d^{2}\bar{\xi}d^{2}\bar{\eta}$
$\displaystyle\times\delta^{2}\left(\bar{\xi}-\left(\dot{\xi}-\frac{\partial
h_{j}}{\partial\eta}\right)\right)\delta^{2}\left(\bar{\eta}-\left(\dot{\eta}+\frac{\partial
h_{j}}{\partial\xi}\right)\right)$
$\displaystyle\times\delta\left(\frac{\partial
r_{c}}{\partial\xi}\cdot\bar{\xi}+\frac{\partial
r_{c}}{\partial\eta}\cdot\bar{\eta}+\\{r_{c},h_{j}\\}-\frac{\partial
H_{j}}{\partial p_{c}}\right)$ $\displaystyle\times\delta\left(\frac{\partial
p_{c}}{\partial\xi}\cdot\bar{\xi}+\frac{\partial
p_{c}}{\partial\eta}\cdot\bar{\eta}+\\{p_{c},h_{j}\\}+\frac{\partial
H_{j}}{\partial r_{c}}\right)$
$\displaystyle\times\delta\left(\frac{\partial\varphi_{c}}{\partial\xi}\cdot\bar{\xi}+\frac{\partial\varphi_{c}}{\partial\eta}\cdot\bar{\eta}+\\{\varphi_{c},h_{j}\\}-\frac{\partial
H_{j}}{\partial l_{c}}\right)$ $\displaystyle\times\delta\left(\frac{\partial
l_{c}}{\partial\xi}\cdot\bar{\xi}+\frac{\partial
l_{c}}{\partial\eta}\cdot\bar{\eta}+\\{l_{c},h_{j}\\}+\frac{\partial
H_{j}}{\partial\varphi_{c}}\right).$ (5\. 14)
The Poisson notation:
$\\{X,h_{j}\\}=\frac{\partial X}{\partial\xi}\frac{\partial
h_{j}}{\partial\eta}-\frac{\partial X}{\partial\eta}\frac{\partial
h_{j}}{\partial\xi}$
was introduced in (5\. 14).
Next, the ”auxiliary” quantity $h_{j}$ have been introduced by following
equalities:
$\displaystyle\\{r_{c},h_{j}\\}-\frac{\partial H_{j}}{\partial
p_{c}}=0,~{}\\{p_{c},h_{j}\\}+\frac{\partial H_{j}}{\partial r_{c}}=0,$
$\displaystyle\\{\varphi_{c},h_{j}\\}-\frac{\partial H_{j}}{\partial
l_{c}}=0,~{}\\{l_{c},h_{j}\\}+\frac{\partial H_{j}}{\partial\varphi_{c}}=0.$
(5\. 15)
Then the functional determinant $\Delta_{c}$ is canceled and
$DM(\xi,\eta)=\delta(E-H_{0})\prod_{t}d^{2}\xi
d^{2}\eta\delta^{2}(\dot{\xi}-\frac{\partial
h_{j}}{\partial\eta})\delta^{2}(\dot{\eta}+\frac{\partial
h_{j}}{\partial\xi}),$ (5\. 16)
It is the desired result of transformation of the measure for given generating
functions $(r_{c},p_{c},\varphi_{c},l_{c})(\xi,\eta)$. In this case the
”Hamiltonian” $h_{j}(\xi,\eta)$ is defined by four equations (5\. 15).
But there is another possibility. Let us assume that
$h_{j}(\xi,\eta)=H_{j}(r_{c},p_{c},\varphi_{c},l_{c})$ (5\. 17)
and the functions $(r_{c},p_{c},\varphi_{c},l_{c})(\xi,\eta)$ are unknown.
Then eqs.(5\. 15) are the equations for this functions. It is not hard to see
that the eqs.(5\. 15) simultaneously with equations fixed by
$\delta$-functions in (5\. 16) are equivalent of incident equations if the
equality (5\. 17) is hold. Indeed, for example,
$\dot{r}_{c}=\frac{\partial r_{c}}{\partial\xi}\cdot\dot{\xi}+\frac{\partial
r_{c}}{\partial\eta}\cdot\dot{\eta}=\\{r_{c},h_{j}\\}=\frac{\partial
H_{j}}{\partial p_{c}},$ (5\. 18)
where (5\. 16) and (5\. 15) was used successively.
So, incident dynamical problem was divided on two parts. First one defines the
trajectory in the $W$ space through eqs.(5\. 15). Second one defines the
dynamics, i.e. the time dependence, through the equations fixed by
$\delta$-functions in the measure (5\. 16).
Therefore, we should consider $r_{c},~{}p_{c},~{}\varphi_{c},~{}l_{c}$ as the
solutions in the $\xi,~{}\eta$ parametrization. The desired parametrization of
classical orbits has the form (one can find it in arbitrary textbook of
classical mechanics):
$r_{c}=\frac{\eta_{1}^{2}(\eta_{1}^{2}+\eta_{2}^{2})^{1/2}}{(\eta_{1}^{2}+\eta_{2}^{2})^{1/2}+\eta_{2}\cos\xi_{1}},~{}p_{c}=\frac{\eta_{2}\sin\xi_{1}}{\eta_{1}(\eta_{1}^{2}+\eta_{2}^{2})^{1/2}},~{}\varphi_{c}=\xi_{1},~{}l_{c}=\eta_{1},$
(5\. 19)
i.e. $r_{c}$ and $p_{c}$ are $\xi_{2}$ independent. At the same time,
$h_{j}=\frac{1}{2(\eta_{1}^{2}+\eta_{2}^{2})^{1/2}}-j_{r}r_{c}-j_{\varphi}\xi_{1}\equiv
h(\eta)-j_{r}r_{c}-j_{\varphi}\xi_{1}.$ (5\. 20)
Noting that the derivatives of $h_{j}$ over $\xi_{2}$ are equal to
zero292929To have the condition (5\. 11) we should assume that $\partial
r_{c}/\partial\xi_{2}\sim\epsilon\neq 0$. We put $\epsilon=0$ completing the
transformation. we find that
$DM(\xi,\eta)=\delta(E-h(T))\prod_{t}d^{2}\xi
d^{2}\eta\delta\left(\dot{\xi}_{1}-\omega_{1}+j_{r}\frac{r_{c}}{\partial\eta_{1}}\right)$
$\times\delta\left(\dot{\xi}_{2}-\omega_{2}+j_{r}\frac{r_{c}}{\partial\eta_{2}}\right)\delta\left(\dot{\eta}_{1}-j_{r}\frac{\partial
r_{c}}{\partial\xi_{1}}-j_{\varphi}\right)\delta(\dot{\eta}_{2}),$ (5\. 21)
where
$\omega_{i}=\partial h/\partial\eta_{i}$ (5\. 22)
are the conserved in classical limit $j_{r}=j_{\varphi}=0$ ”velocities” in the
$W$ space.
### 5.3 Reduction
We see from (5\. 21) that the length of Runge-Lentz vector is not perturbated
by the quantum forces $j_{r}$ and $j_{\varphi}$. To investigate the
consequence of this fact it is useful to project this forces on the axis of
$W$ space. This means splitting of $j_{r},~{}j_{\varphi}$ on
$j_{\xi},~{}j_{\eta}$. The equality
$\prod_{t}\delta\left(\dot{\xi}_{1}-\omega_{1}+j_{r}\frac{r_{c}}{\partial\eta_{1}}\right)=e^{\frac{1}{2i}\int^{T}_{0}dt\hat{j}_{\xi_{1}}\hat{e}_{\xi_{1}}}e^{2i\int^{T}_{0}dtj_{r}e_{\xi_{1}}\partial
r_{c}/\partial\eta_{1}}\prod_{t}\delta(\dot{\xi}_{1}-\omega_{1}+j_{\xi_{1}})$
becomes evident if the Fourier representation of $\delta$-function is used
(see also [26]). The same transformation of arguments of other
$\delta$-functions in (5\. 21) can be applied. Then, noting that the last
$\delta$-function in (5\. 21) is source-free, we find the same representation
as (5\. 4) with
$\hat{\mathbb{K}}(j,e)=\int^{T}_{0}dt(\hat{j}_{\xi_{1}}\hat{e}_{\xi_{1}}+\hat{j}_{\xi_{2}}\hat{e}_{\xi_{2}}+\hat{j}_{\eta_{1}}\hat{e}_{\eta_{1}}),$
(5\. 23)
where the operators $\hat{j}$ are defined by the equality:
$\hat{j}_{X}(t)=\int^{T}_{0}dt^{\prime}\Theta(t-t^{\prime})\hat{X}(t^{\prime})$
(5\. 24)
and $\Theta(t-t^{\prime})$ is the Green function of our perturbation theory
[26].
We should change also
$e_{r}\rightarrow e_{c}=e_{\eta_{1}}\frac{\partial
r_{c}}{\partial\xi_{1}}-e_{\xi_{1}}\frac{\partial
r_{c}}{\partial\eta_{1}}-e_{\xi_{2}}\frac{\partial
r_{c}}{\partial\eta_{2}},~{}~{}e_{\varphi}\rightarrow e_{\xi_{1}}$ (5\. 25)
in the Eq.(5\. 9). The differential measure takes the simplest form:
$DM(\xi,\eta)=\delta(E-h(T))\prod_{t}d^{2}\xi
d^{2}\eta\delta(\dot{\xi}_{1}-\omega_{1}-j_{\xi_{1}})\delta(\dot{\xi}_{2}-\omega_{2}-j_{\xi_{2}})$
$\times\delta(\dot{\eta}_{1}-j_{\eta_{1}})\delta(\dot{\eta}_{2}).$ (5\. 26)
Note now that the $\xi,\eta$ variables are contained in $r_{c}$ only:
$r_{c}=r_{c}(\xi_{1},\eta_{1},\eta_{2}).$
This means that the action of the operator $\hat{j}_{\xi_{2}}$ gives identical
to zero contributions into perturbation theory series. And, since
$\hat{e}_{\xi_{2}}$ and $\hat{j}_{\xi_{2}}$ are conjugate operators, see (5\.
23), we can put
$j_{\xi_{2}}=e_{\xi_{2}}=0.$
This conclusion ends the reduction:
$\hat{\mathbb{K}}(j,e)=\int^{T}_{0}dt(\hat{j}_{\xi_{1}}\hat{e}_{\xi_{1}}+\hat{j}_{\eta_{1}}\hat{e}_{\eta_{1}}),$
(5\. 27) $e_{c}=e_{\eta_{1}}\frac{\partial
r_{c}}{\partial\xi_{1}}-e_{\xi_{1}}\frac{\partial r_{c}}{\partial\eta_{1}}.$
(5\. 28)
The measure has the form:
$DM(\xi,\eta)=\delta(E-h(T))d\xi_{2}(0)d\eta_{2}(0)\prod_{t}d\xi_{1}d\eta_{1}\delta(\dot{\xi}_{1}-\omega_{1}-j_{\xi_{1}})\delta(\dot{\eta}_{1}-j_{\eta_{1}})$
(5\. 29)
since $V=V(r_{c},e_{c},\xi_{1})$ is $\xi_{2}$ independent and
$\int\prod_{t}dX(t)\delta(\dot{X})=\int dX(0).$
### 5.4 Perturbations
One can see from (5\. 29) that the reduction can not solve the H-atom problem
completely: there are nontrivial corrections to the orbital degrees of freedom
$\xi_{1},\eta_{1}$. By this reason we should consider the expansion over
$\hat{\mathbb{K}}$.
Using last $\delta$-functions in (5\. 29) we find, see also [26] (normalizing
$\rho(E)$ on the integral over $\xi_{2}(0)\eta_{2}(0)$):
$\rho(E)=\int^{\infty}_{0}dTe^{-i\hat{\mathbb{K}}(j,e)}\int
dMe^{-iU(r_{c},e)},$ (5\. 30)
where
$dM=\frac{d\xi_{1}d\eta_{1}}{\omega_{2}(E)}.$ (5\. 31)
The operator $\hat{\mathbb{K}}(j,e)$ was defined in (5\. 27) and
$U(r_{c},e_{c})=-s_{0}(r)+$
$+\int^{T}_{0}dt[\frac{1}{((r_{c}+e_{c})^{2}+r_{c}^{2}e_{\xi_{1}}^{2})^{1/2}}-\frac{1}{((r_{c}-e_{c})^{2}+r_{c}^{2}e_{\xi_{1}}^{2})^{1/2}}+2\frac{e_{c}}{r_{c}}]$
(5\. 32)
with $e_{c},~{}e_{\xi_{1}}$ was defined in (5\. 28), (5\. 25) and
$r_{c}(t)=r_{c}(\eta_{1}+\eta(t),\bar{\eta}_{2}(E,T),\xi_{1}+\omega_{1}(t)+\xi(t)),~{}~{}E\equiv
h(\eta_{1}+\eta(T),\bar{\eta}_{2}),$ (5\. 33)
where $\bar{\eta}_{2}(E,T)$ is the solution of equation $E=h$.
The integration range over $\xi_{1}$ and $\eta_{1}$ is as follows:
$0\leq\xi_{1}\leq 2\pi,~{}~{}-\infty\leq\eta_{1}\leq+\infty.$ (5\. 34)
First inequality defines the principal domain of the angular variable
$\varphi$ and second ones take into account the clockwise and anticlockwise
motions of particle on the Kepler orbits.
We can write:
$\rho(E)=\int^{\infty}_{0}dT\int dM:e^{-iV(r_{c},\hat{e})}:$ (5\. 35)
since the operator $\ln\hat{\mathbb{K}}$ is linear over
$\hat{e}_{\xi_{1}},\hat{e}_{\eta_{1}}$. The colons means ”normal product” with
differential operators staying to the left of functions and $U(r_{c},\hat{e})$
is the functional of operators:
$2i\hat{e}_{c}=\hat{j}_{\eta_{1}}\frac{\partial
r_{c}}{\partial\xi_{1}}-\hat{j}_{\xi_{1}}\frac{\partial
r_{c}}{\partial\eta_{1}},~{}~{}2i\hat{e}_{\xi_{1}}=\hat{j}_{\xi_{1}}.$ (5\.
36)
Expanding $U(r_{c},\hat{e})$ over $\hat{e}_{c}$ and $\hat{e}_{\eta_{1}}$ we
find:
$U(r_{c},\hat{e})=-s_{0}(r_{c})+2\sum_{n+m\geq
1}C_{n,m}\int^{T}_{0}dt{\hat{e}_{c}^{2n+1}\hat{e}_{\eta_{1}}^{m}}\frac{1}{r_{c}^{2n+2}},$
(5\. 37)
where $C_{n,m}$ are the numerical constants. We see that the interaction part
presents expansion over $1/r_{c}$ and, therefore, the expansion over $U$
generates an expansion over $1/r_{c}$.
In result, see Sec.4.5,
$\rho(E)=\int^{\infty}_{0}dT\int
dM\\{e^{is_{0}(r_{c})}+B_{\xi_{1}}(\xi_{1},\eta_{1})+B_{\eta_{1}}(\xi_{1},\eta_{1})\\}.$
(5\. 38)
The first term is the pure semiclassical contribution and last ones are the
quantum corrections. The functionals $B$ are the total derivatives:
$B_{\xi_{1}}(\xi_{1},\eta_{1})=\frac{\partial}{\partial\xi_{1}}b_{\xi_{1}}(\xi_{1},\eta_{1}),~{}~{}B_{\eta_{1}}(\xi_{1},\eta_{1})=\frac{\partial}{\partial\eta_{1}}b_{\eta_{1}}(\xi_{1},\eta_{1}).$
(5\. 39)
This means that the mean value of quantum corrections in the $\xi_{1}$
direction are equal to zero:
$\int^{2\pi}_{0}d\xi_{1}\frac{\partial}{\partial\xi_{1}}b_{\xi_{1}}(\xi_{1},\eta_{1})=0$
(5\. 40)
since $r_{c}$ is the closed trajectory independently from initial conditions,
see (5\. 19).
In the $\eta_{1}$ direction the motion is classical:
$\int^{+\infty}_{-\infty}d\eta_{1}\frac{\partial}{\partial\eta_{1}}b_{\eta_{1}}(\xi_{1},\eta_{1})=0$
(5\. 41)
since (i) $b_{\eta_{1}}$ is the series over $1/r_{c}^{2}$ and (ii)
$r_{c}\rightarrow\infty$ when $|\eta_{1}|\rightarrow\infty$. Therefore,
$\rho(E)=\int^{\infty}_{0}dT\int dMe^{is_{0}(r_{c})}.$ (5\. 42)
This is the desired result.
Noting that
$s_{0}(r_{c})=kS_{1}(E),~{}~{}k=\pm 1,\pm 2,...,$
where $S_{1}(E)$ is the action over one classical period $T_{1}$:
$\frac{\partial S_{1}(E)}{\partial E}=T_{1}(E),$
and using the identity [22]:
$\sum^{+\infty}_{-\infty}e^{inS_{1}(E)}=2\pi\sum^{+\infty}_{-\infty}\delta(S_{1}(E)-2\pi
n),$
we find:
$\rho(E)=\pi\Omega\sum_{n}\delta(E+1/2n^{2})$ (5\. 43)
where $\Omega$ is the zero-modes volume.
### 5.5 Conclusions
The demonstrated above mechanism of reduction is universal: one can introduce
from the very beginning the arbitrary number of coordinates $(\xi,\eta)$. But
later on the formalism automatically, through dependence of classical
trajectory on coordinates of $W$, will extract the necessary set of variables
$(\xi,\eta)$. At the same time $\dim(\xi,\eta)=\dim W$ and the integrals over
other ones will give the volume
$V_{0}=\int\prod d\xi(0)d\eta(0),$
see (5\. 29) where $\dim V_{0}=2$.
Notice that appearance of the ”0-dimensional” integral measure
$d\xi_{2}(0)d\eta_{2}(0)$
in (5\. 29) reflects the hidden $O(4)$ symmetry of H-atom problem [23].
Therefore, following our selection rule, we must consider in a first place the
classical trajectory which leads to the maximal value of $\dim V_{0}$, i.e. we
must consider the contributions with maximal number of zero modes.
## 6 Example: sin-Gordon model
### 6.1 Introduction
First of all we will describe ”canonical” transformation in the path-integral
formalism. The method of canonical transformations in spite of its expected
effectiveness is unpopular in quantum theories since on this way exist the
problem: it is necessary to find the transformation from Lagrangian to
Hamiltonian descriptions. This transition in general is very difficult if
$\varphi(x)$ and $\dot{\varphi}(x)=p(x)$ are not the independent quantities
[13]. But we may use following trick. We start from the simplest verse of the
canonical formalism introducing the ”first-order” description303030In other
words, we will still stay in the frame of Lagrangian formalism. and after
transformation come to independent canonically conjugate pares, $(\xi,\eta)$,
i.e. come to Hamiltonian description. It is evident that in general the
transformation
$\varphi_{c}:(\varphi,p)\rightarrow(\xi,\eta)$
will not be canonical. The formalism of present Section is the same as in the
H-atom problem but there is some distinction.
We will continue in this Section description of influence of the phase-space
structure on the result of quantum-mechanical measurements started in previous
Sections. Now we will calculate the expectation value of the ”order parameter”
(mass-shell particles production vertex) $\Gamma(q;u)$ [29]:
$\rho(q)=<|\Gamma(q;u)|^{2}>_{u},$
where $q$ is the mass-shell ($q^{2}=m^{2}$) particles momentum and $<>_{u}$
means averaging over the field $u(x,t)$. Just the procedure of averaging would
be the object of our interest considering the quantum Hamiltonian system with
symmetry $G$. By definition, $\rho$ is the $probability$ to find one mass-
shell particle. Certainly, $\rho(q)=0$ on the sourceless vacuum but, generally
speaking, $\rho(q)\neq 0$ in a field with nonzero energy density.
Calculations will be illustrated by the integrable (1+1)-dimensional model
with non-polynomial Lagrangian
$L=\frac{1}{2}(\partial_{\mu}u)^{2}+\frac{m_{h}^{2}}{\lambda^{2}}[\cos(\lambda
u)-1],$ (6\. 1)
We will consider following formulation of the problem. Formally nothing
prevents to linearize partly our problem considering the Lagrangian
$L=\frac{1}{2}[(\partial_{\mu}u)^{2}-m_{h}^{2}u^{2}]+\frac{m_{h}^{2}}{\lambda^{2}}[\cos(\lambda
u)-1+\frac{\lambda^{2}}{2}u^{2}]\equiv L_{0}(u)-v(u)$ (6\. 2)
to describe creation (and absorption) of the mass $m_{h}$ particles. Then the
last term in (6\. 2),
$v(u)=-\frac{m_{h}^{2}}{\lambda^{2}}[\cos(\lambda
u)-1+\frac{\lambda^{2}}{2}u^{2}],$ (6\. 3)
describes interactions. The corresponding to this theory order parameter is
$\Gamma(q;u)=\int
dxdte^{iqx}(\partial^{2}+m_{h}^{2})u(x,t),~{}~{}~{}q^{2}=m_{h}^{2}.$ (6\. 4)
It will be shown by explicit calculations that
$\rho(q)=0$ (6\. 5)
as the consequence of unbroken $\tilde{sl}(2,C)$ Kac-Moody algebra on which
the solitons of theory (6\. 1) live313131Trivialness of soliton $S$-matrix was
shown in [30], see e.g. [31] and references cited therein323232 It may be
useful at this point to compare our approach with ordinary thermodynamics of
ferromagnetic. The external magnetic field is $\sim<{\mu}>$, where the order
parameter $<\mu>$ is the mean value of the spin, and the phase transition
means that $<\mu>\neq 0$, i.e. $<\mu>=0$ means that corresponding symmetry
stay unbroken. We will suppose that the mean value of $|\Gamma(q,u)|^{2}$,
which is the function of external fields parameter $q$, play the same role for
field theories with symmetry, i.e. $<|\Gamma(q,u)|^{2}>_{u}=0$ means that
corresponding symmetry stay unbroken. Therefore in our approach only the
”external” display of symmetry can be described.. The solution (6\. 5) seems
interesting since it can be interpreted as the explicit demonstration of field
$u(x,t)$ confinement. The main purpose of this paper is to investigate how the
solution (6\. 5) appears.
We will be able to find exact equality (6\. 5) since the model (6\. 1) possess
infinite number of integrals of motion. It is well known that each integral of
motion in involution allows to shrink a number of phase space $\bar{\gamma}$
variables on two units, see e.g. [12]. Resulting phase space $\gamma$ is
called as the reduced phase space [25]. The summation over all reduced phase
space topological classes [27] is assumed.
By this way the field-theoretical problem will reduced to the quantum-
mechanical one. We would consider $\eta$ as the ”particles” generalized
momentum and would introduce $\xi$ as the conjugate to $\eta$ coordinate of
soliton. The $2N$-dimensional phase space (cotangent manifold) $\gamma_{N}$
with local coordinates $(\xi,\eta)$ on it has natural simplectic structure,
and $DM(\gamma_{N})=D^{N}M(\xi,\eta)$ in practical calculations (see
Subsec.6.2). The summation over $N$ is assumed.
The quantum corrections to semiclassical approximation of transformed theory
are simply calculable since $\eta$ are conserved in the classical limit. This
is the particularity of solitons dynamics (solitons momenta is the conserved
quantities). One can consider the developed in this paper formalism as the
path-integral version of nonlinear waves (solitons in our case) quantum theory
(the canonical quantization of sin-Gordon model in the soliton sector was
described also in [14].)
In Subsec.6.3 we will demonstrate Eq.(6\. 5). It will be shown that this
solution is consequence of the previously developed proposition (we would
justify it in Subsec.6.2) that the semiclassical approximation is exact for
sin-Gordon model [11]. The semiclassical approximation in the $\gamma_{N}$
phase space will be considered in Subsec.6.2.
We would not use the complicated algebra to show the reduction procedure
explicitly noting that all solutions of model (6\. 1) are known [24]. Then,
using the $\delta$-likeness of measure $DM(\tilde{\gamma})$, we will find in
Subsec.6.2 $DM(\gamma_{N})$ considering the mapping as an ordinary
transformation to useful variables333333We will apply inverse reduction
procedure. Let $G$ be a group of canonical transformations acting on the
simplectic manifold $\tilde{\gamma}$ and let $\bar{G}$ be the Lie algebra of
$G$ with $G^{*}$ dual of it. Then the momentum [32] mapping
$J:~{}\tilde{\gamma}\rightarrow G^{*}$ introduces the integrals of motion
which reduces the $\tilde{\gamma}$ manifold. Noting that the set of levels
$J^{-1}({\eta})$ (solution of equations $J(\pi)=\eta$, $\pi\in\tilde{\gamma}$)
is a manifold then $\gamma_{\eta}=J^{-1}({\eta})/\bar{G}_{\eta}$ is the
reduced phase space, where $\bar{G}_{\eta}$ is the co-adjoint isotropy
subgroup of $G$. Therefore, the differential measure
$dM=dM(\eta,\gamma_{\eta})$ for reduced phase space. For integrable mechanical
systems (infinite dimensional as well, see e.g. [24]) $\gamma_{\eta}$ shrinks
to the point and in this case $dM=dM(\eta)$ is the measure of momentum
manifold. Just this simplest case would be considered working with Lagrangian
(6\. 1) and more general and interesting case with measure
$DM=DM(\eta,\gamma_{\eta})$, $\gamma_{\eta}\neq\emptyset$, will be considered
later. So, the reduction procedure of our Hamiltonian system with symmetry $G$
looks like canonical transformation [31]. This problem is nontrivial since,
generally speaking, $\dim\tilde{\gamma}$ and $\dim\gamma$ are not the same for
model (6\. 1).. Corresponding perturbation theory, see Subsec.6.3, in the
momentum space $J$ was described in [26]. In Subsec.6.2 the path-integral
definition of $\rho(q)$ will be given.
We would conclude (this is the main result) that a theory in the ”nonlinear
waves” sector may be nontrivial ($\rho\neq 0$) iff the manifold $\gamma$ is
not compact.
### 6.2 Reduction procedure
6.2.1. Introduction into formalism.
Our aim is to calculate the integral:
$\rho(q)=e^{-i\hat{\mathbb{K}}(j,e)}\int
DM(u,p)|\Gamma(q;u)|^{2}e^{iS_{O}(u)-iU(u,e)},$ (6\. 6)
where $\Gamma(q;u)$ was defined in (6\. 4). In this expression the expansion
over operator
$\hat{\mathbb{K}}(j,e)={\rm Re}\int_{C_{+}}dxdt\frac{\delta}{\delta
j(x,t)}\frac{\delta}{\delta e(x,t)}\equiv{\rm
Re}\int_{C_{+}}dxdt\hat{j}(x,t)\hat{e}(x,t)$ (6\. 7)
generates the perturbation theory series. We will assume that this series
exist. The functionals $U(u,e)$ and $S_{O}(u)$ are defined by the equalities:
$\displaystyle V(u+e)-V(u-e)=U(u,e)+\int dxdte(x,t)v^{\prime}(u),$
$\displaystyle S_{0}(u+e)-S_{0}(u-e)=S_{O}(u)+\int
dxdte(x,t)(\partial^{2}+m_{h}^{2})u(x,t).$ (6\. 8)
The action $S_{0}(u)$ corresponds to the free part of Lagrangian (6\. 1) and
$V(u)$ describes interactions. The quantity $S_{O}(u)$ is not equal to zero
since the soliton configurations have nontrivial topological charge (see also
[1]). All time integrals in this expressions were defined on the Mills time
contour [17]:
$2{\rm Re}\int_{C_{+}}=\int_{C_{+}}+\int_{C_{-}}$
and
$C_{\pm}:t\rightarrow t\pm
i\epsilon,~{}~{}~{}\epsilon\rightarrow+0,~{}~{}~{}-\infty\leq t\leq+\infty,$
to avoid the possible light-cone singularities of the perturbation theory. The
variational derivatives in (6\. 7) are defined by the following way:
$\frac{\delta u(x,t\in C_{i})}{\delta u(x^{\prime},t^{\prime}\in
C_{j})}=\delta_{ij}\delta(x-x^{\prime})\delta(t-t^{\prime}),~{}~{}~{}i,j=+,-.$
The auxiliary variables $(j,e)$ must be taken equal to zero at the very end of
calculations.
Considering the first order formalism with new coordinates $(u,p)$ the measure
$DM(u,p)$ has the form:
$DM(u,p)=\prod_{x,t}du(x,t)dp(x,t)\delta\left(\dot{u}-\frac{\delta
H_{j}(u,p)}{\delta p}\right)\delta\left(\dot{p}+\frac{\delta
H_{j}(u,p)}{\delta u}\right)$ (6\. 9)
with the total ”Hamiltonian”
$H_{j}(u,p)=\int
dx\left\\{\frac{1}{2}p^{2}+\frac{1}{2}(\partial_{x}u)^{2}-\frac{m_{h}^{2}}{\lambda^{2}}[\cos(\lambda
u)-1]-ju\right\\}.$ (6\. 10)
The problem will be considered assuming that $u(x,t)$ belongs to Schwartz
space:
$u(x,t)|_{|x|=\infty}=0~{}({\rm{mod}}\frac{2\pi}{\lambda}).$ (6\. 11)
This means that $u(x,t)$ tends to zero $(\rm{mod}\frac{2\pi}{\lambda})$ at
$|x|\rightarrow\infty$ faster then any power of $1/|x|$. Note that
$\dot{u}=p$, i.e. $u$ and $p$ are not the independent quantities.
The measure (6\. 9) allows to perform arbitrary transformations. But, as was
explained in Introduction, we will use the analog of canonical transformation
which conserves the form of equations of motion. Hence, it is sufficient on
this stage of calculations to know only the fact that this transformation
exist [24]. One may propose that in result we should find for $N$-soliton
topology:
$D^{N}M(\xi,\eta)=\prod_{t}d^{N}\xi(t)d^{N}\eta(t)\delta^{(N)}\left(\dot{\xi}-\frac{\partial
h_{j}(\xi,\eta)}{\partial\eta(t)}\right)\delta^{(N)}\left(\dot{\eta}+\frac{\partial
h_{j}(\xi,\eta)}{\partial\xi(t)}\right),$ (6\. 12)
where $h_{j}$ is the ”transformed Hamiltonian”:
$h_{j}=h_{N}(\eta)-\int dxj(x,t)u_{N}(x;\xi,\eta)$ (6\. 13)
and $u_{N}(x;\xi,\eta)$ is the $N$-soliton configuration the time dependence
of which is parameterized by $(\xi,\eta)$. Therefore, the local coordinates
$(\xi,\eta)$ are defined by the equations:
$\dot{\xi}=\frac{\partial
h_{j}}{\partial\eta},~{}~{}~{}\dot{\eta}=-\frac{\partial h_{j}}{\partial\xi},$
(6\. 14)
where $h_{j}$ must obey the Poisson conditions343434See previous Section:
$\\{u_{c}(x,t),h_{j}\\}=\frac{\delta H_{j}}{\delta
p_{c}(x,t)},~{}~{}~{}\\{p_{c}(x,t),h_{j}\\}=-\frac{\delta H_{j}}{\delta
u_{c}(x,t)}.$ (6\. 15)
One can see choosing
$h_{j}(\xi,\eta)=H_{j}(u_{c},p_{c})$ (6\. 16)
that the initial equations have been restored:
$\dot{u}_{c}=\frac{\partial u_{c}}{\partial\xi}\dot{\xi}+\frac{\partial
u}{\partial\eta}\dot{\eta}=\\{u_{c},h_{j}\\}=\frac{\delta H_{j}}{\delta
p_{c}}.$
The same we will have for $\dot{p}_{c}$. Therefore $(u_{c},p_{c})$ are
solutions of equations of motion (6\. 14), if the equality (6\. 16) is hold.
The field theory case in $(1+1)$-dimensional configuration space needs
additional explanations. First of all, the analog of (5\. 10) must be
introduced:
$\Delta(u,p)=\int\prod_{t}d^{N}\xi(t)d^{N}\eta(t)\prod_{x,t}\delta(u(x,t)-u_{c}(x;\xi,\eta))\delta(p(x,t)-p_{c}(x;\xi,\eta))$
(6\. 17)
if the $N$-soliton configuration is considered. Notice that the one-
dimensional $\delta$-functions are introduced in (6\. 17) and $u_{c}$, $p_{c}$
are the functions of sets $(\xi,\eta)$, $\dim(\xi,\eta)=2N$. Introducing (6\.
17) we make the attempt to ”hide” the time dependence entirely into the set of
$independent$ variables $(\xi,\eta)$.
Comparing (6\. 9) and (6\. 12) one can note that $x$ dependence disappeared
and the transformed measure depends on the number $N=1,2,...$ Therefore,
occurs the reduction of the quantum degrees of freedom since the power of the
coordinate set is continuum and the number of solitons $N$ is the countable
set. This means that the proposed transformation to coordinates of solitons
will be unavoidably singular.
Notice then that the $x$ dependence of $\Delta(u,p)$ remain unimportant since
last one always appear under the integrals over all $u(x,t)$ and $p(x,t)$. At
the same time it is important that introduced in previous Section $\Delta_{c}$
disappeared in the final result, if the integral form of Poisson brackets (6\.
15) are hold353535 See the transformation (5\. 12), described in previous
Section. For more confidence one can introduce the appropriate cells in the
$x$ space [24]..
One can try to propose also the local form of canonical commutators (6\. 15),
if the definition (6\. 16) is hold. Indeed, one can find inserting (6\. 16)
into (6\. 15) that:
$\\{u_{c}(x,t),H_{j}(u_{c},p_{c})\\}=\frac{\delta H_{j}(u_{c},p_{c})}{\delta
p_{c}(x,t)},~{}~{}~{}\\{p_{c}(x,t),H_{j}(u_{c},p_{c})\\}=-\frac{\delta
H_{j}(u_{c},p_{c})}{\delta u_{c}(x,t)}.$ (6\. 18)
This equalities must hold for arbitrary $j$. Making use the definition:
$H_{j}(x_{c},p_{c})=\int dy\tilde{H}_{j}(x_{c},p_{c}),$
where $\tilde{H}_{j}$ is the Hamiltonian density, one can write from (6\. 18):
$\int
dy\\{u_{c}(x;\xi,\eta),u_{c}(y;\xi,\eta)\\}\frac{\delta\tilde{H}_{j}}{\delta
u_{c}(y,t)}+$ $+\int
dy(\\{u_{c}(x;\xi,\eta),p_{c}(y;\xi,\eta)\\}-\delta(x-y))\frac{\delta\tilde{H}_{j}}{\delta
p_{c}(y,t)}=0$
and
$\int
dy\\{p_{c}(x;\xi,\eta),p_{c}(y;\xi,\eta)\\}\frac{\delta\tilde{H}_{j}}{\delta
p_{c}(y,t)}-$ $-\int
dy(\\{u_{c}(x;\xi,\eta),p_{c}(y;\xi,\eta)\\}-\delta(x-y))\frac{\delta\tilde{H}_{j}}{\delta
u_{c}(y,t)}=0.$
Then one can propose the solutions of these equations:
$\\{u_{c}(x;\xi,\eta),u_{c}(y;\xi,\eta)\\}=\\{p_{c}(x;\xi,\eta),p_{c}(y;\xi,\eta)\\}=0,$
$\\{u_{c}(x;\xi,\eta),p_{c}(y;\xi,\eta)\\}=\delta(x-y).$ (6\. 19)
But it is interesting that the local commutators (6\. 19) are not
satisfied363636That circumstances was mentioned firstly by V.Voronyuk.. One
can see this inserting the soliton solution into (6\. 19). On the other hand
the integral form (6\. 18) is satisfied. All this means that $u_{c}$ and
$p_{c}$ are not the completely independent variables. It must be stressed that
the local relations (6\. 19) are not the necessary conditions in our
formalism.
In our terms, the quantum force $j(x,t)$ excites the $(\xi,\eta)$ manifold
only, leaving the topology of classical trajectory $(u,p)_{c}$ unchanged. We
can use them immediately since the complete set of canonical coordinates
$(\xi,\eta)$ of sin-Gordon model is known, see e.g. [24].
6.2.3. Perturbation theory on the cotangent bundle.
The classical Hamiltonian $h_{j}$ is the sum:
$h_{j}(\eta)=\int
dp\sigma(r)\sqrt{r^{2}+m_{h}^{2}}+\sum^{N}_{i=1}h(\eta_{i}),$ (6\. 20)
where $\sigma(r)$ is the continuous spectrum and $h(\eta)$ is the soliton
energy. Note absence of interaction energy among solitons.
New degrees of freedom $(\xi,\eta)(t)$ must obey the equations (6\. 14):
$\dot{\xi}_{i}=\Omega(\eta_{i})-\int dxj(x,t)\frac{\partial
u_{N}(x;\xi,\eta)}{\partial\eta_{i}},~{}~{}~{}\Omega(\eta)\equiv\frac{\partial
h(\eta)}{\partial\eta},$ $\dot{\eta}_{i}=\int dxj(x,t)\frac{\partial
u_{N}(\xi,\eta)}{\partial\xi_{i}}.$ (6\. 21)
Hence the sources of quantum perturbations are proportional to the time-local
fluctuations of soliton configurations
$\frac{\partial u_{N}(x;\xi,\eta)}{\partial\eta_{i}},~{}~{}~{}\frac{\partial
u_{N}(x;\xi,\eta)}{\partial\xi_{i}}.$
One can split the Lagrange source onto ”Hamiltonian” ones:
$j(x,t)\rightarrow(j_{\xi},j_{\eta}).$
This gives weight functional $U(u_{N};e_{\xi},e_{\eta})$ and operator
$\hat{\mathbb{K}}(e_{\xi},e_{\eta};j_{\xi},j_{\eta})$. In result:
$\displaystyle\rho(q)=\sum_{N}e^{-i\hat{K}(e_{\xi},e_{\eta};j_{\xi},j_{\eta})}\int
D^{N}M(\xi,\eta)e^{iS_{O}(u_{N})}e^{-iU(u_{N};e_{\xi},e_{\eta})}\times$
$\displaystyle\times|\Gamma(q;u_{N})|^{2}$ (6\. 22)
where, using vector notations,
$\hat{\mathbb{K}}(e_{\xi},e_{\eta};j_{\xi},j_{\eta})=\frac{1}{2}\int
dt\\{\hat{j}_{\xi}(t)\cdot\hat{e}_{\xi}(t)+\hat{j}_{\eta}(t)\cdot\hat{e}_{\eta}(t)\\}.$
(6\. 23)
The measure takes the form:
$D^{N}M(\xi,\eta)=\prod^{N}_{i=1}\prod_{t}d\xi_{i}(t)d\eta_{i}(t)\delta(\dot{\xi}_{i}-\Omega(\eta_{i})-j_{\xi,i}(t))\delta(\dot{\eta}_{i}-j_{\eta,i}(t))$
(6\. 24)
The effective interaction potential
$U(u_{N};e_{\xi},e_{\eta})=-\frac{2m^{2}}{\lambda^{2}}\int dxdt\sin\lambda
u_{N}~{}(\sin\lambda e-\lambda e)$ (6\. 25)
with
$e(x,t)=e_{\xi}(t)\cdot\frac{\partial
u_{N}(x;\xi,\eta)}{\partial\eta(t)}-e_{\eta}(t)\cdot\frac{\partial
u_{N}(x;\xi,\eta)}{\partial\xi(t)}.$ (6\. 26)
Performing the shifts:
$\displaystyle\xi_{i}(t)\rightarrow\xi_{i}(t)+\int
dt^{\prime}g(t-t^{\prime})j_{\xi,i}(t^{\prime})\equiv\xi_{i}(t)+\xi^{\prime}_{i}(t),$
$\displaystyle\eta_{i}(t)\rightarrow\eta_{i}(t)+\int
dt^{\prime}g(t-t^{\prime})j_{\eta,i}(t^{\prime})\equiv\eta_{i}(t)+\eta^{\prime}_{i}(t),$
(6\. 27)
we can move the Green function $g(t-t^{\prime})$ into the operator:
$\hat{\mathbb{K}}(e_{\xi},e_{\eta};{\xi}^{\prime},{\eta^{\prime}})=\frac{1}{2}\int
dtdt^{\prime}g(t-t^{\prime})\\{\hat{\xi}^{\prime}(t^{\prime})\cdot\hat{e}_{\xi}(t)+\hat{\eta}^{\prime}(t^{\prime})\cdot\hat{e}_{\eta}(t)\\}.$
(6\. 28)
Notice that the Green function $g(t-t^{\prime})$ of eqs.(6\. 21) is again the
step function:
$g(t-t^{\prime})=\Theta(t-t^{\prime})$ (6\. 29)
Its imaginary part is equal to zero for real times and this allows to shift
$C_{\pm}$ to the real-time axis (see [26]).
In result:
$D^{N}M(\xi,\eta)=\prod^{N}_{i=1}\prod_{t}d\xi_{i}(t)d\eta_{i}(t)\delta(\dot{\xi}_{i}-\Omega(\eta+\eta^{\prime}))\delta(\dot{\eta}_{i})$
(6\. 30)
with
$u_{N}=u_{N}(x;\xi+\xi^{\prime},\eta+\eta^{\prime}).$ (6\. 31)
The equations:
$\dot{\xi}_{i}=\Omega(\eta_{i}+\eta^{\prime}_{i})$ (6\. 32)
are trivially integrable. In quantum case $\eta^{\prime}_{i}\neq 0$ this
equation describes the motion on nonhomogeneous and anisotropic manifold. So,
the expansion over
$(\hat{\xi^{\prime}},~{}\hat{e}_{\xi},~{}\hat{\eta}^{\prime},~{}\hat{e}_{\eta})$
generates the local in time deformations of $\gamma_{N}$ manifold,
$(\xi,\eta)\in\gamma_{N}$ completely. The weight of this deformations is
defined by $U(u_{N};e_{\xi},e_{\eta})$.
Using the definition:
$\int Dx\delta(\dot{x})=\int dx(0)=\int dx_{0}$
functional integrals are reduced to the ordinary integrals over initial data
$(\xi,\eta)_{0}$. This integrals define the zero modes volume.
### 6.3 Quantum corrections
The proof of (6\. 5) we would divide on two parts. First of all we would
consider the semiclassical approximation (Sec.6.3.1) and in Sec.6.3.2. we will
show that this approximation is exact.
6.1. Introduction and definitions.
The $N$-soliton solution $u_{N}$ depends from $2N$ parameters. Half of them
$N$ can be considered as the position of solitons and other $N$ as the
solitons momentum. Generally at $|t|\rightarrow\infty$ the $u_{N}$ solution
decomposed on the single solitons $u_{s}$ and on the double soliton bound
states $u_{b}$ [24]:
$u_{N}(x,t)=\sum^{n_{1}}_{j=1}u_{s,j}(x,t)+\sum^{n_{2}}_{k=1}u_{b,k}(x,t)+O(e^{-|t|})$
We will see later that main elements of our formalism are the one soliton
$u_{s}$ and two-soliton bound state $u_{b}$ configurations. Its $(\xi,\eta)$
parameterizations, confirmed to eqs.(6\. 15), have the form:
$u_{s}(x;\xi,\eta)=-\frac{4}{\lambda}\arctan\\{\exp(m_{h}x\cosh\beta\eta-\xi)\\},~{}~{}~{}\beta=\frac{\lambda^{2}}{8}$
(6\. 33)
and
$u_{b}(x;\xi,\eta)=-\frac{4}{\lambda}\arctan\\{\tan\frac{\beta\eta_{2}}{2}\frac{m_{h}x\sinh\frac{\beta\eta_{1}}{2}\cos\frac{\beta\eta_{2}}{2}-\xi_{2}}{m_{h}x\cosh\frac{\beta\eta_{1}}{2}\sin\frac{\beta\eta_{2}}{2}-\xi_{1}}\\}.$
(6\. 34)
The $(\xi,\eta)$ parametrization of solitons individual energies $h(\eta)$
takes the form:
$h_{s}(\eta)=\frac{m_{h}}{\beta}\cosh\beta\eta,~{}~{}~{}h_{b}(\eta)=\frac{2m_{h}}{\beta}\cosh\frac{\beta\eta_{1}}{2}\sin\frac{\beta\eta_{2}}{2}\geq
0.$
The bound-states energy $h_{b}$ depends from $\eta_{1}$ and $\eta_{2}$. First
one defines inner motion of two bounded solitons and second one the bound
states center of mass motion. Correspondingly we will call this parameters as
the internal and external ones. Note that the inner motion is periodic, see
(6\. 24).
Performing last integration in (6\. 22) with measure (6\. 30) we find:
$\rho(q)=\sum_{N}\int\prod^{N}_{i=1}\\{d\xi_{0}d\eta_{0}\\}_{i}e^{-i\hat{\mathbb{K}}}e^{iS_{O}(u_{N})}e^{-iU(u_{N};e_{\xi},e_{\eta})}|\Gamma(q;u_{N})|^{2}$
(6\. 35)
where
$u_{N}=u_{N}(\eta_{0}+\eta^{\prime},\xi_{0}+\Omega(t)+\xi^{\prime}).$ (6\. 36)
and
$\Omega(t)=\int
dt^{\prime}\Theta(t-t^{\prime})\Omega(\eta_{0}+\eta^{\prime}(t^{\prime}))$
(6\. 37)
In the semiclassical approximation $\xi^{\prime}=\eta^{\prime}=0$ we have:
$u_{N}=u_{N}(x;\eta_{0},\xi_{0}+\Omega(\eta_{0})t).$ (6\. 38)
Note now that if the surface term
$\int\partial_{\mu}(e^{iqx}\partial^{\mu}u_{N})=0$ (6\. 39)
then
$\int d^{2}xe^{iqx}(\partial^{2}+m_{h}^{2})u_{N}(x,t)=-(q^{2}-m_{h}^{2})\int
d^{2}xe^{iqx}u_{N}(x,t)=0$ (6\. 40)
since $q^{2}$ belongs to mass shell by definition. The condition (6\. 39) is
satisfied since $u_{N}$ belong to Schwartz space (the periodic boundary
condition for $u(x,t)$ do not alter this conclusion). Therefore, in the
semiclassical approximation (6\. 5) is hold.
Expending the operator exponent in (6\. 35) we will find the expansion over
$\rho_{n,m}(q)=\frac{(1/2i)^{n}}{n!}\frac{(1/2i)^{m}}{m!}\lim_{(\xi^{\prime},\eta^{\prime},e_{\xi},e_{\eta})=0}\sum_{N}\int
d^{N}\xi_{0}d^{N}\eta_{0}\times$
$\times\int\prod^{n}_{i=1}\\{dt_{i}dt^{\prime}_{i}\theta(t_{i}-t^{\prime}_{i})\hat{\xi}^{\prime}(t^{\prime}_{i})$
$\times\int\prod^{m}_{i=1}\\{dt_{i}dt^{\prime}_{i}\theta(t_{i}-t^{\prime}_{i})\hat{\eta}^{\prime}(t^{\prime}_{i})\\}e^{iS_{O}(u_{N})}|\Gamma(q;u_{N})|^{2}$
$\times\\{\prod^{n}_{i=1}\hat{e}_{\xi}(t_{i})\prod^{m}_{j=1}\hat{e}_{\eta}(t_{j})e^{-iU(u_{N};e_{\xi},e_{\eta})}\\}|_{e=0},$
(6\. 41)
where $U(u_{N};e_{\xi},e_{\eta})$ was defined in (6\. 25), (6\. 26). Notice
that the action of operators $\hat{\xi}^{\prime}$, $\hat{\eta}^{\prime}$
create terms
$\int d^{2}xe^{iqx}\theta(t-t^{\prime})(\partial^{2}+m^{2})u_{N}(x,t)\neq 0.$
(6\. 42)
6.2. Quantum corrections
Now we will show that The semiclassical approximation is exact in the soliton
sector of (6\. 1), (6\. 11) theory.
The structure of the perturbation theory is readily seen in the ”normal-
product” form:
$\rho(q)=\sum_{N}\int\prod^{N}_{i=1}\\{d\xi_{0}d\eta_{0}\\}_{i}:e^{-iU(u_{N};\hat{j}/2i)}e^{iS_{O}(u_{N})}|\Gamma(q;u_{N})|^{2}:,$
(6\. 43)
where
$\hat{j}=\hat{j}_{\xi}\cdot\frac{\partial
u_{N}}{\partial\eta}-\hat{j}_{\eta}\cdot\frac{\partial
u_{N}}{\partial\xi}=\omega\hat{j}_{X}\frac{\partial u_{N}}{\partial X}$ (6\.
44)
and
$\hat{j}_{X}=\int dt^{\prime}\Theta(t-t^{\prime})\hat{X}(t^{\prime})$ (6\. 45)
with $2N$-dimensional vector $X=(\xi,\eta)$. In Eq. (6\. 44) $\omega$ is the
ordinary simplectic matrix.
The colons in (6\. 43) mean that the operator $\hat{j}$ should stay to the
left of all functions. The structure (6\. 44) shows that each order over
$\hat{j}_{X_{i}}$ is proportional at least to the first order derivative of
$u_{N}$ over conjugate to $X_{i}$ variable.
The expansion of (6\. 43) over $\hat{j}_{X}$ can be written [26] in the form
of total derivatives (omitting the semiclassical approximation):
$\rho(q)=\sum_{N}\int\prod^{N}_{i=1}\\{d\xi_{0}d\eta_{0}\\}_{i}\left\\{\sum^{2n}_{i=1}\frac{\partial}{\partial
X_{0i}}P_{X_{i}}(u_{N})\right\\},$ (6\. 46)
where $P_{X_{i}}(u_{N})$ is the infinite sum of ”time-ordered” polynomials
(see [26]) over $u_{N}$ and its derivatives. The explicit form of
$P_{X_{i}}(u_{N})$ is complicated since the interaction potential is non-
polynomial. But it is enough to know, see (6\. 44), that
$P_{X_{i}}(u_{N})\sim\omega_{ij}\frac{\partial u_{N}}{\partial X_{0j}}.$ (6\.
47)
Therefore,
$\rho(q)=0$ (6\. 48)
since (i) each term in (6\. 46) is the total derivative, (ii) we have (6\. 47)
and (iii) $u_{N}$ belongs to Schwartz space.
We can conclude that the equality (6\. 48) is hold since
$\frac{\partial u_{N}}{\partial X_{0}}=0~{}~{}at~{}~{}X_{0}\in\partial W,$
(6\. 49)
where $\partial W$ is the boundary of $W$.
In our consideration we did not touch the continuous spectrum contributions.
In considered approach this contributions are absent since they are realized
on zero measure: theirs contributions are
$\sim\\{volume~{}of~{}\gamma_{N}\\}^{-1}$.
## 7 Summary
Let as summarize the general results of present and of the previous sections.
1\. The $m$\- into $n$-particles transition (non-normalized) $probability$
$R_{nm}$ would have on the Dirac measure the following symmetrical form:
$\rho_{nm}(p_{1},...,p_{n},q_{1},...,q_{m})=<\prod^{m}_{k=1}|\Gamma(q_{k};u)|^{2}\prod^{n}_{k=1}|\Gamma(p_{k};u)|^{2}>_{u}=$
$=e^{-i\hat{K}(j,e)}\int
DM(u)e^{iS_{O}(u)-iU(u,e)}\prod^{m}_{k=1}|\Gamma(q_{k};u)|^{2}\prod^{n}_{k=1}|\Gamma(p_{k};u)|^{2}\equiv$
$\equiv\hat{\cal
O}(u)\prod^{m}_{k=1}|\Gamma(q_{k};u)|^{2}\prod^{n}_{k=1}|\Gamma(p_{k};u)|^{2}.$
(7\. 50)
Here $p(q)$ are the in(out)-going particle momenta. It should be underlined
that this representation is strict and is valid for arbitrary Lagrange theory
of arbitrary dimensions.
2\. The operator $\hat{\cal O}$ contains three element. The Dirac measure
$DM$, the functionals $S_{O}$, $U(x,e)$ and the operator
$\hat{\mathbb{K}}(j,e)$.
The expansion over the operator
$\hat{\mathbb{K}}(j,e)=\frac{1}{2}{\rm Re}\int_{C_{+}}dxdt\frac{\delta}{\delta
j(x,t)}\frac{\delta}{\delta e(x,t)}\equiv\frac{1}{2}{\rm
Re}\int_{C_{+}}dxdt\hat{j}(x,t)\hat{e}(x,t)$ (7\. 51)
generates the perturbation series. We will assume that this series exist (at
least in Borel sense).
3\. The functionals $U(u,e)$ and $S_{O}(u)$ are defined by the equalities:
$S_{O}(u)=(S_{0}(u+e)-S_{0}(u-e))+2{\rm
Re}\int_{C_{+}}dxdte(x,t)(\partial^{2}+m^{2})u(x,t),$ (7\. 52)
$U(u,e)=V(u+e)-V(u-e)-2{\rm Re}\int_{C_{+}}dxdte(x,t)v^{\prime}(u),$ (7\. 53)
where $S_{0}(u)$ is the free part of the Lagrangian and $V(u)$ describes
interactions. The quantity $S_{O}(u)$ is not equal to zero if $u$ have
nontrivial topological charge.
4\. The measure $DM(u,p)$ has the Dirac form:
$DM(u,p)=\prod_{x,t}du(x,t)dp(x,t)\delta\left(\dot{u}-\frac{\delta
H_{j}(u,p)}{\delta p}\right)\delta\left(\dot{p}+\frac{\delta
H_{j}(u,p)}{\delta u}\right)$ (7\. 54)
with the total Hamiltonian
$H_{j}(u,p)=\int dx\\{\frac{1}{2}p^{2}+\frac{1}{2}(\nabla u)^{2}+v(u)-ju\\}.$
(7\. 55)
This last one includes the energy $ju$ of quantum fluctuations.
5\. Dirac measure contains following information:
a. Only $strict$ solutions of equations
$\dot{u}-\frac{\delta H_{j}(u,p)}{\delta p}=0,~{}\dot{p}+\frac{\delta
H_{j}(u,p)}{\delta u}=0$ (7\. 56)
with $j=0$ should be taken into account. This ”rigidness” of the formalism
means the absence of pseudo-solutions (similar to multi-instanton, or multi-
kink) contribution.
b. $\rho_{nm}$ is described by the $sum$ of all solutions of Eq.(7\. 56),
independently from their ”nearness” in the functional space;
c. $\rho_{nm}$ did not contain the interference terms from various
topologically nonequivalent contributions. This displays the orthogonality of
corresponding Hilbert spaces;
d. The measure (7\. 54) includes $j(x)$ as the external adiabatic source. Its
fluctuation disturbs the solutions of Eq.(7\. 56) and vice versa since the
measure (7\. 54) is strict;
e. In the frame of the adiabatical condition, the field disturbed by $j(x)$
belongs to the same manifold (topology class) as the classical field defined
by (7\. 56) [26].
f. The Dirac measure is derived for $real-time$ processes only, i.e. (7\. 54)
is not valid for tunnelling ones. For this reason, the above conclusions
should be taken carefully.
g. It can be shown that theory on the measure (7\. 54) restores ordinary
(canonical) perturbation theory.
6\. The parameter $\Gamma(q;u)$ plays the role of particle production vertex.
It is connected directly with $external$ particle energy, momentum, spin,
polarization, charge, etc., and is sensitive to the symmetry properties of the
interacting fields system. For the sake of simplicity, $u(x)$ is the real
scalar field. The generalization would be evident.
As a consequence of (7\. 54), $\Gamma(q;u)$ is the function of the external
particle momentum $q$ and is a $linear$ functional of $u(x)$:
$\Gamma(q;u)=-\int dxe^{iqx}\frac{\delta S_{0}(u)}{\delta u(x)}=\int
dxe^{iqx}(\partial^{2}+m^{2})u(x),~{}~{}q^{2}=m^{2},$ (7\. 57)
for the mass $m$ field. This parameter presents the momentum distribution of
the interacting field $u(x)$ on the remote hypersurface $\sigma_{\infty}$ if
$u(x)$ is the regular function. Notice, the operator $(\partial^{2}+m^{2})$
cancels the mass-shell states of $u(x)$.
The construction (7\. 57) means, because of the Klein-Gordon operator and
since the external states being mass-shell by definition [33], the solution
$\rho_{nm}=0$ is possible for a particular topology (compactness and analytic
properties) of $quantum$ field $u(x)$. So, $\Gamma(q;u)$ carries the following
remarkable properties:
– it directly defines the observables,
– it is defined by the topology of $u(x)$,
– it is the linear functional of the actions symmetry group element $u(x)$.
If (7\. 56) have nontrivial solution $u_{c}(x,t)$, then this ”extended
objects” quantization problem arises. We solve it introducing convenient
dynamical variables [34]. Then the measure (7\. 54) admits the transformation:
$u_{c}:~{}(u,p)\rightarrow(\xi,\eta)\in W=G/G_{c}.$ (7\. 58)
and the transformed measure has the form:
$DM(u,p)=\prod_{x,t\it C}d\xi(t)d\eta(t)\delta\left(\dot{\xi}-\frac{\delta
h_{j}(\xi,\eta)}{\delta\eta}\right)\delta\left(\dot{\eta}+\frac{\delta
h_{j}(\xi,\eta)}{\delta\xi}\right),$ (7\. 59)
where $h_{j}(\xi,\eta)=H_{j}(u_{c},p_{c})$ is the transformed Hamiltonian.
It is evident that $(\xi,\eta)$ are parameters of integration of eqs.(7\. 56)
and they form the factor space $W=G/G_{c}$. As a result of mapping of the
perturbation generating operator $\hat{\mathbb{K}}$ on the manifold $W$ the
equations of motion became linearized:
$DM=\prod_{t}\delta\left(\dot{\xi}-\frac{\delta
h(\eta)}{\delta\eta}-j_{\xi}\right)\delta\left(\dot{\eta}-j_{\eta}\right).$
(7\. 60)
If Feynman’s $i\epsilon$-prescription is adopted, then the Green function of
Eq.(7\. 60)
$g(t-t^{\prime})=\Theta(t-t^{\prime})$ (7\. 61)
with boundary property:
$\Theta(0)=1.$
7\. Expansion of $\exp\\{\hat{\mathbb{K}}(j,e)\\}$ gives the ”strong coupling’
perturbation series. Its analysis shows that the action of the integro-
differential operator $\hat{\cal O}$ leads to the following representation:
$\rho_{nm}(p,q)=\int_{W}\\{d\xi(0)\cdot\frac{\partial}{\partial\xi(0)}\rho^{\xi}_{nm}(p,q)+d\eta(0)\cdot\frac{\partial}{\partial\eta(0)}\rho^{\eta}_{nm}(p,q)\\}.$
(7\. 62)
This means that the contributions into $R_{nm}(p,q)$ are accumulated strictly
on the boundary, ”bifurcation manifold”, $\partial W$, i.e. depends directly
on the topology property of $W$.
8\. It was shown that the MP is absent in the frame of Lagrangian (6\. 1). For
this purpose one should modify the sin-Gordon Lagrangian adding for instance
the term:
$\frac{1}{2}(\partial\Phi)^{2}-\frac{1}{2}M^{2}\Phi^{2}-\frac{c}{3}u\Phi^{2}$
(7\. 63)
to describe collision of ”external” field $\Phi$ on the solitons. This model
allows to introduce the nontrivial probabilities $\rho(q_{1},q_{2},...)$
considering creation (and absorption) of the field $\Phi$. Note that field
$u(x)$ is still ”confined” even with this adding.
## 8 Conclusion
The final goal of present approach is to construct the workable at arbitrary
distances, i.e. for arbitrary momenta of produced hadrons, $S$-matrix
formalism for theories with (hidden) symmetry. But this aim remains unachieved
in present paper. In subsequent papers more realistic field models in $4d$
Minkowski space-time metric will be described. But one should not consider the
demonstrated examples of Yang-Mills $S$-matrix as the definite proves since I
am note sure that the used $O(4)\times O(2)$ solution of Yang-Mills equation
in the Minkowski in the situation of general position guarantee the largest
contribution. Moreover, only the $SU(2)$ theory will be considered.
Unfortunately we can not find in the frame of t’Hooft ansatz [35] the solution
for larger $SU(N)$ group [36].
It will be to shown how one or another physical phenomena may be seen in the
field theory with symmetry. Namely,
— no plain waves production exist in theories with symmetry,
i.e. for instance the gluons can not be seen in a free state since simply the
last ones are absent in quantum theory of the symmetry manifolds, or, in other
words, since the gluon states and the ”states” of the symmetry manifold belong
to the orthogonal Hilbert spaces. The quark fields will not be included in
this simplest example. But more realistic model with quarks shows that
— inclusion of matter can not change previous conclusion that the gluons can
not be created.
In the other example we will show how the
— binding potential may arise among quarks.
Here the situation of general position selection rule will be extremely
important: it will be used that the situation when $(q\bar{q})$ potential is
independent from the scale of Yang-Mills fields is mostly probable.
The quantum field theory with constraints will obey following important
property:
— the perturbation theory of quantum systems with symmetry may be free from
any divergences,
i.e. it $may$373737One can not be sure that the approach is universal, can be
used, for instance, in quantum gravity case. be rightful at arbitrary
distances, for VHM case as well. It is the evident consequence of lessening of
the number of dynamical degrees of freedom because of symmetry
constraints383838 And it is unnecessary to have in that case any new
mechanism, such as the supersymmetry for example, to achieve the field theory
without divergences. Possible scenario of such theory will be discussed
later..
Exist also the intriguing question of asymptotic freedom. The point is that
there is no running coupling constants in our strong coupling perturbation
theory without divergences. On the other hand the asymptotic freedom is the
experimental fact. We will show how
— the effect of asymptotic freedom may arise
in our quantum theory of the symmetry manifolds. The main question here is to
find the experimentally observable corrections to the asymptotic freedom law.
In summary, the aim of future publications would be the question: is the
offered approach complete from physical point of view? It is important since
offered quantization scheme in the situation of general position on Dirac
measure must be true for arbitrary distances, since it is free from arbitrary
scale parameters393939That is why I hope that it may give the predictions
acceptable from physical point of view at arbitrary distances..
Acknowledgments
First of all I am thankful to Alexei Sisakian for fruitful conversations
during the work upon the topology conserving perturbation theories ideology.
The offered text was arranged under last, before his sudden death, proposition
to put in order my present-day understanding of the approach. I would like to
note the significant role of E.Levin and L.Lipatov in realization of discussed
formalism. I am grateful to V.Kadyshevski for interest to the discussed in the
paper questions. Various parts of the approach were offered to auditory of
many Institutes and Universities and I am grateful for theirs interest and
comments.
## References
* [1] J.Manjavidze, Sov.J.Nucl.Phys., 45 (1987) 442
* [2] M.V.Fedoryuk, Asymptotics: integrals and series (Nauka, Moscow, 1987)
* [3] G.Parisi and Y.Wu, Ecientia Sinica, 24 (1981) 483, A.A.Migdal and T.A.Kozhamkulov, Yad. Phys., 39 (1984) 1596 [Sov. J. Nucl. Phys. 39 (1984) 1012], L.Lipatov
* [4] A.N.Kolmogorov, DAN SSSR, 98 (1954) 527; V.I.Arnold, Izv. AN SSSR, 25 (1961) 21, V.I.Arnold, UMN, 18 (1963) 81; Yu.Mozer, Math. Phys., Bd.11a (1962) 1
* [5] J.S.Dowker, Ann. Phys.(NY), 62 (1971) 361
* [6] M.S.Marinov, Phys. Rep., 60 (1980) 1
* [7] S.F.Edvards and Y.Guliaev, Proc. Roy. Soc., A279 (1964) 229
* [8] R.P.Feynman and A.R.Hibbs, Quantum Mechanics and Path Integrals, (McGraw-Hill, New York, 1965)
* [9] C.Grosche, Path Integrals, Hyperbolic Spaces, and Selberg Trace Formulae (World Scint., Singapore, New Jersey, london, Hong Kong, 1995)
* [10] H.Kleinert, Path Integrals in Quantum Mechanics, Statistics and Polimer Physics (World Scientific, Singapore, 1989)
* [11] R.Dashen, B.Hasslacher and A.Neveu, Phys. Rev., D10 (1974) 4114
* [12] V.I.Arnold, Mathematical Methods of Classical Mechanics, (Springer Verlag, New York, 1978)
* [13] P.A.M.Dirac, Lectures on quantum mechanics (Yeshiva Univ., New York, !964)
* [14] V.E.Korepin and L.D.Faddeev, Sov. TMF, 25 (1975) 147
* [15] L.Faddeev and V.Korepin, Phys. Rep., 42C (1978) 3; J.Goldstone and R.Jackiw, Phys.Rev., D11 (1975) 1486; V.A.Rubakov, Classical and Gauge Fields (Editorial URSS, Moscow, 1999)
* [16] S.Coleman, Whys in Subnuclear Physics, ed. by Zichichi, Ettore Majorana School, Erice, Italy (1976)
* [17] R.Mills, Propagators of Many-Particles Systems (Gordon & Breach, 1969)
* [18] K.Osterwalder and E.Seiler, Ann. Phys. (N.Y.) 110 (1978) 440
* [19] J.Manjavidze and A.Sissakian, Theor. Math. Phys., 130 (2002) 153
* [20] I.H.Duru, Phys. Rev., D30 (1984) 143
* [21] G.Pocshle and E.Teller, Zs. Phys. 83 (1933) 143
* [22] J.Manjavidze, J.Math.Phys. 41 (2000) 5710
* [23] V.Fock, Zs. Phys., 98 (1935) 145; V.Bargman, Zs. Phys., 99 (1935) 576; V.S.Popov, High Energy Physics and Elementary Particles Theory, (Naukova Dumka, Kiev, 1967)
* [24] L.A.Takhtajan and L.D.Faddeev, Hamiltonian Approach in Solitons Theory (Moskow, Nauka, 1986)
* [25] R.Abraham and J.E.Marsden, Foundations of Mechanics (Benjamin/ Cummings Publ. Comp., Reading, Mass., 1978)
* [26] J.Manjavidze, Perturbation Theory on the Imvariant Subspace, hep-th/9801188
* [27] S.Smale, Inv.Math., 10:4 (1970) 305, ibid., 11:1 (1970) 45
* [28] I.H.Duru and H.Kleinert, Phys. Lett., 84B (1979) 185
* [29] J.Manjavidze and A.Sisakian, Phys. Rep., 346 (2001) 1
* [30] Zamolodchikov, A.B. and A.B.Zamolodchikov, Phys. Lett., 72B, 503 (1978)
* [31] Solitons. (Ed. by R.K.Bullough and J.Caudry, Springer-Verlag, Berlin, Heidelberg, New York, 1980); T.D.Lee, Phys. Scr., 20 (1979) 440; R.Jakiw, Rev. Mod. Phys., 49 (1977) 681; J.Goldstone and R.Jackiw, Phys. Rev., D11, (1975) 1485; R.Rajaraman, Solitons and Instantons (North-Holland Publ. Comp., Amsterdam, New York, Oxford, 1982)
* [32] J.M.Souriae, Structure des Systems Dynamiques (Dunod, Paris, 1970)
* [33] L.Landau and R.Peierls, Zs.Phys., 69 (1931) 56
* [34] J.Manjavidze and A.Sissakian, J. Math. Phys. 42, (2001) 641
* [35] G.t’Hooft, ”Computation of the quantum effects due to a four-dimensional pseudoparticle”, Phys. Rev. D14 (1976) 3432
* [36] J.Manjavidze and V.Voronyuk, Phys.Part.Nucl.Lett. 3, (2006) 391
|
arxiv-papers
| 2011-01-06T11:31:31 |
2024-09-04T02:49:16.180121
|
{
"license": "Public Domain",
"authors": "J. Manjavidze",
"submitter": "Joseph Manjavidze",
"url": "https://arxiv.org/abs/1101.1193"
}
|
1101.1496
|
# On the k-nullity foliations in Finsler geometry and completeness
B. Bidabad111Faculty of Mathematics, Amirkabir University of Technology,
Tehran, Iran.(email:bidabad@aut.ac.ir) and M. Rafie-Rad222Faculty of
Mathematics, Mazandaran University, Babolsar, Iran.(email:
m.rafiei.rad@gmail.com)
###### Abstract
Here, a Finsler manifold $(M,F)$ is considered with corresponding curvature
tensor, regarded as $2$-forms on the bundle of non-zero tangent vectors.
Certain subspaces of the tangent spaces of $M$ determined by the curvature are
introduced and called $k$-nullity foliations of the curvature operator. It is
shown that if the dimension of foliation is constant then the distribution is
involutive and each maximal integral manifold is totally geodesic.
Characterization of the $k$-nullity foliation is given, as well as some
results concerning constancy of the flag curvature, and completeness of their
integral manifolds, providing completeness of $(M,F)$. The introduced
$k$-nullity space is a natural extension of nullity space in Riemannian
geometry, introduced by S. S. Chern and N. H. Kuiper and enlarged to Finsler
setting by H. Akbar-Zadeh and contains it as a special case.
Keywords: Foliation, k-nullity, Finsler manifolds, Curvature operator.
MSC: 2000 Mathematics subject Classification: 58B20, 53C60, 53C12.
## 1 Introduction
Foliations of manifolds occur naturally in various geometric contexts. They
arise in connections with some essential topics as vector fields without
singularities, integrable $m$-dimensional distributions, submersions and
fibrations, actions of Lie groups, direct constructions of foliations such as
Hopf fibrations, Reeb foliations and finally they appear in the existence
study of solution of certain differential equations. In the later case, S.
Tanno in [15] applied the concept of the k-nullity spaces to achieve a
complete proof for the famous Obata Theorem which is a subject of numerous
rigidity results in Riemannian geometry. The nullity space of the Riemannian
curvature tensor was first studied by S. S. Chern and N. H. Kuiper [3] in
1952. They have shown that, if the index of nullity, $\mu$, of a Riemannian
manifold is locally constant, then the manifold admits a locally integrable
$\mu$-dimensional distribution whose integral submanifolds are locally flat.
O. Kowalski and M. Sekizawa have proved that vanishing of the index of nullity
in some senses, results that the tangent sphere bundle is a space of negative
scalar curvature [8].
The concept of nullity spaces are generalized to the ${\bf k}$-nullity spaces
in Riemannian geometry in a number of works such as [4, 7] and [11].
In this work we answer to the following natural questions: Is there any
extension for the concept of k-nullity space in Finsler geometry? Is its
maximal integral manifold totally geodesic? And finally is its maximal
integral manifold complete, provided that $(M,F)$ is complete? Fortunately,
the answer to these questions is affirmative. More precisely, we obtain the
following results.
###### Theorem 1.1.
Let $(M,F)$ be a Finsler manifold for which the index of k-nullity $\mu_{\bf
k}$ be constant on an open subset $U\subseteq M$. Then, the local ${\bf
k}$-nullity distribution on $U$ is completely integrable.
###### Theorem 1.2.
The ${\bf k}$-nullity space of a Finsler manifold $(M,F)$ at a point $x\in M$,
coincides with the kernel of the related curvature operator of $\Omega$.
D. Ferus has proved that the maximal integral manifolds of nullity foliation
are totally geodesic [6]. This result has been extended to the Finsler case by
H. Akbar-Zadeh [2]. Here, we prove the same result for k-nullity foliation in
Finsler manifolds.
###### Theorem 1.3.
Let $(M,F)$ be a Finsler manifold. If the ${\bf k}$-nullity space is locally
constant on the open subset $U$ of $M$, then every $\bf k$\- nullity integral
manifold $N$ in $U$ is an auto-parallel Finsler submanifold with non-negative
constant flag curvature k. Moreover, $(N,\tilde{F})$ is a $P$-symmetric space.
The completeness of the nullity foliations is studied by D. Ferus in [5]. The
similar result is carried out for Finsler manifolds by H. Akbar-Zadeh [2] in
1972.
###### Theorem 1.4.
Let $(M,F)$ be a complete Finsler manifold and $G$ an open subset of $M$ on
which $\mu_{\bf k}$ is minimum. Then, every integral manifold of the k-nullity
foliation in $G$ is a complete submanifold of $M$.
It is worth mentioning that M. Sekizawa and S. Tachibana have studied $k^{th}$
nullity foliations as another generalization of Chern and Kuiper’s nullity in
Riemannian geometry by considering $k^{th}$ consecutive derivative of the
curvature tensor [13, 14].
## 2 Preliminaries and terminologies.
### 2.1 Regular connections and Finsler manifolds.
Let $M$ be a connected differentiable manifold of dimension $n$. We adopt here
the notations and terminologies of [1]. Denote the bundle of tangent vectors
of $M$ by $p:TM\longrightarrow M$, the fiber bundle of non-zero tangent
vectors of $M$ by $\pi:TM_{0}\longrightarrow M$ and the pulled-back tangent
bundle by $\pi^{*}TM\longrightarrow TM_{0}$. Any point of $TM_{0}$ is denoted
by $z=(x,v)$, where $x=\pi z\in M$ and $v\in T_{\pi z}M$. By $TTM_{0}$ we
denote the tangent bundle of $TM_{0}$ and by $\varrho$ the canonical linear
mapping
$\varrho:TTM_{0}\longrightarrow\pi^{*}TM,$
where $\varrho=\pi_{*}$. For all $z\in TM_{0}$, let ${\cal V}_{z}TM$ be the
set of vertical vectors at $z$, that is, the set of vectors which are tangent
to the fiber through $z$. Equivalently, ${\cal V}_{z}TM=\ker\pi_{*}$ where
$\pi_{*}:TTM_{0}\longrightarrow TM$ is the linear tangent mapping.
Let $\nabla$ be a linear connection on the vector bundle
$\pi^{*}TM\longrightarrow TM_{0}$. We define a linear mapping
$\mu:TTM_{0}\longrightarrow\pi^{*}TM,$
by $\mu(\hat{X})=\nabla_{\hat{X}}{\bf v}$ where $\hat{X}\in TTM_{0}$ and ${\bf
v}$ is the canonical section of $\pi^{*}TM$.
The connection $\nabla$ is said to be regular, if $\mu$ defines an isomorphism
between ${\cal V}TM_{0}$ and $\pi^{*}TM$. In this case, there is the
horizontal distribution ${\cal H}TM$ such that we have the Whitney sum:
$TTM_{0}={\cal H}TM\oplus{\cal V}TM.$
This decomposition permits to write a vector $\hat{X}\in TTM_{0}$ into the
form $\hat{X}=H\hat{X}+V\hat{X}$ uniquely. In the sequel we denote all vector
fields on $TM_{0}$ by $\hat{X},\hat{Y}$, etc and the corresponding sections of
$\pi^{*}TM$ by $X=\varrho(X)$, $Y=\varrho(Y)$, etc, respectively, unless
otherwise specified.
The structural equations of the regular connection $\nabla$ are given by:
$\tau(\hat{X},\hat{Y})=\nabla_{\hat{X}}Y-\nabla_{\hat{Y}}X-\varrho[\hat{X},\hat{Y}],$
(1)
$\Omega(\hat{X},\hat{Y})Z=\nabla_{\hat{X}}\nabla_{\hat{Y}}Z-\nabla_{\hat{Y}}\nabla_{\hat{X}}Z-\nabla_{[\hat{X},\hat{Y}]}Z,$
(2)
where $X=\varrho(\hat{X})$, $Y=\varrho(\hat{Y})$, $Z=\varrho(\hat{Z})$ and
$\hat{X}$, $\hat{Y}$ and $\hat{Y}$ are vector fields on $TM_{0}$. The tensors
$\tau$ and $\Omega$ are called Torsion and Curvature tensors of $\nabla$,
respectively. They determine two torsion tensors denoted here by $S$ and $T$
and three curvature tensors denoted by $R$, $P$ and $Q$ defined by:
$S(X,Y)=\tau(H\hat{X},H\hat{Y}),\ \ \ T(\dot{X},Y)=\tau(V\hat{X},H\hat{Y}),$
$R(X,Y)=\Omega(H\hat{X},H\hat{Y}),\ \ P(X,\dot{Y})=\Omega(H\hat{X},V\hat{Y}),\
\ Q(\dot{X},\dot{Y})=\Omega(V\hat{X},V\hat{Y}),$
where $X=\varrho(\hat{X})$, $Y=\varrho(\hat{Y})$, $\dot{X}=\mu(\hat{X})$ and
$\dot{Y}=\mu(\hat{Y})$. The tensors $R$, $P$ and $Q$ are called $hh-$, $hv-$
and $vv-$curvature tensors, respectively. Using the Jacobi identity for three
vector fields ${\hat{X}}$, ${\hat{Y}}$ and ${\hat{Z}}$, one obtains the
Bianchi identities for a regular connection $\nabla$ with curvature 2-forms
$\Omega$, as follows:
$\sigma\Omega({\hat{X}},{\hat{Y}})Z=\sigma\nabla_{{\hat{Z}}}\tau({\hat{X}},{\hat{Y}})+\sigma\tau({\hat{Z}},[{\hat{X}},{\hat{Y}}]),$
(3)
$\sigma\nabla_{{\hat{Z}}}\Omega({\hat{X}},{\hat{Y}})+\sigma\Omega({\hat{Z}},[{\hat{X}},{\hat{Y}}])=0,$
(4)
where, $\sigma$ denotes the circular permutation in the set
$\\{{\hat{X}},{\hat{Y}},{\hat{Z}}\\}$.
Let $(x^{i})$ be a local chart with the domain $U\subseteq M$ and
$(x^{i},v^{i})$ the induced local coordinates on $\pi^{-1}(U)$ where ${\bf
v}=v^{i}\frac{\partial}{\partial x^{i}}\in T_{\pi z}M$, where $i$ run over the
range $1,2,...,n$. A Finsler structure $F$ is defined to be a function $F$ on
$TM_{0}$ satisfying the following conditions: (1)$\ F>0\ \textrm{and}\
C^{\infty}\ \textrm{on}\ TM_{0}$, (2)$\ F(x,\lambda v)=\lambda F(x,v),$ for
every $\lambda>0$ and (3)$\ \
g_{ij}(x,v)=\frac{1}{2}\frac{\partial^{2}F^{2}}{\partial v^{i}\partial v^{j}}$
is positive definite. The pair $(M,F)$ is called a Finsler manifold.
There is a unique regular connection associated to $F$ such that:
$\displaystyle\nabla_{\hat{Z}}g$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle S(X,Y)$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle
g(\tau(V\hat{X},\hat{Y}),Z)$ $\displaystyle=$ $\displaystyle
g(\tau(V\hat{X},\hat{Z}),Y),$
where $X=\varrho(\hat{X})$, $Y=\varrho(\hat{Y})$ and $Z=\varrho(\hat{Z})$ for
all $\hat{X}$, $\hat{Y}$, $\hat{Z}\in TTM_{0}$. The regular connection
$\nabla$ is called the Cartan connection. Given an induced natural coordinates
on $\pi^{-1}(U)$, the coefficients of $\nabla$ can be written as follows:
$\nabla_{\partial_{j}}\partial_{i}=\Gamma^{k}_{\ ij}\partial_{k},\ \
\nabla_{\overset{\bullet}{\partial}_{j}}\partial_{i}=C^{k}_{\
ij}\partial_{k},$
where, $\partial_{i}=\frac{\partial}{\partial x^{i}},\
\overset{\bullet}{\partial}_{i}=\frac{\partial}{\partial v^{i}}$ and
$\Gamma^{k}_{\ ij}$ and $C^{k}_{\ ij}$ are smooth functions defined on
$\pi^{-1}(U)$. One can observe that components of the second torsion tensor
$T$ coincides with components of Cartan tensor $C$ in this coordinates, that
is $T_{ijk}=\frac{1}{2}\overset{\bullet}{\partial}_{k}g_{ij}$, where,
$T_{ijk}=g_{ir}T^{r}_{\ jk}$. It can be shown that the set $\\{\delta_{j}\\}$
defined by $\delta_{j}=\partial_{j}-\Gamma^{k}_{\
0j}\overset{\bullet}{\partial}_{k}$ form a local frame field for the
horizontal space ${\cal H}TM$. Assume that
$\nabla_{\delta_{j}}\partial_{i}=\overset{*}{\Gamma}{{}^{k}_{\
ij}}\partial_{k}$. One can easily see that $\overset{*}{\Gamma}{{}^{k}_{\
ij}}$ is symmetric with respect to the indices $i$ and $j$. The curvature
operator $\Omega(\hat{X},\hat{Y})$ of Cartan connection is anti-symmetric in
the following sense
$g(\Omega(\hat{X},\hat{Y})Z,W)=-g(\Omega(\hat{X},\hat{Y})W,Z),$ (5)
where $\hat{X},\hat{Y}\in{\cal X}(TM_{0})$, $Z=\varrho(\hat{Z})$ and
$W=\varrho(\hat{W})$. The $hv-$curvature tensor $P$ and the $vv-$ curvatures
tensor $Q$ of the Cartan connection $\nabla$ are given respectively by
$P^{i}_{\ jkl}=\nabla^{i}T_{jkl}-\nabla_{j}T^{i}_{\ kl}+T^{i}_{\
kr}\nabla_{0}T^{r}_{\ jl}-T^{r}_{\ kj}\nabla_{0}T^{i}_{\ rl},$ (6) $Q^{i}_{\
jkl}=T^{i}_{\ rl}T^{r}_{\ jk}-T^{i}_{\ rk}T^{r}_{\ jl}.$ (7)
Among the Finsler manifolds, there are some classes determined by non-
Riemannian quantities. One of them which is appeared in the present work is
the $P$-symmetric Finsler manifolds required a kind of partial symmetry in the
indices of $P$. This class of Finsler manifolds, was introduced by M.
Matsumoto an H. Shimada in [9] and [10], and has been extensively studied by
many authors.
The curvature tensor $P^{i}_{\ jkl}$ can be decomposed into the sum of two
symmetric and anti-symmetric tensors with respect to the indices $k$ and $l$,
that is to say $P={{}^{s}P}+{{}^{a}P}$. By means of Eq.(6) the symmetric
tensor ${{}^{s}P}$ can be written in the following form:
${{}^{s}P}^{i}_{\ jkl}=\nabla^{i}T_{jkl}+\frac{1}{2}\\{T^{i}_{\
kr}\nabla_{0}T^{r}_{\ jl}-T^{r}_{\ kj}\nabla_{0}T^{i}_{\ rl}+T^{i}_{\
lr}\nabla_{0}T^{r}_{\ jk}-T^{r}_{\ lj}\nabla_{0}T^{i}_{\ rk}\\}.$ (8)
A Finsler manifold is said to be P-symmetric if $P(X,Y)=P(Y,X),$ $\forall
X,Y\in\Gamma(\pi^{*}TM)$. $P$-symmetric spaces are closely related to the
Finsler manifolds of isotropic sectional curvature. In this relation the
following result is well-known.
###### Proposition 2.1.
[9] A Finsler manifold is $P$-symmetric if and only if $\nabla_{\hat{\bf
v}}Q=0$.
Next we consider the Berwald connection $D$ which is not metric-compatible but
a torsion free regular connection relative to $F$. There is the following
relation between the connections $\nabla$ and $D$
$D_{H\hat{X}}Y=\nabla_{H\hat{X}}Y+(\nabla_{\hat{{\bf v}}}T)(X,Y),\ \ \ \
D_{V\hat{X}}Y=(V\hat{X}.Y^{i})\partial_{i}\,$ (9)
where the vector field $\hat{{\bf v}}=v^{i}\delta_{i}$ is the canonical
geodesic spray of $F$. If we assume $D_{\delta_{j}}\partial_{i}=G^{k}_{\
ij}\partial_{k},\ \ $ then Eqs.(9) can be written in the following local form:
$G^{i}_{\ jk}=\overset{*}{\Gamma}{{}^{i}_{\ jk}}+\nabla_{0}T^{i}_{\ jk},\ \ \
D_{\overset{\bullet}{\partial}_{j}}\partial_{i}=0.$
It is clear from Eq.(9) that the connections $D$ and $\nabla$ associate to the
same geodesic spray, since we have $\nabla_{\hat{X}}{\bf v}=D_{\hat{X}}{\bf
v}$. The metric tensor $g$ related to the Finsler structure $F$ is parallel
along any geodesic of Berwald connection, that is equivalent to $D_{\hat{\bf
v}}g=0$. The Berwald connection $D$ admits the $hh-$ curvature tensors $H$ and
the $hv-$ curvature tensors $G$ with the components $H^{i}_{\ jkl}$ and
$G^{i}_{\ jkl}$. $G^{i}_{\ jkl}$ and $H^{i}_{\ jkl}$ can be determined by
$G^{i}_{\
jkl}=\overset{\bullet}{\partial}_{l}\overset{\bullet}{\partial}_{k}\overset{\bullet}{\partial}_{j}G^{i}=\overset{\bullet}{\partial}_{l}G^{i}_{\
jk},$ $H^{i}_{\ jkl}=(\delta_{k}G^{i}_{\ jl}-G^{i}_{\ ljs}G^{s}_{\
k})-(\delta_{l}G^{i}_{\ jk}-G^{i}_{\ kjs}G^{s}_{\ l})+G^{i}_{\ rk}G^{i}_{\
jl}-G^{i}_{\ rl}G^{i}_{\ jk}.$
Let $z\in TM_{0}$ and ${\cal P}({\bf v},X)\subseteq T_{\pi z}M$ be a plane,
generated by ${\bf v}$ and a linearly independent vector $X$ in $T_{\pi z}M$.
The flag curvature at the point $z\in TM_{0}$ with respect to ${\cal P}({\bf
v},X)$ is denoted by ${\bf K}(z,{\cal P}({\bf v},X))$ and is defined as
follows:
${\bf K}(z,{\cal P}({\bf v},X))=\frac{g(R(X,{\bf v}){\bf
v},X)}{g(X,X)F^{2}-g(X,{\bf v})^{2}},$
where, $R$ denotes the $hh-$curvature of Cartan connection [12]. Note that,
the flag curvature ${\bf K}(z,{\cal P}({\bf v},X))$ does not depend on the
choice of Berwald and Cartan connection, since, after a simple calculation. In
fact, one can easily show that
$H(X,{\bf v}){\bf v}=R(X,{\bf v}){\bf v}.$ (10)
The Finsler manifold $(M,F)$ is said to be of scalar flag curvature at the
point $z\in TM_{0}$ if ${\bf K}(z,P({\bf v},X))$ does not depend on the choice
of the plane ${\cal P}({\bf v},X)$ and it is said to be of scalar flag
curvature if it is of scalar flag curvature at all points $z\in TM_{0}$. In
this case we have:
$R(X,{\bf v}){\bf v}={\bf K}(z)\\{F^{2}X-g(X,{\bf v}){\bf v}\\},\ \ \ \forall
X\in\Gamma(\pi^{*}TM).$
### 2.2 Finsler submanifolds.
Let $S$ be a k-dimensional embedded submanifold of the Finsler manifold
$(M,F)$ defined by embedding ${\bf i}:S\longrightarrow M$. We identify a point
$\tilde{x}\in S$ and a tangent vector $\widetilde{X}\in T_{\tilde{x}}S$ by by
${\bf i}(\tilde{x})$ and ${\bf i}_{*}\widetilde{X}$, respectively. Hence,
$T_{\tilde{x}}S$ can be considered as a subspace of $T_{\tilde{x}}M$. The
embedding ${\bf i}$ induces a map $\tilde{{\bf i}}={\bf
i}_{*}:TS_{0}\longrightarrow TM_{0}$. If we identify a point $\widetilde{z}\in
TS_{0}$ with its image $\tilde{{\bf i}}(\widetilde{z})$, then $TS_{0}$ can be
considered as a sub-fiber bundle of $TM_{0}$. Restricting the map
$\pi:TM_{0}\longrightarrow M$ to $TS_{0}$, we obtain the mapping
$q:TS_{0}\longrightarrow M$. Denote by $\hat{T}S={\bf i}^{*}TM$, the pulled
back bundle of $TM$. The Finsler metric $g$ on $M$ induces a Finsler metric on
$S$ which is denoted by $\widetilde{g}$. Given any point
$\widetilde{x}=q(\widetilde{z})\in S$, where $\widetilde{z}\in TS_{0}$, we
denote by $N_{q(\widetilde{z})}$ the orthogonal complementary subspace of
$T_{q(\widetilde{z})}M$ in $\hat{T}_{q(\widetilde{z})}S$. Therefore we have
the Whitney sum
$\hat{T}_{q(\tilde{z})}S=T_{q(\tilde{z})}S\oplus N_{q(\tilde{z})}.$ (11)
The above decomposition defines the two projection maps ${\bf P}_{1}$ and
${\bf P}_{2}$ as follows:
${\bf P}_{1}:\hat{T}S\longrightarrow TS,$ ${\bf P}_{2}:\hat{T}S\longrightarrow
N,$
where $N=\bigcup_{\tilde{z}\in TS_{0}}N_{q(\tilde{z})}$. We have
$q^{*}\hat{T}S=q^{*}TS\oplus N$. $N$ is called the normal fiber bundle. We
denote by $\rho$ the canonical linear mapping $TTS_{0}\longrightarrow
q^{*}TS$, that is, $\rho=q_{*}$. Let $\widetilde{X}$ and $\widetilde{Y}$ be
two vector fields on $TS_{0}$. Given $\tilde{z}\in TS_{0}$,
$(\nabla_{\widetilde{X}}Y)_{\tilde{z}}$ belongs to $\hat{T}_{q(\tilde{z})}S$.
Therefore, using the decomposition (11), we get
$\nabla_{\widetilde{X}}Y=\widetilde{\nabla}_{\widetilde{X}}Y+\alpha(\widetilde{X},Y),$
(12)
where, $\nabla$ is the Cartan connection, $Y=\rho(\widetilde{Y})$,
$\widetilde{\nabla}_{\widetilde{X}}Y\in T_{q(\tilde{z})}S$ and
$\alpha(\widetilde{X},Y)\in N_{q(\tilde{z})}$. $\alpha$ is called the second
fundamental form of $S$. From Eq.(12), it follows that, $\widetilde{\nabla}$
is a covariant derivative in the vector bundle $q^{*}TS\longrightarrow TS_{0}$
and satisfies $\widetilde{\nabla}\widetilde{g}=0$. $\widetilde{\nabla}$ is
called the tangential covariant derivation.
$\alpha(\widetilde{X},\rho(\widetilde{Y}))$ is a bilinear form possessing its
values in $N$. Let us denote by $\widetilde{\tau}$ the torsion tensor of
$\widetilde{\nabla}$. Then, we have
${\bf
P}_{1}\tau(\widetilde{X},\widetilde{Y})=\widetilde{\tau}(\widetilde{X},\widetilde{Y})=\widetilde{\nabla}_{\widetilde{X}}Y-\widetilde{\nabla}_{\widetilde{Y}}X-\rho[\widetilde{X},\widetilde{Y}],$
${\bf
P}_{2}\tau(\widetilde{X},\widetilde{Y})=\alpha(\widetilde{X},Y)-\alpha(\widetilde{Y},X),$
where $X=\rho(\widetilde{X})$ and $Y=\rho(\widetilde{Y})$. The submanifold $S$
is said to be totally geodesic at a point $\tilde{x}\in S$ if, for every
tangent vector $\widetilde{X}\in T_{\tilde{x}}S$, the geodesic $\gamma(t)$ of
$M$ in the direction of $\widetilde{X}$ lies in $S$ for small values of the
parameter $t$. If $S$ is totally geodesic at every point of $S$, it is called
a totally geodesic submanifold of $M$.
###### Theorem 2.2.
[1] Let $S$ be a submanifold of the Finsler manifold $(M,F)$ with the second
fundamental form $\alpha$. Then, $S$ is a totally geodesic submanifold if and
only if $\alpha(\widetilde{X},{\bf v})=0$, for all $\widetilde{X}\in{\cal
X}(TS_{0})$.
The submanifold $S$ is also said to be an auto-parallel submanifold of $M$ if
the second fundamental form $\alpha$ vanishes identically. Note that, in the
Riemannian manifolds, the concepts of auto-parallel and totally geodesic
submanifolds coincide. Clearly, every auto-parallel submanifold is also
totally geodesic. Notice that, on an auto-parallel submanifold $S$, the
induced connection $\widetilde{\nabla}$ coincides with the Cartan connection
of the induced Finsler structure $\widetilde{F}=\tilde{{\bf i}}^{*}F$.
### 2.3 Nullity space of curvature operator in Finsler geometry.
Let $(M,F)$ be a Finsler manifold and $\nabla$ the Cartan connection related
to $F$. Given any point $z\in TM_{0}$, consider the subspace of ${\cal
H}_{z}TM$ defined by:
$N_{z}:=\\{\hat{X}\in{\cal H}_{z}TM|\ \ \Omega(\hat{X},\hat{Y})=0,\ \
\forall\hat{Y}\in{\cal H}_{z}TM\\},$
where, $\Omega$ is the curvature operator of $\nabla$. For any point $z\in
TM_{0}$ where $\pi z=x$. The subspace ${\cal N}_{x}=\varrho(N_{z})\subset
T_{x}M$ is linearly isomorphic to $N_{z}$. ${\cal N}_{x}$ is called the
nullity space of the curvature operator on the Finsler manifold $(M,F)$ at the
point $x\in M$, while ${\cal N}$ will denote the field of nullity spaces. Its
orthogonal complementary space in $T_{x}M$ is called the co-nullity space at
$x$ and is denoted by $\overset{\perp}{{\cal N}}_{x}$. Every element of ${\cal
N}_{x}$ is called a nullity vector. The non-negative integer valued function
$\mu_{0}:M\longrightarrow I\\!{N}$ defined by $\mu_{0}(p)=\dim{\cal N}_{p}$ is
called the index of nullity and $\mu_{0}(p)$ is called the index of nullity at
the point $p\in M$. Nullity space is called locally constant if given any
$x\in M$, there is a neighborhood $U$ of $x$ such that the function $\mu_{0}$
is constant on $U$. In this case, the correspondence $x\in U\mapsto{\cal
N}_{x}$ is a distribution called the nullity distribution on $U$. In the
sequel we assume $0<\mu_{0}<n$ unless otherwise specified.
Let $\ker_{x}\Omega$ be the kernel of the operator $\Omega$, that is
$\ker_{x}\Omega=\\{Z\in T_{x}M|\ \Omega(\hat{X},\hat{Y})Z=0,\
\forall\hat{X},\hat{Y}\in{\cal H}_{z}TM\\}.$ (13)
Akbar-Zadeh proved that, ${\cal N}_{x}=\ker_{x}\Omega$ and moreover, if the
nullity space is locally constant on $U$, then the nullity distribution on $U$
is completely integrable. This is an extension of the similar result in
Riemannian manifolds, established by Maltz [11] and Gray [7]. Akbar-Zadeh
proved the following result:
###### Theorem 2.3.
Let $(M,F)$ be a complete Finsler manifold and $G$ an open subset in $M$ on
which $\mu_{0}$ is minimum. Then, every nullity manifold is a geodesically
complete submanifold of $M$.
## 3 k-Nullity space of Cartan connection’s curvature operator.
Let $(M,F)$ be an n-dimensional Finsler manifold endowed with the Cartan
connection $\nabla$. The aim of this section is to associate to $(M,F)$ a
k-nullity space of the Cartan connection’s curvature operator. We first
introduce the concept of k-nullity space as a natural extension of nullity
space in Finsler geometry containing nullity space as a special case ${\bf
k}=0$. Furthermore, we study fundamental properties of k-nullity spaces. Given
any non-negative real number k, we define the tensors $\eta^{\bf k}$ and
$\bar{\Omega}$ as follows
$\eta^{\bf k}(\hat{X},\hat{Y})Z={\bf
k}\\{g(Y,Z)X-g(X,Z)Y\\}+{{}^{a}P}(X,\dot{Y})Z,$
$\bar{\Omega}(\hat{X},\hat{Y})Z=\Omega(\hat{X},\hat{Y})Z-\eta^{\bf
k}(\hat{X},\hat{Y})Z,$ (14)
where, $\hat{X},\hat{Y},\hat{Z}\in{\cal X}(TM_{0})$, $X=\varrho(\hat{X})$,
$Y=\varrho(\hat{Y})$, $Z=\varrho(\hat{Z})$ and ${{}^{a}P}$ is the anti-
symmetric part of $hv-$curvature tensor $P(X,Y)$. We refer to $\bar{\Omega}$
as the related curvature operator of $\Omega$. The local representation of
$\bar{\Omega}(H\hat{X},H\hat{Y})$ is given by
$\bar{\Omega}^{i}_{\ jkl}=R^{i}_{\ jkl}-{\bf k}\\{g_{jk}\delta^{i}_{\
l}-g_{jl}\delta^{i}_{\ k}\\},$
and we have from Eq.(14)
$\bar{\Omega}(H\hat{X},V\hat{Y})={{}^{s}P}(X,\dot{Y}),$ (15)
where, $\dot{Y}=\mu(\hat{Y})$. Notice that, Eq.(8) yields
$\bar{\Omega}(H\hat{X},V\hat{Y}){\bf v}={{}^{s}P}(X,\dot{Y}){\bf v}=0,$ (16)
where ${\bf v}$ is the canonical section of $\pi^{*}TM$ given by ${\bf
v}=v^{i}{\partial}_{i}$. Given any point $z\in TM_{0}$, we define the subspace
$N^{\bf k}_{z}$ of ${\cal H}_{z}TM$ by
$N^{{\bf k}}_{z}:=\\{\hat{X}\in{\cal H}_{z}TM|\
\bar{\Omega}(\hat{X},\hat{Y})=0,\ \ \ \forall\hat{Y}\in{\cal H}_{z}TM\\}.$
For any point $z\in TM_{0}$ and $\pi z=x$, we consider the subspace ${\cal
N}^{{\bf k}}_{x}=\varrho(N^{{\bf k}}_{z})\subset T_{x}M$. Clearly, the
subspace ${\cal N}^{\bf k}_{x}=\varrho(N^{\bf k}_{z})\subset T_{x}M$ is
linearly isomorphic to $N^{\bf k}_{z}$, since $\varrho$ is a linear
isomorphism between ${\cal H}TM$ and $\pi^{*}TM$.
Now, we are in position to define a non-Riemannian k-nullity space on Finsler
manifolds.
###### Definition 3.1.
Let $(M,F)$ be a Finsler manifold. ${\cal N}^{\bf k}_{x}$ is called the
k-nullity space of the curvature operator on the Finsler manifold $(M,F)$ at
the point $x\in M$, while ${\cal N}^{\bf k}$ will denote the field of
k-nullity spaces. Its orthogonal complementary space in $T_{x}M$ is denoted by
${\scriptstyle\overset{\perp}{{\cal N}}{{}^{\bf k}_{x}}}$. Every element of
${\cal N}^{\bf k}_{x}$ is called a k-nullity vector. The non-negative integer
valued function $\mu_{\bf k}:M\longrightarrow I\\!{N}$ defined by $\mu_{\bf
k}(p)=\dim{\cal N}^{\bf k}_{p}$ is called the index of k-nullity at the point
$p\in M$. k-nullity space is called locally constant if given any $x\in M$,
there is a neighborhood $U$ of $x$ such that the function $\mu_{\bf k}$ is
constant on $U$. In this case, the correspondence $x\in U\mapsto{\cal N}^{\bf
k}_{x}$ is a distribution called the k-nullity distribution on $U$.
The function $\mu_{\bf k}:M\longrightarrow I\\!\\!{N}$ is upper semi-
continuous. In the sequel we assume that $0<\mu_{\bf k}<n$ unless otherwise
specified.
Observe that, the following relations hold for $\eta^{\bf k}$:
$\sigma\eta^{\bf k}({\hat{X}},{\hat{Y}})Z=0,\ \ \ \nabla_{\hat{Z}}\eta^{\bf
k}=0,\ \ \ \forall\hat{X},\hat{Y},\hat{Z}\in{\cal H}TM.$ (17)
where, $\sigma$ is a circular permutation on the set
$\\{\hat{X},\hat{Y},\hat{Z}\\}$. Thus, it is clear that we have:
$\sigma\bar{\Omega}({\hat{X}},{\hat{Y}})Z=\sigma\Omega({\hat{X}},{\hat{Y}})Z,\
\ \ \forall\hat{X},\hat{Y},\hat{Z}\in{\cal H}TM.$ (18)
The tensor $\bar{\Omega}$ has somehow the same algebraic properties as
$\Omega$. The following properties of $\bar{\Omega}$ are easily verified:
###### Lemma 3.2.
The following statements hold for $\bar{\Omega}$:
$(1)\ \
\sigma\bar{\Omega}({\hat{X}},{\hat{Y}})Z=\sigma\nabla_{{\hat{Z}}}\tau({\hat{X}},{\hat{Y}})+\sigma\tau({\hat{Z}},[{\hat{X}},{\hat{Y}}]),$
$(2)\ \
\sigma\nabla_{{\hat{Z}}}\bar{\Omega}({\hat{X}},{\hat{Y}})+\sigma\bar{\Omega}({\hat{Z}},[{\hat{X}},{\hat{Y}}])=0,$
$(3)\ \
g(\bar{\Omega}(\hat{X},\hat{Y})Z,W)=-g(\bar{\Omega}(\hat{X},\hat{Y})W,Z)$,
where, $\hat{X},\hat{Y},\hat{Z},\hat{W}\in{\cal H}TM$ and $\sigma$ is a
circular permutation in the set $\\{\hat{X},\hat{Y},\hat{Z}\\}$.
###### Proof.
The proof is a simple application of Bianchi identities, Eq.(5), Eq.(17) and
Eq.(18). ∎
Proof of Theorem 1.1. Let $\hat{X}$, $\hat{Y}$ and $\hat{Z}$ be three
horizontal vector fields on $TM_{0}$ such that $\hat{X},\hat{Y}\in N^{\bf
k}_{z}$. Taking into account Eq.(16) and Eq.(10), by a straightforward
computation, we have
$\varrho[\hat{X},\hat{Y}]=[X,Y]_{\pi},$
$\mu([\hat{X},\hat{Y}])=-\Omega(\hat{X},\hat{Y}){\bf v}=-\eta^{\bf
k}(\hat{X},\hat{Y}){\bf v},$ (19)
$H[\hat{X},\hat{Y}]=[\hat{X},\hat{Y}]+\eta^{\bf
k}(\hat{X},\hat{Y})v^{r}\overset{\bullet}{\partial}_{r}.$ (20)
In this case, the relation (2) in Lemma 3.2 reduces to
$\bar{\Omega}(\hat{X},[\hat{Y},\hat{Z}])+\bar{\Omega}(\hat{Y},[\hat{Z},\hat{X}])+\bar{\Omega}(\hat{Z},[\hat{X},\hat{Y}])=0$
The last equation can be written in the following form:
$\bar{\Omega}(\hat{X},V[\hat{Y},\hat{Z}])+\bar{\Omega}(\hat{Y},V[\hat{Z},\hat{X}])+\bar{\Omega}(\hat{Z},[\hat{X},\hat{Y}])=0.$
(21)
Following Eq.(15) and Eq.(19), first and second terms of Eq.(21) become:
$\bar{\Omega}(\hat{X},V[\hat{Y},\hat{Z}])={{}^{s}P}(X,\mu[\hat{Y},\hat{Z}])=-{{}^{s}P}(X,\eta^{\bf
k}(\hat{Y},\hat{Z}){\bf v})$ (22) $={\bf k}g(Z,{\bf v}){{}^{s}P}(X,Y)-{\bf
k}g(Y,{\bf v}){{}^{s}P}(X,Z),$
$\bar{\Omega}(\hat{Y},V[\hat{Z},\hat{X}])={{}^{s}P}(Y,\mu[\hat{Z},\hat{X}])=-{{}^{s}P}(Y,\eta^{\bf
k}(\hat{Z},\hat{X}){\bf v})$ (23) $={\bf k}g(X,{\bf v}){{}^{s}P}(Y,Z)-{\bf
k}g(Z,{\bf v}){{}^{s}P}(Y,X).$
By means of Eq.(22) and Eq.(23) and the symmetry property
${{}^{s}P}(X,Y)={{}^{s}P}(Y,X)$, Eq.(21) can be written in the following form:
$\bar{\Omega}(\hat{Z},[\hat{X},\hat{Y}]+\eta^{\bf
k}(\hat{X},\hat{Y})v^{r}\overset{\bullet}{\partial}_{r})=0,$
Following Eq.(20), the last equation becomes:
$\bar{\Omega}(\hat{Z},H[\hat{X},\hat{Y}])=0,\ \ \ \hat{Z}\in{\cal H}_{z}TM.$
Indeed $H[\hat{X},\hat{Y}]\in N^{\bf k}_{z}$ and
$[X,Y]=\varrho[\hat{X},\hat{Y}]=\varrho(H[\hat{X},\hat{Y}])\in{\cal N}^{\bf
k}_{x}$. Therefore, ${\bf k}$-nullity distribution is involutive or completely
integrable.∎
Considering the kernel of the operator $\bar{\Omega}$
$\ker_{x}\bar{\Omega}=\\{Z\in T_{x}M|\ \bar{\Omega}(\hat{X},\hat{Y})Z=0,\ \
\hat{X},\hat{Y}\in{\cal H}_{z}TM\\},$
we shall show that ${\cal N}^{{\bf k}}_{x}=\ker_{x}\bar{\Omega}$.
Proof of Theorem 1.2. Let $\hat{X}$, $\hat{Y}$ and $\hat{Z}$ be three
horizontal vector fields on $TM_{0}$ such that $\hat{X},\hat{Y}\notin N^{\bf
k}_{z}$ but $\hat{Z}\in N^{\bf k}_{z}$. In this case, the relation (1) in
Lemma 3.2 reduces to
$\bar{\Omega}(\hat{X},\hat{Y})Z=\tau(\hat{X},[\hat{Y},\hat{Z}])+\tau(\hat{Y},[\hat{Z},\hat{X}])+\tau(\hat{Z},[\hat{X},\hat{Y}])$
On the other hand, for every vector field $\hat{W}\in{\cal X}(TM_{0})$, we
have:
$g(\bar{\Omega}(\hat{X},\hat{Y})Z,W)=g(\tau(\hat{X},[\hat{Y},\hat{Z}]),W)+g(\tau(\hat{Y},[\hat{Z},\hat{X}]),W)+g(\tau(\hat{Z},[\hat{X},\hat{Y}]),W).$
(24)
Considering Eq.(19), we have the following relations for the torsion tensor
$\tau$.
$\tau(\hat{X},[\hat{Y},\hat{Z}])=T(\hat{X},\mu[\hat{Y},\hat{Z}])={\bf
k}g(Y,{\bf v})T(X,Z)-{\bf k}g(Z,{\bf v})T(X,Y),$ (25)
$\tau(\hat{Y},[\hat{Z},\hat{X}])=T(\hat{Y},\mu[\hat{Z},\hat{X}])={\bf
k}g(Z,{\bf v})T(Y,X)-{\bf k}g(X,{\bf v})T(Y,Z),$ (26)
$\tau(\hat{Z},[\hat{X},\hat{Y}])=T(\hat{Z},\mu[\hat{X},\hat{Y}])={\bf
k}g(X,{\bf v})T(Z,Y)-{\bf k}g(Y,{\bf v})T(Z,X).$ (27)
Replacing Eqs.(25),(27) and (27), in Eq.(24) we obtain
$g(\bar{\Omega}(\hat{X},\hat{Y})Z,W)=2{\bf k}\\{g(Y,{\bf
v})g(T(X,Z),W)+g(Z,{\bf v})g(T(Y,X),W)$ $+g(X,{\bf v})g(T(Z,Y),W)\\}.$
As a consequence of the relation (3) in Lemma 3.2, the left hand side of the
previous equation is anti-symmetric with respect to $W$ and $Z$. Thus, it
follows that
$2{\bf k}\\{g(Y,{\bf v})g(T(X,Z),W)+g(X,{\bf v})g(T(Z,Y),W)\\}=0.$
Since $W$ is arbitrarily chosen, we have the following relation:
$g(Y,{\bf v})T(X,Z)+g(X,{\bf v})T(Z,Y)=0.$ (28)
From Eq.(19) one can conclude that
$\tau(\hat{Z},[\hat{X},\hat{Y}])=\tau(\hat{Z},V[\hat{X},\hat{Y}])=T(Z,\mu[\hat{X},\hat{Y}])$
$={\bf k}g(X,{\bf v})T(Z,Y)-{\bf k}g(Y,{\bf v})T(Z,X).$
By anti-symmetry property of the tensor $T$ and Eq.(28), we get
$\tau(\hat{Z},V[\hat{X},\hat{Y}])={\bf k}g(X,{\bf v})T(Z,Y)+{\bf k}g(Y,{\bf
v})T(X,Z)=0.$
Plugging Eqs.(25),(26) and (27) into Eq.(24) results
$g(\bar{\Omega}(\hat{X},\hat{Y})Z,W)=g(\tau(\hat{X},[\hat{Y},\hat{Z}]),W)+g(\tau(\hat{Y},[\hat{Z},\hat{X}]),W)+g(\tau(\hat{Z},[\hat{X},\hat{Y}]),W)=0.$
Therefore, we have
$g(\bar{\Omega}(\hat{X},\hat{Y})Z,W)=g(\tau(\hat{Z},[\hat{X},\hat{Y}]),W)=0.$
Finally, since $W$ is arbitrarily chosen, we obtain the following equation,
$\bar{\Omega}(\hat{X},\hat{Y})Z=\tau(\hat{Z},[\hat{X},\hat{Y}])=T(Z,\mu[\hat{X},\hat{Y}])=0.$
(29)
The last equation shows that $Z\in\ker_{x}\bar{\Omega}$, that is ${\cal
N}^{\bf k}_{x}\subseteq\ker_{x}\bar{\Omega}$ and
$\ker\bar{\Omega}^{\perp}\subseteq\overset{\perp}{{\cal N}}{{}^{\bf k}_{x}}$.
Now, let $W\in\overset{\perp}{{\cal N}}{{}^{\bf k}_{x}}$ and $U\in{\cal
N}^{\bf k}_{x}$, we have
$g(\bar{\Omega}(\hat{X},\hat{Y})W,U)=-g(\bar{\Omega}(\hat{X},\hat{Y})U,W)=0.$
The previous equation shows that
$\bar{\Omega}(\hat{X},\hat{Y})W\in\overset{\perp}{{\cal N}}{{}^{\bf k}_{x}}$,
that is $Im_{x}\bar{\Omega}\subseteq\overset{\perp}{{\cal N}}{{}^{\bf
k}_{x}}$. For every k-nullity vector $U\in{\cal N}^{\bf k}_{x}$, Eq.(19)
yields
$g(\mu([\hat{X},\hat{Y}])+\eta^{\bf k}(\hat{X},\hat{Y}){\bf
v},U)=-g(\bar{\Omega}(\hat{X},\hat{Y}){\bf
v},U)=g(\bar{\Omega}(\hat{X},\hat{Y})U,{\bf v})=0.$
By definition of $\eta^{\bf k}$ and the fact that $X,Y\in\overset{\perp}{{\cal
N}}{{}^{\bf k}_{x}}$, we obtain $g(\eta^{\bf k}(\hat{X},\hat{Y}){\bf v},U)=0$.
Therefore,
$\mu([\hat{X},\hat{Y}])\in\overset{\perp}{{\cal N}}{{}^{\bf k}_{x}}.$ (30)
From which $g(\mu([\hat{X},\hat{Y}]),U)=0$. Consider the following
homomorphism of vector spaces:
$\Psi:\frac{T_{x}M}{\ker_{x}\bar{\Omega}}\cong
Im_{x}\bar{\Omega}\longrightarrow\overset{\perp}{{\cal N}}{{}^{\bf k}_{x}},$
defined by $W+\ker_{x}\bar{\Omega}\mapsto\bar{\Omega}(\hat{X},\hat{Y})W$. It
is clear that $\Psi$ is one-to-one and thus it is onto and therefore,
${\scriptstyle\overset{\perp}{{\cal N}}{{}^{\bf
k}_{x}}}=\ker_{x}{\scriptstyle\bar{\Omega}^{\perp}}$ and ${\cal N}^{\bf
k}_{x}=\ker_{x}\bar{\Omega}$. This completes the proof of Theorem. ∎
## 4 Auto-parallel k-nullity maximal integral manifold
Proof of Theorem 1.3. The method used here is inspired by Akbar-Zadeh’s
technic. Let $N$ be an integral manifold of k-nullity distribution in $U$. For
all vector fields $\widetilde{X},\widetilde{W}\in{\cal X}(TN_{0})$ we have by
means of Eq.(12)
$\nabla_{\widetilde{W}}X=\widetilde{\nabla}_{\widetilde{W}}X+\alpha(\widetilde{W},X),$
(31)
where, $\widetilde{\nabla}$ denotes the induced connection on $TN_{0}$,
$X=\rho(\widetilde{X})$ and $\alpha(\widetilde{W},X)$ is the second
fundamental form of $N$.
Let $\widetilde{X},\widetilde{Y}\in{\cal H}TN$ such that
$X,Y\in\overset{\perp}{{\cal N}}{{}^{\bf k}_{x}}$ and $U\in{\cal N}^{\bf
k}_{x}$. By means of Theorem 1.2, we have $\bar{\Omega}(\hat{X},\hat{Y})U=0$.
Suppose that $\widetilde{Z}\in N^{\bf k}_{z}$. It follows immediately from
Eq.(31) that the covariant derivative of $\bar{\Omega}$ along $\widetilde{Z}$
becomes
$\displaystyle(\nabla_{\widetilde{Z}}\bar{\Omega}(\widetilde{X},\widetilde{Y}))U$
$\displaystyle=$
$\displaystyle\nabla_{\widetilde{Z}}\bar{\Omega}(\widetilde{X},\widetilde{Y})U-\bar{\Omega}(\widetilde{X},\widetilde{Y})\nabla_{\widetilde{Z}}U$
$\displaystyle=$
$\displaystyle-\bar{\Omega}(\widetilde{X},\widetilde{Y})\nabla_{\widetilde{Z}}U$
$\displaystyle=$
$\displaystyle-\bar{\Omega}(\widetilde{X},\widetilde{Y})(\widetilde{\nabla}_{\widetilde{Z}}U+\alpha(\widetilde{Z},U))$
$\displaystyle=$
$\displaystyle-\bar{\Omega}(\widetilde{X},\widetilde{Y})\alpha(\widetilde{Z},U).$
Therefore,
$(\nabla_{\widetilde{Z}}\bar{\Omega}(\widetilde{X},\widetilde{Y}))U+\bar{\Omega}(\widetilde{X},\widetilde{Y})\alpha(\widetilde{Z},U)=0.$
Using the identity (2) in Lemma 3.2 and the above equation, we obtain
$\bar{\Omega}(\widetilde{X},\widetilde{Y})\alpha(\widetilde{Z},U)=\bar{\Omega}(\widetilde{Z},[\widetilde{X},\widetilde{Y}])U={{}^{s}P}(\widetilde{Z},\mu[\widetilde{X},\widetilde{Y}])U.$
(32)
If we assume $\mu[\widetilde{X},\widetilde{Y}]=0$, then we have
$\bar{\Omega}(\widetilde{X},\widetilde{Y})\alpha(\widetilde{Z},U)=0$. On the
other hand $\alpha(\widetilde{Z},U)\in\overset{\perp}{{\cal N}}{{}^{\bf
k}_{x}}$, it follows that $\alpha(\widetilde{Z},U)=0$. In this case, the
integral manifold $N$ is an auto-parallel submanifold. Otherwise, assume that
$\mu[\widetilde{X},\widetilde{Y}]\neq 0$. Consider a basis $\\{{\bf
e}_{1},{\bf e}_{2},...,{\bf e}_{n}\\}$ for $T_{x}M$ such that, the first $r$
vectors form a basis for ${\cal N}^{\bf k}_{x}$ and the rest $(n-r)$ vectors
is a basis for $\overset{\perp}{{\cal N}}{{}^{\bf k}_{x}}$. In virtue of
Eq.(30), without loss of generality, one can assume that the vector
$\mu[\widetilde{X},\widetilde{Y}]$ is an element of basis $\\{{\bf
e}_{r-1},...,{\bf e}_{n}\\}$. In the sequel, assume that the following indices
run over the indicated ranges
$a,b=1,2,...,n,\ \ \ \ \alpha,\beta=1,2,...,r,\ \ \ \ i,j=r-1,...,n.$ (33)
Eq.(29) states that, in this basis, we have
$T_{a\alpha j}=0$ (34)
Plugging it into Eq.(8), it results
${{}^{s}P}_{ia\beta j}=\nabla_{i}T_{a\beta j}+\frac{1}{2}\\{T_{i\beta
r}\nabla_{0}T^{r}_{\ aj}-T^{r}_{\ \beta
a}\nabla_{0}T_{irj}+T_{ijr}\nabla_{0}T^{r}_{\ a\beta}-T^{r}_{\
ja}\nabla_{0}T_{ir\beta}\\}=0.$
From the last equation, it results that
${{}^{s}P}(\widetilde{Z},\mu[\widetilde{X},\widetilde{Y}])U=0$. Eq.(32)
implies that
$\bar{\Omega}(\widetilde{X},\widetilde{Y})\alpha(\widetilde{Z},U)=0$, that is
to say $\alpha(\widetilde{Z},U)\in\ker\bar{\Omega}={\cal N}^{\bf k}_{x}$. It
follows $\alpha(\widetilde{Z},U)=0$ and $N$ is an auto-parallel submanifold.
Denote the curvature 2-forms of $\widetilde{\nabla}$ by $\widetilde{\Omega}$.
Since $N$ is an auto-parallel submanifold of $M$, its curvature tensors are
given by
$\widetilde{\Omega}(H\widetilde{X},H\widetilde{Y})Z=\Omega(H\widetilde{X},H\widetilde{Y})Z={\bf
k}\\{\widetilde{g}(Y,Z)X-\widetilde{g}(X,Z)Y\\},$
$\widetilde{\Omega}(H\widetilde{X},V\widetilde{Y})Z=\Omega(H\widetilde{X},V\widetilde{Y})Z={{}^{s}P}(X,\dot{Y})Z,$
$\widetilde{\Omega}(V\widetilde{X},V\widetilde{Y})Z=\Omega(V\widetilde{X},V\widetilde{Y})Z=Q(\dot{X},\dot{Y})Z,$
where, $\widetilde{X},\widetilde{Y}\in{\cal X}(TN_{0})$ and
$Z\in\Gamma(\pi^{*}TN)$. The above relation shows that, $N$ is a $P$-symmetric
space. Indeed components of the $hh-$curvature $\widetilde{R}^{\alpha}_{\
\beta\gamma\theta}$ of $(N,\widetilde{F})$ are given by
$\widetilde{R}^{\alpha}_{\ \beta\gamma\theta}={\bf
k}\\{\widetilde{g}_{\beta\gamma}\delta^{\alpha}_{\
\theta}-\widetilde{g}_{\beta\theta}\delta^{\alpha}_{\ \gamma}\\},$
where, $\widetilde{g}$ denotes the induced metric on $(N,\widetilde{F})$.
Following Eq.(10), we have
$\widetilde{H}(X,{\bf v}){\bf v}=\widetilde{R}(X,{\bf v}){\bf v}={\bf
k}\\{\widetilde{g}({\bf v},{\bf v})Y-\widetilde{g}(Y,{\bf v}){\bf v}\\},$
which shows that $N$ is of constant flag curvature ${\bf k}$. ∎
### 4.1 Completeness of the k-nullity foliation.
Proof of Theorem 1.4. Let $(M,F)$ be an $n$-dimensional Finsler manifold and
$\gamma:[0,c)\longrightarrow N$ a geodesic on the integral manifold $N$ of the
k-nullity foliation in $G$. We shall prove that, $\gamma$ can be extended to a
geodesic $\tilde{\gamma}:[0,\infty)\longrightarrow N$ on $N$. We shall proceed
the proof with the contrary assumption, by supposing that such geodesic does
not exist. Following Theorem 1.3, every ${\bf k}$-nullity manifold is auto-
parallel and hence it is totally geodesic. Therefore, $\gamma$ is a geodesic
on $M$ and has an extension $\tilde{\gamma}:[0,\infty)\longrightarrow M$ such
that $\gamma=\tilde{\gamma}\cap N$. It follows that,
$p=\tilde{\gamma}(c)\notin G$. Suppose that
$p_{0}=\gamma(0)=\tilde{\gamma}(0)$ and put $r_{0}=\mu_{\bf k}(p_{0})$, the
dimension of the k-nullity space at $p_{0}$. The function $\mu_{\bf
k}:M\longrightarrow M$ attains its minimum on $G$ and it results that
$\mu_{\bf k}(p)>r_{0}$. Consider a basis ${\cal B}_{0}=\\{{\bf e}_{1},{\bf
e}_{2},...,{\bf e}_{r_{0}},{\bf e}_{r_{0}+1},...,{\bf e}_{n}\\}$ for
$T_{p_{{}_{0}}}M$ such that $\\{{\bf e}_{1},{\bf e}_{2},...,{\bf
e}_{r_{0}}\\}$ is a basis for ${\cal N}^{\bf k}_{p_{{}_{0}}}$ and ${\bf
e}_{1}$ is the tangent vector to $\gamma$ at the point $p_{0}=\gamma(0)$.
Using the following system of differential equations
$\frac{\nabla{\bf E}_{i}}{dt}=0,\ \ \ \ \ {\bf E}_{i}(0)={\bf e}_{i},$
where $i=1,2,...,n$, one can translate the basis ${\cal B}_{0}$ into the
parallel frame ${\cal B}=\\{{\bf E}_{1},{\bf E}_{2},...,{\bf E}_{r_{0}},{\bf
E}_{r_{0}+1},...,{\bf E}_{n}\\}$ along $\tilde{\gamma}$. There is a
neighborhood $U$ of $p$ on $M$ such the subset $\\{{\bf E}_{1},{\bf
E}_{2},...,{\bf E}_{r_{0}}\\}$ is a basis for the ${\bf k}$-nullity space at
every point $\tilde{\gamma}(t)$ in $G\cap U$. Since $\mu_{\bf k}(p)>r_{0}$,
there is a vector field ${\bf E}_{a}$ along $\tilde{\gamma}$, for a fixed
number $a$ in the range $r_{0}+1,...,n$, such that for every $t\in[0,c)$, we
have ${\bf E}_{a}(t)\in\overset{\perp}{{\cal N}}{{}^{\bf k}_{\gamma(t)}}$ and
${\bf E}_{a}(c)\in{\cal N}^{\bf k}_{p}$. Now, let
$\hat{\tilde{\gamma}}=(\tilde{\gamma},\dot{\tilde{\gamma}})$ be the natural
lift of $\tilde{\gamma}$ to $TM_{0}$ and $\hat{{\cal B}}=\\{\hat{{\bf
E}}_{1},\hat{{\bf E}}_{2},...,\hat{{\bf E}}_{r_{0}},\hat{{\bf
E}}_{r_{0}+1},...,\hat{{\bf E}}_{n}\\}$ the basis for ${\cal
H}_{\hat{\tilde{\gamma}}(t)}TM$ such that $\varrho(\hat{{\bf E}}_{i})={\bf
E}_{i}$. Assume that the coefficients $f_{ija}$ are defined as follows
$\bar{\Omega}(\hat{{\bf E}}_{i},\hat{{\bf E}}_{j}){\bf E}_{a}=f_{ija}.$ (35)
Using the relation (2) in Lemma 3.2, the Cartan horizontal derivative of both
sides of Eq.(35) along $\hat{\tilde{\gamma}}$ in $\pi^{-1}(G\cap U)$ and using
the fact that, $\bar{\Omega}(\hat{{\bf E}}_{1},V[\hat{{\bf E}}_{j},\hat{{\bf
E}}_{i}])=0$, we obtain
$f^{\prime}_{ija}+\bar{\Omega}(\hat{{\bf E}}_{j},[\hat{{\bf E}}_{i},\hat{{\bf
E}}_{1}])+\bar{\Omega}(\hat{{\bf E}}_{i},[\hat{{\bf E}}_{1},\hat{{\bf
E}}_{j}])=0,$ (36)
where, $i,j=r_{0}+1,...,n$. Plugging $\hat{{\bf E}}_{j}$, $\hat{{\bf E}}_{i}$
and $\hat{{\bf E}}_{1}$ instead of $\hat{X}$, $\hat{Y}$ and $\hat{Z}$ into
Eq.(22) and Eq.(23) respectively, we obtain
$\bar{\Omega}(\hat{{\bf E}}_{j},V[\hat{{\bf E}}_{i},\hat{{\bf
E}}_{1}])+\bar{\Omega}(\hat{{\bf E}}_{i},V[\hat{{\bf E}}_{1},\hat{{\bf
E}}_{j}])=0.$
Therefore, Eq.(36) becomes
$f^{\prime}_{ija}+\bar{\Omega}(\hat{{\bf E}}_{j},H[\hat{{\bf E}}_{i},\hat{{\bf
E}}_{1}])+\bar{\Omega}(\hat{{\bf E}}_{i},H[\hat{{\bf E}}_{1},\hat{{\bf
E}}_{j}])=0.$ (37)
But, the horizontal part of $[\hat{{\bf E}}_{j},\hat{{\bf E}}_{1}]$ can be
written in the basis $\hat{{\cal B}}$ in the form
$H[\hat{{\bf E}}_{1},\hat{{\bf E}}_{j}]={\bf W}^{k}_{j}\hat{{\bf E}}_{k}+{\bf
W}^{a}_{j}\hat{{\bf E}}_{a},$
for some functions ${\bf W}^{k}_{j}$ defined on $\hat{\tilde{\gamma}}$ in
$\pi^{-1}(U)$, where the index $k$ runs over the range $1,...,\hat{a},...n$
and the hat over $a$ indicates that the index $a$ is omitted. Plugging the
terms $H[\hat{{\bf E}}_{j},\hat{{\bf E}}_{1}]$ and $H[\hat{{\bf
E}}_{1},\hat{{\bf E}}_{i}]$ into Eq.(37), we obtain the homogenous system of
ODEs
$f^{\prime}_{ija}+{\bf W}^{k}_{i}f_{jka}-{\bf W}^{k}_{j}f_{ika}=0.$
Since ${\bf E}_{a}$ is a ${\bf k}$-nullity vector field at $p$, by means of
Eq.(35), we have clearly for the fixed index $a$, $f_{lma}(c)=0$, where,
$l,m=r_{0}+1,...,n$. Solving the system of ODEs above with initial value
$f_{lma}(c)=0$ implies that $f_{lma}\equiv 0$. Eq.(35), implies that, ${\bf
E}_{a}$ is a ${\bf k}$-nullity vector at every point of $\tilde{\gamma}$ in
$G\cap U$ and specially, it is a ${\bf k}$-nullity vector at every point of
$\gamma$ in $G\cap U$. Obviously, this is merely a contradiction to the
contrary hypothesis and $\gamma$ can be extended to a geodesic
$\tilde{\gamma}:[0,\infty)\longrightarrow N$. ∎
We remark that, relaxing the constant ${\bf k}$ to be zero in the Eq.(14)
leads to a notion of non-Riemannian nullity in Finsler geometry which is a
special case of the nullity space in [2].
## References
* [1] Akbar-Zadeh, H.: Initiation to Global Finslerian Geometry, North Holland, 2006.
* [2] Akbar-Zadeh, H.: Espaces de nullité de cértains opérateurs géométrie des sous-variétés, _C. R. Acad. Sci. Paris_ Sér. A-B, 274 (1972) ,A490–A493.
* [3] Chern S. S. and Kuiper N. H., Some theorems on isometric imbedding of compact Riemann manifolds in Euclidean space, _Ann. of Math._ 56 (1952), 313-316.
* [4] Clifton, Y. and Maltz, H.: The k-nullity space of the curvature operator, _Michigan Math. J._ 17 (1970).
* [5] Ferus, D.: On the completeness of nullity foliation, _Michigan Math. J._ 18 (1971).
* [6] Ferus, D.: Totally geodesic foliation, _Math. Ann._ 188 313-316, (1970).
* [7] Gray, A.: Space of constancy of curvature operator, _Proc. Amer. Math. Soc._ 17 897-902, (1966).
* [8] Kowalski, O. and Sekizawa, M.: On Tangent Sphere Bundles with small or Large Constant Radius, _Ann. Global Anal. Geom._ 18: 207-219, (2000).
* [9] Matsumoto, M. and Shimada, H.: On Finsler spaces with the curvature tensors $P_{hijk}$ and $S_{hijk}$ satisfying special conditions, _Rep. on Math. Phys._ 12 1 ,77-87,(1977).
* [10] Matsumoto, M.: Finsler spaces with the hv-curvature tensor $P_{hijk}$ of a special form, , _Rep. on Math. Phys._ 14 1 ,1-13, (1978).
* [11] Maltz, R.: The nullity space of the curvature operator, _Thesis_ , University of California Los Angeles, (1965).
* [12] Shen, Z.: Lectures on Finsler Geometry, _World Scientific_ , Singapore, 2001.
* [13] Sekizawa, M.: Completeness of the $k$-th nullity foliations, _J. Diff. Geom._ 11 no.3 , 461-465, (1976).
* [14] Tachibana, S. I. and Sekizawa, M.: On the $k$-th nullity space of the Riemannian curvature tensor, _T hoku Math. J._ 2 27, 25-30, (1975).
* [15] Tanno, S.: Some differential equations on Riemannian manifolds, _J. Math. Soc. Japan._ 30, no. 3, (1978).
|
arxiv-papers
| 2011-01-07T18:50:06 |
2024-09-04T02:49:16.204734
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "B. Bidabad, M. Rafie-Rad",
"submitter": "Mehdi Rafie-Rad",
"url": "https://arxiv.org/abs/1101.1496"
}
|
1101.1537
|
# Time-Optimal solutions of Parallel Navigation and Finsler geodesics
M. Rafie-Rad111Email address:_m.rafiei.rad@gmail.com_
Department of Mathematics, Faculty of Sciences,
Mazandaran University, Bablosar, Iran.
###### Abstract
A geometric approach to kinematics in control theory is illustrated. A non-
linear control system is derived for the problem and the Pontryagin maximum
principle is used to find the time-optimal trajectories of the Parallel
navigation. The time-optimal trajectories of the Parallel navigation are
characterized through a geometric formulation. It is notable that the approach
has the advantages using feedback.222 2000 Mathematics subject Classification:
Primary 53C60; Secondary 53B40.
Keywords: Finsler geometry, Parallel navigation, Kinematics, Optimal control,
Pontryagin maximum principle.
## 1 Introduction
The historical development of what became the Calculus of Variations is
closely linked to certain minimization principles in the majority subjects in
mechanics, namely, the principle of least distance, the principle of least
time and ultimately, the principle of least action [7]. To understand solution
of the well-known brachistochrone problem, (i.e finding a curve from point $A$
to point $B$ along which a free-sliding particle will descend more quickly
than on any other $AB$-curve), we are led through Fermat’s principle of least
time: light always takes a path that minimizes travel time.
The Parallel navigation, or briefly P-navigation, is a quiet old problem and
has been studied using several techniques from the viewpoints of kinematics
and dynamics in optimal control theory [17]. The application of Finsler
geometry in Physics, seismology and Biology is a subject of numerous papers
such as [1], [2],[3], [5], [9], [13], [15], [18], etc. Let $O$ be the origin
of an inertial reference frame of coordinates (FOC). The positions of $M$ and
$T$ in this (FOC) are given by the vectors ${\bf r}_{M}=OM$ and ${\bf
r}_{T}=OT$, respectively. In two-point guidance systems, the vector ${\bf
r}={\bf r}_{T}-{\bf r}_{M}$ is conventionally called the range. Its time
derivative $\dot{\bf r}=\dot{\bf r}_{T}-\dot{\bf r}_{M}={\bf v}_{T}-{\bf
v}_{M}$ is the relative velocity between the two objects, and ${\bf v}_{T}$
and ${\bf v}_{M}$ are the velocities of $T$ and $M$, respectively. W always
denote the vectors by bold face and their norms will be shown by the same
normal letter. As an application, it is notable for mariners wishing to
rendez-vous each other at sea. $M$ could be a boat and $T$, a tanker with fuel
for it (or vice-versa). Or, back in history, $T$ could be a merchantman and
$M$ a pirate ship. This rule assumes, of course, constant speeds. Thus, in
most realistic cases, $v_{T}$ and $v_{M}$ are supposed to be constant.
However, it is easy to extend the theory if they are not constant. The closing
velocity, a term often used in the study of guidance, is simply ${\bf
v}_{C}=-\dot{\bf r}$. Notice that, we wish to study the kinematics of
P-navigation in a relative (FOC) rather than a absolute one, i.e., we shall
seek the location of $M$ in a (FOC) attached to $T$. Thus, a trajectory in the
relative (FOC) shows the situation as seen by an observer located at $T$. As
the special cases, we assume that $M=R^{3}$ or $M=R^{2}$. In reality, the
velocity ${\bf v}_{T}$ and ${\bf r}_{T}$ can be detected and reported at any
${\bf r}$ by a grounded radar. Suppose that $\delta({\bf r})$ be the angle
between ${\bf v}_{M}$ and $MT$ and given any $\delta$, there is Finsler metric
$F$ given by:
$F({\bf r},{\bf v},\delta)=\frac{|{\bf v}|^{2}}{v_{M}\cos\delta|{\bf
v}|-\langle{\bf v},{\bf v}_{T}\rangle},$ (1)
where, $|.|$ denotes the Riemannian norm on $M$. A solution of the described
P-navigation is a curve $({\bf r}(t),\delta(t))$ such that respects the
required constraints on velocities.
###### Theorem 1.1
Given any solution $({\bf r},\delta)$ of parallel navigation, the curve ${\bf
r}$ can be reparametrized so that it satisfies $F({\bf r}(t),{\bf
v}(t),\delta(t))=1$.
The indicatrix $S({\bf r},\delta)$ of the metric (1) is the set of unit
tangent vectors ${\bf v}$ with respect to (1) which is defined by $S({\bf
r},\delta)=\\{{\bf v}\in T_{\bf r}M\ |\ F({\bf r},{\bf v},\delta)=1\\}$.
Following Theorem 1.1, at any time $t$ we have $\dot{\bf r}={\bf v}\in S({\bf
r},\delta)$. Hence, at any time $t$, there is a unit vector $f({\bf
r},\delta)\in S({\bf r},\delta)$ such that $\dot{\bf r}={\bf v}=f({\bf
r},\delta)$.
Control problems typically concern finding a (not necessarily unique) control
law $\delta(.)$ , which transfers the system in finite time from a given
initial state $x_{i}={\bf r}(0)$ , to a given final state $x_{f}={\bf
r}(t_{f})$. This transition is to occur along an admissible path, i.e. ${\bf
r}(.)$ and respects all kinematic constraints imposed on it. Let us consider
it as
$\dot{\bf r}=f({\bf r},\delta).$ (2)
We further assume that $\delta(.)$ is admissible, i.e. is piecewise continuous
and belongs to ${\cal U}$ , the admissible control space. Let there now be a
rule which assigns a unique, real-valued number to each of these transfers.
Such a rule can be viewed as the transition cost between $x_{i}$ and $x_{f}$
along an admissible path, completely specified by $\delta(.)$. The Optimal
control concerns specifying this rule and thereby providing a systematic
method for selecting the best , or optimal control law, according to some
prescribed cost functional. One can find an analogue discussion in [5], to
calculate the travel-time along the trajectories of the so called Pure pursuit
navigation. Here, the P-navigation optimal control problem can be founded by
the cost function $C({\bf r},\delta)=F({\bf r},\dot{\bf r},\delta)$ and has
the following form
$\textrm{minimize}\int_{0}^{t_{f}}C({\bf r},\delta)dt,$ (3)
where, $t_{f}\in(0,\infty)$ is the final time which is going to be optimized.
From everyday experience we know that collision courses need not be straight
lines if $T$ changes its speed or direction; so what is exactly the collision
course? It may be curved in some sense. One of our goal in this paper is to
make known the best collision course.
###### Theorem 1.2
Given any time-optimal solution $({\bf r},\delta)$ of P-navigation, the curve
${\bf r}$ is a geodesic of the Finsler metric (1).
The trajectory ${\bf r}_{M}$ can be obtained ${\bf r}_{M}={\bf r}_{T}-{\bf r}$
when ${\bf r}$ is known. One can freely consider ${\bf v}_{M}$ and ${\bf
v}_{T}$ as vector fields alon ${\bf r}$. Now, let $\frac{\nabla}{\ dt}$ be the
covariant derivative defined for any vector field $Y$ along ${\bf r}$ defined
by
$\frac{\nabla Y^{i}}{dt}:=\frac{dY^{i}}{dt}+G^{i}_{jk}({\bf r},\dot{\bf
r},\delta)Y^{j}Y^{k},$
where, $G^{i}_{jk}$ are the connection coefficients of Berwald connection
associated to the Finsler metric (1). As a result of Theorem 1.2, we can
mention the following result:
###### Theorem 1.3
The time-optimal trajectory ${\bf r}_{M}$ of P-navigation satisfies the
following second order ODE:
$\ddot{\bf r}_{M}^{i}+G^{i}_{jk}({\bf r},{\bf v},\delta){\bf v}_{M}^{j}{\bf
v}_{M}^{k}=\frac{\nabla{\bf v}_{T}^{i}}{dt},\ \ \ \ i=1,...,n.$
Our approach is closely related with subjects such as non-holonomic mechanics,
sub-Finslerian geometries, see for a deeper sight [8] and [4]. One may find
various techniques in missile guidance and control in [17].
## 2 Preliminaries
Let $M$ be a n-dimensional $C^{\infty}$ manifold. $T_{x}M$ denotes the tangent
space of M at $x$. The tangent bundle of $M$ is the union of tangent spaces
$TM:=\cup_{x\in M}T_{x}M$. We will denote the elements of $TM$ by $(x,y)$
where $y\in T_{x}M$. Let $TM_{0}=TM\setminus\\{0\\}.$ The natural projection
$\pi:TM_{0}\rightarrow M$ is given by $\pi(x,y):=x$.
A Finsler metric on $M$ is a function $F:TM\rightarrow[0,\infty)$ with the
following properties; (i) $F$ is $C^{\infty}$ on $TM_{0}$, (ii) $F$ is
positively 1-homogeneous on the fibers of tangent bundle $TM$, and (iii) the
$y$-Hessian of $\frac{1}{2}F^{2}$ with elements
$g_{ij}(x,y):=\frac{1}{2}[F^{2}(x,y)]_{y^{i}y^{j}}$ is positive definite on
$TM_{0}$. The pair $(M,F)$ is then called a Finsler space. The Riemannian
metrics are special Finsler metrics. Traditionally, a Riemannian metric is
denoted by $a_{ij}(x)dx^{i}\otimes dx^{j}$. It is a family of inner products
on tangent spaces. Let $\alpha(x,y):=\sqrt{g_{ij}(x)y^{i}y^{j}}$, ${\bf
y}=y^{i}{{\partial}\over{\partial}x^{i}}|_{x}\in T_{x}M$. $\alpha$ is a family
of Euclidean norms on tangent spaces. Throughout this paper, we also denote a
Riemannian metric by $\alpha=\sqrt{a_{ij}(x)y^{i}y^{j}}$.
An $(\alpha,\beta)$-metric is a scalar function on $TM$ defined by
$F:=\Phi(\frac{\beta}{\alpha})\alpha$, where $\phi=\phi(s)$ is a $C^{\infty}$
on $(-b_{0},b_{0})$ with certain regularity.
$\alpha=\sqrt{a_{ij}(x)y^{i}y^{j}}$ is a Riemannian metric and
$\beta=b_{i}(x)y^{i}$ is a 1-form on a manifold $M$. One may find another
important class of $(\alpha,\beta)$-metrics in [16]. The Randers and Matsumoto
metrics are special $(\alpha,\beta)$-metrics defined by $\Phi=1+s$ and
$\Phi=\frac{1}{1-s}$, respectively, i.e, $F=\alpha+\beta$ and
$F=\frac{\alpha^{2}}{\alpha-\beta}$. Randers metrics were introduced by
Randers in 1941 [13] in the context of general relativity. In [6], applying
Fermat’s principle, the authors proved that the time-optimal solutions of the
well-known Zermelo’s navigation-moving that is the motion of a vehicle
equipped with an engine with a fixed power output in presence of a wind
current-are actually the geodesics of a Randers metric. M. Matsumoto gave an
exact formulation of a Finsler surface to measuring the time on the slope of a
hill and introduced the Matsumoto metrics in [9], see also [15].
A Lagrangian on the manifold $M$ is a mapping $L:TM\longrightarrow R$ which is
smooth on $TM_{0}$. A Lagrangian is said to be regular if it has non-
degenerate $y$-Hessian on $TM_{0}$. Thus, given a Finsler metric $F$, the
function $L=\frac{F^{2}}{2}$ is a regular Lagrangian. A large area of
applicability of this geometry is suggested by the connections to Biology,
Mechanics, and Physics and also by its general setting as a generalization of
Finsler and Riemannian geometries [10]. For every smooth curve
$c:[a,b]\longrightarrow R$, the extremal curves of the action integral given
by
$I(c)=\int_{a}^{b}L(c(t),\dot{c}(t))dt,$ (4)
are characterized locally by the Euler-Lagrange equations given as follows:
$\frac{d}{dt}\frac{\partial L}{\partial\dot{x}^{i}}-\frac{\partial L}{\partial
x^{i}}=0,$ (5)
where, $x^{i}(t)$ is a local coordinate expression of $c$. The extremal curves
of the action integral (4) are usually called the geodesics of L. In [1] it is
shown that the Lagrangian and Finslerian approaches are projectively the same.
Given a Finsler manifold $(M,F)$, a globally defined vector field $G$ is
induced by $F$ on $TM_{0}$, which in a standard coordinate $(x^{i},y^{i})$ for
$TM_{0}$ is given by $G=y^{i}{{\partial}\over{\partial
x^{i}}}-2G^{i}(x,y){{\partial}\over{\partial y^{i}}},$ where $G^{i}(x,y)$ are
local functions on $TM_{0}$ satisfying $G^{i}(x,\lambda
y)=\lambda^{2}G^{i}(x,y)\,\,\,,\lambda>0$, see [14]. G is called the
associated spray to $(M,F)$. In local coordinates, a curve $c(t)$ is a
geodesic of $F$ if and only if its coordinates $(c^{i}(t))$ satisfy
$\ddot{c}^{i}+2G^{i}(c,\dot{c})=0$.
### 2.1 The kinematics of Parallel navigation
We shall refer to the target as $T$ and to the pursuer as $M$ and their
velocities as $v_{M}$ and $v_{T}$, respectively. To begin, we set up a
coordinate system called reference frame of coordinates, in which the pursuer
is initially located at the origin $O$. When considering planar motion we
shall use Cartesian coordinates $(x,y)$ or $(x,z)$, and the angles will be
positive if measured counterclockwise. The ray that starts at the pursuer $M$
and is directed at the target $T$ along the positive sense of ${\bf r}$ is
called the line of sight (LOS). The parallel navigation geometrical rule,has
been known since antiquity, mostly by mariners. According to this rule, the
direction of the line of sight, $MT$, is kept constant relative to inertial
space, i.e., the LOS is kept parallel to the initial LOS. In three-dimensional
vector terminology, the rule is very concisely stated as ${\bf
r}\times\dot{\bf r}=0$. Suppose that $\theta$ and $\lambda$ denote,
respectively, the angles between ${\bf v}_{T}$ and ${\bf v}_{M}$ and, ${\bf
v}_{M}$ and the horizontal axis (Figure 1).
,
Figure 1: The range ${\bf r}$, the velocity vectors ${\bf v}_{M}$ and ${\bf
v}_{T}$.
Let us put $r=|{\bf r}|$. The basic rule for moving of the pursuer is
presented by the following two equations [17]:
$\displaystyle\dot{r}$ $\displaystyle=$ $\displaystyle v_{T}\cos\theta-
v_{M}\cos\delta,$ (6) $\displaystyle r\dot{\lambda}$ $\displaystyle=$
$\displaystyle v_{T}\sin\theta-v_{M}\sin\delta.$ (7)
Notice that, in a planar framework, ${\bf v}_{M}$ , ${\bf v}_{T}$ and ${\bf
r}$ being on the same (fixed) plane by definition, therefore, the parallel
navigation geometrical rule can be restated as $\dot{\lambda}=0$. The
requirement $\langle{\bf r},{\bf v}\rangle<0$ must be added in order to ensure
that $M$ should approach $T$ not recede from it. In this case, we have
$\dot{r}<0$, that is $v_{T}\cos\theta<v_{M}\cos\delta$. Let us denote the
projection of any vector ${\bf v}_{T}$ on ${\bf v}$ by $Proj_{{\bf v}}{\bf
v}_{T}$. A solution of the described P-navigation is a curve $({\bf
r}(t),\delta(t))$ such that respects the equations (6) and (7). By the
trajectory of P-navigation, we mean a curve ${\bf r}(t)$ such that $({\bf
r}(t),\delta(t))$ is a solution, for some control $\delta$.
Initiating the process, we have ${\bf r}(0)={\bf r}_{0}$ which shows that, $M$
stands at a point with distance $r_{0}$ from $T$. Through the performance, $r$
decreases by time and hence, $M$ approaches $T$. Therefore, ${\bf r}$ tends to
the origin $O$ and $M$ hits $T$ when ${\bf r}(t_{f})=0$, (Figure 2). It
follows that, P-navigation trajectories are characterized by a curve ${\bf r}$
joining $Q={\bf r}_{0}$ to the origin $O$ (Figure 3). It is of our interests
to find the best $QO$-trajectory. More precisely, the problem is to find a
curve from point $Q$ to point $O$ along which a particle will descend more
quickly than on any other $QO$-curve of P-navigation. In this way, the problem
somehow resembles to a brachistochrone problem.
,
Figure 2: Some possible ranges initiated at the point $Q$.
,
Figure 3: Schematic of exemplary collision courses for $M$.
## 3 The optimal control theory.
A control system of ordinary differential equations is a family of
differential equations in normal form $\frac{d{\bf r}^{i}}{dt}=f^{i}({\bf
r},\delta)$, where ${\bf r}^{i}$ are called state variables, $t$ is the
parameter of evolution (usually the time) and $\delta^{a}$ are the controls.
Geometrically, it can be regarded as a fibred mapping $X:U\longrightarrow TM$,
from a control fiber bundle $(U,\eta,M)$ over the state manifold $M$ to the
tangent bundle $(TM,\pi,M)$, see [11]. Using local coordinates $({\bf
r}^{i}),\ i=1,...,n$ in $M$, adapted coordinates $({\bf r}^{i},\delta^{a}),\
a=1,...,k$ in $U$, and natural coordinates $({\bf r}^{i},{\bf v}^{i})$ in
$TM$, the coordinate expression for $X$ is $X({\bf r},\delta)=f^{i}({\bf
r},\delta)\frac{\partial}{\partial{\bf r}^{i}}$ , or ${\bf v}^{i}=f^{i}({\bf
r},\delta)$, the family of control equations. Admissible curves of the control
system are curves $\gamma:I\subset R\longrightarrow U$ such that $(\eta
o\gamma)^{c}=Xo\gamma$, where c denotes the natural lifting to $TM$ of a curve
in $M$. Interested readers are advised to see [11] for getting familiar to the
geometry of control systems. In Optimal Control Theory, a cost functional
${\cal C}(\gamma)=\int C({\bf r}(t),\delta(t))dt$ is given and the goal is to
obtain admissible curves of the control system, satisfying some boundary
conditions (e.g. $x_{i}={\bf r}(0)$, $x_{f}={\bf r}(t_{f})$) and minimizing
the cost functional. It is therefore a Classical Variational problem with non-
integrable constraints defined by the control equations. Pontryagin maximum
principle [12] provides a set of necessary conditions for a solution $({\bf
r}(t),\hat{\delta}(t))$ to be optimal; introducing a Hamiltonian function
$\displaystyle H({\bf r},{\bf p},\delta)$ $\displaystyle:=$
$\displaystyle\langle{\bf p},X\rangle-C({\bf r},\delta)={\bf p}_{i}f^{i}({\bf
r},\delta)-C({\bf r},\delta),$ $\displaystyle\hat{H}({\bf r},{\bf p})$
$\displaystyle:=$ $\displaystyle\underset{\delta}{\max}\ H({\bf r},{\bf
p},\delta).$
where the variables $({\bf p}_{i})$ are momenta coordinates, the optimal
curves $({\bf r}(t),\hat{\delta}(t))$ must satisfy the control system
equations
${\bf v}^{i}=\frac{\partial\hat{H}}{\partial{\bf p}^{i}}=f^{i}({\bf
r}(t),\hat{\delta}(t))$
and there must exist a solution curve for the adjoint differential equations
$\frac{d{\bf p}_{i}}{dt}=-\frac{\partial\hat{H}}{\partial{\bf r}^{i}},$
Define the Lagrangian $L$ by $L({\bf r},{\bf v})={\bf p}_{i}{\bf
v}^{i}-\hat{H}$. Observe that we have the following relations
$\frac{d{\bf r}}{dt}=\frac{\partial\hat{H}}{\partial{\bf p}}={\bf v},\ \ \ \ \
\ \frac{d{\bf p}}{dt}=-\frac{\partial\hat{H}}{\partial{\bf r}}=\frac{\partial
L}{\partial{\bf r}},\ \ \ \ \ \frac{\partial\hat{H}}{\partial{\bf v}}={\bf
p}-\frac{\partial L}{\partial{\bf v}}=0.$
From the above equations, it results the well-known Euler-Lagrange for $L$
$\frac{d}{dt}\frac{\partial L}{\partial{\bf v}}-\frac{\partial L}{\partial{\bf
r}}=0.$
###### Proposition 3.1
[12] In order for $({\bf r}(t),\hat{\delta}(t))$ to be an optimal solution of
(3), the following are necessary conditions:
(a) There exists a solution curve for the adjoint differential equations
$\frac{d{\bf p}_{i}}{dt}=-\frac{\partial\hat{H}}{\partial{\bf r}^{i}}.$
(b) $\hat{\delta}=\arg\ \underset{\delta}{\max}\ H({\bf r},{\bf p},\delta),\ \
\ \ \forall t\in[0,t_{f}]$.
(c) $\hat{H}({\bf r},{\bf p})=0,\ \ \ \ \forall t\in[0,t_{f}]$.
## 4 Proof of Theorems.
### 4.1 Proof of Theorem 1.1
Let $({\bf r}(t),\delta(t))$ be a pair of the curve ${\bf r}$ and a function
$\delta(t)$. We are going to show that, if $({\bf r}(t),\delta(t))$ be a
solution of P-navigation, then ${\bf t}(t)$ must be reparametrized so that we
we have $F({\bf r}(t),\dot{\bf r}(t),\delta(t))=1$. We notice that, in
P-navigation, ${\bf r}$ and ${\bf v}$ are collinear and $\dot{r}<0$, hence we
have
$\dot{r}=\frac{\langle{\bf r},{\bf v}\rangle}{r}=\pm|Proj_{\bf r}{\bf
v}|=\pm|Proj_{\bf v}{\bf v}|=-|{\bf v}|.$
Now, we summarize (6) in the following relation
$|{\bf v}|=v_{M}\cos\delta-\frac{\langle{\bf v}_{T},{\bf v}\rangle}{|{\bf
v}|}.\\\ $
After simplification, we obtain the following equation
$F({\bf r},{\bf v},\delta)=\frac{|{\bf v}|^{2}}{v_{M}\cos\delta|{\bf
v}|-\langle{\bf v}_{T},{\bf v}\rangle}=1.$
Q.E.D.
### 4.2 Proof of Theorem 1.2
Following Theorem 1.1, at any time $t$ we have $\dot{\bf r}={\bf v}\in S({\bf
r},\delta)$. Hence, at any time $t$, there is a unit vector $X({\bf
r},\delta)\in S({\bf r},\delta)$ such that $\dot{\bf r}={\bf v}=X({\bf
r},\delta)$. Consider the unit canonical vector field $\ell({\bf r},\dot{\bf
r},\delta)=\frac{\dot{\bf r}}{F({\bf r},\dot{\bf r},\delta)}$. We notice that,
in P-navigation framework, we always assume that ${\bf r}$ and $\dot{\bf r}$
are collinear and hence, one can understand $\ell$ as a function of ${\bf r}$
and $\delta$, as well. It follows that, given any trajectory ${\bf r}$ of
P-navigation, $X$ is given by $X({\bf r},\delta)=\ell({\bf r},\dot{\bf
r},\delta)$. Therefore, it is clear that,
$\displaystyle\langle{\bf p},X\rangle$ $\displaystyle=$ $\displaystyle
p_{i}f^{i}({\bf r},\delta)=p_{i}\ell^{i}({\bf r},\dot{\bf r},\delta)=F({\bf
r},\dot{\bf r},\delta),$ $\displaystyle\langle{\bf p},{\bf v}\rangle$
$\displaystyle=$ $\displaystyle p_{i}{\bf v}^{i}=F^{2}({\bf r},\dot{\bf
r},\delta).$
Now, we return to the control system of P-navigation given by (2) with the
cost functional $C({\bf r},\delta)=F({\bf r},\dot{\bf r},\delta)$. It is easy
to verify that, $H=0$, $\hat{H}=0$ and one may consider $\hat{\delta}$ as any
possible control law. The conditions of Proposition 3.1 holds as well and the
Lagrangian $L_{\hat{\delta}}=\langle{\bf p},{\bf v}\rangle-\hat{H}$ is
obtained as
$L_{\hat{\delta}}({\bf r},\dot{\bf r})=F^{2}({\bf r},\dot{\bf
r},\hat{\delta}).$
Therefore, based on Pontryagin maximum principle, the optimal trajectories
${\bf r}(t)$ are geodesics of the Lagrangian $L_{\hat{\delta}}$. Clearly, they
are geodesics of the Finsler metric $F({\bf r},\dot{\bf r},\delta)$.
Now, consider the control-parametric family of Finsler metrics defined by
$F_{\delta}({\bf r},\dot{\bf r}):=F({\bf r},\dot{\bf r},\delta)$. Let ${\cal
L}_{\delta}(\gamma)=\int_{0}^{t_{f}}F_{\delta}(\gamma,\dot{\gamma})dt$ be the
length of any admissible curve $\gamma(t)$ on $(M,F_{\delta})$. A simple
calculation gives the following inequality:
$F_{0}({\bf r},\dot{\bf r})\leq F_{\delta}({\bf r},\dot{\bf r}),\ \ \
\textrm{for all possible controls}\ \delta.$
From that, it follows that the functional ${\cal L}_{\delta}(\gamma)$ takes
its minimum at $\delta=0$, that is
${\cal L}_{0}(\gamma)\leq{\cal L}_{\delta}(\gamma),\ \ \ \textrm{for all
possible controls}\ \delta.$
Therefore, to find a time-optimal solution, one should minimize the cost
functional ${\cal C}(\gamma)=\int F_{0}(\gamma,\dot{\gamma})dt$ and this leads
us to obtain it as a geodesic of $F_{0}$. Q.E.D.
###### Theorem 4.1
The time-optimal trajectory of P-navigation is a geodesic ${\bf r}(t)$ of the
Finsler metric $F_{0}=\frac{|{\bf v}|^{2}}{v_{M}|{\bf v}|-\langle{\bf
v}_{T},{\bf v}\rangle}$.
However, given any control law, one may obtain a geodesic of the metric
$F_{\delta}$ as the time-optimal trajectory. As a remark, we quote that the
target $T$ may not be reachable by the control $\delta=0$.
###### Example 4.1
(Case of plane nonmaneuvering target.) The target $T$ is said to be
nonmaneuvering if ${\bf a}_{T}=0$. In this case, $T$ moves on a straghit line
at velocity $v_{T}$ in the direction with a constant angle $\theta_{0}$ if
measured counterclockwise, see Figure 4. Let us suppose ${\bf
v}_{T}(x^{1},x^{2})=v_{T}\\{\cos\theta_{0}\frac{\partial}{\partial
x^{1}}+\sin\theta_{0}\frac{\partial}{\partial x^{2}}\\}$. Thus, from (7), it
follows that $\delta=\sin^{-1}(\frac{\sin\theta_{0}}{K})$, where, $K$ is the
velocity ratio $K=\frac{v_{M}}{v_{T}}$. Then, $\delta$ is a constant say
$\delta_{0}$. Moreover, ${\bf v}_{T}$ is a parallel vector field and then
$F_{\delta}$ is a Minkowski metric and is flat. Thus, it geodesics are
straight lines. We obtain ${\bf r}(t)={\bf r}_{0}+t{\bf v}_{0}$. But, from
(6), we have $|{\bf v}|=|{\bf
v}_{0}|=v_{M}\cos\delta_{0}-v_{T}\cos\theta_{0}$. Intercept occur when we have
${\bf r}(t_{f})=0$, thus, the total flight time $t_{f}$ is obtained by
$t_{f}=\frac{r_{0}}{v_{M}\cos\delta_{0}-v_{T}\cos\theta_{0}}=\frac{r_{0}}{v_{T}(K\cos\delta_{0}-\cos\theta_{0})}$
and the total range of $M$ equals $r_{0}$ which is the shortest curve joining
${\bf r}_{0}$ to the origin $O$.
,
Figure 4: Collision course for a target moving on a straight line at a
direction with a constant angle $\theta_{0}$.
## References
* [1] O. Amici, B. Casciaro, M. Hashiguchi, On Finsler metrics associated with a Lagrangian, Rep. Fac. Sci., Kagoshima Univ., (Math., Phys. and Chem), No. 20. p. 33-41, 1987.
* [2] P. L. Antonelli, A. Bóna, M. Slawiński, Seismic rays as Finsler geodesics, Nonlinear Analysis: Real World Applications, 4 (2003) 711-722.
* [3] P.L. Antonelli, R.S. Ingarden, M. Matsumoto, The Theory of Sprays and Finsler Spaces with Application in Physics and Biology, Kluwer Academic Publishers, Dordrecht, Boston, London, 1993.
* [4] A. M. Bloch, J. Baillieul, P. Crouch, J. Marsden, Nonholonomic Mechanics and Control, Springer, (2003).
* [5] B. Bidabad, M. Rafie-Rad, Pure pursuit navigation on Riemannian manifolds, Nonlinear Analysis: Real World Applications, 10 (2009), 1265-1269.
* [6] D. Bao, C. Robles, Z. Shen, Zermelo navigation on Riemannian manifolds, J. Diff. Geom., 66 (2004) 377-435
* [7] C. Lanczos, The variational principles of mechanics, University of Toronto Press, Toronto, 1970 (1st ed. 1949).
* [8] C. López, E. Martínez, Sub-Finslerian metric associated to an optimal control system, SIAM J. Control Optim. , to appear.
* [9] M. Matsumoto, A slope of a hill is a Finsler surface with respect to a time measure, J. Math. Kyoto. Univ. 29 (1980) 17 25.
* [10] R. Miron, M. Anastasiei, The Geometry of Lagrange Spaces: Theory and Applications Vol. 59, Fundamental Theories of Physics Series, Kluwer Academic Publishers, Dordrecht, Boston, London, 1994\.
* [11] H. Nijmeijer and A. J. van der Schaft, Nonlinear Dynamical Control Systems, Springer- Verlag, New York (1990).
* [12] L. S. Pontryagin et al., The Mathematical Theory of Optimal Processes, InterScience Pub., New York (1962).
* [13] G. Randers, On an asymmetric metric in the four-space of general relativity, Phys. Rev. 59 (1941) 195-199.
* [14] Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, Dordrecht, Boston, London, 2001.
* [15] H. Shimada, S.V. Sabau, An introduction to Matsumoto metric, Nonlinear Anal. 63 (2005), 165-168.
* [16] C. Shibata, On Finsler spaces with Kropina metric, Rep. Math. Phys. 13 (1978), 117-128.
* [17] N. A. Shneydor, Missile Guidance and Pursuit: Kinematics, Dynamics and Control, Horwood Publishing Chichester, 1998\.
* [18] T. Yajima, H. Nagahama, Zermelo’s condition and seismic ray path, Nonlinear Analysis: Real World Applications 8 (2007) 130-135.
|
arxiv-papers
| 2011-01-07T21:39:05 |
2024-09-04T02:49:16.214183
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Rafie-Rad",
"submitter": "Mehdi Rafie-Rad",
"url": "https://arxiv.org/abs/1101.1537"
}
|
1101.1546
|
arxiv-papers
| 2011-01-07T22:42:52 |
2024-09-04T02:49:16.221544
|
{
"license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/",
"authors": "Therese A. Hart, Gabriel Khan, Mizan R. Khan",
"submitter": "Mizan Khan",
"url": "https://arxiv.org/abs/1101.1546"
}
|
|
1101.1567
|
# Age and mass constraints for a young massive cluster in M31 based on
spectral-energy-distribution fitting
Jun Ma,11affiliation: National Astronomical Observatories, Chinese Academy of
Sciences, Beijing 100012, P. R. China;
majun@nao.cas.cn 22affiliation: Key Laboratory of Optical Astronomy, National
Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P. R.
China Song Wang,11affiliation: National Astronomical Observatories, Chinese
Academy of Sciences, Beijing 100012, P. R. China;
majun@nao.cas.cn 33affiliation: Graduate University, Chinese Academy of
Sciences, Beijing 100039, P. R. China Zhenyu Wu,11affiliation: National
Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P. R.
China;
majun@nao.cas.cn Zhou Fan,11affiliation: National Astronomical Observatories,
Chinese Academy of Sciences, Beijing 100012, P. R. China;
majun@nao.cas.cn Yanbin Yang,11affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100012, P. R. China;
majun@nao.cas.cn Tianmeng Zhang,11affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100012, P. R. China;
majun@nao.cas.cn Jianghua Wu,11affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100012, P. R. China;
majun@nao.cas.cn Xu Zhou,11affiliation: National Astronomical Observatories,
Chinese Academy of Sciences, Beijing 100012, P. R. China;
majun@nao.cas.cn Zhaoji Jiang,11affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100012, P. R. China;
majun@nao.cas.cn and Jiansheng Chen11affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100012, P. R. China;
majun@nao.cas.cn
###### Abstract
VDB0-B195D is a massive, blue star cluster in M31. It was observed as part of
the Beijing-Arizona-Taiwan-Connecticut (BATC) Multicolor Sky Survey using 15
intermediate-band filters covering a wavelength range of 3000–10,000 Å. Based
on aperture photometry, we obtain its spectral-energy distribution (SED) as
defined by the 15 BATC filters. We apply previously established relations
between the BATC intermediate-band and the Johnson-Cousins $UBVRI$ broad-band
systems to convert our BATC photometry to the standard system. A detailed
comparison shows that our newly derived $VRI$ magnitudes are fully consistent
with previous results, while our new $B$ magnitude agrees to within $2\sigma$.
In addition, we determine the cluster’s age and mass by comparing its SED
(from 3000 to 20,000Å, comprising photometric data in the 15 BATC intermediate
bands, optical broad-band $BVRI$, and 2MASS near-infrared $JHK_{\rm s}$ data)
with theoretical stellar population synthesis models, resulting in age and
mass determinations of $60.0\pm 8.0$ Myr and $(1.1-1.6)\times
10^{5}M_{\odot}$, respectively. This age and mass confirms previous
suggestions that VDB0-B195D is a young massive cluster in M31.
###### Subject headings:
galaxies: individual (M31) – galaxies: star clusters – galaxies: stellar
content
††slugcomment: AJ, in press
## 1\. Introduction
Young massive star clusters (YMCs) are among the main objects resulting from
violent star-forming episodes triggered by galaxy collisions, mergers, and
close encounters (see de Grijs & Parmentier, 2007, and references therein).
They are also referred to as ‘young populous clusters,’ a term first coined by
Hodge (1961), who used it to describe 23 clusters containing bright, blue
stars in the Large Magellanic Cloud. In Hodge (1961), the ‘young’ aspect is
demonstrated by the fact that all clusters have main sequences that extend to
absolute magnitudes brighter than $M_{V}=0$, while ‘populous’ describes their
richness (stellar membership). However, YMCs are also observed in quiescent
galaxies (Larsen & Richtler, 1999) and in the disks of isolated spirals,
although higher cluster-formation efficiencies are associated with
environments exhibiting high star-formation rates (see Larsen, 2004; Cao & Wu,
2007, and references therein). It has become clear that, in many ways, YMCs
resemble young versions of the old globular clusters (GCs) associated with all
large galaxies (see Larsen et al., 2004, and references therein). YMCs are
seemingly absent in the Milky Way; possibly the best example of a Galactic YMC
is Westerlund 1, a heavily reddened cluster with an age and mass of 4–5 Myr
(Crowther et al., 2006) and $M_{\rm cl}\sim 10^{5}~{}M_{\odot}$ (Clark et al.,
2005), respectively.
Since the pioneering work of Tinsley (1968, 1972) and Searle (1973),
evolutionary population synthesis modeling has become a powerful tool to
interpret integrated spectrophotometric observations of galaxies and their
components, such as star clusters (e.g., Anders et al., 2004). The evolution
of star clusters is usually modeled by means of the simple stellar population
(SSP) approximation. An SSP is defined as a single generation of coeval stars
formed from the same progenitor molecular cloud (thus implying a single
metallicity), and governed by a given stellar initial mass function (IMF).
Age and metallicity are two basic star cluster parameters. The most direct
method to determine a cluster’s age is by employing main-sequence photometry,
since the absolute magnitude of the main-sequence turnoff is predominantly
affected by age (see Puzia et al., 2002, and references therein). However,
until recently (cf. Perina et al., 2009), this method was only applied to the
star clusters in the Milky Way and its satellites (e.g., Rich et al., 2001),
although Brown et al. (2004) estimated the age of an M31 GC using extremely
deep images observed with the Hubble Space Telescope (HST)’s Advanced Camera
for Surveys. Generally, the ages of extragalactic star clusters are determined
by comparing their observed spectral-energy distributions (SEDs) and/or
spectroscopy with the predictions of SSP models (Williams & Hodge, 2001a, b;
de Grijs et al., 2003a, b, c; Bik et al., 2003; Jiang et al., 2003; Beasley et
al., 2004; Puzia et al., 2005; Ma et al., 2006; Fan et al., 2006; Ma et al.,
2007, 2009; Caldwell et al., 2009; Wang et al., 2010). Nevertheless, SSP
models assume that cluster IMFs are fully populated, i.e., that clusters
contain infinite numbers of stars with a continuous distribution of stellar
masses, and that all evolutionary stages are well sampled. Real clusters,
however, contain a finite number of stars. Therefore, a disagreement between
the observed cluster colors and theoretical colors derived from SSP models may
become apparent (see Piskunov et al., 2009; Popescu & Hanson, 2010, and
references therein). Other limitations inherent to SSP models arise from our
poor understanding of some advanced stellar evolutionary stages, such as the
supergiant and the asymptotic-giant-branch (AGB) phases (see Bruzual &
Charlot, 2003, and references therein).
Located at a distance of $785\pm 25$ kpc, corresponding to a distance modulus
of $(m-M)_{0}=24.47\pm 0.07$ mag (McConnachie et al., 2005), M31 is the
nearest and largest spiral galaxy in the Local Group of galaxies. It has been
the subject of many GC studies and surveys, dating back to the early study of
Hubble (1932). Based on previous publications (Hubble, 1932; Seyfert & Nassau,
1945; Hiltner, 1958; Mayall & Eggen, 1953; Kron & Mayall, 1960), Vetes̆nik
(1962) compiled the first large M31 GC catalog, containing $UBV$ photometric
data of approximately 300 GC candidates. Over the past decades, several major
catalogs of M31 GCs and GC candidates have been published, including major
efforts by the Bologna group (Battistini et al., 1980, 1987, 1993), Barmby et
al. (2000), Galleti et al. (2004, 2005, 2006, 2007), Kim et al. (2007),
Caldwell et al. (2009), and Peacock et al. (2010). Following on from the first
extensive spectroscopic survey of M31 GCs by van den Bergh (1969), a
significant number of authors (e.g., Huchra et al., 1982, 1991; Dubath &
Grillmair, 1997; Federici et al., 1993; Jablonka et al., 1998; Barmby et al.,
2000; Perrett et al., 2002; Galleti et al., 2006; Lee et al., 2008, and
references therein) have studied their spatial, kinematic, and chemical
(metallicity) properties.
M31 is known to host a large number of young star clusters (e.g., Fusi Pecci
et al., 2005; Caldwell et al., 2009; Wang et al., 2010, and references
therein). Fusi Pecci et al. (2005) presented a comprehensive study of 67 very
blue star clusters, which they referred to as ‘blue luminous compact clusters’
(BLCCs). Since they are quite bright ($-6.5\leq M_{V}\leq-10.0$ mag) and very
young ($<2$ Gyr), BLCCs may be equivalent to YMCs (see for details Perina et
al., 2009, 2010). To ascertain their properties, Perina et al. (2009, 2010)
performed an imaging survey of 20 BLCCs in the disk of M31 using the HST’s
Wide Field and Planetary Camera-2 (WFPC2). They obtained the reddening values,
ages, and metallicities of their sample clusters by comparing the observed
color-magnitude diagrams (CMDs) and luminosity functions with theoretical
models.
VDB0-B195D was first detected by van den Bergh (1969). Its color is extremely
blue (e.g., $U-B=-0.48$ mag; van den Bergh, 1969) and it is very bright in
blue bands (e.g., $U=14.66$ mag; van den Bergh, 1969). As a consequence, van
den Bergh (1969) asserted that VDB0-B195D is the brightest open cluster in
M31. He determined an integrated stellar spectral type equivalent to A0, which
implies that the cluster contains massive stars. In addition, VDB0-B195D is
particularly extended and most previous photometric studies did not include
the full extent of the object’s light distribution (see for details Perina et
al., 2009). We will provide an overview of previous studies that included the
cluster in §2.1. It was observed as part of the galaxy calibration program of
the Beijing-Arizona-Taiwan-Connecticut (BATC) Multicolor Sky Survey (e.g., Fan
et al., 1996; Zheng et al., 1999) in 15 intermediate-band filters. Combined
with photometry in optical broad-band $BVRI$ and near-infrared $JHK_{\rm s}$
filters from the Two Micron All Sky Survey (2MASS) taken from Perina et al.
(2009), we obtained the SED of VDB0-B195D in 22 filters, covering the
wavelengh range from 3000 to 20,000 Å.
In this paper, we describe the details of the observations and our approach to
the data reduction in §2. In §3, we determine the age and mass of VDB0-B195D
by comparing observational SEDs with population synthesis models. We discuss
the implications of our results and provide a summary in §4.
## 2\. Optical and near-infrared observations of the YMC VDB0-B195D
### 2.1. Historical overview
VDB0-B195D was first given the designation ‘0’ (i.e., VDB0), the brightest
open cluster in M31, in the catalog of van den Bergh (1969). Battistini et al.
(1987) identified VDB0-B195D independently and called it B195D. In Battistini
et al. (1987), B195D was given a low level of confidence (class D) of being a
genuine cluster (classes A and B were assigned very high and high levels of
confidence, respectively). It was only recently independently confirmed to be
a single object. Caldwell et al. (2009) presented a new catalog containing 670
likely star clusters, stars, possible stars, and galaxies in the field of M31,
all with updated high-quality coordinates accurate to $0.2^{\prime\prime}$,
based on images from either the Local Group Galaxies Survey (LGGS) (Massey,
2006) or the Digitized Sky Survey (DSS). They use the designation VDB0-B195D,
associated with $\rm\alpha_{0}=00^{\rm h}40^{\rm m}29^{\rm s}.43$ and
$\rm\delta_{0}=+40^{\circ}36^{\prime}14^{\prime\prime}.8$ (J2000.0), which are
the coordinates we adopt in this paper. Independently, Perina et al. (2009)
studied the properties of VDB0-B195D in detail based on their HST/WFPC2
imaging survey of young massive GCs in M31. They initially selected VDB0-B195D
as two YMCs in M31, but their WFPC2 images showed unequivocally that these two
sample objects are, in fact, the same cluster. In addition, the HST images
clearly confirmed that VDB0-B195D is a real cluster. However, it is difficult
to establish whether it is more similar to ordinary open clusters, similar to
those in the disk of the Milky Way, than to YMCs that may evolve to become
disk GCs (see for details Perina et al., 2009). Spectral observations of
VDB0-B195D were obtained by van den Bergh (1969)—yielding classification
spectra and the object’s radial velocity—and Perrett et al. (2002), who used
them for determination of its radial velocity and metallicity.
### 2.2. Archival images of the BATC Multicolor Sky Survey
Observations of the YMC VDB0-B195D were obtained with the BATC 60/90cm Schmidt
telescope located at the XingLong station of the National Astronomical
Observatory of China (NAOC). This telescope is equipped with 15 intermediate-
band filters covering the optical wavelength range from 3000 to 10,000 Å. The
filter system was specifically designed to avoid contamination by the
brightest and most variable night-sky emission lines. Descriptions of the BATC
photometric system can be found in Fan et al. (1996). Before February 2006, a
Ford Aerospace 2k$\times$2k thick CCD camera was installed, with a pixel size
of 15 $\mu$m and a field of view of $58^{\prime}\times 58^{\prime}$, yielding
a resolution of $1.7^{\prime\prime}$ pixel-1. Since February 2006, a new E2V
4k$\times$4k thinned CCD with a pixel size of 12 $\mu$m has been in operation,
featuring a resolution of $1.3^{\prime\prime}$ pixel-1. The blue quantum
efficiency of the new, thinned CCD is 92.2% at 4000 Å, which is much higher
than for the old, thick device (see for details Fan et al., 2009). A field
including VDB0-B195D in the $a$–$c$ filters was observed with the thinned CCD,
and in $d$–$p$ bands with the thick CCD. Fig. 1 shows a finding chart of
VDB0-B195D in the BATC $g$ band (centered at 5795 Å), obtained with the NAOC
60/90cm Schmidt telescope. We adopt an aperture with a radius of
$15^{\prime\prime}$ (shown in Fig. 1) for the integrated photometry discussed
in this paper.
Figure 1.— Image of VDB0-B195D in the BATC $g$ band, obtained with the NAOC
60/90cm Schmidt telescope. VDB0-B195D is circled using an aperture with a
radius of $15^{\prime\prime}$. The field of view of the image is
$11^{\prime}\times 11^{\prime}$.
The BATC survey team obtained 61 images of VDB0-B195D in 15 BATC filters
between January 2004 and November 2006. Fan et al. (2009) performed the data
reduction of these images, which formed part of their M31-7 field. Table 1
contains the observation log, including the BATC filter names, the central
wavelength and bandwidth of each filter, the number of images observed through
each filter, and the total observing time per filter. Multiple images through
the same filter were combined to improve image quality (i.e., increase the
signal-to-noise ratio and remove spurious signal).
### 2.3. Intermediate-band photometry of VDB0-B195D
We determined the intermediate-band magnitudes of VDB0-B195D on the combined
images using a standard aperture photometry approach, i.e., the phot routine
in daophot (Stetson, 1987). Calibration of the magnitude zero level in the
BATC photometric system is similar to that of the spectrophotometric AB
magnitude system. For flux calibration, the Oke-Gunn (Oke & Gunn, 1983)
primary flux standard stars HD 19445, HD 84937, BD +26∘2606, and BD +17∘4708
were observed during photometric nights (Yan et al., 2000). VDB0-B195D is
located in the M31-7 field of Fan et al. (2009). The absolute flux of the
M31-7 field was calibrated based on secondary standard transformations using
the M31-1 field, which was calibrated, in turn, by the four Oke-Gunn primary
flux standard stars by Jiang et al. (2003). Since VDB0-B195D is an extended
object, an appropriate aperture size must be adopted to determine its total
luminosity. The (radial) photometric asymptotic growth curve, in all BATC
bands, flattens out at a radius of $\sim 15^{\prime\prime}$. Inspection
ensured that this aperture is adequate for photometry, i.e., VDB0-B195D does
not show any obvious signal beyond this radius. In addition, this aperture is
nearly the same as that adopted by Perina et al. (2009) to determine the
cluster’s photometry in the $BVRI$ bands, based on the M31 imaging survey of
Massey (2006) (see §2.4 below). Therefore, we use an aperture with $r\approx
15^{\prime\prime}$ for integrated photometry, i.e., $r=9$ pixels for the
2k$\times$2k thick CCD camera, and $r=12$ pixels for the 4k$\times$4k thinned
CCD camera. VDB0-B195D is projected onto the disk of M31, where the background
is bright and fluctuates, potentially as a function of distance from the
cluster center. To avoid contamination from background fluctuations, we
adopted annuli for background subtraction spanning between 10 and 15 pixels
for the 2k$\times$2k thick CCD camera, and from 13 to 20 pixels for the
4$\times$4k thinned CCD camera, both corresponding to $\sim 17$–26′′. While
these annuli are spatially as close as possible to the region dominated by
cluster light (so that any differences in background flux are minimized), they
are wide enough to average out any expected background fluctuations. The
calibrated photometry of VDB0-B195D in 15 filters is summarized in column (6)
of Table 1, in conjunction with the $1\sigma$ magnitude uncertainties, which
include uncertainties from the calibration errors of both the M31-1 field
standard stars (see for details Fan et al., 2009; Jiang et al., 2003) and ‘the
secondary standard stars’ in common between the M31-1 and M31-7 fields used
for calculation of the mean magnitude offsets between the standard and
instrumental magnitudes (see for details Fan et al., 2009), as well as those
resulting from our daophot application.
### 2.4. Optical broad-band and near-infrared 2MASS photometry of VDB0-B195D
Four independent sets of photometric data exist for VDB0-B195D. van den Bergh
(1969) obtained $UBV$ photometry using observations of the 200-inch Hale
telescope, Battistini et al. (1987) performed $UBVR$ photometry based on
photographic plates observed with the 152 cm Ritchey-Chrétien $f$/8 telescope
of the University of Bologna in Loiano, King & Lupton (1991) obtained $UBV$
photometry for VDB0-B195D using observations with the University of Hawaii’s
2.2 m telescope on Mauna Kea using the $f$/10 secondary and coronene-coated
$584\times 416$ GEC CCD, and Sharov et al. (1995) performed $UBV$ photometry
based on photo-electric observations with the 2.6 m Shain telescope of the
Crimean Astrophysical Observatory. In addition, in the Revised Bologna
Catalogue (RBC) of M31 GCs published by Galleti et al. (2004), the photometric
data of VDB0-B195D in optical bands are based on Battistini et al. (1987) and
Sharov et al. (1995), and transformed to the reference system of Barmby et al.
(2000) by applying offsets derived from objects in common between the relevant
catalog and the data set of Barmby et al. (2000). In the RBC, VDB0-B195D was
regarded as two objects. We list these photometric data in Table 2 for
comparison. Note that, in the latest RBC incarnation (version 3.5, updated on
27 March 2008), VDB0-B195D is included as a single object.
Galleti et al. (2004) also determined 2MASS $JHK_{\rm s}$ photometric
magnitudes for VDB0-B195D (transformed to the CIT photometric system; Elias et
al., 1982, 1983), which we have included in Table 3. In addition, Perina et
al. (2009) realized that VDB0-B195D is a particularly extended object and that
it is possible that the photometry of Sharov et al. (1995) (compiled in the
RBC) was obtained with apertures that were not large enough to include all of
its flux. Therefore, they redetermined its photometric values in the $BVRI$
bands based on the M31 imaging survey of Massey (2006) using an aperture with
$r=14.4^{\prime\prime}$, which are also listed in Table 3.
From a comparison of the values in Tables 2 and 3, it is clear that the
magnitudes of van den Bergh (1969) are brighter, while the results of the
three other references are consistent. The magnitudes determined by Perina et
al. (2009) are much brighter, however, because of their careful inclusion of
all of the cluster’s flux. To compare our photometric results with previously
published values, we transformed the magnitudes of VDB0-B195D in the BATC
intermediate bands to broad-band $UBVRI$-equivalent photometry based on the
relationships obtained by Zhou et al. (2003). These are also listed in Table
3, and the uncertainties include those originating from the transformation
based on the relationships of Zhou et al. (2003) and their calibration errors
(column 5 of their Table 3). In Fig. 2, we show the result of the comparison.
In general, the other photometric data are fainter than ours and those of
Perina et al. (2009). Fig. 2 and Table 3 show that our new $VRI$ magnitudes
agree with the results of Perina et al. (2009), and that the $B$ magnitude
obtained in this paper is 0.32 mag brighter than that of Perina et al. (2009).
Considering the photometric errors of both Perina et al. (2009) and our
current study, these two $B$-band photometric results are consistent within
$2\sigma$. In addition, we should keep in mind that, although the $VRI$
magnitudes obtained in this paper are consistent with the results of Perina et
al. (2009) within $1\sigma$, the disagreement in $B$ magnitudes at this level
is understandable. This is caused by the fact that the original photometry in
the present paper was obtained in the proprietary BATC filters and transformed
to the $UBVRI$ system using transformation equations. Zhou et al. (2003)
determined these conversions based on the broad-band $UBVRI$ magnitudes of 48
stars from Landolt (1983, 1992) and Galadí-Enríquez et al. (2000) in the
Landolt SA95 field, and their photometric data in the 15 BATC intermediate-
band filters. In addition, the central wavelengths and bandwidths of the BATC
and $UBVRI$ systems differ. In fact, a similar significant disagreement of
$B$-band photometric data for some M31 GCs was reported by Wang et al. (2010),
citing similar arguments.
Figure 2.— Comparison of photometric data from different sources with new
determinations in this paper for VDB0-B195D. The data points shown as black
dots are from Perina et al. (2009).
## 3\. Stellar population of VDB0-B195D
### 3.1. Stellar populations and synthetic photometry
To determine the age and mass of VDB0-B195D, we compared its SED with
theoretical stellar population synthesis models. The SED consists of
photometric data in the 15 BATC intermediate bands obtained in this paper and
optical broad-band $BVRI$ and 2MASS near-infrared $JHK_{s}$ data from Perina
et al. (2009), listed in Table 3. We used the galev SSP models (e.g., Kurth et
al., 1999; Schulz et al., 2002; Anders & Fritze-v. Alvensleben, 2003) for our
comparisons. The galev SSPs are based on the Padova stellar isochrones, with
the most recent versions using the updated Bertelli et al. (1994) isochrones
(which include the thermally pulsing asymptotic giant-branch phase), and a
Salpeter (1955) stellar IMF with lower- and upper-mass limits of 0.10 and
between 50 and 70 $M_{\odot}$, respectively, depending on metallicity. The
full set of models spans the wavelength range from 91Å to 160 $\mu$m. These
models cover ages from $4\times 10^{6}$ to $1.6\times 10^{10}$ yr, with an age
resolution of 4 Myr for ages up to 2.35 Gyr, and 20 Myr for greater ages. The
galev SSP models include five initial metallicities,
$Z=0.0004,0.004,0.008,0.02$ (solar metallicity), and 0.05.
Since our observational data consist of integrated luminosities through the
set of BATC filters, we convolved the galev SSP SEDs with the BATC
intermediate-, optical broad-band $BVRI$, and 2MASS filter-response curves to
obtain synthetic optical and near-infrared photometry for comparison. The
synthetic $i^{\rm th}$ filter magnitude can be computed as
$m=-2.5\log\frac{\int_{\nu}F_{\nu}\varphi_{i}(\nu){\rm
d}\nu}{\int_{\nu}\varphi_{i}(\nu){\rm d}\nu}-48.60,$ (1)
where $F_{\nu}$ is the theoretical SED and $\varphi_{i}$ the response curve of
the $i^{\rm th}$ filter of the BATC, $BVRI$, and 2MASS photometric systems.
Here, $F_{\nu}$ varies with age and metallicity. Since the observed magnitudes
in the $BVRI$ and 2MASS photometric systems are given in the Vega system, we
transformed them to the AB system for our fits.
### 3.2. Reddening and metallicity of VDB0-B195D
To obtain the intrinsic SED of VDB0-B195D, its photometry must be dereddened.
To date, only Perina et al. (2009) obtained reddening values for VDB0-B195D.
They compared the observed CMD with theoretical isochrones and determined
$E(B-V)=0.20\pm 0.03$ mag. Caldwell et al. (2009) were unable to derive the
cluster’s reddening value because of the presence of a foreground field star,
so they adopted $E(B-V)=0.28\pm 0.17$ mag (external rms error), equivalent to
the mean reddening of the young clusters in M31. In this paper, we therefore
adopt the reddening value from Perina et al. (2009).
In addition, cluster SEDs are affected by age and metallicity effects.
Therefore, we can only accurately constrain a cluster’s age if the metallicity
is known. Perina et al. (2009) found that the CMD of VDB0-B195D, based on
their HST/WFPC2 observations, is best reproduced by the solar-metallicity
models of Girardi et al. (2002). We therefore adopt solar metallicity for
VDB0-B195D.
### 3.3. The ‘lowest-luminosity-limit’ test
The lowest-luminosity limit (LLL; Cerviño & Luridiana, 2004) implies that it
is meaningless to compare a cluster with population synthesis models to obtain
its age and mass if its integrated luminosity is lower than the luminosity of
the most luminous star included in the model for the relevant age. The LLL
method states that clusters fainter than this limit cannot be analyzed using
standard procedures such as $\chi^{2}$ minimization of the observed values
with respect to the mean SSP models (see also Barker et al., 2008). Below the
LLL, cluster ages and masses cannot be obtained self-consistently. To take
into account the effects on the integrated luminosities of statistically
sampling the stellar IMF (e.g., Cerviño et al., 2000, 2002; Cerviño &
Luridiana, 2004), we used the theoretical Padova isochrones at
http://stev.oapd.inaf.it/cmd (CMD2.2). This interactive Web interface provides
isochrones for a number of photometric systems, including optical broad-band,
2MASS, and the BATC data used here. We obtained the solar-metallicity
($Z=0.019$) isochrones of Marigo et al. (2008), as recommended by CMD2.2,
based on the (Salpeter, 1955) IMF so as to match the IMF selection for our age
and mass determinations of VDB0-B195D in §3.4 (see §3.1 for details).
Figure 3 shows the LLL values as a function of age for the different filters
used in this paper. These luminosities are obtained by identifying the most
luminous star on each isochrone for the relevant passband. The gray area shows
the cluster’s absolute luminosity, assuming a distance modulus of
$(m-M)_{0}=24.47$ mag (785 kpc) for M31 (McConnachie et al., 2005). The upper
luminosity limit has been corrected for extinction, based on a reddening value
of $E(B-V)=0.20$ mag. The interstellar extinction curve, $A_{\lambda}$, is
taken from Cardelli et al. (1989), $R_{V}=A_{V}/E(B-V)=3.1$.
We see that VDB0-B195D does not lie below the LLL in any of the passbands used
here. This means that, in general, VDB0-B195D can host the most luminous star
that would be present theoretically for the given age of the cluster.
Figure 3.— Lowest-luminosity limit for the filters used in this paper. The
curves indicate the luminosities of the most luminous star on each isochrone
for the relevant passband. The light-gray area shows the absolute magnitudes
of VDB0-B195D based on a reddening value of $E(B-V)=0.20$ mag (Perina et al.,
2009). We used a distance modulus of $(m-M)_{0}=24.47$ mag (785 kpc) for M31
(McConnachie et al., 2005) to calculate the absolute magnitudes.
### 3.4. Fit results
In the previous section, the LLL test proves that the luminosity of VDB0-195D
is higher than the luminosity of its brightest star expected for a given
cluster age, i.e., that using SSP models is not completely meaningless. In
addition, the bright absolute magnitude of VDB0-195D allows us to consider a
possibility that the cluster is massive enough and IMF sampling effects should
not strongly impact the fitting results. So we will determine the cluster’s
age and mass estimates based on direct comparisons with SSP mean values in
this section. However, we should keep in mind that this approach is a
compromise. In fact, the fitting results (Fig. 4 and Table 5) show probable
problem even for relative massive clusters.
We use a $\chi^{2}$ minimization test to determine which galev SSP models are
most compatible with the observed SEDs,
$\chi^{2}=\sum_{i=1}^{22}{\frac{[m_{\nu_{i}}^{\rm intr}-m_{\nu_{i}}^{\rm
mod}(t)]^{2}}{\sigma_{i}^{2}}},$ (2)
where $m_{\nu_{i}}^{\rm mod}(t)$ is the integrated magnitude in the $i^{\rm
th}$ filter of a theoretical SSP at age $t$ (for solar metallicity),
$m_{\nu_{i}}^{\rm intr}$ is the intrinsic, integrated magnitude, and
$\sigma_{i}$ is the magnitude uncertainty, defined as
$\sigma_{i}^{2}=\sigma_{{\rm obs},i}^{2}+\sigma_{{\rm
mod},i}^{2}+(R_{\lambda_{i}}*\sigma_{\rm red})^{2}+\sigma_{{\rm md},i}^{2}.$
(3)
Here, $\sigma_{{\rm obs},i}$ is the observational uncertainty from column (6)
of Table 1 and column (2) of Table 3, $\sigma_{{\rm mod},i}$ is the
uncertainty associated with the model itself, $\sigma_{\rm red}$ is the
uncertainty in the reddening value, and
$R_{\lambda_{i}}=A_{\lambda_{i}}/E(B-V)$, where $A_{\lambda_{i}}$ is taken
from Cardelli et al. (1989), $R_{V}=A_{V}/E(B-V)=3.1$, and $\sigma_{{\rm
md},i}$ is the uncertainty in the distance modulus, for the $i^{\rm th}$
filter. Charlot et al. (1996) estimated the uncertainty associated with the
term $\sigma_{{\rm mod},i}$ by comparing the colors obtained from different
stellar evolutionary tracks and spectral libraries. Following Ma et al. (2007,
2009), we adopt $\sigma_{{\rm mod},i}=0.05$ mag.
Perina et al. (2009) pointed out that VDB0-B195D is a particularly extended
object and that the photometric measurements of van den Bergh (1969),
Battistini et al. (1987), King & Lupton (1991), and Sharov et al. (1995) did
not include all of its flux. Therefore, we adopt the photometry of Perina et
al. (2009) to fit the observed SED with theoretical SSPs for our age
determination. The fit yielding the minimum $\chi^{2}$ value ($\chi^{2}({\rm
min})$) was adopted as the best fit and we adopted the corresponding age
value, $60.0\pm 8.0$ Myr. In addition, our best-fitting age estimate of
$60.0\pm 10.0$ Myr results from using the (redder) $k$–$p$ and $IJHK_{\rm s}$
photometry; using only the blue part of the cluster’s SED ($B,a$–$e$, where
any effects caused by stochasticity may be smaller) yields an age of $72.0\pm
34.0$ Myr. The uncertainty was estimated using confidence limits. If
$\chi^{2}/{\nu}<\chi^{2}({\rm min})/{\nu}+1$, the resulting age is within the
68.3% probability range; here, ${\nu}=21$ is the number of free parameters,
i.e., the number of observational data points minus the number of parameters
used in the theoretical model. Therefore, the accepted age range is derived
from those fits that have $\chi^{2}({\rm
min})/{\nu}<\chi^{2}/{\nu}<\chi^{2}({\rm min})/{\nu}+1$. The best
reduced-$\chi^{2}$—defined as $\chi_{\nu}^{2}({\rm min})=\chi^{2}({\rm
min})/{\nu}$—and age are listed in Table 4. The best fit to the SED of
VDB0-B195D is shown in Fig. 4, where we display the intrinsic cluster SED
(symbols with error bars), as well as the integrated SED (open circles) and
spectrum of the best-fitting model. From Fig. 4, we note that the
observational data in the $b$, $d$, $o$, and $p$ BATC filters and in the
$K_{\rm s}$ band do not match the best-fitting model very well (the difference
is approximately 0.3 mag). Photometric uncertainties in these filters may
cause some differences, although this might not be the main reason for the
discrepancy. As we know, observational star clusters’ SEDs are affected by
age, metallicity and reddening. If the reddening value and metallicity adopted
in this paper are not problematic, discrepancy between our observations and
the best-fitting model may reflect the difficulty in achieving an appropriate
(but formal) fit of an SED of a single, real cluster by SSP models. However,
as we will see below, the reddening value adopted in this paper may be bigger
than the actual reddening of VDB0-B195D. In addition, the differences between
the photometric data and the model in Fig. 4 show a somewhat systematic
behavior with wavelength: in bluer passbands the cluster seems to be more
luminous than predicted by the model, while in redder passbands it is fainter
than the corresponding model predictions. A blue excess and red deficiency in
the observed SED with respect to the model predictions may indicate a shortage
of red giants (RGs), which can occur when the cluster is either younger or
less massive (or both) than the corresponding best-fitting model suggests. In
other words, IMF discreteness may play a role: due to a relatively longer
main-sequence (MS) phase and shorter RG phase, a random young cluster is
typically bluer than predicted by SSP models. At the same time, we find that
the reddening value adopted affects the fitting result greatly. In fact, the
best fit to the SED of VDB0-B195D improves a great deal when adopting a
smaller reddening value such as $E(B-V)=0.1$: $\chi_{\nu}^{2}({\rm
min})=0.73$; the resulting age ($64.0\pm 8.0$ Myr) is nearly the same as one
($60.0\pm 8.0$ Myr) obtained with $E(B-V)=0.2$.
We next determined the mass of VDB0-B195D. The galev models include absolute
magnitudes (in the Vega system) in 77 filters for SSPs of
$10^{6}~{}M_{\odot}$, including 66 filters of the HST, Johnson $UBVRI$ (see
for details Landolt, 1983), Cousins $RI$ (see for details Landolt, 1983), and
$JHK$ (Bessell & Brett, 1988) systems. The difference between the intrinsic
absolute magnitudes and those given by the model provides a direct measurement
of the cluster mass, in units of $10^{6}~{}M_{\odot}$. However, we should keep
in mind that this is only correct for cluster masses above
$10^{6}~{}M_{\odot}$. We estimated the mass of VDB0-B195D using magnitudes in
all of the $BVRI$ and $JHK_{\rm s}$ bands. Therefore, we transformed the 2MASS
$JHK_{\rm s}$ magnitudes to the photometric system of Bessell & Brett (1988)
using the equations given by Carpenter (2001). The resulting mass
determinations for VDB0-B195D are listed in Table 5 with their $1\sigma$
uncertainties including contributions from uncertainties in extinction and
distance modulus. From Table 5, we see that the mass of VDB0-B195D obtained
based on the magnitudes in different filters is very different. (The highest
mass obtained, based on the $B$-band magnitude, is $~{}0.5\times
10^{5}~{}M_{\odot}$ more massive than that obtained using the $K_{s}$
magnitude.) In addition, the mass estimates differ systematically with
filters. Provided that VDB0-B195D is massive enough to be fitted by SSP
models, a systematic trend of masses based on different passbands may indicate
a problem with reddening value adopted for the cluster. If the actual
reddening is smaller than the adopted value, the actual luminosity would be
overestimated. This effect is small in redder filters but strong in bluer
filters. As discussed in age estimation, a smaller reddening value can improve
the fitting result greatly. In fact, a smaller reddening value can reduce the
mass discrepancies based on the magnitudes in different filters. When we
adopted $E(B-V)=0.1$, the mass of VDB0-B195D obtained based on the magnitudes
in different filters is the same within 1$\sigma$. We list these estimates in
Table 7. From Table 5, we know that the mass of VDB0-B195D obtained in paper
is between $(1.1-1.6)\times 10^{5}~{}M_{\odot}$ when the reddening value is
adopted to be $(B-V)=0.2$.
Figure 4.— Best-fitting, integrated theoretical galev SEDs compared to the
intrinsic SED of VDB0-B195D. The photometric measurements are shown as symbols
with error bars (vertical for uncertainties and horizontal for the approximate
wavelength coverage of each filter). Open circles represent the calculated
magnitudes of the model SED for each filter. We used a distance modulus of
$(m-M)_{0}=24.47$ mag (785 kpc) for M31 (McConnachie et al., 2005) to
calculate the absolute magnitudes.
## 4\. Summary and discussion
VDB0-B195D was previously shown to be a massive cluster based on HST/WFPC2
observations. Its color is extremely blue and it is very bright, particularly
in blue bands. In addition, VDB0-B195D is an extended object, and most
previous photometric measurements did not include its full flux distribution
(see Perina et al., 2009, for details).
In this paper, we obtained the cluster’s SED in the 15 BATC intermediate-band
filters. We subsequently determined its age and mass by comparing our
multicolor photometry with theoretical stellar population synthesis models.
Our multicolor photometric data consist of 15 intermediate-band filters
obtained in this paper, and broad-band $BVRI$ and 2MASS $JHK_{\rm s}$ from
Perina et al. (2009), covering a wavelength range from 3000 to 20,000 Å. Our
results show that VDB0-B195D is a genuine YMC in M31.
To understand the real nature of the BLCCs, Perina et al. (2009, 2010)
performed an HST imaging survey of 20 BLCCs in M31’s disk. As a test case,
Perina et al. (2009) presented details of the data-reduction pipeline that
will be applied to all survey data and describe its application to VDB0-B195D.
They estimated the object’s age, by comparison of the observed CMD with
theoretical isochrones from Girardi et al. (2002), at $\simeq 25$ Myr. In
addition, they constrained realistic upper and lower limits to the cluster’s
age, independent of the adopted metallicity, within the relatively narrow
range from 12 to 63 Myr. Using Maraston’s SSP models of solar metallicity
(Maraston, 1998, 2005), Salpeter (1955) and Kroupa (2001) IMFs, and
photometric values in the $V$ and 2MASS $J$, $H$, and $K_{\rm s}$ bands,
Perina et al. (2009) concluded that the mass of VDB0-B195D is $>2.4\times
10^{4}~{}M_{\odot}$, with their best estimates in the range $\simeq(4-9)\times
10^{4}~{}M_{\odot}$.
Caldwell et al. (2009) presented an updated catalog of 1300 objects in M31,
including spectroscopic and imaging surveys, based on images from either the
LGGS or the DSS and spectra taken with the Hectospec fiber positioner and
spectrograph on the 6.5 m MMT. They derived ages and reddening values for 140
young clusters by comparing their observed spectra with model spectra from the
Starburst99 SSP suite (Leitherer et al., 1999). The results show that these
clusters are less than 2 Gyr old, while most have ages between $10^{8}$ and
$10^{9}$ yr (the age of VDB0-B195D they derive is $\log{\rm age/yr}=7.6$). In
addition, Caldwell et al. (2009) also estimated the masses of these young
clusters using $V$-band photometry and model mass-to-light ratios (Leitherer
et al., 1999) corresponding to the derived spectroscopic ages. This resulted
in masses ranging from $2.5\times 10^{2}$ to $1.5\times 10^{5}~{}M_{\odot}$.
The mass of VDB0-B195D obtained by Caldwell et al. (2009) is $\log M_{\rm
cl}/M_{\odot}=5.1$ (no uncertainty quoted).
We compare the various age and mass estimates of VDB0-B195D in Table 6\. Our
newly obtained age is older than the estimates of both Perina et al. (2009)
and Caldwell et al. (2009), while the mass obtained in this paper is higher
than the estimate of Perina et al. (2009) and consistent with the
determination of Caldwell et al. (2009). However, our results are in agreement
with those of both Perina et al. (2009) and Caldwell et al. (2009) within
$3\sigma$. The age and mass obtained in this paper confirms that VDB0-B195D is
genuinely a YMC in M31.
As we know, SSP models describe a very special case of a continuous
distribution of stellar mass (or light) along isochrones. This is well
approximated by clusters with masses larger than $10^{6}~{}M_{\odot}$. Also,
for cluster masses of about $10^{5}~{}M_{\odot}$, SSP models can probably
still be applied since a systematic difference between SSP models and
observations should, on average, be smaller than 0.05 mag for clusters older
than 10 Myr (see Fig. 3 in Piskunov et al. 2009). However, from the results of
this paper, we may conclude that, probably, a formal fitting of SSP models to
observed SEDs cannot be used without caution even for relatively massive (or
apparently massive) clusters, and it is highly doubtful that this approach can
be applied in a routine work providing accurate cluster parameters. The
relative accuracy of 10% for age and 20% found for the mass of VDB0-B195D
seems to be rather formal and not very confident. In addition, observational
star clusters’ SEDs are affected by reddening, an effect that is also
difficult to separate from the combined effects of age and metallicity
(Calzetti 1997; Vazdekis et al. 1997; Origlia et al. 1999). Only the
metallicity and reddening are derived accurately (and, ideally,
independently), these degeneracies are largely (if not entirely) reduced, and
ages can then also be estimated accurately based on a comparison of multicolor
photometry spanning a significant wavelength range (de Grijs et al. 2003b;
Anders et al. 2004) with theoretical stellar population synthesis models. It
is true that the discrepancy between our observations and the best-fitting
model is great, and the mass of VDB0-B195D obtained based on the magnitudes in
different filters is very different. However, when we adopt a smaller
reddening value, the results improve greatly. So, we conclude that the actual
reddening value of VDB0-B195D may be smaller than $E(B-V)=0.2$.
We are indebted to the anonymous referee for very carefully reading our
manuscript, and for many thoughtful comments and insightful suggestions that
improved this paper significantly. We are grateful to Dr. Richard de Grijs for
the help in terms of scientific input and proofreading. This work was
supported by the Chinese National Natural Science Foundation (grants 10873016,
10633020, 10603006, and 10803007) and by the National Basic Research Program
of China (973 Program), No. 2007CB815403.
## References
* Anders & Fritze-v. Alvensleben (2003) Anders, P., & Fritze-v. Alvensleben, U. 2003, A&A, 401, 1063
* Anders et al. (2004) Anders, P., Bissantz, N., Fritze-v. Alvensleben, U., & de Grijs, R. 2004, MNRAS, 347, 196
* Barker et al. (2008) Barker, S., de Grijs, R., & Cerviño, M. 2008, A&A, 484, 711
* Barmby et al. (2000) Barmby, P., Huchra, J., Brodie, J., Forbes, D., Schroder, L., & Grillmair, C. 2000, AJ, 119, 727
* Battistini et al. (1980) Battistini, P., Bònoli, F., Braccesi, A., Fusi Pecci, F., Malagnini, M. L., & Marano, B. 1980, A&AS, 42, 357
* Battistini et al. (1987) Battistini, P., Bònoli F., Braccesi, A., Federici, L., Fusi Pecci, F., Marano, B., & Börngen, F. 1987, A&AS, 67, 447
* Battistini et al. (1993) Battistini, P., Bònoli, F., Casavecchia, M., Ciotti, L., Federici, L., & Fusi Pecci F. 1993, A&A, 272, 77
* Beasley et al. (2004) Beasley, M. A., et al. 2004, AJ, 128, 1623
* Bertelli et al. (1994) Bertelli, G., Bressan, A., Chiosi, C., Fagotto, F., & Nasi, E. 1994, A&AS, 106, 275
* Bessell & Brett (1988) Bessell, M. S., & Brett, J. M. 1988, PASP, 100, 1134
* Bik et al. (2003) Bik, A., Lamers, H. J. G. L. M., Bastian, N., Panagia, N., & Romaniello, M. 2003, A&A, 397, 473
* Bònoli et al. (1987) Bònoli, F., Delpino, F., Federici, L., & Fusi Pecci, F. 1987, A&A, 185, 25
* Brown et al. (2004) Brown, T. M., et al. 2004, ApJ, 613, L125
* Bruzual & Charlot (2003) Bruzual, A. G., & Charlot, S. 2003, MNRAS, 344, 1000
* Caldwell et al. (2009) Caldwell, N., Harding, P., Morrison, H., Rose, J. A., Schiavon, R., & Kriessler, J. 2009, AJ, 137, 94
* Calzetti (1997) Calzetti, D. 1997, AJ, 113, 162
* Cao & Wu (2007) Cao, C., & Wu, H. 2007, AJ, 133, 1710
* Cardelli et al. (1989) Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245
* Carpenter (2001) Carpenter, J. M. 2001, AJ, 121, 2851
* Cerviño et al. (2000) Cerviño, M., Luridiana, V., & Castander, F. J. 2000, A&A, 360, L5
* Cerviño et al. (2002) Cerviño, M., Valls-Gabaud, D., Luridiana, V., & Mas-Hesse, J. M. 2002, A&A, 381, 51
* Cerviño & Luridiana (2004) Cerviño, M., & Luridiana, V. 2004, A&A, 413, 145
* Charlot et al. (1996) Charlot, S., Worthey, G., & Bressan, A. 1996, ApJ, 457, 625
* Clark et al. (2005) Clark, J. S., Negueruela, I., Crowther, P. A., & Goodwin, S. P. 2005, A&A, 434, 949
* Crowther et al. (2006) Crowther, P. A., Hadfield, L. J., Clark, J. S., Negueruela, I., & Vacca, W. D. 2006, MNRAS, 372, 1407
* Dubath & Grillmair (1997) Dubath, P., & Grillmair, C. J. 1997, A&A, 321, 379
* de Grijs & Parmentier (2007) de Grijs, R., & Parmentier, G. 2007, ChJAA, 7, 155
* de Grijs et al. (2003a) de Grijs, R., Bastian, N., & Lamers, H. J. G. L. M. 2003a, MNRAS, 340, 197
* de Grijs et al. (2003b) de Grijs, R., Fritze-v. Alvensleben, U., Anders, P., Gallagher, J. S., Bastian, N., Taylor, V. A., & Windhorst, R. A. 2003b, MNRAS, 342, 259
* de Grijs et al. (2003c) de Grijs, R., Anders, P., Lynds, R., Bastian, N., Lamers, H. J. G. L. M., & O’Neill, E. J. Jr. 2003c, MNRAS, 343, 1285
* Elias et al. (1982) Elias, J. H., Frogel, J. A., Matthews, K., & Neugebauer, G. 1982, AJ, 87, 1029
* Elias et al. (1983) Elias, J. H., Frogel, J. A., Hyland, A. R., & Jones, T. J. 1983, AJ, 88, 1027
* Fan et al. (1996) Fan, X., et al. 1996, AJ, 112, 628
* Fan et al. (2006) Fan, Z., Ma, J., de Grijs, R., Yang Y., & Zhou X. 2006, MNRAS, 371, 1648
* Fan et al. (2009) Fan, Z., Ma, J., & Zhou X. 2009, RAA, 9, 993
* Federici et al. (1993) Federici, L., Bonoli, F., Ciotti, L., Fusi Pecci, F., Marano, B., Lipovetsky, V. A., Neizvestny, S. I., & Spassova, N. 1993, A&A, 274, 87
* Fusi Pecci et al. (2005) Fusi Pecci, F., Bellazzini, M., Buzzoni, A., De Simone, E., Federici, L., & Galleti, S. 2005, AJ, 130, 554
* Galadí-Enríquez et al. (2000) Galadí-Enríquez, D., Trullols, E., & Jordi, C. 2000, A&AS, 146, 169
* Galleti et al. (2004) Galleti, S., Federici, L., Bellazzini, M., Fusi Pecci, F., & Macrina, S. 2004, A&A, 426, 917
* Galleti et al. (2005) Galleti, S., Bellazzini, M., Federici, L., & Fusi Pecci, F. 2005, A&A, 436, 535
* Galleti et al. (2006) Galleti, S., Federici, L., Bellazzini, M., Buzzoni, A., & Fusi Pecci, F. 2006, A&A, 456, 985
* Galleti et al. (2007) Galleti, S., Bellazzini, M., Federici, L., Buzzoni, A., & Fusi Pecci, F. 2007, A&A, 471, 127
* Girardi et al. (2002) Girardi, L., et al. 2002, A&A, 391, 195
* Hiltner (1958) Hiltner, W. A. 1958, ApJ, 128, 9
* Hodge (1961) Hodge, P. W. 1961, ApJ, 133, 413
* Hubble (1932) Hubble, E. P. 1932, ApJ, 76, 44
* Huchra et al. (1982) Huchra, J., Stauffer, J., & van Speybroeck, L. 1982, ApJ, 259, L57
* Huchra et al. (1991) Huchra, J. P., Brodie, J. P., & Kent, S. M. 1991, ApJ, 370, 495
* Jablonka et al. (1998) Jablonka, P., Bica, E., Bonatto, C., Bridges, T. J., Langlois, M., & Carter, D. 1998, A&A, 335, 867
* Jiang et al. (2003) Jiang, L., Ma, J., Zhou, X., Chen, J., Wu, H., & Jiang Z. 2003, AJ, 125, 727
* Kim et al. (2007) Kim, S., et al. 2007, AJ, 134, 706
* King & Lupton (1991) King, J. R., & Lupton, R. H. 1991, ASP Conf. Ser. 13:, The formation and evolution of star clusters, 13, 575
* Kron & Mayall (1960) Kron, G. E., & Mayall, N. U. 1960, AJ, 65, 581
* Kroupa (2001) Kroupa, P. 2001, MNRAS, 322, 231 AJ, 65, 581
* Kurth et al. (1999) Kurth, O. M., Fritze-v. Alvensleben, U., & Fricke, K. J. 1999, A&AS, 138, 19
* Landolt (1983) Landolt, A. U. 1983, AJ, 88, 439
* Landolt (1992) Landolt, A. U. 1992, AJ, 104, 340
* Larsen & Richtler (1999) Larsen, S. S., & Richtler, T. 1999, A&A, 345, 59
* Larsen (2004) Larsen, S. S. 2004, ASP Conf. Ser. 322: The Formation and Evolution of Massive Young Star Clusters, 322, 19
* Larsen et al. (2004) Larsen, S. S., Brodie, J. P., & Hunter, D. A. 2004, AJ, 128, 2295
* Lee et al. (2008) Lee, M. G., Hwang, H. S., Kim, S. C., Park, H. S., Geisler, D., Sarajedini, A., & Harris, W. E. 2008, ApJ, 674, 886
* Leitherer et al. (1999) Leitherer, C., et al. 1999, ApJS, 123, 3
* Ma et al. (2006) Ma, J., de Grijs, R., Yang, Y., Zhou, X., Chen, J., Jiang, Z., Wu, Z., & Wu, J. 2006, MNRAS, 368, 1443
* Ma et al. (2007) Ma, J., et al. 2007, ApJ, 659, 359
* Ma et al. (2009) Ma, J., et al. 2009, AJ, 137, 4884
* McConnachie et al. (2005) McConnachie, A. W., Irwin, M. J., Ferguson, A. M. N., Ibata, R. A., Lewis, G. F., & Tanvir, N. 2005, MNRAS, 356, 979
* Maraston (1998) Maraston, C. 1998, MNRAS, 300, 872
* Maraston (2005) Maraston, C. 2005, MNRAS, 362, 799
* Marigo et al. (2008) Marigo, P., Girardi, L., Bressan, A., Groenewegen, M. A. T., Silva,L., & Granato, G. L. 2008, A&A, 482, 883
* Massey (2006) Massey, P., Olsen, K. A. G., Hodge, P. W., Strong, S, B., Jacoby, G. H., Schlingman, W., & Smith, R. C. 2006, AJ, 131, 2478
* Mayall & Eggen (1953) Mayall, N. U., & Eggen, O. J. 1953, PASP, 65, 24
* Origlia et al. (1999) Origlia, L., Goldader, J. D., Leitherer, C., Schaerer, D., & Oliva, E. 1999, ApJ, 514, 96
* Oke & Gunn (1983) Oke, J. B., & Gunn, J. E. 1983, ApJ, 266, 713
* Peacock et al. (2010) Peacock, M., et al. 2010, MNRAS, 402, 803
* Perina et al. (2009) Perina, S., et al. 2009, A&A, 494, 933
* Perina et al. (2010) Perina, S., et al. 2010, A&A, 511, 23
* Perrett et al. (2002) Perrett, K. M., Bridges, T. J., Hanes, D. A., Irwin, M. J., Brodie, J. P., Carter, D., Huchra, J. P., & Watson, F. G. 2002, AJ, 123, 2490
* Piskunov et al. (2009) Piskunov, A. E., Kharchenko, N. V., Schilbach, E., Röser, S., Scholz, R. D., & Zinnecker, H. 2009, A&A, 507L, 5
* Popescu & Hanson (2010) Popescu, B., & Hanson, M. M. 2010, ApJ, 713L, 21
* Puzia et al. (2002) Puzia, T. H., Zepf, S. E., Kissler-Patig, M., Hilker, M., Minniti, D., & Goudfrooij, P. 2002, A&A, 391, 453
* Puzia et al. (2005) Puzia, T. H., Perrett, K. M., & Bridges, T. J. 2005, A&A, 434, 909
* Racine (1991) Racine, R. 1991, AJ, 101, 865
* Rich et al. (2001) Rich, R. M., Shara, M. M., & Zurek, D. 2001, AJ, 122, 842
* Salpeter (1955) Salpeter, E. E. 1955, ApJ, 121, 161
* Schulz et al. (2002) Schulz, J., Fritze-v. Alvensleben, U., Möller, C. S., & Fricke, K. J. 2002, A&A, 392, 1
* Searle (1973) Searle, L., Sargent, W. L. W., & Bagnuolo, W. G. 1973, ApJ, 179, 427
* Seyfert & Nassau (1945) Seyfert, C. K., & Nassau, J. J. 1945, ApJ, 102, 377
* Sharov et al. (1995) Sharov, A. S., Lyutyi, V. M., & Esipov, V. F. 1995, SvAL, 21, 240
* Stetson (1987) Stetson, P. B. 1987, PASP, 99, 191
* Tinsley (1968) Tinsley, B. M. 1968, ApJ, 151, 547
* Tinsley (1972) Tinsley, B. M. 1972, ApJ, 178, 319
* Vazdekis et al. (1997) Vazdekis, A., Peletier, R. F., Beckman, J. E., & Casuso, E. 1997, ApJS, 111, 203
* van den Bergh (1969) van den Bergh, S. 1969, ApJS, 19, 45
* Vetes̆nik (1962) Vetes̆nik, M. 1962, Bull. Astron. Inst. Czech., 13, 182
* Wang et al. (2010) Wang, S., Fan, Z., Ma, J., de Grijs, R., & Zhou, X. 2010, AJ, 139, 1438
* Williams & Hodge (2001a) Williams, B. F., & Hodge, P. W. 2001a, ApJ, 548, 190
* Williams & Hodge (2001b) Williams, B. F., & Hodge, P. W. 2001b, ApJ, 559, 851
* Yan et al. (2000) Yan, H. J., et al. 2000, PASP, 112, 691
* Zheng et al. (1999) Zheng, Z. Y., et al. 1999, AJ, 117, 2757
* Zhou et al. (2003) Zhou, X., et al. 2003, A&A, 379, 361
Table 1BATC photometry of the M31 YMC VDB0-B195D. Filter | Central wavelength | Bandwidth | Number of images | Exposure time | Magnitude
---|---|---|---|---|---
| (Å) | (Å) | | (hours) |
$a$ | 3360 | 222 | 6 | 2.0 | $15.51\pm 0.14$
$b$ | 3890 | 187 | 6 | 2.0 | $14.73\pm 0.11$
$c$ | 4210 | 185 | 3 | 0.8 | $14.70\pm 0.07$
$d$ | 4550 | 222 | 3 | 1.0 | $14.49\pm 0.10$
$e$ | 4920 | 225 | 3 | 1.0 | $14.67\pm 0.05$
$f$ | 5270 | 211 | 3 | 1.0 | $14.65\pm 0.05$
$g$ | 5795 | 176 | 3 | 1.0 | $14.59\pm 0.04$
$h$ | 6075 | 190 | 3 | 1.0 | $14.56\pm 0.02$
$i$ | 6660 | 312 | 3 | 1.0 | $14.50\pm 0.02$
$j$ | 7050 | 121 | 5 | 1.7 | $14.51\pm 0.06$
$k$ | 7490 | 125 | 3 | 1.0 | $14.47\pm 0.05$
$m$ | 8020 | 179 | 3 | 1.0 | $14.43\pm 0.07$
$n$ | 8480 | 152 | 6 | 2.0 | $14.49\pm 0.05$
$o$ | 9190 | 194 | 6 | 2.0 | $14.49\pm 0.05$
$p$ | 9745 | 188 | 6 | 2.0 | $14.50\pm 0.05$
Table 2Comparison of broad-band photometry of VDB0-B195D.
Filter | ${\rm Mag}^{a}$ | ${\rm Mag}^{b}$ | ${\rm Mag}^{c}$ | ${\rm Mag}^{d}$ | ${\rm Mag}^{e}$ | ${\rm Mag}^{f}$
---|---|---|---|---|---|---
$U$ | 14.66 | 15.11 | $15.12\pm 0.012$ | $14.97\pm 0.01$ | 15.110 | 15.140
$B$ | 15.14 | 15.39 | $15.49\pm 0.010$ | $15.31\pm 0.01$ | 15.410 | 15.510
$V$ | 14.94 | 15.19 | $15.32\pm 0.013$ | $15.06\pm 0.01$ | 15.190 | 15.280
$R$ | | 15.27 | | | 14.920 |
avan den Bergh (1969); bBattistini et al. (1987); cKing & Lupton (1991),
uncertainties are the median uncertainties in the mean for all sample cluster
measurements; dSharov et al. (1995); ePhotometry from Galleti et al. (2004),
based on Battistini et al. (1987); fPhotometry from Galleti et al. (2004),
based on Sharov et al. (1995).
Table 3Recently determined photometry for VDB0-B195D.
Filter | ${\rm Mag}^{a}$ | ${\rm Mag}^{b}$ | ${\rm Mag}^{c}$
---|---|---|---
$U$ | | | $14.37\pm 0.22$
$B$ | $14.94\pm 0.09$ | | $14.62\pm 0.13$
$V$ | $14.67\pm 0.05$ | | $14.67\pm 0.05$
$R$ | $14.45\pm 0.11$ | | $14.60\pm 0.06$
$I$ | $14.01\pm 0.11$ | | $14.19\pm 0.10$
$J$ | $13.26\pm 0.07$ | $13.78\pm 0.03$ |
$H$ | $12.76\pm 0.12$ | $13.15\pm 0.04$ |
$K_{s}$ | $12.77\pm 0.15$ | $12.96\pm 0.03$ |
aPerina et al. (2009); bGalleti et al. (2004); cThis paper.
Table 4Age estimate of VDB0-B195D based on the the galev models. Age | log (Age) | $\chi_{\nu}^{2}({\rm min})$
---|---|---
(Myr) | [yr] | (per degree of freedom)
$60.0\pm 8.0$ | $7.78\pm 0.05$ | 2.2
Table 5Mass estimates (and uncertainties) of VDB0-B195D based on the galev models. $B$ | $V$ | $R$ | $I$ | $J$ | $H$ | $K_{\rm s}$
---|---|---|---|---|---|---
| | | Mass $(10^{5}~{}M_{\odot})$ | | |
$1.6\pm 0.18$ | $1.6\pm 0.13$ | $1.4\pm 0.17$ | $1.4\pm 0.16$ | $1.4\pm 0.13$ | $1.3\pm 0.16$ | $1.1\pm 0.17$
Table 6Comparison of age and mass estimates of VDB0-B195D.
Age1 | Age2 | log (Age)3 | log (Age)2 | Mass1 | Mass2 | log (Mass)3 | log (Mass)2
---|---|---|---|---|---|---|---
(Myr) | (Myr) | [yr] | [yr] | ($10^{4}~{}M_{\odot}$) | ($10^{5}~{}M_{\odot}$) | [$M_{\odot}$] | [$M_{\odot}$]
25 | $60.0\pm 8.0$ | 7.6 | $7.78\pm 0.05$ | $4-9$ | $1.1-1.6$ | $5.1$ | $5.0-5.2$
1Perina et al. (2009); 2This paper; 3Caldwell et al. (2009).
Table 7Mass estimates (and uncertainties) of VDB0-B195D based on the galev models with $E(B-V)=0.1$. $B$ | $V$ | $R$ | $I$ | $J$ | $H$ | $K_{\rm s}$
---|---|---|---|---|---|---
| | | Mass $(10^{5}~{}M_{\odot})$ | | |
$1.1\pm 0.12$ | $1.2\pm 0.10$ | $1.1\pm 0.14$ | $1.2\pm 0.14$ | $1.3\pm 0.12$ | $1.2\pm 0.16$ | $1.1\pm 0.17$
|
arxiv-papers
| 2011-01-08T03:10:28 |
2024-09-04T02:49:16.226184
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jun Ma (1,2), Song Wang (1,3), Zhenyu Wu (1), Zhou Fan (1), Yanbin\n Yang (1), Tianmeng Zhang (1) and Jianghua Wu (1) et al. ((1) National\n Astronomical Observatories, Chinese Academy of Sciences, (2) Key Laboratory\n of Optical Astronomy, National Astronomical Observatories, Chinese Academy of\n Sciences)",
"submitter": "Jun Ma",
"url": "https://arxiv.org/abs/1101.1567"
}
|
1101.1569
|
Vol.0 (200x) No.0, 000–000
11institutetext: 1National Astronomical Observatories, Chinese Academy of
Sciences, Beijing, 100012, P. R. China
2Key Laboratory of Optical Astronomy, National Astronomical Observatories,
Chinese Academy of Sciences, Beijing, 100012, China
11email: majun@nao.cas.cn
# Detailed study of B037 based on HST images
Jun Ma 1122
(Received 2001 month day; accepted 2001 month day)
###### Abstract
B037 is of interest because it is both the most luminous and the most highly
reddened cluster known in M31. Images of deep observations and of highly
spatial resolutions with the Advanced Camera for Surveys on the Hubble Space
Telescope (HST) firstly show that this cluster is crossed by a dust lane.
Photometric data in the F606W and F814W filters obtained in this paper provide
that, colors of ($\rm{F606W-F814W}$) in the dust lane are redder $\sim 0.4$
mags than ones in the other regions of B037. The HST images show that, this
dust lane seems to be contained in B037, not from the M31 disk or the Milky
Way. As we know, the formation of dust requires gas with a rather high
metallicity. However, B037 has a low metallicity to be $\rm[Fe/H]=-1.07\pm
0.20$. So, it seems improbable that the observed dust lane is physically
associated with B037. It is clear that the origin of this dust lane is worthy
of future study. In addition, based on these images, we present the precise
variation of ellipticity and position angle, and of surface brightness
profile, and determine the structural parameters of B037 by fitting a single-
mass isotropic King model. In the F606W filter, we derive the best-fitting
scale radius, $r_{0}=0.56\pm 0.02\arcsec~{}(=2.16\pm 0.08~{}\rm{pc})$, a tidal
radius, $r_{t}=8.6\pm 0.4\arcsec~{}(=33.1\pm 1.5~{}\rm{pc})$, and a
concentration index $c=\log(r_{t}/r_{0})=1.19\pm 0.02$. In the F814W filter,
we derive $r_{0}=0.56\pm 0.01\arcsec~{}(=2.16\pm 0.04~{}\rm{pc})$,
$r_{t}=8.9\pm 0.3\arcsec~{}(=34.3\pm 1.2~{}\rm{pc})$, and
$c=\log(r_{t}/r_{0})=1.20\pm 0.01$. The extinction-corrected central surface
brightness is $\mu_{0}=13.53\pm 0.03~{}{\rm mag~{}arcsec^{-2}}$ in the F606W
filter, and $12.85\pm 0.03~{}{\rm mag~{}arcsec^{-2}}$ in the F814W filter,
respectively. We also calculate the half-light radius, at $r_{h}=1.05\pm
0.03\arcsec(=4.04\pm 0.12~{}\rm{pc})$ in the F606W filter and $r_{h}=1.07\pm
0.01\arcsec(=4.12\pm 0.04~{}\rm{pc})$ in the F814W filter, respectively. In
addition, we derived the whole magnitudes of B037 in $V$ and $I$ bands by
transforming the magnitudes from the ACS system to the standard system, which
are in very agreement with the previous ground-based broad-band photometry.
###### keywords:
galaxies: evolution – galaxies: individual (M31) – globular cluster:
individual (B037)
## 1 Introduction
Globular clusters (GCs) are effective laboratories for studying stellar
evolution and stellar dynamics, and they are ancient building blocks of
galaxies which can help us to understand the formation and evolution of their
parent galaxies. In addition, GCs exhibit surprisingly uniform properties,
suggesting a common formation mechanism.
The closest other populous GC system beyond the halo of our Galaxy is that of
M31. The study of M31 has been and continues to be a keystone of extragalactic
astronomy (Barmby et al., 2000), and the study of GCs in M31 can be traced
back to Hubble (1932). M31 GC B327 (B for ‘Baade’) or Bo37 (Bo for ‘Bologna’,
see Battistini 1987), which, in the nomenclature introduced by Huchra et al.
(1991) is referred to as B037, a designation from the Revised Bologna
Catalogue (RBC) of M31 GCs and candidates (Galleti et al. 2004, 2006, 2007),
which is the main catalog used in studies of M31 GCs. The extremely red color
of B037 was firstly noted by Kron & Mayall (1960), who suggested that this
cluster must be highly reddened. Two years later, Vetes̆nik (1962a) determined
magnitudes of 257 M31 GC candidates including B037 in the $UBV$ photometric
system, and then Vetes̆nik (1962b) studied the intrinsic colors of M31 GCs,
and found that B037 was the most highly reddened with $E(B-V)=1.28$ in his
sample of M31 GC candidates based on the photometric catalog of Vetes̆nik
(1962a). With low-resolution spectroscopy, Crampton et al. (1985) also found
that B037 is the most highly reddened GC candidate in M31 to have
$E(B-V)=1.48$. Based on a large database of multicolor photometry, Barmby et
al. (2000) determined the reddening value for each individual M31 GC including
B037 using the correlations between optical and infrared colors and
metallicity by defining various “reddening-free” parameters, and the reddening
value of B037 is $E(B-V)=1.38\pm 0.02$ (which is kindly given us by P.
Barmby). Again, Barmby et al. (2002b) derived the reddening value for this
cluster to be $E(B-V)=1.30\pm 0.04$, using the spectroscopic metallicity to
predict the intrinsic colors. Ma et al. (2006a) also determine the reddening
of B037 by comparing independently obtained multicolor photometry with
theoretical stellar population synthesis models to be $E(B-V)=1.360\pm 0.013$,
which is in good agreement with the other results. Following the methods of
Barmby et al. (2000), Fan et al. (2008) (re-)determined reddening values for
443 clusters and cluster candidates including B037, and the redding value of
B037 obtained by Fan et al. (2008) is $E(B-V)=1.21\pm 0.03$, which is a little
smaller than the previous determinations.
The brightest GCs in M31 are more luminous than the most brightest Galactic
cluster, $\omega$ Centauri. Among these are B037 (van den Bergh, 1968) and G1
(see details from Barmby et al., 2002b). These two clusters are both
considered as the possible remnant core of a former dwarf galaxy which lost
most of its envelope through tidal interactions with M31 (Meylan & Heggie,
1997; Meylan et al., 2001; Mackey & van den Bergh, 2005; Ma et al., 2006b,
2007).
In this paper, we will present the photometric data of B037 using its deep
images obtained with the Advanced Camera for Survey (ACS) on the HST. The deep
images of highly spatial resolutions showed that this cluster is crossed by a
dust lane. Our results provide that colors of F814W$-$F606W in the dust lane
are redder $\sim 0.4$ mags than ones of the other regions. In addition, we
studied structures of B037 in detail based on these images.
## 2 Observations and Data Reduction
### 2.1 HST images of B037
We searched the HST archive and found B037 to have been observed with the ACS-
Wide Field Channel (WFC) in the F606W and the F814W filters, which were
observed on 2004 August 2 and on 2004 July 4, respectively. The exposure time
is 2370.0 seconds for both bands. The HST ACS-WFC resolution is $0.05\arcsec$
per pixel. The images in F606W and F814W both show that B037 is crossed by a
dust lane. Fig. 1 clearly shows the dust lane, which crosses B037. If the dust
lane is true, its color should be different from ones of the other regions.
If not otherwise stated, the magnitudes are always on the VEGAMAG scale as
defined by Sirianni et al. (2005). The relevant zero-point for this system is
26.398 and 25.501 for WFC F606W and WFC F814W, respectively. A distance to M31
of 780 kpc ($1{\arcsec}$ subtends 3.85 pc) is adopted in this paper.
Figure 1: The images of GC B037 observed in the F606W and F814W filters of
ACS/HST. The images clearly show that the cluster is crossed by a dust lane.
The image size is $17.5\arcsec\times 17.5\arcsec$ for each panel.
### 2.2 Color difference between the dust lane and the other regions
In order to study whether the color difference between the dust lane and the
other regions in B037 exists, we select nine points, three of which (No. 7, 8
and 9) are located in the dust lane, the other six are randomly located in the
other regions (see Figure 2). For each sample point, the PHOT routine in
DAOPHOT (Stetson, 1987) is used to obtain magnitude. We adopt an aperture of a
diameter of 4 pixels. The photometric data for these nine sample points are
given in Table 1, in conjunction with the $1\sigma$ magnitude uncertainties
from daophot. Column 4 gives the color of ($\rm{F606W-F814W}$). From Table 1,
we can see that colors of ($\rm{F606W-F814W}$) in the dust lane are redder
$\sim 0.4$ mags than ones of the other regions.
Figure 2: The sample positions of photometry (black circles) are showed in the image of GC B037 observed in the F606W filter of ACS/HST. An aperture of radii of 2 pixels is adopted for photometry. The image size is $17.5\arcsec\times 17.5\arcsec$. Table 1: Photometric data for B037 Source | F606W | F814W | F606W$-$F814W
---|---|---|---
No. | (mag) | (mag) | (mag)
1 | $23.29\pm 0.26$ | $21.35\pm 0.16$ | $1.94$
2 | $23.03\pm 0.23$ | $21.14\pm 0.15$ | $1.89$
3 | $22.95\pm 0.22$ | $21.00\pm 0.14$ | $1.95$
4 | $23.22\pm 0.25$ | $21.25\pm 0.15$ | $1.97$
5 | $23.17\pm 0.25$ | $21.36\pm 0.16$ | $1.81$
6 | $21.15\pm 0.10$ | $19.10\pm 0.06$ | $2.05$
7 | $22.93\pm 0.22$ | $20.53\pm 0.11$ | $2.40$
8 | $23.41\pm 0.28$ | $20.94\pm 0.13$ | $2.47$
9 | $24.66\pm 0.49$ | $22.31\pm 0.25$ | $2.35$
### 2.3 Surface brightness profiles
We used the iraf task ellipse to obtain F606W and F814W surface brightness
profiles for B037. B037 center position was fixed at a value derived by object
locator of ellipse task, however an initial center position was determined by
centroiding. Elliptical isophotes were fitted to the data, with no sigma
clipping. We ran two passes of ellipse task, the first pass was run in the
usual way, with ellipticity and position angle allowed to vary with the
isophote semimajor axis. In the second pass, surface brightness profiles on
fixed, zero-ellipticity isophotes were measured, since we choose to fit
circular models for the intrinsic cluster structure and the point spread
function (PSF) as Barmby et al. (2007) did (see §2.4 for details). The
background value was derived as the mean of a region of $100\times 100$ pixels
in “empty” areas far away from the cluster.
#### 2.3.1 Ellipticity and position angle
Tables 2 and 3 give the ellipticity, $\epsilon=1-b/a$, and the position angle
(P.A.) as a function of the semi-major axis length, $a$, from the center of
annulus in the F606W and F814W filter bands, respectively. These observables
have also been plotted in Figures 3 and 4, respectively; the errors were
generated by the iraf task ellipse, in which the ellipticity errors are
obtained from the internal errors in the harmonic fit, after removal of the
first and second fitted harmonics. From Table 3, and Figs. 3 and 4, we can see
that, the values of ellipticity and position angle cannot be obtained within
$0.1448\arcsec$ in the F814W filter because of very high ellipticity ($>1.0$).
Ma et al. (2006b) analyzed the same F606W image of B037 used here, fitting a
King (1962) model to a surface brightness profile made from a PSF-deconvolved
image. They also plotted the distributions of ellipticity and the position
angle as a function of the semi-major axis length. Comparison of Fig. 2 of Ma
et al. (2006b) and Figs. 3 and 4 shows that, the general trend of the
cluster’s ellipticity as a function of semimajor axis radius is similar
between Ma et al. (2006b) and the present paper. The comparison also shows
that uncertainties in the exact value of the PA are only of secondary
importance for the general trend in ellipticity observed, given that the PA
determination between Ma et al. (2006b) and the present paper differs somewhat
greatly. There are a number of possible reasons for the offsets in PA observed
between these two studies. The main reason is that, Ma et al. (2006b) used the
PSF-deconvolved image. Other reasons include those related to the positions of
the centering of isophotes and the different geometrical parameters set when
fitting. In addition, Fig. 3 shows that the ellipticity varies significantly
with position along the semimajor axis radius, especially smaller than
$0.5\arcsec$. In the F814W filter band, the ellipticity is larger than 1.0
along the semimajor axis radius smaller than $0.1448\arcsec$.
Table 2: B037: Ellipticity, $\epsilon$, and position angle (P.A.) as a function of the semimajor axis, $a$, in the F606W filter of HST ACS-WFC $a$ | $\epsilon$ | P.A. | $a$ | $\epsilon$ | P.A.
---|---|---|---|---|---
(arcsec) | | (deg) | (arcsec) | | (deg)
0.0260 | $0.638\pm 0.228$ | $92.9\pm 15.5$ | 0.3757 | $0.177\pm 0.031$ | $69.8\pm 5.6$
0.0287 | $0.638\pm 0.229$ | $93.2\pm 15.6$ | 0.4132 | $0.151\pm 0.029$ | $64.7\pm 5.9$
0.0315 | $0.639\pm 0.230$ | $93.4\pm 15.7$ | 0.4545 | $0.090\pm 0.027$ | $60.3\pm 9.1$
0.0347 | $0.640\pm 0.232$ | $93.7\pm 15.8$ | 0.5000 | $0.005\pm 0.025$ | $172.6\pm 30.0$
0.0381 | $0.642\pm 0.233$ | $94.0\pm 15.9$ | 0.5500 | $0.060\pm 0.020$ | $155.5\pm 9.8$
0.0420 | $0.643\pm 0.235$ | $94.4\pm 16.0$ | 0.6050 | $0.117\pm 0.015$ | $156.3\pm 3.9$
0.0461 | $0.645\pm 0.236$ | $94.7\pm 16.0$ | 0.6655 | $0.174\pm 0.012$ | $157.2\pm 2.2$
0.0508 | $0.647\pm 0.182$ | $95.2\pm 12.3$ | 0.7321 | $0.233\pm 0.011$ | $157.2\pm 1.5$
0.0558 | $0.599\pm 0.159$ | $96.8\pm 11.2$ | 0.8053 | $0.278\pm 0.011$ | $159.1\pm 1.3$
0.0614 | $0.546\pm 0.142$ | $98.5\pm 10.6$ | 0.8858 | $0.322\pm 0.011$ | $160.8\pm 1.1$
0.0676 | $0.503\pm 0.127$ | $100.2\pm 10.0$ | 0.9744 | $0.358\pm 0.012$ | $162.1\pm 1.2$
0.0743 | $0.458\pm 0.099$ | $102.2\pm 8.3$ | 1.0718 | $0.380\pm 0.021$ | $164.5\pm 2.1$
0.0818 | $0.400\pm 0.059$ | $104.3\pm 5.5$ | 1.1790 | $0.367\pm 0.022$ | $168.0\pm 2.2$
0.0899 | $0.410\pm 0.050$ | $101.0\pm 4.6$ | 1.2969 | $0.343\pm 0.025$ | $169.2\pm 2.7$
0.0989 | $0.428\pm 0.044$ | $98.7\pm 3.9$ | 1.4266 | $0.319\pm 0.025$ | $166.8\pm 2.8$
0.1088 | $0.437\pm 0.046$ | $97.6\pm 3.9$ | 1.5692 | $0.252\pm 0.022$ | $165.1\pm 3.0$
0.1197 | $0.428\pm 0.028$ | $96.6\pm 2.5$ | 1.7261 | $0.239\pm 0.021$ | $165.7\pm 2.9$
0.1317 | $0.410\pm 0.027$ | $96.5\pm 2.5$ | 1.8987 | $0.211\pm 0.026$ | $163.6\pm 4.0$
0.1448 | $0.400\pm 0.031$ | $96.3\pm 3.0$ | 2.0886 | $0.201\pm 0.029$ | $152.8\pm 4.6$
0.1593 | $0.364\pm 0.023$ | $95.1\pm 2.3$ | 2.2975 | $0.188\pm 0.037$ | $150.1\pm 6.3$
0.1752 | $0.352\pm 0.027$ | $94.8\pm 2.8$ | 2.5272 | $0.182\pm 0.033$ | $149.9\pm 5.8$
0.1928 | $0.337\pm 0.027$ | $93.2\pm 2.9$ | 2.7800 | $0.180\pm 0.034$ | $145.6\pm 6.0$
0.2120 | $0.311\pm 0.027$ | $92.5\pm 3.0$ | 3.0580 | $0.191\pm 0.031$ | $137.5\pm 5.2$
0.2333 | $0.287\pm 0.026$ | $90.6\pm 3.1$ | 3.3638 | $0.143\pm 0.034$ | $125.4\pm 7.2$
0.2566 | $0.258\pm 0.026$ | $88.6\pm 3.3$ | 3.7001 | $0.180\pm 0.041$ | $121.1\pm 7.1$
0.2822 | $0.233\pm 0.027$ | $85.4\pm 3.8$ | 4.0701 | $0.257\pm 0.033$ | $121.6\pm 4.1$
0.3105 | $0.207\pm 0.029$ | $81.2\pm 4.5$ | 4.4772 | $0.233\pm 0.048$ | $121.6\pm 6.5$
0.3415 | $0.189\pm 0.030$ | $75.3\pm 5.0$ | 4.9249 | $0.237\pm 0.063$ | $116.1\pm 8.5$
Table 3: B037: Ellipticity, $\epsilon$, and position angle (P.A.) as a function of the semimajor axis, $a$, in the F814W filter of HST ACS-WFC $a$ | $\epsilon$ | P.A. | $a$ | $\epsilon$ | P.A.
---|---|---|---|---|---
(arcsec) | | (deg) | (arcsec) | | (deg)
0.0260 | | | 0.4132 | $0.031\pm 0.030$ | $77.8\pm 28.3$
0.0287 | | | 0.4545 | $0.031\pm 0.029$ | $20.2\pm 27.5$
0.0315 | | | 0.5000 | $0.044\pm 0.027$ | $161.2\pm 17.6$
0.0347 | | | 0.5500 | $0.084\pm 0.023$ | $150.3\pm 8.3$
0.0381 | | | 0.6050 | $0.127\pm 0.021$ | $150.3\pm 5.0$
0.0420 | | | 0.6655 | $0.175\pm 0.019$ | $153.6\pm 3.4$
0.0461 | | | 0.7321 | $0.220\pm 0.016$ | $157.4\pm 2.3$
0.0508 | | | 0.8053 | $0.247\pm 0.013$ | $161.8\pm 1.7$
0.0558 | | | 0.8858 | $0.251\pm 0.014$ | $167.8\pm 1.8$
0.0614 | | | 0.9744 | $0.263\pm 0.017$ | $170.0\pm 2.1$
0.0676 | | | 1.0718 | $0.293\pm 0.034$ | $170.8\pm 4.0$
0.0743 | | | 1.1790 | $0.297\pm 0.035$ | $172.5\pm 4.1$
0.0818 | | | 1.2969 | $0.230\pm 0.028$ | $171.5\pm 4.0$
0.0899 | | | 1.4266 | $0.216\pm 0.025$ | $166.7\pm 3.7$
0.0989 | | | 1.5692 | $0.198\pm 0.031$ | $165.5\pm 5.1$
0.1088 | | | 1.7261 | $0.198\pm 0.025$ | $169.5\pm 4.0$
0.1197 | | | 1.8987 | $0.188\pm 0.029$ | $167.5\pm 5.0$
0.1317 | | | 2.0886 | $0.139\pm 0.031$ | $166.9\pm 6.9$
0.1448 | | | 2.2975 | $0.117\pm 0.031$ | $117.8\pm 8.0$
0.1593 | $0.908\pm 0.117$ | $89.9\pm 6.9$ | 2.5272 | $0.100\pm 0.034$ | $148.1\pm 10.2$
0.1752 | $0.878\pm 0.026$ | $90.9\pm 1.5$ | 2.7800 | $0.118\pm 0.051$ | $141.0\pm 13.3$
0.1928 | $0.827\pm 0.151$ | $90.8\pm 9.6$ | 3.0580 | $0.094\pm 0.043$ | $115.8\pm 13.7$
0.2120 | $0.749\pm 0.034$ | $90.3\pm 2.3$ | 3.3638 | $0.094\pm 0.035$ | $132.7\pm 11.1$
0.2333 | $0.731\pm 0.037$ | $89.4\pm 2.6$ | 3.7001 | $0.103\pm 0.056$ | $121.3\pm 16.4$
0.2566 | $0.695\pm 0.041$ | $87.1\pm 2.9$ | 4.0701 | $0.127\pm 0.061$ | $120.1\pm 14.6$
0.2822 | $0.624\pm 0.031$ | $84.4\pm 2.3$ | 4.4772 | $0.162\pm 0.031$ | $115.7\pm 5.9$
0.3105 | $0.546\pm 0.035$ | $80.4\pm 2.6$ | 4.9249 | $0.091\pm 0.054$ | $131.4\pm 17.7$
0.3415 | $0.401\pm 0.035$ | $72.4\pm 3.2$ | 5.4174 | $0.150\pm 0.065$ | $135.5\pm 13.3$
0.3757 | $0.258\pm 0.029$ | $63.4\pm 3.8$ | 5.9591 | $0.188\pm 0.044$ | $161.8\pm 7.4$
Figure 3: Ellipticity as a function of the semimajor axis in the F606W and
F814W filters of ACS/HST. Figure 4: P.A. as a function of the semimajor axis
in the F606W and F814W filters of ACS/HST.
### 2.4 Point spread function
At a distance of 780 kpc, the ACS/WFC has a scale of
$\rm{0.05~{}arcsec=0.19~{}pc~{}pixel^{-1}}$, and thus M31 clusters are clearly
resolved with it. Their observed core structures, however, are still affected
by the PSF. We chose not to deconvolve the data, instead fitting structural
models after convolving them with a simple analytic description of the PSF as
Barmby et al. (2007) did. To estimate the PSF for the WFC, Barmby et al.
(2007) used the iraf task ellipse with circular symmetry enforced to produce
intensity profiles out to radii of about $2^{\prime\prime}$ (40 pixels) for a
number of isolated stars on a number of images, and combined them to produce a
single, average PSF. This was done separately for the F606W and F814W filters.
They originally tried to fit these with simple Moffat profiles (with
backgrounds added), but found that a better description was given by a
function of the form below. For the combination of the WFC and F606W filter,
$I_{\rm PSF}=I_{0}\left[1+\left(R/0\farcs 0686\right)^{3}\right]^{-1.23}\ ,$
(1)
which has a full width at half-maximum of ${\rm FWHM}=0\farcs 125$, or about
2.5 px; for the combination of the WFC and F814W filter,
$I_{\rm PSF}=I_{0}\left[1+\left(R/0\farcs 0783\right)^{3}\right]^{-1.19}\ ,$
(2)
which has a full width at half-maximum of ${\rm FWHM}=0\farcs 145$, or about
2.9 px. In addition, since this PSF formula is radially symmetric and the
models of King (1966) we fit are intrinsically spherical, the convolved models
to be fitted to the data are also circularly symmetric.
### 2.5 Extinction
When we fit models to the brightness profiles of B037, we will correct the
inferred magnitude parameters for extinction. The reddening law from Cardelli
et al. (1989) is employed in this paper. The effective wavelengths of the ACS
F606W and F814W filters are $\lambda_{\rm eff}=5918$ and 8060 Å (Sirianni et
al., 2005), so that from Cardelli et al. (1989), $A_{\rm{F606W}}\simeq
2.8\times E(B-V)$ and $A_{\rm{F814W}}\simeq 1.8\times E(B-V)$ (see Barmby et
al., 2007; McLaughlin et al., 2008, for details). The reddening value of
$E(B-V)=1.360\pm 0.013$ from Ma et al. (2006a) is adopted in this paper.
### 2.6 Magnitudes of B037 in F606W and F814W filters
We derived the total flux of B037 in F606W and F814W filter bands using the
iraf task phot in dapphot as below: measuring aperture magnitudes in
concentric apertures with an interval of $0.1\arcsec$, drawing magnitude
growth curves, and paying attention to where the flux does not increase. At
last, we obtained the magnitudes of B037 in F606W and F814W to be $16.21\pm
0.010$ and $14.16\pm 0.006$, respectively. In the photometry, we derived the
background value as the mean of a region far away from the cluster (see §2.3
for details). We use VEGAMAG photometric system. In order to allow a
meaningful comparison with the previous ground-based broad-band photometry of
Barmby et al. (2000), we transformed the magnitudes from the ACS system to the
standard broad-band photometric system by following the transformation
equations and coefficients of Table 22 of Sirianni et al. (2005). The results
are $m_{V}{(\rm ACS)}=16.83$ (this paper) versus $m_{V}=16.82$ (Barmby et al.,
2000), and $m_{I}{(\rm ACS)}=14.15$ (this paper) versus $m_{I}=14.16$ (Barmby
et al., 2000). Our results are in good agreement with Barmby et al. (2000).
## 3 Models and Fits
### 3.1 Structural models
After elliptical galaxies, GCs are the best understood and most thoroughly
modelled class of stellar systems. For example, a large majority of the $\sim
150$ Galactic GCs have been fitted by the simple models of single-mass,
isotropic, lowered isothermal spheres developed by Michie (1963) and King
(1966) (hereafter “King models”), yielding comprehensive catalogs of cluster
structural parameters and physical properties (see McLaughlin & van der Marel,
2005, and references therein). For extragalactic GCs, HST imaging data have
been used to fit King models to a large number of GCs in M31 (e.g., Barmby et
al., 2002a, 2007, and references therein), in M33 (Larsen et al., 2002), and
in NGC 5128 (e.g., Harris et al., 2002; McLaughlin et al., 2008, and
references therein). In this paper, we fit the usual King models to the
density profile of B037 observed with ACS/WFC.
### 3.2 Observed data
Tables 4 and 5 list the surface brightness, $\mu$, of B037, and its integrated
magnitude, $m$, as a function of radius in the F606W and F814W filters,
respectively. The errors in the surface brightness were also generated by the
iraf task ellipse, in which they are obtained directly from the root mean
square scatter of the intensity data along the zero-ellipticity isophotes. In
addition, the surface photometries at radii where the ellipticity and position
angle cannot be measured, are obtained based on the last ellipticity and
position angle as the iraf task ellipse is designed.
Table 4: B037: Surface brightness, $\mu$, and integrated magnitude, $m$, as a function of the radius in the F606W filter of HST ACS-WFC $R$ | $\mu$ | $m$ | $R$ | $\mu$ | $m$
---|---|---|---|---|---
(arcsec) | (mag) | (mag) | (arcsec) | (mag) | (mag)
0.0260 | $17.327\pm 0.007$ | 23.827 | 0.3757 | $17.792\pm 0.040$ | 18.456
0.0287 | $17.328\pm 0.008$ | 23.827 | 0.4132 | $17.863\pm 0.046$ | 18.264
0.0315 | $17.328\pm 0.008$ | 23.827 | 0.4545 | $17.944\pm 0.049$ | 18.123
0.0347 | $17.329\pm 0.009$ | 23.827 | 0.5000 | $18.040\pm 0.049$ | 17.962
0.0381 | $17.330\pm 0.010$ | 23.827 | 0.5500 | $18.148\pm 0.048$ | 17.827
0.0420 | $17.331\pm 0.011$ | 23.827 | 0.6050 | $18.267\pm 0.048$ | 17.672
0.0461 | $17.331\pm 0.012$ | 23.827 | 0.6655 | $18.389\pm 0.053$ | 17.549
0.0508 | $17.332\pm 0.014$ | 22.086 | 0.7321 | $18.495\pm 0.061$ | 17.414
0.0558 | $17.334\pm 0.015$ | 22.086 | 0.8053 | $18.598\pm 0.070$ | 17.295
0.0614 | $17.338\pm 0.016$ | 22.086 | 0.8858 | $18.716\pm 0.077$ | 17.165
0.0676 | $17.341\pm 0.018$ | 22.086 | 0.9744 | $18.854\pm 0.079$ | 17.039
0.0743 | $17.346\pm 0.020$ | 21.452 | 1.0718 | $19.006\pm 0.077$ | 16.927
0.0818 | $17.351\pm 0.022$ | 21.452 | 1.1790 | $19.193\pm 0.107$ | 16.822
0.0899 | $17.356\pm 0.024$ | 21.452 | 1.2969 | $19.440\pm 0.105$ | 16.731
0.0989 | $17.363\pm 0.027$ | 21.452 | 1.4266 | $19.721\pm 0.116$ | 16.642
0.1088 | $17.372\pm 0.027$ | 21.062 | 1.5692 | $20.001\pm 0.113$ | 16.570
0.1197 | $17.382\pm 0.029$ | 20.552 | 1.7261 | $20.293\pm 0.116$ | 16.505
0.1317 | $17.395\pm 0.028$ | 20.552 | 1.8987 | $20.597\pm 0.122$ | 16.451
0.1448 | $17.411\pm 0.028$ | 20.370 | 2.0886 | $20.872\pm 0.128$ | 16.401
0.1593 | $17.429\pm 0.029$ | 19.964 | 2.2975 | $21.224\pm 0.105$ | 16.358
0.1752 | $17.450\pm 0.028$ | 19.964 | 2.5272 | $21.457\pm 0.112$ | 16.320
0.1928 | $17.475\pm 0.026$ | 19.764 | 2.7800 | $21.719\pm 0.138$ | 16.283
0.2120 | $17.504\pm 0.026$ | 19.528 | 3.0580 | $22.082\pm 0.154$ | 16.251
0.2333 | $17.538\pm 0.025$ | 19.337 | 3.3638 | $22.603\pm 0.164$ | 16.225
0.2566 | $17.578\pm 0.026$ | 19.091 | 3.7001 | $23.042\pm 0.225$ | 16.206
0.2822 | $17.624\pm 0.026$ | 19.009 | 4.0701 | $23.694\pm 0.467$ | 16.191
0.3105 | $17.675\pm 0.030$ | 18.802 | 4.4772 | $24.509\pm 0.571$ | 16.182
0.3415 | $17.732\pm 0.034$ | 18.634 | 4.9249 | $25.173\pm 1.342$ | 16.172
Table 5: B037: Surface brightness, $\mu$, and integrated magnitude, $m$, as a function of the radius in the F814W filter of HST ACS-WFC $R$ | $\mu$ | $m$ | $R$ | $\mu$ | $m$
---|---|---|---|---|---
(arcsec) | (mag) | (mag) | (arcsec) | (mag) | (mag)
0.0260 | $15.301\pm 0.010$ | 21.800 | 0.4132 | $15.772\pm 0.032$ | 16.190
0.0287 | $15.302\pm 0.011$ | 21.800 | 0.4545 | $15.863\pm 0.036$ | 16.048
0.0315 | $15.303\pm 0.012$ | 21.800 | 0.5000 | $15.967\pm 0.038$ | 15.889
0.0347 | $15.303\pm 0.013$ | 21.800 | 0.5500 | $16.078\pm 0.037$ | 15.754
0.0381 | $15.304\pm 0.015$ | 21.800 | 0.6050 | $16.190\pm 0.033$ | 15.598
0.0420 | $15.305\pm 0.016$ | 21.800 | 0.6655 | $16.300\pm 0.036$ | 15.474
0.0461 | $15.306\pm 0.018$ | 21.800 | 0.7321 | $16.407\pm 0.046$ | 15.338
0.0508 | $15.308\pm 0.019$ | 20.061 | 0.8053 | $16.543\pm 0.053$ | 15.218
0.0558 | $15.310\pm 0.021$ | 20.061 | 0.8858 | $16.706\pm 0.054$ | 15.094
0.0614 | $15.313\pm 0.024$ | 20.061 | 0.9744 | $16.847\pm 0.054$ | 14.974
0.0676 | $15.317\pm 0.026$ | 20.061 | 1.0718 | $16.995\pm 0.061$ | 14.868
0.0743 | $15.321\pm 0.030$ | 19.430 | 1.1790 | $17.185\pm 0.105$ | 14.767
0.0818 | $15.326\pm 0.033$ | 19.430 | 1.2969 | $17.459\pm 0.086$ | 14.681
0.0899 | $15.331\pm 0.037$ | 19.430 | 1.4266 | $17.717\pm 0.107$ | 14.596
0.0989 | $15.337\pm 0.041$ | 19.430 | 1.5692 | $18.006\pm 0.101$ | 14.527
0.1088 | $15.347\pm 0.043$ | 19.039 | 1.7261 | $18.279\pm 0.102$ | 14.464
0.1197 | $15.360\pm 0.043$ | 18.531 | 1.8987 | $18.567\pm 0.086$ | 14.411
0.1317 | $15.367\pm 0.045$ | 18.531 | 2.0886 | $18.833\pm 0.092$ | 14.359
0.1448 | $15.375\pm 0.048$ | 18.350 | 2.2975 | $19.167\pm 0.095$ | 14.316
0.1593 | $15.396\pm 0.046$ | 17.940 | 2.5272 | $19.460\pm 0.094$ | 14.277
0.1752 | $15.411\pm 0.046$ | 17.940 | 2.7800 | $19.649\pm 0.156$ | 14.240
0.1928 | $15.428\pm 0.045$ | 17.738 | 3.0580 | $20.075\pm 0.137$ | 14.207
0.2120 | $15.447\pm 0.046$ | 17.496 | 3.3638 | $20.538\pm 0.103$ | 14.181
0.2333 | $15.466\pm 0.048$ | 17.301 | 3.7001 | $21.002\pm 0.138$ | 14.160
0.2566 | $15.501\pm 0.046$ | 17.046 | 4.0701 | $21.399\pm 0.203$ | 14.142
0.2822 | $15.531\pm 0.043$ | 16.960 | 4.4772 | $21.964\pm 0.228$ | 14.129
0.3105 | $15.580\pm 0.037$ | 16.745 | 4.9249 | $22.519\pm 0.240$ | 14.117
0.3415 | $15.632\pm 0.031$ | 16.573 | 5.4174 | $23.311\pm 0.588$ | 14.106
0.3757 | $15.695\pm 0.029$ | 16.388 | 5.9591 | $23.508\pm 0.782$ | 14.096
### 3.3 Fits
Our fitting procedure involves computing in full large numbers of King
structural models, spanning a wide range of fixed values of the appropriate
shape parameter $W_{0}$ (see McLaughlin & van der Marel, 2005, in detail). And
then the models are convolved with the ACS/WFC PSF for the F606W and F814W
filters of equations of (1) and (2):
$\widetilde{I}_{\rm
mod}^{*}(R|r_{0})=\int\\!\\!\\!\int_{-\infty}^{\infty}\widetilde{I}_{\rm
mod}(R^{\prime}/r_{0})\times\widetilde{I}_{\rm
PSF}\left[(x-x^{\prime}),(y-y^{\prime})\right]\ dx^{\prime}\,dy^{\prime}\ ,$
(3)
where $\widetilde{I}_{\rm mod}\equiv I_{\rm mod}/I_{0}$; and
$\widetilde{I}_{\rm PSF}$ is the PSF profile normalized to unit total
luminosity (see McLaughlin et al., 2008, in detail). We changed the luminosity
density to surface brightness $\widetilde{\mu}_{\rm
mod}^{*}=-2.5\,\log\,[\widetilde{I}_{\rm mod}^{*}]$ before fitting them to the
observed surface-brightness profile of B037,
$\mu=\mu_{0}-2.5\,\log\,[I(R/r_{0})/I_{0}]$, finding the radial scale $r_{0}$
and central surface brightness $\mu_{0}$ which minimize $\chi^{2}$ for every
given value of $W_{0}$. The $(W_{0},r_{0},\mu_{0})$ combination that yields
the global minimum $\chi_{\rm min}^{2}$ over the grid used defines the best-
fit model of that type:
$\chi^{2}=\sum_{i}{\frac{\left[\mu_{\rm obs}(R_{i})-\widetilde{\mu}_{\rm
mod}^{*}(R_{i}|r_{0})\right]^{2}}{\sigma_{i}^{2}}},$ (4)
in which $\sigma_{i}$ is the error in the surface brightness. Estimates of the
one-sigma uncertainties on these basic fit parameters follow from their
extreme values over the subgrid of fits with $\chi^{2}/\nu\leq\chi_{\rm
min}^{2}/\nu+1$, here $\nu$ is the number of free parameters. Figure 5 shows
our best King fits to B037. In Fig. 5, open squares are ellipse data points
included in the least-squares model fitting, and the asterisks are points not
used to constrain the fit. These observed data points shown by asterisks are
included in the radius of $R<2~{}\rm{pixels}=0\farcs 1$, and the isophotal
intensity is dependent on its neighbors. As Barmby et al. (2007) pointed out
that, the ellipse output contains brightnesses for 15 radii inside 2 pixel,
but they are all measured from the same 13 central pixels and are not
statistically independent. So, to avoid excessive weighting of the central
regions of B037 in the fits, we only used intensities at radii $R_{\rm min}$,
$R_{\rm min}+(0.5,1.0,2.0~{}{\rm pixels})$, or $R>2.5~{}{\rm pixels}$ as
Barmby et al. (2007) used. Table 6 summarizes the results obtained in this
paper.
Figure 5: Surface brightness profile of B037 measured in the F606W and F814 filters. Dashed curve (blue) trace the PSF intensity profiles and solid (red) curves are the PSF-convolved best-fit models. Open squares are ellips data points included in the $\chi^{2}$ model fitting, and the asterisks are points not used to constrain the fits (see the text in detail). Table 6: Structural parameters of B037 Parameters | F606W | F814W
---|---|---
$r_{0}$ | $0.56\pm 0.02\arcsec~{}(=2.16\pm 0.08~{}\rm{pc})$ | $0.56\pm 0.01\arcsec~{}(=2.16\pm 0.04~{}\rm{pc})$
$r_{t}$ | $8.6\pm 0.4\arcsec~{}(=33.1\pm 1.5~{}\rm{pc})$ | $8.9\pm 0.3\arcsec~{}(=34.3\pm 1.2~{}\rm{pc})$
$c=\log(r_{t}/r_{0})$ | $1.19\pm 0.02$ | $1.20\pm 0.01$
$r_{h}$ | $1.05\pm 0.03\arcsec(=4.04\pm 0.12~{}\rm{pc})$ | $1.07\pm 0.01\arcsec(=4.12\pm 0.04~{}\rm{pc})$
$\mu_{0}$ (${\rm mag~{}arcsec^{-2}}$) | $13.53\pm 0.03$ | $12.85\pm 0.03$
### 3.4 Comparison to previous results
Ma et al. (2006b) analyzed the same F606W image of B037 used here, fitting a
King (1962) model to a surface brightness profile derived from a PSF-
deconvolved image. They derived the scale radius $r_{0}=0\farcs 72$ (it is
called the core radius in Ma et al. (2006b)), half-light radius $r_{h}=1\farcs
11$, concentration index $c=0.91$, and central surface brightness
$\mu(0)=17.21~{}{\rm mag~{}arcsec^{-2}}$ (using the value for extinction
adopted in this paper, this becomes $\mu_{0}=13.40~{}{\rm
mag~{}arcsec^{-2}}$). Comparing the results of Ma et al. (2006b) with Table 6
of this paper, we find that our model fits produce a somewhat higher
concentration and smaller scale radius. These differences come from: (i) using
different models (King (1962) vs King (1966)), (ii) the observed data are
obtained with different ways. In (ii), Ma et al. (2006b) derived the surface
brightness profile from a PSF-deconvolved image; in addition, Ma et al.
(2006b) derived the surface brightness profile with ellipticity and position
angle allowed to vary with the isophote semimajor axis, however, in this
paper, we derived the surface brightness profile on fixed, zero-ellipticity
isophotes, since we choose to fit circular models for the intrinsic cluster
structure and the PSF as Barmby et al. (2007) did (see §2.4 for details). In
fact, from Fig. 5 of this paper and Fig. 3 of Ma et al. (2006b), we can see
that, the observed data are somewhat different between Ma et al. (2006b) and
this paper.
Barmby et al. (2007) analyzed the same F606W and F814W images of B037 used
here with nearly the same observed data and method. The results of comparison
are listed in Table 7 (Table 5 of Barmby et al. (2007) in the electronic
edition did not list the results of B037 in the F814W filter.), from which we
can see that the results obtained in this paper are in good agreement with
ones of Barmby et al. (2007) (about the central surface brightnesses, we have
corrected them using the value for extinction adopted in this paper).
Table 7: Results of comparison | F606W | F814W
---|---|---
Parameters | Barmby et al. (2007) | This paper | Barmby et al. (2007) | This paper
$r_{0}$ | 056 | 056 | 059 | 056
$c=\log(r_{t}/r_{0})$ | 1.23 | 1.19 | 1.18 | 1.20
$r_{h}$ | 109 | 105 | | 107
$\mu_{0}$ (${\rm mag~{}arcsec^{-2}}$) | 13.45 | 13.53 | 12.75 | 12.85
## 4 Discussion and summary
As discussed in §3.1, it is impossible that the dust lane comes from the Milky
Way. Another possibility is that the dust lane is contained in B037 itself. As
we know, the formation of dust requires gas with a rather high metallicity.
Perrett et al. (2002) presented metallicities for more than 200 GCs in M31
including B037, using the Wide Field Fibre Optic Spectrograph at the 4.2 m
William Herschel Telescope in La Palma, Canary Islands, which provides a total
spectral coverage of $\sim$ 3700-5600 Å with two gratings. One grating (H2400B
2400 line) yielded a dispersion of 0.8 Å${\rm~{}pixel^{-1}}$ and a spectral
resolution of 2.5 Å over the range 3700-4500 Å covering the CN feature at 3883
Å, the H and K lines of calcium, $\rm H\delta$, the CH G band and the 4000 Å
continuum break, and the other grating (R1200R 1200 line) presented a
dispersion of 1.5 Å${\rm~{}pixel^{-1}}$ and a spectral resolution of 5.1 Å
over the range 4400-5600 Å to add absorption features such as $\rm H\beta$,
the Mg $b$ triplet, and two iron lines near 5300 Å. Then, Perrett et al.
(2002) calculated 12 absorption-line indices based on the prescription of
Brodie & Huchra (1990). By the comparison of the line indices with the
published M31 GC [Fe/H] values from the previous literature (Bònoli et al.,
1987; Brodie & Huchra, 1990; Barmby et al., 2000), the results of linear fits
were obtained. Final cluster metallicities were determined from an unweighted
mean of the [Fe/H] values calculated from the CH (G), Mg $b$, and Fe53 line
strengths. For B037, Perrett et al. (2002) obtained its metallicity to be
$\rm[Fe/H]=-1.07\pm 0.20$. It is clear that B037 has a low metallicity. So, it
is intricate that where is the dust lane from?
In this paper, using the images of deep observations and of highly spatial
resolutions with the ACS/HST, we firstly present that the GC B037 in M31 is
crossed by a dust lane. Photometric data in the F606W and F814W bands provide
that, colors of ($\rm{F606W-F814W}$) in the dust lane are redder $\sim 0.4$
mags than ones in the other regions of B037. From the HST images, this dust
lane seems to be contained in B037, not from the Milky Way. However, the
formation of dust requires gas with a rather high metallicity. So, it seems
impossible that the observed dust lane is physically associated with B037
itself, which has a low metallicity to be $\rm[Fe/H]=-1.07\pm 0.20$ from
Perrett et al. (2002). So, that the observed dust lane in the view of B037 is
from B037 itself or from the Milky Way needs to be confirmed in the future. In
addition, based on these images, we present the precise variation of
ellipticity and position angle, and of surface brightness profile, and
determine the structural parameters of B037 by fitting a single-mass isotropic
King model.
###### Acknowledgements.
I am indebted to Daming Chen, Zhou Fan, Tianmeng Zhang and Song Wang for their
helps in finishing this paper. I am also grateful to the referee for the
important comments. This work was supported by the Chinese National Natural
Science Foundation grands No. 10873016, and 10633020, and by National Basic
Research Program of China (973 Program), No. 2007CB815403.
## References
* Barmby et al. (2002a) Barmby, P., Holland, S., & Huchra, J. 2002a, AJ, 123, 1937
* Barmby et al. (2000) Barmby, P., Huchra, J., Brodie, J., Forbes, D., Schroder, L., & Grillmair, C. 2000, AJ, 119, 727
* Barmby et al. (2007) Barmby P., McLaughlin D. E., Harris W. E., Harris G. L. H., & Forbes D. A. 2007, AJ, 133, 2764
* Barmby et al. (2002b) Barmby, P., Perrett, K. M., & Bridges, T. J. 2002b, MNRAS, 329, 461
* Battistini et al. (1987) Battistini, P., Bònoli, F., Braccesi, A., Federici, L., Fusi Pecci F., Marano, B., & Börngen, F. 1987, A&AS, 67, 447
* Bònoli et al. (1987) Bònoli, F., Delpino, F., Federici, L., & Fusi Pecci, F. 1987, A&A, 185, 25
* Brodie & Huchra (1990) Brodie, J. P., & Huchra, J. P. 1990, ApJ, 362, 503
* Cardelli et al. (1989) Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245
* Crampton et al. (1985) Crampton, D., Cowley, A. P., Schade, D., & Chayer, P. 1985, ApJ, 288, 494
* Fan et al. (2008) Fan, Z., Ma, J., de Grijs, R., & Zhou, X. 2008, MNRAS, 385, 1973
* Galleti et al. (2004) Galleti, S., Federici, L., Bellazzini, M., Fusi Pecci, F., & Macrina, S. 2004, A&A, 416, 917
* Galleti et al. (2006) Galleti, S., Federici, L., Bellazzini, M., Buzzoni, A., & Fusi Pecci, F. 2006, A&A, 456, 985
* Galleti et al. (2007) Galleti, S., Bellazzini, M., Federici, L., Buzzoni, A., & Fusi Pecci, F. 2007, A&A, 471, 127
* Harris et al. (2002) Harris W. E., Harris G. L. H., Holland S. T., & McLaughlin D. E. 2002, AJ, 124, 1435
* Hubble (1932) Hubble, E. P. 1932, ApJ, 76, 44
* Huchra et al. (1991) Huchra, J. P., Brodie, J. P., & Kent, S. M. 1991, ApJ, 370, 495
* King (1962) King, I. R. 1962, AJ, 67, 471
* King (1966) King, I. R. 1966, AJ, 71, 64
* Kron & Mayall (1960) Kron, G. E., & Mayall, N. U. 1960, AJ, 65, 581
* Larsen et al. (2002) Larsen, S. S., Brodie, J. P., Sarajedini, A., & Huchra, J. P. 2002, AJ, 124, 2615
* Ma et al. (2007) Ma, J., et al. 2007, MNRAS, 376, 1621
* Ma et al. (2006a) Ma, J., de Grijs, R., Yang, Y., Zhou, X., Chen, J., Jiang, Z., Wu, Z., & Wu, J. 2006a, MNRAS, 368, 1143
* Ma et al. (2006b) Ma, J., van den Bergh, S., & Wu, H., et al. 2006b, ApJ, 636, L93
* Mackey & van den Bergh (2005) Mackey, A., & van den Bergh, S. 2005, MNRAS, 360, 631
* McLaughlin et al. (2008) McLaughlin, D. E., Barmby, P., Harris, W. E., Forbes, D. A., & Harris, G. L. H. 2008, MNRAS, 384, 563
* McLaughlin & van der Marel (2005) McLaughlin, D. E., & van der Marel, R. P. 2005, ApJS, 161, 304
* Meylan & Heggie (1997) Meylan, G., & Heggie, D. 1997, A&A Rev., 8, 1
* Meylan et al. (2001) Meylan, G., Sarajedini, A., Jablonka, P., Djorgovski, S., Bridges, T., & Rich, R. 2001, AJ, 122, 830
* Michie (1963) Michie, R. W. 1963, MNRAS, 125, 127
* Perrett et al. (2002) Perrett, K. M., et al. 2002, AJ, 123, 2490
* Sirianni et al. (2005) Sirianni, M., et al. 2005, PASP, 117, 1049
* Stetson (1987) Stetson, P. B. 1987, PASP, 99, 191
* van den Bergh (1968) van den Bergh, S. 1968, The Observatory, 88, 168
* Vetes̆nik (1962a) Vetes̆nik, M. 1962a, BAC, 13, no. 5, p. 180
* Vetes̆nik (1962b) Vetes̆nik, M. 1962b, BAC, 13, no. 6, p. 218
|
arxiv-papers
| 2011-01-08T03:21:55 |
2024-09-04T02:49:16.235370
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jun Ma (1,2) ((1) National Astronomical Observatories, Chinese Academy\n of Sciences, (2) Key Laboratory of Optical Astronomy, National Astronomical\n Observatories, Chinese Academy of Sciences)",
"submitter": "Jun Ma",
"url": "https://arxiv.org/abs/1101.1569"
}
|
1101.1580
|
# Infall and outflow motions in the high-mass star forming complex G9.62+0.19
Tie Liu11affiliation: Department of Astronomy, Peking University, 100871,
Beijing China; liutiepku@gmail.com, ywu@pku.edu.cn , Yuefang Wu11affiliation:
Department of Astronomy, Peking University, 100871, Beijing China;
liutiepku@gmail.com, ywu@pku.edu.cn , Sheng-Yuan Liu22affiliation: Institute
of Astronomy and Astrophysics, Academia Sinica, Taipei, Taiwan , Sheng-Li
Qin33affiliation: I. Physikalisches Institut, Universität zu Köln, Zülpicher
Str. 77, 50937 Köln, Germany 44affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, Beijing, 100012 , Yu-Nung
Su22affiliation: Institute of Astronomy and Astrophysics, Academia Sinica,
Taipei, Taiwan , Huei-Ru Chen55affiliation: Institute of Astronomy and
Department of Physics, National Tsing Hua University, Hsinchu, Taiwan
22affiliation: Institute of Astronomy and Astrophysics, Academia Sinica,
Taipei, Taiwan and Zhiyuan Ren11affiliation: Department of Astronomy, Peking
University, 100871, Beijing China; liutiepku@gmail.com, ywu@pku.edu.cn
###### Abstract
We present the results of a high resolution study with the Submillimeter Array
towards the massive star forming complex G9.62+0.19. Three sub-mm cores are
detected in this region. The masses are 13, 30 and 165 M☉ for the northern,
middle and southern dust cores, respectively. Infall motions are found with
HCN (4-3) and CS (7-6) lines at the middle core (G9.62+0.19 E). The infall
rate is $4.3\times 10^{-3}~{}M_{\odot}\cdot$yr-1. In the southern core, a
bipolar-outflow with a total mass about 26 M☉ and a mass-loss rate of
$3.6\times 10^{-5}~{}M_{\odot}\cdot$yr-1 is revealed in SO ($8_{7}-7_{7}$)
line wing emission. CS (7-6) and HCN (4-3) lines trace higher velocity gas
than SO ($8_{7}-7_{7}$). G9.62+0.19 F is confirmed to be the driving source of
the outflow. We also analyze the abundances of CS, SO and HCN along the
redshifted outflow lobes. The mass-velocity diagrams of the outflow lobes can
be well fitted by a single power law. The evolutionary sequence of the cm/mm
cores in this region are also analyzed. The results support that UC Hii
regions have a higher blue excess than their precursors.
Massive core:pre-main sequence-ISM: molecular-ISM: kinematics and dynamics-
ISM: jets and outflows-stars: formation
††slugcomment: Accepted to ApJ
## 1 Introduction
High-mass stars play a major role in the evolution of the Galaxy. They are the
principal sources of heavy elements and UV radiation (Zinnecker & Yorke,
2007). However, the formation and evolution of high-mass stars are still
unclear. A possible evolution sequence of high-mass stars from infrared dark
clouds to classic Hii regions has been suggested (Van der Tak & Menten, 2005).
But one of the major topics whether high-mass stars form through accretion-
disk-outflow, like low-mass ones (Shu, Adams & Lizano, 1987), or form via
collision-coalescence (Wolfire, & Cassinelli, 1987; Bonnell et al., 1998) is
still far from solved.
Yet more and more observations at various resolutions seem to support the
accretion-disk-outflow models rather than collision-coalescence models. Disks
are detected in several high-mass star forming regions (Patel et al., 2005;
Jiang et al., 2005; Sridharan, Williams, & Fuller, 2005). Outflows are found
with a high detection rate as in low-mass cores in single-dish surveys (Wu et
al., 2004; Zhang et al., 2005; Qin et al., 2008a). High resolution studies
have also confirmed that molecular outflows are common in high-mass star
forming regions (Su, Zhang, & Lim, 2004; Qiu et al., 2007; Qin et al., 2008b,
c; Qiu et al., 2009). Searching for inflow motions also has made large
progress in recent years (Wu & Evans, 2003; Fuller, Williams, & Sridharan,
2005; Wyrowski et al., 2006; Klaassen, & Wilson, 2007; Wu et al., 2007, 2009;
Furuya, Cesaroni, & Shinnaga, 2011). Both infall and outflow motions in the
massive core JCMT 18354-0649S are detected (Wu et al., 2005), and further
confirmed by higher resolution observations (Liu et al., 2011). Although
accretion-disk-outflow systems are found in high-mass star forming regions,
there may be differences between low- and high-mass formation.
The infall motion can be detected via ”blue profile”, a double-peaked profile
with the blueshifted peak being stronger for optically thick lines and a
single peak at the absorption part of optically thick lines for optically thin
lines, which is caused by self absorption of the cooler outer infalling gas
towards the warmer central region (Zhou et al., 1993). In contrast, the ”red
profile” where the redshifted peak of a double-peaked profile being stronger
for optically thick lines is suggested as indicators for outflow motions.
Mardones et al. (1997) defined the ”blue excess” in a survey, E, as E = (NB-
NR)/NT (Mardones et al. 1997), where NT is number of sources, NB and NR mark
the number of sources with blue and red profiles, respectively. The blue
excess seems to be no significant differences among the low-mass cores in
different evolutionary phases. However, using the IRAM 30 m telescope, Wu et
al. (2007) found that UC Hii regions show a higher blue excess than their
precursors, indicating fundamental differences between low- and high-mass-star
forming conditions. The searches need to be expanded.
Located at a distance of 5.7 kpc (Hofner et al., 1994), G9.62+0.19 is a well
studied high-mass star forming region containing a cluster of Hii regions,
which are probably at different evolutionary stages. Multiwavelength VLA
observations have identified nine radio continuum sources (denoted from A-I)
(Garay et al., 1993; Testi et al., 2000), and components C-I are very compact
($<5\arcsec$ in diameter) (Garay et al., 1993; Testi et al., 2000). As
revealed in NH3 (4,4), (5,5) and CH3CN (J=6-5), component F is a hot molecular
core (HMC) and hence likely the youngest source in the region (Cesaroni et
al., 1994; Hofner et al., 1994, 1996b). G9.62+0.19 E is a young massive star
surrounded by a very small UC Hii region and a dusty envelope (Hofner et al.,
1996b), while G9.62+0.19 D a small cometary UC Hii region excited by a B0.5
ZAMS star (Hofner et al., 1996b; Testi et al., 2000). Both G9.62+0.19 E and
G9.62+0.19 D seem to be at a more evolved stage than G9.62+0.19 F. Thus G9.62
complex is an ideal sample to examine massive star forming activities
including outflow and infall motions.
Maser emissions of NH3, H2O, OH, and CH3OH, as well as the strong thermal NH3
emissions were detected along a narrow region with projected length
$20\arcsec$ and width$\leq 2\arcsec$ (Hofner et al., 1994). A possible
explanation for this alignment is compression of the molecular gas by shock
front originating from an even more evolved Hii region to the west of the
star-forming front (Hofner et al., 1994). High-velocity molecular outflows
also have been detected in this region, and G9.62+0.19 F is believed to be the
driving source (Gibb, Wyrowski, & Mundy , 2004; Hofner, Wiesemeyer, & Henning
, 2001; Su et al., 2005). However, most of previous work was carried out at
low frequencies, probing low excitation conditions. To exam the hot dust/gas
environment and dynamical processes in this region, higher resolution studies
at high frequencies are needed. In this paper we report the results of the
Submillimeter Array (SMA111Submillimeter Array is a joint project between the
Smithsonian Astrophysical Observatory and the Academia Sinica Institute of
Astronomy and Astrophysics and is funded by the Smithsonian Institution and
the Academia Sinica.) observations toward G9.62+0.19 region at 860 $\micron$.
## 2 Observations
The observations of G9.62+0.19 with the SMA were carried out on July 9th, 2005
with seven antennas in its compact configuration at 343 GHz for the lower
sideband (LSB) and 353 GHz for the upper sideband (USB). The Tsys ranges from
210 to 990 K with a typical value of 380 K at both sidebands during the
observations. The observations had two fields for the G9.62+0.19 complex to
cover the entire region with emissions. One phase reference center was
RA(J2000) = 18h06m14.21s and DEC(J2000) = -$20\arcdeg 31\arcmin 46.2\arcsec$,
and the other was RA(J2000) = 18h06m15.00s and DEC(J2000) = -$20\arcdeg
31\arcmin 34.20\arcsec$. Uranus and Neptune were observed for antenna-based
bandpass calibration. QSOs 1743-038 and 1911-201 were employed for antenna-
based gain correction. Neptune was used for flux-density calibration. The
frequency spacing across the spectra band was 0.8125 MHz, corresponding to a
velocity resolution of $\sim$0.7 km s-1.
MIRIAD was employed for calibration and imaging (Sault et al., 1995). The
imaging was done to each field separately and the mosaic continuum map was
made using a linear mosaicing algorithm (task ”linmos” in MIRIAD). The 860
$\micron$ continuum data was acquired by averaging over all the line-free
channels in both sidebands. The spectral cubes were constructed using the
continuum-subtracted spectral channels smoothed into a velocity resolution of
1 km s-1. Additional self-calibration with models of the clean components from
previous imaging process was performed on the continuum data in order to
remove residual errors due to phase and amplitude problems, and the gain
solutions obtained from the continuum data were applied to the line data. The
synthesized beam size of the continuum emission with robust weighting of 0.5
is $2.76\arcsec\times 1.88\arcsec$ (P.A.=$21.4\arcdeg$).
## 3 Results
### 3.1 Continuum emission
The 860 $\micron$ continuum image combining the visibility data from both
sidebands is shown in Figure 1 . Three sub-mm cores are detected. The known cm
and mm continuum components (Testi et al., 2000) of B, C, D, E, F, G, H, and I
are marked by plus signs. Water masers (Hofner, & Churchwell, 1996a) are
marked by open squares and methanol masers (Norris et al., 1993) by triangles.
The near-IR sources (Persi et al., 2003; Testi et al., 1998; Linz et al.,
2005) are marked by filled circles. IRAC sources are taken from the database
of Galactic Legacy Infrared Mid-Plane Survey Extraordinaire (GLIMPSE)
222http://irsa.ipac.caltech.edu/data/SPITZER/GLIMPSE/ and labeled with
asterisks. The northern core is located at south-east of G9.62+0.19 C, and the
middle core is associated with G9.62+0.19 E. The 860 $\micron$ continuum
emission at the southern core is concentrated on the hot molecular core
G9.62+0.19 F and extends to G9.62+0.19 D in the south and to G9.62+0.19 G in
the north.
Gaussian fits were made to the continuum. The northern core seems to be a
point-like source. The middle core is very compact with a deconvolved size of
$\sim 1.4\arcsec$. The southern core is found to be elongated from north to
south with an average size of $2.4\arcsec$, containing at least three sources,
D, F, and G. F is at its peak position. The peak positions, sizes, peak
intensities and total fluxes of these three sub-mm cores are listed in Column
2-5 in Table 1. The physical properties of these cores will be further
discussed in section 4.2.
### 3.2 Line emission
Tens of molecular transitions including hot molecular lines CH3OH, HCOOCH3,
and CH3OCH3 are detected toward both the middle and southern sub-mm cores,
indicating these two cores are hot and dense (Qin et al., 2010). Figure 2
presents the full LSB and USB spectra in the UV domain over the shortest
baseline. The strongest lines are identified and labeled on the plots. Only
HCN (4-3) and CS (7-6) line emissions are detected towards the northern core
with our sensitivity. Thus we mainly focus on the middle and southern cores in
this paper. The systemic velocities of 2.1 km s-1 for the middle core and 5.2
km s-1 for the southern core are obtained by averaging the Vlsr of multiple
singly peaked lines. Six transitions of the thioformaldehyde (H2CS) and
molecular transitions SO ($8_{7}$-77), CS (7-6), HC15N (4-3) and HCN (4-3) are
analyzed here, while the others will be discussed in another paper. We have
made gaussian fits to the beam-averaged spectra, and present the observed
parameters of these lines in Table 2.
#### 3.2.1 Line emission at the middle sub-mm core
The integrated intensity maps of four transitions of H2CS towards the middle
core are shown in the upper panels of Figure 3. From (a) to (d), the upper
level energy of H2CS transitions varies from $\sim 90$ K to $\sim 400$ K. The
H2CS emission is spatially coincident with continuum emission of the middle
core very well. The Position-Velocity (P-V) diagram and first moment map of
H2CS (102,8-92,7) emission are presented in Figure 4. The P-V diagram is
constructed across the peak of the continuum along the N-S direction. From P-V
diagram two emission peaks are clearly revealed. The velocities of the two
emission peaks are at 1 and 3 km s-1 with $1.5\arcsec$ spatial separation,
indicating a velocity gradient in N-S direction. The first moment map also
shows velocity changes in N-S direction. The small velocity gradient detected
in H2CS (102,8-92,7) emission may indicate a disk with a low inclination along
the line of sight, which requires further confirmation with higher angular
resolution observations and other molecular line tracers.
The spectra and integrated intensity maps of HC15N (4-3) and SO ($8_{7}$-77)
are presented in Figure 5. The two spectra seem to be symmetric, and their
cores are associated with that of the continuum emission very well.
Figure 6 presents the spectra and P-V diagrams of HCN (4-3) and CS (7-6)
emissions of the middle core. HCN (4-3) and CS (7-6) show asymmetric profile.
The blue and red emission peaks of HCN (4-3) are around 0 km s-1 and 6 km s-1,
respectively. The blueshifted emission of CS (7-6) peaks around 1 km s-1,
while the redshifted around 4 km s-1. We can see the blueshifted emission of
both HCN (4-3) and CS (3-2) is always stronger than the redshifted emission
and the absorption is also redshifted, which are blue profiles (see Sect. 1).
Besides the ”blue profile”, some weak absorption dips are found around 10 km
s-1 in both the spectra and P-V diagrams of HCN (4-3) and CS (7-6), and
further observations are needed to determine the properties of these
absorption dips. In this paper we only pay attention to the ”blue-profile”
found in CS (7-6) and HCN (4-3) emission.
The integrated intensity maps of HCN (4-3) and CS (7-6) towards the middle
core are presented in Figure 7. The HCN (4-3) and CS (7-6) are associated with
the dust emission.
#### 3.2.2 Line emission at the southern core
The integrated intensity maps of four transitions of H2CS at the southern core
are shown in the lower panels of Figure 3. The upper level energy of H2CS
transitions varies from $\sim 90$ K to $\sim 400$ K from panel (e) to panel
(h). As the upper level energy increases, the emission peak of the different
transitions of H2CS moves from S-E to N-W, indicating a temperature gradient
in the southern core.
Averaged spectra of SO ($8_{7}$-77), HC15N (4-3), HCN (4-3) and CS (7-6) at
the southern core are presented in Figure 8. The spectra of SO ($8_{7}$-77)
and HC15N are averaged over a region of 4$\arcsec$, while HCN (4-3) and CS
(7-6) are averaged over a region of 6$\arcsec$. SO ($8_{7}$-77) emission has a
total velocity extent of larger than 20 km s-1. From gaussian fit to the
spectrum, the peak velocity of SO emission is $5.1\pm 0.1$ km s-1, coincident
very well with the systemic velocity 5.2 km s-1. HC15N (4-3) has a velocity
extent of about 15 km s-1. The velocity extents of CS (7-6) and HCN (4-3) are
as high as 40 km s-1 and 60 km s-1, respectively. Emission wings are clearly
detected from the spectra of the four lines. A ”red-profile” is significantly
exhibited in the spectra of CS (7-6) and HCN (4-3), of which the redshifted
emission is always stronger than the blueshifted emission with an absorption
dip at the blueshifted side of the systemic velocity (5.2 km s-1). This
profile is caused by absorption of the colder blueshifted gas in front of the
hot core, indicating outflow motions. The ”red-profile” is consistent with
that detected using single dish observations (see Figure 6 of Hofner,
Wiesemeyer, & Henning (2001)).
The integrated intensity maps of HC15N (4-3) and SO ($8_{7}$-77) at the
southern core are presented in Figure 9. To avoid the influence of outflow
motions, both the maps are integrated from 2 km s-1 to 8 km s-1. Both the
emission of HC15N (4-3) and SO ($8_{7}$-77) coincides with the cm/mm component
F, and extends from D to G.
As shown in the left panels of Figure 10, the high velocity gas of HC15N (4-3)
and SO ($8_{7}$-77) can be identified by the vertically dashed lines in the
P-V diagrams. For SO ($8_{7}$-77), we integrate from -4 km s-1 $\leq$ V $\leq$
0 km s-1 for the blue wing and 10 km s-1 $\leq$ V $\leq$ 14 km s-1 for the red
wing, and present the contour map in (c) of Figure 10. For HC15N (4-3), only
the red wing emission is presented in (d) of Figure 10. The high velocity
emission of both HC15N (4-3) and SO ($8_{7}$-77) is associated with core F,
indicating that core F is the driven source of the outflow. The blue and red
wings of SO ($8_{7}$-77) overlap to a large extent in the contour maps, and
hence the molecular outflow revealed by SO ($8_{7}$-77) is observed close to
its flow axis.
Figure 11 presents the channel maps of CS (7-6) emission. The redshifted high-
velocity gas seems to be elongated from north-east to south-west, while the
blueshifted high-velocity gas from north to south. The high velocity gas
revealed by CS (7-6) is also very obvious in the P-V diagram in Figure 13(d).
As shown in the P-V diagram, the blueshifted high-velocity gas extends about
8$\arcsec$ from north to south. The high-velocity emission integrated over the
wings (-12 km s-1 $\leq$ V $\leq$ -5 km s-1 for the blue wing and 15 km s-1
$\leq$ V $\leq$ 22 km s-1 for the red wing) is presented in Figure 13(e).
Figure 12 is the channel maps of HCN (4-3) emission. The maximum of
absorptions appears at around 0 km s-1. The redshifted high-velocity gas seems
to be elongated from west to east, while the blueshifted high-velocity gas
from north to south. At very high velocity channels (V$\leq$ -16 km s-1), the
blueshifted emission is totally located at south-east. By comparing the
channel maps and P-V diagrams (see Figure 13) of HCN (4-3) and CS (7-6) at
velocity intervals -12 km s-1 $\leq$ V $\leq$ -5 km s-1 and 15 km s-1 $\leq$ V
$\leq$ 22 km s-1, we find similar structures in CS (7-6) and HCN (4-3)
emissions. The high-velocity emission of HCN (4-3) integrated from -12 km s-1
to -5 km s-1 for the blue wing and from 15 km s-1 to 22 km s-1 for the red
wing is presented in the panel (b) of Figure 13\. As of CS (7-6), the
blueshifted gas revealed by HCN (4-3) is elongated from north to south with
the emission center located between G9.62+0.19 F and G9.62+0.19 D, while the
redshifted gas from north-east to south-west. Two clumps are found in the
blueshifted high-velocity emission of HCN (4-3), which locate at north-west
and south-east of F, respectively.
In order to reveal the very high velocity emission traced by HCN (4-3) but not
CS (7-6), we integrate over the wings at much higher velocities (-20 km s-1
$\leq$ V $\leq$ -13 km s-1 for the blue wing and 23 km s-1 $\leq$ V $\leq$ 39
km s-1 for the red wing), and present the integrated emission map in Figure
13(c). The redshifted emission is elongated from north-east to south-west with
the emission center located between G9.62+0.19 F and G9.62+0.19 G, while the
blueshifted emission center located between G9.62+0.19 F and G9.62+0.19 D. It
is clearly seen that the high velocity gas traced by SO ($8_{7}$-77), CS (7-6)
and HCN (4-3) have different spatial distributions, which should be caused by
the complicated interactions between the outflow and the ambient gas. It may
also indicate a change of the outflow axis. The change of outflow axis is also
found in IRAS 20126+4104 (Su et al., 2007) and JCMT 18354-0649S (Liu et al.,
2011).
From the integrated emission maps of SO ($8_{7}$-77), HCN (4-3) and CS (7-6)
high velocity gas, it is clearly seen that G9.62+0.19 F is located at the
middle of the redshifted and blueshifted lobes, suggesting G9.62+0.19 F is the
outflow driving source.
## 4 Discussion
### 4.1 Rotational temperature of H2CS transitions
Six transitions of H2CS have been detected in the middle and southern cores,
enabling us to estimate the rotational temperature. Under the assumptions that
the gas is optically thin under local thermodynamic equilibrium and the gas
emission fills the beam, the rotation temperature and beam-averaged column
density can be estimated using the Rotational Temperature Diagram (RTD) by
(Cummins, Linke, &Thaddeus, 1986; Turner et al., 1991; Liu, et al., 2002)
$\textrm{ln}(\frac{N_{u}}{g_{u}})=\textrm{ln}(\frac{N_{T}}{Q_{rot}})-\frac{E_{u}}{T_{rot}}=\textrm{ln}[2.04\times
10^{20}\frac{\int~{}I(Jy~{}beam^{-1})dv(km~{}s^{-1})}{\theta_{a}\theta_{b}(arcsec^{2})g_{I}g_{K}\nu^{3}(GHz^{3})S\mu^{2}(debye^{2})}]$
(1)
where Nu is the observed column density of the upper energy level, gu is the
degeneracy factor in the upper energy level, NT is the total beam-averaged
column density, Qrot is the rotational partition function, Eu is the upper
level energy in K, Trot is the rotation temperature, $\int$ I dv is the
integrated intensity of the specific transition, $\theta_{a}$ and $\theta_{b}$
are the FWHM beam size, gK is the K-ladder degeneracy, gI is the degeneracy
due to nuclear spin, $\nu$ is the rest frequency, and S is line strength and
$\mu$ the permanent dipole moment. For H2CS, the interchangeable nuclei are
spin $\frac{1}{2}$, leading to ortho- and para-forms with gI equaling
$\frac{3}{4}$ and $\frac{1}{4}$, respectively (Blake et al., 1987; Turner et
al., 1991). The partition function Qrot of H2CS is (Blake et al., 1987)
$Q_{rot}=2[\frac{\pi(kT_{rot})^{3}}{h^{3}ABC}]^{\frac{1}{2}}$ (2)
where k and h are the Boltzmann and Planck constants, respectively, and A, B,
and C are the rotation constants. Thus the rotation temperature Trot and total
column density NT can be estimated by least-squares fitting to the multiple
transitions. We applied the RTD method towards D, E, F, G (see Figure 14), and
the fitting results are listed in the second and third columns of Table 3\.
The rotational temperature of the middle core (E) is 83$\pm$21 K. In the
southern core, the rotational temperature estimated decreases from G (91 K) to
F (83 K) and D (43 K), suggesting the temperature gradient in the southern
core. The total column density of H2CS ranges from 1.3$\times$1015 (G) to
3.8$\times$1015 cm-2 (D).
However, the filling factor and the optical depth correction were not taken
account of in the RTD method. To investigate their effect we applied the
Population Diagram (PD) analysis (Goldsmith, & Langer, 1999; Wang et al.,
2010). In the PD analysis, we have
$\textrm{ln}(\frac{\hat{N_{u}}}{g_{u}})=\textrm{ln}(\frac{N_{T}}{Q_{rot}})-\frac{E_{u}}{T_{rot}}+\textrm{ln}(f)-\textrm{ln}(\frac{\tau}{1-e^{-\tau}})$
(3)
where $\hat{N_{u}}$ is the inferred column density of the upper energy level
from the PD analysis, f is the source filling factor and $\tau$ is the optical
depth. The optical depth $\tau$ can be expressed by (Remijan et al., 2004)
$\tau=\frac{8\pi^{3}S\mu^{2}\nu}{3k\Delta\textrm{v}T_{rot}}\frac{N_{T}}{Q_{rot}}e^{-\frac{E_{u}}{T_{rot}}}$
(4)
where $\Delta$v is the FWHM line width. Under LTE, the upper-level
populations, $\hat{N_{u}}$, can be predicted according to the right-hand side
of Equation (3) for a given set of total column density, NT, rotational
temperature, Trot, and source filling factor, f. The expected $\hat{N_{u}}$
were evaluated for the parameter space of Trot = 10-500 K, NT = 1014-1017
cm-2, and f between 0.01 and 1.0. To compare the observed $N_{u}$ and the
inferred $\hat{N_{u}}$, we calculate the $\chi^{2}$ as:
$\chi^{2}=\sum(\frac{N_{u}-\hat{N_{u}}}{\delta~{}N_{u}})^{2}$ (5)
where $\delta~{}N_{u}$ is the 1 $\sigma$ error of observed upper-state column
density. Although the $\chi^{2}$ is a good representation of the goodness of
fit, the parameter set with the lowest $\chi^{2}$ may not actually represent
physical parameters very well due to the uncertainties of the observed data.
In order to find a representative parameter set, we compute a weighted mean
and standard deviation for all the parameters, with the weights being the
inverse of the $\chi^{2}$. All the parameter sets where the inferred upper-
level population $\hat{N_{u}}$ corresponds with the observed upper-level
population $N_{u}$ within 3 $\sigma$ are used to compute the weighted means
and standard deviations. The derived rotational temperature, total column
density and filling factor of each component are list in the [3-5] columns of
Table 3. The inferred optical depths of each line transition are listed in the
last six columns of Table 3. The rotational temperatures of D, E, F, G are
estimated to be 42$\pm$34, 92$\pm$74, 51$\pm$23 and 105$\pm$37 K,
respectively. A temperature gradient in the southern core is also revealed as
in the RTD method. The four components D, E, F, G has similar total column
densities as high as 4$\times 10^{16}$ cm-2, about an order of magnitude
higher than those obtained from RTD method, which are mainly due to the small
source filling factor ($<$0.5). The optical depths of H2CS (100,10-90,9) at
the four components are all much larger than one, while the other transitions
are always optically thin except H2CS (102,9-92,8) line at G.
### 4.2 Core properties
In the optically thin case, the total dust and gas masses of the three sub-mm
cores can be obtained with the formula
$M=S_{\nu}D^{2}/\kappa_{\nu}RB_{\nu}(T_{d})$ (Hildebrand , 1983), where
$S_{\nu}$ is the flux at 860 $\micron$, D is the distance, R=0.01 is the mass
ratio of dust to gas, and $\kappa_{\nu}$ is dust opacity per unit dust mass.
$B_{\nu}(T_{d})$ is the Planck function at a dust temperature of Td. We assume
that Td equals the rotational temperature of H2CS. For the northern core,
since only CS (7-6) (upper energy Eu = 65.8 K) and HCN (4-3) (Eu = 42.5 K)
exhibit strong emission lines, we assume Td to be 50 K. Together with the
measurements at centimeter and millimeter wavelengths, Su et al. (2005)
extrapolated the ionized gas emission at mm/submm wavelengths, and found that
the 0.85 mm continuum associated with components D, E, and F are dominated by
thermal dust emission. They have derived opacity index $\beta$ of components E
and F to be 1.2, and 0.8, respectively. For the northern sub-mm core,
${\beta}=1.5$ is assumed. Using the above dust opacity indexes, we adopt
$\kappa_{\nu}$=2.0, 1.8, and 1.5 cm2g-1 for the northern, middle and southern
cores, respectively (Ossenkopf & Henning, 1994). At the distance of 5.7 kpc ,
we get the total dust and gas masses for these three cores, and list all the
parameters in Table 1. The deduced masses for the northern, middle and
southern cores are 13, 30, 165 M☉, respectively. The column density of H2 are
1.2$\times 10^{24}$ and 2.1$\times 10^{24}$ cm-2 for the middle and southern
sub-mm cores, respectively.
### 4.3 Infall properties in the middle core
In the middle core, both CS(7-6) and HCN(4-3) emission exhibits ”blue profile”
feature, indicating infall motions of the gas envelope toward the central star
(Keto, Ho,& Haschick, 1988; Zhou et al., 1993; Zhang, Ho, & Ohashi, 1998; Wu &
Evans, 2003; Wu et al., 2005, 2007; Fuller, Williams, & Sridharan, 2005;
Wyrowski, 2007; Sun, & Gao, 2008). The velocity difference (0.9 km s-1)
between the absorption dip in CS (7-6) spectrum (3 km s-1) and the systemic
velocity (2.1 km s-1) is taken as the infall velocity $V_{in}$. Since both HCN
(4-3) and CS (7-6) emissions are not resolved towards the middle core, we
simply take the dust core size as the radius of the infall region, which may
underestimate the infall rate derived below. The kinematic mass infall rate
can be calculated using dM/dt=$4{\pi}R_{in}^{2}nmV_{in}$. n=1.5$\times
10^{7}$cm-3 is the number density of this dust core. Taking Helium into
account, the mean molecular mass m is 1.36 times of H2 molecule mass. The
infall rate calculated is $4.3\times 10^{-3}~{}M_{\odot}\cdot$yr-1. For
comparison, the $V_{in}$ from pure free-infall assumption is also derived with
the formula $V_{in}^{2}=2GM/R_{in}$. The pure free-infall velocity is
$V_{in}=3.6~{}$km s-1 and thus the ”gravitational” mass infall rate is
$1.7\times 10^{-2}~{}M_{\sun}\cdot$yr-1, which is larger than the kinematic
infall rate.
### 4.4 Outflow properties in the southern core
#### 4.4.1 Shock chemistry in the outflow region of the southern core
Observations have suggested that there are important differences in molecular
abundances in different outflow regions (Bachiller et al., 1997; Choi et al.,
2004; Jörgensen, Schöier, & van Dishoeck, 2004; Codella et al., 2005).
Significant abundance enhancements are found in the shocked region for sulfur-
bearing molecules (Bachiller et al., 1997; Jörgensen, Schöier, & van Dishoeck,
2004), and the abundance of HCN in outflow regions is related to atomic carbon
abundance (Choi, 2002). However, previous studies of the chemical impact of
outflows are confined to the well collimated outflows around Class 0 sources,
while such studies especially high resolution studies on massive outflows are
rare (Bachiller et al., 1997; Jörgensen, Schöier, & van Dishoeck, 2004; Arce
et al., 2007).
A red and bright IRAC source is found to be associated with the southern core.
The magnitudes of the IRAC source at 3.6 $\micron$, 4.5 $\micron$ and 5.8
$\micron$ are $10.102\pm 0.093$, $8.361\pm 0.108$ and $7.778\pm 0.302$ mag,
respectively. The [3.6-4.5] color is as large as 1.74, indicating shocked
emission in the southern core (Takami et al., 2010). Maser emissions of NH3,
H2O, OH, and CH3OH, as well as the strong thermal NH3 emissions also uncover
the existence of the shocked gas (Hofner et al., 1994). Outflows can be
revealed from shocked H2 emission probed by the strong and extended emission
at the 4.5 $\micron$ band (Qiu et al., 2008; Takami et al., 2010). Thus the
massive outflow in the southern core of G9.62 complex provides an ideal sample
to study shock chemistry.
The fractional abundance of a certain molecule is defined as
$\chi=N_{T}/N_{H_{2}}$, where $N_{T}$ is the total column density of a
specific molecule and $N_{H_{2}}$ is the H2 column density. Assuming that the
gas is optically thin and the emission fills the beam, the beam-averaged total
column density of a specific molecule can be obtained from:
$N_{T}=2.04\times
10^{20}\frac{\int~{}I(Jy~{}beam^{-1})dv(km~{}s^{-1})Q_{rot}e^{E_{u}/T_{rot}}}{\theta_{a}\theta_{b}(arcsec^{2})g_{I}g_{K}\nu^{3}(GHz^{3})S\mu^{2}(debye^{2})}$
(6)
Assuming that Trot of HC15N equals to that of H2CS and the gas is optically
thin, $N_{T}$ of HC15N is calculated to be $3.0\times 10^{13}$ cm-2 at the
core region. At the galactocentric distance of 3 kpc for G9.62+0.19 (Scoville
et al.1987,Hofner et al.1994), the abundance ratio
[14N]/[15N]$~{}\approx~{}350$ (Wilson and Rood. 1994). Thus the total column
density of HCN at the core region should be $1.1\times 10^{16}$ cm-2.
Therefore, the fractional abundance of HCN relative to H2 at the core region
is $5.2\times 10^{-9}$. HCN appears to be greatly enhanced in the outflow
regions of the L1157 (Bachiller et al., 1997), while has similar abundances in
the outflow region and the ambient cloud of NGC 1333 CIRAS 2A (Jörgensen,
Schöier, & van Dishoeck, 2004). Owing to the lack of a direct estimation of
the H2 column density towards the outflow region, the fractional abundance of
HCN in the outflow region is also assigned to $5.2\times 10^{-9}$ in
calculating the outflow parameters. Since the HC15N emission traces outflowing
gas at much lower velocity than HCN, perhaps HCN could be more enhanced in the
high velocity component. With the possibility of higher opacity and the lack
of direct H2 column density measurement, the derived fractional abundance
perhaps is a lower limit anyway. Su et al. (2007) estimate an HCN abundance of
$\sim 1-2\times 10^{-8}$ in the massive outflow lobes of IRAS 20126+4104,
which is comparable to our estimation here.
Since the blueshifted outflow gas traced by CS (7-6) and HCN (4-3) suffers
self-absorption, the abundance ratios among SO ($8_{7}-7_{7}$), CS (7-6), and
HCN (4-3) were inferred from the beam-averaged spectra taken from the
redshifted outflow lobe. The abundance ratio as a function of flow velocity
(the outflow velocity relative to the systemic velocity) of [CS/SO] is
obtained assuming five different excitation temperatures in the left panel of
Figure 15\. It can be seen that the abundance ratio of [CS/SO] increases with
the excited temperature. At each excitation temperature, the abundance ratio
of [CS/SO] has lower values at flow velocities less than 6 km s-1, and higher
values when Vflow larger than 8 km s-1, whereas the abundance ratio seems to
be constant at flow velocities between 6 km s-1 and 8 km s-1. There are two
reasons for the lower abundance ratio when V${}_{flow}~{}<$ 6 km s-1: first,
the flux missing of CS (7-6) due to the interferometer is more serious than SO
($8_{7}-7_{7}$); second, CS (7-6) may be more optically thick at lower flow
velocities than SO ($8_{7}-7_{7}$). As shown in the P-V diagrams, the emission
region of CS (7-6) is much larger than SO ($8_{7}-7_{7}$) at high velocities.
The higher abundance ratio when V${}_{flow}~{}>$ 8 km s-1 is due to the
smaller filling factor of SO ($8_{7}-7_{7}$) emission. We propose the mean
observed value between 6 km s-1 and 8 km s-1 can represent the actual
abundance ratio of [CS/SO]. Assuming a typical excitation temperature of
Tex=30 K (Wu et al., 2004), the abundance ratio of [CS/SO] at the redshifted
lobe is inferred as 0.7. Nilsson et al. (2000) find that the [SO/CS] abundance
ratios are strongly enhanced in the Orion A and NGC 2071 outflows where the
[SO/CS] ratios are estimated to be about 24 and 2.2, respectively. However,
the [SO/CS] abundance ratio in the outflow of G9.62+0.19 is found to be 1.4,
much lower than that found in Orion A outflow.
As shown in the right panel of Figure 15, the abundance ratio of [CS/HCN]
decreases linearly with the flow velocity. To avoid the missing flux
difficulty, the abundance ratio is calculated at high flow velocities larger
than 7 km s-1. The decreasing of the abundance ratio with velocity is because
that the emission region traced by CS (7-6) is always smaller than HCN (4-3),
leading to smaller filling factor for CS (7-6), which can be verified easily
by comparing the channel maps between CS (7-6) in Figure 11 and HCN (4-3) in
Figure 12 at high velocities. We fitted the observed data with a linear
function, and adopted the value at flow velocity of 10 km s-1 as the actual
abundance ratio of [CS/HCN] in the outflow region, which is [CS/HCN]=1.2.
Since HCN fractional abundance is $5.2\times 10^{-9}$, the fractional
abundances of CS and SO are deduced to be $6.2\times 10^{-9}$ and $8.9\times
10^{-9}$, respectively.
#### 4.4.2 Properties of the bipolar-outflow traced by SO ($8_{7}-7_{7}$)
emission
The SO ($8_{7}-7_{7}$) emission in the southern core shows line wings,
suggesting outflow motions. From the integrated intensity map in Figure 10(c),
we find the outflow lobes revealed by SO ($8_{7}-7_{7}$) emission peak at
different position with different position angle compared with previously
reported H2S ($2_{2,0}-2_{1,1}$) (Gibb, Wyrowski, & Mundy , 2004) and HCO+
(1-0) data (Hofner, Wiesemeyer, & Henning , 2001). But in the same sense, the
blue- and red-lobes revealed by SO overlap to a large extent as well as HCO+
(1-0) and H2S ($2_{2,0}-2_{1,1}$) data, consistent with the argument of the
outflow being viewed pole-on (Hofner, Wiesemeyer, & Henning , 2001).
The total mass of each outflow lobe is given by:
$M_{flow}=1.04~{}\times~{}10^{-4}D^{2}\frac{Q_{rot}e^{E_{u}/T_{rot}}}{\chi\nu^{3}S\mu^{2}}\int\frac{\tau}{1-e^{-\tau}}S_{\nu}dv$
(7)
where Mflow, D, Sν, $\chi$, and $\tau$ are the outflow gas mass in M☉, source
distance in kpc, line flux density in Jy, relative abundance to H2, and
optical depth. The other parameters have the same units as in equation (1).
The fractional abundance of SO is taken as $8.9\times 10^{-9}$ (see
Sec.4.4.1). Assuming an excitation temperature of 30 K and the outflowing gas
is optically thin, the inferred outflow masses are 13 M☉ for each of red and
blueshifted lobes. Thus, the momentum can be calculated by $P=\sum$M(v)dv, and
the energy by $E=\sum{\frac{1}{2}}$M(v)v2dv, where $v$ is the flow velocity.
The derived parameters are listed in Table 4. The momentum and energy of the
red lobe are 82 M${}_{\sun}\cdot$km s-1 and $5.4\times 10^{45}$ erg. For the
blue lobe, the momentum and energy are calculated to be 86 M${}_{\sun}\cdot$km
s-1 and $5.8\times 10^{45}$ erg. The dynamical timescale tdyn is estimated as
R/Vchar, where R ($\sim$ 0.06 pc) is adopted as the mean size of the outflow
lobes assuming a collimation factor of unity, and Vchar ($\sim$ 5.5 km s-1) is
assumed as the mass weighted mean velocity. Thus, the dynamic timescale is
estimated to be $1\times 10^{4}$ year, which may be underestimated due to the
uncertainty of the outflow scale. The mechanical luminosity L, and the mass-
loss rate $\dot{M}$ are calculated as L=E/t, $\dot{M}=P/(tV_{w})$, where the
wind velocity Vw is assumed to be 500 km s-1 (Lamers et al., 1995). The
mechanical luminosity L and the total mass-loss rate are estimated to be 9.3
L☉ and $3.6\times 10^{-5}$ M☉$\cdot$yr-1, respectively.
#### 4.4.3 Very high-velocity gas detected in CS (7-6) emission
The CS (7-6) emission at the southern core shows ”red-profile” with wide
wings. We take $6.2\times 10^{-9}$ as the fractional abundance of CS relative
to H2 along the outflow lobes. Assuming Tex = 30 K, we derive the parameters
for the CS outflow (Table 4) with the same method used for SO ($8_{7}-7_{7}$).
The outflow masses at very high velocities (v${}_{flow}~{}>~{}$10 km s-1) are
3.7 M☉ and 5.5 M☉ for the blueshifted and redshifted lobes, respectively. The
momentum and energy of the blueshifted lobe at very high velocities are
calculated to be 47 M${}_{\sun}\cdot$km s-1 and $6.0\times 10^{45}$ erg. For
the redshifted lobe, the momentum and energy at extremely high velocities are
calculated to be 68 M${}_{\sun}\cdot$km s-1 and $8.7\times 10^{45}$ erg, which
are similar to the blueshifted lobe.
#### 4.4.4 Very high-velocity gas detected in HCN (4-3) emission
As discussed before, HCN (4-3) has a velocity extent of at least 60 km s-1,
which traces extremely high-velocity (EHV) gas. Adopting an excited
temperature of 30 K, and an HCN-to-H2 abundance ratio of $5.2\times 10^{-9}$,
the parameters of the outflow are calculated and listed in Table 4. The
outflow mass at very high velocities (v${}_{flow}~{}>~{}$10 km s-1) are 5.2 M☉
and 17.6 M☉ for the blueshifted and redshifted lobes, respectively. The
momentum and energy of the blueshifted lobe at very high velocities are 85
M${}_{\sun}\cdot$km s-1 and $1.4\times 10^{46}$ erg. For the redshifted lobe,
the momentum and energy at very high velocities are 294 M${}_{\sun}\cdot$km
s-1 and $5.5\times 10^{46}$ erg, which are larger than the blueshifted lobe.
#### 4.4.5 Mass-Velocity diagrams
A broken power law, $dM(v)/dv\propto v^{-\gamma}$ usually exhibits in
molecular outflows near young stellar objects (Chandler et al., 1996; Lada, &
Fich, 1996; Ridge, & Moore, 2001; Su, Zhang, & Lim, 2004; Qiu et al., 2007,
2009). The slope, $\gamma$, typically ranging from 1 to 3 at low outflow
velocities, and often steepens at velocities larger than 10 km s-1 — with
$\gamma$ as large as 10 in some cases (Arce et al., 2007). Assuming optically
thin, the mass-velocity diagrams of the outflow at the southern core of
G9.62+0.19 complex are shown in Figure 16. SO ($8_{7}-7_{7}$), CS (7-6), HCN
(4-3) results were all used in the mass spectra. We calculate the outflow mass
traced by CS (7-6) and HCN (4-3) from Vflow of 10 km s-1 to avoid the
absorption of the spectra. Instead of broken power law appearance, the mass-
velocity diagram of blueshifted lobe can be well fitted by a single power law
with a power indexes of $2.28\pm 0.23$. The mass-velocity diagram of
redshifted lobe can be well fitted by a single power law with a power indexes
of $1.70\pm 0.17$ even though the mass drops more rapidly after 25 km s-1. As
marked by the dashed ellipse in the right panel, the outflow mass revealed by
CS (7-6) is much lower than that revealed by HCN (4-3) at very high
velocities. Despite the CS data, the mass-velocity diagram of redshifted lobe
at velocities smaller than 25 km s-1 can be fitted by a single power law with
a much smaller power indexes of $1.08\pm 0.09$. However, no significant slope
changes are found in both the red- and blue-shifted lobes of the outflow at
the southern core, which are very different from those previous works.
### 4.5 Different evolutionary stages of the three dust cores
The northern core has the smallest diameter and mass among the three cores. It
seems likely to be a point source after deconvolution. It is located south of
the nominal radio UC Hii region G9.62+0.19 C. In this region, eight near-IR
sources are detected in a diffuse near-IR nebulosity at the west of the radio
emission peak (Persi et al., 2003). The reddest one c7
(18h06m14.34s,-20$\arcdeg$31$\arcmin$25.0$\arcsec$) is located within
$1\arcsec$ of the radio peak, while the faintest one c8
(18h06m14.42s,-20$\arcdeg$31$\arcmin$27.4$\arcsec$) seems to be associated
with the sub-mm core detected in SMA observation. Source c8 is too faint to be
detected even at H band and also shows no emission at 12.5 $\micron$. In
contrast to the bright, rich molecular spectrum forest in the middle and
southern sub-mm cores, the northern sub-mm core lacks strong molecular
emissions. There is also no other early star forming signature such as masers
associated with it. Since it is with near-IR emission and at the edge of the
UC Hii region G9.62+0.19 C, the northern core may be just a remnant core in
the envelope of UC Hii region G9.62+0.19 C, which needs further observations.
The middle core is associated with the hyper-compact Hii region G9.62+0.19 E
(Garay et al., 1993; Kurtz, & Franco, 2002). OH, H2O, and NH3 (5,5) masers
have been detected near the radio emission peak (Forster & Caswell, 1989;
Hofner et al., 1994; Hofner, & Churchwell, 1996a). Periodic class II methanol
masers are also found in G9.62+0.19 E (van der Walt, Goedhart, & Gaylard,
2009; Goedhart, Gaylard, & van der Walt, 2005; Norris et al., 1993). Methanol
masers are believed to be a good tracer of young massive star forming regions
at stages earlier than relatively evolved UC Hii regions (Longmore et al.,
2007). No infrared source coincides with G9.62+0.19 E (Persi et al., 2003).
Hot molecular CH3CN lines are detected in this region, and a kinematic
temperature of Tk = 108 K was obtained from CH3CN emission with LVG model
(Hofner et al., 1996b), which is coincident with the rotational temperature
(Trot = 92 K) obtained from H2CS emission. A spectra forest including hot
molecular lines, such as CH3OH, is detected towards G9.62+0.20 E, suggesting
this core is in a hot phase. Infall motions are traced by CS (7-6) and HCN
(4-3) lines, indicating active star forming in this region. All above suggest
that G9.62+0.20 E is forming a massive young star.
The 860 $\micron$ dust emission of the southern core peaks at G9.62+0.19 F,
and extends from north to south. A hump structure is found to the southeast of
the emission peak, indicating another possible sub-mm core. The previously
recognized mm/cm cores (G9.62+0.19 D, G) are at the edges of the southern
core. G9.62+0.19 G is a weak radio source (Testi et al., 2000), while
G9.62+0.19 D is consistent with an isothermal UC Hii region excited by a B0.5
star (Hofner et al., 1996b). Weaker radio emission was found at core F. H2O
and OH masers are found across the whole sub-mm core from north to south
(Forster & Caswell, 1989; Hofner, & Churchwell, 1996a). A near-IR source with
large NIR excess is found to be associated with G9.62+0.19 F (Testi et al.,
1998; Persi et al., 2003). With higher resolution observations Linz et al.
(2005) found four near-IR objects (F1-F4) in this core. F4 is with little
emission at K band but becomes redder at longer wavelengths, which seems to
correspond to the bright IRAC source with large excess at 4.5 $\micron$. This
object is the dominating and closest associated source of core F. Core F is
also confirmed to be the driving source of an active outflow. All of above
imply that G9.62+0.19 F is a very young massive star forming region.
### 4.6 Blue excess in high-mass star forming regions
Wu et al. (2007) found that UC Hii regions show a higher blue excess than UC
Hii precursors with the IRAM 30 m telescope. Wyrowski et al. (2006) also
detected large blue excess in UC Hii regions. ”Blue profile” was detected with
CS (7-6) and HCN (4-3) lines in UC Hii region G9.62+0.19 E, while ”red
profile” in hot molecular core G9.62+0.19 F, which coincides with their
argument. The detection of infall signature in G9.62+0.19 E also coincides the
interpretation that material is still accreted during the UC Hii phase (Wu et
al., 2007; Keto, 2002). Around younger cores, the outflow is more active and
cold than UC Hii regions, which leads to more ”red profile”. While in UC Hii
regions, the outflows become weak. The surrounding gas of UC Hii regions is
thermalized and the temperature gradient towards the central star is more
likely to cause ”blue profile”, which results in the higher blue excess than
UC Hii precursors.
## 5 Summary
We have observed the G9.62+0.19 complex with the Submillimeter Array (SMA)
both in the 860 $\micron$ continuum and molecular lines emission. The main
results of this study are as follows:
1\. Dust continuum at 860 $\micron$ reveals three sub-mm cores in G9.62+0.19
star forming complex. With H2CS as the rotational temperature prober, the
temperatures of E and F are estimated to be 92$\pm$74 and 51$\pm$23 K,
respectively. The mass calculated are 13, 30, and 165 M☉ for the northern,
middle and southern core.
2\. In the middle core, HCN (4-3) and CS (7-6) spectra exhibit infall
signature. The infall rate calculated is $4.3\times 10^{-3}$
M${}_{\sun}\cdot$yr-1. The detection of infall signature in G9.62+0.19 E
coincides the interpretation that material is still accreted after the onset
of the UC Hii phase (Wu et al., 2007).
3\. In the southern core, high-velocity gas is detected in SO ($8_{8}-7_{7}$),
CS (7-6) and HCN (4-3) lines. A bipolar-outflow with a total mass about 26 M☉
and a mass-loss rate of $3.6\times 10^{-5}$ M${}_{\sun}\cdot$yr-1 is revealed
in SO ($8_{8}-7_{7}$) line wing emission. G9.62+0.19 F is confirmed to be the
driving source of the outflows in the southern sub-mm core. The abundance
ratios of [CS/SO] and [CS/HCN] in the outflow region are found to be 0.7 and
1.2, respectively. The abundance ratio [CS/HCN] decreases with the flow
velocity, indicating smaller outflow regions revealed by CS (7-6) than that
revealed by HCN (4-3). The mass-velocity diagrams of the blueshifted and
redshifted outflow lobes can be well fitted by a single power law. The power
indexes for the blueshifted and redshifted lobes are $2.28\pm 0.23$ and
$1.70\pm 0.17$. No significant slope changes are found in the mass-velocity
diagrams.
4\. The evolutionary sequence of the cm/mm cores in this region are also
analyzed. The northern core may be just a remnant core in the envelope of UC
Hii region G9.62+0.19 C, which needs further observations. The middle core
(G9.62+0.19 E) is in a hyper-compact Hii region. Core G9.62+0.19 F is
confirmed to be a hot molecular core.
5\. The detection of blue profiles at the hyper-compact Hii region E and the
red profiles at the hot molecular core F supports the results of single-dish
observations that UC Hii regions have a higher blue excess than their
precursors.
## Acknowledgment
We are grateful to the SMA staff making the observations. This work is funded
by Grants of NSFC No 10733030 and 10873019.
## References
* Arce et al. (2007) Arce, H. G., Shepherd, D., Gueth, F., Lee, C.-F., Bachiller, R., Rosen, A., Beuther, H., 2007, Protostars and Planets V, p. 245
* Bachiller et al. (1997) Bachiller, R., Perez Gutiérrez, M. 1997, ApJ, 487, L93
* Blake et al. (1987) Blake, G. A., Sutton, E. C., Masson, C. R., & Phillips, T. G., 1987, ApJ, 315, 621
* Bonnell et al. (1998) Bonnell, I. A., Bate, M. R., & Zinnecker, H., 1998, MNRAS, 298, 93
* Cesaroni et al. (1994) Cesaroni, R., Churchwell, E., Hofner, P., Walmsley, C. M., & Kurtz, S., 1994, A&A, 288, 903
* Chandler et al. (1996) Chandler, C. J., Terebey, S., Barsony, M., Moore, T. J. T., & Gautier, T. N., 1996, ApJ, 471, 308
* Choi (2002) Choi, M., 2002, ApJ, 575, 900
* Choi et al. (2004) Choi, M., Kamazaki, T., Tatematsu, K., & Panis, J.-F., 2004, ApJ, 617, 1157
* Codella et al. (2005) Codella, C., Bachiller, R., Benedettini, M., Caselli, P., Viti, S., & Wakelam, V., 2005, MNRAS, 361, 244
* Cummins, Linke, &Thaddeus (1986) Cummins, S. E., Linke, R. A., Thaddeus, P., 1986, ApJS, 60, 819
* Forster & Caswell (1989) Forster, J. R. & Caswell, J. L., 1989, A&A, 213, 339
* Fuller, Williams, & Sridharan (2005) Fuller, G. A., Williams, S. J., & Sridharan, T. K., 2005, A&A, 442, 949
* Furuya, Cesaroni, & Shinnaga (2011) Furuya, R. S., Cesaroni, R., & Shinnaga, H., 2011, A&A, 525, 72
* Garay et al. (1993) Garay, G., Rodriguez, L. F., Moran, J. M., & Churchwell, E., 1993, ApJ, 418, 368
* Gibb, Wyrowski, & Mundy (2004) Gibb, A. G., Wyrowski, F., & Mundy, L. G., 2004, ApJ, 616, 301
* Goedhart, Gaylard, & van der Walt (2005) Goedhart, S., Minier, V., Gaylard, M. J., & van der Walt, D. J., 2005, MNRAS, 356, 839
* Goldsmith, & Langer (1999) Goldsmith, P. F., & Langer, W. D. 1999, ApJ, 517, 209
* Hildebrand (1983) Hildebrand, R. H. 1983, QJRAS, 24, 267
* Hofner et al. (1994) Hofner, P., Kurtz, S., Churchwell, E., Walmsley, C. M., & Cesaroni, R., 1994, ApJ, 429, L85
* Hofner, & Churchwell (1996a) Hofner, P., & Churchwell, E., 1996a, A&AS, 120, 283
* Hofner et al. (1996b) Hofner, P., Kurtz, S., Churchwell, E., Walmsley, C. M., & Cesaroni, R., 1996b, ApJ, 460, 359
* Hofner, Wiesemeyer, & Henning (2001) Hofner, P., Wiesemeyer, H., & Henning, T., 2001, ApJ, 549, 425
* Jiang et al. (2005) Jiang, Z., Tamura, M., Fukagawa, M., Hough, J., Lucas, P., Suto, H., Ishii, M., Yang, J., 2005, Nature, 437, 112
* Jörgensen, Schöier, & van Dishoeck (2004) Jörgensen, J. K., Hogerheijde, M. R., Blake, G. A., van Dishoeck, E. F., Mundy, L. G., Schöier, F. L., 2004, A&A, 415, 1021
* Keto, Ho,& Haschick (1988) Keto, E. R., Ho, P. T. P., & Haschick, A. D., 1988, ApJ. 324, 920
* Keto (2002) Keto, E., 2002, ApJ, 280, 580
* Klaassen, & Wilson (2007) Klaassen, P. D., & Wilson, C. D., 2007, ApJ, 663, 1092
* Kurtz, & Franco (2002) Kurtz, S., & Franco, J., 2002, RMxAC, 12, 16
* Lada, & Fich (1996) Lada, C. J. & Fich M., 1996, ApJ, 459, 638
* Lamers et al. (1995) Lamers, H. J. G. L. M., Snow, T. P., Lindholm, D. M. 1995, ApJ, 455, 269
* Linz et al. (2005) Linz, H., Stecklum, B., Henning, T., Hofner, P., Brandl, B., 2005, A&A, 429, 903
* Liu, et al. (2002) Liu, S., Girart, J. M., Remijan, A., & Snyder, L. E. 2002, ApJ, 576, 255
* Liu et al. (2011) Liu, T., Wu, Y., Zhang, Q., Ren, Z., Guan, X., & Zhu, M., 2011, ApJ, 727, 1
* Longmore et al. (2007) Longmore, S. N., Burton, M. G., Barnes, P. J., Wong, T., Purcell, C. R., & Ott, J., 2007, IAUS, 242, 125
* Mardones et al. (1997) Mardones, D., Myers, P. C., Tafalla, M., Wilner, D. J., Bachiller, R., & Garay, G., 1997, ApJ, 489, 719
* Nilsson et al. (2000) Nilsson, A., Hjalmarson, ${\AA}$., Bergman, P., Millar, T. J., 2000, A&A, 358, 257
* Norris et al. (1993) Norris, R. P., Whiteoak, J. B., Caswell, J. L., Wieringa, M. H., & Gough, R. G., 1993, ApJ, 412, 222
* Ossenkopf & Henning (1994) Ossenkopf, V., Henning, T. 1994, A&A, 291, 943
* Patel et al. (2005) Patel, N. A., et al., 2005, Nature, 437, 109
* Persi et al. (2003) Persi, P., Tapia, M., Roth, M., Marenzi, A. R., Testi, L., & Vanzi, L., 2003, A&A, 397, 227
* Qin et al. (2008a) Qin, S., Wang, J., Zhao, G., Miller, M., & Zhao, J., 2008a, A&A, 484, 361
* Qin et al. (2008b) Qin, S., et al., 2008b, ApJ, 677, 353
* Qin et al. (2008c) Qin, S.-L., Huang, M., Wu, Y., Xue, R., Chen, S., 2008c, ApJ, 686, L21
* Qin et al. (2010) Qin, S.-L., Wu, Y., Huang, M., Zhao, G., Li, D., Wang, J.-J., Chen, S., 2010, ApJ, 711, 399
* Qiu et al. (2007) Qiu, K., Zhang, Q., Beuther, H., & Yang, J., 2007, ApJ, 654, 361
* Qiu et al. (2008) Qiu, Keping., 2008, ApJ, 685, 1005
* Qiu et al. (2009) Qiu, K., Zhang, Q., Wu, J., & Chen, H.-R., 2009, ApJ, 696, 66
* Remijan et al. (2004) Remijan, A., Sutton, E. C., Snyder, L. E., Friedel, D. N., Liu, S.-Y., & Pei, C.-C., 2004, ApJ, 606, 917
* Ridge, & Moore (2001) Ridge, N. A., & Moore, T. J. T., 2001, A&A, 378, 495
* Sault et al. (1995) Sault, R. J., Teuben, P. J., & Wright, M. C. H. 1995, in ASP Conf. Ser. 77, Astronomical Data Analysis Software and Systems IV, ed. R. A. Shaw, H. E. Payne, & J. J. E. Hayes (San Francisco, CA: ASP), 433
* Shu, Adams & Lizano (1987) Shu, F. H., Adams, F. C., & Lizano, S., 1987, ARA&A, 25, 23
* Sridharan, Williams, & Fuller (2005) Sridharan, T. K., Williams, S. J., & Fuller, G. A., 2005, ApJ, 631, L73
* Su, Zhang, & Lim (2004) Su, Y.-N., Zhang, Q., & Lim, J., 2004, ApJ, 604, 258
* Su et al. (2005) Su Y.-N., Liu S.-Y., Lim J., Chen H.-R., 2005, in Protostars and Planets V Submillimeter Observations of the High- Mass Star Forming Complex G9.62+0.19. pp 8336$-+$
* Su et al. (2007) Su, Y-N., Liu, S.-Y.., Chen, H.-R., Zhang, Q., Cesaroni, R., 2007, ApJ, 671,571
* Sun, & Gao (2008) Sun, Y., & Gao, Y., 2008, MNRAS, 392, 170
* Takami et al. (2010) Takami, M., Karr, J. L., Koh, H., Chen, H.-H., Lee, H.-T., 2010, ApJ, 720, 155
* Testi et al. (1998) Testi, L., Felli, M., Persi, P., & Roth, M., 1998, A&A, 329, 233
* Testi et al. (2000) Testi, L., Hofner, P., Kurtz, S., & Rupen, M., 2000, A&A, 359, L5
* Turner et al. (1991) Turner B. E., 1991, ApJS, 76, 617
* van der Walt, Goedhart, & Gaylard (2009) van der Walt, D. J., Goedhart, S., & Gaylard, M. J., 2009, MNRAS, 298, 961
* Van der Tak & Menten (2005) Van der Tak, F. F. S., & Menten, K. M., 2005, A&A, 437, 947
* Wang et al. (2010) Wang, K.-S., Kuan, Y.-J., Liu, S.-Y., Charnley, S. B., 2010, ApJ, 713, 1192
* Wolfire, & Cassinelli (1987) Wolfire, M. G., & Cassinelli, J. P., 1987, ApJ, 319, 850
* Wu & Evans (2003) Wu, J., & Evans N. J. II., 2003, ApJ, 592, L79
* Wu et al. (2004) Wu, Y., Wei, Y., Zhao, M., Shi, Y., Yu, W., Qin, S., Huang, M., 2004, A&A, 426, 503
* Wu et al. (2005) Wu, Y., Zhu, M., Wei, Y., Xu, D., Zhang, Q., & Fiege, J. D., 2005, ApJ, 628, L57
* Wu et al. (2007) Wu, Y., Henkel, C., Xue, R., Guan, X., & Miller, M., 2007, ApJ, 669, L37
* Wu et al. (2009) Wu, Y., Qin, S.-L., Guan, X., Xue, R., Ren, Z., Liu, T., Huang, M., Chen, S., 2009, ApJ, 697, L116
* Wyrowski et al. (2006) Wyrowski, F., Heyminck, S., Gsten, R., & Menten, K. M., 2006, A&A, 454, L95
* Wyrowski (2007) Wyrowski, F., 2007, ASPC, 387, 3W
* Zhang, Ho, & Ohashi (1998) Zhang, Q., Ho, P. T. P., & Ohashi, N., 1998, ApJ, 494, 636
* Zhang et al. (2005) Zhang, Q, Hunter, T. R., Brand, J., Sridharan, T. K., Cesaroni, R., Molinari, S., Wang, J., Kramer, M., 2005, ApJ, 625, 864
* Zhou et al. (1993) Zhou, S., Evans, N. J. II., Koempe, C., & Walmsley, C. M., 1993, ApJ, 404, 232
* Zinnecker & Yorke (2007) Zinnecker, H., & Yorke, H. W., 2007, ARA&A, 45, 481
Figure 1: The 860 $\micron$ continuum emission image. The contour levels are
from 0.03 Jy beam-1 (3$\sigma$) in steps of 0.06 Jy beam-1 (6$\sigma$). The
known cm and mm continuum components (Testi et al., 2000) of B, C, D, E, F, G,
H, and I are marked by plus signs. Water masers (Hofner, & Churchwell, 1996a)
are marked by open squares and methanol masers (Norris et al., 1993) by
triangles. The near-IR sources (Persi et al., 2003; Testi et al., 1998; Linz
et al., 2005) are marked by filled circles. IRAC sources are marked with
asterisks. Figure 2: The full LSB and USB spectra in the UV domain over the
shortest baseline. The strongest lines are identified and labeled on the
plots. Figure 3: Integrated intensity maps of four transitions of H2CS at the
middle (upper panels) and southern cores (lower panels). The known cm and mm
continuum components are marked by plus signs as in the continuum map. The
contour levels in all the panels are from 3$\sigma$ in steps of 3$\sigma$. The
rms levels are 0.3, 0.3, 0.3 and 0.2 Jy beam${}^{-1}\cdot$km s-1 for H2CS
(100,10-90,9) in panels (a) and (e), H2CS (102,8-92,7) in panels (b) and (f),
H2CS (103,7-93,6) in panels (c) and (g), and H2CS (105-95) in panels (d) and
(h), respectively. Figure 4: The P-V diagram (left) and First moment map
(right) of H2CS (102,8-92,7) emission at the middle core. (a) The contours of
the P-V diagram are from 0.6 to 1.4 in steps of 0.2 Jy beam-1 (1$\sigma$). (b)
contour plot of H2CS (102,8-92,7) integrated intensity image overlayed on the
first moment map. The contours are from 0.9 (3$\sigma$) in steps of 0.9 Jy
beam${}^{-1}\cdot$km s-1. The First moment map is constructed from the data
after imposing a cutoff of 3$\sigma$. Figure 5: Spectra and integrated
intensity maps of HC15N (4-3) (upper panels) and SO ($8_{7}-7_{7}$) (lower
panels) at the middle core. The systemic velocity is marked with the thick
vertical dashed lines at the spectra panels. The known cm and mm continuum
components of C, and E are marked by plus signs at the integrated maps as the
continuum map. (a) the beam-averaged spectrum of HC15N (4-3) at E, (b) the
integrated intensity map of HC15N (4-3). The contour levels are -1.2
(6$\sigma$), 1.2, 2.4, 4.2, 6.6, 9.6 Jy beam${}^{-1}\cdot$km s-1, (c) the
beam-averaged spectrum of SO ($8_{7}-7_{7}$) at E. (d) the integrated
intensity map of SO ($8_{7}-7_{7}$). The contour levels are -1.2 (6$\sigma$),
1.2, 2.4, 4.2, 6.6, 9.6, 13.2, 17.4, 22.2 Jy beam${}^{-1}\cdot$km s-1. Figure
6: Beam-averaged spectra and Position-Velocity (P-V) diagrams of HCN (4-3)
(upper panels) and CS (7-6) (lower panels) at the middle core. The P-V
diagrams are cut along a position angle of 0$\arcdeg$. (a) the beam-averaged
spectrum of HCN (4-3) at E, (b) the P-V diagram of HCN (4-3). The contour
levels are -1.5 (5$\sigma$), -0.9, 0.9, 1.5, 2.1, 2.7, 3,3, 3.9, 4.5 Jy
beam-1, (c) the beam-averaged spectrum of CS (7-6) at E. (d) the P-V diagram
of CS (7-6). The contour levels are -1.5 (5$\sigma$), -0.9, 0.9, 1.5, 2.1,
2.7, 3,3, 3.9, 4.5 Jy beam-1. Figure 7: Integrated intensity maps of HCN (4-3)
(left) and CS (7-6) (right) at the middle core. The contour levels in both
maps are from 1.5 (5$\sigma$) in steps of 3 Jy beam${}^{-1}\cdot$km s-1. HCN
(4-3) is integrated from -3 to 7 km s-1, while CS (7-6) from -1 to 6 km s-1
Figure 8: Averaged spectra of SO ($8_{7}-7_{7}$) (upper-left), HC15N (4-3)
(lower-left), HCN (4-3) (upper-right) and CS (7-6) (lower-right) at the
southern core. The spectra of SO ($8_{7}-7_{7}$) and HC15N (4-3) are averaged
over a region of 4$\arcsec$, while HCN (4-3) and CS (7-6) are averaged over a
region of 6$\arcsec$. HCN $\nu 2=1$ (4-3) emission is marked by the arrow in
the upper-right panel, which can be clearly distinguished from the red wing of
HCN (4-3). Figure 9: The integrated intensity maps of HC15N (4-3) (left panel)
and SO ($8_{7}-7_{7}$) (right panel) at the southern core. To avoid the
influence of outflow motions, both the maps are integrated from 2 km s-1 to 8
km s-1. The contour levels are (a) -1.2 (6$\sigma$), 1.2, 2.4, 4.2, 6.6, 9.6,
13.2, 17.4 Jy beam${}^{-1}\cdot$km s-1 for HC15N (4-3), (b) -1.2 (6$\sigma$),
1.2, 2.4, 4.2, 6.6, 9.6, 13.2, 17.4, 22.2, 27.6, 33.6 Jy beam${}^{-1}\cdot$km
s-1 for SO ($8_{7}-7_{7}$) Figure 10: P-V diagrams and integrated intensity
maps of SO ($8_{7}-7_{7}$) (upper panels), and HC15N (4-3) (lower panels) at
the southern core. The P-V diagrams are cut along N-S direction. The vertical
solid line in P-V diagrams labels the systemic velocity. The dashed and solid
contours in the right panels show the red- and blue-shifted emission,
respectively. The integral velocity intervals are marked by thick dashed lines
in the P-V diagrams. For both SO ($8_{7}-7_{7}$) and HC15N (4-3), the blue-
shifted emission is integrated from -4 km s-1 to 0 km s-1, while the red-
shifted emission from 10 km s-1 to 14 km s-1 in the integrated intensity maps.
(a) P-V diagram of SO ($8_{7}-7_{7}$). The contours are from 0.6 (3$\sigma$)
in steps of 0.6 Jy beam-1. (b) P-V diagram of HC15N (4-3). The contours are
from 0.6 (3$\sigma$) in steps of 0.4 Jy beam-1. (c) Integrated intensity maps
of SO ($8_{7}-7_{7}$) at line wings. The contours are from 1 (5$\sigma$) in
steps of 1 Jy beam${}^{-1}\cdot$km s-1 for both red- and blue-shifted
emission. (d) Integrated intensity maps of HC15N (4-3) at red wing. The
contours are 0.6 (3$\sigma$), 1.2, 2, 3 Jy beam${}^{-1}\cdot$km s-1. Figure
11: CS (7-6) channel maps at the southern core, which is smoothed to a
velocity resolution of 3 km s-1. The contours are -0.6 (3$\sigma$), 0.6, 1.2,
2.4, 4.8, 7.2, 9.6 Jy beam-1. Figure 12: HCN (4-3) channel maps at the
southern core, which is smoothed to a velocity resolution of 4 km s-1. The
contours are -0.6 (3$\sigma$), 0.6, 1.2, 2.4, 3.6, 4.8, 7.2 Jy beam-1. Figure
13: P-V diagrams and integrated intensity maps of HCN (4-3) (upper panels),
and CS (7-6) (lower panels) at the southern core. The P-V diagrams are cut
along N-S direction. The vertical solid line in P-V diagrams labels the
systemic velocity. The dashed and solid contours in the right panels show the
red- and blue-shifted emission, respectively. The blue- and red-shifted
emission in the integrated maps are integrated from -12 km s-1 to -5 km s-1
and 15 km s-1 to 22 km s-1, respectively in (b) and (e) panels. (a) P-V
diagram of HCN (4-3). The contours are from 0.9 (3$\sigma$) in steps of 1.2 Jy
beam-1. (b) Integrated intensity maps of HCN (4-3) at line wings. The contours
are 1.5 (5$\sigma$), 4.5, 7.5, 10.5 Jy beam${}^{-1}\cdot$km s-1. (c) The
integrated intensity maps of HCN (4-3) at extremely high velocities. The blue-
and red-shifted emission in the integrated maps are integrated from -20 km s-1
to -13 km s-1 and 23 km s-1 to 39 km s-1, respectively. The contours are from
1.5 (5$\sigma$) in steps of 3 Jy beam${}^{-1}\cdot$km s-1 for both blue- and
red-shifted emission. (d) P-V diagram of CS (7-6). The contours are from 0.9
(3$\sigma$) in steps of 1.2 Jy beam-1. (e) Integrated intensity maps of CS
(7-6) at line wings. The contours are 1.5 (5$\sigma$), 4.5, 7.5, 10.5 Jy
beam${}^{-1}\cdot$km s-1
Figure 14: Population diagrams of H2CS towards four cm/mm cores. The names of
the cores are labeled on the upper-right corner of each panel. Open circles in
blue represent the observed data. The vertical bars present 3$\sigma$ errors
of ln(Nu/gu) due to the uncertainties of integrated intensities. The solid
line shows the linear least-squares fitting using the Rotational Temperature
Diagram method. Crosses in red mark the weighted mean results from Population
Diagram analysis. The inferred parameters from the Population Diagram analysis
are presented on the upper-right corners of each panel.
Figure 15: Abundance ratios of [CS/SO] (left) and [CS/HCN] (right) versus flow
velocity along the redshifted lobe. We range the excitation temperature from
10 K to 50 K to derive the abundance ratios of [CS/SO]. The excitation
temperature in calculation of abundance ratios of [CS/HCN] is assumed to be 30
K. The solid line in the right panel is the linear least-squares fitting, and
the fitting results are presented in the upper-right corner.
Figure 16: Mass-Velocity relationships for the outflow lobes. Left :
blueshifted lobe; right: redshifted lobe. The solid lines in both panels show
the power law fit towards all the data. The dashed line in the right panel
shows the power law fit towards the HCN and SO data up to Vflow = 25 km s-1.
The fitting results are presented in the lower-left corners.
Table 1: Parameters of 860 $\micron$ continuum emission | R.A. | Decl. | Deconvolution sizes | Ipeak | Sν | TdaaThe dust temperature is assumed to be the same as the rotational temperature of H2CS transitions | $\beta$aaThe dust temperature is assumed to be the same as the rotational temperature of H2CS transitions | Mass | N${}_{H_{2}}$
---|---|---|---|---|---|---|---|---|---
Name | (J2000) | (J2000) | ($\arcsec~{}\times~{}\arcsec$) | (Jy beam-1) | (Jy) | (K) | | (M☉) | ($10^{24}$ cm-2)
Northern core | 18:06:14.447 | -20:31:28.253 | Point source | 0.20$\pm$0.02 | 0.26 | 50 | 1.5 | 13 |
Middle core | 18:06:14.668 | -20:31:31.830 | $1.48\arcsec\times 1.29\arcsec$ (P.A.=$-37.8\arcdeg$) | 0.76$\pm$0.04 | 1.07 | 92 | 1.2 | 30 | 1.2
Southern core | 18:06:14.889 | -20:31:40.149 | $4.71\arcsec\times 1.26\arcsec$ (P.A.=$-20.2\arcdeg$) | 0.95$\pm$0.12 | 2.52 | 51 | 0.8 | 165 | 2.1
Table 2: Observed parameters of the lines
Molecule | Transition | Frequency | Eu | rms | VlsrbbThe opacity index $\beta$ is obtained from Su et al. (2005) | IntensitybbThe Vlsr, Intensity and FWHM of each transition are derived from single gaussian fit towards the beam-averaged spectra. | FWHMbbThe Vlsr, Intensity and FWHM of each transition are derived from single gaussian fit towards the beam-averaged spectra.
---|---|---|---|---|---|---|---
| | (GHz) | (K) | (Jy beam-1) | (km s-1) | (Jy beam-1) | (km s-1)
| | | | | D | E | F | G | D | E | F | G | D | E | F | G
H2CS | 100,10-90,9 | 342.946 | 90.6 | 0.3 | 5.5$\pm$0.2 | 2.7$\pm$0.2 | 5.9$\pm$0.2 | 6.0$\pm$0.5 | 1.9$\pm$0.3 | 1.4$\pm$0.2 | 1.7$\pm$0.1 | 0.9$\pm$0.2 | 2.7$\pm$0.4 | 3.1$\pm$0.5 | 4.3$\pm$0.6 | 3.8$\pm$1.1
| 102,9-92,8ccfootnotemark: | 343.322 | 143.3 | 0.3 | 4.0$\pm$0.5 | 2.2$\pm$1.1 | | 4.9$\pm$0.3 | 0.9$\pm$0.2 | 1.4$\pm$0.3 | | 1.1$\pm$0.3 | 4.0$\pm$0.6 | 4.4$\pm$1.6 | | 1.9$\pm$0.6
| 102,8-92,7 | 343.813 | 143.3 | 0.3 | 6.0$\pm$0.3 | 2.5$\pm$0.4 | 4.9$\pm$0.5 | | 1.3$\pm$0.2 | 0.9$\pm$0.1 | 0.8$\pm$0.2 | | 3.0$\pm$0.7 | 5.5$\pm$0.9 | 3.9$\pm$1.3 |
| 103,8-93,7ddBlended with H2CS (103,7-93,6) | 343.410 | 209.1 | 0.3 | 6.2$\pm$0.8 | 2.4$\pm$0.7 | 5.0$\pm$0.3 | | 1.1$\pm$0.7 | 1.1$\pm$0.1 | 2.1$\pm$0.3 | | 3.3$\pm$0.7 | 4.1$\pm$0.5 | 2.7$\pm$0.6 |
| 103,7-93,6eeBlended with H2CS (103,8-93,7) | 343.414 | 209.1 | 0.3 | 5.8$\pm$2.8 | 2.6$\pm$0.2 | 5.5$\pm$0.3 | 5.8$\pm$0.2 | 0.7$\pm$0.2 | 2.0$\pm$0.2 | 2.4$\pm$0.3 | 1.5$\pm$0.3 | 5.9$\pm$0.8 | 4.0$\pm$0.5 | 2.9$\pm$0.6 | 2.3$\pm$0.5
| 105,6/5-95,5/4ffThe two transitions of H2CS (105,6-95,6) and (105,6-95,6) have the same frequency, line strength and permanent dipole moment. Therefore they has same contributions to the observed line profile. | 343.203 | 419.2 | 0.2 | | 1.8$\pm$0.4 | 5.9$\pm$0.3 | 5.9$\pm$0.2 | | 0.6$\pm$0.2 | 0.7$\pm$0.2 | 0.9$\pm$0.2 | | 3.2$\pm$1.0 | 2.4$\pm$0.8 | 1.5$\pm$0.4
SO | 88-77 | 344.311 | 87.5 | 0.2 | 4.9$\pm$0.1 | 2.2$\pm$0.1 | 5.0$\pm$0.1 | 4.4$\pm$0.1 | 2.9$\pm$0.1 | 4.0$\pm$0.1 | 5.5$\pm$0.1 | 3.5$\pm$0.1 | 3.9$\pm$0.2 | 5.3$\pm$0.2 | 9.0$\pm$0.2 | 8.2$\pm$0.3
HC15N $\nu$=0 | 4-3 | 344.200 | 41.3 | 0.2 | 6.1$\pm$0.4 | 2.6$\pm$0.1 | 5.2$\pm$0.1 | 4.8$\pm$0.3 | 0.6$\pm$0.1 | 2.1$\pm$0.1 | 2.8$\pm$0.1 | 1.1$\pm$0.1 | 3.6$\pm$1.0 | 4.8$\pm$0.3 | 8.0$\pm$0.3 | 7.5$\pm$0.8
CS | 7-6 | 342.883 | 65.8 | 0.3 | | | | | | | | | | | |
HCN $\nu$=0 | 4-3 | 354.505 | 42.5 | 0.3 | | | | | | | | | | | |
aafootnotetext: Not all the detected lines are listed in this table. The
others will be presented in another paper.
bbfootnotetext: Blended with H${}_{2}^{13}$CO (51,4-41,4) at 343.325713 GHz.
Table 3: The physical parameters of H2CS transitions obtained with Rotational
Temperature Diagram (RTD) method and Population Diagram (PD) analysis
Core | RTD | | PD
---|---|---|---
| Trot (K) | Ntot (1015 cm-2) | | Trot (K) | Ntot (1016 cm-2) | f | $\tau$
| | | | | | | (100,10-90,9) | (102,9-92,8) | (102,8-92,7) | (103,8-93,7) | (103,7-93,6) | (105,6/5-95,6/4)
D | 43$\pm$9 | 3.8$\pm$2.9 | | 42$\pm$34 | 4.2$\pm$2.9 | 0.46$\pm$0.24 | 6.4$\pm$4.4 | 0.7$\pm$0.5 | 0.9$\pm$0.6 | 0.4$\pm$0.5 | 0.2$\pm$0.3 |
E | 83$\pm$21 | 2.5$\pm$1.6 | | 92$\pm$74 | 3.6$\pm$3.0 | 0.26$\pm$0.23 | 4.1$\pm$4.3 | 0.7$\pm$0.7 | 0.6$\pm$0.6 | 0.7$\pm$0.8 | 0.7$\pm$0.8 | 0.1$\pm$0.1
F | 83$\pm$7 | 2.6$\pm$0.6 | | 51$\pm$23 | 4.0$\pm$2.9 | 0.34$\pm$0.23 | 3.9$\pm$3.1 | | 1.1$\pm$0.9 | 1.1$\pm$1.2 | 1.1$\pm$1.2 | 0.0$\pm$0.1
G | 91$\pm$17 | 1.3$\pm$0.7 | | 105$\pm$37 | 3.7$\pm$3.1 | 0.12$\pm$0.18 | 2.5$\pm$2.7 | 2.4$\pm$2.3 | | | 2.4$\pm$2.1 | 0.3$\pm$0.3
aafootnotetext: The rotational temperature and total column density of H2CS
transitions derived from RTD analysis are presented in the second and third
columns, while those derived from PD analysis are shown in the forth and fifth
columns. The sixth column gives the filling factor of each source inferred
from PD analysis. The last six columns exhibit the optical depth of each
transition using PD analysis.
Table 4: Outflow parameters of the southern core Molecule | Velocity interval | M | P | E
---|---|---|---|---
| (km s-1) | (M☉) | (M${}_{\sun}~{}\cdot~{}km~{}s^{-1}$) | ($10^{45}$erg)
Component | Blue | Red | Blue | Red | Blue | red | Blue | Red
SO | [-4,0] | [10,14] | 13 | 13 | 86 | 82 | 5.8 | 5.4
CS | [-12,-5] | [15,22] | 3.7 | 5.5 | 47 | 68 | 6.0 | 8.7
HCN | [-20,-5] | [15,39] | 5.4 | 17.6 | 85 | 294 | 14.1 | 54.6
|
arxiv-papers
| 2011-01-08T09:31:31 |
2024-09-04T02:49:16.244323
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tie Liu, Yuefang Wu, Sheng-Yuan Liu, Sheng-Li Qin, Yu-Nung Su, Huei-Ru\n Chen and Zhiyuan Ren",
"submitter": "Tie Liu",
"url": "https://arxiv.org/abs/1101.1580"
}
|
1101.1682
|
# Detecting gross alignment errors in the Spoken British National Corpus
###### Abstract
The paper presents methods for evaluating the accuracy of alignments between
transcriptions and audio recordings. The methods have been applied to the
Spoken British National Corpus, which is an extensive and varied corpus of
natural unscripted speech. Early results show good agreement with human
ratings of alignment accuracy. The methods also provide an indication of the
location of likely alignment problems; this should allow efficient manual
examination of large corpora. Automatic checking of such alignments is crucial
when analysing any very large corpus, since even the best current speech
alignment systems will occasionally make serious errors. The methods described
here use a hybrid approach based on statistics of the speech signal itself,
statistics of the labels being evaluated, and statistics linking the two.
Index Terms: ASPA, HMM, phonetic, transcription, label, segment, alignment,
accuracy, quality assessment, error detection
## 1 Introduction
In linguistic and phonetic research, increasing emphasis is being placed on
the analysis of very large corpora, to give more general, stronger
conclusions. However, the analysis of such databases often requires aligned
textual and audio data, and performing such alignments manually is extremely
time-consuming and hence expensive. Automatic alignment can be performed by
transcribing the words spoken in the corpus and then aligning them to the
speech via standard HMM techniques (e.g. [1]). These Automatic Speech-to-
Phoneme Alignment (ASPA) systems can produce accurate estimates of word and
phoneme locations, but often fail in cases such as:
1. 1.
The speech has been recorded in an environment with non-stationary background
noise, competing un-transcribed speech, distortion, and/or reverberation.
2. 2.
The phonemic transcription of the speech is not accurate. The phonemic
transcription is usually obtained by a dictionary look-up process. Citation
forms are often used, and are often unrealistic, especially when dealing with
spontaneous speech. In natural, unscripted speech, people will sometimes talk
simultaneously so there may be no sequence of words or phonemes that can
correctly represent the audio. Even if the words can be identified and
transcribed individually, they cannot be organised into a simple sequence, and
so require more complex techniques (e.g. [2]), which are impractical for large
corpora.
3. 3.
The word-level transcription is inaccurate and/or inconsistent in its handling
of nonspeech sounds, backchannels, mumbles, and speech-like noises (e.g. dog
barks).
4. 4.
The speech is only available in long continuous recordings: the accumulation
of HMM probabilities over extended periods of time can introduces numerical
errors which distort the alignment process [3].
As a result, automatic alignment of large quantities of spontaneous
unconstrained speech is invariably error-prone. When labelling ``speech in the
wild'' such as the Spoken British National Corpus (BNC) [4], the above four
conditions frequently occur, and lead to failures of the alignment system.
Identifying where such an alignment succeeds and where it fails allows bad
regions to be avoided or realigned.
Previous work on this topic is sparse. Traditionally, aligner accuracy is
assessed by comparison with manually estimated labels in one way or another
[5]. However, the task here is fundamentally different in that we operate
under conditions where any aligner may fail, regardless of whether it is
generally accurate or not. [6] have addressed our problem in a different
context (clean speech that is intended for use in TTS systems) with some
success. They flagged 24% of the segments as suspicious and detected 43% of
the total errors, which would have led to a modest reduction in the effort
required to verify a corpus. [7] looked at the related problem of finding
errors in the lexicon used in the alignment process. Some text-to-speech
system builders may also have used similar ideas (e.g. [8]) but details are
lacking. Some preliminary work was done by [9], using ideas related to our
improbable and unexpected features.
The work described here is part of a project [10] to label speech from the
Spoken BNC, originally recorded on analogue cassette tapes between 1991 and
1994. The data consists of recently-digitised recordings with an associated
word-level transcription of the audio data. The recordings are mostly of
unscripted, spontaneous speech, and include a diverse range of recording
conditions, accents, and microphone positions relative to the speakers. Each
track of the original cassette recordings has been digitised to a single file,
so the data to be aligned is generally just over either 45 or 60 minutes long.
Background noise varies widely, both in terms of amplitude and character
(competing speech, mechanical noise, music and speech from television, radio
or other sources, microphone-handling noise, etc.).
Our earlier work [11, 12] investigated alignment errors by comparing the
alignments produced by a large number of ASPA systems. However, in that work
we were attempting to assess the general ability of aligners to identify
particular phone-transitions, and the general quality of alignments produced
by a specific system, respectively.
## 2 Methods
We used human evaluations of overall alignment quality for each of a set of
recordings, to construct algorithms that should be able to identify suspicious
regions, then to compare these to the human evaluations.
### 2.1 Alignment Procedures
To evaluate our methods, we used the Penn Phonetics Lab Forced Aligner,
P2FA[13] to analyse 46 recordings taken from the BNC, each consisting of a
full audio session recorded on one side of an audio tape without a break.
P2FA is an automatic phonetic aligner based on HTK, and developed at the
Phonetics Laboratory of the University of Pennsylvania. It employs monophone
Gaussian Mixture Model based HMMs which were trained using 39 perceptual
linear prediction (PLP) coefficients. We used the 39 phone set from the
Carnegie-Mellon University Pronouncing Dictionary, CMUdict [14], with the
addition of OH for British English // distinct from //, including lexical
stress marking for the vowels. The current CMU Pronouncing Dictionary was
extended to include all the out-of-vocabulary words and to include a range of
common British English word pronunciations. This extension was performed using
semi-automatic methods by experienced phoneticians.
### 2.2 Human Evaluations
We observed that some of the alignments were much more successful than others,
and we quantified that with a rating procedure. One of the authors examined
5-second long regions, approximately once every 60 seconds throughout the
files, and checked whether that region was correctly aligned at a word level.
The overall score of the file was subjective on a scale between 0 (very poor)
and 10 (very good), but was intended to reflect the number and magnitude of
alignment problems.
### 2.3 Algorithms
We developed five algorithms to indicate potential problems with an alignment,
listed below. Each one takes the aligner output (segment label times, and log-
probabilities) and optionally the audio file, and identifies a list of
suspicious locations. In these descriptions, $L_{i}$ is the aligner's HMM log-
probability value for phoneme instance $i$, $p_{i}$ is the phoneme (i.e. /a/,
/t/, //, …), and $\delta_{i}$ is the duration. The algorithms were developed
without reference to the human evaluations, except for the setting of each
algorithm's threshold.
Unexpected Log(P): This algorithm builds a prediction of the aligner's log-
probability score per unit length from the corpus as a whole (except the data
file under analysis). It then computes the difference between
$L_{i}/\delta_{i}$ and the prediction111We drop phones with $\delta_{i}=0$ or
with $\delta_{i}\geq 1s$ on the grounds that the scaling of $L_{i}$ with
$\delta_{i}$ may not be accurate on these phones.. It operates on the
assumption that most of the audio in the corpus is correctly aligned so that
its predictions correspond to good alignment. Thus, when the aligner is doing
worse than usual, and $L_{i}$ is low, the difference will be substantially
negative. We have observed that when the aligner fails, it typically fails for
a relatively large region: a word or more. To make use of this knowledge, we
smooth the difference over a 1 second long region. When this smoothed
difference of $log(P)$ is more negative than a threshold, the algorithm has
identified a suspicious region.
The predictor for $L_{i}$ starts with the median value for that phoneme,
$\lambda(p_{i})=\mathrm{median}(L_{j}/\delta_{j}\;\mathrm{if}\;p_{j}=p_{i})$.
It then adds in a 5-term linear prediction. The independent variable in the
first term is $log(\delta_{i}/D(p_{i}))$, where $D$ is the median duration of
a phoneme class, and
$D(p_{i})=\mathrm{median}(\delta_{j}\;\mathrm{if}\;p_{j}=p_{i})$. The
remaining four terms capture some information on the phoneme sequence. The
second captures the typical difference in duration between the phone class
under consideration and the previous phone class: $log(D(p_{i})/D(p_{i-1}))$.
The third captures the typical difference in $log(P)$ between the phoneme
class under consideration and the preceding phoneme class:
$\lambda(p_{i})-\lambda(p_{i-1})$. The fourth and fifth are the same, except
they refer to the succeeding phoneme. The predictor (along with the medians)
is trained on phonemes with $0.04s\leq\delta_{i}\leq 0.18s$. The output of
this algorithm becomes the ``unexpected'' feature.
Word Log(P): we consider words with four or more phonemes to avoid variations
due to the vagaries of individual phonemes. The individual $L_{i}$ values for
each phoneme are summed over the respective word, then normalised by dividing
by the word duration. This log probability per unit time provides a stable
indication of the goodness of fit of the observed data to the HMM. The final
stage of of the Word Log(P) method compares the normalised log probability
with a fixed threshold, yielding the ``improbable'' feature. This method is
complemementary to the Unexpected Log(P) method, above, in that it combines
data from a whole word, whereas the Unexpected Log(P) method utilises features
at the individual phoneme level, before smoothing them, i.e. the Unexpected
Log(P) algorithm normalises $L_{i}$ by duration over a larger unit.
In the Unexpected Log(P) algorithm, the threshold was set by experiment, to
identify about 100 suspicious regions per hour on files that had substantial
alignment problems. The Word Log(P) threshold was set to identify a similar
number of events on poorly aligned files, and typically fewer on well-aligned
ones.
Extremes of Amplitude: Many alignment systems will produce erroneous
alignments when several consecutive speech labels bunch-up into a short
region, with the remaining speech labelled as an extended silence. This
misidentification of speech and silence can be detected by examining the
amplitude of signals in each labelled phoneme. A contiguous region of quiet,
of a length comparable to a short word within a segment labelled as speech,
indicates that an error may have occurred. Similarly, an error is likely if
there is a word-length region of high amplitude within a segment labelled as
silence. High amplitude ``silence'' regions should be marked for human
inspection even if they do not contain speech, because they represent high-
amplitude background noise, which is itself a potential cause of problems in
real-world data.
The thresholds for ``quiet'' and ``high amplitude'' were set as the
$3^{\mathrm{rd}}$ and the $97^{\mathrm{th}}$ percentile of the amplitudes
observed over the whole of the recording. The nominal length of a ``short
word'' was set to $\nicefrac{{1}}{{4}}$ second, assuming four phonemes with an
average duration of $\nicefrac{{1}}{{16}}$ second each. These parameters were
estimated by experiment, and chosen to give a relatively small number of false
positives. This algorithm produces two factors (``loud'', and ``quiet'') as it
reports the extremes separately.
Word Duration: this algorithm simply examines the durations of the segments,
and if there are any which are unexpectedly long or short, indicates an error.
It is difficult (simply from their duration) to detect periods of silence
which have become extended or merged due to an alignment error. But the
durations of segments labelled as speech can be of great help.
This algorithm takes a word-based approach to detecting unusual segment
durations. Individual phoneme durations are not reliable indicators because of
variabilities in pronunciation due either to dialect, style of speaking, or
the effects of transient background noise. Thus we take all words with four or
more phonemes, calculate the duration of the region labelled as the word,
normalise it by dividing by the number of phonemes, and compare it with two
thresholds representing the largest and the smallest average phoneme duration.
Any result outside the range $\nicefrac{{1}}{{32}}\;\mathrm{s}<\mbox{mean
duration}<\nicefrac{{1}}{{8}}\;\mathrm{s}$ is flagged. The lower threshold is
just above the minimum possible duration of a phoneme label for our HMMs
(which use 3 left-to-right states per phone). This algorithm yields two
features (``short'' and ``long''), as it reports the two extremes separately.
Duration Mismatch: This algorithm builds a duration model for phonemes and
then measures how far each phoneme222We do not compute a result for phonemes
with $\delta_{i}=0$ or with $\delta_{i}\geq 1s$ on the grounds that they are
outside the range of validity of the duration model, and are almost all
silences, anyway. However, these phones may be used as neighbours in the
computation of other phones. deviates from the model. Regions are identified
as suspicious if the smoothed absolute value of the deviation is large enough
to exceed a threshold.
The duration model predicts the log of the phoneme duration as $d_{i}$. It
starts with a value typical of that phone:
$\Delta_{i}=\mathrm{median}(\log(\delta_{j})\;\mathrm{if}\;p_{i}=p_{j})$. It
then adds on a 25-term linear predictor: the constant term captures a constant
offset from $\Delta_{i}$. Then, twelve terms capture the length of nearby
phonemes relative to their median durations (six neighbours on each side), via
factors that are $\mathrm{q}(D_{i}\delta_{i+k},D_{i+k}\delta_{i}),$ where k
specifies which neighbour333In practice, these first 12 terms amount to a
normalisation of the duration for changes in the local speech rate: the model
adjusts the phone duration by 36% of the average change in nearby durations.
The $\mathrm{q}(a,b)=\begin{Bmatrix}2(a/b)^{0.5}-2,\;\mathrm{if}\;a\leq b\\\
2-\mathrm{q}(b,a),\;\mathrm{else}\end{Bmatrix}$ function is a sigmoid whose
domain is $[0,\infty]$, and it is well-behaved at the endpoints, an important
property since some phoneme durations are zero. The final 12 terms similarly
capture the differences between the typical durations of neighbouring phones.
The are represented by features $\mathrm{q}(D_{i},D_{i+k})$. This duration
model is trained to match $\mathrm{log}(\delta_{i})$ as in the Unexpected
Log(P) algorithm.
Finally, each phone is scored by $S_{i}=|\delta_{i}-d_{i}|/m_{i}$, where
$m_{i}=\mathrm{median}(|\log(\delta_{j})-\Delta_{i}|\;\mathrm{if}\;p_{j}=p_{i})$
is the mean absolute deviation of the log duration. The scores are then
smoothed and thresholded as in the Unexpected Log(P) algorithm. This produces
the ``badlength'' feature.
## 3 Results
Figures 1 and 2 show two examples of regions correctly identified as
misaligned. Many such identifications are correct. We combined the results for
each audio file to give an overall score based on the total number or duration
of suspected of regions. The outputs of the above seven features were then
correlated with the evaluations using a linear regression, via the ``glm''
method of the R software package [15].
The lengths of the audio files varied, as did the amount of speech, so we
devised three ways to define a score, and since we had no clear criterion to
pick one over the other, we computed separate linear regressions with each.
The first, $s^{nd}$, is the number of identified regions divided by the
duration of the audio file; the second, $s^{nw}$, is the number of regions
divided by the number of words in the audio file (as determined from the BNC
transcriptions); and the third, $s^{dd}$, is the total duration of the regions
divided by the duration of the audio file.
Figure 1: Quiet segment error: the ``quiet'' detector has identified a region
(shaded), labelled as the word ``it's''. The vowel is omitted from the
labelled region, and the end of the region extended into silence. The whole
region is very quiet. The top tier is the spectrogram, then phoneme and word
labels, respectively.
Figure 2: Long word error. Displayed as per Figure 1, it shows where the
``long'' detector has identified a region (shaded in the Figure) that was
aligned as a single word, ``wasn't'', but actually included several words.
The distributions of the evaluation variable and the various $s$ variables
were strongly non-Gaussian, with a maxima at one edge. Transforming the
independent variables by raising them to the power 0.3 made the distribution
subjectively more normal, as did squaring the evaluations. However, these
transforms were not more than partially successful, so we elected to regress
both with and without the transforms. This led to four regressions for each
choice of $s$, or 12 regressions in all.
Of those 12 regressions, short was statistically significant on 10 ($P<0.01$),
badlength was significant on 6, loud on 4.444Note that with 12 regressions, we
expect 3 false significances at the $P<0.05$ level and one at the $P<0.01$
level. Therefore, of the 9 significances reported at the $P<0.05$ level (4 for
badlength, 3 for loud, 2 for long), half are probably spurious. For
simplicity, we will ignore them all. At least one of those factors was
significant at the $P<0.01$ level in each regression. Pearson's $R^{2}$
averaged 0.66 with a standard deviation of 0.13 over the regressions,
indicating that a combination of the algorithms was reasonably effective at
matching the human judgements of overall alignment accuracy.
Of the best three fits ($R^{2}=0.81$, $0.81$, $0.87$), short was significant
on each at $P<0.01$, along with badlength and loud once each. Of these, one
used $s^{dd}$ and did not transform the data at all; the other two used
$s^{nw}$ and did not transform the independent variables.
In the second analysis, the correlations between the various algorithms were
calculated for each recording. Figure 3 shows the results for one recording.
Each pair of algorithms scored a point when they identified a pair of regions
within 5 s of each other. We compared the number of such pairs to the number
of accidental pairs that would occur if the algorithm's outputs were
uncorrelated with each other.
Figure 3: Identified regions: each row shows where an algorithm flagged a
potential alignment error. The audio file was rated 8 for overall alignment
success. NB: the improbable detector never fired on this file.
There is a pattern of correlation between some of the detectors in Figure 3.
Several pairs of algorithms were strongly correlated: notably badlength and
unexpected, badlength and long, and long and unexpected. These coincided 3.6
to 5.1 times more often than chance, with statististical significances well
beyond $P<0.001$. Several of the pairs of algorithms were anticorrelated,
notably loud vs. short, loud vs. badlength, and loud vs. long. This is due to
the designs of the algorithms: specifically, loud triggers only on silences,
while the others trigger only on speech sounds. As a result, they never pick
the same phoneme, and only occasionally pick phonemes within 5 s of each
other. The remaining pairs were either nearly independent (loud & unexpected,
badlength & short, long & short, short & unexpected) or did not have enough
occurrences to draw any reliable conclusion.
Duration-based measurements seem to perform best (i.e. short and badlength).
One of the most useful indicators of a gross alignment error was also the
simplest: the short algorithm detected a sequence of phonemes whose durations
were at the minimum allowed by their state topology (here, 3 states or 30
milliseconds). The relative lack of success of the improbable and unexpected
algorithms was unexpected: good and bad alignments have similar distributions
of Log(P) scores.
## 4 Conclusions
We have shown that automated techniques can usefully identify regions of bad
alignment. The regions identified by some of our algorithms correlate well
with human evaluations of the overall quality of the alignment. In general it
appears that the most reliable method for judging the quality is simply to
consider the statistics of the segment durations, either over a fixed time
window, or a linguistic unit (e.g. a word). This research should allow semi-
automatic evaluation of the alignment of large speech corpora, which will be
important for their future use in speech research.
## 5 Acknowledgements
We thank JISC (in the UK) and NSF (in the USA) for their support of Mining a
Year of Speech, under the Digging into Data programme. This work is also
partly supported by the UK ESRC (awards RES-062-23-2566, RES-062-23-1172, and
RES-062-23-1323). We thank John Coleman and Ranjan Sen for the dictionaries,
and John Coleman for his comments.
## References
* [1] Sjölander, K., ``An HMM-based system for automatic segmentation and alignment of speech'', Umeå University, Department of Philosophy and Linguistics, _PHONUM 9_ , pp. 93-96, 2003.
* [2] M. Cooke, J. R. Hershey, S. J. Rennie, ``Monaural speech separation and recognition challenge'', _Computer Speech and Language_ 24(1), January, 2010
* [3] Toth, A. R., ``Forced Alignment for Speech Synthesis Databases Using Duration and Prosodic Phrase Breaks'', in _Proc. 5th ISCA Speech Synthesis Workshop_ , Pittsburgh, June, 2004.
* [4] ``The British National Corpus'', http://www.natcorp.ox.ac.uk/
* [5] De Villiers, E., ``Automatic alignment and error detection for phonetic transcriptions in the African speech technology project databases'', MScEng (Electrical and Electronic Engineering) Thesis, University of Stellenbosch, 2006.
* [6] Barnard, E and Davel, M., ``Automatic error detection in alignments for speech synthesis'', _17th Annual Symposium of the Pattern Recognition Association of South Africa (PRASA)_ , Parys, South Africa, 29 Nov - 1 Dec, pp. 53-56, 2006.
* [7] Davel, M. and Barnard, E., ``Bootstrapping Pronunciation Dictionaries: Practical Issues'', in _Proc. Interspeech 2005_ , Lisboa, Portugal, pp. 1561–1564, 2005.
* [8] Huang, X., Acero, A., Adcock, J., Hon, H.-w., Goldsmith, J., Liu, J., and Plumpe, M., ``Whistler: A Trainable Text-To-Speech System'', _Proc. ICSLP 96_ , Philadelphia, PA, October 1996, pp. 2387-2390.
* [9] Das, R., Izak, J., Yuan, J., Liberman, M., ``Forced Alignment Under Adverse Conditions'', University of Pennsylvania, CIS Dept. Senior Design Project Report, 2010.
* [10] Coleman, J., Liberman, M., Kochanksi, G., Burnard, L., and Yuan. J., ``Mining a Year of Speech,'' _New Tools and Methods for Very-Large-Scale Phonetics Research Workshop_ , University of Pennsylvania, January 28-31, 2011.
* [11] Baghai-Ravary, L., Kochanski, G. and Coleman J.,``Precision of Phoneme Boundaries Derived using Hidden Markov Models'', _Proc. Interspeech 2009_. ISSN 1990-9772, Brighton, UK, pp. 2879-2882, 2009.
* [12] Baghai-Ravary, L., Kochanski, G., and Coleman, J., ``Objective Optimisation of Automatic Speech-to-Phoneme Alignment Systems'', in _Human Language Technologies as a Challenge for Computer Science and Linguistics_ , Vetulani, Z., (ed.), 2009.
* [13] Yuan, J. and Liberman, M., ``Speaker identification on the SCOTUS corpus'', in _Proc. Acoustics '08_ , pp. 5687-5690, 2008.
* [14] Carnegie Mellon University Pronouncing Dictionary, available from http://www.speech.cs.cmu.edu/cgi-bin/cmudict
* [15] ``The R Reference Index'', http://www.r-project.org/
|
arxiv-papers
| 2011-01-09T23:02:52 |
2024-09-04T02:49:16.258494
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ladan Baghai-Ravary, Sergio Grau, Greg Kochanski",
"submitter": "Greg P. Kochanski",
"url": "https://arxiv.org/abs/1101.1682"
}
|
1101.1703
|
# Identifying and Characterizing Nodes Important to Community Structure Using
the Spectrum of the Graph
Yang Wang, Zengru Di, Ying Fan111yfan@bnu.edu.cn Department of Systems
Science, School of Management and Center for Complexity Research, Beijing
Normal University, Beijing 100875, China
###### Abstract
Background: Many complex systems can be represented as networks, and how a
network breaks up into subnetworks or communities is of wide interest.
However, the development of a method to detect nodes important to communities
that is both fast and accurate is a very challenging and open problem.
Methodology/Principal Findings: In this manuscript, we introduce a new
approach to characterize the node importance to communities. First, a
centrality metric is proposed to measure the importance of network nodes to
community structure using the spectrum of the adjacency matrix. We define the
node importance to communities as the relative change in the eigenvalues of
the network adjacency matrix upon their removal. Second, we also propose an
index to distinguish two kinds of important nodes in communities, i.e.,
“community core” and “bridge”.
Conclusions/Significance: Our indices are only relied on the spectrum of the
graph matrix. They are applied in many artificial networks as well as many
real-world networks. This new methodology gives us a basic approach to solve
this challenging problem and provides a realistic result.
###### pacs:
89.75.Hc, 89.75.-k, 89.75.Fb
## I Introduction
Networks, despite their simplicity, represent the interaction structure among
components in a wide range of real complex systems, from social relationships
among individuals, to interactions of proteins in biological systems, to the
interdependence of function calls in large software projects. The network
concept has been developed as an important tool for analyzing the relationship
of structure and function for many complex systems in the last
decadesRev.Mod.Phys.74 ; SIAM Rev.45 ; Science 286 ; Nature 393 ; Phys. Rep.
424 . Many real-world systems show the existence of structural modules that
play significant and defined functional roles, such as friend groups in social
networks, thematic clusters on the world wide web, functional groups in
biochemical or neural networksPNAS99 . Exploring network communities is
important for the reasons listed belowPlos1 : 1) communities reveal the
network at a coarse level, 2) communities provide a new aspect for
understanding dynamic processes occurring in the network and 3) communities
uncover relationships among the nodes that, although they can typically be
attributed to the function of the system, are not apparent when inspecting the
graph as a whole. As a result, it is not surprising that recent years have
witnessed an explosion of research on community structure in graphs, and a
huge number of methods or techniques have been designedPNAS99 ; Phys. Rev. E
74 ; Physics Reports486 ; Eur.Phys.J.B.38 ; Phys. Rev. E 72 ; Proc. Natl.
Acad. 103 ; Phys. Rev. E 72 ; arXiv:1002.2007v1 ; arXiv:0902.3331v1 ; Rhys.
Rev. E 77 ; arXiv:0907.3708 (seePhysics Reports486 as a review).
It is believed that community structure is important to the function of a
systemProc. Natl. Acad. Sci.100 ; Physica A 384 ; Europhys. Lett.72 . In many
situations, it might be desirable to control the function of modular networks
by adjusting the structure of communities. For example, in biological systems,
one might like to identify the nodes that are key to communities and protect
them or disrupt them, such as in the case of lung cancerPhysica A 384 . In
epidemic spreading, one would like to find the important nodes to understand
the dynamic processes, which could yield an efficient method to immunize
modular networksEurophys. Lett.72 . Such strategies would greatly benefit from
a quantitative characterization of the node importance to community structure.
Some important work related to this topic has been proposed. In 2006, Newman
proposed a community-based metric called “Community Centrality” to measure
node importance to communitiesPhys. Rev. E 74 . His basic idea relies on the
modularity function $Q$. Those vertices that contribute more to $Q$ are more
important for the communities than those vertices that contribute less. Kovacs
et al. also proposed an influence function to measure the node importance to
communitiesPLOSONE .
In fact, the important nodes can have distinct functions with respect to
community structure. Some previous studies have also revealed such
classifications. Guimera et al. have proposed a classification of the nodes
based on their roles within communities, using their within-module degree and
their participation coefficientNature433 . They divided the hubs into three
categories: provincial hubs, connector hubs and kinless hubs. Other approaches
have also been suggested to discuss the connection between nodes and
modularity in biological networks, by dividing hub nodes into two categories
called “party hubs” and “date hubs”Nature430 ; PLoS Bioo ; PLoS Bio . When
removed from the network, party and date hubs have strikingly distinct effects
on the overall topology of the network. Recently, Kovacs et al. proposed an
interesting approach. They introduced an integrative method family to detect
the key nodes, overlapping communities and “date” and “party” hubsPLOSONE . In
a very recent work, the authors mentioned that modular networks naturally
allow the formation of clusters, and hubs connecting the modules would enhance
the integration of the whole network, such as in the case of neuron
networksPRE82 . As a result, it is intuitive that nodes that are important to
communities can be divided into “community cores” and “bridges”. However,
there is one problem. Before using the participation coefficient and the
influence function to distinguish these two kinds of vertices, the exact
communities of the network must first be given. In contrast, it is interesting
to characterize node importance to communities before the division of the
network.
It is understood that the adjacency matrix contains all the information of the
network. Developing methods based only on the adjacency matrix of the network
to detect important nodes to communities and then distinguish them as either
“community core” or “bridge” is an interesting and important problem in
network research. In this manuscript, based only on the adjacency matrix of
the network, we try to access the fundamental questions: how to evaluate the
node importance to communities and how to distinguish different kinds of
important nodes? It is implied that in many cases the spectrum of the
adjacency matrix gives an indication of the community structure in the
networkPRE80 . If the network has $c$ strong communities, the $c$ largest
eigenvalues of the adjacency matrix are significantly larger than the
magnitudes of all the other eigenvalues. These large eigenvalues are key
quantities to the community structure. For this reason, we suggest a basic
approach to solve the above open problem using the spectrum of the graph. We
define the importance of nodes to communities as the relative change in the
$c$ largest eigenvalues of the network adjacency matrix upon their removal.
Furthermore, using the eigenvectors of the graph Laplacian, we divide the
important nodes into community cores and bridges. We apply our method to many
networks, including artificial networks and real-world networks. This new
methodology gives us a basic approach to solve this challenging problem and
provides a realistic result.
The organization of this paper is as follows. In section II, the centrality
metric identifying the important nodes to communities is proposed using the
spectrum of the adjacency matrix. An index to distinguish the two kinds of
important nodes using the corresponding eigenvector of the graph Laplacian is
introduced in section III. In section IV, our method is applied to artificial
networks and some real-world networks, and we obtain some interesting results.
In section V, we extend our method into weighted networks. Finally, concluding
remarks are presented in section V.
## II Centrality Metric Based on the Spectrum of the Adjacency Matrix
We consider a binary network $G=(V,E)$ with $N$ nodes. The adjacency matrix
$A$ is the matrix with elements $A_{ij}=1$ if there is an edge joining
vertices $i$ and $j$, otherwise $0$. We denote each eigenvalue of $A$ by
$\lambda$ and the corresponding eigenvector by v, such that
$A\textit{{v}}=\lambda\textit{{v}}$. The eigenvector is orthogonal and
normalized. The eigenvalues are ordered by decreasing magnitude:
$\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}$. It is easy to show that
$A$ is symmetric and the eigenvalues of $A$ are real. Consider the case of
networks that have $c$ communities. It is implied that when these communities
are disconnected, each one has its own largest eigenvalues. With proper
labeling of the nodes, the matrix $A$ will have a block matrix structure with
$c\times c$ blocks. Blocks on the diagonal correspond to the adjacency
matrices of the individual communities, while the off-diagonal blocks
correspond to the edges between communities; in other words, we can consider
them as a perturbation. Therefore, $A$ can be written as
$A=A_{0}+\delta A,$ (1)
where $A_{0}$ is a matrix whose diagonal block elements are the diagonal block
elements of $A$ and whose off-diagonal block elements are zeros, while $\delta
A$ is a matrix with zeros on its diagonal blocks and with the off-diagonal
blocks of $A$ as its off-diagonal block elements. Chauhan et al.PRE80 have
proved that if the perturbation strength is small, the largest eigenvalues of
disconnected communities are perturbed more weakly than the perturbation
applied. The spectrum of the adjacency matrix of a network gives a clear
indication of the number of communities in the network. If the network has $c$
strong communities, the $c$ largest eigenvalues are well separated from
others. These eigenvalues are key quantities to the community structure.
For this reason, we define the importance of node $k$ to communities as the
relative change in the $c$ largest eigenvalues of the network adjacency matrix
upon its removal:
$I_{k}=-\sum\limits_{i=1}^{c}{\frac{{\Delta\lambda_{i}}}{{\lambda_{i}}}},$ (2)
where $c$ is the number of communities. To avoid the computational cost, we
use perturbation theory to provide approximations of $I_{k}$ in terms of the
corresponding eigenvector v. Let us denote the matrix before the removal of
the node by $A$ and the matrix after the removal by $A+\Delta A$; the
eigenvalue of this matrix is $\lambda+\Delta\lambda$, and the corresponding
eigenvector is $\textit{{v}}+\Delta\textit{{v}}$. For large matrices, it is
reasonable to assume that the removal of a node has a small effect on the
whole matrix and the spectral properties of the network, so that $\Delta A$
and $\Delta\lambda$ are small. We obtain
$(A+\Delta
A)(\textit{{v}}+\Delta\textit{{v}})=(\lambda+\Delta\lambda)(\textit{{v}}+\Delta\textit{{v}}).$
(3)
The effect on the adjacency matrix $A$ of removing node $k$ is given by
$(\Delta A)_{ij}=-A_{ij}(\delta_{ik}+\delta_{jk})$. We cannot assume that the
$\Delta\textit{{v}}$ is small because $\Delta v_{k}=-v_{k}$, so we set
$\Delta\textit{{v}}=\delta\textit{{v}}-v_{k}\widehat{e}_{k}$ where
$\delta\textit{{v}}$ is small and $\widehat{\textit{{e}}}$ is the unit vector
for the $k$ component. Left multiplying (3) by $\textit{{v}}^{T}$ and
neglecting second order terms $\textit{{v}}^{T}\Delta A\delta\textit{{v}}$ and
$\textit{{v}}^{T}\Delta\lambda\delta\textit{{v}}$, we obtain
$\Delta\lambda=\frac{{\textit{{v}}^{T}\Delta
A\textit{{v}}-\textit{{v}}^{T}v_{k}\Delta
A\widehat{e}_{k}}}{{\textit{{v}}^{T}\textit{{v}}-v_{k}^{2}}}.$ (4)
For a large network ($N\gg 1$), we know that $\textit{{v}}^{T}\textit{{v}}\gg
v_{k}^{2}$; therefore, we can write
$\Delta\lambda\approx\frac{{\textit{{v}}^{T}\Delta
A\textit{{v}}-\textit{{v}}^{T}\textit{{v}}_{k}\Delta
A\widehat{\textit{{e}}}_{k}}}{{\textit{{v}}^{T}\textit{{v}}}}$ (5)
Because $(\Delta A)_{ij}=-A_{ij}(\delta_{ik}+\delta_{jk})$, we obtain
$\textit{{v}}^{T}\Delta A\textit{{v}}=-2\lambda
v_{k}^{2},\textit{{v}}^{T}v_{k}\Delta A\widehat{e}_{k}=-\lambda v_{k}^{2}.$
(6)
Finally, the importance of node $k$ to the community structure is obtained by
$I_{k}=-\sum\limits_{i=1}^{c}{\frac{{\Delta\lambda_{i}}}{{\lambda_{i}}}}\approx\sum\limits_{i=1}^{c}{\frac{{v_{ik}^{2}}}{{\textit{{v}}_{i}^{T}\textit{{v}}_{i}}}},$
(7)
where $c$ is the number of communities, $v_{ik}$ is the kth element of
$\textit{{v}}_{i}$ and $I_{k}$ lies in the interval $[0,1]$. If $I_{k}$ is
large, node k is important to the community structure; otherwise, $k$ is on
the periphery of the community.
Using this metric $I$, we can quantify the node importance to the community
structure. If the node is important to the community structure, when we remove
it from the network, the relative changes of the $c$ largest eigenvalues are
large; otherwise, the changes are small. Before applying $I$, the value of $c$
needs to be determined. The determination of the number of communities is an
important but challenging question in community analysis. Here we use the
method proposed by Ref.PRE80 . This method is based on the properties of the
spectrum of the graph and is independent of the partition algorithms, so our
metric is quite convenient to use.
## III distinguish two kinds of important nodes
As mentioned above, there are two kinds of nodes that are important to
communities. One is the “community core”, and the other is the “bridge”
between communities. Each will affect communities deeply upon its removal.
When we remove the “community core”, the community structure in the network
will become fuzzy, while the community structure will become clear when we
remove the “bridge”. See Fig. 1 for an example. Vertices 1 and 8 are the
“community cores”, and they organize their respective communities. Meanwhile,
node 15 is the “bridge” between the two communities. The “community core” is
the leader in the community, and it can organize the function of each
community. In contrast, the “bridge” connects the modules and can enhance the
integration of the whole network. It is believed that a combination of both
segregation and integration, as in neural systems, is crucialPRE82 . It is
clear that effectively disconnected and fully non-synchronous regions cannot
allow collective or integrative action of the elements. Similarly, a fully
synchronized regime does not allow separated or segregated performance of the
elements. Therefore, both situations are biologically unrealistic, as can be
seen from the existence of related conditions, such as epileptic seizures
(collective phenomena) and Parkinson’s disease (segregated phenomena)Neuro .
For this reason, both the “community core” and the “bridge” are important to
communities, but they play different roles. The metric we proposed in
SectionII can determine the nodes that are important to communities, but now a
method to distinguish these two kinds of important nodes is needed.
In agreement with earlier findingsPLOSONE ; Nature430 ; PLoS Bioo ; PLoS Bio ,
we assumed that bridge nodes should have more inter-modular positions than
community cores. The existence of bridge nodes often leads to some inter-
modular edges. Given a graph, the simplest and most direct way to construct a
partition of the graph is to solve the mincut problem (minimize the number of
edges between communities $R$)CMJ23 . In practice, however, this method often
does not lead to satisfactory partitions. The problem is that, in many cases,
the solution of mincut simply separates one individual vertex from the rest of
the graph. Of course, this is not what we want to achieve in clustering, as
clusters should be reasonably large groups of points. Due to this shortcoming
in the mincut problem, one common objective function to encode the desired
information is RatioCutRatiocut :
$RatioCut(C_{1},\cdots
C_{c})\buildrel\textstyle.\over{=}\sum\limits_{i=1}^{c}{\frac{{R(C_{i},\bar{C}_{i})}}{{|C_{i}|}}},$
(8)
where $|C_{i}|$ is the size of community $C_{i}$. If the sizes of the
communities are almost the same, the RatioCut problem reduces to the mincut
problem.
### III.1 The Condition of $c=2$
If the network is divided into only two communities ($c=2$), we define an
index vector s with $N$ elements:
$s_{i}=\left\\{\begin{array}[]{l}\sqrt{{{|\bar{C}|}\mathord{\left/{\vphantom{{|\bar{C}|}{|C|}}}\right.\kern-1.2pt}{|C|}}}\quad\quad{\rm{if\quad
vertex}}\quad i\in C,\\\
-\sqrt{{{|C|}\mathord{\left/{\vphantom{{|C|}{|\bar{C}|}}}\right.\kern-1.2pt}{|\bar{C}|}}}\quad{\rm{if\quad
vertex}}\quad i\in\bar{C}.\\\ \end{array}\right.$ (9)
Then the RatioCut function is obtained as followsTutorial :
$RatioCut(C,\bar{C})=\frac{1}{{|V|}}\textit{{s}}^{T}\textit{{L}}\textit{{s}},$
(10)
where $|V|$ is the number of vertices in the network and L is the graph
Laplacian. L is defined as $L_{ij}=-A_{ij}$ for $i\neq j$ and $L_{ii}=k_{i}$,
where $k_{i}$ is the degree of node $i$. We also have two constraints on s:
$\sum\limits_{i=1}^{n}{s_{i}}=0$ and $\sum\limits_{i=1}^{n}{s_{i}^{2}}=n$.
Here the partition problem is equal to the problem
$\min{\rm{}}\textit{{s}}^{T}\textit{{L}}\textit{{s}};\ {\rm{subject\ to}}\
\sum\limits_{i=1}^{n}{s_{i}}=0,\sum\limits_{i=1}^{n}{s_{i}^{2}}=n.$ (11)
If the components of the vector s are allowed to take arbitrary values, it can
be seen immediately that the solution of this problem is given by the vector s
that is the eigenvector corresponding to the second-smallest eigenvalue of L,
denoted by $\textit{{u}}_{2}$. So we can approximate a minimizer of RatioCut
by the second eigenvector of L. Unfortunately, the components of s are only
allowed to take two particular values.
Thus, the simplest solution is achieved by assigning vertices to one of the
groups according to the sign of the eigenvector $\textit{{u}}_{2}$. In other
words, we assign vertices as follows: if $\textit{{u}}_{2}^{i}>0$, we assign
vertex $i$ to community $C$; otherwise, we assign it to $\bar{C}$. Assignation
priority begins with the most positive and the most negative; the node with
the most positive magnitude is first to be assigned to $C$, then the second
and so on, while the node with the most negative magnitude is similarly the
first to be assigned to $\bar{C}$. If a node’s corresponding element is close
to zero, it may have nearly equal membership in both communities, and we can
assign it to both communities. In conclusion, if the network is divided into
only two communities, we can use this method to characterize which are the
“community cores” and which are the “bridge” between communities. If node $i$
is a “community core”, $|\textit{{u}}_{2}^{i}|$ is relatively large;
otherwise, $|\textit{{u}}_{2}^{i}|$ is near zero.
### III.2 The Condition of $c>2$
Consider the division of a network into $c$ nonoverlapping communities, where
$c$ is the number of communities. We define an $n\times c$-index matrix S with
one column for each community,
$\textit{{S}}=(\textit{{s}}_{1}|\textit{{s}}_{2}|\cdots|\textit{{s}}_{c})$, by
$s_{i,j}=\left\\{\begin{array}[]{l}{{\rm{1}}\mathord{\left/{\vphantom{{\rm{1}}{\sqrt{|C_{j}|}}}}\right.\kern-1.2pt}{\sqrt{|C_{j}|}}}\quad{\rm{if\quad
vertex}}\quad i\in C_{j},\\\ {\rm{0\quad otherwise}}.\\\ \end{array}\right.$
(12)
Following the previous section, we obtain
$RatioCut=Tr(\textit{{S}}^{T}\textit{{L}}\textit{{S}}),$ (13)
where $Tr$ is the trace of a matrix and $\textit{{S}}^{T}$ is the transpose
matrix of S. L is a semi-positive and symmetric matrix. We can write
$\textit{{L}}=\textit{{U}}\textit{{D}}\textit{{U}}^{T}$, where U is the
eigenvector of L,
$\textit{{U}}=(\textit{{u}}_{1}|\textit{{u}}_{2}|\cdots|\textit{{u}}_{n})$ and
D is the diagonal matrix of eigenvalues $D_{ii}=\beta_{i}$. We therefore
obtain
$RatioCut=\sum\limits_{j=1}^{n}{\sum\limits_{k=1}^{c}{\beta_{j}(u_{j}^{T}s_{k})^{2}}}.$
(14)
It can also be written as
$RatioCut=\sum\limits_{k=1}^{c}{\sum\limits_{j=1}^{n}{\beta_{j}[\sum\limits_{i=1}^{n}{U_{ij}S_{ik}}]^{2}}}.$
(15)
Now we define the vertex vector of $i$ as $r_{i}$, and let
$[r_{i}]_{j}=U_{ij}.$ (16)
If the network has almost equal-sized communities, then equation (15) can be
written as
$RatioCut\approx\frac{{\sum\limits_{k=1}^{c}{\sum\limits_{j=1}^{n}{\beta_{j}[\sum\limits_{i\in
G_{k}}{[r_{i}]_{j}}]^{2}}}}}{{|C|}},$ (17)
where $G_{k}$ is the set of vertices belonging to community $k$ and $|C|$ is
the community size.
Minimizing the RatioCut can be equated with the task of choosing the
nonnegative quantities so as to place as much of the weight as possible in the
terms corresponding to the low eigenvalues and as little as possible in the
terms corresponding to the high eigenvalues. This equates to the following
maximization problem:
$Max\ \sum\limits_{k=1}^{c}{\sum\limits_{j=1}^{p}{\beta_{j}[\sum\limits_{i\in
G_{k}}{[r_{i}]_{j}}]^{2}}},$ (18)
where $p$ is a parameter. We could choose $p=c$ if the community structure was
clear. To this end, we propose an easy way to distinguish two kinds of
important nodes using the theory of the graph Laplacian. If the community
structure is quite clear, we focus on the vertex vector magnitude $|r_{i}|$ in
the first $p$ terms, denoted by the $w$-score:
$w_{i}=\sqrt{\sum\limits_{j=1}^{p}{[r_{i}]_{j}^{2}}}.$ (19)
If the $w$-score of a given vertex is close to zero, we believe that this
vertex has nearly equal membership in more than one community, and it is
likely to be the “bridge” of these communities. This discrimination process
equates to the “fuzzy” division of the network into communities. In many
cases, this type of fuzzy division could result in a more accurate picture of
real-world networks.
## IV Results
Now we test the validity of our indices introduced in section II and section
III in various artificial networks and real-world networks.
### IV.1 Artificial Networks
First, we consider a sketch composed of 15 nodes (see Fig. 1) forming two
communities. It is intuitive that vertices 1, 8 and 15 are important to the
community structure in this sketch. Vertices 1 and 8 are the so-called
“community cores”, and they form both the communities. Vertex 15 is the
“bridge” between communities, and it connects these two communities. As we
discussed before, removing vertex 1 or 8 will make the community structure
fuzzy, and removing vertex 15 will make it clear.
Figure 1: Sketch of a network composed of 15 nodes. The diameter of one vertex
is proportional to the centrality metric $I$. Moreover, the color of one
vertex is related to the index $w$-score. Red vertices behave like
“overlapping” nodes or “bridges” between communities, and yellow vertices
often lie inside their own communities.
Here we use the index $H$ proposed by Hu et al.arXiv:1002.2007v1 to measure
the significance of communities:
$H=\frac{n}{{\bar{k}\sum\limits_{j=c+1}^{n}{\frac{1}{{|\overline{\beta}-\beta_{j}|}}}}},$
(20)
where $\beta$ is the eigenvalue of the graph Laplacian, $\overline{\beta}$ is
the average value of $\beta_{2}$ through $\beta_{c}$, $\bar{k}$ is the average
degree of the network and $n$ is the number of vertices in the network. In
networks with strong communities (many links are within communities with very
sparse connections outside), $H$ is always large. Here we focus on the change
of $H$ due to the removal of vertices, denoted by $\Delta H$. We also use the
centrality metric proposed by NewmanPhys. Rev. E 74 , which we denote here by
$M$. The results are shown in Tab. 1. Through $\Delta H$, it is implied that
vertices 1 and 8 are more important than other vertices because the magnitude
of $\Delta H$ is relatively larger than others. Moreover, their removal makes
the communities fuzzy, while vertex 15 acts like a ”bridge” between the
communities, and its removal makes the communities clear. We can see that our
centrality metric performs quite well; it can identify not only the “community
cores”, but also the “bridge” between communities. $M$ can also identify the
“community cores”, but it has some problems. One issue is that its values tend
to span a rather small dynamic range from largest to smallest. Moreover, in
some cases (such as this sketch), $M$ cannot recognize important vertices
among communities. In calculating the index $H$, we need to go through every
vertex in the network, incurring significant computational cost. In contrast,
our method provides a more efficient way, requiring less computational cost,
and yields the correct answer.
Table 1: Centrality metrics of the example sketched in Fig. 1. Vertex Label | $I$ | $M$ | $\Delta H$ | $w$-score
---|---|---|---|---
1 | 0.32 | 0.758 | -0.145 | 0.2405
8 | 0.32 | 0.758 | -0.145 | 0.2405
15 | 0.173 | 0.69 | 0.116 | 0.00
2,7,9,14 | 0.09 | 0.704 | 0.04 | 0.198
3,6,10,13 | 0.1 | 0.7535 | -0.021 | 0.285
4,5,11,12 | 0.105 | 0.7327 | -0.054 | 0.3175
Here we use the classical GN benchmark presented by Girvens and Newman to test
the measurementsProc. Natl. Acad. 103 . Each network has $N=128$ nodes that
are divided into four communities (c = 4) with 32 nodes each. Edges between
two nodes are introduced with different probabilities, which depend on whether
the two nodes belong to the same community or not. Each node has $<k_{in}>$
links on average with its fellows in the same community and $<k_{out}>$ links
with the other communities, and we impose $<k_{in}>+<k_{out}>=16$. The
communities become fuzzier and thus more difficult to identify as $k_{out}$
increases. Because the GN benchmark is a homogenous network, there should not
be any nodes that are important to the community structure. To check whether
our conjecture is correct or not, we let $<k_{in}>=12$ so that the community
structure is quite clear and average the result for the GN benchmark over 100
configurations of networks. From the result, about 120 nodes’ importances lie
in the interval $[0.03,0.04]$, while others lie in the interval [0.02,0.03].
The mean value of $I$ is 0.0312, and the standard deviation is 0.0014. It can
be concluded that, in the GN benchmark, there are no nodes that are important
to the community structure.
We may also test the method on the more challenging LFR benchmark presented by
Lancichinetti et al.Phys. Rev. E 78 . In the LFR benchmark, the degree
distribution obeys a power-law distribution $p(k)\propto k^{-\alpha}$, and the
sizes of the communities are also taken from a power-law distribution with an
exponent $\gamma$. Moreover, each node shares a fraction $1-\mu$ of its links
with other nodes of its own community and a fraction $\mu$ with others in the
rest of the network. The community structure can be adjusted by the mixing
parameter $\mu$. Without loss of generality, we let
$\alpha=2.5,\gamma=1.0,\mu=0.25$ and the size of the network $N=1000$. Our
numerical results in the LFR benchmark are shown in Fig. 2. In this case,
there is no “bridge” between communities because $\mu=0.25$. We may also
calculate the $w$-score, of which the mean value is 0.1736 and the standard
deviation is 0.0292. Moreover, the centrality metric is positively correlated
with node degree ($r^{2}=0.7329$), but some vertices have quite high
centrality while having relatively low degree, and thus the correlation index
is not very high.
Figure 2: (a) The Zipf plot of the nodes’ centrality to communities. (b) The
centrality metric we propose is correlated with node degree. The parameters in
the LFR benchmark are as follows: $\alpha=2.5,\gamma=1.0,\mu=0.25$ and the
size of the network $N=1000$.
### IV.2 Real-world Networks
We apply our method to some real-world networks, such as the Zachary club
networkJAR33 , the word association networkSOUTH , the scientific
collaboration networkwebsite , and the C. elegans neural networkTRS .
First, we consider a famous example of a social network, the Zachary’s karate
club network. This network represents the pattern of friendships among members
of a karate club at a North American university. It contains 34 vertices, and
the links between vertices are the friendships between people. The nodes
labeled as 1 and 34 correspond to the club instructor and the administrator,
respectively. They had a conflict which resulted in the breakup of the club.
Most other nodes have a relationship with node 1, node 34, or both. In this
network, $c=2$. The numerical results are shown in Fig. 3 and Fig. 4. In Fig.
3(a), we can see that nodes 1 and 34 are the most important nodes in the
communities. Our method to distinguish important nodes are shown in Fig. 3(b).
From the result, we can see that nodes 1 and 34 are the so-called “community
cores”, and they have many connections in their own communities. Furthermore,
we compare our method with Newman’s. This result is also shown in Fig. 3(a),
and the two metrics are normalized by
$x_{nor}=\frac{{x-<x>}}{{\sigma_{x}}},$ (21)
where $<x>$ is the average value of each index and $\sigma_{x}$ is the
standard deviation of each index. It is implied that these two methods have
some differences. In our method, nodes 1 and 34 are absolutely more important
than other nodes, while in Newman’s method, nodes 2 and 33 are also quite
important, even more than node 1. In this network, the modularity function $Q$
reaches its maximum value when the network is divided into 4 communities; this
fact may be the cause of the differences between the results of these two
methods. The visualization of the karate network with our two measurements is
sketched in Fig. 4. The diameter of each vertex is proportional to the
centrality metric $I$. A large diameter indicates an important vertex.
Additionally, the color of each vertex is related to the index $w$-score. Red
vertices behave like “overlapping” nodes or “bridges” between communities, and
yellow vertices often lie inside their own communities.
Figure 3: It is shown that our method works quite well in the Zachary’s karate
club network. Nodes 1 and 34 are the instructor and the administrator,
respectively. In Fig. 3(a), we can see that these two nodes are more important
to the community structure than other nodes. We also compare our method with
Newman’s and find that the two methods exhibit some differences. In Fig. 3(b),
we shown that nodes 1 and 34 are the so-called “community cores”. Figure 4:
The Zachary’s karate club network, which is composed of 34 vertices. Vertex
diameters indicate the community centrality $I$. The color of each vertex is
proportional to the index $w$-score.
Second, we analyze the word association network starting from the word
“Bright”. This network was built on the University of South Florida Free
Association NormsSOUTH . An edge between words A and B indicates that some
people associate the word B to the word A. The graph displays four
communities, corresponding to the categories Intelligence, Astronomy, Light,
Colors. The word Bright is related to all of them by construction. We applied
our method to this network, and the results are shown in Fig. 5. From the
results, we can observe that our method considers Bright, Sun, Smart, Moon as
important nodes to the community structure. It may be inferred from the result
that Moon and Smart are the “community cores”, while Bright and Sun are the
“bridges” between communities. Indeed, our metric yields the correct answer.
For example, Smart is the core of the community Intelligence, while Moon is
the core of the community Astronomy. Meanwhile, the $w$-score of node Bright
is 0.08, which is close to zero. We would therefore conclude that it is a
“bridge” between communities, and Bright is in fact the “bridge” among these
four communities, as the network was originally derived from it.
Moreover, we may investigate the effect of node removal on the modularity
function $Q$. “Community cores” and “bridges” have different effects on
community structure. When a “community core” is removed, the communities
become clear. For example, the removal of the node “bright” makes the
modularity function $Q$ increase by 0.03, which is the largest increase caused
by the removal of any single node, while the removal of node “Moon” causes $Q$
to decrease by 0.015. These results are averaged over 20 trials. We can see
from our results that important nodes (i.e., nodes with large $I$) affect the
communities considerably. For example, the removal of the node “Smart”
decreases $Q$ by 0.0152, while the removal of the node “Gifted”, which seems
to be a peripheral node, decreases $Q$ by only 0.0048.
Figure 5: Index $I$ and $\omega$-score for the nodes of the word association
network. The node importance versus vertex rank is shown in (a). In (b), we
distinguish “community cores” and “bridges” using the index $w$-score. Figure
6: The centrality metric $I$ and $w$-score for the scientist collaboration
network (a,b). The centrality metric $I$ and $w$-score are also calculated in
the C. elegans neural network (c,d).
We may also apply our method to social networks, such as the scientist
collaboration networkwebsite , and neural networks, such as the C. elegans
neural networkTRS . We analyzed the largest connected component of each
network. The scientist collaboration network represents scientists whose
research centers on the properties of networks of one kind or another. There
are 379 vertices, representing scientists who are divided into 12 communities.
Edges are placed between scientists who have published at least one paper
together. The neural network of C. elegans contains 302 neurons and 2,359
links. This network is divided into 3 communities, with each node representing
a neuron and each link representing a synaptic connection between neurons.
Here we consider the C. elegans neural network to be undirected. The results
are shown in Fig. 6.
In the scientist collaboration network, our centrality metric $I$ identifies
“group leaders”, such as M. Newman, S. Boccaletti, and A. Barabasi. Their
$w$-scores are not very large because they often have some collaboration
between scientists outside their own communities. We can also find so-called
“community cores” based on our method, such as R. Sole, and “bridge” vertices
among some communities, such as B. Kahng. As we know, the C. elegans neural
networks are composed of sensory neurons, interneurons and motor neurons. The
neurons with high centrality metrics often have the most important functions,
and all of them are interneurons, such as $AVA$, $AVB$, $AVD$, and $AVE$.
These classes, which synapse onto motor neurons in the ventral cord, are among
the most prominent neurons in the whole nervous system. They generally have
larger-diameter processes than other neurons and have many synaptic
connectionsTRS ; JN . As a result, they have larger $I$ than other vertices,
while the typical $w$-score in these classes is quite small (smaller than
0.05). In the C. elegans neural network, connection between communities is
more necessary and frequent due to some special functions.
## V Applications in Weighted networks
Our method can be generalized to weighted networks because the adjacency
matrix in an undirected weighted network is real and symmetric. Thus, in
weighted networks, the importance of a node and its role in communities are
also characterized by its $I$ and $w$-score. Let us first consider an
artificial weighted network. We use similarity weight in this weighted
network. A higher weight means a closer relationship between vertices. At
first, 10 nodes form a complete network and are divided into two communities
with 5 nodes each. We assign vertices 4 and 9 as the core of each community,
each of which has links with weight 2 connecting to vertices within its
community and weight 0.2 connecting to outside vertices. All other intra-
connections have weight 1, and all other interconnections have weight 0.2.
Then we introduce vertex 11 as the bridge between the two communities. It
connects to all 10 nodes with weight 1. The index $I$ and $w$-score for each
node are given in Tab. 2. The results indicate that vertices 4, 9 and 11 are
more important than the other vertices, while vertex 11 is a “bridge” between
these two communities. Our method works quite well in this small artificial
weighted network.
Table 2: Centrality metrics $I$ and $w$-score in a complete weighted network. Vertex Label | I | | $w$-score
---|---|---|---
4 | 0.295 | | 0.316
9 | 0.295 | | 0.316
11 | 0.16 | | 0.00
others | 0.156 | | 0.316
Figure 7: Sketch of the SFI scientific collaboration network as a weighted,
undirected network. It has 118 scientists. Vertex diameters indicate the
community centrality $I$. The color of each vertex is proportional to the
index $w$-score.
As an example of a real-world weighted network, we investigate the
collaboration network among scientists working at the Santa Fe Institute (the
SFI network). Here we consider it as a weighted, undirected network.
Collaboration events between the scientists can be repeated again and again,
and a higher frequency of collaboration usually indicates a closer
relationship. Furthermore, weights can be assigned to the scientists’
collaboration quite naturally: an article with $n$ authors corresponds to a
collaboration act of weight $\frac{1}{{n-1}}$ between every pair of its
authorsPRE73 . The results for the SFI collaboration network are sketched in
Fig. 7. Vertex diameters indicate the community centrality $I$. The color of
each vertex is proportional to the index w-score. Red vertices behave like
“overlapping” nodes or “bridges” between communities, and yellow vertices
often lie inside their own communities. We do not know the specific names;
however, we observe that the positions of the large vertices are just like the
“group leaders”. Vertices 2, 12 and 24 are so-called “community cores” in
communities because their $w$-scores are quite large. In fact, they are the
group leaders in the fields of Mathematical Ecology, Statistical Physics and
Structure of RNA, respectively. However, vertices 1, 9 and 11 are the
“bridges” between communities, and they have relative small $w$-scores.
Interestingly, the result in the weighted network is different from the one in
the corresponding unweighted network. It can be concluded that the edge weight
may affect the result. For example, vertex 9 and vertex 11 collaborate quite
often; this makes both of them quite important in a weighted network, while in
an unweighted network, neither of them is very important to the community
structure.
## VI Conclusion And Discussion
In this paper, we characterize the node importance to community structure
using the spectrum of the graph. The eigenspectrum of the adjacency matrix
gives a clear indication of the number of “dominant” communities in a
networkPRE80 . We give a centrality metric based on the spectrum of the
adjacency matrix of the graph, and it can identify the nodes important to the
community structure in many cases. In addition, we propose an index to
distinguish the two kinds of important nodes that we term “community cores”
and “bridges” using the spectrum of the graph Laplacian.
We demonstrate a variety of applications of our method to both artificial and
real-world networks representing social and neural networks. Our method works
well in many cases without knowing the exact community structure, although the
number of communities should be known. However, a limitation of this method
arises when one or more of the communities is much smaller than the largest
community, or when a community has very sparse intra-community connections
compared to other communities. This may happen when
$N_{small}^{2}<N_{large}$PRE80 . Even in the absence of perturbation, the
maximum eigenvalue of a smaller community can lie inside the cloud of non-
Perron-Frobenius eigenvalues of the largest community. But, with the
understanding that the intent of our method is to find the important nodes in
the community structure, the nodes in very small communities may be ignored.
Even so, if the community structure is so fuzzy that we cannot identify the
number of communities, our method is not accurate.
Our method can also be used in weighted networks. From our result in the SFI
network, it can be inferred that edge weight may affect the result.
Furthermore, it may generalize to directed networks because the Perron-
Frobenius eigenvalues are often real and positiveSIAMR . We have yet to treat
the case of directed networks. The identification of such key nodes is
important and could potentially be used to identify the organizer of the
community in social networks, to develop an immunization strategy in an
epidemic process, to identify key nodes in biological networks and so on. We
hope our results may be helpful to future research.
## ACKNOWLEDGEMENTS
The authors thank Di Huan, An Zeng, and Hongzhi You for their helpful
suggestions. This work is supported by the NSFC under grants No. 70771011 and
No. 60974084, NCET-09-0228, and fundamental research funds for the Central
Universities of Beijing Normal University.
## References
* (1) Albert R, Barabasi A-L (2002), Statistical mechanics of complex networks. Rev. Mod. Phys.74: 47-97.
* (2) Newman MEJ, The structure and function of complex networks (2003). SIAM Rev. 45: 167-256.
* (3) Barabasi A-L, Albert R (1999), Emergence of Scaling in Random Networks. Science 286: 509-512.
* (4) Watts DJ, Strogatz SH (1998), Collective dynamics of ‘small-world’ networks. Nature 393: 440-442.
* (5) Boccaletti S, Latora V, Moreno Y, Chavez M and Hwang D-U (2006), Complex networks: Structure and dynamics. Physics Reports 424: 175-308.
* (6) Girvan M, Newman MEJ (2002), Community structure in social and biological networks. Proc. Natl. Acad. Sci. 99: 7821-7826.
* (7) Lancichinetti A, Kivela M, Saramaki J, Fortunato S (2010), Characterizing the Community Structure of Complex Networks. PloS ONE, 5: e11976.
* (8) Newman MEJ (2006), Finding community structure in networks using the eigenvectors of matrices. Phys. Rev. E 74: 036104.
* (9) Fortunato S (2009), Community detection in graphs. Physics Reports 486: 75-174.
* (10) Wu F and Huberman BA (2004), Finding communities in linear time: A physics approach. Eur.Phys.J.B.38: 331-338.
* (11) Duch J and Arenas A (2005), Community detection in complex networks using extremal optimization. Phys. Rev. E 72: 027104.
* (12) Newman MEJ (2006), Modularity and community structure in networks. Proc. Natl. Acad. 103: 8577-8582.
* (13) Gfeller D, Ghappelier J-C and Los Rios P de (2005), Finding instabilities in the community structure of complex networks. Phys. Rev. E 72: 056135.
* (14) Hu Y, Ding Y, Fan Y and Di Z (2010), How to Measure Significance of Community Structure in Complex Networks. arXiv:1002.2007v1.
* (15) Hu Y, Nie Y, Yang H, Cheng J, Fan Y and Di Z (2010), Measuring the significance of community structure in complex networks. Phys. Rev. E 82: 066106.
* (16) Karrer B, Levina E and Newman MEJ (2008), Robustness of community structure in networks. Rhys. Rev. E 77: 046119.
* (17) Lancichinetti A, Radicchi F, Ramasco JJ (2010), Statistical significance of communities in networks. Phys. Rev. E 81: 046110.
* (18) Spirin V, Mirny LA (2003), Protein complexes and functional modules in molecular networks. Proc. Natl. Acad. Sci.(100); 12123-12128.
* (19) Sun L, Li M, Jiang L, Tan L (2007), Comparative analysis of the gene co-regulatory network of normal and cancerous lung. Physica A 384: 739-746.
* (20) Liu Z and Hu B (2005), Epidemic spreading in community networks. Europhys. Lett.72: 315.
* (21) Guimera R, Amaral LAN (2005), Functional cartography of complex metabolic networks. Nature 433: 895-900.
* (22) Kovacs IA, Palotai R, Szalay MS, Csermely P (2010), Community landscapes: an integrative approach to determine overlapping network module hierarchy, identify key nodes and predict network dynamics. PLoS ONE 5: e12528.
* (23) Han JD, Bertin N, Hao T, Goldberg DS, Berriz GF (2004), et al. Evidence for dynamically organized modularity in the yeast protein Cprotein interaction network. Nature, 430: 88-93.
* (24) Batada NN, Reguly T, Breitkreutz A, Boucher L, Breitkreutz BJ (2006), et al. Stratus not altocumulus: A new view of the yeast protein interaction network. PLoS Biol 4: e317.
* (25) Batada NN, Reguly T, Breitkreutz A, Boucher L, Breitkreutz BJ, Hurst LD, Tyers M (2007), Still stratus not altocumulus: further evidence against the date/party hub distinction. PLoS Biol 5: e154.
* (26) Zhao M, Zhou C, Chen Y, Hu B, Wang B (2010), Complexity versus modularity and heterogeneity in oscillatory networks: Combining segregation and integration in neural systems. Physical Review E 82: 046225.
* (27) Chauhan S, Girvan M, Ott E (2009), Spectral properties of networks with community structure. Physical Review E, 80: 056114.
* (28) Stam CJ (2005), Nonlinear dynamical analysis of EEG and MEG: Review of an emerging field. Clin. Neurophysiol. 116: 2266-2301.
* (29) Fiedler M (1973), Algebraic connectivity of graphs. Czech. Math. J. 23: 298-305.
* (30) Hagen L, Kahng A (1992), New spectral methods for ratio cut partitioning and clustering. IEEE Trans. Computer-Aided Design, 11: 1074-1085.
* (31) Luxburg UV (2007), A Tutorial on spectral clustering. Statistics and Computing, 17: 395-416.
* (32) Lancichinetti A, Fortunato F and Radicchi F (2008), Benchmark graphs for testing community detection algorithms. Phys. Rev. E 78: 046110.
* (33) Zachary WW (1977), An information flow model for conflict and fission in small groups. Journal of Anthropological Research 33: 452-473.
* (34) Nelson DL, McEvoy CL, Schreiber TA (1998), The university of south florida word association, rhyme, and word fragment norms.
* (35) http://www-personal.umich.edu/ mejn/netdata/
* (36) White JG et al. (1986), The structure of the nervous system of the nematode caenorhabditis elegans. Philos. Trans. R. Soc. London, Ser. B 314: 1-340.
* (37) Tsalik EL and Hobert OL, Neurobiol J (2003). Functional Mapping of Neurons That Control Locomotory Behavior in Caenorhabditis elegans. 56: 178-197.
* (38) Ramasco JJ and Morris SA (2006), Social inertia in collaboration networks. Phys. Rev. E 73: 016122.
* (39) MacCluer CR (2000), The many proofs and applications of Perron’s Theorem. SIAM Rev. 42: 487-498.
|
arxiv-papers
| 2011-01-10T03:31:15 |
2024-09-04T02:49:16.265875
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yang Wang, Zengru Di, Ying Fan",
"submitter": "Wang Yang",
"url": "https://arxiv.org/abs/1101.1703"
}
|
1101.1927
|
# A Finite State Model for Time Travel
Hwee Kuan Lee Bioinformatics Institute, 30 Biopolis Street, #07-01, Matrix,
Singapore 138671
###### Abstract
A time machine that sends information back to the past may, in principle, be
built using closed time-like curves. However, the realization of a time
machine must be congruent with apparent paradoxes that arise from traveling
back in time. Using a simple model to analyze the consequences of time travel,
we show that several paradoxes, including the grandfather paradox and
Deutsch’s unproven theorem paradox, are precluded by basic axioms of
probability. However, our model does not prohibit traveling back in time to
affect past events in a self-consistent manner.
## I Background
The possibility of building a time machine has been proposed by many authors
friedman ; gott ; godel ; bonnor ; morris ; politzer ; boulware ; hartie ;
politzer2 ; deutsch ; novikov ; lloyd ; pegg ; svetlichny . Two common
approaches are through closed time-like curves (CTC) friedman ; gott ; godel ;
bonnor ; morris ; politzer ; boulware ; hartie ; politzer2 and quantum
phenomena deutsch ; lloyd ; pegg ; svetlichny . Although the general theory of
relativity allows for CTCs, it is not clear if the laws of physics permit
their existence hawking ; carroll ; deser ; carroll2 . Hence the possibility
of traveling back to the distant past remains an open question. Paradoxical
thought experiments have been devised to suggest that traveling back in time
may lead to violations of causality, and hence is not possible. The most
famous paradox is the grandfather paradox, in which an agent travels back in
time to kill his grandfather before his father was conceived. In this case,
the agent will not exist at the current time and hence cannot travel back in
time to kill his grandfather. An alternative version of the grandfather
paradox is autoinfanticide, where an agent travels back in time to kill
himself as an infant. This paradox plays a central role in the argument
against traveling back in time. Another paradox is the Deutsch’s unproven
theorem paradox lloyd , in which an agent travels back in time to reveal the
proof of a mathematical theorem. The proof is then recorded in a document that
the agent reads in future time. Another version of Deutsch’s unproven paradox
is what we call the chicken-and-egg paradox. A hen travels back in time to lay
an egg. The egg hatches into the hen herself. Without the egg, the hen would
not exist but without the hen traveling back in time, the egg would not be
laid.
In this paper, a simple model is used in an attempt to solve time travel
paradoxes and help set the logical foundations of traveling back in time. Our
approach is quite different from approaches that focus on how a time machine
can be built (in principle) lloyd . We suppose that a time machine can be
built and then analyze what could be possible (or impossible) in time travel.
We use a simple directed cyclic graph to explain causal relationships in
different scenarios of time travel. Our conclusion is that, assuming traveling
back in time is feasible, an agent who travels back in time is unable to kill
himself although he may be able to alter the past in other ways; in a self-
consistent manner.
The self-consistency principle was proposed by Wheeler and Feynman feynman ,
Novikov et al novikov and Lloyd et al lloyd . It states that traveling back
in time may be possible, but it cannot happen in a way that violates
causality. Causality in this case includes events that happen in the future
affecting the past. This principle precludes time travel paradoxes but does
not forbid traveling back in time. Due to space limitations, the reader is
referred to feynman ; novikov ; lloyd for detailed discussion of the self-
consistency principle.
## II Model
Our model can be considered as a simple case of graphical models. Graphical
models have been extensively studied and are applicable in many fields such as
in econometric models, social sciences, artificial intelligence and even in
medical studies. Publications on graphical models are so numerous that we can
only provide a non-exhaustive list richardson1997 ; richardson1996 ; schmidt ;
spirtes ; lacerda ; pearlbk ; rebane1987 ; lauritzen ; lauritzen1 ; salmon ;
morgan ; spirtesbk ; cooper1991 . Although directed acyclic graphs have been
at the center stage of graphical models, directed cyclic graphical models have
also received significant attention schmidt ; richardson1997 ; richardson1996
; spirtes ; lacerda . Two important components in graphical models are
intervention and the do calculus. The theory of graphical models has few
constraints built in on what is physically possible. This leaves the theory
very general.
$\sigma_{1}\rightarrow\sigma_{2}\rightarrow\sigma_{3}\rightarrow\sigma_{4}\cdots\rightarrow\sigma_{i}\cdots\rightarrow\sigma_{k}\cdots\rightarrow\sigma_{n}$
Figure 1: A simple graphical model for a Markov Chain
We use a simple directed cyclic graph to study traveling back in time. First,
we build constraints into our model as follows. Consider physical states
evolving on a timeline as shown in Fig. 1. The graph is a one dimensional
chain, and branching is excluded. Traveling back in time introduces a loop as
in Fig. 2. We do not include intervention and do calculus because this enables
us to simplify our analysis, while capturing the important physics for a
closed system.
$\sigma_{1}\rightarrow\sigma_{2}\rightarrow\sigma_{3}\rightarrow\sigma_{4}\cdots\rightarrow\sigma_{i}\cdots\rightarrow\sigma_{k}\cdots\rightarrow\sigma_{n}$
Figure 2: A simple cyclic graph to model traveling back in time from $t=k$ to
$t=i$.
At each time $t$, the state of the system $\sigma_{t}$ is a random variable.
Time is also discretized and the arrows connect events at neighboring times
$\sigma_{t}\rightarrow\sigma_{t+1}$. The probability of transition from
$\sigma_{t}$ to $\sigma_{t+1}$ is given by $T_{t+1}(\sigma_{t+1}|\sigma_{t})$.
In this case, the conditional probabilities can be interpreted as a transition
matrix, and the graph as a Markov Chain. The following assumptions are used
based on physical considerations:
1. 1.
The statistical time flows in the same direction as the physical time.
2. 2.
Local normalization constraint is enforced, i.e.
$\sum_{\sigma_{t+1}}T_{t+1}(\sigma_{t+1}|\sigma_{t})=1$. Given that the system
is in a state $\sigma_{t}$ at time $t$, the system has to take on a state at
$t+1$. In general, we can condition on more than one variable, e.g.
$T_{t+1}(\sigma_{t+1}|\sigma_{i},\sigma_{j},\cdots)$, then the local
normalization condition is
$\sum_{\sigma_{t+1}}T_{t+1}(\sigma_{t+1}|\sigma_{i},\sigma_{j},\cdots)=1$.
3. 3.
Basic probability axioms are satisfied. Let $A_{i}$ be a set of states and
$P(A_{i})$ be its probability measure, then,
$0\leq P(A_{i})\leq 1$ (1) $P(\Omega)=1,\mbox{\hskip 14.22636pt}$ (2)
$P(A_{i}\cup A_{j})=P(A_{i})+P(A_{j}),$ (3)
$\Omega$ is the set of all possible states and $A_{i}$ and $A_{j}$ are
mutually exclusive. Clearly, for discrete events if $\sigma_{i}\in\Omega$ and
$\sigma_{j}\in\Omega$, $\sigma_{i}\neq\sigma_{j}$, then
$P(\sigma_{i}\cup\sigma_{j})=P(\sigma_{i})+P(\sigma_{j})$. Here, we use a
shorthand notation $\sigma_{i}\equiv\\{\sigma_{i}\\}$.
A sequence of states $\pi_{n}$ is shown in Fig. 1. If the set of all possible
states is given by $\Omega$, then the set of all possible sequences is given
by $\mathbf{\Pi}=\Omega^{n}$. The probability of obtaining $\pi_{n}$ is,
$P_{mc}(\pi_{n})=p(\sigma_{1})T_{2}(\sigma_{2}|\sigma_{1})T_{3}(\sigma_{3}|\sigma_{2})\cdots
T_{n}(\sigma_{n}|\sigma_{n-1})$ (4)
$p(\sigma_{1})$ is the probability of sampling the initial state $\sigma_{1}$.
The conditional probabilities encode the physics of how the system evolve from
state to state. It can be shown that for $P_{mc}(\pi_{n})$, basic axioms of
probabilities hold.
In the case of traveling back in time, the causal relationship has an arrow
that loops back into the past (Fig. 2). To model traveling back in time, we
condition on two states instead of one,
$\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},\sigma_{k})$ where $\sigma_{k}$ is an
event in the future with respect to time $i$. In this case,
$P(\pi_{n})=p(\sigma_{1})T_{2}(\sigma_{2}|\sigma_{1})\cdots\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},\sigma_{k})\cdots
T_{n}(\sigma_{n}|\sigma_{n-1})$ (5)
All the conditional probabilities $T_{j}(\sigma_{j}|\sigma_{j-1})$ are the
same as in Eq. (4) except for
$\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},\sigma_{k})$. Making such a
generalization is non-trivial because we need to check that the basic axioms
of probabilities continue to hold. At this point, we would like to emphasize
some key points that are important in this paper,
1. 1.
Time travel consists of sending a signal back to the past. The signal causes
an effect only at one time point $t=i$ as in Fig. 2. The signal could contain
a set of instructions to carry out some tasks or be an agent that travels back
in time.
2. 2.
The conditional probabilities $T_{j}$, $j=1,2,\cdots$, $j\neq i$ in Eq. (4)
are determined by the physics of how the system evolves forward in time.
3. 3.
The term $\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},\sigma_{k})$ is special as it is
the only term in Eq. (21) that encodes the effects of traveling back in time.
4. 4.
Our framework is probabilistic, in which many sequences of states can happen
with non-zero probability, in contrast to a deterministic view where only one
sequence is possible. Given any sequence $\pi_{n}$, its probability of
occurrence can be calculated using Eq. (21).
5. 5.
A paradox be represented by many different sequences of states. Our objective
is to show that either all these sequences happen with zero probability, or
they result in violation of the basic axioms of probability.
Consider $\hat{T}_{i}$ to be a function of three discrete variables,
$\sigma_{i-1},\sigma_{i}$ and $\sigma_{k}$. This function has to satisfy,
$0\leq\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},\sigma_{k})\leq 1$ (6)
$\sum_{\\{\pi_{n}\\}}P(\pi_{n})=1$ (7)
$\sum_{\sigma_{i}}\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},\sigma_{k})=1$ (8)
The first two conditions are analogous to Eq. (1) and (2). The last condition
is the local normalization condition. Eq. (7) can be reduced to,
$\sum_{\sigma_{i},\sigma_{k}}\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},\sigma_{k})V(\sigma_{k}|\sigma_{i})=1$
(9)
$V(\sigma_{k}|\sigma_{i})$ is the conditional probability of $\sigma_{k}$
given $\sigma_{i}$ summed over all possible intermediate states
$\sigma_{i+1}\cdots\sigma_{k-1}$. Detailed derivation of Eq. (9) is given in
Appendix A. This is an important equation. We will use this equation together
with Eq. (6) and (8) to show that the grandfather paradox, Deutsch’s unproven
theorem paradox and chicken-and-egg paradox have to be precluded in time
travel.
### II.1 Two-state system
For a two-state system, $\sigma$ takes the values $\\{0,1\\}$. Using Eq. (9)
and (8) and summing over four combinations
$\sigma_{i+1},\sigma_{k}\in\\{0,1\\}$, we obtain,
$[\hat{T}_{i}(0|\tilde{\sigma}_{i-1},1)-\hat{T}_{i}(0|\tilde{\sigma}_{i-1},0)][V(1|0)-V(1|1)]=0$
(10)
We must have $V(1|0)=V(1|1)$ or
$\hat{T}_{i}(0|\tilde{\sigma}_{i},1)=\hat{T}_{i}(0|\tilde{\sigma}_{i},0)$. For
the case when $V(1|0)\neq V(1|1)$, the transition matrix $\hat{T}_{i}$ does
not depend on $\sigma_{k}$. In this case, the backward loop in Fig. 2 has no
effect. We can’t change the probability distribution of the past. For the case
$V(1|0)=V(1|1)$, we could have
$\hat{T}_{i}(0|\tilde{\sigma}_{i-1},1)\neq\hat{T}_{i}(0|\tilde{\sigma}_{i-1},0)$
and the transition probabilities at $t=i$ could be affected by a signal from
future time ($t=k$).
### II.2 Grandfather paradox in a two-state system
The grandfather paradox can be used to illustrate the physical implications of
Eq. (10). The basic assumptions we will use are (i) resurrection is
impossible, and (ii) basic axioms of probabilities must be satisfied.
Consider an agent sending a signal back in time to kill himself. Let us denote
the dead state as $\sigma=0$ and alive state as $\sigma=1$. No resurrection
implies that $V$ is of the form, $V=\left(\begin{array}[]{cc}1&\beta^{*}\\\
0&\beta\end{array}\right),$ $\beta^{*}=1-\beta$. Let
$\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},1)=S(\sigma_{i}|\tilde{\sigma}_{i-1})$
be the transition probabilities for the scenario in which the agent sends a
signal from the future to kill himself. Let
$\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},0)=N(\sigma_{i}|\tilde{\sigma}_{i-1})$
be the transition probabilities for the sequences of events the agent is dead
at $t=k$ and hence no signal is sent from the future to kill himself. Hence
$S$ (the “killing” matrix) and $N$ are of the form,
$S=\left(\begin{array}[]{cc}1&1\\\ 0&0\end{array}\right)\mbox{\hskip
17.07182pt}N=\left(\begin{array}[]{cc}1&b^{*}\\\ 0&b\end{array}\right)$ (11)
$b^{*}=1-b$ is the probability of dying at $t=i$. Substituting values of $N$,
$S$ and $V$ into Eq. (10), we obtain $[1-b^{*}]\beta=0$. Either $b^{*}=1$ or
$\beta=0$. When $b^{*}=1$ then $N=S$, the agent dies at $t=i$ with probability
1. If $\beta=0$, the agent dies sometime between $t=i$ and $t=k$ with
probability 1. In either case, the scenario in which the agent is alive at
$t=k$ and thus able to send the signal occurs with zero probability. Note that
we are analyzing probabilities rather than specific events.
The conclusion comes about because resurrection is impossible ($V(1|0)=0$).
Suppose resurrection is possible, $V(1|0)=\alpha^{*}<1$, the paradox goes away
when $\alpha^{*}=\beta$. Intuitively, if we allow resurrection, the agent
could send a signal back in time from $t=k$ to kill himself at $t=i<k$.
Between the time $t=i$ and $t=k$, the agent is resurrected and hence could
again send the signal at $t=k$. There is no contradiction in this case.
Another way to resolve the paradox is to relax the assumption that the agent
always succeeds to kill himself. In this case, the matrix $S$ is
$\left(\begin{array}[]{cc}1&\lambda^{*}\\\ 0&\lambda\end{array}\right)$,
$\lambda>0$. Eq. (10) gives, $\beta(\lambda-b)=0$. If $\beta=0$, then the
agent dies sometime between $t=i$ and $t=k$. If $\lambda=b$ then $S=N$, the
signal from the future could not change the transition probability at $t=i$.
The agent cannot change his own fate by sending a signal to the past.
### II.3 Deutsch’s unproven theorem paradox
An agent sends a signal containing the proof of a mathematical theorem back in
time. The signal is encoded in a document that the agent reads in future time.
Denote the existence of the proof as $\sigma=0$ and absence of the proof as
$\sigma=1$. A general form of $V$ is,
$V=\left(\begin{array}[]{cc}\alpha&\beta^{*}\\\
\alpha^{*}&\beta\end{array}\right),$ $\alpha^{*}=1-\alpha$,
$\beta^{*}=1-\beta$. The basic assumptions we use are (i) the transition from
$\sigma=1$ to $\sigma=0$ (transition of absence of proof to existence of
proof) happens solely through the signal traveling back in time, and (ii) the
transition from $\sigma=0$ to $\sigma=1$ happens with zero probability (once
the proof is obtained, it never gets lost). Hence $\beta=1$ and
$\alpha^{*}=0$. The transition probabilities are
$\hat{T}_{i}(\sigma_{i}=0|\tilde{\sigma}_{i-1}=1,\sigma_{k}=1)=0$ representing
no signal sent if proof does not exist at $t=k$ ($\sigma_{k}=1$).
$\hat{T}_{i}(\sigma_{i}=0|\tilde{\sigma}_{i-1}=1,\sigma_{k}=0)=1$ represents a
signal being sent when the proof exists at $t=k$. These basic assumptions
contradict with Eq. (10),
$[\hat{T}_{i}(0|1,1)-\hat{T}_{i}(0|1,0)](\alpha^{*}-\beta)=(0-1)(0-1)\neq 0$.
Hence the assumptions are false and Deutsch’s unproven theorem paradox is
precluded.
The paradox can be resolved if we relax the assumptions. Suppose we allow the
possibility that the proof can get lost ($\alpha^{*}\geq 0$) and that the
proof can be derived by some brilliant mathematician $\beta\leq 1$. Then Eq.
(10) can be satisfied if $\alpha^{*}=\beta$. There is no paradox here because
the proof can be sent back in time and subsequently be lost. It can be re-
derived again and be sent back to the past.
$N(1|\tilde{\sigma}_{i})-S(1|\tilde{\sigma}_{i})$$N(2|\tilde{\sigma}_{i})-S(2|\tilde{\sigma}_{i})$$-a/b$
Figure 3: Shaded region shows the possible values of
$N(1|\tilde{\sigma}_{i},0)-S(1|\tilde{\sigma}_{i})$, (x-axis) and
$N(2|\tilde{\sigma}_{i},0)-S(2|\tilde{\sigma}_{i})$, (y-axis).
### II.4 Three-state system
For a three-state system, $\sigma$ takes the values $\\{0,1,2\\}$. For
simplicity, let
$\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},0)=N(\sigma_{i}|\tilde{\sigma}_{i-1})$
and
$\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},1)=\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},2)=S(\sigma_{i}|\tilde{\sigma}_{i-1})$.
Using Eq. (8) and (9),
$\displaystyle[N(1|\tilde{\sigma}_{i-1})-S(1|\tilde{\sigma}_{i-1})][V(0|1)-V(0|0)]+\mbox{}$
(12)
$\displaystyle[N(2|\tilde{\sigma}_{i-1})-S(2|\tilde{\sigma}_{i-1})][V(0|2)-V(0|0)]=$
$\displaystyle 0$
this is an equation of the form $xa+yb=0$ given $a=[V(0|1)-V(0|0)]$ and
$b=[V(0|2)-V(0|0)]$, $x=[N(1|\tilde{\sigma}_{i-1})-S(1|\tilde{\sigma}_{i-1})]$
and $y=[N(2|\tilde{\sigma}_{i-1})-S(2|\tilde{\sigma}_{i-1})]$ can be solved.
There are in general infinitely many solutions. From Eq. (6), the range of
$[N(1|\tilde{\sigma}_{i-1})-S(1|\tilde{\sigma}_{i-1})]$ and
$[N(2|\tilde{\sigma}_{i-1})-S(2|\tilde{\sigma}_{i-1})]$ is bounded by the
shaded region in Fig. 3. Given $a$ and $b$, the set of solutions for $x$ and
$y$ contains all the points on the line shown in Fig. 3. The slope of the line
is given by $-a/b$. $N\neq S$ implies that transition to the state
$\sigma_{i}$ depends on future state $\sigma_{k}$, that is, signals from the
future can affect the probability distribution of the past.
### II.5 The grandfather paradox in a three-state system
Consider the three states represent healthy ($\sigma=2$), sick ($\sigma=1$)
and dead ($\sigma=0$). First, we lay down our assumptions,
1. 1.
Assume resurrection is impossible so that transition from $\sigma=0$ to
$\sigma\neq 0$ happens with zero probability. Then the matrix $V$ is of the
form,
$V=\left(\begin{array}[]{ccc}1&\alpha_{0}&\beta_{0}\\\
0&\alpha_{1}&\beta_{1}\\\ 0&\alpha_{2}&\beta_{2}\end{array}\right)$ (13)
with $\alpha_{0}+\alpha_{1}+\alpha_{2}=1$ and
$\beta_{0}+\beta_{1}+\beta_{2}=1$.
2. 2.
The agent is able to send a signal back in time to kill himself only if he is
not dead at $t=k$.
$\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},1)$ and
$\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},2)$ are the conditional probabilities
that the agent is alive and sends a signal back in time to kill himself. Let,
$\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},1)=\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},2)=S(\sigma_{i}|\tilde{\sigma}_{i-1})$.
$\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},0)$ is the conditional probability that
the agent is dead at $t=k$ and can not send a signal back in time to kill
himself. Let
$\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},0)=N(\sigma_{i}|\tilde{\sigma}_{i-1})$.
Hence $S$ (the “killing” matrix) $N$ are,
$S=\left(\begin{array}[]{ccc}1&1&1\\\ 0&0&0\\\
0&0&0\end{array}\right)\mbox{\hskip
8.5359pt}N=\left(\begin{array}[]{ccc}1&a_{0}&b_{0}\\\ 0&a_{1}&b_{1}\\\
0&a_{2}&b_{2}\end{array}\right)$ (14)
We have from Eq. (12),
$\displaystyle a_{1}(1-\alpha_{0})+a_{2}(1-\beta_{0})$ $\displaystyle=$
$\displaystyle 0$ (15) $\displaystyle b_{1}(1-\alpha_{0})+b_{2}(1-\beta_{0})$
$\displaystyle=$ $\displaystyle 0$
There are four cases in which Eq. (15) is satisfied.
1. 1.
$\alpha_{0}=1$ and $\beta_{0}=1$. Then $V=S$ which means the agent is dead at
$t=k$ with probability 1 (recall that $S$ is the killing matrix).
2. 2.
$\alpha_{0}=1$ and $\beta_{0}<1$. To satisfy Eq. (15), $a_{2}=b_{2}=0$. In
this case the agent is dead at $t=k$ with probability 1 (see Appendix B for
the proof).
3. 3.
$\alpha_{0}<1$ and $\beta_{0}=1$. To satisfy Eq. (15), $a_{1}=b_{1}=0$. In
this case the agent is dead at $t=k$ with probability 1 (see Appendix B for
the proof).
4. 4.
$\alpha_{0}<1$ and $\beta_{0}<1$. Then $a_{1}=a_{2}=b_{1}=b_{2}=0$ and $N=S$
which means the agent is dead at $t=i$ with probability 1.
In all cases, the agent is dead with probability 1 at $t=k$ and hence never
has a chance to send a signal back in time to kill himself. Suppose $S$ is not
the killing matrix (Eq. (14)) or resurrection is possible, then this argument
does not hold, and the agent is able to alter his fate by changing the
probability of being healthy, sick or dead.
### II.6 The chicken-and-egg paradox
Consider the chicken-and-egg paradox in which at time $t=k$, a hen travels
back in time to $t=i$ to lay an egg. The egg hatches into the hen herself. At
this time point, there are two copies of the hen, the older self and the
younger self (the chick). As both copies travel to time $t=k$, the chick grow
older and travels back in time to lay the egg. This paradox seems “self-
consistent” in the sense that there is no contradiction in existence of the
hen and chick from one time point to another. However the problem is the hen
seems to pop out from nowhere.
There are three possible states, hen and chick ($\sigma=0$), hen only
($\sigma=1$) and no hen and no chick ($\sigma=2$). We exclude the state of
chick only, otherwise we would need four states.
There are no hen and no chick initially, hence $\tilde{\sigma}_{i-1}=2$. Let
$\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},1)=\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},2)=N(\sigma_{i}|\tilde{\sigma}_{i-1})$.
This is the case when no chick travels back in time and hence there remains no
hen and no chick at $t=i$. Let
$\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},0)=S(\sigma_{i}|\tilde{\sigma}_{i-1})$,
the chick travels back in time from $t=k$ to $t=i$. The matrices $S$ and $N$
are,
$S=\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&1\\\
0&0&0\end{array}\right)\mbox{\hskip
11.38092pt}N=\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\
0&0&1\end{array}\right)$ (16)
The matrix $V$ is of the form,
$V=\left(\begin{array}[]{ccc}\alpha_{0}&\beta_{0}&0\\\
\alpha_{1}&\beta_{1}&0\\\ \alpha_{2}&\beta_{2}&1\end{array}\right)$ (17)
The first two columns are general expressions with
$\sum_{j=0}^{2}\alpha_{j}=1$ and $\sum_{j=0}^{2}\beta_{j}=1$. The last column
is $(0,0,1)^{T}$ because when there is no hen and no chick at time $t=i$, then
there will be no hen and no chick at $t=k$. Now consider the probability,
$P(\tilde{\sigma}_{i-1},\sigma_{i},\sigma_{k})=p(\tilde{\sigma}_{i-1})\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},\sigma_{k})V(\sigma_{k}|\sigma_{i})$
(18)
$p(\tilde{\sigma}_{i-1})$ is the probability of sampling the state
$\tilde{\sigma}_{i-1}$. Since $\tilde{\sigma}_{i-1}=2$,
$p(\tilde{\sigma}_{i-1})=\delta_{\tilde{\sigma}_{i-1},2}$. We remind the
reader that the probability distribution $V$ is the sum of probabilities over
all possible intermediate sequences. The chicken-and-egg paradox requires both
hen and chick to be present at $t=k$ ($\sigma_{k}=0$) and the chick to appear
at $t=i$ ($\sigma_{i}=1$), all intermediate states can take arbitrary values.
Reading off entries from matrices $S$ and $V$,
$P(\tilde{\sigma}_{i-1}=2,\sigma_{i}=1,\sigma_{k}=0)=\beta_{0}$ (19)
Using Eq. (12) we can calculate what $\beta_{0}$ should be,
$\displaystyle[1-0](\beta_{0}-\alpha_{0})-[0-1]\alpha_{0}$ $\displaystyle=$
$\displaystyle 0$ (20) $\displaystyle\Rightarrow\beta_{0}$ $\displaystyle=$
$\displaystyle 0$
The sum of probabilities of all possible sequences of states that represent
the chicken-and-egg paradox equals zero. Therefore the chicken-and-egg event
happens with zero probability.
## III Discussion
We have shown, using a graphical model with a loop back into the past, that
the grandfather paradox, Deutsch’s unproven theorem paradox and the chicken-
and-egg paradox are precluded in time travel. We have also demonstrated that
changing the probability distributions of the past is possible when no
contradicting events are present. For the paradoxes we discussed in this
paper, we gave scenarios in which the paradoxes are resolved. Our analysis is
based on isolated two- and three-state systems.
For future work, it would be useful to generalize our formalism to arbitrary
systems. Lastly, in cases when the causal relationship between events at
different times are very complex, the existence of time travel paradoxes in
these cases may be very subtle. We hope that our mathematical framework can be
used to uncover new time travel paradoxes, especially those that are embedded
in complex interactions of events and are not obvious.
The author would like to thank Mui Leng Seow and Ivana Mihalek for
proofreading this article.
## IV Appendix A: Derivation of Eq. (9)
Probability of a sequence $\pi_{n}$ is given by,
$P(\pi_{n})=p(\sigma_{1})T_{2}(\sigma_{2}|\sigma_{1})\cdots\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},\sigma_{k})\cdots
T_{n}(\sigma_{n}|\sigma_{n-1})$ (21)
Summing over all sequences,
$\displaystyle\sum_{\\{\pi_{n}\\}}P($ $\displaystyle\pi_{n})=$ (22)
$\displaystyle\sum_{\sigma_{1},\sigma_{2},\cdots,\sigma_{n}}$ $\displaystyle
p(\sigma_{1})T_{2}(\sigma_{2}|\sigma_{1})\cdots\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},\sigma_{k})\cdots
T_{n}(\sigma_{n}|\sigma_{n-1})$
Since $\sum_{\sigma_{j}}T_{j}(\sigma_{j}|\sigma_{j-1})=1\forall j$, summation
can be evaluated recursively between $\sigma_{k+1}$ and $\sigma_{n}$. That is,
$\sum_{\sigma_{k+1},\cdots\sigma_{n}}T_{k+1}(\sigma_{k+1}|\sigma_{k})\cdots
T_{n}(\sigma_{n}|\sigma_{n-1})=1$ (23)
Next define,
$U(\sigma_{i-1})=\sum_{\sigma_{1},\cdots\sigma_{i-2}}p(\sigma_{1})T_{2}(\sigma_{2}|\sigma_{1})\cdots
T_{i-1}(\sigma_{i-1}|\sigma_{i-2})$ (24)
$V(\sigma_{k}|\sigma_{i})=\sum_{\sigma_{i+1},\cdots\sigma_{k-1}}T_{i+1}(\sigma_{i+1}|\sigma_{i})\cdots
T_{k}(\sigma_{k}|\sigma_{k-1})$ (25)
Then Eq. (22) becomes,
$\sum_{\\{\pi_{n}\\}}P(\pi_{n})=\sum_{\sigma_{i-1},\sigma_{i},\sigma_{k}}U(\sigma_{i-1})\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},\sigma_{k})V(\sigma_{k}|\sigma_{i})$
(26)
The objective is to find the conditions in which
$\sum_{\\{\pi_{n}\\}}P(\pi_{n})=1$. $U(\sigma_{i-1})$ is the probability of
sampling the state $\sigma_{i-1}$, it depends on the conditional probabilities
$T_{j}$, $j\leq i-1$ and the initial condition $p(\sigma_{1})$. We therefore
have the freedom to choose $U$ for example, by choosing different initial
conditions. Holding $T$ and $V$ fixed, we require $\sum P(\pi_{n})=1$ for
different choices of $U$, in which we arrive at,
$\sum_{\sigma_{i},\sigma_{k}}\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},\sigma_{k})V(\sigma_{k}|\sigma_{i})=1$
(27)
## V Appendix B: The grandfather paradox in a three-state system
We present the proof that for the grandfather paradox in a three-state system,
the probability that the agent is dead at $t=k$ is one. We consider cases II
and III in which Eq. (15) is satisfied,
### V.1 Case II: $\alpha_{0}=1$ and $\beta_{0}<1$
In this case, $a_{2}=b_{2}=0$ and,
$V=\left(\begin{array}[]{ccc}1&1&\beta_{0}\\\ 0&0&\beta_{1}\\\
0&0&\beta_{2}\end{array}\right)$ (28)
$N=\left(\begin{array}[]{ccc}1&a_{0}&b_{0}\\\ 0&a_{1}&b_{1}\\\
0&0&0\end{array}\right)$ (29)
We calculate the probability that the agent is dead,
$\displaystyle P(\sigma_{k}=0)$ $\displaystyle=$
$\displaystyle\sum_{\sigma_{1},\cdots,\sigma_{k-1}}p(\sigma_{1})T_{2}(\sigma_{2}|\sigma_{1})\cdots$
(30) $\displaystyle=$
$\displaystyle\sum_{\sigma_{i-1},\sigma_{i}}U(\sigma_{i-1})\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},0)V(0|\sigma_{i})$
$\displaystyle=$
$\displaystyle\sum_{\sigma_{i-1},\sigma_{i}}U(\sigma_{i-1})N(\sigma_{i}|\sigma_{i-1})V(0|\sigma_{i})$
Reading off the entries of matrices $V$ and $N$ in Eq. (28) and (29), we get
$\sum_{\sigma_{i}}N(\sigma_{i}|\sigma_{i-1})V(0|\sigma_{i})=1$ for all
$\sigma_{i-1}$. Hence $P(\sigma_{k}=0)=1$.
### V.2 Case III: $\alpha_{0}<1$ and $\beta_{0}=1$
In this case, $a_{1}=b_{1}=0$ and,
$V=\left(\begin{array}[]{ccc}1&\alpha_{0}&1\\\ 0&\alpha_{1}&0\\\
0&\alpha_{2}&0\end{array}\right)$ (31)
$N=\left(\begin{array}[]{ccc}1&a_{0}&b_{0}\\\ 0&0&0\\\ 0&a_{2}&b_{2}\\\
\end{array}\right)$ (32)
We calculate the probability that the agent is dead, using Eq. (30) and
reading off the entries of matrices $V$ and $N$ in Eq. (31) and (32), we get
$\sum_{\sigma_{i}}N(\sigma_{i}|\sigma_{i-1})V(0|\sigma_{i})=1$ for all
$\sigma_{i-1}$. Hence $P(\sigma_{k}=0)=1$.
## References
* (1) J. Friedman, M. S. Morris, I. D. Novikov, F. Echeverria, G. Klinkhammer, K. S. Thorne, U. Yurtsever, Phys. Rev. D 42, 1915 (1990)
* (2) J. R. Gott, Phys. Rev. Lett. 66, 1126 (1991)
* (3) K. Godel, Rev. Mod. Phys. 21, 447-450 (1949)
* (4) W. B. Bonnor, J. Phys. A 13, 2121 (1980)
* (5) M. S. Morris, K. S. Thorne, U. Yurtsever, Phys. Rev. Lett. 61, 1446-1449 (1988)
* (6) H. D. Politzer, Phys. Rev. D 49, 3981 (1994)
* (7) D. G. Boulware, Phys. Rev. D 46, 4421-4441 (1992)
* (8) J. B. Hartle, Phys. Rev. D 49, 6543-6555 (1994)
* (9) H. D. Politzer, Phys. Rev. D 46, 4470-4476 (1992)
* (10) D. Deutsch, Phys. Rev. D 44, 3197-3217 (1991)
* (11) S. Lloyd, L. Maccone et al, arXiv:1005.2219, arXiv:1007.2615
* (12) D. T. Pegg, arXiv:quant-ph/0506141v1.
* (13) G. Svetlichny, arXiv:0902.4898v1.
* (14) I. D. Novikov, Phys. Rev. D 45, 1989 (1992)
* (15) S. M. Carroll, E. Farhi, A. H. Guth, Phys. Rev. Lett. 68, 263 (1992)
* (16) S. W. Hawking, Phys. Rev. D 46, 603 (1992)
* (17) S. Deser, R. Jackiw, G. ’t Hooft, Phys. Rev. Lett. 68, 267-269 (1992)
* (18) S. M. Carroll, E. Farhi, A. H. Guth, K. D. Olum, Phys. Rev. D 50, 6190-6206 (1994)
* (19) J. A. Wheeler, R. P. Feynman, Rev. Mod. Phys. 21, 425 (1949)
* (20) J. Pearl, Causality (Cambridge University Press 2000)
* (21) P. Sprites, C. Glymour, R. Scheines, Causation, Prediction and Search (MIT Press, Cambridge 2000)
* (22) W. C. Salmon, Scientific Explanation and the Causal Structure of the World (Princeton University Press, Princeton, 1984).
* (23) S. L. Lauritzen, Graphical Models (Oxford University Press, New York, 1996).
* (24) S. L. Morgan and C. Winship, Counterfactuals and Causal Inference: Methods and Principles for Social Research (Cambridge University Press, Cambridge, England, 2007).
* (25) G. Rebane, J. Pearl, Proc. Uncertainty in Artificial Intelligence Conf. 222-228 (1987)
* (26) S. Lauritzen, In Complex Stochastic Systems (eds. Barndorff-Nielsen, Cox and Kluppelberg) (63-107 CRC Press 2000)
* (27) G. F. Cooper, E. Herskovits, Proc. Uncertainty in Artificial Intelligence Conf. 86-94 (1991)
* (28) T. Richardson, Int. J. Approx. Reasoning 17, 107-162 (1997)
* (29) T. Richardson, Proc. Uncertainty in Artificial Intelligence Conf. 462-469 (1996)
* (30) M. Schmidt, K. Murphy Proc. Uncertainty in Artificial Intelligence Conf. (2009)
* (31) P. Spirtes Proc. Uncertainty in Artificial Intelligence Conf. (1995)
* (32) G. Lacerda, P. Spirtes, J. Ramsey, P. Hoyer, Proc. Uncertainty in Artificial Intelligence Conf. (2008)
|
arxiv-papers
| 2011-01-10T19:13:01 |
2024-09-04T02:49:16.281471
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hwee Kuan Lee",
"submitter": "Hwee Kuan Lee",
"url": "https://arxiv.org/abs/1101.1927"
}
|
1101.1942
|
arxiv-papers
| 2011-01-10T20:27:45 |
2024-09-04T02:49:16.288116
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mehdi Rafie-Rad",
"submitter": "Mehdi Rafie-Rad",
"url": "https://arxiv.org/abs/1101.1942"
}
|
|
1101.2094
|
On Integrability of Type 0A Matrix model in the presence of D brane
Chandrima Paul 111mail:chandrima@phy.iitb.ac.in
_Department of Physics,
Indian Institute of Technology Bombay_,
_Mumbai 400 076, India
_
We consider type 0A matrix model in the presence of spacelike D brane which is
localized in matter direction at any arbitrary point. In string theory, the
boundary state which in matrix model corresponds to the Laplace transform of
the macroscopic loop operator, is known to obey the operator constraints
corresponding to open string boundary condition. When we analyze MQM as well
as the respective collective field theory and compare it with dual string
theory it appears that consistency of the theory requires a condition
equivalent to a constraint on the matter part that needed to be imposed in the
matrix model. We identified this condition and observed that this has only
effect into constraining the macroscopic loop operator so that it projects the
Hilbert space generated by the operator to its physical sector at the point of
insertion while keeping the bulk matrix model remains unaffected, thereby
describing a situation parallel to string theory. We analyzed the theory with
uncompactified time and have shown explicitly that the matrix model
predictions are in good agreement with the relevant string theory. Next we
considered the theory with compactified time, analyzed MQM on a circle in the
presence of D brane. We evaluated the partition function along with the
constrained macroscopic loop operator in the grand canonical ensemble and
showed the free energy corresponds to that of a deformed Fermi surface. We
have compared the matrix model features with that of the relevant string
theory. We have also shown that the path integral in the presence of D brane
can be expressed as the Fredholm determinant. We have studied the fermionic
scattering in a semiclassical regime. Finally we considered the compactified
theory in the presence of the D brane with tachyonic background. From the
collective field theory analysis we have predicted the right structure of the
theory in the presence of D brane. We evaluated the free energy in the grand
canonical ensemble. We have shown the integrable structure of the respective
partition function and it corresponds to the tau function of Toda hierarchy.
We have also analyzed the dispersionless limit.
###### Contents
1. 1 Introduction
2. 2 Type 0A MQM in the presence of the D-brane
1. 2.1 Type 0A MQM
2. 2.2 Type 0A MQM in the presence of the D brane : The constraint from dual string theory
3. 2.3 String theoretical interpretation
3. 3 Type 0A MQM on a circle in the presence of D brane
1. 3.1 Evaluation of the free energy of Type 0A matrix model on a circle
2. 3.2 Free energy of Type 0A matrix model on a circle with D brane
3. 3.3 Evaluation of the thermal partition function
4. 3.4 String theoretical interpretation
4. 4 Fermionic scattering and semiclassical analysis
5. 5 Perturbation by momentum modes
1. 5.1 Collective field theory analysis
2. 5.2 Lax Formalism
3. 5.3 String theory on a circle with the D brane in the presence of tachyonic background
4. 5.4 Lax formalism for Type 0A MQM in the presence of D brane
5. 5.5 Representation in terms of a bosonic field
6. 5.6 The dispersionless (quasiclassical) limit
6. 6 Conclusion
7. A Appendix
1. A.1 Orthogonality Condition
2. A.2 Biorthogonality Relation
## 1 Introduction
The two dimensional string theory (see e.g. [1], [2], [3] for reviews) is a
very instructive model when we would like to understand the nature of string
theory as a complete theory of quantum gravity. This theory has a powerful
dual description of $c=1$ matrix model defined by the simple quantum mechanics
of a Hermitian matrix $\Phi$ with the inverse harmonic oscillator potential
$U(\Phi)=-\Phi^{2}$ after the double scaling limit.Matrix model is
successfully used to describe 2D string theory in the simplest linear dilaton
background as well as to incorporate perturbations. In last decade the $c=1$
matrix quantum mechanics has received lots of attention because of its new
interpretation as the decoupled world volume theory of unstable D0-branes [4,
6, 5]. The matrix model dual to type 0 string theories were also proposed in
[8, 9]. In particular, the matrix model dual of the two dimensional type 0
string gives a non-perturbatively well-defined formulation. For example, the
type 0B model is defined by the hermitian matrix model with two Fermi
surfaces. The type 0B matrix quantum mechanics (MQM) describes open string
tachyons living on the unstable D0-branes, whereas the type 0A MQM describes
tachyonic open strings stretched between stable D0- and anti-D0-branes. Upon
compactification on Euclidean time, these two matrix models are conjectured to
be T-dual to each other. The exact agreement in free energy was found in [9].
Matrix model dual of type 0 string in the flux background was explored in [29,
30]. However, unlike $c=1$ matrix model which can be derived from discretizing
the Polyakov action on the string world sheet, such a derivation is not known
for type 0 matrix models. An attempt was made in [13] to obtain the exact form
of the macroscopic loop operator in Type 0 string theory. If we consider the
bosonic string partition function
$\int
D{\phi}DX\,\,exp\left[-\int{d^{2}}z[\,{\frac{1}{4\pi}}({\partial{X}\partial{X}}+{\partial{\phi}\partial{\phi}})+QR{\phi}+{\mu}e^{2b\phi}]-\,\int_{\partial\Sigma}{d\xi}\,\,[{\frac{Qk\phi}{2\pi}}+{\mu_{B}}e^{b\phi}\,]\,\,\right],$
(1.1)
the macroscopic loop operator inserts an operator
${W(t,l)\sim\delta\left(\int_{\partial\Sigma}e^{\phi}-l\right)\cdot\delta(X^{0}-t)},$
(1.2)
within the path integral [10, 11].
$\langle W(l)\rangle=Z(l)=\int DXD\phi D[\,{\rm
ghost}\,]\,\,\delta\left(\int_{\partial\Sigma}e^{\phi}-l\right)\cdot\delta(X^{0}-t)\,f(x,\phi)\,Z(\phi(\sigma),X(\sigma),[{\rm
ghost}]),$ (1.3)
where f is some wave function for matter ghost and Liouville. The physical
meaning of this operator in two dimensional string theory is the presence of a
‘Euclidean D-brane’ localized in the time direction. To be more precise after
we take the Laplace transformation $\int d\phi e^{-\mu_{B}e^{\phi}}$, we get a
D-brane with the Neumann boundary condition in the Liouville direction and the
Dirichlet one in the time direction
$\int{\frac{dl}{l}}e^{-\mu_{B}l}\
W_{bos}(t,l)\simeq|B_{(FZZT)}(\mu_{B})\rangle_{\phi}\otimes|D\rangle_{X^{0}}.$
(1.4)
when we impose the condition that boundary Liouville term is zero.
${\partial_{n}}{\phi}+{\mu_{B}}{e^{b\phi}}=0$ (1.5)
Where ${\partial_{n}}{\phi}$ denotes the Liouville momentum normal to boundary
while along the boundary we have ${\partial_{t}}{X^{o}}=0$. Now consider 2D
superstring action obtained from extending the bosonic fields to their
superspace and expanding the 2D superspace action in terms of the component
field,
$S={\frac{1}{2\pi}}\int{d^{2}}z[{\delta_{\mu\nu}}({\partial}{X^{\mu}}{\overline{\partial}}{X^{\nu}}+{\psi^{\mu}}{{\overline{\partial}}{\psi^{\nu}}}+{\overline{\psi^{\mu}}}{\partial}{\overline{\psi^{\nu}}})+{\frac{Q}{4}}R{X^{1}}]+2i\mu{b^{2}}\int{d^{2}}z({\psi^{1}}{\overline{\psi^{1}}}+2\pi\mu{e^{\phi}}):e^{\phi}:$
(1.6)
We can also consider the macroscopic loop operator which is the superspace
analogue of $W_{bos}(t,l)$, inserts the boundary condition on the fermionic
coordinate ${\overline{\psi}}(\overline{z})={\eta}{\psi}(z)$ where ${\eta}=\pm
1$ describes the RR and NS NS sector. Laplace transform of this operator
inside the string path integral describes the boundary states NS NS and RR
sector. However depending on helicity, in each sector we have two types of
boundary states ${\epsilon=\pm}$ so that we have four types of macroscopic
loop operator given by ${W_{NS}^{+}},{W_{NS}^{-}},{W_{R}^{+}},{W_{R}^{-}}$.
The parameter ${\mu_{B}}$ corresponds to the boundary cosmological constant in
the boundary state. Indeed we can show this relation [33, 34] by computing one
point function on the brane or equally annulus amplitude as shown in [12]. For
$c=1$ matrix model the expression of these operators were obtained and its
equivalence to string theory is verified in [10, 11, 12]. Author of [13]
obtained the expressions of macroscopic loop operator in Type 0B matrix model
and also for ${NS}$ sector of Type 0A matrix model which was verified by
calculating the one point function. Now once we understand the duality between
noncritical string theory in the linear dilaton background and Matrix model,
its natural to ask whether we can understand the string theory with nontrivial
background which has an obvious realization in matrix model by adding
perturbations which survive in the double scaling limit. There are two ways to
change the background of string theory: either to consider strings propagating
in a non-trivial target space or to introduce the perturbations . In the first
case one arrives at a complicated sigma-model. Not many examples are known
when such a model turns out to be solvable. Besides, it is extremely difficult
to construct a matrix model realization of a general sigma-model since not
much known about matrix operators explicitly perturbing the metric of the
target space. Thus, we lose the possibility to use the powerful matrix model
machinery to tackle our problems. On the other hand, following the second way,
we find that the integrability of the theory in the trivial background is
preserved by the perturbations. Also when we study the theory in a nontrivial
background in most of the cases the target space metric of such backgrounds is
curved and often it incorporates the black hole singularities. In the
superstring theories, the supersymmetry allows for some interesting nontrivial
solutions which are stable and exact. But the string theory on such
backgrounds is usually an extremely complicated sigma-model, very difficult
even to formulate it explicitly, not to mention studying quantitatively its
dynamics. The two-dimensional bosonic string theory as well as Type 0 theory
are the rare cases of sigma-model where such a dynamics is integrable, at
least for some particular backgrounds, including the dilatonic black hole
background. A physically transparent way to study the perturbative (one loop)
string theory around such a background is provided by the CFT approach.
However once we try to understand higher loops or multipoint correlators, we
have to address ourselves to the matrix model approach to the 2D string theory
. The 2D string theory has been constructed as the collective field theory
[25],[28], in which the only excitation, the massless tachyon, was related to
the eigenvalue density of the matrix field. Now consequence of the deformation
in eigenvalue density corresponding to deformation in string background at
classical limit was studied in [26]. $C=1$ string theory perturbed by
tachyonic mode studied in [32]. Vortex perturbation and its equivalence to
sine -Liouville theory was studied in [36],[37]. Its shown that partition
function is integrable and have Toda structure. Toda structure and Lax
formalism in the context of matrix model described in [31]. Many more works in
this direction was done in [15, 16, 17, 18, 19, 21, 22, 23].
Now its an interesting question to ask that can we study this nontrivial
background in the context of dual matrix model in the presence of D brane
which are just the Laplace transform of the macroscopic loop operator as we
discussed. Open close duality predicts that partition function must have
integrable structure. So we consider Type 0A MQM with simplest macroscopic
loop operator, which is the operator in NS sector(as prescribed in [13]),
which is localized in time direction and with it we show that partition
function indeed have an integrable structure.We obtain the string equation.
The plan of this work is in section 2 we are going to consider basic Type 0A
MQM in the presence of D brane, with uncompactified time, We are going to
introduce a no leakage condition to matrix model which is equivalent to some
constraint to boundary state of string theory. We are going to explain its
origin as well as its string theoretical interpretation. In section 3 we
consider MQM compactified on a circle in the presence of D brane. From path
integral approach we are going to show that the partition function can be
expressed as Fredholm determinant. We have explicitly evaluated the thermal
partition function and have shown that without application of this constraint
the partition function diverge. In section 4 we have discussed scattering in
semiclassical regime. In section 5 we have considered string theory in the
presence of momentum modes and have shown that the partition function in this
background have an integrable structure if we apply this constraint.
## 2 Type 0A MQM in the presence of the D-brane
### 2.1 Type 0A MQM
Let us start with the MQM of type 0A theory in two dimensions, which is the
decoupled world volume theory of (stable) D0 –brane and anti D0–branes. A
spacelike D0 –$\overline{D0}$ pair, i.e with Neumann boundary condition in
Liouville direction and Dirichlet in matter direction, gives a macroscopic
loop observable of the matrix model after Laplace transformation [9]. We are
going to consider Type 0A MQM in the presence of this operator and study the
relevant physics. Here is a brief review of the Type 0A MQM. In the background
with no net D0-brane charges, the matrix model has $U(N)\times U(N)$ gauge
symmetry. This is the case we are going to consider. We have the $U(N)\times
U(N)$ gauge field $A_{0}$ and bifundamental tachyon $\Phi$,
${A_{o}}=\left(\begin{array}[]{cc}A&0\\\ 0&\tilde{A}\\\ \end{array}\right),$
(2.1) ${\Phi}=\left(\begin{array}[]{cc}0&M\\\ {M^{\dagger}}&0\\\
\end{array}\right).$ (2.2)
The action is
$\int
dtTr\left[{({D_{o}}M)}^{\dagger}{{D_{o}}M}+{\frac{1}{2\alpha^{\prime}}}{M^{\dagger}}M\right],$
(2.3)
where
${D_{o}}M={\partial_{o}}M+iAM-iM{\overline{A}}.$ (2.4)
As M is a complex matrix so we denote M by Z, ${M^{\dagger}}=\overline{Z}$
$\displaystyle{D_{o}}Z={\partial_{o}}Z+iAZ-iZ{\tilde{A}}$
$\displaystyle\quad;\quad$
$\displaystyle{({D_{o}}M)}^{\dagger}={{\overline{D}}_{o}}{\overline{Z}}={\partial_{o}}\overline{Z}+i{\tilde{A}}\overline{Z}-i\overline{Z}A.$
(2.5)
Now define
${{Z}_{\pm}}={\frac{1}{\sqrt{2\alpha^{\prime}}}}Z\pm{D_{o}}Z\quad;\quad{{\overline{Z}}_{\pm}}={\frac{1}{\sqrt{2\alpha^{\prime}}}}\overline{Z}\pm{\overline{D}_{o}}\overline{Z}.$
(2.6)
The Type 0A matrix model action in terms of the light cone variable
$S=\int
dtTr\left[{\overline{Z}_{+}}{D_{A}}{Z_{-}}+Z_{+}\overline{({D_{A}}Z)}+{\frac{1}{2}}(\overline{Z}_{-}Z_{+}+\overline{Z}_{+}Z_{-})\right].$
(2.7)
The gauge field A acts as a lagrange multiplier which projects the theory onto
singlet wave functions. Its shown in [15, 9] that type 0A MQM when projected
to singlet sector can be represented by non-relativistic free fermions moving
in a two dimensional upside-down harmonic oscillator potential
$\hat{H}={\frac{1}{2}}(\hat{p}_{x}^{2}+\hat{p}_{y}^{2})-{\frac{1}{4\alpha^{\prime}}}(\hat{x}^{2}+\hat{y}^{2}).$
(2.8)
The theory has different independent sectors labeled by net D0-brane charge q,
which is the same as the angular momentum
$\hat{J}=\hat{x}\hat{p}_{y}-\hat{y}\hat{p}_{x}$ [9]. Here we will consider the
case where there is no net D0 –brane charge, namely the $J=0$ sector. Now with
$z,{\overline{z}}=x\pm iy;$ and light cone variable are as defined in (2.6 )
we have the hamiltonian
$\displaystyle{H_{o}}$ $\displaystyle=$
$\displaystyle-{\frac{1}{2}}({\hat{z}_{+}}{\hat{\overline{z}}_{-}}+{\hat{\overline{z}}_{+}}{\hat{z}_{-}}-{\frac{2i}{\sqrt{2{\alpha^{\prime}}}}})$
(2.9) $\displaystyle=$
$\displaystyle{\mp}{\frac{i}{\sqrt{2{\alpha^{\prime}}}}}[{z_{\pm}}{\frac{\partial}{\partial{z_{\pm}}}}+{\overline{z}_{\pm}}{\frac{\partial}{\partial{\overline{z}_{\pm}}}}+1].$
The commutation relation satisfied by these operators
$\displaystyle[{\hat{z}_{+}},{\hat{\overline{z}}_{-}}]=[{\hat{\overline{z}}_{+}},{\hat{z}_{-}}]=2{\frac{i}{\sqrt{2{\alpha^{\prime}}}}},$
$\displaystyle[{\hat{z}_{+}},{\hat{z}_{-}}]=[{\hat{\overline{z}}_{+}},{\hat{\overline{z}}_{-}}]=0,$
(2.10)
so that
${\hat{\overline{z}}_{+}}=-{\frac{\partial}{\partial{z_{-}}}}{\quad\quad;\quad\quad}{\hat{z}_{-}}={\frac{\partial}{\partial{\overline{z}_{+}}}}.$
(2.11)
We have Schrodinger equation
$\displaystyle i{\frac{\partial}{\partial
t}}{\Psi}({\overline{z}_{+}},{z_{+}},t)$ $\displaystyle=$
$\displaystyle{\mp}{\frac{i}{\sqrt{2{\alpha^{\prime}}}}}[({z_{+}}{\frac{\partial}{\partial{z_{+}}}}+{\overline{z}_{+}}{\frac{\partial}{\partial{\overline{z}_{+}}}}+1]{\Psi}({z_{+}},{\overline{z}_{+}},t).$
(2.12)
Note, here we have absorbed the Vandermonde determinant in the wave function
so that the wave function ${\psi}$ in (2.12) describes a fermion.
### 2.2 Type 0A MQM in the presence of the D brane : The constraint from
dual string theory
Consider the type 0A matrix model in the presence of D brane which arises when
we insert an operator ${e^{\int dtW(t)\delta(t-{t_{o}})}}$ in the matrix model
path integral where $W(t)$ is the Laplace transform of the macroscopic loop
operator([13], [18]). In the dual two dimensional type 0A theory this means
that there is one Euclidean $D0\textendash\overline{D0}$ brane is localized at
time $t_{0}$. The branes extends along the Liouville direction after the
Laplace transformation. The macroscopic operators can be divided into NSNS and
RR sector part such that they correspond to the NSNS and RR sector part of the
D-brane boundary state. Moreover,since we know that there are two types of
(FZZT-like) boundary states $|B(\epsilon)\rangle$ according to the spin
structures there should be two macroscopic operators $W^{(\epsilon)}$ with
$\epsilon=\pm$ in each sector. First consider the simplest expression of
macroscopic loop operator which is the one in NS NS sector as prescribed in
[13] and expressed as ${e^{-l{M^{\dagger}}M({t_{o}})}}$. Now, consider the
Laplace transform of the operator
$\displaystyle\int{\frac{dl}{l}}e^{-{\mu_{B}^{2}}l}{e^{-l{M^{\dagger}}M}}$
$\displaystyle=$ $\displaystyle-
Tr\log(1+{\frac{{M^{\dagger}}M}{\mu_{B}^{2}}})$ (2.13) $\displaystyle=$
$\displaystyle-Tr\log(1+{\frac{\overline{Z}Z}{\mu_{B}^{2}}})$ $\displaystyle=$
$\displaystyle-\sum\log(1+{\frac{\overline{z}z}{\mu_{B}^{2}}})$
$\displaystyle=$
$\displaystyle-\sum\log(1+{\frac{({{z}_{+}}+{{z}_{-}})({{{\overline{z}}}_{+}}+{{{\overline{z}}}_{-}})}{\mu_{B}^{2}}})$
$\displaystyle=$
$\displaystyle-\sum\log(1+{\frac{{{{\overline{z}}}_{+}}{{z}_{+}}+{{{\overline{z}}}_{-}}{{z}_{-}}+{{{\overline{z}}}_{+}}{{z}_{-}}+{{{\overline{z}}}_{-}}{{z}_{+}}}{\mu_{B}^{2}}})$
$\displaystyle=$ $\displaystyle
W({\overline{z}_{+}},{z_{+}},{\overline{z}_{-}},{z_{-}}).$ (2.15)
(Here $\sum$ implies sum over the eigenvalues ). Now the macroscopic loop
operator for ${{NS}^{-}}$ sector can be expressed as
$\displaystyle W$ $\displaystyle=$
$\displaystyle-\sum\log(1+{\frac{{{{\overline{z}}}_{+}}{{z}_{+}}+{{{\overline{z}}}_{-}}{{z}_{-}}+{{{\overline{z}}}_{+}}{{z}_{-}}+{{{\overline{z}}}_{-}}{{z}_{+}}}{\mu_{B}^{2}}})$
(2.16) $\displaystyle=$
$\displaystyle-\sum\\{{\displaystyle\sum_{n=1}^{\infty}}{\frac{{(-1)}^{n}}{n}}{{[{\frac{{{{\overline{z}}}_{+}}{{z}_{+}}+{{{\overline{z}}}_{-}}{{z}_{-}}+{{{\overline{z}}}_{+}}{{z}_{-}}+{{{\overline{z}}}_{-}}{{z}_{+}}}{\mu_{B}^{2}}}]}^{n}}\\}.$
The path integral over the Euclidean time in the presence of D brane is
expressed as
$\int\prod{dZ_{+}}d{Z_{-}}d{\overline{Z}_{+}}d{\overline{Z}_{-}}dAd{\tilde{A}}{e^{-\int
dt[\beta
L(Z_{+},\overline{Z}_{+},Z_{-},\overline{Z}_{-},A,{\tilde{A}})-W\delta(t-{t_{o}})]}},$
(2.17)
apparently implies a shift222Note that as
$W({\overline{z}_{+}},{z_{+}},{\overline{z}_{-}},{z_{-}})$ in any sector
expressed in light cone variable, so does not involve any derivative. Hence we
can just add it to hamiltonian or lagrangian as a potential localized in
$t_{o}$ 333Note, when we are adding the term $W\delta(t-{t_{o}})$ in the
expression of the hamiltonian from the operator ${e^{W({t_{o}})}}$ it is
supposed to add in the hamiltonian the terms like
$[\beta\int_{{t_{o}}-\epsilon}^{{t_{o}}+\epsilon}dtH,\int_{{t_{o}}-\epsilon}^{{t_{o}}+\epsilon}d{t^{\prime}}W\delta({t^{\prime}}-{t_{o}})]$
\+ ….higher commutators
$=e^{-W(t_{o})}[\,\beta\int_{{t_{o}}-\epsilon}^{{t_{o}}+\epsilon}dtH\,]e^{W(t_{o})}+...O({\beta\epsilon})^{2}..\,$.
However as around the ${\delta-{\rm function}}$, $\epsilon$ can be made
arbitrarily small i.e $\epsilon<<{\frac{1}{\beta}}$, so upto quasiclassical
limit one can just put these terms to zero while in the classical limit these
terms are trivially zero. in the free single fermion hamiltonian $\beta
H\rightarrow\beta H+W\delta(t-{t_{o}})$ (in Euclidean time). Note here we are
going to consider complete quantum theory along with the macroscopic loop
operator W. However before proceeding note that the operator $e^{\int
dtW\delta(t-{t_{o}})}$ which is localized at $t={t_{o}}$, in general breaks
the time translation symmetry of MQM action. So as far as matrix model action
in the presence of brane is concerned (as considered in [13]) this is
describing leakage of energy exactly at $t_{o}$. Now to be more precise
consider MQM path integral in the presence of an operator $e^{W(t_{o})}$, for
which under any infinitesimal variation in time $t\rightarrow t+\epsilon(t)$,
Ward identity implies 444Here we have used the fact that equation of motion in
the presence of an insertion $e^{W(t_{o})}$ is satisfied
$\displaystyle\delta(t-{t_{o}})\delta\langle e^{W(t_{o})}\rangle$
$\displaystyle=$
$\displaystyle-\partial_{t}\langle{H_{o}}(t)\,e^{W(t_{o})}\rangle$ (2.18)
$\displaystyle\Rightarrow$ $\displaystyle\langle\,\delta
W({t_{o}})\,e^{W(t_{o})}\,\rangle+\lim_{\epsilon\rightarrow
0}\int_{t_{o}-\epsilon}^{t_{o}+\epsilon}\partial_{t}\langle{H_{o}}(t)\,e^{W(t_{o})}\,\rangle,$
where $\delta W$ is the variation of W due to infinitesimal time translation
at fixed time ${t_{o}}$. The above identity arises when we consider the first
order(in $\epsilon$) variation. However $e^{W(t_{o})}$ is a coherent source of
the operator $W(t_{o})$ and on expansion generates infinitely many source W in
the path integral. So in principle we sum up the contribution from every order
of $\epsilon$ this gives rise an operator in the path integral
$\langle{e^{\left[\epsilon\partial_{t}{H_{o}}(t)+\epsilon\delta
W(t)\delta(t-{t_{o}})+\\{W(t_{o})+\epsilon[\partial_{t}{H_{o}}(t),W(t_{o})]+...{\rm
higher}\,{\rm commutators}\\}\right]}}\rangle$
(with proper time ordering of operators). However integrating the argument of
exponential over an infinitesimally small interval ${t_{o}}-\epsilon$ to
${t_{o}}+\epsilon$,time translation invariance implies
$\int_{t_{o}-\epsilon}^{t_{o}+\epsilon}dt\,\partial_{t}\langle\,{H_{o}}(t)\,\rangle+\delta\langle\,W({t_{o}})\,\rangle=0.$
(2.19)
(one can verify the commutator terms in this integration will give zero
because of time ordering) which is an operator constraint. Note the term
$W(t_{o})$ which is completely localized at a point $t_{o}$ essentially
creates the effect of boundary in the matrix model action which is defined on
infinite real line in the time direction. So any variation of the expectation
value of the hamiltonian (from (2.18)) exactly at $t_{o}$ due to the
interaction $[{H_{o}},W\delta(t-{t_{o}})]$ with the source $W(t_{o})$ is the
signal of leakage of MQM hamiltonian ${H_{o}}$ exactly at $t_{o}$. So we
conclude that the time translation invariance of the path integral
implies555Note the effect of leakage is observed within an interval
$t_{o}-\epsilon$ to $t_{o}+\epsilon$. Once we move slightly away from $t_{o}$
system will evolve according to the conserved hamiltonian $H_{o}$ that in an
infinitesimal small interval around $t_{o}$ we have
$\langle\,{H_{o}}(t)\,\rangle{|_{t_{o}-\epsilon}^{t_{o}+\epsilon}}+\delta\langle
W(t_{o})\,\rangle=0$, where $\delta
W({t_{o}})=[\int_{t_{o}-\epsilon}^{t_{o}+\epsilon}[{H_{o}},W(t)\delta(t-{t_{o}})]$
is the variation of $W(t_{o})$ due to time translation $t_{o}\rightarrow
t_{o}+\epsilon(t_{o})$, which indicates the leakage of energy of the fermionic
system. However the inclusion of an operator $W(t_{o})$ in MQM action has an
interpretation in dual string theory is to create a boundary to string world
sheet by insertion of a macroscopic loop localized at $X^{o}\equiv i{t_{o}}$
or presence of a boundary state in closed string channel. So the above
phenomenon in string theory implies that closed string hamiltonian is
undergoing a leakage while being scattered from the boundary state localized
at $X={X^{o}}=i{t_{o}}$ ! This means that energy from the bulk is flowing out
across the boundary or in other words the bulk hamiltonian is not conserved in
the presence of boundary! This effect can be visualized in matrix model from
the consideration of collective field theory.
Note the path integral with the operator $e^{W(t_{o})}$ does have an
interpretation that free fermionic state are getting scattered from an
operator $e^{W(t_{o})}$. The free fermionic states after being scattered
becomes permanently changed due to the action of an operator which is the
function of the leakage factor $\int dt[\delta W(t)]\delta(t-{t_{o}})$ at
$t\geq{t_{o}}$. Clearly the scattered state will differ from the incoming
state with a term which is function of $\delta W(t_{o})$. As the term $\delta
W(t_{o})$ is not present in the effective hamiltonian 666which is given by
$W\delta(t-{t_{o}})+e^{-W(t_{o})}[\,\beta\int_{{t_{o}}-\epsilon}^{{t_{o}}+\epsilon}dtH\,]e^{W(t_{o})}\sim\beta{H_{o}}+W\delta(t-{t_{o}})$
as we mentioned or cannot arise in by the time evolution of
$\langle{H_{o}}\rangle$ w.r.t the complete hamiltonian 777as
$W(t_{o})\rightarrow W(t_{o})+\delta W(t_{o})$ is an instantaneous process so
it describes a leakage. These deformed states although evolve according to the
free fermionic hamiltonian ${H_{o}}$ but they can be considered as the
superposition of the states which are stationary w.r.t a hamiltonian deformed
from ${H_{o}}$ where the deformation is caused by the leakage as we discussed.
Now in collective field theory the fluctuation of collective field from its
static value gives a field which corresponds to 2D spacetime tachyon. The
action for this fluctuation gives the propagator of a 2D massless scalar. Now
in the presence of the operator $e^{W(t_{o})}$, the wave function of this
scalar field above $t\geq{t_{o}}$ although evolve according to the hamiltonian
of a 2D massless scalar but will be deformed from the one at $t\leq{t_{o}}$ by
the action of an operator which is function of the leakage factor $\delta
W(t_{o})$. As this operator is not present in the path integral so the
deformation of the wave function above $t\geq{t_{o}}$ must show up as the
modification of the propagator from that of a 2D massless scalar ! This can be
easily seen from the canonical quantization of 2D massless field and
considering the deformation of the wave function.
So we see that although the time translation invariance is maintained by
extending Ward’s principle to every order but its not giving the right string
theory picture! This is because the closed strings which are getting scattered
from D brane the scattered states remain the same on shell states w.r.t the
hamiltonian same as that for incoming one! So it appears that we must need to
impose a constraint in matrix model side in order to extract right string
theory from it. Lets briefly go through the string theory scenario and try to
understand string theoretical origin of such constraint. Note as far as the
open string world sheet is concerned Dirichlet boundary condition which fixes
the matter coordinate ${X}={X^{o}}\equiv{it_{o}}$ at the boundary implies
nonconservation of the momentum associated with the matter direction. However
the open string action ${S_{\rm open}}$ and the open string path integral
$Z_{\rm open}$ remains invariant under an infinitesimal variation of X which
is ensured from the boundary condition
${\delta{X}}({X^{o}})=0\quad\Rightarrow\quad{\delta_{X}}{S_{\rm
open}}=0\quad,\quad{\delta_{X}}[Z_{\rm open}]=0,$ (2.20)
where ${\delta_{X}}$ implies infinitesimal variation in X at every point of
the world sheet. Also the conservation of the string hamiltonian is associated
with the boundary condition
$T(z)-\overline{T}(\overline{z})=0,$ (2.21)
which ensures there is no leakage of energy at the boundary 888 To explain a
bit more, in the presence of the D brane we know the energy of the incoming
state is (associated with closed string )is not same as that of the outgoing
state in the direction with Dirichlet boundary condition as D brane act as a
source. However (2.20) implies we can consider the incoming and the outgoing
state as the separate conserved system evolve according to same conserved
hamiltonian (but different state) with none suffering any leakage at the
boundary [40, 39]. Moreover we have the constraint from the current algebra
and superpartners of all the above conditions which ensures the conservation
of the symmetry generators. Now the string path integral with a Laplace
transformed macroscopic loop (which creates the boundary localized at
$X={X^{o}}$ with the imposed wave function giving right string one point
function ) must obey the conditions (2.20,2.21) where the condition (2.20)
follows from the function $\delta(X_{\rm boundary}-{X^{o}})$ present in the
macroscopic loop functional. Hence these conditions must show up in the dual
matrix model with a D brane. To state more precisely the 2D path integral on a
manifold with the macroscopic loop (for the bosonic case which is given by
(1.3) ) corresponds to a physical state where the respective wave function
satisfy WdW equation [10]. WdW equation implies invariance of the wave
function under the action of the generator of $\tau$(worldsheet time)
translation. However the conservation of these symmetry generators follows
from these conditions and hence the respective state must have information
about it. So to gain the insight about what these constraints correspond in
the matrix model let us look at the closed string channel and express the
constraints in terms of the boundary state. Now in the minisuperspace
approximation only the zero mode part of the constraints will be relevant and
can be expressed as
$(L_{o}-\overline{L}_{o})|B\rangle=0\quad;\quad\delta X_{\rm
boundary}|B\rangle=0.$ (2.22)
Note that both the above constraints are followed by their superpartners
however as far as zero mode is concerned we have already applied such
constraints when we classified macroscopic boundary state according to their
spin structure [9]. The second condition essentially describes the zero mode
condition
$(\hat{X}-{X^{o}})|B\rangle=0\quad\Rightarrow\quad|B\rangle\equiv\delta(\hat{X}-{X^{o}})|0\rangle.$
(2.23)
In closed siring channel path integral with a Laplace transformed macroscopic
loop corresponds to boundary states [39]. This can be seen by expressing the
path integral functional $\Psi(X,\phi)$ as a sum over operators(in Minkowskian
signature) ${O_{i}}$ by state operator mapping where we know that these
operators corresponds to Ishibashi states and the wave function
$\langle{\Psi}|O_{i}\rangle$ gives the one point function. So $|\Psi\rangle$
must be annihilated by the constraints from (2.23) [42]. So the same
constraints must be imposed on the state associated with matrix model path
integral in the presence of D brane. This is because macroscopic loop operator
in matrix model can equivalently be expressed in terms of operators along with
the respective wave function where each component is in one to one
correspondence with the one in string theory side. So we conclude that the
kind of leakage we discussed at the beginning of the section is caused due to
absence of any condition equivalent to (2.22) and must be cured once we impose
an equivalent condition to matrix model. Lets try to find out the constraint
in matrix model. From the first condition in (2.22) along with the one from
the zero mode part of the current algebra in Dirichlet boundary state for
matter just emphasizes the fact that 2D boundary state or the macroscopic loop
operator will correspond to the superposition of primary states/operators
expressed in terms of the momentum modes(i.e no winding modes)[39] with a
reflection symmetry $P\rightarrow-P$ in matter as well as Liouville part [33,
39] which is already known in the matrix model [2, 28]. The reason we obtain
Liouville one point function in exact form from matrix model, is that in
minisuperspace approximation this condition is trivially satisfied and
consequently not going to impose any constraint in matrix model side. However
the second condition in (2.22), (2.23) or (2.20) is not yet properly
understood in the matrix model. More precisely in the presence of macroscopic
operator the state from the path integral is represented by an wave function
expressed as a functional of bulk d.o.f $\Psi=\Psi(\\{X\\},\\{\phi\\}...)$ and
under any infinitesimal transformation $X\rightarrow X+\delta X$ we must have
$\delta\Psi=\int_{\rm boundary}\delta{\hat{X}}J(\hat{X})\,\Psi=0$ (2.24)
which ensures the bulk conformal invariance and implication of (2.23). Note
$\Psi$ is the wave function $\Psi=\Psi(X_{\rm boundary},\phi_{\rm boundary})$
which is an eigenfunction of complete string hamiltonian, representing BRST
invariance. Similarly its discussed in [4, 6] that matrix model path integral
in the presence of an operator $e^{W(t_{o})}$ ( which arises by including a
probe eigenvalue) is an wave function
$\psi=\psi({\overline{z}_{\pm}}{z_{\pm}}(t_{o})\,)$which satisfied the
Schrodinger equation. However considering the fact that this operator also
creates an effect of boundary and $\psi$ is a functional of MQM variables
$\psi=\psi(\\{{\overline{z}_{\pm}}{z_{\pm}}(t)\\})$ we must have an condition
equivalent to (2.24) in matrix model which ensures conservation of MQM
hamiltonian in presence of such operator. Naturally no such constraint arises
from Liouville d.o.f for the reason as we discussed. Here first we will find
such constraint in the matrix model from somewhat intuitive way, solve it in
the context of the matrix model path integral and show its consequence.
Finally with the help of it we will come to more rigorous analogy between the
string theory and the matrix model scenario in the next subsection. First note
that in the matrix model the matter coordinate X is getting mapped to
time(Minkowskian) coordinate, $X\rightarrow it$. So we can guess that string
theory boundary condition must be reflected in MQM as an overall invariance of
the path integral under time translation i.e $\delta X\equiv\delta t$ with no
leakage. Note when there is no leakage, under infinitesimal time translation
$t\rightarrow t+\epsilon(t)$ the variation of path integral is given by
$\langle{\delta}e^{\int dtW({t})\delta(t-{t_{o}})}\rangle$, where $\delta$
defines the variation of the operator due to infinitesimal time translation at
fixed time $t={t_{o}}$. So string theory boundary condition (2.20) must be
reflected in the following constraint in matrix model
$\delta\langle e^{\int dtW({t})\delta(t-{t_{o}})}\rangle=\langle\delta e^{\int
dtW({t})\delta(t-{t_{o}})}\rangle=0.$ (2.25)
Indeed in the string theory path integral if we expand the macroscopic loop in
terms of operators which corresponds to Ishibashi state one can verify that
this is the consequence of Ward identity under an infinitesimal transformation
$X\rightarrow X+\delta X$ which arises on application of the second condition
in (2.20) and we have already mentioned it in an alternative way in (2.24). In
next subsection we will show that the consequence of this condition are in
exact agreement with that of string theory. To understand the impact of this
condition in MQM first we need to write down the Schrodinger equation and
study the Hilbert space. We have the time dependent Schrodinger equation for a
single fermion 999When we consider the insertion of $\beta$ factor, for the
macroscopic loop operator we have the expression ${\frac{1}{\beta}}{\hat{W}}$,
however when we rewrite the Schrodinger equation in terms of the eigenvalues
x,y which corresponds to the real and imaginary part of eigenvalue z we have
the Schrodinger equation
$[{\frac{1}{\beta^{2}}}{\frac{\partial^{2}}{\partial{x^{2}}}}+{x^{2}}+{\frac{1}{\beta^{2}}}{\frac{\partial^{2}}{\partial{y^{2}}}}+{y^{2}}+{\frac{1}{\beta}}{\hat{W}}\delta(t-{t_{o}})]\psi=E\psi$
In the double scaling limit we take
$x,y\rightarrow{\sqrt{\beta}}x,{\sqrt{\beta}}y$ [1], which gives the
Schrodinger equation (2.12 ) in Minkowskian time
$\displaystyle[i{\frac{\partial}{{\partial}t}}$ $\displaystyle-$
$\displaystyle
i{\delta}(t-{t_{0}})W({\overline{z}_{+}},{z_{+}},{\overline{z}_{-}},{z_{-}})]{\Psi}({\overline{z}_{\pm}}{z_{\pm}},t)$
(2.26) $\displaystyle=$
$\displaystyle{\mp}{\frac{i}{\sqrt{2{\alpha^{\prime}}}}}[({z_{+}}{\frac{\partial}{\partial{z_{\pm}}}}+{\overline{z}_{+}}{\frac{\partial}{\partial{\overline{z}_{\pm}}}}+1{]}{\Psi}({z_{\pm}},{\overline{z}_{\pm}},t).$
For t away from ${t_{o}}$ we have time independent Schrodinger equation
$\displaystyle i{\frac{\partial}{\partial
t}}{\psi}({\overline{z}_{\pm}}{z_{\pm}},t)$ $\displaystyle=$
$\displaystyle{\mp}{\frac{i}{\sqrt{2{\alpha^{\prime}}}}}[({z_{\pm}}{\frac{\partial}{\partial{z_{\pm}}}}+{\overline{z}_{\pm}}{\frac{\partial}{\partial{\overline{z}_{\pm}}}}+1]{\Psi}({\overline{z}_{\pm}}{z_{\pm}},t)=E{\psi}({\overline{z}_{\pm}}{z_{\pm}},t).$
(2.27)
The free fermion solution with energy E is
${\psi^{E}_{o\pm}}({z_{\pm}},t)={e^{-iEt}}{e^{\mp
i{\frac{\phi_{o}(E)}{2}}}}{{({\overline{z}_{\pm}}{z_{\pm}})}^{{\pm}iE-{\frac{1}{2}}}},$
(2.28)
where we have chosen ${\alpha^{\prime}}=2$ and ${\phi_{o}}(E)$ is determined
from biorthogonal property (discussed in the Appendix) and given by
$e^{i{\phi_{0}(E)}}={\Gamma(iE+{\frac{1}{2}})\over\Gamma(-iE+{\frac{1}{2}})}.$
(2.29)
Now consider the commutation relation
$\displaystyle[{H_{o}},{\hat{z}_{+}}]$ $\displaystyle=$
$\displaystyle-i{\quad\quad;\quad\quad}[{H_{o}},{\hat{\overline{z}}_{+}}]=-i,$
$\displaystyle[{H_{o}},{\hat{z}_{-}}]$ $\displaystyle=$ $\displaystyle\quad
i{\quad\quad;\quad\quad}[{H_{o}},{\hat{\overline{z}}_{-}}]=i.$ (2.30)
This implies ${\hat{\overline{z}}_{+}}{\hat{z}_{+}}$ and
${\hat{\overline{z}}_{-}}{\hat{z}_{-}}$ when acts on a state $|E\rangle$
expressed as ${\hat{\overline{z}}_{+}}{\hat{z}_{+}}|E\rangle=|E-i\rangle$ and
${\hat{\overline{z}}_{-}}{\hat{z}_{-}}|E\rangle=|E+i\rangle$. These states can
be represented as $|E\pm ni\rangle$. These describe a different Hilbert space
[20] which can be understood as their inner product with the state $|E\rangle$
either diverge or zero. These states actually can be identified with the
discrete tachyonic states over the matrix model ground state [28, 20]. Now to
the meaning of the constraint. First we will find the expression for the
constraint and then solve it in the context of the matrix model path integral
for the macroscopic loop operator we considered or in the more complicated
case to reach to the right expression free energy. Consider the v.e.v of the
operator in single fermionic state which is given by
$\langle e^{W({t_{o}})}\rangle=\langle e^{{\rm
log}(1+{\frac{{\overline{z}_{+}}{z_{+}}+{\overline{z}_{-}}{z_{-}}-2{H_{o}}}{\mu_{B}^{2}}})}\rangle=\langle(1+{\frac{{\overline{z}_{+}}{z_{+}}+{\overline{z}_{-}}{z_{-}}-2{H_{o}}}{\mu_{B}^{2}}})\rangle.$
(2.31)
(Here we have shown the expectation value w.r.t the single fermionic state.
This we could do because the theory is projected singlet sector and N
fermionic state is just the direct product of each). So (2.25) along with
(2.30) gives the constraint
$\langle{\overline{z}_{+}}{z_{+}}(t_{o})-{\overline{z}_{-}}{z_{-}}(t_{o})\rangle=0.$
(2.32)
As the constraint is exclusively on the variable associated with W so it must
has effect in constraing the Hilbert space created by W at $t_{o}$. Note for
the macroscopic loop operator in any other sector, in general the variation of
$\langle e^{W({t_{o}})}\rangle$ can be expressed as
$\langle e^{W({t_{o}})}\delta
W({t_{o}})\rangle=0\quad\quad\Rightarrow\quad\quad\langle
e^{W({t_{o}})}[{\overline{z}_{+}}{z_{+}}(t_{o})-{\overline{z}_{-}}{z_{-}}(t_{o})]\rangle=0.$
(2.33)
As the path integral with the uncompactified time essentially describes the
transition amplitude from the initial state
$|{\overline{z}_{\pm}}{z_{\pm}}({t_{i}})\rangle$ to the final state
$|{\overline{z}_{\pm}}{z_{\pm}}({t_{f}})\rangle$, so the condition (2.25)
remain true for any arbitrary variation in time implies that we have
$\langle{\psi_{f}}|\delta e^{{W}({t_{o}})}|\psi_{i}\rangle=0,$ (2.34)
between any two physical states $|\psi_{i}\rangle$, $|{\psi_{f}}\rangle$ which
according to the Hilbert space described by (2.30) will be of either form
$|{\overline{z}_{+}}{z_{+}},t\rangle$ or
$|{\overline{z}_{-}}{z_{-}},t\rangle$. Now recall the expression of the
inserted macroscopic loop operator $W(t_{o})$ in (2.15) which is expressed in
the form (2.31). While the ${\hat{H}_{o}}$ in the expression of W keeps the
underlying state invariant the ${\hat{\overline{z}}_{+}}{\hat{z}_{+}}$ and
${\hat{\overline{z}}_{-}}{\hat{z}_{-}}$ act on the vacuum to create the states
$|E=-i\rangle$ , $|E=i\rangle$ or in other words the macroscopic loop operator
(2.15) due to its $\overline{z}z$ field acts on the vacuum to create a left
moving as well as right moving state. So the constraint essentially relates
the two exactly at $t_{o}$ by putting the following constraint inside the path
integral,
${\overline{z}_{+}}(t_{o})-{\overline{z}_{-}}(t_{o})=0\quad;\quad{z_{+}}(t_{o})-{z_{-}}(t_{o})=0,$
(2.35)
The physical meaning of the above condition is that the quantum fluctuations
of the variables ${z_{+}}({\overline{z}_{+}})$ and
${z_{-}}({\overline{z}_{-}})$ around its classical value appears to be
identical at $t_{o}$. This in turn implies that the wave functions of the
states which are being created by the action of
${\hat{\overline{z}}_{+}}{\hat{z}_{+}}$ and
${\hat{\overline{z}}_{-}}{\hat{z}_{-}}$ over the ground state in
${\overline{z}_{+}},{z_{+}}$ and ${\overline{z}_{-}}.{z_{-}}$ representations
respectively , appeared to be indistinguishable exactly at $t_{o}$. Note that
in the above constraint (2.35 ), the l.h.s will not remain invariant once we
move away from $t_{o}$. In otherwords if we express them in terms of the
respective operators(which acts on vacuum), we see that l.h.s does not commute
with the free hamiltonian ${H_{o}}$ (2.9). So once we move away from the point
$t={t_{o}}$ we have our physical Hilbert space described by the free
hamiltonian ${H_{o}}$ and the relation (2.30) so that again the left and right
moving states created by $\hat{W}$ will appear to be distinguishable. In next
few steps we will see that the condition (2.35)when acts inside the path
integral it just has a meaning to project the operator W and the Hilbert space
generated by it to its physical sector while keeping the bulk physics
unaffected! So with the constraint we can express the path integral somewhat
schematically as 101010above identity can arise by inserting
$\delta({\overline{z}_{+}}(t_{o})-{\overline{z}_{-}}(t_{o}))\delta({z_{+}}(t_{o})-{z_{-}}(t_{o}))$
in the path integral, operating
${\overline{z}_{-}}{z_{-}}={\frac{\partial}{\partial{z_{+}}}}{\frac{\partial}{\partial{\overline{z}_{+}}}}$
on the inner product
$\langle{\overline{z}_{+}}{z_{+}}(t_{o}+dt_{o})|{\overline{z}_{+}}{z_{+}}(t_{o})\rangle$.
Its important to note that due to $\delta(t-{t_{o}})$, inside the path
integral we can make the time interval around ${t_{o}}$ arbitrarily small i.e
$dt<<{\frac{1}{\beta}}$ so that around $t_{o}$,
$e^{-\beta\int_{{t_{o}}-\epsilon}^{{t_{o}}+\epsilon}dtL}$ will just be
identity and hence the effect the constraint is only to modify W without
affecting the MQM lagrangian L and hence no boundary condition will be imposed
on the variables in original lagrangian at $t={t_{o}}$. Finally integrating
over ${z_{-}}(t_{o}),{{\overline{z}_{-}}}(t_{o})$ where the $\delta-{\rm
function}$ gives ${\overline{z}_{-}},{z_{-}}={\overline{z}_{+}},{z_{+}}$,
leads to the above expression of W. As W becomes independent of
${\overline{z}_{-}},{z_{-}}$ so we can just express the integral as a
continuous integral. Also note in the expression of
$W({\hat{\overline{z}}_{+}}{\hat{z}_{+}},{H_{o}})$,the action of $H_{o}$ on
any intermediate state in the path integral gives the constant and it can be
alternatively given by ${\frac{\partial}{\partial t}}$. So it will not be
affected by the constraint. 111111note that here path integral is expressed in
a schematic way omitting the angular factors. One can verify that even the
inclusion of angular factor will not change the picture
$\displaystyle\int
d{\overline{z}_{+}}d{z_{+}}d{\overline{z}_{-}}d{z_{-}}{e^{-\int_{-\infty}^{\infty}dt[\beta
L-W({\overline{z}_{+}},{z_{+}},{\overline{z}_{-}},{z_{-}};{t_{o}})]}}$
$\displaystyle=$
$\displaystyle\int\displaystyle\prod_{t\leq{t_{o}}}d{\overline{z}_{+}}d{z_{+}}d{\overline{z}_{-}}d{z_{-}}{e^{-\int_{t_{o}}^{\infty}dtL}}$
$\displaystyle\int
d{\overline{z}_{+}}(t_{o})d{z_{+}}(t_{o})d{\overline{z}_{-}}(t_{o})d{z_{-}}(t_{o})$
$\displaystyle\delta({\overline{z}_{+}}({t_{o}})-{\overline{z}_{-}}({t_{o}}))\delta({z_{+}}({t_{o}})-{z_{-}}({t_{o}}))e^{W({\overline{z}_{+}}{z_{+}},{\overline{z}_{-}}{z_{-}},H_{o};\,{t_{o}})}$
$\displaystyle\int\displaystyle\prod_{t\geq{t_{o}}}d{\overline{z}_{+}}d{z_{+}}d{\overline{z}_{-}}d{z_{-}}{e^{-\int_{-\infty}^{t_{o}}dtL}}$
$\displaystyle=$ $\displaystyle\int
d{\overline{z}_{+}}d{z_{+}}d{\overline{z}_{-}}d{z_{-}}{e^{-\int_{-\infty}^{\infty}dt[\beta
L-W({\overline{z}_{\pm}}{z_{\pm}}(t_{o}),H_{o})]}}.$ (2.36)
Its important to note that the effect of the constraint is only to project W
at its physical sector without imposing any boundary condition to original
lagrangian which happens due to $\delta(t-t_{o})$ factor as we explained.
Projection implies we can get same path integral expression or the transition
amplitude by expressing W either in ${\overline{z}_{+}}{z_{+}}$ or in
${\overline{z}_{-}}{z_{-}}$ mode which happens due to the fact that wave
function associated with the either mode appears to be same at $t_{o}$. Its
also important to note that when we consider the original expression of W
(2.15), the quantum fluctuations of the varibles ${z_{+}}({z_{-}})$ and
${\overline{z}_{+}}({\overline{z}_{-}})$ are constrained by (2.35) and in that
sense when we consider the complete Hilbert space as described by (2.30).
${\hat{z}_{+}}({\hat{z}_{-}})$ and
${\hat{\overline{z}}_{+}}({\hat{\overline{z}}_{-}})$ are not ordinary
operators. However the implication of (2.36) is that we can replace the theory
with the expression of W (as in (2.15) ) by the projected one $W\rightarrow
W_{\rm proj}=W({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}},H_{o})$ and in the
theory with $W_{\rm proj}$ these operators act as ordinary operators and one
can evaluate the partition function in a formal way in MQM by using $W_{\rm
proj}$ instead of W to give the right transition amplitude as in the original
theory with the constraint(2.35). Now we will show that how the constraint is
solved in the context of the path integral leads to the right expression of
the partition function. Note in (2.34) $|\psi_{i},t_{i}\rangle$ can be
expressed in the free fermionic Hilbert space $|E\rangle$ while the constraint
(2.25) implies the macroscopic loop operator which has expression
$W({\overline{z}_{+}}{z_{+}}(t_{o}),{H_{o}})$ or
$W({\overline{z}_{-}}{z_{-}}(t_{o}),{H_{o}})$ leads to the most general
expression of the final state $|\psi_{f}\rangle$ is
$\sum{c_{o;E}}|E\rangle+\sum{c_{n;E}}|E+ni\rangle$ or
$\sum{c_{o;E}}|E\rangle+\sum{c_{-n;E}}|E-ni\rangle$. As the tachyonic states
have imaginary energy so the out state with indefinite number of tachyons will
contribute to the path integral. So (2.34) now can be expressed as a trivial
identity
$\displaystyle\sum_{n=0}^{\infty}{c_{-n;E}}\langle E-ni;f|{\rm
exp}[W({\overline{z}_{+}}{z_{+}},{H_{o}},(t_{o}))]|E\rangle=\displaystyle\sum_{n=0}^{\infty}{c_{n;E}}\langle
E+ni;f|{\rm exp}[W({\overline{z}_{-}}{z_{-}},{H_{o}},(t_{o}))]|E\rangle.$
(2.37)
As $\psi({\overline{z}_{+}}{z_{+}},E)$ and $\psi({\overline{z}_{-}}{z_{-}},E)$
are the same wave function in different representation so we see in the above
l.h.s and r.h.s are the same expression, expressed in different
representation. Now before coming to the string theoretical interpretation of
all the matrix model events first we will derive the expression of the wave
function(physical) for $t\geq{t_{o}}$. Integrating (2.26) over the
infinitesimally small interval around $t={t_{o}}$ and following the constraint
from (2.35,2.36) we have
$\displaystyle
i{\Psi}({z_{\pm}},{\overline{z}_{\pm}},t){|_{{t_{0}}+{\epsilon}}}-i{\Psi}({z_{\pm}},{\overline{z}_{\pm}},t){|_{{t_{0}}-{\epsilon}}}$
$\displaystyle=$ $\displaystyle iW_{\rm
proj}\Psi({z_{\pm}},{\overline{z}_{\pm}},{t_{0}})$ (2.38) . $\displaystyle=$
$\displaystyle-\sum
i\log(1+{\frac{2{\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}}+{\hat{\overline{z}}_{+}}{\hat{z}_{-}}+{\hat{\overline{z}}_{-}}{\hat{z}_{+}}}{\mu_{B}^{2}}}){\Psi}({z_{\pm}},{\overline{z}_{\pm}},{t_{o}})$
So exactly at $t={t_{o}}$ the wave function is given by
$\lim_{\epsilon\to
0}{\Psi_{>}}({z_{\pm}},{\overline{z}_{\pm}},t_{o})=(1-W({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}},{H_{o}})){\Psi_{o}}({z_{\pm}},{\overline{z}_{\pm}},t_{o})$
(2.40)
So for the macroscopic loop operator in the NS NS sector we have the physical
wave function for $t\geq{t_{o}}$
${\Psi_{>}}({\overline{z}_{\pm}}{z_{\pm}},t_{o})=[1-{\rm
log}(1-{\frac{2{\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}}(t_{o})-2{H_{o}}}{\mu_{B}^{2}}})]{\Psi_{o}}({\overline{z}_{\pm}}{z_{\pm}},t)$
(2.41)
Now we need to find the wave function at $t>{t_{o}}$. In order to do so first
note that for $t\geq{t_{o}}$ wave function evolves according to the free
hamiltonian ${H_{o}}$. So at the first sight it appears that the wave function
at $t\geq{t_{o}}$ is given by the one obtained from the time evolution from
${\Psi_{>}}({\overline{z}_{\pm}}{z_{\pm}},t_{o})$. It is given by
${\Psi_{>}}({\overline{z}_{\pm}}{z_{\pm}},t)=[1-{\rm
log}(1-{\frac{2{\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}}(t)-2{H_{o}}}{\mu_{B}^{2}}})]{\Psi_{o}}({\overline{z}_{\pm}}{z_{\pm}},t)$
(2.42)
Now although exactly at $t={t_{o}}$, $\hat{W}$ is expressed in the form
$W({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}},{H_{o}})$ but once we move above
$t_{o}$, W can in principle be expressed in terms of both
${\hat{\overline{z}}_{+}}{\hat{z}_{+}}$ and
${\hat{\overline{z}}_{-}}{\hat{z}_{-}}$ as both of them are related by the
constraint (2.35) at ${t_{o}}$. More precisely these states are given by
$|{\psi_{f}}\rangle=\sum{c_{mn}}(E){({\hat{\overline{z}}_{+}}{\hat{z}_{+}})}^{m}{({\hat{\overline{z}}_{-}}{\hat{z}_{-}})}^{n}|E\rangle,$
(2.43)
So following the previous discussion $|{\psi_{f}}\rangle$ is expected to be
given by
$(1-W({\hat{\overline{z}}_{+}}{\hat{z}_{+}}(t),{\hat{\overline{z}}_{+}}{\hat{z}_{+}}(t){H_{o}})),{\Psi_{o}}({z_{\pm}},{\overline{z}_{\pm}},t).$
(2.44)
We can express these states as the one time evoluted from the wave function
from $t={t_{o}}$. Although it appears that exactly at $t={t_{o}}$,
$|{\psi_{f}}\rangle$ and $|{\Psi_{>}}\rangle$ are of similar structure but
they are not same as time evolution property of ${\hat{\overline{z}}_{+}}$ and
${\hat{z}_{+}}$ are different and once we move away from ${t_{o}}$ we have our
original Hilbert space. Finally note that at $t={t_{o}}$ fermionic states are
being converted from ${\psi_{o}}$ to $\psi_{>}$ so the change of fermion
number N must be measured from the variation of the transition amplitude at
$t={t_{o}}$ i.e in terms of the macroscopic loop operator described in (2.15)
we have
${\frac{dN}{dt}}={\frac{d}{dt}}\langle{\psi_{>}}({t_{f}})|{\psi_{o}({t_{i}})}\rangle=\langle{\frac{dW}{dt}}\rangle{|_{t={t_{o}}}},$
(2.45)
Where ${t_{i}}\rightarrow-\infty$ and ${t_{f}}\rightarrow\infty$. The average
$\langle{\frac{dW}{dt}}\rangle{|_{t={t_{o}}}=0}$ described above is w.r.t the
path integral. So in order to have the fermion number unchanged we must need
to impose the condition
$\langle{\frac{dW}{dt}}\rangle{|_{t={t_{o}}}}=0.$ (2.46)
Note that this is just a condition parallel to (2.25) and have an effect to
project W as described above. In the next part of this section we are going to
see that all these matrix model events are exactly in one to one
correspondence to the string theory.
### 2.3 String theoretical interpretation
In this section we will see that the constraint (2.25,2.35) we imposed leads
to right string theoretical result. First The boundary state of 2D spacelike
brane is expressed as
$|B_{\rm(SuperFZZT)}(\mu_{B})\rangle_{\phi}\otimes|D\rangle_{X^{0}}$ where
$|D\rangle_{X^{0}}$ for NS NS sector [39] is expressed as ${\cal
N}\int_{-\infty}^{\infty}dPe^{iP{X^{o}}}|P\rangle$ ( ${\cal N}$ is the
normalization factor) which in terms of the vertex operator can be written as
$|B{\rangle_{X}}={\cal
N}\int_{-\infty}^{\infty}d{P_{m}}[e^{i{P_{m}}(X-{X^{o}})}]|0\rangle+{\rm
descendants}.$ (2.47)
The string endpoints are localized at $X=X^{o}$ and the matter part of the
boundary state form the representation of $\delta(\hat{X}-{X^{o}})|0\rangle$,
satisfying (2.22). Boundary state of Super Liouville NS NS sector is given by
$|B{\rangle_{L}}={\cal
N}_{l}{\int_{0}^{\infty}}dPU(P_{l})|P_{l}\rangle\quad;\quad
U(P_{l})={\frac{\pi{\rm cos}2\pi s{P_{l}}}{{\rm
sin}h(\pi{P_{l}})}}\quad;\quad|P_{l}\rangle=(1+{\frac{L_{-1}\tilde{L}_{-1}}{P_{l}^{2}+M^{2}}}+.....)|v_{P_{l}}\rangle$
(2.48)
where $|v_{P_{l}}\rangle$ primary macroscopic state associated with a vertex
$e^{({\frac{Q}{2}}+i{P_{l}})\phi}$, M is the mass of intermediate propagating
mode.
The boundary state is the direct product of the matter and Liouville part
along with the ghost factors. Setting ${P_{m}}={P_{l}}=P$ we have the primary
part of the boundary state without ghost excitation mode, which is the
superposition of the tachyonic field can be expressed in terms of the
operators from the state operator mapping as
$\int
dPU(P)[e^{{iP(X-{X^{o}})}}+e^{{-iP(X-{X^{o}})}}]{e^{({\frac{Q}{2}}+iP)\phi}}.$
(2.49)
Now in the 2nd quantized matrix model we can express the macroscopic loop
operator as
$W(l,t)=\int e^{l{\overline{z}{z}}}{\psi^{\dagger}}{\psi}=\int d\tau e^{l{\rm
cosh}^{2}\tau}{\partial_{\tau}}\eta(l,t),$ (2.50)
where $\tau$ is the time of flight coordinate obtained from reparameterization
${\overline{z}{z}}={r^{2}}\quad;\quad{r^{2}}\sim{\rm cos}^{2}{h\tau}$ ,
${\psi^{\dagger}}{\psi}\sim{\partial\tau}\eta(l,t)$ and $\eta$ is the massless
bosonic field $({\partial_{t}^{2}}-{\partial_{\tau}^{2}})\eta(t,\tau)=0$ which
corresponds to the tachyon in the string theory at asymptotic $\tau$[25, 2,
28]. $\eta$ corresponds to the fluctuation of the collective field
$\phi={\phi_{o}}+{\partial_{\tau}}\eta$. $\eta$ satisfies Dirichlet boundary
condition in $\tau$ direction [11], [28]. However note that when associated
with the Laplace transformed macroscopic loop operator operator W which
correspond to D brane boundary state, $\eta$ is no longer an ordinary state
but it corresponds to an Ishibashi state. Here we show that the constraint we
imposed (2.25,2.35) leads to the right matter one point function from $\eta$
what is expected from string theory. Consider
$\eta(\tau,t)=\int_{-\infty}^{\infty}{\frac{dp}{p}}{\tilde{\eta}}(p)(a(p)e^{-ipt}+b(p)e^{ipt}){\rm
sin}(p\tau).$ (2.51)
In order to find the exact t dependence consider the fact that the implication
of (2.46) in the context of collective field theory is
${\delta_{t}}{\phi}{|_{t=t_{o}}}=0,$ (2.52)
where as before $\delta_{t}$ implies variation of the collective field $\phi$
due to infinitesimal variation in t, at fixed t.
So (2.52) implies in (2.51) we have
$a=e^{ipt_{o}}\quad;\quad b=e^{-ipt_{o}}$ (2.53)
So (2.51) is expressed as
$\eta(\tau,t)=\int_{-\infty}^{\infty}{\frac{dp}{p}}{\tilde{\eta}}(p)(e^{-ip(t-{t_{o}})}+e^{ip(t-{t_{o}})}){\rm
sin}p\tau$. Now integrating over $\tau$ we have
$W(p,t)={\frac{e^{-l\mu}}{2}}p{K_{ip}}(\mu
l){\tilde{\eta}}(p)(e^{-ip(t-{t_{o}})}+e^{ip(t-{t_{o}})})$ where ${K_{ip}}(\mu
l)$ is the Bessel’s function which is the macroscopic wave function satisfying
WdW equation [13]. This on Laplace transform
$\int{\frac{dl}{l}}e^{-\mu_{B}^{2}l}$ can be expressed as
$W(p,t)=U(p)\tilde{\eta}(p,t)\quad;\quad U(p)={\frac{\pi{\rm cos}2\pi sp}{{\rm
sin}h(\pi p)}}$ (2.54)
where we have
$\mu_{B}^{2}=2\sinh^{2}(\pi s)|\mu|\ \ \ (\epsilon\cdot{\rm sign}(\mu)<0),\ \
\ \ \ \mu_{B}^{2}=2\cosh^{2}(\pi s)|\mu|\ \ \ (\epsilon\cdot{\rm
sign}(\mu)>0).$ (2.55)
Note (2.25) and (2.52)are just the same constraint expressed in the 1rst and
2nd quantized formalism. Given the form of $\eta$, on Laplace transform of
(2.50) and on inverse Fourier transform of (2.54) we can express W as
$W_{p}(\tau,t)=U(p){\tilde{\eta}}(p)[e^{ip(t-{t_{o}})}+e^{-ip(t-{t_{o}})}]=U(p)[{\partial_{\tau}}{\eta_{p}}(\tau+(t-{t_{o}}))+{\partial_{\tau}}{\eta_{p}}(\tau-(t-{t_{o}}))]$
(2.56)
where the suffix p implies pth component. We know $\eta$ corresponds to the
space time tachyonic field in the asymptotic $\tau$ region which in the string
theory side describes asymptotics of Liouville field $\phi$ which describes
vanishing Liouville wall. So we see that each component with momentum p in the
expansion of W describes the tachyonic operator ${\tilde{\eta}}(p)$ dressed
with Liouville and matter wave function (2.47,2.48) where W is symmetric under
$P\rightarrow-P$. So these are just in one to one correspondence with the
state obtained in the expansion of the boundary state in (2.49). So (2.56) is
just the same as (2.49) when we identify the physical d.o.f. More
interestingly if we try to extrapolate $\eta$ in the region $P_{m}\neq P_{L}$
with the same $\tau$ dependence as in (2.51), from (2.52,2.51,2.56) we have
$\displaystyle\int{\frac{dl}{l}}{e^{-\mu_{B}l}}W(l)$ $\displaystyle=$
$\displaystyle[\int_{0}^{\infty}dP_{l}U(P_{l}){\eta_{l}}(P_{l})]\times[\int_{-\infty}^{\infty}d{P_{m}}e^{i{P_{m}}(t-{t_{o}})}]$
(2.57) $\displaystyle=$
$\displaystyle[\int_{0}^{\infty}dP_{l}U(P_{l}){\eta_{l}}(P_{l})]\times\delta(t-{t_{o}}),$
where $\eta_{l}$ implies Liouville part of $\eta$. This has a very similar
structure with that of 2D boundary state
$|B\rangle=|B_{\rm Liouville}\rangle\otimes|B_{\rm
matter}\rangle=\int_{0}^{\infty}dP_{l}U(P_{l})|P\rangle\otimes\delta(\hat{X}-{X^{o}})|0\rangle.$
(2.58)
However this is a mere extrapolation without any physical justification.
Finally we see the effect of the constraint (2.25) is only to give the correct
form of the matter wave function while keeping the Liouville wave function
unchanged as we discussed in section 2.2. Finally lets explain the meaning of
the projection $W\rightarrow W_{\rm proj}$ as described in (2.36),(2.37). Note
both Liouville as well as the matter part of the boundary state is symmetric
under interchange of $P\rightarrow-P$ (2.48). So projecting the theory to
either right moving or left moving part i.e in the boundary state keeping
either the left moving or the right moving part while expressing the
superposition of left and right moving state as only left or only right moving
state (i.e flipping the sign of p where necessary)in the incoming or the
outgoing sector yield the same transition amplitude as the original one.
Overall time translation invariance of the projected theory follows from the
consideration of either left or the right moving part as the image of the
other. Lets come to a precise analogy between MQM, 2nd quantized matrix model
or collective field theory and string theory, which arises as a consequence of
the constraint. We know the macroscopic loop operator can be expanded in terms
of microscopic operators. To see that in dual matrix model consider $W_{\rm
proj}({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}},H_{o})$,acting on ground state
${\rm
log}(1+{\frac{{\hat{\overline{z}}_{+}}{\hat{z}_{+}}+{\hat{\overline{z}}_{-}}{\hat{z}_{-}}-2{H_{o}}}{\mu_{B}^{2}}})|\mu\rangle=\sum\displaystyle{a_{mn}}(\mu)[{\frac{{\hat{\overline{z}}_{+}}{\hat{z}_{+}}}{\mu_{B}^{2}}}{]^{n}}[{\frac{{\hat{\overline{z}}_{-}}{\hat{z}_{-}}}{\mu_{B}^{2}}}{]^{m}}|\mu\rangle$
(2.59)
Essentially the above which is described in Minkowskian time corresponds to a
coherent state. Also the above is an expansion in terms of microscopic
operator. This is evident from the correspondence between the operator
${({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}})}^{n}$ and the vertex operator
for discrete tachyonic states.[28]. From the previous discussion it follows
that when we consider the space time wave function for each component in W, we
found an effect of boundary exactly at $X={X^{o}}({\rm or,}\,t={t_{o}}$). More
precisely at this point the state associated with each component of boundary
state in (2.49) or the states appear on expansion of W (2.56) in collective
field theory, expressed with an wave function which has only Liouville part.
So the left and right moving state appears to be identical at $X={X^{o}}({\rm
or}\,t={t_{o}}$). Now in order to understand this effect in MQM note that the
macroscopic loop operator
$W({\hat{\overline{z}}_{+}}{\hat{z}_{+}},{\hat{\overline{z}}_{-}}{\hat{z}_{-}},{H_{o}})$
acting on ground state, essentially create a left moving as well as right
moving state (2.59). The constraint (2.35) inside the path integral have an
interpretation in MQM that at $t={t_{o}}$ we have
$[{{{\hat{\overline{z}}_{+}}{\hat{z}_{+}}}}{]^{n}}||\mu\rangle\equiv[{\hat{\overline{z}}_{-}}{\hat{z}_{-}}{]^{n}}|\mu\rangle.$
(2.60)
i.e the left and right moving states which are created by the action of
$\hat{W}$ on MQM ground state appear to be exactly identical at $t_{o}$ (but
different when we move away from $t_{o}$). This leads to the projection
$W\rightarrow{W_{\rm
proj}}=W({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}}(t_{o}),{H_{o}})$ in the
path integral (2.36). More explicitly the created states obey
${\psi_{+}}({\overline{z}_{+}}{z_{+}}(t_{o});E=ni)={\psi_{-}}({\overline{z}_{-}}{z_{-}}(t_{o});E=-ni)$.
This is supported from (2.35) and give the justification for (2.37). Again
lets emphasize on the fact that these constraints no way affects the free
fermionic states as well as the discrete tachyonic states of the original Type
0A matrix model. This distinction just corresponds to that of the closed
string states associated with the boundary states and the free closed string
states in the bulk. Now once move away from $t_{o}$ we are back to our
original scenario described by (2.30) with the distinguishable states
described by (2.41,2.44). An important point to note here is that when a bulk
operator approaches to the boundary we encounter a singularity [40]. Similarly
in MQM we have the ordering ambiguity of the operators arises when we move
from the region $t={t_{o}}$ to $t\neq{t_{o}}$ on time evolution of (2.40) to
(2.44). If we consider the entire space of N fermions $N\rightarrow\infty$
this leads to singularity. Now in order to find the exact coefficient of
superposition of the states from (2.41) and (2.43) above $t_{o}$ we must
remember the fact that these states just show up as indistinguishable at
$t_{o}$ while away from $t_{o}$ they are distinct. So the coherent state above
$t_{o}$ must be given by (2.44). From the discussion of [11, 28, 2] these
operators corresponds to special tachyonic states and higher Virasoro
primaries. Now in this context we need to identify the open string operator or
the boundary operator. Note that exactly at $t={t_{o}}$ for the
states/operators from the string theory(2.47), collective field theory (2.56)
or from the MQM (2.60) (when we set the origin of time at ${t_{o}}$), they do
not show any explicit time dependence exactly at $t_{o}$. These operators can
either be viewed as the one obtained from the one at $t\neq{t_{o}}$ by time
evolution as we discussed throughout or the one without any explicit time
dependent part. The time independent one must correspond to the
states/operators which are extended in Liouville direction but localized in
matter direction. Hence they should be identified with the open string
operators. Finally let us briefly tell about matrix model string theory
correspondence of the complete scenario we just obtained. Theory describes
free fermionic state as well as coherent state where the coherent state is
strongly localized at $t={t_{o}}$. The closed string state turned into a
coherent state at $t_{o}$ as described by (LABEL:scrh2), which follows the
view of [44, 5]. The transition amplitude can be read from collective field
theory.
As the collective field correspond to a spacetime tachyon so from (2.56) we
obtain the closed string emission/absorption amplitude, which is giving the
one point function in mini superspace approximation
$\langle\hat{W}\,\hat{V}(P)\rangle=U(P)e^{-iP{t_{o}}.}$ (2.61)
where V(P) is the operator representing tachytonic vertex and the above
relation can be viewed on evaluation of the above in 2nd quantized field
theory [2]. Now the state in $(\ref{ps})$ while show up as time dependent
state w.r.t the free hamiltonian ${H_{o}}$, they are stable w.r,t an effective
hamiltonian $H_{\rm eff}={e^{-W}}{H_{o}}{e^{W}}$. The effect of the
macroscopic loop operator is to cause transition of a fermion from Fermi level
to above. At the double scaling limit $\beta\rightarrow\infty$ the transition
amplitude for a single fermion is given from (2.28,2.40) by
$\langle{\psi_{>}}({E^{\prime}},t_{f})|{\psi}(E,t_{i})\rangle\sim\langle{\psi_{o}}(E^{\prime})|W|{\psi_{o}}(E)\rangle,$
(2.62)
with ${t_{i}}(t_{f})={-\infty}(\infty)$. The presence of D brane will change
the Fermi level $\mu\rightarrow\mu^{\prime}$. So from the 2nd quantized theory
we can see that the transition amplitude
$|\mu\rangle\rightarrow|\mu^{\prime}\rangle$ is in accordance with [4],[38].
$\langle{\mu^{\prime}}|W(l,t)|\mu\rangle=e^{-i\delta(\mu-{\mu^{\prime}})}e^{i(\mu-{\mu^{\prime}})t}K_{i(\mu-{\mu^{\prime}})}(\sqrt{\mu
l}).$ (2.63)
Note the transition amplitude is time independent which is evident from matter
one point function (2.56) and it signifies the fact that we have stable D
brane.
## 3 Type 0A MQM on a circle in the presence of D brane
### 3.1 Evaluation of the free energy of Type 0A matrix model on a circle
We consider Type 0A matrix model compactified on a circle of radius R. As
considered in the previous section, there is no net background D0 brane charge
and hence it is described with $U(N)\times U(N)$ gauge symmetry. The partition
function in terms of the light cone variables is given by
$\displaystyle\int
d{Z_{+}}d{Z_{-}}d{\overline{Z}_{+}}d{\overline{Z}_{-}}dAd{\tilde{A}}e^{{-\beta\int_{0}^{2{\pi}R}}dtTr\left[{\overline{Z}_{+}{D_{A}}}{Z_{-}}+{Z_{+}\overline{({D_{A}}Z_{-})}}+{\frac{1}{2}}(\overline{Z}_{-}Z_{+}+\overline{Z}_{+}Z_{-})\right]},$
(3.1)
where ${D_{A}}Z={\partial_{t}}Z+i[{A}Z-Z{\tilde{A}}]$ and A is as given in
(2.1). Now we fix the gauge ${\partial_{t}}A=0$, which sets $A$ and
$\tilde{A}$ to their zero modes $A^{(0)}\equiv X/2\pi\alpha^{\prime}$ and
$\tilde{A}^{(0)}\equiv\tilde{X}/2\pi\alpha^{\prime}$, where in the T-dual
theory X and ${\tilde{X}}$ corresponds to collective coordinate of D0 and anti
D0 brane [9]. As before the gauge fixing introduces the FP determinant [15]
$\int db\,dc\exp({\rm
Tr}b\partial_{t}D_{t}c)={\prod_{i<j}}{\left({\frac{\sin[(x_{i}-x_{j})R/2]}{(x_{i}-x_{j})R/2}}\right)}^{2}{\left({\frac{\sin[(\tilde{x}_{i}-\tilde{x}_{j})R/2]}{(\tilde{x}_{i}-\tilde{x}_{j})R/2}}\right)}^{2},$
(3.2)
where $x_{i}$ and $\tilde{x}_{i}$ are the eigenvalues of $X$ and $\tilde{X}$
respectively. Now the denominator gets canceled with the respective
Vandermonde determinant so that the usual measure factor
$\Delta(x)^{2}\Delta(\tilde{x})^{2}$ is converted to the measure for unitary
matrices
${\prod_{i<j}\sin^{2}({(x_{i}-x_{j})R\over
2})\sin^{2}({(\tilde{x}_{i}-\tilde{x}_{j})R\over 2})}.$ (3.3)
Note these are the measure for unitary matrices
$U=e^{\frac{iXR}{2}},~{}\tilde{U}=e^{\frac{i\tilde{X}R}{2}}$.which are
holonomy factors (we chose ${\alpha^{\prime}=2}$). Therefore the natural
variables to be integrated over are the “holonomies” $U=e^{2i\pi
A^{(0)}R},~{}\tilde{U}=e^{2\pi i\tilde{A}^{(0)}R}$. Once we gauge fix A and
$\tilde{A}$ The partition function depends on the gauge field only through the
global holonomy factor, given by the unitary matrix
$\Omega=\hat{T}e^{i\int_{0}^{2\pi
R}A(t)dt}\quad;\quad\tilde{\Omega}=\hat{T}e^{i\int_{0}^{2\pi
R}\tilde{A}(t)dt}.$ (3.4)
In the $A=const$ gauge, in the path integral, the constant modes of A can be
absorbed by redefining the fields ${Z_{-}},{\overline{Z}_{-}}$ as
$Z_{-}(t)\rightarrow{e^{-iAt}Z_{-}(t)e^{i\tilde{A}t}}\quad;\quad\overline{Z}_{-}(t)\rightarrow{e^{-i\tilde{A}t}\overline{Z}_{-}(t)e^{iAt}},$
(3.5)
which replaces the periodic boundary condition
${Z_{\pm}}(2{\pi}R)={Z_{\pm}}(0)\quad;\quad{\overline{Z}_{\pm}}(2{\pi}R)={\overline{Z}_{\pm}}(0)$
by a $SU(N)$-twisted one [14],[17]
$\displaystyle Z_{+}(2{\pi}R)={Z_{\pm}}(0)$
$\displaystyle\quad;\quad\overline{Z}_{+}(2{\pi}R)=\overline{Z}_{+}(0)$
$\displaystyle Z_{-}(2{\pi}R)={\Omega}Z_{-}(0){\tilde{\Omega}^{-1}}$
$\displaystyle\quad;\quad\overline{Z}_{-}(2{\pi}R)={\tilde{\Omega}}\overline{Z}_{-}(0){\Omega^{-1}},$
(3.6)
So in the constant A gauge integration with respect to the fields
${Z_{\pm}}(x),{\overline{Z}_{\pm}}(x)$ is Gaussian with the determinant of the
quadratic form equal to one. Therefore it is reduced to the integral with
respect to the initial values
${Z_{\pm}},{\overline{Z}_{\pm}}={Z_{\pm}},{\overline{Z}_{\pm}}(0)$ of the
action evaluated along the classical trajectories, which satisfy the twisted
periodic boundary condition (3.6). Therefore the canonical partition function
of the matrix model can be reformulated as an ordinary matrix integral with
respect to the hermitian matrices $Z_{+}$ , $Z_{-}$ ; $\overline{Z}_{+}$ ,
$\overline{Z}_{-}$ and the unitary matrices $\Omega,\tilde{\Omega}$, :
${\cal{Z}}_{N}=\int dZ_{+}dZ_{-}d\overline{Z}_{+}d\overline{Z}_{-}d\Omega
d{\tilde{\Omega}}e^{i\beta
Tr(\overline{Z}_{+}Z_{-}+Z_{+}\overline{Z}_{-}-q\overline{Z}_{-}\Omega
Z_{+}\tilde{\Omega}^{-1}-qZ_{-}\tilde{\Omega}\overline{Z}_{+}\tilde{\Omega})},$
(3.7)
where we denote
$q=e^{2i\pi R}.$ (3.8)
Now note the above expression can be written as
$\displaystyle\cal{Z}_{N}$ $\displaystyle=$ $\displaystyle\int
dZ_{+}dZ_{-}d\overline{Z}_{+}d\overline{Z}_{-}d\Omega
d{\tilde{\Omega}}e^{i\beta{\rm
Tr}(\overline{Z}_{+}Z_{-}+Z_{+}\overline{Z}_{-}-q\overline{Z}_{+}\Omega
Z_{-}\tilde{\Omega}^{-1}-qZ_{+}\tilde{\Omega}\overline{Z}_{-}{{\Omega}^{-1}})}$
(3.9) $\displaystyle=$ $\displaystyle\int
dZ_{+}dZ_{-}d\overline{Z}_{+}d\overline{Z}_{-}d\Omega
d{\tilde{\Omega}}e^{i\beta{\rm
Tr}(\overline{Z}_{+}Z_{-}+Z_{+}\overline{Z}_{-}-q\overline{Z}_{+}(\Omega
Z_{-}{\Omega}^{-1})\Omega\tilde{\Omega}^{-1}-qZ_{+}\tilde{\Omega}{\Omega}^{-1}({\Omega}\overline{Z}_{-}{{\Omega}^{-1}})}$
$\displaystyle=$ $\displaystyle\int
dZ_{+}dZ_{-}d\overline{Z}_{+}d\overline{Z}_{-}d\Omega
d{\tilde{\Omega}}e^{i\beta{\rm
Tr}(\overline{Z}_{+}Z_{-}+Z_{+}\overline{Z}_{-}-q\Omega\tilde{\Omega}^{-1}\overline{Z}_{+}(\Omega
Z_{-}{\Omega}^{-1})-qZ_{+}\tilde{\Omega}{\Omega}^{-1}({\Omega}\overline{Z}_{-}{{\Omega}^{-1}})}$
$\displaystyle=$ $\displaystyle\int
dZ_{+}^{\prime}dZ_{-}d\overline{Z}_{+}^{\prime}d\overline{Z}_{-}d\Omega
d{\tilde{\Omega}}e^{i\beta{\rm
Tr}(\tilde{\Omega}{\Omega}^{-1}{\overline{Z}_{+}^{\prime}}Z_{-}+{\overline{Z}_{-}}{Z_{+}^{\prime}}\Omega\tilde{\Omega}^{-1}-q\overline{Z}_{+}^{\prime}\Omega
Z_{-}{\Omega}^{-1}-qZ_{+}^{\prime}{\Omega}\overline{Z}_{-}{{\Omega}^{-1}}})$
$\displaystyle=$ $\displaystyle\int
dZ_{+}^{\prime}dZ_{-}d\overline{Z}_{+}^{\prime}d\overline{Z}_{-}d\Omega
d{\Omega^{\prime}}e^{i\beta{\rm
Tr}(\Omega^{\prime}{\overline{Z}_{+}^{\prime}}Z_{-}+{\overline{Z}_{-}}{Z_{+}^{\prime}}{{\Omega^{\prime}}^{-1}}-q\overline{Z}_{+}^{\prime}\Omega
Z_{-}{\Omega}^{-1}-qZ_{+}^{\prime}{\Omega}\overline{Z}_{-}{{\Omega}^{-1}}}),$
where we define
$Z_{+}\tilde{\Omega}{\Omega}^{-1}=Z_{+}^{\prime}\quad;\quad\Omega\tilde{\Omega}^{-1}\overline{Z}_{+}={\overline{Z}_{+}^{\prime}}$
; ${{\Omega^{\prime}}}=\tilde{\Omega}{\Omega}^{-1}$.The last expression
implies replacing $\tilde{X}$ by $\tilde{X}-X$, both running over the infinite
real line. The redefinition of the variables will keep the measure invariant.
So by generalizing Harishchandra-Itzykson-Zuber integral we can write the
above partition function as 121212$(\int
dUe^{iTrUX{U^{-1}}Y}=Const.{\frac{\det{e^{i{x_{k}}{y_{l}}}}}{\Delta(x)\Delta(y)}}$where
${x_{k}}$ and ${y_{l}}$ are eigenvalues of X and Y and ${\Delta(x)}$
${\Delta(y)}$ are Vandermonde determinant given by
$\Delta(x)={\displaystyle\prod_{i\leq j}}({x_{k}}-{x_{l}})$ , 131313To express
the part involving ${\Omega^{\prime}}$ we used the fact that in the integral
$(\int dUe^{i{\rm Tr}UX{U^{-1}}DY}$ where D is a complex diagonal matrix with
${D^{-1}}={D^{\dagger}}$ and $Y=VyV^{-1}$ where y is the eigenvalue of Y and V
is the unitary matrix diagonalizing Y. Now we can write
$DY={V^{\prime}}d\,y{{V^{\prime}}^{-1}}$ for some other diagonalizing matrix
${V^{\prime}}$ which exploits the fact that ${\rm
Diag}({{(DY)}^{\dagger}}DY)={\rm Diag}({{Y}^{\dagger}}Y)={\rm
Diag}({V}{y^{*}}y{V^{\dagger}})$ which implies the above expression (d is
eigenvalue of D).So following the formal derivation of the integral we can
write $(\int
dUe^{iTrUX{U^{-1}}DY}=Const.{\frac{\det{e^{i{x_{k}}{d_{kl}}{y_{l}}}}}{\Delta(x)\Delta(y)}},$
which is nonzero only when $k=l$. for any diagonal matrix D.Note,we are not
summing over k and l.Denominators gets canceled with the Vandermonde
determinants appearing from
$\overline{Z}_{+},Z_{+},\overline{Z}_{-},Z_{-}$.[14].
$\displaystyle{\cal{Z}}_{N}(t)$ $\displaystyle=$
$\displaystyle\int\limits_{-\infty}^{\infty}\prod_{k=1}^{N}[d{z_{+}}_{k}][d{z_{-}}_{k}][d{\overline{z}_{+}}_{k}][d{\overline{z}_{-}}_{k}][d{\Omega^{\prime}}_{kk}]$
(3.10)
$\displaystyle[\det_{jk}\left(e^{i{{\Omega^{\prime}}_{jk}}{\overline{z}_{+}}_{j}{z_{-}}_{k}}\right){\rm
det}_{jk}\left(e^{-iq{\overline{z}_{+}}_{j}{z_{-}}_{k}}\right){\rm
det}_{jk}\left(e^{i{{\Omega^{\prime}}^{-1}}_{jk}{\overline{z}_{-}}_{j}{z_{+}}_{k}}\right)$
$\displaystyle{\rm
det}_{jk}\left(e^{-iq{z_{+}}_{j}{{\overline{z}_{-}}_{k}}}\right)],$
where ${\Omega^{\prime}_{jk}}$ which has only diagonal elements nonzero. Now
we show that the grand canonical partition function can be written as a
Fredholm determinant
$Z(\mu,t)={\rm Det}(1+e^{-2{\pi}R\beta\mu}K_{+}K_{-}),$ (3.11)
where
$\displaystyle[{K_{+}}f]({{\overline{z}_{-}}{z_{-}}})$ $\displaystyle=$
$\displaystyle\int[d{\overline{z}_{+}}][d{z_{+}}]dt{e^{i(t{\overline{z}_{+}}{z_{-}}+{t^{-1}}{\overline{z}_{-}}{z_{+}})}}f({{\overline{z}_{+}}{z_{+}}}),$
$\displaystyle[{K_{-}}f]({{\overline{z}_{+}}{z_{+}}})$ $\displaystyle=$
$\displaystyle\int[d{\overline{z}_{-}}][d{z_{-}}]{e^{-iq({\overline{z}_{+}}{z_{-}}+{\overline{z}_{-}}{z_{+}})}}f({{\overline{z}_{-}}{z_{-}}}).$
(3.12)
${K_{+}}{K_{-}}f({\overline{z}_{+}}{z_{+}})=\int[d{\overline{z}_{-}}][d{z_{-}}]dte^{i(t{\overline{z}_{+}}{z_{-}}+{t^{-1}}{\overline{z}_{-}}{z_{+}})}\int[d{\overline{z}_{+}}^{\prime}][d{z_{+}}^{\prime}]e^{-iq({\overline{z}_{+}}^{\prime}{z_{-}}+{\overline{z}_{-}}{z_{+}}^{\prime})}f({{\overline{z}_{+}}^{\prime}}{{z_{+}}^{\prime}}).$
(3.13)
Note that t and ${t^{-1}}$ denote the diagonal elements of ${\Omega^{\prime}}$
and ${{\Omega^{\prime}}^{-1}}$ corresponding to
${\overline{z}_{\pm}},{z_{\pm}}$. Now note when we evaluate the determinant in
a diagonalizable basis which is naturally given by
$f({\overline{z}_{\pm}}{z_{\pm}})$,
${K_{+}}{K_{-}}f({\overline{z}_{\pm}}{z_{\pm}})$ will be independent of t i.e
$\int dt$ will come out as an overall factor. So following the analysis of
[14] the grand canonical partition function
$\displaystyle\sum_{N}{e^{-2{\pi}R\beta N\mu}{\cal{Z}}_{N}}$ can be expressed
as the Fredholm determinant (3.11) which is same as that of $c=1$ matrix
model. Now following [24] we can express the partition function as
$Trexp[-2{\pi}R\beta H]$. The gauge field A project the theory to singlet
sector so that in the absence of perturbation, the grand canonical partition
function is given by the Fredholm determinant
${\cal{Z}}(\mu)={\rm Det}(1+e^{-2{\pi}R\beta(\mu+H_{0})}),$ (3.14)
which must be same as (3.11). This can be interpreted as the grand canonical
finite-temperature partition function for a system of non-interacting fermions
in the inverse Gaussian potential. The Fredholm determinant can be computed
once we know a complete set of eigenfunctions for the one-particle Hamiltonian
${H_{o}}$. Now in order to evaluate the free energy we need to find the
density of states, it is conventional to introduce a cutoff $\Lambda$.
There is no momentum flow through the wall
$\overline{z}z={x^{2}}+{y^{2}}=|{\Lambda}|^{2}$ is implied by the condition
$({\hat{x}}{\hat{p}_{y}}+{\hat{x}}{\hat{p}_{y}}){\psi_{\pm}}(x,y){|_{({x^{2}}+{y^{2}}=|{\Lambda}|^{2})}}=({\hat{\overline{z}}_{+}}{\hat{z}_{+}}-{\hat{\overline{z}}_{-}}{\hat{z}_{-}}){\psi_{\pm}}({\overline{z}},z){|_{({\overline{z}}z=|{\Lambda}|^{2})}}=0$,
which has a solution
$\psi^{E}_{+}({\Lambda})=\psi^{E}_{-}({\Lambda}).$ (3.15)
This condition is satisfied for a discrete set of energies $E_{n}(n\in Z)$
defined by
$\phi_{0}(E_{n})-E_{n}\log\Lambda+2\pi n=0.$ (3.16)
From (3.16) we can find the density of the energy levels in the confined
system
$\rho(E)={\frac{{\rm
log}\Lambda}{2\pi\beta}}-{\frac{1}{2\pi\beta}}{\frac{d\phi_{0}(E)}{dE}},$
(3.17)
as derived in [14] Now we can calculate free energy ${\cal{F}}(\mu,R)={\rm
log}{\cal{Z}}(\mu,R)$ as
${\cal
F}(\mu,R)=\int_{-\infty}^{\infty}dE\,\rho(E)\log\left[1+e^{-2{\pi}R\beta(\mu+E)}\right],$
(3.18)
with the density (3.17). Integrating by parts in and dropping out the
$\Lambda$-dependent piece, we get
${\cal{F}}(\mu,R)=-{1\over{2\pi\beta}}\int
d\phi_{0}(E)\log\left(1+e^{-2{\pi}R\beta(\mu+E)}\right)=-R\int_{-\infty}^{\infty}dE{\phi_{0}(E)\over
1+e^{2{\pi}R\beta(\mu+E)}}..$ (3.19)
We close the contour of integration in the upper half plane and take the
integral as a sum of residues. This gives for the free energy
${\cal{F}}=-i\sum_{r=n+{\frac{1}{2}}>0}\phi_{o}\left(ir/R-\mu\right).$ (3.20)
As the Fredholm determinant is similar to that of $c=1$ MQM so following the
analysis of [14], [15] we can see From (3.20) it follows that [19]
$2{\rm sin}{\frac{\partial_{\mu}}{2{\beta R}}}\cdot{\cal
F}(\mu)=\phi_{o}(-\mu).$ (3.21)
Also its shown that the free energy can be expressed as
${\cal F}_{\rm
pert}(\mu)_{\\{t_{k}=0\\}}=-\frac{R}{2}\mu^{2}\log\frac{\mu}{\Lambda}-\frac{R+{1\over
R}}{24}\log\frac{\mu}{\Lambda}+R\sum\limits_{h=2}^{\infty}\mu^{2-2h}c_{h}(R),$
(3.22)
### 3.2 Free energy of Type 0A matrix model on a circle with D brane
In this section we will consider the type 0A matrix model path integral in the
presence of a D brane and show the grand canonical partition function can be
expressed as the Fredholm determinant. We consider the brane in the NS NS
sector and show that how to generalize the analysis for the brane in any other
sector. Consider the path integral in the presence of the macroscopic loop
operator localized at ${t_{o}}$, which is the generalization of (3.1). The
classical action will remain periodic even in the presence of D brane so we
can express the
$\displaystyle\int
d{Z_{+}}d{Z_{-}}d{\overline{Z}_{+}}d{\overline{Z}_{-}}dAd{\tilde{A}}e^{{-\beta\int_{0}^{2{\pi}R}}dt{\rm
Tr}\left[{\overline{Z}_{+}{D_{A}}}{Z_{-}}+Z_{+}{D_{A}}{\overline{Z}_{-}}+{\frac{1}{2}}(\overline{Z}_{-}Z_{+}+\overline{Z}_{+}Z_{-})\right]+{\rm
Tr}{W(t_{o})}},$ (3.23)
The macroscopic loop operator depends on diagonal elements only, so the
partition function (3.7) can be expressed as
$\displaystyle{{\cal{Z}}_{N}}(t)=\int\limits_{-\infty}^{\infty}\prod_{k=1}^{N}$
$\displaystyle[d{z_{+}}_{k}][d{z_{-}}_{k}][d{\overline{z}_{+}}_{k}][d{\overline{z}_{-}}_{k}][dt_{k}]{\rm
det}_{jk}\left(e^{i{t^{-1}_{jk}}{z_{-}}_{j}{\overline{z}_{+}}_{k}}\right){\rm
det}_{jk}{\left(e^{-iq{z_{-}}_{j}{\overline{z}_{+}}_{k}}\right)}{\rm
det}_{jk}\left(e^{i{t_{jk}}{\overline{z}_{-}}_{j}{z_{+}}_{k}}\right)$
$\displaystyle{\rm
det}_{jk}{\left(e^{-iq{\overline{z}_{-}}_{j}{z_{+}}_{k}}\right)}$
$\displaystyle
exp[\displaystyle\sum_{i}log(1+{\frac{{\overline{z}_{+}}_{i}{z_{+}}_{i}+{\overline{z}_{-}}_{i}{z_{-}}_{i}+{\overline{z}_{+}}_{i}{z_{-}}_{i}+{\overline{z}_{-}}_{i}{z_{+}}_{i}}{\mu_{B}^{2}}})],$
(3.24)
(when we have off-diagonal t is zero , also we are not summing over j,k)
where $\displaystyle\sum_{i}$ is coming from Trace and again above can be
expressed as
$\displaystyle{\cal{Z}}_{N}(t)$ $\displaystyle=$
$\displaystyle\int\limits_{-\infty}^{\infty}\prod_{k=1}^{N}[d{z_{+}}_{k}][d{z_{-}}_{k}][d{\overline{z}_{+}}_{k}][d{\overline{z}_{-}}_{k}]{\rm
det}_{jk}\left(e^{i{t^{-1}_{jk}}{z_{-}}_{j}{\overline{z}_{+}}_{k}}\right){\rm
det}_{jk}\left(e^{-iq{z_{-}}_{j}{\overline{z}_{+}}_{k}}\right){\rm
det}_{jk}\left(e^{i{t_{jk}}{\overline{z}_{-}}_{j}{z_{+}}_{k}}\right)$ (3.25)
$\displaystyle{\rm
det}_{jk}\left(e^{-iq{\overline{z}_{-}}_{j}{z_{+}}_{k}}\right)\prod_{r=1}^{N}(1+{\frac{{\overline{z}_{+}}_{r}{z_{+}}_{r}+{\overline{z}_{-}}_{r}{z_{-}}_{r}+{\overline{z}_{+}}_{r}{z_{-}}_{r}+{\overline{z}_{-}}_{r}{z_{+}}_{r}}{\mu_{B}^{2}}}).$
Now in order to write the above expression we have used the fact that the
classical action is periodic even in the presence of the macroscopic loop
operator $Z(2\pi R)=Z(0)$. However at the quantum level there is a
discontinuity of state
$|\psi(2{\pi}R-\epsilon)\rangle\neq|\psi(0)+\epsilon\rangle$. This causes the
absence of the vortex d.o.f. Now if $f({{\overline{z}_{\pm}}{z_{\pm}}})$ is
the function which form the representation of $K_{+}$ and $K_{-}$ (3.12,
3.13), the action of W on f is given by
${\hat{W}}f({{\overline{z}_{\pm}}{z_{\pm}}})=(1+{\frac{2{\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}}+{\hat{\overline{z}}_{+}}{\hat{z}_{-}}+{\hat{\overline{z}}_{-}}{\hat{z}_{+}}}{\mu_{B}^{2}}})f({{\overline{z}_{\pm}}{z_{\pm}}}).$
(3.26)
Now, the operator ${\hat{W}}$ does not introduce any interaction between the
fermions, so no off diagonal terms from W. Now from (3.11) and (3.24) we can
write the grand canonical partition $\sum_{N}{e^{-2{\pi}R\beta
N}}{\cal{Z}}_{N}$ as
${\cal{Z}}(\mu)={\rm det}(1+e^{-2\pi R\beta\mu}WK).$ (3.27)
Now in order to evaluate (3.27) following (3.13) a representation of K is
formed by the basis $f({{\overline{z}_{+}}_{i}}{{z_{+}}_{i}})$ with i runs
from 1 to N. Also from (3.13) it follows that
$f({\overline{z}_{+}}{z_{+}})\sim{{({\overline{z}_{+}}{z_{+}})}^{n}}$. So when
we evaluate the expectation value of WK in this basis, in the expression of W
we see that $\langle{\overline{z}_{\pm}}{z_{\pm}}\rangle=0$ as on a closed
contour the angular integral will vanish. in the inner product, the other term
${\overline{z}_{+}}{z_{-}}+{z_{+}}{\overline{z}_{-}}$ expresses nothing but
the hamiltonian of which f is an eigenfunction. So if ${\psi_{n}}$ are the set
of functions which diagonalizes K we can write (3.27) as
$\displaystyle\sum_{n}{\rm log}\langle{\psi_{n}}|(1+e^{-2\pi
R\beta\mu}\hat{W}\hat{K})|{\psi_{n}}\rangle,$ (3.28)
where
$\displaystyle\hat{W}{K_{+}}{K_{-}}f({\overline{z}_{+}}{z_{+}})$
$\displaystyle=$
$\displaystyle\int[d{\overline{z}_{-}}][d{z_{-}}]e^{i({\overline{z}_{+}}{z_{-}}+{\overline{z}_{-}}{z_{+}})}[d{\overline{z}_{+}}^{\prime}][d{z_{+}}^{\prime}]e^{-iq({\overline{z}_{+}}^{\prime}{z_{-}}+{\overline{z}_{-}}{z_{+}}^{\prime})}$
(3.29)
$\displaystyle(1+{\frac{{\overline{z}_{+}}^{\prime}{z_{-}}+{\overline{z}_{-}}{z_{+}}^{\prime}}{\mu_{B}^{2}}})f({{\overline{z}_{+}}^{\prime}}{{z_{+}}^{\prime}}).$
As the expression depends on
$({\overline{z}_{+}}{z_{-}}+{\overline{z}_{-}}{z_{+}})$ which is the
expression for free hamiltonian ${H_{o}}$ so comparison with (3.14), Fredholm
determinant is expected to be given by
${\cal{Z}}(\mu)={\rm
det}(1+e^{-2{\pi}R\beta(\mu+H_{0})-\log(1-{\frac{2{H_{o}}}{\mu_{B}^{2}}})}).$
(3.30)
This is,we are going to analyze in the next part of this section.
### 3.3 Evaluation of the thermal partition function
In this section we are going to study type 0A MQM in the presence of D brane
with time t compactified on a circle, evaluate and analyze the free energy. In
the absence of the brane when we compactify string theory on a circle of
radius R, in the dual MQM the Schrodinger equation have periodic solution i.e
${\psi}(t)={\psi}(t+2{\pi}R)$, which implies $E={\frac{n}{R}}$. Now consider
the theory with D brane which can be accomplished by including a macroscopic
loop operator localized at $t={t_{o}}=0\equiv 2{\pi}R$ (say) to the action.
From previous discussion it follows that in the presence of the operator
Schrodinger equation will have well defined solution only in the region $0\leq
t\leq 2{\pi}R$ when discontinuity occur at the respective point and we have
$\psi({2{\pi}R-\epsilon})\neq\psi({2{\pi}R+\epsilon})$ in the limit
$\epsilon\rightarrow 0$. This is consistent with the fact that the presence of
a spacelike brane breaks the winding symmetry and apparently the theory
correspond to that of an open string. At the end we will see how the closed
string scenario arise in this picture. Now in a compact time we must have the
condition ${\psi}(t)={\psi}(2{\pi}R+t)$. So effectively we can view the theory
as MQM on a line of length ${2{\pi}R}$ with two ${\delta}\textendash{\rm
potential}$ along with the operator $\hat{W}$ (where one is the image of the
other, superimposed) placed at its two ends. When we cross the boundary on
either side, situation repeats 141414note it never implies periodicity, its
just similar to the situation of an open string in 2D with Dirichlet boundary
condition in compact direction and identification of the matter direction with
t. It winds along the circle m times although ends are not identified. The
open string which wraps m times a circle of length $2\pi{R^{\prime}}$ with
$2\pi m{R^{\prime}}=2{\pi}R$, we can define same theory on either of the
slices $2\pi(n-1)R\leq t\leq 2\pi nR$, crossing the boundary of the slice
implies going back from that end of the string to the other and hence the
situation repeats, i.e we can define the theory on any of the slices
$2\pi(n-1)R\leq t\leq 2\pi nR$. So effectively we have the time dependent
Schrodinger equation with double delta potential well as:
$\displaystyle[\,\,i{\frac{\partial}{{\partial}t}}$ $\displaystyle-$
$\displaystyle\\{{\delta}(t)+{\delta}(t-2{\pi}R)\\}W({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}}(t),{H_{o}})]{\Psi}({\overline{z}_{\pm}}{z_{\pm}},t)$
(3.31) $\displaystyle=$
$\displaystyle{\mp}i\left[{z_{\pm}}{\frac{\partial}{\partial{z_{\pm}}}}+{{\overline{z}_{\pm}}}{\frac{\partial}{\partial{\overline{z}_{\pm}}}}+1\right]{\Psi}({\overline{z}_{\pm}}{z_{\pm}},t),$
We have the discontinuity
$\displaystyle{\psi}({\epsilon})-{\psi}({-\epsilon})$ $\displaystyle=$
$\displaystyle{\hat{W}}{\psi_{o}}(t=0)$
$\displaystyle{\psi}({2{\pi}R}+{\epsilon})-{\psi}({2{\pi}R-\epsilon})$
$\displaystyle=$ $\displaystyle{\hat{W}}{\psi_{o}}(t={2{\pi}R}).$ (3.32)
In order to evaluate the partition function we must need to know what is the
right Hilbert space describe the wave function $\psi$ on the circle. This is
because we know that the Hilbert space $\\{|E\rangle\\}$ and $\\{|E\pm
ni\rangle\\}$ cannot be mapped to each other. So, to answer this note that
when we define the Schrodinger equation in the double delta potential well in
an uncompactified direction we have one type of the solution inside the well,
while the solution at the left and the rightside of the well differs from the
same, decided by the discontinuity (3.32). Now compactification on a circle of
length $2{\pi}R$ imply the outside region of the well is just squeezed to a
point $t=0\equiv 2{\pi}R$ and the wave function at the left and right side of
the double delta-well are given by the wave function at the right and left
$\epsilon\textendash{\rm neighborhood}$ of that point. Now in uncompactified
time we had free fermion wave function (2.28) in the region $-\infty$ to
${t_{o}}$ while from ${t_{o}}$ to $\infty$ the wave function is described by
(2.44). So in the compactified time we have the ambiguity that which one
should describe the fermionic wave function in the region $0\leq t\leq
2{\pi}R$. To resolve recall the macroscopic loop operator $\hat{W}$ ( 2.15)
and the constraint (2.25) are symmetric under
${\overline{z}_{+}},{z_{+}}\rightarrow{\overline{z}_{-}},{z_{-}}$ . So in the
Schrodinger equation(2.12) we have the symmetry,
$t\rightarrow{2{\pi}R-t};\quad\hat{W}\rightarrow-\hat{W},$ (3.33)
which takes an wave function
${\overline{z}_{+}},{z_{+}}\rightarrow{\overline{z}_{-}},{z_{-}}$
representation151515 this is from the definition of
${\overline{z}_{\pm}},{z_{\pm}}$ (2.6) and the reversal of the sign of W is
explained from (3.32),the transformation changes the sign in the r.h.s of
(3.32) because the time interval $\epsilon\rightarrow-\epsilon$ under the
transformation and hence the relation will remain unchanged.Note this
transformation is also associated with the reversal of the sign of the gauge
field A. but we chose axial gauge $a=\tilde{a}$,where $a,\tilde{a}$ are the
zero modes of gauge fields $A,\tilde{A}$, so this is not affected along with
reversal of the sign of the energy $E\rightarrow-E$ in the free fermionic wave
function as introduced in (2.28). As both ${\overline{z}_{+}},{z_{+}}$ and
${\overline{z}_{-}},{z_{-}}$ describes same wave function in different
representation so it must be a symmetry inside the double delta well (note
this is never a symmetry outside the well where the fermion can see only one
potential barrier). The condition (2.25) remains unaffected by this symmetry
and we can project the wave function $t={t_{o}}=0\equiv 2{\pi}R$ to the
physical sector. Now under the transformation (3.33) the free fermion wave
function (2.28) is just changed by a phase ${e^{2iE\pi R}}$ whereas according
to (2.28,2.30,2.41) the wave function describes a state $|E\pm ni\rangle$ goes
from $e^{-iEt\pm nt}\rightarrow e^{iE(2{\pi}R-t)\mp(2\pi nR-nt)}$. Although
the transformation (3.33) take the wave function from ${z_{+}}$ to ${z_{-}}$
representation but the both have the same time dependent part. So when we
consider the wave function (2.41) we see it does not respect the symmetry.
Note unlike the closed string momentum modes which respects the symmetry in
Euclidean time because of their periodicity on the circle,
$W({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}},{H_{o}})$ generates discrete
shift in energy $E\pm ni$ at any R, which leads to the violation of symmetry.
So we conclude the wave function inside the well which is our compact time
$0\leq t\leq 2{\pi}R$ must corresponds to that of a free fermion (2.28). We
will come to the string theoretical interpretation in the next subsection. The
wave function at $t=0\equiv 2{\pi}R$ corresponds to (2.41) so the partition
function corresponds to the transition amplitude
$\displaystyle\lim_{\epsilon\to
0}\,\langle{\psi_{o}}{(\epsilon)}|\psi_{>}(2{\pi}R-\epsilon)\rangle$
$\displaystyle=$
$\displaystyle\langle{\psi_{o}}[{e^{-{\beta\int_{0}^{2{\pi}R}}dt{\hat{H}_{o}}}}{e^{{\hat{W}}({t_{o}})}}]{\psi_{o}}\rangle$
(3.34) $\displaystyle=$ $\displaystyle{{\rm
Tr}_{\psi_{o}^{E}}}[{e^{-{2{\pi}R\beta}{\hat{H}_{o}}}}[{e^{{\hat{W}}({t_{o}})}}]\
]$ $\displaystyle=$ $\displaystyle{{\rm
Tr}_{\psi_{o}^{E}}}\left[{e^{-{2{\pi}R\beta}[{\hat{H}_{o}}-{\frac{1}{2{\pi}R\beta}}\hat{W}({t_{o}})]}}\right],$
where ${{\rm Tr}_{\psi_{o}^{E}}}$ implies the summation over all free fermion
eigenfunctions and $\psi_{>}$ corresponds to (2.41). As
$\beta\rightarrow\infty$ at double scaling limit, so inside the partition
function we can replace it by $\psi_{>}\rightarrow
e^{-W{(t_{o})}}{\psi_{o}}\,$ 161616In order to reach from the 1rst to 2nd step
in (3.34)we utilize the fact that we can scale the time $t\rightarrow\beta t$
so that the term with the macroscopic loop operator $\int
dtW(t)\delta(t-{t_{o}})$ will get a factor ${\frac{1}{\beta}}$. Hence in the
double scaling limit where $\beta\rightarrow\infty$ and with Euclidean time,
we can lift up the term to the exponential and the exponent gives an exact
expression what we have obtained from the path integral (3.23) . Also
following the discussion of section 2 we can directly add $W(t_{o})$ in the
expression of hamiltonian in Euclidean time to get the expression (3.34) .
Note when we consider Schrodinger equation, the contribution from the term
${\frac{1}{2{\pi}R\beta}}\hat{W}({t_{o}})$ in the expression of hamiltonian
(3.34) at double scaling limit will not be negligible due to a transformation
of variable which leads to the physics at the vicinity of the top of the
potential, as we discussed section $2.1$ and can be found [1]. From (2.15) we
know that in a single fermionic state the presence of D brane implies implies
the insertion of the following operator
${e^{{\hat{W}}({t_{o}})}}=1+{\frac{{\hat{\overline{z}}_{+}}{\hat{z}_{+}}+{\hat{\overline{z}}_{-}}{\hat{z}_{-}}+{\hat{\overline{z}}_{-}}{\hat{z}_{+}}+{\hat{\overline{z}}_{+}}{\hat{z}_{-}}}{\mu_{B}^{2}}}.$
(3.35)
Here first we will evaluate the partition function for single fermionic d.o.f
in order to understand the behaviour of the system in the presence of a brane.
Next we will derive the grand canonical partition function. Now following the
discussion in section 2 we can represent the wave function (2.40) at
$t=2{\pi}R-\epsilon$ in (3.34) with the operator $\hat{W}$ expressed either in
terms of ${\hat{\overline{z}}_{+}},{\hat{z}_{+}}$ or
${\hat{\overline{z}}_{-}},{\hat{z}_{-}}$ representation. We have shown in the
Appendix that
$\langle{\overline{z}_{+}}{z_{+}}|{\hat{\overline{z}}_{+}}{\hat{z}_{+}}|{\overline{z}_{+}}{z_{+}}\rangle$
and
$\langle{\overline{z}_{-}}{z_{-}}|{\hat{\overline{z}}_{-}}{\hat{z}_{-}}|{\overline{z}_{-}}{z_{-}}\rangle$
diverge. So we must express $\hat{W}$ as
$W({{\hat{\overline{z}}_{-}}{\hat{z}_{-}},H_{o}})$ for the basis
$|{\overline{z}_{+}}{z_{+}},E\rangle$ basis and vice versa. So applying (A.7)
we can write (3.34) as:
${{\rm
Tr}_{\psi_{o}}}\left[{e^{-{2{\pi}R\beta}[{\hat{H}_{o}}-{\frac{1}{2{\pi}R\beta}}{\rm
log}(1-{\frac{2\hat{H}_{o}}{\mu_{B}^{2}}})]}}\right]={{\rm
Tr}_{\psi_{o}}}\left[e^{-2{\pi}R\beta H_{o}^{\prime}}\right],$ (3.36)
Where
$\hat{H}_{o}^{\prime}={\hat{H}_{o}}-{\frac{1}{2{\pi}R\beta}}{\rm
log}(1-{\frac{2\hat{H}_{o}}{\mu_{B}^{2}}})\quad;\quad{E_{o}^{\prime}}={E}-{\frac{1}{2{\pi}R}}{\rm
log}(1-{\frac{2{E}}{\mu_{B}^{2}}}),$ (3.37)
with $E_{o}^{\prime}$ is the eigenvalue of $H_{o}^{\prime}$(note that we
omitted $\beta$ factor from the expression of ${E_{o}^{\prime}}$ following the
discussion of section 2, which can be done at double scaling limit by
redefinition of the variable). Before any further analysis let us make the
comment that here we have considered the macroscopic loop operator in the NS
NS sector. For any other sector we can use analysis of section 2, expressing W
in terms of
${\hat{\overline{z}}_{-}}.{\hat{z}_{-}}({\hat{\overline{z}}_{+}},{\hat{z}_{+}})$
in ${\overline{z}_{+}},{z_{+}}({\overline{z}_{-}},{z_{-}})$ representation and
using (A.7) to express $W({t_{o}})$ complete in terms of ${H_{o}}$ within the
trace. Although we will have a very different expression of ${H_{o}^{\prime}}$
but the analysis will remain same. Also if we did not apply this condition
(2.25), note we will have the term
$\langle{\hat{\overline{z}}_{-}}{\hat{z}_{-}}\rangle_{-}(\,\,\langle{\hat{\overline{z}}_{+}}{\hat{z}_{+}}\rangle_{+}\,\,)$
in the partition function from the expression of $\hat{W}$. This gives rise to
an infinite contribution to the partition function
$\lim_{r\to\infty}r{e^{i\phi(E-i)}}$ (where $\phi$ is the phase of the wave
function) and so the partition function diverge. This is the signature of the
presence of an unphysical degree of freedom leads to instability of the system
due to leakage. Now according to the discussion of section 2, at the double
scaling limit the partition function (3.36) can be expressed as the sum over
${E_{o}^{\prime}}$the eigenvalue of ${H_{o}^{\prime}}$ as:
${{\rm Tr}_{\psi_{o}}}\left[e^{-2{\pi}R\beta
H_{o}^{\prime}}\right]=\sum_{E}\left[{e^{-{2{\pi}R\beta}[{E}-{\frac{1}{2{\pi}R}}{\rm
log}(1-{\frac{2{E}}{\mu_{B}^{2}}})]}}\right]$ (3.38)
Now note that ${E_{o}^{\prime}}$,the eigenvalue of ${H_{o}^{\prime}}$, has
branch cut at $E={\frac{\mu_{B}^{2}}{2}}$ so we need to subtract a small cut-
off ${\rm log}\epsilon$ in order to have an well defined expression of the
energy and after subtraction $E\rightarrow{E^{\prime}}$ is an one to one
mapping, Also note at the singular point, $E={\frac{1}{2}}{\mu_{B}^{2}}$,
$e^{-2{\pi}R\beta{E^{\prime}}}$ is trivially zero and so it will not
contribute any pole to integrand. Now the string theory compactified at time
interval $2{\pi}R$ is described by the grand canonical partition function of
fermion at finite temperature ${\frac{1}{2{\pi}R}}$ and chemical potential
${\mu}$. So the free energy ${{\cal F}}={\rm log}{\cal{Z}}$ in the presence of
Dbrane is given by
${{\cal
F}}({\mu})={\int_{\infty}^{\infty}}dE{\rho}(E)log(1+{e^{-{\beta}({\mu}+{E^{\prime}}(E))}}),$
(3.39)
where$\rho(E)$ is given in (3.17)
$e^{i\phi_{0}(E)}=R(E)={\frac{\Gamma(iE+1/2)}{\Gamma(-iE+1/2)}}.$ (3.40)
Now we can calculate free energy ${\cal{F}}(\mu,R)={\rm log}{\cal{Z}}(\mu,R)$
as.
${\cal{F}}(\mu,R)=\int_{-\infty}^{\infty}dE\rho(E)\log[1+e^{-\beta(\mu+{E^{\prime}}(E))}].$
(3.41)
with the density (3.17),and from (3.39,3.37) the free energy is given by
$\displaystyle{\cal F}(\mu,R)$ $\displaystyle=$
$\displaystyle-{\frac{1}{2\pi}}\int d{\phi_{0}}(E){\rm
log}\left(1+{e^{-2{\pi}R\beta({\mu}+{E^{\prime}}(E))}}\right)$ (3.42)
$\displaystyle=$
$\displaystyle-R{\int_{-\infty}^{\infty}}d{E^{\prime}}{\frac{\phi_{0}(E({E^{\prime}}))}{1+e^{-2{\pi}R\beta(\mu+{E^{\prime}}(E)))}}}$
$\displaystyle=$ $\displaystyle-i{\sum_{r=n+{\litfont{1\over
2}}>0}\phi_{o}(E({E^{\prime}}={\frac{ir}{\beta R}}-\mu))}$ $\displaystyle=$
$\displaystyle-i{\sum_{r=n+{\litfont{1\over
2}}>0}}{\phi_{o}^{\prime}}({\frac{ir}{R}}-\mu),$
where we have
${\phi_{0}^{\prime}}({E^{\prime}})={\phi_{0}}(E)$ (3.43)
So from the expression of the density of the energy eigenstates (3.17) (ignore
the $\Lambda$ factor ) this implies the number of energy eigenstates between E
to E+dE is same as that of between ${E^{\prime}}$ to
${E^{\prime}}+d{E^{\prime}}$. So the partition function on a circle in the
presence of D brane corresponds to a deformation in static Fermi sea where the
deformation is expressed as $E=-\mu\,\Rightarrow\,E^{\prime}=-\mu$ with all
the energy eigenstates are in one to one mapping. Note that the partition
function is getting contribution only from the deformation of Fermi surface
instead of excitation modes. This is because we are in compact dimension and
its only the trace over energy eigenstates contribute, which we will explain
more in the next subsection. Finally the expression of the free energy ${\cal
F}(\mu,R)$ in (3.42) along with (3.43) suggests the following
$\displaystyle{{\rm
Tr}_{\psi_{o}}}\left[{e^{-{2{\pi}R\beta}[{\hat{H}_{o}}-{\frac{1}{2{\pi}R\beta}}{\hat{W}}({t_{o}})]}}\right]$
$\displaystyle=$ $\displaystyle{{\rm
Tr}_{\psi_{o}}}[{e^{-{2{\pi}R\beta}\\{{H_{o}}-{\frac{1}{2{\pi}R\beta}}{\log}(1-{\frac{2{H_{o}}}{\mu_{B}^{2}}})\\}}}]$
(3.44) $\displaystyle=$ $\displaystyle{{\rm
Tr}_{\psi_{o}}}[{e^{-{2{\pi}R\beta}\\{{H_{o}}-{\frac{1}{2{\pi}R\beta}}f({H_{o}})\\}}}]$
$\displaystyle=$ $\displaystyle{{\rm
Tr}_{\psi_{o}}}[{e^{-{2{\pi}R\beta}({H_{o}^{\prime}})}}]$ $\displaystyle=$
$\displaystyle{{\rm Tr}_{\psi_{o}^{\prime}}}[{e^{-{2{\pi}R\beta}({H_{o}})}}],$
where
$f({H_{o}})={\log}(1-{\frac{2{H_{o}}}{\mu_{B}^{2}}}){\quad;\quad}{H_{o}^{\prime}}={H_{o}}-{\frac{1}{2{\pi}R\beta}}f({H_{o}}).$
(3.45)
and in the last step we made a transformation from the basis
${\psi^{\pm}_{o}}(E)={e^{\mp{i\phi_{o}}(E)}}{e^{-iEt}}{{({\overline{z}_{\pm}}{z_{\pm}})}^{\pm
iE-{\frac{1}{2}}}}\rightarrow{\psi^{\pm}_{o}}({E^{\prime}})={e^{\mp{i\phi_{o}^{\prime}}({E^{\prime}})}}{e^{-i{E^{\prime}}t}}{{({\overline{z}_{\pm}}{z_{\pm}})}^{\pm
i{E^{\prime}}-{\frac{1}{2}}}}$ with
${E^{\prime}}=E-{\frac{1}{2{\pi}R}}{\log}(1-{\frac{2E}{\mu_{B}^{2}}})$ and
also ${\phi_{o}^{\prime}}({E^{\prime}})={\phi_{o}}(E)$. Note as we discussed
in section 2, although the contribution from the macroscopic loop operator has
a factor ${\frac{1}{\beta}}$ however in the double scaled hamiltonian it
cannot be ignored and the shifted energy will be given by $E^{\prime}$. Above
expression implies that the Type 0A MQM on a circle in the presence of a D
brane can be viewed as a free theory with the free hamiltonian ${H_{o}}$ with
the wave function replaced by the above one. This point will be relevant in
section 5. Note the relation (3.21,3.22) can be expressed as
$2{\rm sin}{\frac{\partial_{\mu}}{2\beta R}}\cdot{\cal
F}(\mu)=\phi_{o}^{\prime}(-\mu).$ (3.46)
Also note that from nonlinear relation between the effective hamiltonian
${H_{o}^{\prime}}$ and the free hamiltonian ${H_{o}}$ its evident that in the
genus expansion of free energy in the relation (3.22) will have both odd and
even powers of $\mu$( this is because the new free energy corresponds to
replacing $\mu\rightarrow E(E^{\prime}){|_{E^{\prime}=\mu}}$ which follows
from (3.39)). This is the signature of the presence of surface with boundary
which is the implication from MQM/string theory duality.
### 3.4 String theoretical interpretation
Let us very briefly say about the string theory side of the above story. The
matrix model partition function (3.44) corresponds to the disk amplitude
$\langle B|q^{(L_{o}+\overline{L}_{o})}|I\rangle$ (3.47)
Where $|B\rangle$ stands for boundary state and the above expression resembles
(3.34) on the matrix model side. The invariance of theory under the symmetry
(3.33) is related to the symmetry of the boundary state (2.47,2.48) under
interchange of the left and right moving tachyonic components in W
$U(k)e^{ik(X+\phi)-\sqrt{2}\phi}\leftrightarrow
U(-k)e^{ik(X-\phi)-\sqrt{2}\phi}$ (3.48)
as well as other Virasoro primaries obtained from discrete shift of matter and
Liouville momentum of tachyonic modes. One can see that in absence of D brane
this is a symmetry in MQM. The symmetry is implemented from the interchange
$X\rightarrow 2{\pi}R-X$ followed by a sign reversal of momentum
$k\rightarrow-k$ which is in accordance (3.33) with $k={\frac{n}{R}}$ and
$X\rightarrow it$, U(k) is the wave function. This keeps the matter part (
temporal part in matrix model) of the operator/state invariant. The reason for
this symmetry in matrix model is that the tachyonic states obtained from the
2nd quantized free fermionic theory (ground state) or the collective field
theory are actually in one to one to one correspondence with the above
operators(3.48), as we explained in section 2.3. Consequently we found that
the symmetry (3.33) projects the Hilbert space between $0<t<2{\pi}R$ to free
fermionic ground state. This is because the discrete tachyonic modes arises
from Minkowskian theory with shift $X\rightarrow iX\,;\,k\rightarrow
ik\,\,\Rightarrow e^{ikX}(e^{ikt})\rightarrow e^{ikX}(e^{ikt})$. However the
states created by the action of the operator
${({\hat{\overline{z}}_{-}}{\hat{\overline{z}}_{+}})}^{n}{({\hat{\overline{z}}_{-}}{\hat{\overline{z}}_{+}})}^{m}$
on ground state, although describes a state with imaginary energy but remains
in Minkowskian time (unperiodic) and so projected out by the symmetry within
$0<t<2{\pi}R$. However these states arise on expansion of
$W({\hat{\overline{z}}_{-}}{\hat{z}_{-}},{\hat{\overline{z}}_{+}}{\hat{z}_{+}},{H_{o}})$
which actually causes the excitation of a fermion over free fermionic ground
state,creates a coherent state. So we have only free fermionic ground state in
the region $0<t<2{\pi}R$ which in string theory indicates that we have only
free closed string modes in the region $0<X<2{\pi}R$. Coherent states are
strongly localized at the point of insertion of $\hat{W}$. In matrix model
partition function, the absence of the excitation modes has an explanation in
the fact that on a circle in the presence of a brane, partition function
corresponds to transition $\psi\rightarrow{\hat{W}}\psi$ exactly at
$t={t_{o}}=0\equiv 2{\pi}R$, where $|\psi\rangle$ is the free fermionic ground
state. So the operators from W which generates excitation naturally will not
contribute in the trace. This, alongwith the condition (2.25)(which ensures
the conservation of fermion number) implies that the partition function will
corresponds to that of a deformed Fermi surface. The poles of the partition
function (3.42) corresponds to free closed string tachyons in the bulk. In
section 5 we will see that same features will be reflected even when we turn
on the tachyonic background.
## 4 Fermionic scattering and semiclassical analysis
In this section we will study scattering of fermions in the presence of D
brane and tachyonic background at quasiclassical limit. The scattering
amplitude is given by
$S=\langle\beta,{t\rightarrow\infty}|\alpha,{t\rightarrow-\infty}\rangle,$
(4.1)
where ${\alpha}$ and ${\beta}$ denotes the incoming and outgoing state. As the
single incoming and outgoing state is given by
$|{\overline{z}_{+}}{z_{+}}\rangle$ and $|{\overline{z}_{-}}{z_{-}}\rangle$
respectively [28], so
$S=\langle{\overline{z}_{-}}{z_{-}},{\rm out}|{\overline{z}_{+}}{z_{+}},{\rm
in}\rangle.$ (4.2)
Now note that ${\overline{z}_{+}}{z_{+}}$ and ${\overline{z}_{-}}{z_{-}}$
representations are related by a unitary operator $\hat{S}$, which in our case
is nothing but the Fourier transformation on the complex plane. Recall, the
energy eigenstates in absence of the D brane are given by (2.28). The wave
functions in $(z_{+},\bar{z}_{+})$ and $(z_{-},\bar{z}_{-})$ representations
are related by
$\displaystyle\psi_{-}(z_{-},\bar{z}_{-})$ $\displaystyle=$
$\displaystyle\hat{S}{\psi_{+}}(z_{-},\bar{z}_{-})$ (4.3) $\displaystyle=$
$\displaystyle\int
dz_{+}d\bar{z}_{+}K(\bar{z}_{-},z_{+})K(z_{-},\bar{z}_{+})\psi_{+}(z_{+},\bar{z}_{+}),$
where $K(z_{-},z_{+})={1\over\sqrt{2\pi}}e^{iz_{-}z_{+}}$. Acting on energy
eigenstates, we have
$\hat{S}\psi_{+}^{E}={\cal{R}}(E)\psi_{-}^{E},~{}~{}~{}~{}{\cal{R}}(E)={\frac{\Gamma(iE+{\frac{1}{2}})}{\Gamma(-iE+{\frac{1}{2}})}}.$
(4.4)
The factor ${\cal R}(E)$ is a pure phase
$\overline{{\cal R}(E)}{\cal R}(E)={\cal R}(-E){\cal R}(E)=1,$ (4.5)
which proves the unitarity of the operator $\hat{S}$. Now in absence of the D
brane wave function (2.28) evolve according to free hamiltonian ${H_{o}}$ so
from orthonormality of the wave functions (4.2) is given by ${\cal
R}(E){\delta}({E_{+}}-{E_{-}})$. The operator $\hat{S}$ relates the incoming
and the outgoing waves and therefore can be interpreted as the fermionic
scattering matrix. The factor ${\cal R}(E)$ is identical to the the reflection
coefficient. This condition can also be expressed as the orthonormality of in
and out eigenfunctions
$\langle\Psi^{{}_{E_{{-}}}}_{-}|K|\Psi^{{}_{E_{{+}}}}_{+}\rangle=\delta(E_{+}-E_{-}),$
(4.6)
with respect to the scalar product. We usually absorb the factor${\cal R}(E)$
in phase by defining
$e^{i{\phi(E)}}={\cal R}(E)$ (4.7)
to make the wave function biorthogonal, where ${\phi(E)}$ is the phase of the
incoming and the outgoing wave function(2.28). Now consider the presence of D
brane. For a single fermionic state, (4.2) is given by
$\langle{\overline{z}_{+}}{z_{+}},t=\infty|{e^{\hat{W}(t)}}|{\overline{z}_{-}}{z_{-}},t=\infty\rangle=\langle{\overline{z}_{+}}{z_{+}},{\rm
out}|(1-{\frac{{\overline{z}_{+}}{z_{+}}+{\overline{z}_{-}}{z_{-}}-2{H_{o}}}{\mu_{B}^{2}}})|{\overline{z}_{-}}{z_{-}},{\rm
in}\rangle.$ (4.8)
Now according to (2.32) W will be expressed either in
${\overline{z}_{+}}{z_{+}}$ or ${\overline{z}_{-}}{z_{-}}$ mode
$\displaystyle\langle{z_{+}},E_{+},{\rm
out}|e^{{\hat{W}}({t_{o}})}|{{z_{-}}},{E_{-}},{\rm in}\rangle$
$\displaystyle=$
$\displaystyle\langle{{z_{+}}},E_{+}|(1+{\frac{2{\overline{z}_{-}}{z_{-}}-2{H_{o}}}{\mu_{B}^{2}}}{)_{t={t_{o}}}}|{z_{-}},{E_{-}}{\rm
in}\rangle$ (4.9) $\displaystyle=$
$\displaystyle\langle{z_{+}},E_{+}|(1+{\frac{-2{H_{o}}}{\mu_{B}^{2}}})|{z_{-}},{E_{-}}\rangle$
$\displaystyle=$
$\displaystyle\langle{z_{+}},E_{+}|(1-{\frac{2E}{\mu_{B}^{2}}})|{z_{-}},{E_{-}}\rangle$
$\displaystyle=$ $\displaystyle{\cal
R}(E_{+}){e^{log(1-{\frac{2E}{\mu_{B}^{2}}})}}\delta(E_{+}-{E_{-}}),$
where in the 2nd step we have $\langle{{\overline{z}_{+}}{z_{+}}},E_{+};{\rm
out}|{\hat{\overline{z}}_{-}}{\hat{z}_{-}}|{{\overline{z}_{-}}{z_{-}}},{E_{-}},{\rm
in}\rangle=0$ from (A.6) . Now the presence of the D brane will modify the
phase of the outgoing state over the incoming, which is given by the factor
${e^{log(1-{\frac{2E}{\mu_{B}^{2}}})}}$. So for the change of phase
$\delta\phi(E)$ we can write
$e^{-i\frac{\delta\phi(E)}{2}}={e^{log(1-{\frac{2E}{\mu_{B}^{2}}})}}$ (4.10)
Now in (4.10) using the relation (4.7) will leave us with the amplitude
$e^{-i\frac{\delta\phi(E)}{2}}$. In [43] its explained that the complex phase
in the wave function is the signature of tunneling and we can presume the
above factor accounts for the same. Also note that instead of the above
macroscopic loop operator in NS sector, if we took the macroscopic loop
operator in some other sector given by
${W^{\prime}}({\frac{{\overline{z}_{+}}{z_{+}},{\overline{z}_{-}}{z_{-}},{H_{o}}}{\mu_{B}^{2}}})$
according to the discussion in Appendix we will have the phase shift of the
outgoing state
${\frac{\phi(E)}{2}}+i{W^{\prime}(1-{\frac{2E}{\mu_{B}^{2}}})}$. Lets consider
scattering of a tachyonic state (which are being created from the action of
${({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}})}^{n}$ on fermionic ground state)
from D brane. This is better understood from the collective field theory where
scattering to a single tachyonic state with energy E is given by $\sim U(E)$
where U(E) is given by (2.48). This supports the fact that the D brane act as
a coherent source of closed strings. Now we consider the classical limit
$\beta\rightarrow\infty$. At this limit the ground state of MQM is obtained by
filling all energy levels up to some fixed Fermi energy which we choose to be
$E_{F}=-\mu$. Quasiclassically every energy level corresponds to a certain
trajectory in the phase space of
${\overline{z}_{+}}{z_{+}},{\overline{z}_{-}}{z_{-}}$ variables and they are
separated by a factor ${\frac{1}{\beta}}$. The Fermi sea can be viewed as a
stack of all classical trajectories with $E\leq E_{F}$ and the ground state is
completely characterized by the curve representing the trajectory of the
fermion with highest energy $E_{F}$. For the Hamiltonian ${H_{o}}$ all
trajectories are hyperboles
${\overline{z}_{+}}{z_{-}}+{\overline{z}_{-}}{z_{+}}=-E$ and the profile of
the Fermi sea is given by
${\overline{z}_{+}}{z_{-}}+{\overline{z}_{-}}{z_{+}}=-\mu.$ (4.11)
First consider the theory without D brane. Then the low lying collective
excitations are represented by deformations of the Fermi surface,
${\overline{z}_{+}}{z_{-}}+{\overline{z}_{-}}{z_{+}}=M({\overline{z}_{+}}{z_{-}}+{\overline{z}_{-}}{z_{+}}).$
(4.12)
In order to study the scattering with such deformed background we will follow
the analysis of [16]. The perturbed wave functions are related to the old ones
by a phase factor
${\psi^{E}_{\pm}}({\overline{z}_{\pm}}{z_{\pm}})=e^{\mp
i{\varphi_{\pm}}({\overline{z}_{\pm}}{z_{\pm}};E)}{\psi^{E}_{\pm}}({\overline{z}_{\pm}}{z_{\pm}}),$
(4.13)
whose asymptotics at large ${\overline{z}_{\pm}}{z_{\pm}}$ characterizes the
incoming/outgoing tachyon state. We split the phase into three terms
$\varphi_{\pm}({\overline{z}_{\pm}}{z_{\pm}};E)=V_{\pm}({\overline{z}_{\pm}}{z_{\pm}})+{\frac{1}{2}}\phi(E)+v_{\pm}({\overline{z}_{\pm}}{z_{\pm}};E),$
(4.14)
where the potentials $V_{\pm}$ are fixed smooth functions vanishing at
${\overline{z}_{\pm}}{z_{\pm}}=0$, while the term $v_{\pm}$ vanishing at
infinity and the constant $\phi$ are to be determined. Now in order to
understand the time-dependent profile of Fermi sea first consider the
situation in the absence of the brane as described in [16].
$\langle\Psi^{{}_{E_{-}}}_{-}|\Psi^{{}_{E_{+}}}_{+}\rangle={\cal
N}{e^{-i\phi}}\int_{0}^{\infty}{\frac{d{\overline{z}_{+}}d{\overline{z}_{-}}d{z_{-}}d{z_{+}}}{\sqrt{{\overline{z}_{+}}{z_{+}}}\sqrt{{\overline{z}_{-}}{z_{-}}}}}{e^{i({\overline{z}_{+}}{z_{-}}+{z_{+}}{\overline{z}_{-}})}}e^{-i{\varphi_{+}}({z_{+}})-i{\varphi_{-}}({z_{-}})}{({\overline{z}_{+}}{z_{+}})}^{iE_{-}}{({\overline{z}_{-}}{z_{-}})}^{iE_{+}},$
(4.15)
where ${\cal N}$ is the normalization. At $\beta\rightarrow\infty$ Fermi
profile can be obtained from saddle point approximation which is given by
${\overline{z}_{+}}{z_{-}}+{z_{+}}{\overline{z}_{-}}=-E_{\pm}+({z_{\pm}}\partial_{\pm}+{\overline{z}_{\pm}}{\overline{\partial}_{\pm}})\varphi_{\pm}({\overline{z}_{\pm}}{z_{\pm}}).$
(4.16)
So following [16] it appears the perturbed state will be an eigenstate of the
deformed hamiltonian $H={H_{o}}+{H_{p}}$ where $H_{p}$ is given by
${H_{p}}=({z_{\pm}}\partial_{\pm}+{\overline{z}_{\pm}}{\overline{\partial}_{\pm}})\varphi_{\pm}({\overline{z}_{\pm}}{z_{\pm}};H)$
(4.17)
Now in the presence of the D brane scattering matrix element will be given by
$\langle{\overline{z}_{+}}{z_{+}},E_{+},{\rm
out}|e^{\hat{W}({t_{o}})}|,{E_{-}},{\overline{z}_{-}}{z_{-}},{\rm in}\rangle$
. S–matrix element is expressed as
$\displaystyle S_{perturb}$ $\displaystyle=$ $\displaystyle{e^{-i\phi}{\cal
N}\int\limits_{0}^{\infty}{\frac{d{\overline{z}_{+}}d{\overline{z}_{-}}d{z_{-}}d{z_{+}}}{\sqrt{{\overline{z}_{+}}{z_{+}}}\sqrt{{\overline{z}_{-}}{z_{-}}}}}{e^{i({\overline{z}_{+}}{z_{-}}+{z_{+}}{\overline{z}_{-}})}}e^{-i{\varphi}{({\overline{z}_{-}}{z_{-}})}}{({\overline{z}_{-}}{z_{-}})}^{iE_{-}}}$
$\displaystyle[e^{-i{\int_{t_{o}}^{\infty}}{\hat{H}_{o}}}[e^{-i{W_{\rm
proj}}({t_{o}})}]e^{i{\int_{-\infty}^{t_{o}}}{\hat{H_{o}}}}]e^{-i{\varphi_{+}}({\overline{z}_{+}}{z_{+}})}{({\overline{z}_{+}}{z_{+}})}^{iE_{+}}$
$\displaystyle\sim$ $\displaystyle e^{-i\phi}{\cal
N}\int\limits_{0}^{\infty}{\frac{d{\overline{z}_{+}}d{\overline{z}_{-}}d{z_{-}}d{z_{+}}}{{\sqrt{{\overline{z}_{+}}{z_{+}}}}\sqrt{{\overline{z}_{-}}{z_{-}}}}}{e^{i({\overline{z}_{+}}{z_{-}}(t)+{z_{+}}{\overline{z}_{-}}(t))}}$
$\displaystyle
e^{-i{\varphi}({\overline{z}_{-}}{z_{-}}(t))}{({\overline{z}_{-}}{z_{-}}(t))}^{iE_{-}}[e^{iW(t)({\overline{z}_{\pm}}{z_{\pm}},{H_{o}})}]e^{-i{\varphi_{+}}({\overline{z}_{+}}{z_{+}}(t))}{({\overline{z}_{+}}{z_{+}}(t))}^{iE_{+}}$
So in the presence of the D brane, in the classical regime from (LABEL:pertu)
we can write the Fermi profile in the presence of D brane as
${\overline{z}_{+}}{z_{-}}+{z_{+}}{\overline{z}_{-}}=-E_{\pm}+({z_{\pm}}\partial_{\pm}+{\overline{z}_{\pm}}{\overline{\partial}_{\pm}})\varphi_{\pm}({\overline{z}_{\pm}}{z_{\pm}})+({z_{\pm}}\partial_{\pm}+{\overline{z}_{\pm}}{\overline{\partial}_{\pm}})W({\overline{z}_{\pm}}{z_{\pm}},E)$
(4.19)
The perturbed hamiltonian for the deformed state is given by
${H_{p}^{\prime}}={H_{p}}+({z_{\pm}}\partial_{\pm}+{\overline{z}_{\pm}}{\overline{\partial}_{\pm}})W({\overline{z}_{\pm}}{z_{\pm}};H)$
(4.20)
So essentially the Dbrane act as a source at $t_{o}$. So as we see in the
presence of D brane Fermi profile develops instability.
## 5 Perturbation by momentum modes
### 5.1 Collective field theory analysis
In this subsection we will consider type 0A matrix model with the time t
compactified on a circle of radius R, perturbed by momentum modes
${V_{\frac{n}{R}}}$ in the presence of D brane. We know that the presence of D
brane change the tachyonic background [18] so that we need to go through
collective field theory analysis to understand how the MQM wave function in a
perturbed background with a D brane is related to the one without a brane.
Here first we briefly review the scenario without D brane and then study what
happens when we consider the theory in the presence of D brane. The tachyon
modes of the closed string theory are the asymptotic states of collective
field theory [25]. The discrete tachyonic operator ${\cal T}_{n}\sim\int_{\rm
worldsheet}e^{\pm inx/R}e^{(|n|/R-2)\phi}$ corresponds to the following
operator in matrix model[28, 26],
${V_{{\pm}n/R}}=e^{-{\frac{n}{R}}t}{{({\overline{z}_{\pm}}{z_{\pm}})}^{n/R}}.$
(5.1)
These operators creates a discrete tachyonic state of momenta ${\frac{n}{R}}$
over the matrix model ground state and are periodic in Euclidean time.
$[{H_{o}},{V_{{\pm}n/R}}]=\mp i{\frac{n}{R}}{V_{{\pm}n/R}}\quad;\quad k\geq
1.$ (5.2)
So ${V_{{\pm}n/R}}$ shift the energy $E\rightarrow E\mp i{\frac{n}{R}}$ cause
a time-dependent perturbation to Fermi sea. The perturbed state in general can
be expressed as
$\Psi_{\pm}^{E}=e^{\mp
i\varphi(z_{\pm}\bar{z}_{\pm};E)}\psi_{o\pm}^{E}\equiv{\cal
W}_{\pm}\psi_{o\pm}^{E},$ (5.3)
where the phases $\varphi_{\pm}$ have Laurent expansion
${\varphi_{\pm}}({z_{\pm}}\overline{z}_{\pm};E)={\frac{1}{2}}\phi(E)+R\sum_{k\geq
1}t_{\pm k}{(z_{\pm}\overline{z}_{\pm})}^{k/R}-R\sum_{k\geq
1}{\frac{1}{k}}v_{\pm k}({z_{\pm}}\overline{z}_{\pm})^{-k/R}..$ (5.4)
$t_{\pm k}$ parametrize the asymptotic perturbation by momentum modes of NS-NS
scalars, corresponding to the operator introduced (5.1), Note the above wave
function asymptotically behave as
$\Psi^{E}_{\pm}({\overline{z}_{\pm}}{z_{\pm}})\sim{({\overline{z}_{\pm}}{z_{\pm}})}^{\pm
iE-{\litfont{1\over 2}}}\ e^{\mp{\litfont{1\over 2}}i\phi(E)}\
e^{iU_{\pm}({\overline{z}_{\pm}}{z_{\pm}})}\quad;\quad
U_{\pm}({\overline{z}_{\pm}}{z_{\pm}})=\sum_{k\geq
1}{|{\overline{z}_{\pm}}{z_{\pm}}|}^{\frac{k}{R}}.$ (5.5)
From the above its evident that tachyonic perturbation can be achieved by
deforming the integration measures $d[{\overline{z}_{\pm}}{z_{\pm}}]$ to [14]
$[d{\overline{z}_{\pm}}{z_{\pm}}]\rightarrow[d{\overline{z}_{\pm}}{z_{\pm}}]{\rm
exp}\left(\pm i{U_{\pm}}({\overline{z}_{\pm}}{z_{\pm}})\right).$ (5.6)
Extending the discussions of section 3, these wave functions (5.3)
diagonalizes the deformed kernel (5.6). While the perturbed wave function
evolves in time with ${H_{o}}$, but it can be seen as the stationary state
w.r.t an effective hamiltonian $H={H_{o}}+{H_{p}}(H)$, where the expressions
for perturbed hamiltonian ${H_{p}}$ from semiclassical analysis is obtained in
[16] as we reviewed in section 4. The partition function is given by ${\rm
Tr}e^{-2{\pi}R\beta H}$, following section 3 which can also be expressed as
Fredholm determinant. We have the free energy${\cal F}=-i\sum_{r\geq
1/2}\phi\left(ir/R-\mu\right)$ where $\phi(E)$ is the phase described by
(5.4). It satisfies the equation
$\phi(-\mu)=2{\rm sin}\left({\frac{\partial_{\mu}}{2\beta R}}\right){\cal
F}(\mu,R)..$ (5.7)
Having given a brief review of type 0A MQM perturbed by the momentum modes
lets consider the theory with D brane. Note that type 0A MQM without any net
D0 brane background charge, in the double scaling limit can be viewed as a
pair of noninteracting fermions moving in inverted harmonic oscillator
potential in x and y direction respectively (2.8). So from the analysis of
[25] its apparent that the respective collective field theory will be the
generalization of the one for $c=1$ case to two (target space ) dimension(x,y)
and the collective field expressed as $\phi(x,y,t)=\phi(z,\overline{z},t)$,
give the eigenvalue density in two dimension with appropriate normalization.
Collective field hamiltonian will be the sum of the hamiltonian for
$\phi(x,t)$ and $\phi(y,t)$. The fluctuation of the collective field over the
static value $\phi={\phi_{o}}+{\partial_{\tau}}\eta({\tau},t)$,where
$\pi\phi_{o}={p_{o}}=\sqrt{\mu_{F}-x^{2}-y^{2}}$ with
$p_{o}^{2}=p_{ox}^{2}+p_{oy}^{2}\,;\,p_{ox}={\frac{dx}{d\tau}}\,;\,p_{oy}={\frac{dy}{d\tau}}$
and ${\partial_{\tau}}\eta({\tau},t)={\psi^{\dagger}}\psi$ where $\psi$
corresponds to the 2nd quantized fermionic field. Now the presence of D brane
implies, inclusion of a macroscopic loop operator W (Laplace transformed,
localized in time) to the collective field theory action which essentially
creates a coherent state over MQM ground state. In linearized approximation we
have $W(z,\overline{z},t_{o})$ $\sim\int dt\int
d\tau{e^{-l\overline{z}z}}{\partial_{\tau}}{\eta}(\tau,t)\delta(t-t_{o})$. So
the presence of the D brane implies a field independent source term in the
collective field equation of motion (which can be viewed as the back-reaction
due to the D-brane). Consequently we obtain the solution for collective field
at $t\geq{t_{o}}$ as $\phi={\phi_{\rm free}}+{\phi_{\pm{\rm perturbed}}}$
where ${\phi_{\rm perturbed}}=\int d\tau j(t_{o},\tau)G({t_{o}},\tau)$ with
$j(t_{o})$ is the current associated with the macroscopic loop operator and G
is the Green’s function. As ${\phi_{\rm perturbed}}$ is independent of $\phi$
so we see that the effect of macroscopic loop operator in the action is to
change the momentum associated with $\eta$ by the external current or in other
words the classical solution for $\eta$ will get a field independent(but
profile ($x(t_{o}),y(t_{o})$) dependent) shift. The stationary field
$\phi_{o}$ will also be shifted due to the change of the potential.
Essentially from string theory side we can just identify the current j
(transformed from $\tau$ to $\phi$ space,and Fourier transformed to momentum
space) with the overlap amplitude $\langle V(p)|B\rangle$ and the interaction
term introduced in the collective field action $\int
dtd\tau\,j\,{\partial_{\tau}}\eta\delta(t-{t_{o}})$ as $\int\phi_{\rm
cl}(p)\langle V|B\rangle$ where $\phi_{\rm cl}$ can be viewed as closed string
field. On this identification we see that quadratic action for $\eta$ [25, 28]
along with the source term resembles the closed string field action in the
presence of D brane $S=S_{\rm closed}(\phi_{\rm cl})+\phi_{\rm
cl}(X,\phi)\langle V(X,\phi)|B\rangle$ where $\phi_{\rm cl}$ is the closed
string field. Change in momentum of $\eta$ due to interaction with the
localized source has an explanation in the fact that closed string momentum is
not conserved in the direction of spacelike boundary condition of D brane. Now
the meaning of the constraint (2.25) is that we have to ensure the fact that
while the collective field is in interaction with the localized source $L_{\rm
int}=\int d\tau d{t}\,j\,(\partial_{\tau}\eta)(t.\tau)\,\delta(t-{t_{o}})$,
the hamiltonian for $\eta$ will always remain conserved which implies time
translation invariance of complete action with no leakage from the bulk by
making $\delta_{t}L_{\rm int}|_{\rm t={t_{o}}}\,=0$ . So the quantized action
for $\eta$ always gives the propagator have a pole corresponding to 2D
massless scalar [25] i.e resembles the tachyon. So from string theoretical
point of view we see that the constraint act as a no leakage condition ensures
bulk conformal invariance as we mentioned in section 2.
. Now lets find out the exact form of MQM wave function in the presence of D
brane in the background perturbed by momentum modes from Collective field
theory(which in the absence of Dbrane is given by (5.3) ). Lets proceed in the
following way. The collective field equation of motion implies the classical
solution for left and right moving field $\alpha_{x\pm}\,,\,\alpha_{y\pm}\,$
(given by
$\phi(x)+{\alpha^{\prime}}\partial_{x}\Pi(x)\,\,,\,\,\phi(y)+{\alpha^{\prime}}\partial_{y}\Pi(y)$
where $\Pi$ is the collective field momentum), correspond to fermionic
momentum density at the edge of the Fermi sea [26, 28]. From the expression of
the hamiltonian (2.8), Fermi surface is described by
${\frac{1}{2}}({p}_{x}^{2}+{p}_{y}^{2})-{\frac{1}{4\alpha^{\prime}}}(\hat{x}^{2}+\hat{y}^{2})=-\mu.$
(5.8)
The collective field equation corresponds to two separate equation for two
fermions described by x,y as no interaction exists among them. So the momentum
density ${p_{\pm x}},{p_{\pm y}}$ at the edge of the Fermi surface evolves
with time t as [26]
$\partial_{t}{p_{\pm x}}+{p_{\pm x}}{\partial_{x}}{p_{\pm
x}}-x=0\quad;\quad\partial_{t}{p_{\pm y}}+{p_{\pm y}}{\partial_{y}}{p_{\pm
y}}-y=0.$ (5.9)
If we consider the fluctuation of collective field around its classical
solution $p_{o}\,(p_{ox},p_{oy})$.
$\alpha_{\pm}\rightarrow{p_{o}}+{\epsilon_{\pm}}$ defining
${\epsilon_{\pm}}={\frac{1}{p_{o}}}{\xi_{\pm}}$, ${\xi_{\pm}}$ is shown to
correspond the right and left moving tachyonic fluctuations [28]. Now at
classical limit the macroscopic loop operator contributes a source term to the
above
$\partial_{t}{p_{\pm x}}+{p_{\pm x}}{\partial_{x}}{p_{\pm
x}}-x=\left[\,{\frac{\partial W(x,y)}{\partial
x}}{|_{y}}\,\right]{\delta(t-{t_{o}})}\quad;\quad\partial_{t}{p_{\pm
y}}+{p_{\pm y}}{\partial_{y}}{p_{\pm y}}-y=\left[\,{\frac{\partial
W(x,y)}{\partial y}}{|_{x}}\,\right]{\delta(t-{t_{o}})}.$ (5.10)
This essentially gives a discontinuity
$p_{\pm{x_{i}}}{|_{{t_{o}}+\epsilon}}-p_{\pm{x_{i}}}{|_{{t_{o}}-\epsilon}}={\frac{\partial
W(x,y)}{\partial{x_{i}}}}{|_{t_{o}}}\quad,$ (5.11)
where in (5.11) $x_{i}$ stands for x,y for $i=1,2$. So the effect of D brane
is to change the Fermi profile above $t\geq{t_{o}}$ to
${p_{o}}\rightarrow{p_{o}^{\prime}}={p_{o}}+{\frac{\partial
W(x,y)}{\partial{x_{i}}}}({t_{o}})$. As above $t_{o}$, $p_{\pm}$ evolves
according to free hamiltonian ${H_{o}}$ so we can express the fluctuation of
the collective field for $t\geq{t_{o}}$ as
${\epsilon^{\prime}_{\pm}}={\frac{1}{p_{o}^{\prime}}}{\xi^{\prime}_{\pm}}$ to
see ${\xi^{\prime}_{\pm}}$ corresponds to tachyonic fluctuation mode. However
the redefinition above ${t_{o}}$ implies a nonlinear shift of collective field
momenta $p_{o}^{\prime}=p_{o}^{\prime}(p_{o},x,y)$ and the fluctuation
$\xi^{\prime}=\xi^{\prime}(\xi,p_{o},x,y)$. ${\epsilon^{\prime}_{\pm x}}$
,${\epsilon^{\prime}_{\pm y}}$ combined to give complex tachyonic field. So
from the viewpoint of collective field scenario lets find out the picture in
MQM. The shift of fluctuation mode above $\xi\rightarrow\xi^{\prime}$ implies
a shift in the perturbing phase (5.4) .
${\psi^{\prime}_{p\pm>}}^{E}=e^{\mp
i\varphi_{w}(z_{\pm}\overline{z}_{\pm};E)}\psi_{o\pm}^{E}={{\cal{W}}^{\prime}}\psi_{o\pm}^{E},$
(5.12)
where
$\displaystyle{\varphi_{w\pm}}({z_{\pm}}\overline{z}_{\pm};E)$
$\displaystyle=$ $\displaystyle{\frac{1}{2}}\phi(E)+R\sum_{k\geq 1}t_{\pm
k}(t_{m\pm},v_{n\pm},{\frac{r}{R}},E){f_{tk}}(E,{\overline{z}_{\pm}}{z_{\pm}})\,{\left[z_{\pm}\overline{z}_{\pm}\right]}^{k/R}$
(5.13) $\displaystyle-$ $\displaystyle R\sum_{k\geq 1}{\frac{1}{k}}v_{\pm
k}(t_{m\pm},v_{n\pm},{\frac{r}{R}},E){f_{vk}}(E,{\overline{z}_{\pm}}{z_{\pm}})\,\left[{z_{\pm}}\overline{z}_{\pm}\right]^{-k/R},$
${f_{vk}}$ and ${f_{tk}}$is the extra factor arises due to the nonlinear shift
in $\xi$ from ${\overline{z}_{\pm}}{z_{\pm}}$ factor which arises due to the
action of W. These factors give nonperiodic shift to momentum by integer
numbers i.e ${\frac{k}{R}}\rightarrow{\frac{k}{R}}+n$ with $t_{\pm k}=t_{\pm
k}(t_{m\pm},v_{n\pm},E)\,,\,v_{\pm k}=v_{\pm k}(t_{m\pm},v_{n\pm},E)$. Note
that once we try to interpret the consequence of (5.11) in MQM we need to
replace $W\rightarrow W_{\rm proj}$ (2.41). The fact that in absence of
$\hat{W}$ we get back our original wave function so (5.3) implies that the
shifted dressing operator ${\cal W}^{\prime}$ is of the form ${\cal
W}^{\prime}=F(\hat{W}){\cal W}$ with $F\rightarrow 1$ in absence of $\hat{W}$.
Exploiting the fact that (LABEL:scrh2) is just the quantum version of (5.11)
and the indication of the semiclassical analysis (4.19) implies ${\cal
W}^{\prime}$ can be expressed as :
$\Psi^{\prime E}_{p\pm>}=e^{\mp
i\varphi_{w}({\overline{z}_{\pm}}{z_{\pm}};E)}\psi_{o}^{E}=(1\mp\hat{W}_{\rm
proj})e^{\mp
i\varphi({\overline{z}_{\pm}}{z_{\pm}};E)}\psi_{o\pm}^{E}=(1\mp\hat{W}_{\rm
proj}){\cal{W}}\psi_{o\pm}^{E},$ (5.14)
This is because at double scaling limit $\beta\rightarrow\infty$ the wave
function is effectively given by $e^{-\hat{W}}{\cal{W}}\psi_{o\pm}^{E}$(as we
have seen in (3.34) ). This is the solution from (2.40) if the initial free
fermionic wave function (2.28) is replaced by the dressed one (5.3). Finally
as before we will express $W_{\rm proj}$ in terms of
${\hat{\overline{z}}_{-}}{\hat{z}_{-}}\,({\hat{\overline{z}}_{+}}{\hat{z}_{+}})$
in ${\overline{z}_{+}}{z_{+}}\,({\overline{z}_{-}}{z_{-}})$ basis as this will
express (5.13) in the following form
$\displaystyle{\varphi_{wp\pm}}({z_{\pm}}\overline{z}_{\pm};E)$
$\displaystyle=$ $\displaystyle{\frac{1}{2}}\phi(E)+R\sum_{k\geq 1}t_{\pm
k}(t_{m\pm},v_{n\pm},{\frac{r}{R}},E){f_{tk}}(E,{a_{r}}{({\overline{z}_{\pm}}{z_{\pm}})}^{-r})\,{\left[z_{\pm}\overline{z}_{\pm}\right]}^{k/R}$
(5.15) $\displaystyle-$ $\displaystyle R\sum_{k\geq 1}{\frac{1}{k}}v_{\pm
k}(t_{m\pm},v_{n\pm},{\frac{r}{R}},E){f_{vk}}(E,{b_{r}}{({\overline{z}_{\pm}}{z_{\pm}})}^{-r})\,\left[{z_{\pm}}\overline{z}_{\pm}\right]^{-k/R},`$
where r is a positive integer. In next subsection we will see that this
expression of ${\varphi^{\prime}}$ lead to convergence of the partition
function. The tachyonic perturbation can be introduced in the path integral by
deforming the kernel as in (5.6) and consequently the string partition
function can be expressed as the Fredholm determinant as in (3.27). We can
evaluate the Fredholm determinant with a set of diagonalizing wave function
which is given by (5.12). In the next part of this section we will evaluate
the gran canonical partition function in the hamiltonian formalism. We will
show that the tachyonic deformation in the presence of the D brane is
generated by a system of commuting flows $H_{n}$ associated with the coupling
constants $t_{\pm n}$. The associated integrable structure of the partition
function is that of a constrained Toda Lattice hierarchy. Now in order to see
the Toda structure of the partition function we need to review Lax formalism.
### 5.2 Lax Formalism
Here we will briefly review the Lax formalism in the context of Type 0A matrix
model. Consider the operator $(\hat{z}_{\pm}\hat{\bar{z}}_{\pm})$ which can be
represented as shift operators $\hat{\omega}^{\pm 1}$, where $\hat{\omega}$
acts on energy eigenstates as $\hat{\omega}^{\pm
1}\psi_{\pm}^{E}=\psi_{\pm}^{E\mp i}$. We have
$\hat{\omega}=e^{-i\partial_{E}}$ shifts the variable E by $i$. The operators
$\hat{\omega}$ and $\hat{E}$ satisfy the Heisenberg-Weyl commutation relation
$[\hat{\omega},-\hat{E}]=i\hat{\omega},\qquad[\hat{\omega}^{-1},-\hat{E}]=-i\hat{\omega}^{-1}.$
(5.16)
Now let us consider the representation of these commutation relations in the
perturbed theory. The dressing operators ${\cal W}_{\pm}$ (5.5)are now
exponents of series in $\hat{\omega}$ with $\hat{E}$-dependent coefficient
$\hat{\cal W}_{\pm}=e^{iR\sum_{n\geq 1}t_{\pm n}\hat{\omega}^{n/R}}\ e^{\mp
i\phi(E)}e^{iR\sum_{n\geq 1}v_{\pm n}({E})\ \hat{\omega}^{-n/R}}.$ (5.17)
The operators
$\displaystyle L_{+}$ $\displaystyle=$ $\displaystyle{\cal
W}_{+}\hat{\omega}{\cal W}^{-1}_{+},\quad L_{-}={\cal
W}_{-}\hat{\omega}^{-1},{\cal W}^{-1}_{-},$ $\displaystyle M_{+}$
$\displaystyle=$ $\displaystyle-{{\cal W}_{+}}\hat{E}{{\cal
W}^{-1}_{+}}\quad{M_{-}}=-{{\cal W}_{-}}{\hat{E}},{\cal W}^{-1}_{-}.$ (5.18)
known as Lax and Orlov-Schulman operators satisfy the same commutation
relations as the operators $\hat{\omega}$ and $\hat{E}$
$[L_{+},M_{+}]=iL_{+}\quad,\quad[L_{-},M_{-}]=-iL_{-}.$ (5.19)
The Lax operators $L_{\pm}$ represent the canonical coordinates
${\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}}$ in the basis of perturbed wave
functions
$\langle E|e^{\pm i{\frac{\phi_{0}}{2}}}\hat{\cal
W}_{\pm}L_{\pm}|{\overline{z}_{\pm}}{z_{\pm}}\rangle=\langle E|e^{\pm
i{\frac{\phi_{0}}{2}}}\hat{{\cal
W}}_{\pm}{\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}}|{\overline{z}_{\pm}}{z_{\pm}}\rangle,$
(5.20)
while the Orlov-Shulman operators $M_{\pm}$ represent hamiltonian
$H_{0}=-{\frac{1}{2}}({\hat{\overline{z}}_{+}}{\hat{z}_{-}}+{\hat{\overline{z}}_{-}}\hat{\hat{z}_{+}})$.
Therefore the L and M operators are related also by
$M_{+}=M_{-},~{}~{}~{}[L_{+},L_{-}]=2iM_{\pm},~{}~{}~{}\\{L_{+},L_{-}\\}=2M_{\pm}^{2}-{1\over
2}.$ (5.21)
The last identity is not satisfied automatically in the Toda system and
represent an additional constraint analogous to the string equations. The
operators $M_{\pm}$ can be expanded as infinite series of the $L$-operators.
Indeed, as they act to the dressed wave functions as
$\displaystyle\langle E|$ $\displaystyle e^{\pm i\phi_{0}}\hat{\cal
W}_{\pm}\;{M_{\pm}}|{\overline{z}_{\pm}}{z_{\pm}}\rangle=\pm
i({z_{\pm}}\partial_{{z_{\pm}}}+{\overline{z}_{\pm}}\partial_{{\overline{z}_{\pm}}}+1)\Psi^{E}_{\pm}({\overline{z}_{\pm}}{z_{\pm}})$
(5.22) $\displaystyle=$ $\displaystyle\left(\sum_{k\geq 1}kt_{\pm
k}{({\overline{z}_{\pm}}{z_{\pm}})}^{k/R}+\mu+\sum_{k\geq 1}v_{\pm
k}{{\overline{z}_{\pm}}{z_{\pm}}}^{-k/R}\right)\Psi^{E}_{\pm}(({\overline{z}_{\pm}}{z_{\pm}})).$
we can write
$M_{\pm}=\sum_{k\geq 1}kt_{\pm k}L_{\pm}^{k/R}+\mu+\sum_{k\geq 1}v_{\pm
k}L_{\pm}^{-k/R}.$ (5.23)
In order to exploit the Lax equations and the string equations we need the
explicit form of the two operators. It follows from that $L_{\pm}$ can be
represented as series of the form
$\displaystyle{L_{+}}$ $\displaystyle=$ $\displaystyle
e^{-i\phi/2}\left(\omega+\sum_{k\geq 1}a_{k}\omega^{1-n/R}\right)e^{i\phi/2},$
$\displaystyle{L_{-}}$ $\displaystyle=$ $\displaystyle
e^{i\phi/2}\left(\omega^{-1}+\sum_{k\geq
1}a_{-k}\omega^{-1+n/R}\right)e^{-i\phi/2}.$ (5.24)
Recall that the dressing operators ${\cal W}_{\pm}$ in terms of $\hat{E}$ and
$\hat{\omega}$ are of the for
${\cal{W}}_{\pm}=e^{\mp i\phi/2}\left(1+\sum_{k\geq 1}w_{\pm
k}\hat{\omega}^{\mp k/R}\right)e^{\mp iR\sum_{k\geq 1}t_{\pm
k}\hat{\omega}^{\pm k/R}}$ (5.25)
Studying the evolution laws of the Orlov–Shulman operators, one can find that
[20]
${\partial v_{k}\over\partial t_{l}}={\partial v_{l}\over\partial t_{k}}.$
(5.26)
It means that there exists a generating function $\tau_{s}[t]$ of all
coefficients $v_{\pm k}$
$v_{k}(s)={({\frac{1}{\beta}})^{2}}\,{\partial\log\tau_{s}[t]\over\partial
t_{k}}.$ (5.27)
It is called $\tau$-function of Toda hierarchy. It also allows to reproduce
the zero mode $\phi$ and, consequently, the first coefficient in the expansion
of the Lax operators
$e^{\beta\phi(s)}={\tau_{s}\over\tau_{s+{\frac{1}{\beta}}}},\qquad
r^{2}(s-{\frac{1}{\beta}})={\tau_{s+{\frac{1}{\beta}}}\tau_{s-{\frac{1}{\beta}}}\over\tau_{s}^{2}}.$
(5.28)
We are going to show that the partition function coincides with
$\tau$–function (5.27). Finally note as the partition function is described in
terms of the Fermi level ${\mu}$. So in the description of Lax formalism we
will replace E by ${\mu}$. Now let us discuss about the integrable flow. Let
us identify the integrable flows associated with the coupling constants
$t_{n}$.
$\partial_{t_{n}}L_{\pm}=[H_{n},L_{\pm}],$ (5.29)
where from (5.18), the operators $H_{n}$ are related to the dressing operators
as
$H_{n}=({\partial_{t_{n}}}{\cal W}_{+}){\cal
W}_{+}^{-1}=({\partial_{t_{n}}}{\cal W}_{-}){{\cal W}_{-}^{-1}}.$ (5.30)
it is clear that $H_{n}=W_{+}\hat{\omega}^{n/R}W_{+}^{-1}+$ negative powers of
$\hat{\omega}^{1/R}$, which implies expression of ${H_{n}}$ can be given by
[14]
$H_{\pm
n}=(L_{\pm}^{n/R})_{{}^{>}_{<}}+{\frac{1}{2}}(L_{\pm}^{n/R})_{0},\qquad n>0,$
(5.31) $\partial_{t_{m}}H_{n}-\partial_{t_{n}}H_{m}-[H_{m},H_{n}]=0.$ (5.32)
Equations (5.32,5.30,5.31) imply that the perturbed theory possesses the Toda
lattice integrable structure. The Toda structure implies an infinite hierarchy
of PDE’s for the coefficients $v_{n}$ of the dressing operators, the first of
which is the Toda equation for the phase $\phi(\mu)\equiv\phi(E=-\mu)$
$i{\frac{\partial}{\partial t_{1}}}{\frac{\partial}{\partial
t_{-1}}}\phi(\mu)=e^{i\phi(\mu)-i\phi(\mu-i/R)}-e^{i\phi(\mu+i/R)-i\phi(\mu)}.$
(5.33)
### 5.3 String theory on a circle with the D brane in the presence of
tachyonic background
In this section we are going to evaluate the free energy of type 0A MQM in the
presence of D brane and with tachyonic background, in the grand canonical
ensemble and try to understand the relevant string theory. Recall in section 3
we have seen that in the absence of the momentum modes, within the time circle
$0\leq t\leq 2\pi R$ the solution of the Schrodinger equation corresponds to
the free fermionic wave function. This in the string theory side giving a
picture that we have free closed string states along the circle and the
coherent states are strongly localized at $t={t_{o}}\equiv i{X^{o}}$ . So with
the same view in the presence of tachyonic background, within the circle
$0\leq t\leq 2{\pi}R$ the wave function must be given by (5.3). The perturbed
wave function, while time dependent w.r.t the free hamiltonian ${H_{o}}$ it is
stationary w.r.t an effective hamiltonian H, similar as discussed in section
5.1. So lets consider the perturbed MQM with the effective hamiltonian
$H={H_{o}}+{H_{p}}(H)$ in the presence of D brane. First consider the
partition function (3.7) where now we replace the integration kernels with the
deformed measures (5.6). So as a generalization of (3.7), in the perturbed
background, the Matrix model partition function in the presence of D brane
(3.23, 3.24,3.25 ) will be with the deformed kernel as
$\displaystyle{\cal{Z}}_{N}(t)$ $\displaystyle=$
$\displaystyle\int\limits_{-\infty}^{\infty}\prod_{k=1}^{N}[d{z_{+}}_{k}][d{z_{-}}_{k}][d{\overline{z}_{+}}_{k}][d{\overline{z}_{-}}_{k}][d{t_{k}}]{e^{i{t_{n\pm}}{({\overline{z}_{\pm}}{z_{\pm}})}^{{\frac{n}{R}}}}}{\rm
det}_{jk}\left(e^{i{t_{jk}}{\overline{z}_{+}}_{j}{z_{-}}_{k}}\right)$ (5.34)
$\displaystyle{\rm
det}_{jk}\left(e^{-iq{z_{+}}_{j}{\overline{z}_{-}}_{k}}\right){\rm
det}_{jk}\left(e^{i{t^{-1}_{jk}}{\overline{z}_{+}}_{j}{z_{-}}_{k}}\right)\
{\rm det}_{jk}\left(e^{-iq{z_{+}}_{j}{\overline{z}_{-}}_{k}}\right)$
$\displaystyle{\rm exp}[\displaystyle\sum_{i}{\rm
log}(1+{\frac{{\overline{z}_{+}}_{i}{z_{+}}_{i}+{\overline{z}_{-}}_{i}{z_{-}}_{i}+{\overline{z}_{+}}_{i}{z_{-}}_{i}+{\overline{z}_{-}}_{i}{z_{+}}_{i}}{\mu_{B}^{2}}})].$
The partition function will be given by Fredholm determinant ${\rm
Det}(1+e^{-\beta\mu}WK))$ (3.27) where in order to evaluate the determinant we
need to choose the basis which diagonalizes K (i.e (3.13) with deformed
measure as given in (5.34) ) and we evaluate the expectation value of
${\hat{W}}$ in the same. In order to evaluate free energy in the presence of D
brane we will proceed in the following way. First consider the scenario
without D brane. Recall the expression of free energy which is expressed in
terms of the phase of wave function [14, 15](which is of the same form of
(3.42), expressed in the absence of brane). In a perturbed background the
phase $\phi(E)$ will be replaced by that of the perturbed wave function (5.3)
in the expression of free energy[14]. So for the effective hamiltonian H (
$H={H_{o}}+{H_{p}}(H)$ where $H_{p}$in the semiclassical limit obtained in
(4.17) ) of which (5.3) is an eigenfunction, the analysis of section 3.3
implies that free energy of the perturbed system in grand canonical ensemble
is given by ${\cal F}={\rm log}{\cal{Z}}$ with
${\cal{Z}}=\displaystyle\sum_{N=0}^{\infty}e^{-2{\pi}R\beta\mu N}\\{{\rm
Tr}e^{-2{\pi}R\beta H}{\\}_{N}}={\rm Det}\,(1+e^{-2{\pi}R\beta(\mu+H)})$. This
is supported from the view of [16] where in the semiclassical regime the
explicit expression of ${\cal{Z}}$ is obtained in this form. The Fredholm
determinant (3.13) in a perturbed background is given by ${\cal{Z}}$ [14]. In
the presence of D brane we have the Fredholm determinant (3.27) which in
perturbed background is expressed in (5.34). So as in section 3.3 free energy
must be obtained from the thermal partition function in the presence of D
brane i.e by insertion of the operator $e^{W(t_{o})}$ in the partition
function and evaluating the expectation value. So here in hamiltonian
formalism we will evaluate the grand canonical partition function ${\rm
Det}(1+e^{-\beta(H+\mu)})$ in the basis (5.3) with the insertion of the
operator and it must be same as the Fredholm determinant (5.34). Here we are
going to show that if we consider the prtojected theory as described in
section 2, the above mentioned grand canonical partition function have the
integrable structure of tau function of Toda hierarchy. Now the partition
function with the momentum modes in the presence of the D brane is given by
the transition amplitude from the initial state ${\cal{W}}{\psi_{o}}$ to the
final state${\cal{W}}^{\prime}\psi_{o>}$, where $\psi_{o>}$ is given in (2.41)
and they represent the fermionic wave function before and after being
scattered from the D brane and the corresponding dressing operator is
${\cal{W}}^{\prime}$ .Now note that in a compact dimension just before being
scattered, the wave function at $t=2{\pi}R-\epsilon$ must be given by the one
at $t=\epsilon$ with a time evolution $2{\pi}R$. This leads to the identity
${\cal{W}}^{\prime}\psi_{o>}(t_{o})={\cal{W}}^{\prime}(1-W({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}},{H_{o}})){\psi_{o}}(t_{o})=\left(1-W({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}},{H_{o}})\right){\cal{W}}{\psi_{o}}(t_{o}),$
(5.35)
where ${t_{o}}=0\equiv 2{\pi}R$. The above relation can also be viewed from
(2.40) in the presence of tachyonic background, if we replace the initial
fermionic wave function (2.28) by the dressed one (5.3). Hence the partition
function on the circle corresponds to the transition amplitude
$\displaystyle{\cal{Z}}$ $\displaystyle=$ $\displaystyle\lim_{\epsilon\to
0}\,\langle{{\cal{W}}\psi_{o}}{(\epsilon)}|{\cal{W}}^{\prime}\psi_{o>}(2{\pi}R-\epsilon)\rangle$
(5.36) $\displaystyle=$ $\displaystyle\lim_{\epsilon\to
0}\,\langle{{\cal{W}}\psi_{o}}{(\epsilon)}|(1-W({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}},{H_{o}}))|{\cal{W}}\psi_{o}(2{\pi}R-\epsilon)\rangle$
$\displaystyle=$ $\displaystyle{\rm
Tr}_{{\cal{W}}\psi_{o}}\\{e^{{-\beta}[{\int_{\epsilon}^{2{\pi}R-{\epsilon}}}dtH+{\int_{-\epsilon}^{\epsilon}}dtH]+{\int_{-\epsilon}^{\epsilon}}dtW{\delta}(t)\\}}\\}$
$\displaystyle=$ $\displaystyle{\rm
Tr}_{{\cal{W}}\psi_{o}}\\{e^{{-\beta}[{\int_{\epsilon}^{2{\pi}R-{\epsilon}}}dtH]}e^{W(t=0)}\\}$
$\displaystyle=$ $\displaystyle{\rm
Tr}_{{\cal{W}}\psi_{o}}\\{e^{{-2{\pi}R\beta}H}e^{W(t=0)}\\}.$
Where the partition function is evaluated in Euclidean time and ${\rm
Tr}_{{\cal{W}}\psi_{o}}$ denotes the trace taken w.r.t (5.3)171717In order to
reach from the 2nd to 3rd step in (5.36)we utilize the fact that we can scale
the time $t\rightarrow\beta t$ so that the term with the macroscopic loop
operator $\int dtW(t)\delta(t-{t_{o}})$ will get a factor ${\frac{1}{\beta}}$
so that in the double scaling limit where $\beta\rightarrow\infty$ and with
Euclidean time, we can lift up the term to the exponential and the exponent
gives an exact expression what we have obtained from the path integral (2.36).
Grand canonical Partition function will be given by the following expression
where we will have the contribution from singlet states only
${\displaystyle\prod_{E}}\\{1+e^{{-2\beta}{\pi
R}(\mu+E)}\langle{\psi_{p}^{E}}|e^{\hat{W}(t=0)}|{\psi_{p}^{E}}\rangle\\}\\\
={\displaystyle\prod_{E}}\\{1+e^{{-2\beta}{\pi
R}(\mu+E)}\langle{\psi_{p}^{E}}|e^{\hat{W}}|{\psi_{p}^{E}}\rangle\\},$
Note, we could write the above expression for grand canonical partition
function only because $\hat{W}$ can be expressed as the direct product of the
operators for the single fermionic states. Now following (2.15, 2.41) the
partition function (5.36) can be expressed as
$\displaystyle{\displaystyle\prod_{E}}$ $\displaystyle[1+e^{{-2\beta}{\pi
R}(\mu+E)}\langle{\psi_{p}^{E}}|e^{{\rm
log}(1+{\frac{2{\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}}-2{H_{o}}}{\mu_{B}^{2}}})}|{\psi_{p}^{E}}\rangle]$
(5.37) $\displaystyle=$
$\displaystyle{\displaystyle\prod_{E}}[1+e^{{-2\beta}{\pi
R}(\mu+E)}\langle{\psi_{p}^{E}}|(1+{\frac{{\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}}-2{H_{o}}}{\mu_{B}^{2}}})|{\psi_{p}^{E}}\rangle].$
Now we have shown in the appendix that
$\langle{\overline{z}_{+}}{z_{+}}|{\hat{\overline{z}}_{+}}{\hat{z}_{+}}|{\overline{z}_{+}}{z_{+}}\rangle$
and
$\langle{\overline{z}_{-}}{z_{-}}|{\hat{\overline{z}}_{-}}{\hat{z}_{-}}|{\overline{z}_{-}}{z_{-}}\rangle$
diverge. So we must express $\hat{W}$ as
$W({{\hat{\overline{z}}_{-}}{\hat{z}_{-}},H_{o}})$ for the basis
$|{\overline{z}_{+}}{z_{+}},E\rangle$ basis and vice versa. Now note according
to the commutation relation (2.30) and from the form of the wave function
(5.3)
${\hat{\overline{z}}_{-}}{\hat{z}_{-}}={e^{-i{\frac{\varphi{(\hat{E+i})}}{2}}}}{{\frac{\partial}{\partial{\overline{z}_{+}}}}{\frac{\partial}{\partial{z_{+}}}}}{e^{i{\frac{\varphi{(\hat{E})}}{2}}}}$
. So according to the analysis Appendix,
$\langle{\psi_{p}^{E}}|{\frac{2{\hat{\overline{z}}_{-}}{\hat{z}_{-}}}{\mu_{B}^{2}}}|{\psi_{p}^{E}}\rangle=0$
except when R is an integer181818This is because we can write the integral as
$\langle{\psi_{o}}|{({\overline{z}_{+}}{z_{+}})}^{{\frac{n}{R}}-1}|{\psi_{o}}\rangle$
which following the analysis of Appendix-A contributes only at pole.. However
for R an integer it contributes a constant term independent of $\phi,E$, in
the partition function and can be ignored by subtracting out an overall
constant from the hamiltonian which amounts to multiplying the partition
function by an overall factor. For the macroscopic loop operator in any other
sector, we can proceed in the same way . So we can write the partition
function as
$\displaystyle{\prod_{E}}[1$ $\displaystyle+$
$\displaystyle\langle{\psi_{p}^{E}}|(1+{\frac{{\hat{\overline{z}}_{+}}{\hat{z}_{-}}+{\hat{\overline{z}}_{-}}{\hat{z}_{+}}}{\mu_{B}^{2}}})e^{{-2\beta}{\pi
R}(\mu+E)})|{\psi_{p}^{E}}\rangle]$ (5.38) $\displaystyle=$
$\displaystyle{\prod_{E}}[1+\langle{\psi_{p}^{E}}|e^{{\rm
log}(1-2{\frac{H_{0}}{\mu_{B}^{2}}})}e^{{-2\beta}{\pi
R}(\mu+E)})|{\psi_{p}^{E}}\rangle]$ $\displaystyle=$
$\displaystyle{\prod_{E}}[1+\langle{\psi_{p}^{E}}|e^{{-2\beta}{\pi
R}(\mu+H-{\frac{1}{2{\pi}{\beta}R}}{\rm
log}(1-2{\frac{H_{o}}{\mu_{B}^{2}}})}|{\psi_{p}^{E}}\rangle]$ $\displaystyle=$
$\displaystyle{\prod_{E}}[1+\langle{\psi_{p}^{E}}|e^{{-2\beta}{\pi
R}(\mu+{H_{o}^{\prime}}+{H_{p}})}|{\psi_{p}^{E}}\rangle]$ $\displaystyle=$
$\displaystyle{\rm Tr}_{\psi_{p}^{E}}[e^{{-2\beta}{\pi
R}(\mu+{H_{o}^{\prime}}+{H_{p}})}]$
where $H_{o}^{\prime}$ is as discussed in (3.45), given by
${H_{o}^{\prime}}={H_{o}}-{\frac{1}{2{\pi}R\beta}}{\rm
log}(1-{\frac{2H_{o}}{\mu_{B}^{2}}})$;
${H_{p}}={H_{p}}({\overline{z}_{\pm}}{z_{\pm}},H)$ is the effective
perturbation in the presence of momentum modes 191919 In order to reach from
2nd to 3rd step we used the same tricks of section 3 which implies that around
a delta function in time, we can make the time interval infinitesimally small
so that we can ignore the commutator terms ($[{H_{o}^{\prime}},{H_{o}}]$
+….higher commutators) what can arise on exponential as a consequence of Baker
Hausdorff formula .
First note that $|{\psi_{p}^{E}}\rangle$, the eigenstate of
$H={H_{o}}+{H_{p}}$ does not diagonalize the complete effective hamiltonian
$H_{\rm eff}=H-{\frac{1}{2{\pi}{\beta}R}}{\rm
log}(1-2{\frac{H_{o}}{\mu_{B}^{2}}})$. This is the indication from collective
field theory as we discussed in section 5.1 that the effect of putting a
macroscopic loop operator in MQM action is to deform the static Fermi sea as
well as the tachyonic background which shows up as a nonlinear shift of the
perturbing phase $\varphi\rightarrow\varphi_{wp}$ (5.15,5.14). Clearly the
eigenfunction which diagonalizes the complete effective hamiltonian
${H_{o}^{\prime}}+{H_{p}}$ will be given by a shift ${\cal W}\rightarrow{\cal
W}^{\prime}=e^{i{\frac{\phi^{\prime}}{2}}}$ where ${i\phi^{\prime}}$ will be
of similar form of (5.15) and so we will denote it by $\phi_{wp}$. Now we can
express ${\cal W}^{\prime}$ as the product ${\cal
W}^{\prime}=U({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}},H_{o}){\cal W}$ where
U is the factor responsible for the presence of macroscopic loop operator so
that in absence of the operator we should have ${\cal W}^{\prime}={\cal W}$.
Now in order to diagonalize lets recall the tricks we used in (3.44). If
${\cal W}^{\prime}{\psi_{o}}$ diagonalize the deformed hamiltonian in (5.38)
then we can replace the partition function with deformed hamiltonian
$H={H_{o}}^{\prime}+{H_{p}^{\prime}}(H,{\overline{z}_{\pm}}{z_{\pm}})$
evaluated in the basis ${\cal W}^{\prime}{\psi_{o}}(E)$ wlth the one in the
shifted basis ${\cal W}^{\prime}{\psi_{o}}(E)\rightarrow{\cal
W}{\psi_{o}}(E^{\prime})$ evaluated w.r.t the hamiltonian
$H={H_{o}}+{H_{p}}(H,{\overline{z}_{\pm}}{z_{\pm}})$. By this shift we can
identify $U({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}},H_{o})$ with an operator
which has an effect to shift $E^{\prime}\rightarrow E$ in the eigenfunction
$\psi^{p}({E^{\prime}})$. So the operator U will be of the form
$U(\sum_{n}{a_{n}}{({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}})}^{in}..){U_{1}}$,
where ${U_{1}}$ is an operator making unitary transformation to ${\cal W}$.
Clearly ${\cal W}$ and ${\cal W}^{\prime}$ are not related by unitary
transformation and so ${\cal W}\psi_{o}$ and ${\cal W}^{\prime}\psi_{o}$
defines the basis of completely different Hilbert space. So in the presence of
the Brane we need to evaluate the partition function in the basis ${\cal
W}^{\prime}\psi_{o}$ which we can view as deformed tachyonic background caused
by the presence of the brane. One can verify the expectation value of
macroscopic loop operator $W({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}},H_{o})$
in this basis effectively gets contribution from its hamiltonian part i.e
$W(H_{o})$ as in the case for undeformed basis (5.3) and following similar
steps we will get the partition function (5.38) with a shifted basis ${\cal
W}{\psi_{o}}\rightarrow{\cal W}^{\prime}{\psi_{o}}$. The perturbing phase
$\phi^{\prime}$ of the shifted basis in principle can have a complex part for
which we need to choose appropriate normalization. However in order to
evaluate the partition function we will follow (3.44). This partition function
is exactly given by the one with a shift $\psi^{\prime E}_{p}={\cal
W}^{\prime}{\psi_{o}}(E)\,\,\Rightarrow\,\,\psi_{p}^{E^{\prime}}={\cal
W}{\psi_{o}}(E^{\prime})$ and the perturbing phase
$\phi_{wp}(E)\rightarrow\phi(E(E^{\prime}))=\phi^{\prime}(E^{\prime})$
evaluated w.r.t the effective hamiltonian
$H={H_{o}}+{H_{p}}(H,{\overline{z}_{\pm}}{z_{\pm}})$ but without insertion of
the macroscopic loop operator W, where $\phi_{wp}(E)$ is as introduced in
(5.15) and $\psi^{\prime E}_{p}$ is the basis which diagonalizes the deformed
hamiltonian ${H_{o}^{\prime}}+{H_{p}}$ , as we discussed. So following (3.44)
we can evaluate the partition function (5.38)in the shifted basis
$\displaystyle{\psi_{p}^{\prime E}}$ $\displaystyle\rightarrow$
$\displaystyle{\psi_{p}^{E^{\prime}}}={\cal{W}}_{s}{\psi_{o}^{E^{\prime}}}=e^{{\frac{1}{2}}\phi(E({E^{\prime}}))+R\sum_{k\geq
1}t_{\pm k}{(z_{\pm}\bar{z}_{\pm})}^{k/R}+\sum_{k\geq 1}{1\over k}v_{\pm
k}(E({E^{\prime}})){(z_{\pm}\bar{z}_{\pm})}^{-k/R}}$
$\displaystyle{\psi_{o}^{E^{\prime}}}=$ $\displaystyle
e^{{\frac{1}{2}}(\phi^{\prime})({E^{\prime}})+R\sum_{k\geq 1}t_{\pm
k}(z_{\pm}\bar{z}_{\pm})^{k/R}-R\sum_{k\geq 1}{1\over k}{v^{\prime}}_{\pm
k}({E^{\prime}})(z_{\pm}\bar{z}_{\pm})^{-k/R}}{\psi_{o}^{E^{\prime}}},$ (5.39)
where the shifted dressing operator is given by
${\cal{W}}_{s}=e^{{\frac{1}{2}}\phi(E({E^{\prime}}))+R\sum_{k\geq 1}t_{\pm
k}(z_{\pm}\bar{z}_{\pm})^{k/R}+\sum_{k\geq 1}{1\over k}v_{\pm
k}(E({E^{\prime}})){(z_{\pm}\bar{z}_{\pm})}^{-k/R}}$
and $\phi(E)$ is the pase for perturbed wave function (5.4). So following
(3.42) we have the free energy given by
$\displaystyle{\cal{F}}(\mu,R)$ $\displaystyle=$
$\displaystyle\phi(E({E^{\prime}}={\frac{ir}{\beta R}}-\mu))$ (5.40)
$\displaystyle=$ $\displaystyle-i{\sum_{r=n+{\frac{1}{2}}\geq
0}}{\phi^{\prime}}({\frac{ir}{\beta R}}-\mu),$
where we have
${\phi^{\prime}}({E^{\prime}})={\phi}(E)$ (5.41)
So we see the partition function in the presence of D brane in a background
perturbed by momentum modes with compactified time is obtained from the one
without D brane by the shift
$E\rightarrow{E^{\prime}}\quad;\quad{\cal{W}}\rightarrow{\cal{W}}_{s}$
which defines a deformed Fermi surface..
### 5.4 Lax formalism for Type 0A MQM in the presence of D brane
Our lesson from the previous discussion is that Toda structure for Type 0A MQM
perturbed by tachyonic modes ,in the presence of D brane can be obtained when
we replace
$\displaystyle{\cal{W}}$ $\displaystyle\rightarrow$
$\displaystyle{\cal{W}}_{s}=e^{iR\sum_{n\geq 1}t_{\pm n}\omega^{n/R}}\ e^{\mp
i{\phi^{\prime}}(E^{\prime})}\ e^{iR\sum_{n\geq 1}{v_{\pm
n}^{\prime}}(E^{\prime})\ \omega^{-n/R}}.$ $\displaystyle\psi_{o}^{\prime E}$
$\displaystyle\rightarrow$
$\displaystyle\psi_{o}^{E^{\prime}}={\psi_{o}}{(E-{\frac{1}{2\pi
R}}\log(1-{\frac{2E}{\mu_{B}^{2}}}))},$ (5.42) $\displaystyle{L^{\prime}}_{+}$
$\displaystyle=$
$\displaystyle{{\cal{W}}_{s}}_{+}\omega{{\cal{W}}_{s}}^{-1}_{+},\quad
L_{-}={{\cal{W}}_{s-}}\omega^{-1}{{\cal{W}}_{s-}}^{-1}_{-},$
$\displaystyle{M^{\prime}}_{+}$ $\displaystyle=$
$\displaystyle{{\cal{W}}_{s+}}{\hat{E}}{{\cal{W}}_{s+}}^{-1},\quad
M_{-}={{\cal{W}}_{s-}}{\hat{E}}{{\cal{W}}_{s-}}.$ (5.43)
Note the operator algebra (5.19) remains same.
$\langle E|e^{\pm
i{\phi^{\prime}}}{\hat{\cal{W}}_{s\pm}}{L^{\prime}}_{\pm}|{\overline{z}_{\pm}}{z_{\pm}}\rangle=\langle
E|e^{\pm
i{\phi^{\prime}}_{0}}\hat{\cal{W}}_{s\pm}{\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}}|{\overline{z}_{\pm}}{z_{\pm}}\rangle,$
(5.44)
where ${\phi^{\prime}}=\phi({E^{\prime}})$. Expression of ${M^{\prime}}$ is as
described in (5.21)
$\displaystyle\langle E|$ $\displaystyle e^{\pm i{\phi^{\prime}}}\hat{\cal
W}_{\pm}\;{{M^{\prime}}_{\pm}}|{\overline{z}_{\pm}}{z_{\pm}}\rangle=\pm
i({z_{\pm}}\partial_{{z_{\pm}}}+{\overline{z}_{\pm}}\partial_{{\overline{z}_{\pm}}}+1)\Psi^{E^{\prime}}_{\pm}({\overline{z}_{\pm}}{z_{\pm}})$
(5.45) $\displaystyle=$ $\displaystyle\left(\sum_{k\geq 1}k\ t_{\pm k}\
{({\overline{z}_{\pm}}{z_{\pm}})}^{k/R}+{E^{\prime}}+\sum_{k\geq
1}{v^{\prime}}_{\pm k}\
{({\overline{z}_{\pm}}{z_{\pm}})}^{-k/R}\right)\Psi^{E^{\prime}}_{\pm}(({\overline{z}_{\pm}}{z_{\pm}})).$
As the partition function described in terms of Fermi level ${\mu}$
${M^{\prime}_{\pm}}=\sum_{k\geq 1}kt_{\pm
k}{L^{\prime}}_{\pm}^{k/R}+\hat{\mu}+\sum_{k\geq 1}{v^{\prime}}_{\pm
k}{L^{\prime}_{\pm}}^{-k/R}.$ (5.46)
The structure of the integrable flow remain same. The Toda flow equation will
be given by ${\phi^{\prime}}(\mu)\equiv\phi({E^{\prime}}=-\mu)$
$i{\partial\over\partial t_{1}}{\partial\over\partial
t_{-1}}{\phi^{\prime}}(\mu)=e^{i{\phi^{\prime}}(\mu)-i{\phi^{\prime}}(\mu-i/R)}-e^{i{\phi^{\prime}}(\mu+i/R)-i\phi(\mu)}.$
(5.47)
Now in order to see that partition function is a tau function of Toda lattice
hierarchy first note that
${\cal{Z}}(\mu,t)=\prod\limits_{n\geq
0}\exp\left[{i\beta}\phi\left(i{\frac{1}{\beta}}{n+{\litfont{1\over 2}}\over
R}-\mu\right)\right].$ (5.48)
with Fermi level ${E^{\prime}}=-\mu$. Now on the other hand, the zero mode of
the perturbing phase is actually equal to the zero mode of the dressing
operators (5.42). Hence it is expressed through the $\tau$-function as in
(5.28). Since the shift in the discrete parameter n is equivalent to an
imaginary shift of the chemical potential $\mu$, so (5.48) implies
$e^{{i\beta}\phi(-\mu)}={\frac{{{\cal{Z}}}(\mu+{\frac{i}{2R\beta}})}{{{\cal{Z}}}(\mu-{\frac{i}{2R\beta}})}}.$
(5.49)
However from (5.28) we have
$e^{{i\beta}\phi(-\mu)}={\frac{{\tau_{o}}(\mu+{\frac{i}{2R\beta}})}{{\tau_{o}}(\mu-{\frac{i}{2R\beta}})}}$
(5.50)
So one concludes that
${\cal{Z}}(\mu,t)=\tau_{0}(\mu,t).$ (5.51)
### 5.5 Representation in terms of a bosonic field
Here we will study the classical limit following the analysis of [14] The
momentum modes can be described as the oscillator modes of a bosonic field
$\varphi({\overline{z}_{+}}{z_{+}},{\overline{z}_{-}}{z_{-}})=\varphi_{+}({\overline{z}_{+}}{z_{+}})+\varphi_{-}({\overline{z}_{-}}{z_{-}})$.
The bosonization formula is
$\Psi^{{E^{\prime}}=-\mu-i}_{\pm}({\overline{z}_{\pm}}{z_{\pm}})={\cal{Z}}^{-1}e^{\pm
i\varphi_{\pm}({\overline{z}_{\pm}}{z_{\pm}})}\cdot{\cal{Z}}.$ (5.52)
(Note here in the presence of FZZT brane ${\mu}$ corresponds to the deformed
Fermi surface) where ${\cal{Z}}$ is the partition function and
$\varphi_{\pm}({\overline{z}_{\pm}}{z_{\pm}})=+R\sum_{k\geq
1}t_{k}{({\overline{z}_{\pm}}{z_{\pm}})}^{k/R}+{1\over
R}\partial_{\mu}+\mu\log{\overline{z}_{\pm}}{z_{\pm}}-R\sum_{k\geq 1}{1\over
k}{({\overline{z}_{\pm}}{z_{\pm}})}^{-k/R}{\partial\over\partial t_{k}}.$
(5.53)
Then from (5.45) the operators $M_{\pm}$ are represented by the currents
${\overline{z}_{\pm}}{z_{\pm}}\partial_{\pm}\varphi$
${M_{\pm}^{\dagger}}\Psi_{\pm}^{E}({\overline{z}_{\pm}}{z_{\pm}})|_{E=-\mu-i}={\cal{Z}}^{-1}{\overline{z}_{\pm}}{z_{\pm}}\partial_{\pm}\varphi\cdot{\cal{Z}}.$
(5.54)
### 5.6 The dispersionless (quasiclassical) limit
We consider the quasiclassical limit $\beta\rightarrow\infty$. In this limit
the integrable structure described above reduces to the dispersionless Toda
hierarchy where the operators $\hat{\mu}$ and $\hat{\omega}$ can be considered
as a pair of classical canonical variables with Poisson bracket
$\\{\omega,\mu\\}=\omega$ (5.55)
Similarly, all operators become $c$-functions of these variables. The Lax
operators can be identified with the classical phase space coordinates
${\overline{z}_{\pm}}{z_{\pm}}$, which satisfy
$\\{{\overline{z}_{+}},{z_{-}}\\}=\\{{z_{+}},{\overline{z}_{-}}\\}=1$ (5.56)
The shape of the Fermi sea is determined by the classical trajectory
corresponding to the Fermi level ${E^{\prime}}=-\mu$. So we have
${\overline{z}_{+}}{z_{-}}+{z_{+}}{\overline{z}_{-}}-{\frac{1}{2{\pi}R\beta}}\log(1-{\frac{2({\overline{z}_{+}}{z_{-}}+{z_{+}}{\overline{z}_{-}})}{\mu_{B}^{2}}})-\log{\epsilon}=-\mu.$
(5.57)
Where $\log{\epsilon}$ is the cut-off cancelling the singular contribution
from the point
$(1-{\frac{2({\overline{z}_{+}}{z_{-}}+{z_{+}}{\overline{z}_{-}})}{\mu_{B}^{2}}})=0$.
In the perturbed theory the classical trajectories are of the form
${\overline{z}_{\pm}}{z_{\pm}}={L^{\prime}}_{\pm}(\omega,\mu).$ (5.58)
where the functions $L_{\pm}$ are of the form
${L^{\prime}}_{\pm}(\omega,\mu)=e^{{\frac{1}{2}}\partial_{\mu}{\phi^{\prime}}}\
\omega^{\pm 1}\left(1+\sum_{k\geq 1}{a^{\prime}}_{\pm k}(\mu)\ \omega^{\mp
k/R}\right).$ (5.59)
The flows $H_{n}$ become Hamiltonians for the evolution along the ‘times’
$t_{n}$. The unitary operators ${\cal W}_{\pm}$ becomes a pair of canonical
transformations between the variables $\omega,\mu$ and $L_{\pm},M_{\pm}$.
Their generating functions are given by the expectation values
$S_{\pm}={\cal{Z}}^{-1}\ \varphi_{\pm}({\overline{z}_{\pm}}{z_{\pm}})\
\cdot{\cal{Z}}$ of the chiral components of the bosonic field $\phi$
$S_{\pm}=\pm R\sum_{k\geq 1}t_{\pm k}\
{({\overline{z}_{\pm}}{z_{\pm}})}^{k/R}+\mu\log({\overline{z}_{\pm}}{z_{\pm}})-{\phi^{\prime}}\pm
R\sum_{k\geq 1}{1\over k}{v_{k}^{\prime}}\
{({\overline{z}_{\pm}}{z_{\pm}})}^{-k/R},$ (5.60)
where $v_{k}=\partial{\cal F}/\partial t_{k}$. The differential of the
function $S_{\pm}$ is
${dS}_{\pm}=M_{\pm}d{\rm log}({\overline{z}_{\pm}}{z_{\pm}})+{\rm log}\omega\
d\mu+R\sum_{n\neq 0}{H_{n}}{dt_{n}}.$ (5.61)
If we consider the coordinate $\omega$ as a function of either
${\overline{z}_{+}}{z_{+}}$ or ${\overline{z}_{-}}{z_{-}}$, then
$\omega=e^{{\partial_{\mu}}{S_{+}}({\overline{z}_{+}}{z_{+}})}=e^{{\partial_{\mu}}{S_{-}}({\overline{z}_{-}}{z_{-}})}.,$
(5.62)
The classical string equation
${\overline{z}_{+}}{z_{-}}+{z_{+}}{\overline{z}_{-}}-{\frac{1}{2\pi
R}}\log(1-{\frac{2({\overline{z}_{+}}{z_{-}}+{z_{+}}{\overline{z}_{-}})}{\mu_{B}^{2}}})={M_{+}}={M_{-}},$
(5.63)
can be written as
$\displaystyle{\overline{z}_{+}}{z_{-}}$ $\displaystyle+$
$\displaystyle{\overline{z}_{-}}{z_{+}}-{\frac{1}{2\pi R}}{\rm
log}(1-{\frac{2({\overline{z}_{+}}{z_{-}}+{z_{+}}{\overline{z}_{-}})}{\mu_{B}^{2}}})$
(5.64) $\displaystyle=$ $\displaystyle\sum_{k\geq
1}kt_{k}{({\overline{z}_{+}}{z_{+}})}^{k/R}+\mu+\sum_{k\geq
1}v_{k}{({\overline{z}_{+}}{z_{+}})}^{-k/R}.$
## 6 Conclusion
Here we have studied Type 0A matrix model in the presence of spacelike D brane
which are localized in matter direction. In matrix model this is expressed by
insertion of an operator $e^{W(t_{o})}$ into the path integral. When we
studied the respective MQM we found by application of Ward identity that the
time translation invariance of the path integral in the presence of such
operator gives the signal of leakage of MQM hamiltonian. However in dual
string theory this phenomenon has a meaning that closed string hamiltonian is
undergoing a leakage when the string is getting scattered from Dbrane! in
order to obtain right string theory picture we impose a constraint (2.25) on
matrix model path integral in the presence of D brane. We explained that this
condition has an effect to constrain the Hilbert space generated by
macroscopic loop operator while keeping type 0A MQM unaffected. We have shown
that when we impose the constraint we get the matter one point function from
collective field theory. We have further shown that exactly at the point of
insertion of the brane ( which in string theory correspond to the point where
open string ends are localized) the wave function for the right and left
moving component of boundary state with any momentum appears to be identical
which can be seen in matrix model as a consequence of this constraint. We also
found right transition amplitude from a free fermionic state to coherent
state. Next we consider type 0A MQM with the time t compactified on a circle.
We have shown matrix model path integral in the presence of Dbrane can be
expressed as Fredholm determinant. We evaluated the thermal partition function
in grand canonical ensemble. As the theory is defined on a circle so the
partition function correspond to that of a deformed Fermi surface. We have
further shown that in absence of any such constraint, the partition function
diverges. Theory on the circle also posses a symmetry (3.33), which is
parallel to string theory as we discussed in section 2.4. This symmetry
clearly indicates that coherent states are strongly localized at the point of
insertion of macroscopic loop operator. Finally we considered type 0A MQM in
the background of momentum modes. First in section 4 we made a semiclassical
analysis, studied fermionic scattering in the presence of D brane. We found
the effective hamiltonian in the perturbed background from semiclassical
analysis. Its known that the presence of D brane change the tachyonic
background. So from collective field theory analysis we found the right
expression of MQM wave function i.e exact modification of the dressing
operator in the presence of Dbrane. We derived the grand canonical partition
function in the perturbed background in the presence of D brane. We have shown
the partition function corresponds to tau function of Toda hierarchy. We have
also analyzed the theory in dispersionless limit. Its interesting to study T
duality between type 0A and type 0B MQM in the presence of D brane. One can
also study the theory in the presence of flux background and see the
consequence of the constraint.
Acknowledgments
The author wishes to express her deep sense of gratitude to Raghava Varma for
constant support, motivation and encouragement to pursue the research at IIT
Bombay. Its a pleasure to thank Satchidananda Naik for useful discussions
which lead to the finding of this problem and valuable suggestions during the
progress of the work. The author is greatly indebted to P. Ramadevi for her
generous support and encouragement, necessary advices, and comments on the
draft. The author is grateful to Sunil Mukhi and Koushik Ray for valuable
discussions and important comments. The author would also like to thank S. Uma
Sankar and Soumitra Sengupta for their inputs. Special thanks to Ankhi Roy and
Rajeeb R. Mallick for their sincere cooperation during the work in IIT Bombay.
The author is also thankful to Partha Pratim Pal and Colina Dutta for proof-
reading the manuscript. Finally, the author wants to thank all the research
scholars in the High energy Physics group at IIT Bombay, in particular,
Reetanjali Moharana, Sushant k. Raut, Sasmita Mishra,Amal Sarkar, Ravi
Manohar, Neha Shah, Himani Bhatt, Kabita Chandwani, Suprabh Prakash for
discussions and help. This work is supported by funds from IRCC, IITB and the
research development fund of Prof. Raghava Varma.
Appendix
## Appendix A Appendix
Here we will show that type 0A MQM wave functions in the presence of D brane
satisfy orthogonality and biorthogonality conditions.
### A.1 Orthogonality Condition
The wave function is expected to show orthonormal property.
1\. For $t<{t_{o}}$ we have the wave function ${\psi_{o}}$ given in (2.28).
One can check the orthonormality property by considering the contour integral
,and it is given by [15]
$\langle{\psi_{E^{\prime}}}|{\psi}_{E}\rangle={\delta}(E-{E^{\prime}}).$ (A.1)
For ${t>{t_{o}}}$, first consider the wave function
${\psi_{+}}({z_{+}},{\overline{z}_{+}},t;E)$ given in (2.42). we have
$\displaystyle\langle{\psi}({E^{\prime}},t){\psi}(E,t)\rangle$
$\displaystyle=$ $\displaystyle\int d{{z_{+}}}d{{\overline{z}_{+}}}[\\{1-{{\rm
log}(1+{\frac{W({\overline{z}_{-}}{z_{-}},2{H_{o}})}{\mu_{B}^{2}}})\\}{e^{i{E^{\prime}}(t-{t_{o}})}}{e^{i{{\frac{\phi_{o}}{2}}}(E^{\prime})}}{({\overline{z}_{-}}{z_{-}})}^{-i{E^{\prime}}-{\frac{1}{2}}}}]$
(A.2) $\displaystyle[\\{1-{\rm
log}(1+{\frac{W({\overline{z}_{-}}{z_{-}},2{H_{o}})}{\mu_{B}^{2}}})\\}{e^{-iE(t-{t_{o}})}}{e^{-i{\frac{\phi_{o}}{2}}(E)}}{({\overline{z}_{+}}{z_{+}})}^{iE-{\frac{1}{2}}}].$
Now in order to see the orthonormal property first recall the commutation
relation (2.30)
Note that ${\hat{z}_{+}}$ and ${\hat{z}_{-}}$ shifts E by -i and +i
respectively.So we can write
${\hat{\overline{z}_{+}}}{\hat{z}_{+}}={e^{\frac{-i\phi_{o}}{2}}}{e^{-i{\partial_{E}}}}{e^{\frac{i\phi_{o}}{2}}}$
(A.3)
Similarly
${\hat{\overline{z}}_{-}}{\hat{z}_{-}}={e^{\frac{-i\phi_{o}}{2}}}{e^{i{\partial_{{E}}}}}{e^{\frac{i\phi_{o}}{2}}}.$
(A.4)
Now to evaluate (A.2) first consider the expression
$\displaystyle\langle{\psi_{+}}|[{\hat{\overline{z}}_{-}}{\hat{z}_{-}}{]^{m}}|{\psi_{+}}\rangle$
$\displaystyle=$ $\displaystyle\int
d{z_{+}}d{\overline{z}_{+}}\\{{e^{i{E^{\prime}}t}}{e^{{\frac{i}{2}}{\phi_{o}}(E^{\prime})}}{{({z_{+}}{\overline{z}_{+}})}^{-i{E^{\prime}}-{\frac{1}{2}}}}\\}$
(A.5)
$\displaystyle[{\frac{\partial}{\partial{\overline{z}_{+}}}}{\frac{\partial}{\partial{z_{+}}}}{]^{m}}\\{{e^{-iEt+mt}}{e^{-{\frac{i}{2}}{\phi_{o}}(E-mi)}}{{({z_{+}}{\overline{z}_{+}})}^{iE-{\frac{1}{2}}}}\\}.$
For $E\neq{E^{\prime}}$ this can just be written as
$\displaystyle\langle{\psi_{+}}|[{\hat{\overline{z}}_{-}}{\hat{z}_{-}}{]^{m}}|{\psi_{+}}\rangle$
$\displaystyle=$
$\displaystyle{e^{-{\frac{i}{2}}{\phi_{o}}}}{e^{im{\partial_{E}}}}{e^{{{\frac{i}{2}}\phi_{o}}}}\int
d{z_{+}}d{\overline{z}_{+}}\\{{e^{i{E^{\prime}}t}}{e^{{\frac{i}{2}}{\phi_{o}}(E^{\prime})}}{{({z_{+}}{\overline{z}_{+}})}^{-i{E^{\prime}}-{\frac{1}{2}}}}\\}$
(A.6)
$\displaystyle\\{{e^{-iEt}}{e^{-{\frac{i}{2}}{\phi_{o}}(E)}}{{({z_{+}}{\overline{z}_{+}})}^{iE-{\frac{1}{2}}}}\\}$
$\displaystyle=$
$\displaystyle{e^{-{\frac{i}{2}}{\phi_{o}}}}{e^{im{\partial_{E}}}}{e^{{\frac{i}{2}}{\phi_{o}}}}\langle{\psi_{+}}({E^{\prime}})|{\psi_{+}}(E){\rangle_{E\neq{E^{\prime}}}}$
$\displaystyle=$ $\displaystyle 0.$
For $E={E^{\prime}}$ (A.2) takes the form
$\displaystyle\langle{\psi_{+}}|[{\hat{\overline{z}}_{-}}{\hat{z}_{-}}{]^{m}}|{\psi_{+}}\rangle$
$\displaystyle=$
$\displaystyle\\{{e^{i{\phi_{o}}(E)}}{e^{-i{\phi_{o}}(E-mi)}}\\}\int
d{z_{+}}d{\overline{z}_{+}}[{\overline{z}_{+}}{z_{+}}{]^{-m-1}}{e^{-mt}}.$
(A.7)
From contour integral which is 0 for $m\geq 1$. Also we conclude that
$\langle{\psi_{+}}|[{\hat{\overline{z}}_{+}}{\hat{z}_{+}}{]^{m}}|{\psi_{+}}\rangle$
and
$\langle{\psi_{-}}|[{\hat{\overline{z}}_{-}}{\hat{z}_{-}}{]^{m}}|{\psi_{-}}\rangle$
diverge. Here before going to show the orthogonality lets consider the
situation when m is not an integer. This kind of integration we had in the
section 5 in the expression
$\langle{\psi_{o}}|{({\overline{z}_{+}}{z_{+}})}^{{\frac{n}{R}}-1}|{\psi_{o}}\rangle$.
We can consider this as the product over two branh cut integrals ${z_{+}}$ and
${\overline{z}_{+}}$ and the each branch cut integral can be expressed as the
sum of two standard contour integral with the cut on right and left side of
the real axis $0\geq x\geq\infty$ and $-\infty\geq x\geq 0$ respectively and
with a pole at $x=0$. One can see that the integral turns out to be zero for
any noninteger ${\frac{m}{R}}$. We have nonzero contribution only when R is an
integer and the contributing term is m=R.This term corresponds to a pure pole
and give a constant contribution to the integral, Now back to the question of
orthogonality. So using (A.4 ,A.7), we can be written (A.2) as
$\displaystyle\langle{\psi}({E^{\prime}},t)|{\psi}(E,t)\rangle$
$\displaystyle=$ $\displaystyle\int d{z_{+}}d{\overline{z}_{+}}[\\{1-{\rm
log}(1+{\frac{f(0,2{H_{o}})}{\mu_{B}^{2}}})\\}{e^{i{E^{\prime}}(t-{t_{o}})}}{e^{{\frac{i}{2}}{\phi_{o}}(E^{\prime})}}{{({z_{+}}{\overline{z}_{+}})}^{-i{E^{\prime}}-{\frac{1}{2}}}}]$
(A.8) $\displaystyle[\\{1-{\rm
log}(1+{\frac{f(0,2{H_{o}})}{\mu_{B}^{2}}})\\}{e^{-iE(t-{t_{o}})}}{e^{-{\frac{i}{2}}{\phi_{o}}(E)}}{{({z_{+}}{\overline{z}_{+}})}^{iE-{\frac{1}{2}}}}]$
$\displaystyle=$ $\displaystyle\int d{z_{+}}d{\overline{z}_{+}}[\\{1-{\rm
log}(1+{\frac{f(0,2E)}{\mu_{B}^{2}}})\\}{e^{i{E^{\prime}}(t-{t_{o}})}}{e^{{\frac{i}{2}}{\phi_{o}}(E^{\prime})}}{{({z_{+}}{\overline{z}_{+}})}^{-i{E^{\prime}}-{\frac{1}{2}}}}]$
$\displaystyle[\\{1-{\rm
log}(1+{\frac{f(0,2E)}{\mu_{B}^{2}}})\\}{e^{-iE(t-{t_{o}})}}{e^{-{\frac{i}{2}}{\phi_{o}}(E)}}{{({z_{+}}{\overline{z}_{+}})}^{iE-{\frac{1}{2}}}}].$
Now write ${\phi_{o}}={\phi_{oRe}}+i{\phi_{oIm}}$. Shifting
${\phi_{oIm}}\rightarrow{\phi_{oIm}}-{\frac{i}{2}}{\rm log}[1-{\rm
log}(1+{\frac{f(0,2E)}{\mu_{B}^{2}}})]$ we get the orthogonal property (A.1)
with
${\psi_{+}}(E,t)={e^{iE(t-{t_{o}})}}{e^{-{\frac{i}{2}}{\phi_{0+}}(E)}}{{({z_{+}}{\overline{z}_{+}})}^{iE-{\frac{1}{2}}}},$
(A.9)
where
${\phi_{o+}}(E)={\phi_{o}}(E)-i{\rm
log}[1-log(1+{\frac{f(0,2E}{\mu_{B}^{2}}})].$ (A.10)
Similarly for the wave function in ${z_{-}}$ representation we find
${\phi_{o-}}(E)={\phi_{o}}(E)+ilog[1-log(1+{\frac{f(0,2E}{\mu_{B}^{2}}})]$
(A.11)
So we see the consequence of the insertion of macroscopic loop operator is
that the phase of the wave function develops an imaginary part which is
associated with tunneling.
### A.2 Biorthogonality Relation
The wave function is expected to satisfy the following biorthogonality
condition which has a consequence in the evaluation of scattering amplitude
[14, 15, 16].
$\int
d{{z_{+}}}d{{z_{-}}}d{{\overline{z}_{+}}}d{{\overline{z}_{-}}}{\overline{\psi^{E}_{+}}}({\overline{z}_{+}}{z_{+}},t){e^{i({\overline{z}_{+}}{z_{-}}+{\overline{z}_{-}}{z_{+}})}}{\psi^{E^{\prime}}_{-}}({\overline{z}_{-}}{z_{-}},t)={\delta}(E-{E^{\prime}})$
(A.12)
Now for $t\leq{t_{o}}$ we have ${\psi}={\psi_{o}}$ and for which
biorthogonality relation is already derived in [14],[15] giving
$e^{i\phi_{0}(E)}={\frac{\Gamma(iE+1/2)}{\Gamma(iE+1/2)}}$ [15]. For
$t\geq{t_{o}}$ , .Biorthogonality relation takes the form
$\displaystyle\int
d{{z_{+}}}d{{z_{-}}}d{{\overline{z}_{+}}}d{{\overline{z}_{-}}}$
$\displaystyle{\overline{\psi}^{E}_{+}}({\overline{z}_{+}}{z_{+}},t)\\{1-{\hat{W}}({{\hat{z}_{+}},{\hat{\overline{z}}_{+}},{H_{o}},t})\\}{e^{i({\overline{z}_{+}}{z_{-}}+{z_{+}}{\overline{z}_{-}})}})(1-{\hat{W}}({{\hat{z}_{+}},{\hat{\overline{z}}_{+}},{H_{o}},t})){\psi^{E^{\prime}}_{-}}$
(A.13) $\displaystyle=$ $\displaystyle{\delta}(E-{E^{\prime}}),$
In order to show this note that
$\displaystyle\int
d{{z_{+}}}d{{z_{-}}}d{{\overline{z}_{+}}}d{{\overline{z}_{-}}}$
$\displaystyle{\overline{\psi^{E}_{+}}}({{z_{+}}},{{\overline{z}_{+}}},t)\\{1-{\hat{W}}({{\hat{z}_{-}},{\hat{\overline{z}}_{-}},{H_{o}},t})\\}{e^{i({z_{+}}{z_{-}}+{\overline{z}_{+}}{\overline{z}_{-}})}})(1-{\hat{W}}({{\hat{z}_{-}},{\hat{\overline{z}}_{-}},{H_{o}},t})){\psi^{E^{\prime}}_{-}}$
$\displaystyle=$ $\displaystyle\int
d{{z_{+}}}d{{\overline{z}_{+}}}{\overline{\psi^{E}_{+}}}({{z_{+}}},{{\overline{z}_{+}}},t)\\{1-{\hat{W}}({{\hat{z}_{-}},{\hat{\overline{z}}_{-}},{H_{o}},t})\\}(1-{\hat{W}}({{\frac{\partial}{\partial{z_{+}}}},{\frac{\partial}{\partial{\overline{z}_{+}}}},{H_{o}},t}))$
$\displaystyle\int
d{{z_{-}}}d{{\overline{z}_{-}}}{e^{i({\overline{z}_{+}}{z_{-}}+{\overline{z}_{-}}{z_{+}})}}){\psi^{E^{\prime}}_{-}}({{\overline{z}_{-}}{z_{-}}},t)$
$\displaystyle=$ $\displaystyle\int
d{{z_{+}}}d{{\overline{z}_{+}}}{\overline{\psi^{E}_{+}}}({{z_{+}}},{{\overline{z}_{+}}},t)\\{1-{\hat{W}}({{\hat{z}_{-}},{\hat{\overline{z}}_{-}},{H_{o}},t})\\}(1-{\hat{W}}({{\hat{z}_{-}},{\hat{\overline{z}}_{-}},{H_{o}},t}))$
$\displaystyle\int
d{{z_{-}}}d{{\overline{z}_{-}}}{e^{i({z_{+}}{z_{-}}+{\overline{z}_{+}}{\overline{z}_{-}})}}){\psi^{E^{\prime}}_{-}}({{z_{-}}},{{\overline{z}_{-}}},t)$
$\displaystyle=$ $\displaystyle R(E)\int
d{{z_{+}}}d{{\overline{z}_{+}}}{\overline{\psi^{E}_{+}}}({{z_{-}}},{{\overline{z}_{-}}},t)\\{1-{\hat{W}}({{z_{-}},{\overline{z}_{-}}{H_{o}},t})\\}(1-{\hat{W}}({{z_{-}},{\overline{z}_{-}},{H_{o}},t}))$
$\displaystyle{\psi^{E^{\prime}}_{-}}({{z_{+}}},{{\overline{z}_{+}}},t)$
(A.14) $\displaystyle=$
$\displaystyle{R}(E){e^{i{\phi_{o+}}}}{\delta}(E-{E^{\prime}}).$
Where in order to come from 2nd to 3rd step we used the fact that in ${z_{+}}$
representation we have
${\hat{z}_{-}},{\hat{\overline{z}}_{-}}={\frac{\partial}{\partial{z_{+}}}},{\frac{\partial}{\partial{\overline{z}_{+}}}}$
The integral in the 4th step, we have evaluated in (A.8) leads to the last
step. In order to get the biorthogonality relation(A.13) we must need to set
$e^{i{\phi_{o+}}(E)}=R(E)$. Compared to the $t\leq{t_{o}}$ case note the shift
of ${\phi_{o}}(E)$ due to the insertion of macroscopic loop operator.
$R(E){\psi^{E^{\prime}}_{+}}({z_{+}},{\overline{z}_{+}},t)={\int_{-\infty}^{\infty}}d{{z_{-}}}d{{\overline{z}_{-}}}{e^{i({z_{+}}{z_{-}}+{\overline{z}_{+}}{\overline{z}_{-}})}}){\psi^{E^{\prime}}_{-}}({z_{-}},{\overline{z}_{-}},t)$
(A.15)
R(E) getting absorbed to decide ${\phi_{o}}$ and shifting ${\phi_{o}}$
according to (A.11) we get the above result.
## References
* [1] Igor R. Klebanov, String Theory in Two Dimensions, hep-th/9108019
* [2] P. Ginsparg, Gregory Moore, Lectures on 2D gravity and 2D string theory, (TASI 1992), hep-th/9304011
* [3] Joseph Polchinski, What is String Theory , hep-th/9411028
* [4] John McGreevy, Herman Verlinde, Strings from Tachyons, hep-th/0304224
* [5] Igor R. Klebanov, Juan Maldacena, Nathan Seiberg, D-brane Decay in Two-Dimensional String Theory, hep-th/0305159
* [6] John McGreevy, Joerg Teschner, Herman Verlinde, Classical and Quantum D-branes in 2D String Theory , hep-th/0305194
* [7] Xi Yin, Matrix Models, Integrable Structures, and T-duality of Type 0 String Theory, hep-th/0312236
* [8] Tadashi Takayanagi, Nicolaos Toumbas, A Matrix Model Dual of Type 0B String Theory in Two Dimensions, hep-th/0307083
* [9] M. R. Douglas, I. R. Klebanov, D. Kutasov, J. Maldacena, E. Martinec, N. Seiberg, A New Hat For The c=1 Matrix Model , hep-th/0307195
* [10] Gregory W. Moore, Nathan Seiberg, Matthias Staudacher, From loops to states in 2-D quantum gravity, Nucl.Phys.B362:665-709,1991
* [11] Gregory W. Moore, Nathan Seiberg, From loops to fields in 2-D quantum gravity, Int.J.Mod.Phys.A7:2601-2634,1992
* [12] Emil J. Martinec, The Annular report on noncritical string theory, hep-th/0305148
* [13] Tadashi Takayanagi, Notes on D-branes in 2-D type 0 string theory, hep-th/0402196
* [14] Ivan K. Kostov, Integrable flows in c = 1 string theory, hep-th/0208034
* [15] Xi Yin, Matrix models, integrable structures, and T duality of type 0 string theory, hep-th/0312236
* [16] Sergei Yu. Alexandrov, Vladimir A. Kazakov, Time dependent backgrounds of 2-D string theory, hep-th/0205079
* [17] Jaemo Park, Takao Suyama, Type 0A matrix model of black hole, integrability and holography, hep-th/0411006
* [18] Ian Ellwood, Akikazu Hashimoto, Open/closed duality for FZZT branes in c=1, hep-th/0512217
* [19] Sergei Yu. Alexandrov, Ivan K. Kostov, Time-dependent backgrounds of 2-D string theory: Non-perturbative effects, hep-th/0412223
* [20] Sergei Alexandrov, Matrix quantum mechanics and two-dimensional string theory in nontrivial backgrounds, hep-th/0311273
* [21] Sergei Yu. Alexandrov, Vladimir A. Kazakov, David Kutasov, Nonperturbative effects in matrix models and D-branes, hep-th/0306177
* [22] Sergei Yu. Alexandrov, Vladimir A. Kazakov, Thermodynamics of 2-D string theory, hep-th/0210251
* [23] Anindya Mukherjee, Sunil Mukhi, Noncritical string correlators, finite-N matrix models and the vortex condensate, hep-th/0602119
* [24] David J. Gross, Igor R. Klebanov, One-Dimensional String Theory On A Circle, Nucl.Phys.B344:475-498,1990
* [25] Sumit R. Das, Antal Jevicki, String Field Theory And Physical Interpretation Of D = 1 Strings, Mod.Phys.Lett.A5:1639-1650,1990.
* [26] Joseph Polchinski, Classical limit of (1+1)-dimensional string theory, Nucl.Phys.B362:125-140,1991
* [27] Robbert Dijkgraaf, Gregory W. Moore, Ronen Plesser, The Partition function of 2-D string theory, hep-th/9208031
* [28] Antal Jevicki. Development in 2-d string theory, hep-th/9309115
* [29] Juan Martin Maldacena, Nathan Seiberg, Flux-vacua in two dimensional string theory. hep-th/0506141
* [30] Sergei Gukov, Tadashi Takayanagi, Flux backgrounds in 2-D string theory, hep-th/0312208
* [31] Kanehisa Takasaki, Integrable hierarchies and dispersionless limit, hep-th/9405096
* [32] Ivan K. Kostov, String equation for string theory on a circle, hep-th/0107247
* [33] B. Ponsot, J. Teschner, Boundary Liouville field theory: Boundary three point function , : hep-th/0110244
* [34] Alexander B. Zamolodchikov, Alexei B. Zamolodchikov, Liouville field theory on a pseudosphere, hep-th/0101152
* [35] Joshua L. Davis, Finn Larsen, Ross O’Connell, Diana Vaman, Integrable Deformations of hat c =1 Strings in Flux Backgrounds, hep-th/0607008
* [36] Dmitri Boulatov , Vladimir Kazakov, One-dimensional string theory with vortices as the upside down matrix oscillator, hep-th/0012228
* [37] Vladimir Kazakov , Ivan K. Kostov , David Kutasov, A Matrix model for the two-dimensional black hole, hep-th/0101011
* [38] V. Kazakov, I. Kostov, Loop Gas Model for Open Strings , hep-th/9205059
* [39] A. Recknagel, V. Schomerus, D-branes in Gepner models, hep-th/9712186
* [40] A. Recknagel, V. Schomerus, Boundary Deformation Theory and Moduli Spaces of D-Branes, hep-th/9811237
* [41] Nathan Seiberg, Notes on quantum Liouville theory and quantum gravity , Prog.Theor.Phys.Suppl.102:319-349,1990.
* [42] A.M. Polyakov, Gauge Fields and Space-Time , hep-th/0110196
* [43] Sergei Alexandrov, (m,n) ZZ branes and the c=1 matrix model, hep-th/0310135
* [44] Neil Lambert, Hong Liu, Juan Maldacena, Closed strings from decaying D-branes, hep-th/0303138
|
arxiv-papers
| 2011-01-11T11:09:26 |
2024-09-04T02:49:16.301618
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Chandrima Paul",
"submitter": "Chandrima Paul",
"url": "https://arxiv.org/abs/1101.2094"
}
|
1101.2130
|
020010 2010 V. Lakshminarayanan S. Roy, Dayalbagh Educational Institute, Agra,
India. 020010
Live cell imaging using metallic nanoparticles as tags is an emerging
technique to visualize long and highly dynamic processes due to the lack of
photobleaching and high photon rate. However, the lack of excited states as
compared to fluorescent dyes prevents the use of resonance energy transfer and
recently developed super resolution methods to measure distances between
objects closer than the diffraction limit. In this work, we experimentally
demonstrate a technique to determine subdiffraction distances based on the
near field coupling of metallic nanoparticles. Due to the symmetry breaking in
the scattering cross section, not only distances but also relative
orientations can be measured. Single gold nanoparticles were prepared on
glass, statistically yielding a small fraction of dimers. The sample was
sequentially illuminated with two wavelengths to separate background from
nanoparticle scattering based on their spectral properties. A novel total
internal reflection illumination scheme in which the polarization can be
rotated was used to further minimize background contributions. In this way,
radii, distance and orientation were measured for each individual dimer, and
their statistical distributions were found to be in agreement with the
expected ones. We envision that this technique will allow fast and long term
tracking of relative distance and orientation in biological processes.
# Experimental determination of distance and orientation of metallic
nanodimers by polarization dependent plasmon coupling
H. E. Grecco [inst1, inst2] O. E. Martínez[inst1] E-mail: hgrecco@df.uba.arE-
mail: oem@df.uba.ar
($22$ April 2010; 2 December 2010)
††volume: 2
99 inst1 Laboratorio de Electrónica Cuántica, Universidad de Buenos Aires.
Buenos Aires, Argentina. inst2 Current address: Department of Systemic Cell
Biology Max Planck Institute of Molecular Physiology. Dortmund, Germany
## 1 Introduction
Microscopy is an example of the ongoing symbiotic relationship between physics
and biology: as early microscopes allowed fundamental discoveries like
microorganisms or DNA; the need to see smaller, faster and deeper has pushed
the development of a plethora of optical concepts and microscopy techniques.
Today, fluorescence microscopy is an essential tool in biology as it can
visualize the spatio-temporal dynamics of intracellular processes. However,
many important mechanisms, like protein interaction, clustering or
conformational changes, occur at length scales smaller than the resolution
limit of conventional microscopy and therefore cannot be assessed by standard
imaging. Unraveling the dynamics of such inter- and intramolecular mechanisms
that provides function richness to molecules and molecular complexes is
essential to understand key biological processes such as cellular signal
propagation.
Subdiffraction distances have been determined by exploiting quantum and near
field properties of the interaction between light and matter in the nanometer
scale. For example, Fluorescence/Förster Resonance Energy Transfer (FRET) [5,
6, 7, 14] has proven to be a valuable technique as it provides an optical
signal directly related to the proximity of the molecules. The desire to
extend this technique to other biological systems with different time and
length scales has been hindered by the inherent limitations of fluorescent
dyes (i.e. lack of photostability, low brightness and short range of
interaction). Super resolution techniques such as STED [23] or PALM [2] have
recently gained momentum to directly observe fluorescent molecules spaced
closer than the diffraction limit. Although much work has been done to
increase the total acquisition time and frame rate, these methods are still
limited by the lack of photostability and the need to image a single
resolvable structure per diffraction limited spot at a time.
In the past, it has been shown that scattering microscopy using metallic
nanoparticles can complement its fluorescence sibling as it uses an
everlasting tag with no rate-limited amount of photons [10]. Metallic
nanoparticles are stable, biocompatible and easy to synthesize and conjugate
to biological targets and thereby ideal as contrast agents. A landmark example
of the biological application of such techniques was the direct observation of
receptors hopping across previously unknown membrane domains. This provided
valuable insight into the spatial regulation of signaling complexes and closed
a 30-year controversy about the diffusion coefficients of membrane proteins
[21]. While previous experiments using fluorescent tags yielded a diffusion
coefficient in biological membranes much slower than the one observed in
synthetic membranes, the fast acquisition speed (40 103 frames per second)
enabled by scattering microscopy showed that this is the result of a fast
diffusion and a slow hopping rate between domains [9].
In addition, the presence of a plasmon, i.e. a collective oscillation of the
free electrons within the nanoparticle, converts metallic nanoparticles into
very effective scatterers when illuminated at their resonance optical
frequency. The resulting strong electromagnetic enhancement in the vicinity of
the particle provides a near field effect that can be used to sense
information about their surroundings such as the effective index of refraction
or the presence of other scatterers [8, 16]. For example, it has been
experimentally shown that the shift in the plasmon resonance can be used to
determine the length of DNA molecules attached to a metallic nanoparticle
[13]. Moreover, the coupling between two nanoparticles in close proximity
produces an alteration of the plasmon spectra. This alteration has been used
as a nanometric ruler to determine the distance between them [20, 19]. In a
previous work, [4] we theoretically showed that the coupling between two
nanoparticles is highly sensitive to the polarization of the external field.
The scattering cross section ($C_{sca}$) is maximum when the incident
polarization is parallel to the dimer orientation due to the reinforcement of
the external field by the induced dipoles [Fig. 1(a)]. As the coupling
decreases monotonically with the distance between nanoparticles, so does the
average $C_{sca}$ over all polarizations ($v_{m})$ [Fig. 1(b)] and the
anisotropy [Fig. 1(c)] defined as:
$\displaystyle\eta=\frac{C_{sca}^{\parallel}-C_{sca}^{\perp}}{C_{sca}^{\parallel}+C_{sca}^{\perp}}.$
(1)
We have proposed, in our previous work, that by measuring the scattering cross
section as a function of the incident polarization angle, the axis of the
dimer and the distance between nanoparticles could be determined. In this
work, we provide experimental evidence supporting this concept by measuring
gold nanodimers on a glass surface and we introduce a novel total internal
reflection experimental setup that provides polarized illumination with a high
NA objective.
Figure 1: Conceptual idea of the technique. (a) Two metallic nanoparticles of
radii $a$ located at a distance $d$ are illuminated with a linearly polarized
electromagnetic field. (b) Theoretical results as a function of the
interparticle distance for the average $C_{sca}$ over all polarizations
($v_{m}$) and (c) the anisotropy. Parameters of the calculation: $a$ = 20 nm,
wavelength of the light: 532 nm.
## 2 Materials and methods
### 2.1 Sample preparation
Coverslips were cleaned by sonication at 50∘ for 20 minutes in Milli-Q water,
and then sequentially immersed for 5 seconds in HFL 5%, sodium bicarbonate and
acetone (analytic level). After the cleaning process, coverslips were dried
and stored in a chamber overpressurized with nitrogen until further use.
Before sample preparation, a Parafilm chamber was assembled on top of the
coverslip. To create a hydrophilic surface, bovine serum albumina (BSA) in
phosphate buffer solution (PBS) was incubated for 15 min and then rinsed with
PBS. Fluorescein-streptavidin in PBS (50 mg/ml) was then incubated for 30 min
and rinsed with PBS, to obtain an adsorbed layer that was verified using
confocal fluorescence microscopy. Finally, a solution of biotinylated gold
nanoparticles, nominal radius ($20\pm 5$) nm (GB-01-40. EY Laboratories, USA),
was incubated for 15 min and then rinsed by washing 5 times with PBS. The
concentration and incubation time where empirically chosen to provide a
concentration about 1 nanoparticle/10 $\mu$m2. As the particles are randomly
distributed, it is expected to find many monomers, some dimers, and very few
trimers and higher n-mers. A negative control sample was prepared in the same
way but omitting the incubation of gold nanoparticles.
### 2.2 Dual color scheme
Spurious reflections and scattering centers other than gold will produce
unwanted bright spots in the images. Even thresholding the image taken at the
resonance peak (532 nm) will result in many false positive regions. The
presence of a plasmon resonance in the scattering spectrum of gold
nanoparticles was used as a signature to distinguish them. The ratio between
the scattering cross section at 532 nm and 473 nm was found to be larger than
1.4 for gold monomers [Fig. 2(a), solid thin line] using Mie theory [3] and
even larger for dimers (dashed line) as calculated using GMMie, a
multiparticle extension of the Mie theory [11, 12]. In contrast, non metallic
scattering centers lack of a plasmon resonance and therefore yield a smaller
ratio between 532 nm and 473 nm $C_{sca}$ (solid thick line). Therefore, by
imaging at these two wavelengths and thresholding the ratio image above 1.4,
the pixels containing gold nanoparticles were further segmented.
Figure 2: Experimental Setup. (a) Comparison of spectra averaging over all
polarizations. While the spectrum of dielectric particles (thick solid line)
decreases monotonically, the spectra of metallic monomers and dimers (thin
lines) show a plasmon resonance. (b) The polarization of the refracted wave is
dependent on the direction of the incident polarization with respect to the
displacement in the back focal plane (BFP) of the objective which defines the
plane of incidence. (c) The sample is illuminated in Total Internal Reflection
and imaged using a cooled CCD. Laser light is tightly focused off-center in
the back focal plane of the objective and the angle $\beta$ is moved using a
pair of galvanometer scanners moving in orthogonal directions.
### 2.3 Polarization control in Total Internal Reflection
We used Total Internal Reflection (TIR) microscopy [1] to restrict the
illumination to the surface of the coverslip using an evanescent wave. In
objective-based TIR, the beam is focused off-axis in the back focal plane
(BFP) of the objective to achieve critical illumination. The components of
evanescent field are defined by the angle of incidence ($\theta$) and the
incident polarization with respect to the plane of incidence. Indeed, rotating
the excitation polarization before entering the microscope does not produce a
constant intensity in the sample plane as the transmission efficiency for the
parallel polarization [Fig. 2(b), left] will be much smaller than for the
perpendicular one [Fig. 2(b), center]. A circularly polarized beam before the
objective results in an “elliptically”111The electromagnetic field in the
sample cannot be said to be strictly elliptically polarized as an evanescent
field (not a propagating beam) is generated after the interface. Nevertheless,
an elliptical rotation of the electric field is achieved. polarized field
which has the minor axis in the plane of incidence [Fig. 2(b), right]. The
ratio between the major and minor axis of this evanescent “elliptical” beam
depends on $\theta$ and if the plane of incidence is changed, the ellipse will
rotate with it. We therefore modified a wide-field inverted microscope (IX71.
Olympus, Japan) using a TIRF objective (Olympus TIRFM 63X/1.45 PlanApo Oil) to
allow rotating the plane of incidence [Fig. 2(c)] by changing the position in
which the beam is focused in the BFP. Two lasers were used: one near the gold
particle plasmon resonance (532 nm, Compass C315M. Coherent Inc., USA) and
another shifted towards shorter wavelengths (473 nm, VA-I-N-473. Viasho
Technology, China). The power of the lasers after the objective was 13 $\mu$W.
Circularly polarized light was achieved at the BFP by inserting a quarter and
a half wave plate in the beam path adjusted to precompensate for the
polarization dependent transmission of the beam splitter, filters and mirrors.
The beam was expanded and filtered to achieve a diffraction limited spot in
the BFP. In order to displace the beam in the BFP and therefore change the
plane of incidence, a pair of computer controlled galvanometer scanners
(SC2000 controller, Minisax amplifier and M2 galvanometer. GSI Group, USA)
were used. The polarization distortion due to the change in the angle of
incidence onto the mirrors of the scanners (while moving) was verified to be
negligible. Images of the sample were acquired using a cooled monochrome CCD
camera (Alta U32. Apogee Instruments, USA. $2148\times 1472$ pixels each
$6.8\times 6.8$ $\mu$m2) through a dichroic filter for the fluorescence sample
(XF2009 550DCLP. Chroma Technology, USA) or a 30/70 beam splitter for the gold
nanoparticles (21009. Chroma Technology, USA).
### 2.4 Intrinsic anisotropy determination
To assure a constant ratio between the two polarizations of the beam and a
uniform intensity, the beam needed to be moved on the BFP in a circle centered
in the optical axis. Failure to do this would have reduced the dynamic range
of the system by introducing an intrinsic anisotropy. To minimize this value,
the path of the beam was iteratively modified while measuring the anisotropy
(see below) of a diluted solution of Rhodamine 101. The emission of such a
sample is independent of the excitation polarization and thus the measured
anisotropy can be assigned only to the system. After optimization, the
obtained anisotropy for the 532 and 473 channels was 0.06 and 0.05 in a region
of $50\times 50$ $\mu$m2 ($440\times 440$ pixel2). It is worth noting than
these values are five times smaller than the expected anisotropy for a 20 nm
homodimer.
### 2.5 Image acquisition and processing
The acquisition process consisted in sequentially imaging at 473 nm and 532 nm
while changing the angle $\beta$ in 20 discrete steps over $2\pi$ to sample
different polarizations. An image with both lasers off was also acquired to
account for ambient light and dark counts of the camera. Each image was
background corrected and normalized by the excitation power and detection
efficiency at the corresponding wavelength. Mean images ($m_{473}$ and
$m_{532}$) were obtained by averaging over all polarizations and, from these,
the ratio image $m=m_{532}/m_{473}$ was calculated.
The scattering image at the resonance peak ($m_{532}$) was segmented by Otsu’s
thresholding and masked with the ratio image thresholded above 1.4 to detect
pixels containing gold. A connected region analysis was performed to keep only
those regions with area between 3 and 15 pixels. The upper bound was chosen to
be slightly bigger than the airy diffraction limited spot for the system, but
still much smaller than the mean distance between gold nanoparticles. Regions
containing single gold nanoparticles should have a constant intensity over the
stack of frames acquired for different polarization orientation, while dimers
should provide an oscillating signal with period $\pi$. Therefore, a Fourier
analysis [Eq. (2)] was performed. For each pixel, the following coefficients
were calculated:
$\displaystyle\tilde{c}_{2}$
$\displaystyle=\frac{2}{\sum_{\beta}I(\beta)}\sum_{\beta}I(\beta)cos(2\beta)$
(2a) $\displaystyle\tilde{s}_{2}$
$\displaystyle=\frac{2}{\sum_{\beta}I(\beta)}\sum_{\beta}I(\beta)sin(2\beta)$
(2b) $\displaystyle\eta$
$\displaystyle=\sqrt{\tilde{c}_{n}^{2}+\tilde{s}_{n}^{2}}$ (2c) $\displaystyle
tan(\delta)$ $\displaystyle=\frac{\tilde{s}_{2}}{\tilde{c}_{2}}$ (2d)
This was done for each wavelength obtaining values for the anisotropy
($\eta_{473}$ and $\eta_{532}$) and orientation ($\delta_{473}$ and
$\delta_{532}$) in each pixel. The acquisition and analysis process were
repeated for 30 and 10 fields of view of the sample and negative control
sample respectively. Retrieval uncertainty was estimated by performing the
same numerical analysis on simulated data calculated by adding two terms to
the theoretical response for different dimers. The first, an oscillating term
with $2\pi$ periodicity, emulated a small misalignment that produced a non-
constant illumination while rotating the beam in the BFP. The amplitude of
this term was obtained from the Rhodamine calibration. The second term
simulated coherent background and its values for each pixel were drawn from a
Gaussian distribution obtained from the control images.
Figure 3: Experimental results (a) Representative images of a field of view
for single wavelength (left) and a ratio (right) imaging. Some points
(circles) are bright in both images (gold) while others squares faint in the
ratio image (not gold). Gold containing regions were segmented by finding
bright pixels in both images. Images size: 50x50 ${\mu m}^{2}$. 440x440
pixels. (b) Anisotropy vs. mean value. A strong correlation is observed
between anisotropy and mean value for both wavelengths as expected. The 473 nm
data shows a plateau due to the intrinsic anisotropy of the system.
## 3 Results and discussion
The need for a two color approach is evident when comparing single wavelength
with ratio images: while regions with and without nanoparticles [Fig. 3(a),
circles and squares respectively] were bright due to the high background in
the single channel image, only gold nanoparticles were above the threshold in
the ratio image. Importantly, in the negative control stack, no pixel was
found above the a priori defined threshold. For the 35 regions identified as
gold monomers/dimer, a strong correlation between anisotropy and mean value
was found as expected [Fig. 3(b)]. The variance over each region for all
values was below the corresponding retrieval uncertainty. A plateau was
observed for the 473 nm channel due to the intrinsic anisotropy of the system.
In this set of candidates for dimers, eight points presented an unexpected
high individual anisotropy and hence were rejected. Although the exact origin
of this eight scattering centers could not be established, it is worthwhile
noting that it is extremely relevant in a tracking experiment to avoid false
positives that would severely distort the retrieved information, and this
ability to reject scattering centers based on their response is an additional
advantage of the technique.
Figure 4: Comparison between experimental data and homodimer model. The mean
value ratio is plotted against the anisotropy ratio for the regions segmented
from the images (blue dots). Theoretical calculations for different homodimers
are also plotted. The vertical lines show the results keeping constant the
surface to surface distance ($d_{ss}$) while changing the radii ($a$). The
opposite is shown in the horizontal lines. Remarkably, experimental data
distributed close to the curve for 20 nm homodimers (solid line) as expected
as this is in fact the mean radii of the particles used. Notice that this is
not a fit (no free parameters), but the predictions from the homodimer model
superimposed to the experimental data.
The recovered scattering parameters were compared with the expected results
for a homodimer configuration (Fig. 4) obtaining a good correspondence with
the nominal size of the nanoparticles used (20 nm). Indeed, the mesh shown in
Fig. 4 was calculated using only the photophysical and geometrical properties
of the dimer (no fitted parameters). The ability of the technique to blindly
recover the correct size of the particles was a cross-check for its
reliability.
The actual configuration of each dimer was obtained by fitting the theoretical
model to the experimental values. The in-plane orientation was directly
obtained as a weighted average of these values. To fit the radii of each
particle and the distance between them, the values of $\eta_{473}$,
$\eta_{532}$ and $m$ were used. The values were first fitted using the
analytical solution of a homodimer configuration in the dipole-dipole
approach, in which the induced dipole moment $\vec{p}$ of each particle in an
incident field $\vec{E}_{inc}$ can be expressed as:
$\displaystyle\vec{p}_{\parallel}=\frac{1}{1-\frac{\alpha}{2d^{3}\pi}}\epsilon_{m}\alpha\vec{E}_{inc}$
(3a)
$\displaystyle\vec{p}_{\perp}=\frac{1}{2+\frac{\alpha}{2d^{3}\pi}}\epsilon_{m}\alpha\vec{E}_{inc}$
(3b)
$\epsilon_{m}$ being the dielectric constant of the medium and $\alpha$ the
polarizability of the sphere which is proportional to the cube of its radius
[4, 3]. This homodimer configuration was used as an initial value in the time
consuming iterative process of finding a heterodimer configuration compatible
with the experimental data using GMMie calculation. The traveling wave
approximation of the evanescent field was used as the particles are small
compared to the decay length of the field [18, 22] and the collection
efficiency is much smaller for the dipole induced in the optical axis
orientation than for the one perpendicular to it.
In this way, the two radii, orientation and distance for each dimer were
obtained (Fig. 5). The distribution of radii was found to be centered in 20
nm, compatible with the nominal size of the particles. For the interparticle
distances, the distribution showed an increase as expected but then a decrease
for distances at which the anisotropy is close to the intrinsic anisotropy of
the system. This mismatch at larger distances is due to the conservative
criterion to separate dimers from monomers that fails to identify correctly
nanoparticles that couple weakly. The dimer orientation was uniformly
distributed between $-\pi$ and $\pi$, as expected. To further test this, we
compared the experimental and simulated distributions using a Kolmogorov-
Smirnov [15] statistical test. The level of significance set at the usual
value of 5%. The expected distributions (Fig. 5, right column) were obtained
from the nominal radius of the nanoparticles and Monte Carlo simulation of the
adsorption process. The experimental and simulated distributions for radii and
orientation were found in close agreement. For the interparticle distance, the
distributions were found compatible when compared up to 70 nm.
Figure 5: Fit to a heterodimer configuration using GMMie. For each dimer
(left column, x axis), the radii (top), the distance (middle) and the angle
(bottom) were obtained. Experimental and simulated histograms are shown for
each magnitude (middle and right columns).
## 4 Conclusions
We have experimentally shown that distance between two nanoparticles, as well
as their individual radii, can be obtained by measuring the intensity of spot
as a function of the incident polarization. Additionally, the in-plane
orientation of the dimer was obtained with less than 10∘ uncertainty. The
presented method strongly exploits the particular spectroscopic properties of
metallic nanoparticles to sense their environment. It is worth noting that the
distance in which the technique is sensitive scales with the radii of the used
particles. By using nanoparticles with radius between 4 and 20 nm, the gap
between FRET and standard super-resolution techniques (10 nm to 50 nm) could
be bridged. This fact, together with the ability to recover orientation, makes
this approach unique.
A non-uniform anisotropic illumination is the major source of uncertainty as
it will mask the anisotropy of the dimers, specially for those in which the
distance is much larger than the radius of the particles. This should be
properly controlled by measuring an isotropic sample as it was done in this
work. Additionally, it is important to mention that various factors such as a
non-monodisperse or non-spherical population of particles will have an
incidence in the recovery of dimer distance, size and orientation from model
based fittings. However, having a multiparametric readout (i.e. $m_{532}$,
$m_{473}$, $\eta_{473}$ and $\eta_{532}$) with a non-trivial dependence of the
physical parameters (i.e. distance, size, shape) provides a way to control for
this and exclude points that do no match the expected relations between
photophysical properties. Such conservative criterion would be recommended for
tracking experiments where false negatives have minor impact as they only
reduce the amount of information gathered per frame. If the yield of dimers
can be raised and several dozens of dimers can be imaged in the same field of
view, we expect that the presented technique will be useful to add information
about the relative movement of the two particles to already existing tracking
assays. Numerical simulations showed that if coherent background can be
diminished, an order of magnitude (i.e. by the use of broader band excitation
source), distance and orientation could be tracked at 100 Hz. If just the
rotation and the movement of the center of mass is desired, the retrieval can
be performed much faster as only one wavelength (532 nm) would be necessary
after an initial identification of the dimers is made by the two color method.
As other scattering based techniques, the lack of photobleaching constitutes a
major advantage of this approach. Moreover, the absence of saturation in the
light scattering of metallic nanoparticles, as compared to the absorption of
fluorescent molecules, provides an acquisition rate only limited by the
detector speed. The combination of these two aspects means that scattering
based microscopy does not need to make compromises between experiment length
and temporal resolution.
We have also demonstrated that the use of two color imaging can provide an
efficient way to detect scattering centers that have plasmon resonances.
Recent work by Olk et al. [17] has shown that upon illumination with a
wideband light source, a modulation of the spectra due to far-field effects
can be observed as a function of the incident polarization. The combination of
the two techniques could lead to a more robust detection of both, orientation
and distance.
Finally, the novel illumination setup introduced in this work provides a
robust way to change the polarization in TIR, allowing the implementation of
anisotropy based techniques in fluorescence and scattering microscopy.
Additionally, the same scheme permits a fast switching between TIR and
standard wide-field as well as sweeping of multiple evanescent field
penetration depths.
###### Acknowledgements.
HEG was funded by the Universidad de Buenos Aires.
## References
* [1] D Axelrod, T P Burghardt, N L Thompson, Total internal reflection fluorescence, Annu. Rev. Biophys. Bio. 13, 247 (1984).
* [2] E Betzig, G H Patterson, R Sougrat, O W Lindwasser, S Olenych, J S Bonifacino, M W Davidson, J Lippincott-Schwartz, H F Hess, Imaging intracellular fluorescent proteins at nanometer resolution, Science 313, 1642 (2006).
* [3] C F Bohren, D R Huffman, Absorption and scattering of light by small particles, Whiley (1983).
* [4] H E Grecco, O E Martínez, Distance and orientation measurement in the nanometric scale based on polarization anisotropy of metallic dimers, Opt. Express 14, 8716 (2006).
* [5] F G Haj, P J Verveer, A Squire, B G Neel, P I H Bastiaens, Imaging sites of receptor dephosphorylation by PTP1B on the surface of the endoplasmic reticulum, Science 295, 1708 (2002).
* [6] E A Jares-Erijman, T M Jovin, FRET imaging, Nat. Biotechnol. 21, 1387 (2003).
* [7] Z Kam, T Volberg, B Geiger, Mapping of adherens junction components using microscopic resonance energy transfer imaging, J. Cell Sci. 108, 1051 (1995).
* [8] K L Kelly, E Coronado, L L Zhao, G C Schatz, The optical properties of metal nanoparticles: The influence of size, shape, and dielectric environment, J. Phys. Chem. B 107, 668 (2003).
* [9] A Kusumi, C Nakada, K Ritchie, K Murase, K Suzuki, H Murakoshi, R S Kasai, J Kondo, T Fujiwara, Paradigm shift of the plasma membrane concept from the two-dimensional continuum fluid to the partitioned fluid: High-speed single-molecule tracking of membrane molecules, Annu. Rev. Bioph. Biom. 34, 351 (2005).
* [10] D Lasne, G Blab, S Berciaud, M Heine, L Groc, D Choquet, L Cognet, B Lounis, Single nanoparticle photothermal tracking (SNaPT) of 5-nm gold beads in live cells, Biophys. J. 91, 4598 (2006).
* [11] Y lin Xu, Electromagnetic scattering by an aggregate of spheres, Appl. Optics 34, 4573 (1995).
* [12] Y lin Xu, Electromagnetic scattering by an aggregate of spheres: Far field, Appl. Optics 36, 9496 (1997).
* [13] G L Liu, et al., A nanoplasmonic molecular ruler for measuring nuclease activity and dna footprinting, Nat. Nanotechnol. 1, 47 (2006).
* [14] N P Mahajan, K Linder, G Berry, G W Gordon, R Heim, B Herman, Bcl-2 and bax interactions in mitochondria probed with green fluorescent protein and fluorescence resonance energy transfer, Nat. Biotechnol. 16, 547 (1998).
* [15] F J Massey, The Kolmogorov–Smirnov test for goodness of fit, J. Am. Stat. Assoc. 46, 68 (1951).
* [16] A D McFarland, R P V Duyne, Single silver nanoparticles as real-time optical sensors with zeptomole sensitivity, Nano Lett. 3, 1057 (2003).
* [17] P Olk, J Renger, M T Wenzel, L M Eng, Distance dependent spectral tuning of two coupled metal nanoparticles, Nano Lett. 8, 1174 (2008).
* [18] M Quinten, A Pack, R Wannemacher, Scattering and extinction of evanescent waves by small particles, Appl. Phys. B: Lasers O. 68, 87 (1999).
* [19] B M Reinhard, M Siu, H Agarwal, A P Alivisatos, J Liphardt, Calibration of dynamic molecular rulers based on plasmon coupling between gold nanoparticles, Nano Lett. 5, 2246 (2005).
* [20] C Sönnichsen, B M Reinhard, J Liphardt, A P Alivisatos, A molecular ruler based on plasmon coupling of single gold and silver nanoparticles, Nature Biotech. 23, 741 (2005).
* [21] K Suzuki, K Ritchie, E Kajikawa, T Fujiwara, A Kusumi, Rapid hop diffusion of a g-protein-coupled receptor in the plasma membrane as revealed by single-molecule techniques, Biophys. J. 88, 3659 (2005).
* [22] R Wannemacher, A Pack, M Quinten, Resonant absorption and scattering in evanescent fields, Appl. Phys. B: Lasers O. 68, 225 (1999).
* [23] V Westphal, S O Rizzoli, M A Lauterbach, D Kamin, R Jahn, S W Hell, Video-rate far-field optical nanoscopy dissects synaptic vesicle movement, Science 320, 246 (2008).
|
arxiv-papers
| 2011-01-11T14:07:56 |
2024-09-04T02:49:16.323120
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. E. Grecco, O. E. Mart\\'inez",
"submitter": "Luis Ariel Pugnaloni",
"url": "https://arxiv.org/abs/1101.2130"
}
|
1101.2181
|
# CLASSIFICATION OF 5-DIMENSIONAL MD-ALGEBRAS HAVING NON-COMMUTATIVE DERIVED
IDEALS
Le Anh ${\bf Vu}^{*}$, Ha Van ${\bf Hieu}^{**}$ and Tran Thi Hieu ${\bf
Nghia}^{***}$
∗Department of Mathematics and Economic Statistics, University of Economics
and Law
Vietnam National University - Ho Chi Minh City, Viet nam
E-mail: vula@uel.edu.vn
∗∗ E-mail: havanhieu88@gmail.com
∗∗∗ E-mail: hieunghiatoan1a@gmail.com
###### Abstract
The paper presents a subclass of the class of MD5-algebras and MD5-groups,
i.e. five dimensional solvable Lie algebras and Lie groups such that their
orbits in the co-adjoint representation (K-orbits) are orbits of zero or
maximal dimension. The main result of the paper is the classification up to an
isomorphism of all MD5-algebras having non-commutative derived ideals.
AMS Mathematics Subject Classification: Primary 22E45, Secondary 46E25, 20C20.
Key words: Lie group, Lie algebra, MD5-group, MD5-algebra, K-orbits.
### INTRODUCTION
In 1962, studying theory of representations, A. A. Kirillov introduced the
Orbit Method (see [2]). This method quickly became the most important method
in the theory of representations of Lie groups. Using the Kirillov’s Orbit
Method, we can obtain all the unitary irreducible representations of solvable
and simply connected Lie Groups. The importance of Kirillov’s Orbit Method is
the co-adjoint representation (K-representation). Therefore, it is meaningful
to study the K-representation in the theory of representations of Lie groups.
After studying the Kirillov’s Orbit Method, Do Ngoc Diep in 1980 suggested to
consider the class of Lie groups and Lie Algebras MD such that the
$C^{*}-algebras$ of them can be described by using KK-functors (see [1]). Let
G be an n-dimensional real Lie group. G is called an MDn-group if and only if
its orbits in the K-representation (i.e. K-orbits) are orbits of dimension
zero or maximal dimension. The corresponding Lie algebra of G is called an
MDn-algebra. Thus, classification and studying of K-representation of the
class of MDn-groups and MDn-algebras is the problem of interest. Because all
Lie algebras of n dimension (with $n\leq 3$) were listed easily, we have to
consider MDn-groups and MDn-algebras with $n\geq 4$. In 1990, all MD4-algebras
were classified up to an isomorphism by Vu - the first author (see [5]).
Recently, Vu and some his colleagues have continued studying MD5-groups and
MD5-algebras having commutative derived ideals (see [6], [7], [8]). In 2008, a
classification of all MD5-algebras having commutative derived ideals was given
by Vu and Kar Ping Shum (see [9]).
In this paper, we shall give the classification up to an isomorphism of all
MD5-algebras $\mathcal{G}$ whose derived ideals
${\mathcal{G}}^{1}:=[\mathcal{G},\mathcal{G}]$ are non-commutative. This
classification is the main result of the paper.
The paper is organized as follows: The first section deals with some
preliminary notions, section 2 is devoted to the discussion of some results on
MDn-algebras, in particular, the main result of the paper is given in this
section.
## 1 PRELIMINARIES
We first recall in this section some preliminary results and notations which
will be used later. For details we refer the reader to the book [2] of A. A.
Kirillov and the book [1] of Do Ngoc Diep.
### 1.1 The K-representation and K-orbits
Let G be a Lie group, $\cal G$ =Lie(G) be the corresponding Lie algebra of G
and ${\cal G}^{*}$ be the dual space of $\cal G$. For every $g\in G$, we
denote the internal automorphism associated with g by $A_{(g)}$, and whence,
$A_{(g)}:G\to G$ can be defined as follows $A_{(g)}:=g.x.g^{-1},\forall x\in
G$. This automorphism induces the following map
${A_{\left(g\right)}}_{*}:{\cal G}\to{\cal G}$ which is defined as follows
${A_{\left(g\right)}}_{*}\left(X\right):=\frac{d}{{dt}}\left[{g.\exp\left({tX}\right).g^{-1}}\right]\left|{{}_{t=0}}\right.$,
$\forall X\in{\cal G}.$
This map is called the tangent map of $A_{(g)}$. We now formulate the
definitions of K-representation and K-orbit.
###### Definition 1.1.1.
The action
$K:G\longrightarrow Aut(\mathcal{G}^{*})$ $g\longmapsto K_{(g)}$
such that
$\left\langle{K_{\left(g\right)}(F),X}\right\rangle:=\left\langle{F,{A_{\left(g^{-1}\right)}}_{*}\left(X\right)}\right\rangle,\forall
F\in{\cal G}^{*},\,\forall X\in{\cal G}$
is called the co-adjoint representation or K-representation of G in
$\mathcal{{G}^{*}}$.
###### Definition 1.1.2.
Each orbit of the co-adjoint representation of G is called a K-orbit of G.
We denote the K-orbit containing F by $\Omega_{F}$. Thus, for every $F\in{\cal
G}^{*}$, we have $\Omega_{F}:=\left\\{{K\left(g\right)(F)|g\in G}\right\\}$.
The dimension of every K-orbit of an arbitrary Lie group G is always even. In
order to define the dimension of the K-orbits $\Omega_{F}$ for each F from the
dual space ${\cal G}^{*}$ of the Lie algebra $\cal G$ = Lie(G), it is useful
to consider the following skew-symmetric bilinear form $B_{F}$ on $\cal G$:
$B_{F}\left({X,Y}\right)=\left\langle{F,\left[{X,Y}\right]}\right\rangle,\forall
X,Y\in{\cal G}$. We denote the stabilizer of F under the co-adjoint
representation of G in ${\cal G}^{*}$ by $G_{F}$ and ${\cal G}_{F}:=$
Lie($G_{F}$).
We shall need in the sequel of the following result.
###### Proposition 1.1.3 (see [2, Section 15.1]).
$KerB_{F}={\cal{G}}_{F}$ and
$dim{\Omega}_{F}=dim{\mathcal{G}}-dim{\mathcal{G}}_{F}=rankB_{F}.$ $\square$
### 1.2 MD-groups and MD-algebras
###### Definition 1.2.1 (see [1, Chapter 2]).
An MD-group is a real solvable Lie group of finite dimension such that its
K-orbits are orbits of dimension zero or maximal dimension (i.e. dimension k,
where k is some even constant and no more than the dimension of the considered
group). When the dimension of considered group is n (n is a some positive
integer), the group is called an MDn-group. The Lie algebra of an MD-group
(MDn-group, respectively) is called an MD-algebra (MDn-algebra, respectively).
The following proposition gives a necessary condition for a Lie algebra
belonging to the class of MD-algebras.
###### Proposition 1.2.2 (see [3, Theorem 4]).
Let $\mathcal{G}$ be an MD-algebra. Then its second derived ideal
${\mathcal{G}}^{2}:=[[\mathcal{G},\mathcal{G}],[\mathcal{G},\mathcal{G}]]$ is
commutative. $\square$
We point out here that the converse of the above result is in general not
true. In other words, the above necessary condition is not a sufficient
condition. So, we now only consider the real solvable Lie algebras having
commutative second derived ideals. Thus, they could be MD-algebras.
###### Proposition 1.2.3 (see [1, Chapter 2, Proposition 2.1]).
Let $\cal{G}$ be an MD-algebra with F (in ${\cal{G}}^{*}$) is not vanishing
perfectly in ${\cal{G}}^{1}:=[\mathcal{G},\mathcal{G}]$, i.e. there exists
$U\in{\cal{G}}^{1}$ such that $\langle F,U\rangle\neq 0.$ Then the K-orbit
${\Omega}_{F}$ is one of the K-orbits having maximal dimension. $\square$
## 2 THE CLASS OF MD5-ALGEBRAS
HAVING NON-COMMUTATIVE
DERIVED IDEALS
### 2.1 Some Results on the Class of MD-algebras
In this subsection, we shall present some results on general MDn-algebras
($n\geq 4$).
Firstly, we consider a real solvable Lie algebra $\cal{G}$ of dimension $n$
such that $dim{\cal G}^{1}=n-k$ ($k$ is some integer constant, $1\leq k<n$),
${\cal G}^{2}$ is non - trivial commutative and $dim{\cal G}^{2}=dim{\cal
G}^{1}-1=n-k-1$. Without loss of generality, we may assume that
${\cal G}\,\,\,=gen\left(X_{1},X_{2},...,X_{n}\right),\,(n\geq 4)$,
${{\cal G}^{1}}=gen\left(X_{k+1},X_{k+2},...,X_{n}\right),\,(n>k\geq 1)$,
${{\cal G}^{2}}=gen\left(X_{k+2},...,X_{n}\right)$,
with the Lie brackets are given by
$\left[X_{i},X_{j}\right]=\sum\limits_{l=k+1}^{n}{C_{ij}^{l}{X_{l}}},\,1\leq
i<j\leq n,$
where $C_{ij}^{l}\left({1\leq i<j\leq n},k+1\leq l\leq n\right)$ are
constructional constants of $\cal{G}$.
###### Theorem 2.1.1.
There is no MD-algebra $\cal G$ such that its second derived ideal ${\cal
G}^{2}$ is not trivial and less than its first derived ideal ${\cal G}^{1}$ by
one dimension: $dim{\cal G}^{2}=dim{\cal G}^{1}-1.$
In order to prove this theorem, we need some lemmas.
###### Lemma 2.1.2.
The operator $ad_{X_{k+1}}$ restricted on ${\cal G}^{2}$ is an automorphism.
###### Proof.
Since ${\cal G}^{2}$ is commutative,
$\left[{{X_{i}},{X_{j}}}\right]=0,\,\,\forall i,j\geq k+2$. Hence,
$\begin{array}[]{l}\quad\,gen\left({X_{k+2}},{X_{k+3}},\cdots,{X_{n}}\right)={{\cal
G}^{2}}=\left[{{{\cal G}^{1}},{{\cal G}^{1}}}\right]\\\
=gen\left(\left[X_{k+1},X_{k+2}\right],\left[X_{k+1},X_{k+3}\right],\cdots,\left[X_{k+1},X_{n}\right]\right)\\\
=gen\left(ad_{X_{k+1}}\left(X_{k+2}\right),\cdots,ad_{X_{k+1}}\left(X_{n}\right)\right).\end{array}$
It follows that $ad_{X_{k+1}}$ restricted on ${\cal G}^{2}$ is automorphic. ∎
###### Lemma 2.1.3.
Without any restriction of generality, we can always suppose right from the
start that $\left[X_{i},X_{k+1}\right]=0$ for all indices $i$ such that $1\leq
i\leq k$.
###### Proof.
Firstly, we remark that $\left[X_{1},X_{k+1}\right]\in{\cal G}^{1}$, so there
exists $X\in{\cal G}^{2}$ such that
$\left[X_{1},X_{k+1}\right]=C_{1,k+1}^{k+1}X_{k+1}+X$. Since ${ad_{X_{k+1}}}$
restricted on ${\cal G}^{2}$ is automorphic, there exists $Y\in{\cal G}^{2}$
such that $ad_{X_{k+1}}\left(Y\right)=X$.
By changing $X^{\prime}_{1}=X_{1}+Y$, we get
$\left[X^{\prime}_{1},X_{k+1}\right]=C_{1,k+1}^{k+1}{X_{k+1}}$. Using the
Jacobi identity for $X^{\prime}_{1},X_{k+1}$, and an arbitrary element
$Z\in{\cal G}^{2}$ we obtain
$ad_{X_{1}}ad_{X_{k+1}}-ad_{X_{k+1}}ad_{X_{1}}=\alpha ad_{X_{k+1}}$, where
$\alpha$ is some real constant. Since $ad_{X_{k+1}}$ is automorphic on ${\cal
G}^{2}$, $\alpha$ must be zero. Therefore, $C_{1,k+1}^{k+1}=0$, i.e.
$\left[X^{\prime}_{1},X_{k+1}\right]=0$. So, we can suppose that
$\left[X_{1},X_{k+1}\right]=0$.
By the same way, we can suppose
$\left[X_{2},X_{k+1}\right]=\cdots=\left[X_{k},X_{k+1}\right]=0.$ ∎
###### Lemma 2.1.4.
$\left[X_{i},X_{j}\right]=C_{ij}^{k+1}X_{k+1}$ for all pairs of indices i, j
such that $1\leq i<j\leq k$.
###### Proof.
Consider an arbitrary pair of indices $i,j$ such that $1\leq i<j\leq k$. Note
that
$\left[X_{i},X_{j}\right]=\sum\limits_{l=k+1}^{n}{C_{ij}^{l}{X_{l}}}=C_{ij}^{k+1}X_{k+1}+\sum\limits_{l=k+2}^{n}{C_{ij}^{l}{X_{l}}}$.
By using the Jacobi identity, we have
$\left[X_{i},\left[X_{j},X_{k+1}\right]\right]+\left[X_{j},\left[X_{k+1},X_{i}\right]\right]+\left[X_{k+1},\left[X_{i},X_{j}\right]\right]=0$
$\Longrightarrow\left[X_{k+1},\left[X_{i},X_{j}\right]\right]=\left[X_{k+1},\,C_{ij}^{k+1}X_{k+1}+\sum\limits_{l=k+2}^{n}{C_{ij}^{l}{X_{l}}}\right]=0$
$\Longrightarrow
ad_{X_{k+1}}{\left(\sum\limits_{l=k+2}^{n}{C_{ij}^{l}{X_{l}}}\right)}=0$
$\Longrightarrow\sum\limits_{l=k+2}^{n}{C_{ij}^{l}{X_{l}}}=0$ (because
$ad_{X_{k+1}}$ is automorphic on ${\cal{G}}^{2}$)
$\Longrightarrow\left[{{X_{i}},{X_{j}}}\right]=C_{ij}^{k+1}{X_{k+1}};1\leq
i<j\leq k$.
∎
We now prove Theorem 2.1.1. Namely, we will prove that if $\cal G$ is a real
solvable Lie algebra such that ${\cal G}^{2}$ is non - trivial commutative and
$dim{\cal G}^{2}=dim{\cal G}^{1}-1=n-k-1,\,1\leq k<n$, then $\cal G$ is not an
MD-algebra.
###### Proof of Theorem 2.1.1.
According to above lemmas, we can choose a suitable basis
$\left(X_{1},X_{2},\cdots,X_{n}\right)$ of $\cal G$ which satisfies the
following conditions:
$\left[X_{i},X_{j}\right]=C_{ij}^{k+1}X_{k+1},\,\,1\leq i<j\leq k$;
$\left[X_{i},X_{k+1}\right]=0,\,\,1\leq i\leq k$;
$\left[X_{i},X_{j}\right]=\sum\limits_{l=k+2}^{n}{C_{ij}^{l}{X_{l}}},\,\,1\leq
i\leq k+1,\,k+2\leq j\leq n$.
Moreover, the constructional constants $C_{ij}^{k+1}$ can not concomitantly
vanish and the matrix $A={\left(C_{j,k+1}^{l}\right)}_{k+2\leq j,l\leq n}$ is
invertible because $ad_{X_{k+1}}$ restricted on ${\cal{G}}^{2}$ is
automorphic.
Since A is invertible, there exist
$\alpha_{k+2},\cdots,\alpha_{n}\in{\mathbb{R}}$, which are not concomitantly
vanished, such that
$A\left[\begin{array}[]{l}\alpha_{k+2}\\\ \,\,\,\,\vdots\\\
\alpha_{n}\end{array}\right]=\left[\begin{array}[]{l}1\\\ 0\\\ \vdots\\\
0\end{array}\right]\in{\mathbb{R}}^{n-k-1}.$
Let $\left(X_{1}^{*},X_{2}^{*},\cdots,X_{n}^{*}\right)$ is the dual basis in
${\cal G}^{*}$ of $\left(X_{1},X_{2},\cdots,X_{n}\right)$. We choose
${F_{1}}=X_{k+1}^{*}$ and
${F_{2}}=X_{k+1}^{*}+\alpha_{k+2}X_{k+2}^{*}+\cdots+{\alpha_{n}}X_{n}^{*}$ in
${\cal G}^{*}$ . It can easily be seen that $F_{1},F_{2}$ are not perfectly
vanishing in ${\mathcal{G}}^{1}$. In the view of Proposition 1.2.3, if $\cal
G$ is an MD-algebra then $\Omega_{F_{1}},\Omega_{F_{2}}$ are orbits of maximal
dimension, in particular we have
$rankB_{F_{2}}=dim\Omega_{F_{1}}=dim\Omega_{F_{2}}=rankB_{F_{2}}.$
But it is easy to verify that $rankB_{F_{2}}\geq rankB_{F_{1}}+2$. This
contradiction proves that $\cal G$ is not an MD-algebra and the proof of
Theorem 2.1.1 is therefore complete. $\square$
Now we consider an arbitrary real solvable Lie algebra $\cal{G}$ of dimension
$n$ ($n\geq 5$) such that $dim{\cal G}^{1}=n-1$. It is obvious that we can
choose one basis $\left(X_{1},X_{2},\cdots,X_{n}\right)$ of $\cal G$ such that
${\cal G}^{1}=gen\left(X_{2},X_{3},\cdots,X_{n}\right)$, $\,{\cal
G}^{2}\subset gen\left(X_{3},\cdots,X_{n}\right)$ and ${\cal G}^{2}$ is
commutative. Let $C_{ij}^{l}\left({1\leq i<j\leq n},2\leq l\leq n\right)$ are
constructional constants of $\cal{G}$. Then the Lie brackets are given by the
following formulas
$\left[{{X_{i}},{X_{j}}}\right]=\sum\limits_{l=2}^{n}{C_{ij}^{l}{X_{l}}}\left({1\leq
i<j\leq n}\right).$
###### Theorem 2.1.5.
Let ${\cal G}$ be a real solvable Lie algebra of dimension $n$ such that its
first derived ideal ${\cal G}^{1}$ is $(n-1)$-dimensional ($n\geq 5$) and its
second derived ideal ${\cal G}^{2}$ is commutative. Then ${\cal G}$ is MDn-
algebra if and only if $\,{\cal G}^{1}$ is commutative.
In order to prove this theorem, once again, we also need some lemmas.
###### Lemma 2.1.6.
If $\,\cal G$ is an MD-algebra of dimension $n\,$ ($n\geq 5$) such that
$\,\dim{\cal G}^{1}=n-1$ then $\dim{\Omega_{F}}\in\\{0,2\\}\,$ for every
$F\in{\cal G}^{*}$.
###### Proof.
Let $ad_{X_{1}}=\left(a_{ij}\right)_{n-1}\in End\left({\cal G}^{1}\right)$.
With $F_{0}=X_{2}^{*}\in{\cal G}^{*}$, the matrix of the bilinear form
$B_{F_{0}}$ in the chosen basis as follows
$B_{F_{0}}=\left[\begin{array}[]{l}0\,\,\,\,\,\,\,-{a_{12}}\,\,\,\,\,\,-{a_{13}}\,\,\,....\,\,-{a_{1n}}\\\
{a_{12}}\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,....\,\,\,\,\,\,\,\,0\\\
{a_{13}}\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,....\,\,\,\,\,\,\,\,0\\\
....\,\,\,\,\,\,\,\,\,\,\,....\,\,\,\,\,\,\,\,\,\,\,....\,\,\,\,\,\,\,\,....\,\,\,\,\,\,....\\\
{a_{1n}}\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,....\,\,\,\,\,\,\,\,0\end{array}\right].$
It is plain that $rank\,B_{F_{0}}=2$. Since ${\cal G}$ is an MD-algebra, we
get $\dim{\Omega_{F}}\in\\{0,2\\}\,$ for every $F\in{\cal G}^{*}$. ∎
###### Lemma 2.1.7.
Suppose that ${\cal G}=gen\left(X_{1},X_{2},\cdots,X_{n}\right)$ is a real
solvable Lie algebra of dimension n such that
${\cal G}^{1}=gen\left(X_{2},X_{3},\cdots,X_{n}\right)$ and $\,{\cal
G}^{2}=gen\left(X_{k+1},\cdots,X_{n}\right)$, $k>1$.
Let
$A=\left({\begin{array}[]{*{20}{c}}{C_{12}^{2}}&\ldots&{C_{1k}^{2}}\\\
\vdots&\ddots&\vdots\\\ {C_{12}^{k}}&\cdots&{C_{1k}^{k}}\end{array}}\right)$
be the matrix established by the constructional constants ${C_{1j}^{l}}$
$\left(2\leq j,l\leq k\right)$ of $\cal G$. Then A is invertible.
###### Proof.
Since ${\cal G}^{1}=\left[\cal G,\cal G\right]$, there exist real numbers
$\alpha_{ij},\,1\leq i<j\leq n$, such that
${X_{2}}=\sum\limits_{1\leq i<j\leq
n}{\alpha_{ij}}\left[{{X_{i}},{X_{j}}}\right]$
$=\sum\limits_{j=2}^{k}{{\alpha_{1j}}\left[{{X_{1}},{X_{j}}}\right]}+\sum\limits_{j=k+1}^{n}{{\alpha_{1j}}\left[{{X_{1}},{X_{j}}}\right]}+\sum\limits_{2\leq
i<j\leq n}{{\alpha_{{ij}}}\left[{{X_{i}},{X_{j}}}\right]}$
$=\sum\limits_{j=2}^{k}{{\alpha_{1j}}\left[{{X_{1}},{X_{j}}}\right]+LC_{1}\left({{{\cal
G}^{2}}}\right)}$
$=\sum\limits_{j=2}^{k}{{\alpha_{1j}}{\left(\sum\limits_{l=2}^{n}{C_{1j}^{l}{X_{l}}}\right)}+LC_{1}\left({\cal
G}^{2}\right)}$
$=\sum\limits_{j=2}^{k}{{\alpha_{1j}}{\left(\sum\limits_{l=2}^{k}{C_{1j}^{l}{X_{l}}}+\sum\limits_{l=k+1}^{n}{C_{1j}^{l}{X_{l}}}\right)}+LC_{1}\left({\cal
G}^{2}\right)}$
$=\sum\limits_{l=2}^{k}{\sum\limits_{j=2}^{k}{C_{1j}^{l}{\alpha_{1j}}X_{l}}+LC_{2}\left({\cal
G}^{2}\right)},$
where $LC_{1}\left({\cal G}^{2}\right),\,LC_{2}\left({\cal G}^{2}\right)$ are
linear combinations of some definite vectors from the chosen basis of ${\cal
G}^{2}.$ This implies that there exists columnar vector
${Y_{2}}\in{{\mathbb{R}}^{k-1}}$ such that
$AY_{2}=\left[\begin{array}[]{l}1\\\ 0\\\ \vdots\\\
0\end{array}\right]\in{\mathbb{R}}^{k-1}.$
Similarly, there exist columnar vectors
$Y_{3},\cdots,{Y_{k}}\in{{\mathbb{R}}^{k-1}}$ such that
$AY_{3}=\left[\begin{array}[]{l}0\\\ 1\\\ 0\\\ \vdots\\\
0\end{array}\right],\,\cdots,\,AY_{k}=\left[\begin{array}[]{l}0\\\ 0\\\
\vdots\\\ 0\\\ 1\end{array}\right]\in{\mathbb{R}}^{k-1}.$
Thus, there is a real matrix $P$ such that $A.P=I$, where $I$ is the identity
$(k-1)$-matrix. So $A$ is invertible and Lemma 2.1.7 is proved completely. ∎
### Proof of Theorem 2.1.5
Firstly, we shall prove that, if ${\cal
G}=gen\left(X_{1},X_{2},\cdots,X_{n}\right)\,\left(n\geq 5\right)$ such that
${\cal G}^{1}=gen\left(X_{2},X_{3},\cdots,X_{n}\right)$ is non-commutative,
then ${\cal G}$ is not an MD-algebra. Let ${{\cal
G}^{2}}=gen\left(X_{k+1},\cdots,X_{n}\right),\,2\leq k<n$.
We need consider some cases which contradict each other as follows.
* 1.
$k=2$. Then, $\dim{\cal G}^{2}=\dim{\cal G}^{1}-1$. According to Theorem
2.1.1, G is not an MD-algebra.
* 2.
$k=3$. That means that ${\cal G}^{2}=gen\left(X_{4},\cdots,X_{n}\right)$.
Assume that $\cal G$ is an MD-algebra. Remember that
$\left[X_{1},X_{2}\right]=\sum\limits_{l=2}^{n}{C_{12}^{l}{X_{l}}},\,\left[X_{1},X_{3}\right]=\sum\limits_{l=2}^{n}{C_{13}^{l}{X_{l}}},\,\left[X_{i},X_{j}\right]=\sum\limits_{l=4}^{n}{C_{ij}^{l}{X_{l}}}\scriptsize{\footnotesize\textbullet},$
for all $j>4$ when $i=1$, $j\geq 3$ when $i=2$ and $j>3$ when $i=3$. According
to Lemma 2.1.7, the matrix
$P=\left[{\begin{array}[]{*{20}{c}}{C_{12}^{2}}&{C_{12}^{3}}\\\
{C_{13}^{2}}&{C_{13}^{3}}\end{array}}\right]$ is invertible.
Let
$F={\alpha_{1}}X_{1}^{*}+{\alpha_{2}}X_{2}^{*}+\cdots+{\alpha_{n}}X_{n}^{*}$
be an arbitrary element of ${\cal G}^{*}$, where
$\alpha_{1},\alpha_{2},\cdots,\alpha_{n}\in\mathbb{R}$. The matrix of the
bilinear form ${B_{F}}$ is
${B_{F}}=\left[{\begin{array}[]{*{20}{c}}0&{-F\left({\left[{{X_{1}},{X_{2}}}\right]}\right)}&\cdots&{-F\left({\left[{{X_{1}},{X_{n}}}\right]}\right)}\\\
{F\left({\left[{{X_{1}},{X_{2}}}\right]}\right)}&0&\cdots&{-F\left({\left[{{X_{2}},{X_{n}}}\right]}\right)}\\\
\cdots&\cdots&\cdots&\cdots\\\
{F\left({\left[{{X_{1}},{X_{n}}}\right]}\right)}&{F\left({\left[{{X_{2}},{X_{n}}}\right]}\right)}&\cdots&0\end{array}}\right].$
Now we consider the $4$-submatrices of ${B_{F}}$ established by the elements
which are on the rows and the columns of the same numbers 1, 2, 3, i ($i>3$).
Because $\cal G$ is an MD-algebra, so according to Lemma 2.1.6, we get
$rank(B_{F})\in\\{0,2\\}$, this implies that the determinants of these
$4$-submatrices are zero for any $F\in{{\cal G}^{*}}$. By direct computations,
using the following obvious result of Linear Algebra: the determinant of any
skew-symmetric real $4$-matrix $\left(a_{ij}\right)_{4}$ is equal to zero if
and only if ${a_{12}}.{a_{34}}-{a_{13}}.{a_{24}}+{a_{14}}.{a_{23}}=0$, we get
$C_{2i}^{l}=C_{3i}^{l}=0,l\geq 4$. This implies
$\left[X_{2},X_{i}\right]=\left[X_{3},X_{i}\right]=0,i\geq 4$. Note that
${\cal G}^{2}$ is commutative. So we have
${\cal G}^{2}=\left[{\cal G}^{1},{\cal
G}^{1}\right]=gen\left(X_{4},\cdots,X_{n}\right)$
$=gen\left(\left[X_{i},X_{j}\right];i,j\geq 2\right)$
$=gen\left(\left[X_{2},X_{3}\right]\right)$.
Thus, $n-3=\dim{\cal G}^{2}\leq 1$, i.e. $n\leq 4$. This contradicts the
hypothesis $n\geq 5$. That means $\cal G$ is not an MD-algebra.
* 3.
$k\geq 4$. By an argument analogous to that used above, we also prove that
$\cal G$ is not an MD-algebra.
Conversely, assume that ${\cal G}$ is a real solvable Lie algebra of dimension
$n$ such that its first derived ideal is $(n-1)$-dimensional and commutative,
i.e. ${{\cal
G}^{1}}\equiv{\mathbb{R}}.{X_{2}}\oplus{\mathbb{R}}.{X_{3}}\oplus...\oplus{\mathbb{R}}.{X_{n}}\equiv{{\mathbb{R}}^{n-1}}$.
We need show that $\cal G$ is an MD-algebra.
Let
$F={\alpha_{1}}X_{1}^{*}+{\alpha_{2}}X_{2}^{*}+\cdots+{\alpha_{n}}X_{n}^{*}\equiv\left(\alpha_{1},\alpha_{2},\cdots,\alpha_{n}\right)\in{\mathbb{R}}^{n}$
be an arbitrary element from ${\cal G}^{*}\equiv{\mathbb{R}}^{n}$, where
$\alpha_{1},\alpha_{2},\cdots,\alpha_{n}\in{\mathbb{R}}$. By simple
computation, we can see that the matrix of the bilinear form ${B_{F}}$ is
${B_{F}}=\left[{\begin{array}[]{*{20}{c}}0&{-F\left({\left[{{X_{1}},{X_{2}}}\right]}\right)}&\cdots&{-F\left({\left[{{X_{1}},{X_{n}}}\right]}\right)}\\\
{F\left({\left[{{X_{1}},{X_{2}}}\right]}\right)}&0&\cdots&0\\\
\cdots&\cdots&\cdots&\cdots\\\
{F\left({\left[{{X_{1}},{X_{n}}}\right]}\right)}&0&\cdots&0\end{array}}\right].$
It is clear that $rankB_{F}\in\left\\{{0,2}\right\\}$. Hence, ${\cal G}$ is an
MDn-algebra and Theorem 2.1.5 is proved completely. $\square$
### 2.2 Classification of MD5-algebras having
non-commutative derived ideals
The following theorem is the main result of the paper. It gives the
classification up to an isomorphism of MD5-algebras having non-commutative
derived ideals.
###### Theorem 2.2.1.
Let ${\cal G}$ be an MD5-algebra such that the first derived ideal ${\cal
G}^{1}=\left[{\cal G},{\cal G}\right]$ is non-commutative. Then the following
assertions hold.
* (i)
If ${\cal G}$ is decomposable, then ${\cal G}\cong{\cal H}\oplus{\cal K}$,
where ${\cal H}$ and ${\cal K}$ are MD-algebras of dimensions which are no
more than 4.
* (ii)
If ${\cal G}$ is indecomposable, then we can choose a suitable basis
$(X_{1},X_{2},X_{3},$
$X_{4},X_{5})$ of ${\cal G}$ such that ${\cal
G}^{1}=gen\left(X_{3},X_{4},X_{5}\right),\,\left[X_{3},X_{4}\right]=X_{5}$;
operators $ad_{X_{1}},ad_{X_{2}}$ act on ${\cal G}^{1}$ as the following
endomorphisms
$ad_{X_{1}}=\left({\begin{array}[]{*{20}{c}}1&0&0\\\ 0&1&0\\\
0&0&2\end{array}}\right),\,\,ad_{X_{2}}=\left({\begin{array}[]{*{20}{c}}0&-1&0\\\
1&0&0\\\ 0&0&0\end{array}}\right)$
and the other Lie brackets are trivial.
We need to prove some lemmas before we prove Theorem 2.2.1.
###### Lemma 2.2.2.
Let ${\cal G}$ be a real solvable Lie algebra. For any $Z\in{\cal G}$ we
consider $ad_{Z}$ as an operator acted on ${\cal G}^{1}$. Then we have
$Trace\left(ad_{Z}\right)=0$ for all $Z\in{\cal G}^{1}$.
###### Proof.
Using the Jacobi identity for $X,Y\in{\cal G}$ and an arbitrary element
$Z\in{{\cal G}^{1}}$, we have
$\left[{X,\left[{Y,Z}\right]}\right]+\left[{Y,\left[{Z,X}\right]}\right]+\left[{Z,\left[{X,Y}\right]}\right]=0$.
So, $a{d_{X}}\circ a{d_{Y}}-a{d_{Y}}\circ a{d_{X}}=a{d_{\left[{X,Y}\right]}}$.
This implies $Trace\left({a{d_{\left[{X,Y}\right]}}}\right)=0$. Note that
${{\cal G}^{1}}=\left[{{\cal G},{\cal G}}\right]$ and $a{d_{Z}}$ is a linear
map. So we get $Trace\left({a{d_{Z}}}\right)=0$ for all $Z\in{{\cal G}^{1}}$.
∎
###### Lemma 2.2.3.
If ${\cal G}$ is a real solvable Lie algebra with $\dim{\cal G}^{1}=2$ then
${{\cal G}^{1}}$ is commutative.
###### Proof.
We choose a basis $(X,Y)$ of ${{\cal G}^{1}}$. Assume that
$\left[X,Y\right]=aX+bY$. So we have
$ad_{X}=\left({\begin{array}[]{*{20}{c}}0&a\\\
0&b\end{array}}\right),ad_{Y}=\left({\begin{array}[]{*{20}{c}}{-a}&0\\\
{-b}&0\end{array}}\right)\in End({\cal G}^{1})$. According to Lemma 2.2.2, we
get $a=b=0$. Hence, ${{\cal G}^{1}}$ is commutative. ∎
Now we are ready to prove Theorem 2.2.1 - The main result of the paper.
###### Proof of Theorem 2.2.1.
It is clear that assertion (i) of Theorem 2.2.1 holds obviously. We only need
to prove assertion (ii). Let ${\cal G}$ be an indecomposable MD5-algebra with
the first derived ideal ${\cal G}^{1}=\left[{\cal G},{\cal G}\right]$ is non -
commutative and the second derived ideal ${\cal G}^{2}=\left[{\cal
G}^{1},{\cal G}^{1}\right]$ is commutative. According to Theorems 2.1.1, 2.1.5
and Lemma 2.2.3, the dimensions of ${\cal G}^{1}$ and ${\cal
G}^{2}=\left[{\cal G}^{1},{\cal G}^{1}\right]$ must be 3 and 1, respectively.
We choose a basis $\left(X_{1},X_{2},X_{3},X_{4},X_{5}\right)$ such that
${\cal G}^{1}=gen\left(X_{3},X_{4},X_{5}\right)$ and ${\cal
G}^{2}=gen\left(X_{5}\right)$ with the Lie brackets are given by
$\begin{array}[]{l}\left[X_{1},X_{2}\right]={a_{3}}{X_{3}}+{a_{4}}{X_{4}}+{a_{5}}{X_{5}},\\\
\left[X_{1},X_{3}\right]={b_{3}}{X_{3}}+{b_{4}}{X_{4}}+{b_{5}}{X_{5}},\\\
\left[X_{1},X_{4}\right]={c_{3}}{X_{3}}+{c_{4}}{X_{4}}+{c_{5}}{X_{5}},\\\
\left[X_{1},X_{5}\right]={d_{3}}{X_{3}}+{d_{4}}{X_{4}}+{d_{5}}{X_{5}},\\\
\left[X_{2},X_{3}\right]={e_{3}}{X_{3}}+{e_{4}}{X_{4}}+{e_{5}}{X_{5}},\\\
\left[X_{2},X_{4}\right]={f_{3}}{X_{3}}+{f_{4}}{X_{4}}+{f_{5}}{X_{5}},\\\
\left[X_{2},X_{5}\right]={k_{3}}{X_{3}}+{k_{4}}{X_{4}}+{k_{5}}{X_{5}},\\\
\left[X_{3},X_{4}\right]={g_{5}}{X_{5}},\left[X_{3},X_{5}\right]={h_{5}}{X_{5}},\left[X_{4},X_{5}\right]={l_{5}}{X_{5}},\end{array}$
where $a_{i},b_{i},c_{i},d_{i},e_{i},f_{i},k_{i}\,(i=3,4,5)$ and
$g_{5},h_{5},l_{5}$ are the definite real numbers.
Now we give some useful remarks as follows.
* a.
According to Lemma 2.2.2,
$Trace\left(ad_{X_{3}}\right)=Trace\left(ad_{X_{4}}\right)=0$. That means
${h_{5}}={l_{5}}=0$.
* b.
$g_{5}\neq 0$ because ${\cal G}^{2}=gen(X_{5})$. By changing ${X_{3}}$ with
${X^{\prime}_{3}}={\frac{1}{g_{5}}}X_{3}$, we get
$\left[X^{\prime}_{3},X_{4}\right]=X_{5}$. So, we can suppose right from the
start that $\left[X_{3},X_{4}\right]=X_{5}$, i.e. $g_{5}=1$.
* c.
${d_{3}}={d_{4}}={k_{3}}={k_{4}}=0$ because ${\cal G}^{2}=\mathbb{R}.X_{5}$ is
an ideal of ${\cal G}$. So, we get
$\left[X_{1},X_{5}\right]={d_{5}}{X_{5}},\,\left[X_{2},X_{5}\right]={k_{5}}{X_{5}}.$
If $k_{5}\neq 0$, by changing
$X^{\prime}_{2}=X_{1}-{\frac{d_{5}}{k_{5}}}X_{2}$ we get
$\left[X^{\prime}_{2},X_{5}\right]=0$. So, we can always assume that
$k_{5}=0$.
* d.
By changing ${X_{1}}$ with
${X^{\prime}_{1}}={X_{1}}-{c_{5}}{X_{3}}+{b_{5}}{X_{4}}$ and ${X_{2}}$ with
${X^{\prime}_{2}}={X_{2}}-{f_{5}}{X_{3}}+{e_{5}}{X_{4}}$, we get
$\left[X^{\prime}_{1},X_{3}\right]={b_{3}}{X_{3}}+{b_{4}}{X_{4}}$,
$\left[X^{\prime}_{1},X_{4}\right]={c_{3}}{X_{3}}+{c_{4}}{X_{4}}$,
$\left[X^{\prime}_{2},X_{3}\right]={e_{3}}{X_{3}}+{e_{4}}{X_{4}}$,
$\left[X^{\prime}_{2},X_{4}\right]={f_{3}}{X_{3}}+{f_{4}}{X_{4}}$.
Thus, we can suppose right from the start that
${b_{5}}={c_{5}}={e_{5}}={f_{5}}=0$.
Using the Jacobi identity for triads $X_{1},X_{2},X_{i}\,(i=3,4,5)$, we obtain
$(I)\left\\{\begin{array}[]{l}{a_{3}}={a_{4}}=0,\\\
{e_{4}}{c_{3}}={b_{4}}{f_{3}},\\\
{e_{3}}{b_{4}}+{e_{4}}{c_{4}}={b_{3}}{e_{4}}+{b_{4}}{f_{4}},\\\
{f_{3}}{b_{3}}+{f_{4}}{c_{3}}={c_{3}}{e_{3}}+{c_{4}}{f_{3}},\\\
{b_{3}}+{c_{4}}={d_{5}},\\\ {e_{3}}+{f_{4}}=0.\end{array}\right.$
So we can reduce the Lie brackets as follows
$\begin{array}[]{l}\left[{{X_{1}};{X_{2}}}\right]=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{a_{5}}{X_{5}},\\\
\left[{{X_{1}};{X_{3}}}\right]={b_{3}}{X_{3}}+{b_{4}}{X_{4}},\\\
\left[{{X_{1}};{X_{4}}}\right]={c_{3}}{X_{3}}+{c_{4}}{X_{4}},\\\
\left[{{X_{1}};{X_{5}}}\right]=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left({{b_{3}}+{c_{4}}}\right){X_{5}},\\\
\left[{{X_{2}};{X_{3}}}\right]={e_{3}}{X_{3}}+\,{e_{4}}{X_{4}},\\\
\left[{{X_{2}};{X_{4}}}\right]={f_{3}}{X_{3}}-{e_{3}}{X_{4}},\\\
\left[{{X_{3}};{X_{4}}}\right]=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{X_{5}}.\end{array}$
Thus, Relations $(I)$ can be rewritten as follows
$(II)\left\\{\begin{array}[]{l}{e_{4}}{c_{3}}={b_{4}}{f_{3}},\\\
2{e_{3}}{b_{4}}={e_{4}}\left({{b_{3}}-{c_{4}}}\right),\\\
2{c_{3}}{e_{3}}={f_{3}}\left({{b_{3}}-{c_{4}}}\right).\end{array}\right.$
Now we need consider the following cases which contradict each other.
* Case 1:
${e_{3}}={e_{4}}=0.$
$(II)\Leftrightarrow\left\\{\begin{array}[]{l}{b_{4}}{f_{3}}=0,\\\
{f_{3}}\left({{b_{3}}-{c_{4}}}\right)=0.\end{array}\right.$
* 1.1.
Assume that ${f_{3}}=0$. Then, Relations $(II)$ is automatically satisfied.
* By choosing $F_{1}=X_{3}^{*}\in{\cal G}^{*}$, we get $rank{B_{{F_{1}}}}=2$.
* Now we choose $F_{2}=X_{5}^{*}\in{\cal G}^{*}$. By simple computation, we obtain
${B_{{F_{2}}}}=\left[{\begin{array}[]{*{20}{c}}0&{-{a_{5}}}&0&0\\\
{{a_{5}}}&0&0&0\\\ 0&0&0&{-1}\\\ {\begin{array}[]{*{20}{c}}0\\\
{{b_{3}}+{c_{4}}}\end{array}}&{\begin{array}[]{*{20}{c}}0\\\
0\end{array}}&{\begin{array}[]{*{20}{c}}1\\\
0\end{array}}&{\begin{array}[]{*{20}{c}}0\\\
0\end{array}}\end{array}\,\,\,\,\,\,\,\begin{array}[]{*{20}{c}}{-\left({{b_{3}}+{c_{4}}}\right)}\\\
0\\\ 0\\\ {\begin{array}[]{*{20}{c}}0\\\ 0\end{array}}\end{array}}\right]$
Since ${\cal G}$ is an MD-algebra, this implies that
$rank{B_{{F_{2}}}}=rank{B_{{F_{1}}}}=2$. That fact implies that
${a_{5}}={b_{3}}+{c_{4}}=0$, so ${\cal G}$ is decomposable, which is a
contradiction. Thus, this case cannot happen.
* 1.2.
Now we assume that ${f_{3}}\neq 0$. Then $b_{4}=0,b_{3}=c_{4}$.
By changing $X_{1}$ and $X_{2}$ with
${X^{\prime}_{1}}={X_{1}}-{c_{3}}{X^{\prime}_{2}}$ and
$X^{\prime}_{2}={\frac{1}{f_{3}}X_{2}}$ we can suppose right from the start
that $f_{3}=1,c_{3}=0$. Because of the dimension of ${\cal G}^{1}$ is 3,
$b_{3}\neq 0$. By changing $X_{1}$ with
$X^{\prime}_{1}={\frac{1}{b_{3}}}{X_{1}}$, we can always assume that
$b_{3}=1$. By changing $X_{2}$ with
$X^{\prime}_{2}=X_{2}-\frac{a_{5}}{2}{X_{5}}$, we can assume that $a_{5}=0$.
Now, we choose $F_{3}=X_{3}^{*}$ and $F_{4}=X_{4}^{*}$ from ${\cal G}^{*}$, we
get $rank{B_{{F_{3}}}}=2,\,rank{B_{{F_{4}}}}=4$. This cannot happen because
${\cal G}$ is an MD-algebra.
* Case 2:
${e_{4}}=0,{e_{3}}\neq 0$
$\left({II}\right)\Leftrightarrow\left\\{\begin{array}[]{l}{b_{4}}{f_{3}}=0,\\\
{b_{4}}{e_{3}}=0,\\\
2{c_{3}}{e_{3}}={f_{3}}\left({{b_{3}}-{c_{4}}}\right).\end{array}\right.\Leftrightarrow\left\\{\begin{array}[]{l}{b_{4}}=0,\\\
2{c_{3}}{e_{3}}={f_{3}}\left({{b_{3}}-{c_{4}}}\right).\end{array}\right.$
By changing $X_{2}$ with $X^{\prime}_{2}={\frac{1}{e_{3}}}{X_{2}}$, we can
assume that $e_{3}=1$. Now Relations $(II)$ can be rewritten as follows
$(III)\left\\{\begin{array}[]{l}{b_{4}}=0,\\\
2{c_{3}}={f_{3}}\left({{b_{3}}-{c_{4}}}\right).\end{array}\right.$
By changing $X_{4}$ with $X^{\prime}_{4}=X_{4}-\frac{f_{3}}{2}{X_{3}}$ and
$X_{1}$ with $X^{\prime}_{1}=X_{1}-{b_{3}}{X_{2}}$, we can suppose that
$b_{3}=f_{3}=0$. From Relations (III) we get $b_{4}=c_{3}=0$. Let
$F=\alpha\,X_{1}^{*}+\beta X_{2}^{*}+\gamma X_{3}^{*}+\delta X_{4}^{*}+\sigma
X_{5}^{*}\equiv\left({\alpha,\beta,\gamma,\delta,\sigma}\right)$ be an
arbitrary element from ${{\cal G}^{*}}\equiv{\mathbb{R}^{5}}$;
$\alpha,\beta,\gamma,\delta,\sigma\in\mathbb{R}$. By simple computation, we
obtain the matrix of the bilinear form ${B_{F}}$ as follows
${B_{F}}=\left[{\begin{array}[]{*{20}{c}}0&{-{a_{5}}\sigma}&0&{-{c_{4}}\delta}\\\
{{a_{5}}\sigma}&0&{-\gamma}&\delta\\\ 0&\gamma&0&{-\sigma}\\\
{\begin{array}[]{*{20}{c}}{{c_{4}}\delta}\\\
{{c_{4}}\sigma}\end{array}}&{\begin{array}[]{*{20}{c}}\delta\\\
0\end{array}}&{\begin{array}[]{*{20}{c}}\sigma\\\
0\end{array}}&{\begin{array}[]{*{20}{c}}0\\\
0\end{array}}\end{array}\,\,\,\,\,\,\,\begin{array}[]{*{20}{c}}{-{c_{4}}\sigma}\\\
0\\\ 0\\\ {\begin{array}[]{*{20}{c}}0\\\ 0\end{array}}\end{array}}\right]$
Since ${\cal G}$ is an MD-algebra, $rank{B_{F}}$ only get two values zero or
two. Hence, it is easy to prove that $c_{4}=a_{5}=0$. But this implies that
${\cal G}$ is decomposable. This is a contradiction. Thus, this case cannot
happen.
* Case 3:
${e_{4}}\neq 0$.
By changing $X_{4}$ with $X^{\prime}_{4}={e_{4}}{X_{4}}+{e_{3}}{X_{3}}$, we
can assume that $e_{3}=0,e_{4}=1$. Now, Relations $(II$) can be rewritten as
follows
$\left\\{\begin{array}[]{l}{c_{3}}={b_{4}}{f_{3}},\\\
{b_{3}}={c_{4}}.\end{array}\right.$
By changing $X_{1}$ with $X^{\prime}_{1}=X_{1}-{b_{4}}{X_{2}}$, we can assume
that $b_{4}=0$. Putting this into the relation above, we get $c_{3}=0$.
Then, the Lie brackets in $\cal G$ can be reduced as follows
$\begin{array}[]{l}\left[X_{1},X_{2}\right]=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\lambda{X_{5}},\\\
\left[X_{1},X_{3}\right]=\mu{X_{3}},\\\
\left[X_{1},X_{4}\right]=\,\,\,\,\,\,\,\,\,\mu{X_{4}},\\\
\left[X_{1},X_{5}\right]=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\mu{X_{5}},\\\
\left[X_{2},X_{3}\right]=\,\,\,\,\,\,\,\,\,\,\,\,\,{X_{4}},\\\
\left[X_{2},X_{4}\right]=\,\theta{X_{3}},\\\
\left[X_{3},X_{4}\right]=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{X_{5}}.\end{array}$
Let $F=\alpha\,X_{1}^{*}+\beta X_{2}^{*}+\gamma X_{3}^{*}+\delta
X_{4}^{*}+\sigma X_{5}^{*}\in{\cal G}^{*}$ be an arbitrary element from ${\cal
G}^{*}\equiv{\mathbb{R}^{5}}$. Then by simple computation, we see that
${B_{F}}=\left[\begin{array}[]{l}0\,\,\,\,\,-\lambda\sigma\,\,\,\,-\mu\gamma\,\,\,\,-\mu\delta\,\,\,\,-2\mu\sigma\\\
\lambda\sigma\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,-\delta\,\,\,\,\,\,-\theta\gamma\,\,\,\,\,\,\,\,\,\,\,\,0\\\
\mu\gamma\,\,\,\,\,\,\,\,\delta\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,-\sigma\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\\\
\mu\delta\,\,\,\,\,\,\,\,\theta\gamma\,\,\,\,\,\,\,\,\,\,\,\,\sigma\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\\\
2\mu\sigma\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\end{array}\right].$
* 3.1.
Assume $\mu\neq 0$.
* When $\sigma\neq 0$, we have $rank{B_{F}}=4$. Because ${\cal G}$ is an MD5-algebra, we get $rank{B_{F}}\in\\{0,4\\}$ for all $\alpha,\beta,\delta,\gamma,\sigma\in\mathbb{R}$. But this only can happen if $\theta<0$. By changing $X_{1},X_{2},X_{4},X_{5}$ with $X^{\prime}_{1}=\mu{X_{1}},X^{\prime}_{2}=\sqrt{-\theta}{X_{2}},X^{\prime}_{4}=\sqrt{-\theta}{X_{4}},X^{\prime}_{5}=\sqrt{-\theta}{X_{5}}$, we can assume that $\theta=-1$. By changing $X_{2}$ with $X^{\prime}_{2}=X_{2}-\frac{\lambda}{2}{X_{5}}$, we can assume that $\lambda=0$. Hence, the Lie brackets of $\cal G$ can be reduced as follows
$\begin{array}[]{l}\left[{{X_{1}};{X_{3}}}\right]={X_{3}},\\\
\left[{{X_{1}};{X_{4}}}\right]=\,\,\,\,\,\,\,\,\,{X_{4}},\\\
\left[{{X_{1}};{X_{5}}}\right]=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2{X_{5}},\\\
\left[{{X_{2}};{X_{3}}}\right]=\,\,\,\,\,\,\,\,\,\,\,\,{X_{4}},\\\
\left[{{X_{2}};{X_{4}}}\right]=\,-{X_{3}},\\\
\left[{{X_{3}};{X_{4}}}\right]=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{X_{5}}.\end{array}$
* 3.2
Assume $\mu=0$. By the same way, we consider the matrix of the bilinear form
${B_{F}}$ and obtain $\lambda=0$. But this shows that ${\cal G}$ is
decomposable. This construction show that this case can not happen.
The theorem 2.2.1 is proved completely. So there is only one MD5-algebra
having non-commutative derived ideal.$\hfill\square$
### CONCLUDING REMARK
Let us recall that each real Lie algebra $\cal G$ defines only one connected
and simply connected Lie group G such that Lie(G) = $\cal G$. Therefore we
obtain only one connected and simply connected MD5-group corresponding to the
MD5-algebra given in Theorem 2.2.1. In the next paper, we shall describe the
geometry of K-orbits of considered MD5-group, describe topological properties
of MD5-foliation formed by the generic K-orbits of this MD5-group and give the
characterization of the Connes’s $C^{*}$-algebra associated to this
MD5-foliation.
## References
* [1] Do Ngoc Diep, Method of Noncommutative Geometry for Group $C^{*}$-algebras, Chapman and Hall / CRC Press Reseach Notes in Mathematics Series, # 416, 1999.
* [2] A.A. Kirillov, Elements of the Theory of Representations, Springer-Verlag, Berlin – Heidenberg – New York, 1976.
* [3] Vuong Manh Son et Ho Huu Viet, Sur La Structure Des $C^{*}-alg{\grave{e}}bres$ D’une Classe De Groupes De Lie, J. Operator Theory, 11 (1984), 77-90.
* [4] Kristopher Tapp, Matrix Groups for Undergraduates, AMS, 2005.
* [5] Le Anh Vu, On the foliations formed by the generic K-orbits of the MD4-groups, Acta Math. Vietnam, $N^{0}$ 2 (1990), 39 - 45.
* [6] Le Anh Vu, On a Subclass of 5-dimensional Lie Algebras Which have 3-dimensional Commutative Derived Ideals, East-West J. of Mathematics, Vol.7, $N^{0}$ 1 (2005), 13 - 22.
* [7] Le Anh Vu, Classification of 5-dimensional MD-algebras Which have 4-dimensional Commutative Derived Ideals, Scientific Journal of University of Pedagogy of Ho Chi Minh City, $N^{0}$ 12 (46) (2007), 3 - 15 (In Vietnamese).
* [8] Le Anh Vu and Duong Quang Hoa, The Geometricaly Picture of K-orbits of Connected and Simply Connected MD5-groups such that their MD5-algebras have 4-dimensional Commutative Derived Ideals, Scientific Journal of University of Pedagogy of Ho Chi Minh City, $N^{0}$ 12 (46) (2007), 16 - 28 (In Vietnamese).
* [9] Le Anh Vu and Kar Ping Shum, On a Subclass of 5-dimentional Solvable Lie Algebras Which Have Commutative Derived Ideal, Advances in Algebra and Combinatorics, World Scientific Publishing Co. (2008), pp 353 - 371.
|
arxiv-papers
| 2011-01-11T19:11:39 |
2024-09-04T02:49:16.331659
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Le Anh Vu, Ha Van Hieu and Tran Thi Hieu Nghia",
"submitter": "Vu Le Anh",
"url": "https://arxiv.org/abs/1101.2181"
}
|
1101.2314
|
††thanks: Corresponding author
# Decoherence in Attosecond Photoionization
Stefan Pabst Center for Free-Electron Laser Science, DESY, Notkestrasse 85,
22607 Hamburg, Germany Department of Physics,University of Hamburg,
Jungiusstrasse 9, 20355 Hamburg, Germany Loren Greenman Department of
Chemistry and The James Franck Institute, The University of Chicago, Chicago,
Illinois 60637, USA Phay J. Ho Argonne National Laboratory, Argonne,
Illinois 60439, USA David A. Mazziotti Department of Chemistry and The James
Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA
Robin Santra robin.santra@cfel.de Center for Free-Electron Laser Science,
DESY, Notkestrasse 85, 22607 Hamburg, Germany Department of
Physics,University of Hamburg, Jungiusstrasse 9, 20355 Hamburg, Germany
###### Abstract
The creation of superpositions of hole states via single-photon ionization
using attosecond extreme-ultraviolet pulses is studied with the time-dependent
configuration interaction singles (TDCIS) method. Specifically, the degree of
coherence between hole states in atomic xenon is investigated. We find that
interchannel coupling not only affects the hole populations, it also enhances
the entanglement between the photoelectron and the remaining ion, thereby
reducing the coherence within the ion. As a consequence, even if the spectral
bandwidth of the ionizing pulse exceeds the energy splittings among the hole
states involved, perfectly coherent hole wave packets cannot be formed. For
sufficiently large spectral bandwidth, the coherence can only be increased by
increasing the mean photon energy.
###### pacs:
32.80.Aa, 42.65.Re, 03.65.Yz
The typical time scale of electronic motion in atoms, molecules, and condensed
matter systems ranges from a few attoseconds ($1~{}\text{as}=10^{-18}$ s) to
tens of femtoseconds ($1~{}\text{fs}=10^{-15}$ s) Krausz and Ivanov (2009);
Zewail (2000); Cavalieri et al. (2007). In the last decade the remarkable
progress in high harmonic generation Doumy et al. (2009); Dudovich et al.
(2006); López-Martens et al. (2005); Gibson et al. (2004); Schafer et al.
(1993) made it possible to generate attosecond pulses as short as 80 as
Goulielmakis et al. (2008). Attosecond pulses have opened the door to real-
time observations of the most fundamental processes on the atomic scale Krausz
and Ivanov (2009); Pfeifer et al. (2008). For instance, the generation of
attosecond pulses was utilized to determine spatial structures of molecular
orbitals Haessler et al. (2010); an interferometric technique using attosecond
pulses was used to characterize attosecond electron wave packets Mauritsson et
al. (2010); and attosecond pulse trains Singh et al. (2010) and isolated
attosecond pulses Sansone et al. (2010), in combination with an intense few-
cycle infrared pulse, enabled the control of electron localization in
molecules. Attosecond technology demonstrated the ability to follow, on a
subfemtosecond time scale, processes such as photoionization Skantzakis et al.
(2010), Auger decay Drescher et al. (2002), and valence electron motion driven
by relativistic spin-orbit coupling Goulielmakis et al. (2010). Furthermore,
the availability of attosecond pulses fuelled a broad interest in exploring
charge transfer dynamics following photoexcitation or photoionization Sansone
et al. (2010).
In this Letter, we analyze the creation of hole states via single-photon
ionization using a single extreme-ultraviolet attosecond pulse. We investigate
the impact of the freed photoelectron on the remaining ion and demonstrate
that the interaction between the photoelectron and the ion cannot be neglected
for currently available state-of-the-art attosecond pulses. In particular, the
interchannel coupling of the initially coherently excited hole states greatly
enhances the entanglement between the photoelectron and the ionic states.
Interchannel coupling is mediated by the photoelectron and mixes different
ionization channels, i.e., hole configurations, with each other. Consequently,
the degree of coherence among the ionic states is strongly reduced, making it
impossible to describe the subsequent charge transfer in the ion with a pure
quantum mechanical state. Experiments on photosynthetic systems Lee et al.
(2007); Sarovar et al. (2010); Collini et al. (2010); Harel et al. (2010) have
revealed a correlation between highly efficient energy transport and coherent
dynamics in molecules (nuclear and electronic dynamics in this case).
Similarly, high degrees of coherence in nonstationary hole states may be
necessary for efficient charge transport within molecules.
In the last decade, much work has been done in the realm of hole migration
Breidbach and Cederbaum (2003); Nest et al. (2008); Kuleff et al. (2010). It
was shown that electronic motion can be triggered solely by electron
correlation Breidbach and Cederbaum (2003). Charge transfers mediated by
electronic correlations typically take place in a few femtoseconds and are
thus faster than electronic dynamics initiated by nuclear motion Kröner et al.
(2007); Muskatel et al. (2009). Recent experiments Weinkauf et al. (1997);
Schlag et al. (2007) have demonstrated that electronically excited ionic
states can modify site-selective reactivity within tens of femtoseconds,
making hole migration processes a promising tool to control chemical
reactions. Up to now, theoretical calculations Breidbach and Cederbaum (2003);
Kuleff et al. (2010) investigating hole migration phenomena have neglected the
interaction between the parent ion and the photoelectron and assumed a
perfectly coherent hole wave packet. As long as the photoelectron departs
sufficiently rapidly from the parent ion, this assumption is appropriate
Cederbaum et al. (1986). However, for attosecond pulses with large spectral
bandwidths, the enhanced production of slow photoelectrons will affect (mainly
via interchannel coupling) both the final hole populations and the coherence
among these hole states. Furthermore, recent results in high harmonic
spectroscopy suggest that interchannel coupling may be the missing link to
understand hole dynamics occurring in high harmonic generation processes
before the ejected electron recombines with the parent ion Mairesse et al.
(2010).
We investigate the creation of hole states via attosecond photoionization
using the implementation of the time-dependent configuration-interaction
singles (TDCIS) approach described in Ref. Greenman et al. (2010) (see also
Schlegel et al. (2007); Klinkusch et al. (2009)). TDCIS allows us to study
ionization dynamics beyond the single-channel approximation and to understand
systematically the relevance of interchannel coupling in the hole creation
process. The TDCIS wave function for the entire system is
$\displaystyle\left|\Psi(t)\right>=\alpha_{0}(t)\,\left|\Phi_{0}\right>+\sum_{a,i}\alpha^{a}_{i}(t)\,\left|\Phi^{a}_{i}\right>,$
(1)
where $\left|\Phi_{0}\right>$ is the Hartree-Fock ground state and
$\left|\Phi^{a}_{i}\right>=\hat{c}^{\dagger}_{a}\hat{c}_{i}\left|\Phi_{0}\right>$
is a one-particle–one-hole excitation ($\hat{c}^{\dagger}_{a}$ and
$\hat{c}_{i}$ are creation and annihilation operators for an electron in
orbitals $a$ and $i$, respectively). The corresponding coefficients
$\alpha_{0}(t)$ and $\alpha^{a}_{i}(t)$, respectively, are functions of time
and describe the dynamics of the system. Throughout, indices $i,j,$ are used
for occupied orbitals in $\left|\Phi_{0}\right>$; indices $a,b,$ stand for
unoccupied orbitals. We focus our discussion on the case where single-photon
ionization is the dominant effect and higher order processes can be neglected.
Our model system is atomic xenon. The corresponding Hamiltonian (neglecting
spin-orbit coupling) is
$\displaystyle\hat{H}(t)$ $\displaystyle=$
$\displaystyle\hat{H}_{0}+\hat{H}_{1}+E(t)\,\hat{z},$ (2a) where $E(t)$ is the
electric field, $\hat{z}$ the dipole operator, and $\hat{H}_{0}$ is the mean-
field Fock operator, which is diagonal with respect to the basis used in Eq.
(1). The residual Coulomb interaction, $\displaystyle\hat{H}_{1}$
$\displaystyle=$ $\displaystyle\hat{V}_{c}-\hat{V}_{\text{MF}},$ (2b)
is defined such that $\hat{H}_{0}+\hat{H}_{1}$ gives the exact nonrelativistic
Hamiltonian for the electronic system in the absence of external fields
($\hat{V}_{c}$ is the electron–electron interaction). We study the impact of
different approximations for $\hat{H}_{1}$ on the hole state as follows. The
Coulomb-free model, the simplest approximation, removes the residual Coulomb
interaction ($\hat{H}_{1}=0$) between the excited electron and the parent ion.
In this approximation, the excited electron always sees a neutral atom via the
$\hat{V}_{\text{MF}}$ potential Szabo and Ostlund (1996). A more realistic
approximation is the intrachannel model including direct and exchange
contributions of the Coulomb interaction only within a given channel. In this
second model, the excited electron can only interact with the occupied orbital
from which it originates. Interactions between different occupied orbitals are
neglected, i.e. we set
$\left<\Phi^{a}_{i}\right|\hat{H}_{1}\left|\Phi^{b}_{j}\right>=0$ for $i\neq
j$. The third and final model describes the Coulomb interaction exactly within
the TDCIS framework. We refer to this as the full model. Note that the exact
nonrelativistic Hamiltonian $\hat{H}_{0}+\hat{H}_{1}$ is diagonal with respect
to the ionic one-hole states
$\left|\Phi_{i}\right>=\hat{c}_{i}\left|\Phi_{0}\right>$. In the full model,
the photoelectron can couple the hole states, as $\hat{H}_{1}$ in the
particle-hole space is not diagonal with respect to the hole index (i.e.,
$\left<\Phi^{a}_{i}\right|\hat{H}_{1}\left|\Phi^{b}_{j}\right>$ generally
differs from zero). This type of photoelectron-mediated interaction is called
interchannel coupling Starace (1980). As a consequence, in the full model the
hole index is not a good quantum number, whereas in the Coulomb-free and
intrachannel models, excited eigenstates of $\hat{H}_{0}+\hat{H}_{1}$ are
characterized by a well-defined hole index. To describe the hole states of the
remaining ion, we employ the ion density matrix Greenman et al. (2010)
$\displaystyle\hat{\rho}^{\text{IDM}}_{i,j}(t)$ $\displaystyle=$
$\displaystyle\text{Tr}_{a}[\left|\Psi(t)\right>\left<\Psi(t)\right|]_{i,j}=\sum_{a}\left<\Phi^{a}_{i}|\Psi(t)\right>\left<\Psi(t)|\Phi^{a}_{j}\right>,$
(3)
where $\text{Tr}_{a}$ stands for the trace over the photoelectron. The
properties of the ion density matrix can be measured using attosecond
transient absorption spectroscopy Goulielmakis et al. (2010). A description of
the cationic eigenstates in terms of one-hole configurations is a physically
meaningful approximation for noble-gas atoms such as xenon Buth et al. (2003).
Figure 1: (color online) The $4d_{0}$ [panel (a)] and $5s$ [panel (b)] hole
populations of xenon as a function of time are shown for three different
residual Coulomb interaction approximations: (1) the full model (red solid
line), (2) the intrachannel model (green dotted line), and (3) the Coulomb-
free model (blue dash-dotted line). The attosecond pulse has a peak field
strength of 25 GV/m, a pulse duration of 20 as, a (mean) photon energy of 136
eV, and is centered at $t=0$ as.
In Fig. 1 the hole populations $\rho^{\text{IDM}}_{5s,5s}(t)$ and
$\rho^{\text{IDM}}_{4d_{0},4d_{0}}(t)$ of the xenon $5s$ and $4d_{0}$
orbitals, respectively, are shown for all three interaction models ($4d_{0}$
stands for the $4d$ orbital with $m=0$). The ionizing, gaussian-shaped
attosecond pulse is linearly polarized and has a peak field strength of 25
GV/m, a pulse duration of $\tau=20$ as, and a (mean) photon energy of
$\omega_{0}=136$ eV. The hole dynamics of the Coulomb-free and intrachannel
models are alike. In both cases, the population is constant after the pulse,
since the hole index is a good quantum number within these models. The
extension to the exact Coulomb interaction changes the situation. Interchannel
coupling causes the hole populations to remain nonstationary as long as the
photoelectron remains close to the ion. As the distance between the
photoelectron and the ion increases, the interchannel coupling weakens and the
populations $\rho^{\text{IDM}}_{i,i}(t)$ become stationary (see Fig. 1). We
confine our discussion to the first hundreds of attoseconds after the pulse,
allowing us to neglect decay processes, which start to take place after a few
femtoseconds.
As we will see in the following, interchannel coupling not only affects the
hole populations but also the coherence between the created hole states. The
degree of coherence between $\left|\Phi_{i}\right>$ and
$\left|\Phi_{j}\right>$ is given by
$\displaystyle
g_{i,j}(t)=\frac{|\rho^{\text{IDM}}_{i,j}(t)|}{\sqrt{\rho^{\text{IDM}}_{i,i}(t)\rho^{\text{IDM}}_{j,j}(t)}}.$
(4)
Totally incoherent statistical mixtures result in $g_{i,j}(t)=0$. The fact
that the density matrix is positive semidefinite implies the Cauchy-Schwarz
relations
$|\rho^{\text{IDM}}_{i,j}(t)|^{2}\leq\rho^{\text{IDM}}_{i,i}(t)\rho^{\text{IDM}}_{j,j}(t)$,
which bound the maximum achievable (perfect) coherence ($g_{i,j}(t)=1$). To
investigate the effect of interchannel coupling on the coherence between the
orbitals $4d_{0}$ and $5s$ in xenon, we restrict the definition of the
$4d_{0}$ hole population to the events where the photoelectron has angular
momentum $l=1$. The other possible angular momentum for the $4d_{0}$
photoelectron, $l=3$, does not contribute to the coherence, since the
photoelectron from $5s$ can only have $l=1$. For a similar reason, it is
impossible to create a coherent superposition of $5p$ and $5s$ (or $4d$) hole
states via one-photon absorption in the electric dipole approximation.
Figure 2: (color online) The time evolution of the coherence between the
$4d_{0}$ and $5s$ hole states in xenon is shown for the full Coulomb
interaction model. The photon energy is 136 eV and the pulse duration varies
from 5–60 as.
Figure 2 illustrates the time evolution of the coherence between $4d_{0}$ and
$5s$ in xenon for different pulse durations and fixed photon energy
($\omega_{0}=136$ eV). Here, we use the full interaction model. Directly after
the ionizing pulse is over, the initial degree of coherence (at $t\approx 0$
as) rises with decreasing pulse duration, i.e., increasing spectral bandwidth,
and converges to a value close to unity. (The difference of the ionization
potentials, $\varepsilon_{5s}-\varepsilon_{4d_{0}}$, is $\approx 50$ eV.) At
$t\approx 0$ as, the photoelectron is still in immediate contact with the
parent ion. Therefore, the coherence properties of the system of interest—the
parent ion—are affected by its interaction with the bath represented by the
photoelectron. The system–bath interaction leads to a reduction in the
coherence of the system Breuer and Petruccione (2002), which can be seen by
the rapid drops in all curves in Fig. 2 within tens of attoseconds after the
pulse. With time, as the photoelectron departs from the ion, the Coulomb
(“system-bath”) interaction becomes less important and the coherence converges
to a stationary value. The maximum for this stationary value is obtained with
a 25 as pulse ($g_{4d_{0},5s}\approx 0.6$). For pulses shorter than 25 as,
oscillations in $g_{4d_{0},5s}$ occur that persist for hundreds of
attoseconds, and the final degree of coherence reached falls below 0.6. The
spin-orbit dynamics associated with the fine-structure within the $4d$ shell
is slow in comparison to the time scale of the decoherence between $4d_{0}$
and $5s$, and is, therefore, not considered here.
Figure 3: (color online) The time evolution of the coherence between the
$4d_{0}$ and $5s$ hole states, calculated with the full Coulomb interaction
model, is shown for different photon energies. The pulse duration is in all
cases 20 as.
We see in Fig. 3 that when holding the pulse duration fixed ($\tau=20$ as),
the degree of coherence rises with increasing $\omega_{0}$. The magnitude of
the oscillations decreases as the final coherence (at $t\approx 1$ fs)
increases. This trend indicates less system-bath interactions occur with
higher photoelectron energies keeping the degree of coherence among the hole
states high.
In Fig. 4 we compare the impact of the different Coulomb approximations on the
final coherence. The drops in coherence that occur for the full model for
short pulses [Fig. 4(a)] and low photon energies [Fig. 4(b)] cannot be seen in
the Coulomb-free and intrachannel models—which both neglect interchannel
coupling. Hence, the decay of coherence is solely driven by the interchannel
coupling due to the slow photoelectron. As a comparison to the Coulomb-free
model shows, intrachannel coupling affects the coherence in an insignificant
way.
Figure 4: (color online) The dependence of the coherence between the $4d_{0}$
and $5s$ hole states as function of the pulse duration (a) and as function of
the photon energy (b) are shown for all three interaction approximations.
In the limit of long pulse durations (small spectral bandwidths), the
coherence vanishes for all models, since photoelectrons from the $4d_{0}$ and
$5s$ become energetically distinguishable and cannot contribute to a coherent
statistical mixture of hole states. The slight drop in the coherence for the
Coulomb-free and intrachannel models with increasing $\omega_{0}$ [Fig. 4(b)]
is related to the reduced factorizability of the numerator of Eq. (4). In
contrast, the trend in the full model for increasing $\omega_{0}$ is dominated
by the gain in coherence due to higher photoelectron energy resulting in less
system-bath interaction.
In conclusion, we demonstrated that the coherence of the ionic states produced
via attosecond photoionization is not solely determined by the bandwidth of
the ionizing pulse, but greatly depends on the kinetic energy of the
photoelectron, which can be controlled by the (mean) photon energy.
Interchannel coupling leads to an enhanced entanglement between the
photoelectron and the parent ion resulting in a reduced coherence in the ionic
states. This reduction can be mitigated with higher photon energies, thereby
sacrificing high photon cross sections and the possibility of controlling
independently the relative populations of the various hole states in the
statistical mixture.
Our results have far-reaching consequences beyond the atomic case. Molecules
will be even more strongly affected by interchannel coupling due to the
reduced symmetry and smaller energy splittings between the cation many-
electron eigenstates. Interchannel coupling is also likely to be significant
for inner-valence hole configurations in molecules, which show strong mixing
to configurations outside the TDCIS model space. The present study suggests
that interchannel coupling accompanying the hole creation process will affect
attosecond experiments investigating charge transfer processes in photoionized
systems. The control of decoherence requires widely tunable attosecond
sources, thus offering a new opportunity for x-ray free-electron lasers
Zholents and Fawley (2004).
###### Acknowledgements.
P.J.H. was supported by the Office of Basic Energy Sciences, U.S. Department
of Energy under Contract No. DE-AC02-06CH11357. L.G. thanks Martha Ann and
Joseph A. Chenicek and their family. D.A.M. gratefully acknowledges the NSF,
the Henry-Camille Dreyfus Foundation, the David-Lucile Packard Foundation, and
the Microsoft Corporation for their support.
## References
* Krausz and Ivanov (2009) F. Krausz and M. Ivanov, Rev. Mod. Phys. 81, 163 (2009).
* Zewail (2000) A. H. Zewail, J. Phys. Chem. A 104, 5660 (2000).
* Cavalieri et al. (2007) A. L. Cavalieri et al., Nature 449, 1029 (2007).
* Doumy et al. (2009) G. Doumy et al., Phys. Rev. Lett. 102, 093002 (2009).
* Dudovich et al. (2006) N. Dudovich et al., Nature Phys. 2, 781 (2006).
* López-Martens et al. (2005) R. López-Martens et al., Phys. Rev. Lett. 94, 033001 (2005).
* Gibson et al. (2004) E. A. Gibson et al., Phys. Rev. Lett. 92, 033001 (2004).
* Schafer et al. (1993) K. J. Schafer et al., Phys. Rev. Lett. 70, 1599 (1993).
* Goulielmakis et al. (2008) E. Goulielmakis et al., Science 320, 1614 (2008).
* Pfeifer et al. (2008) T. Pfeifer et al., Chem. Phys. Lett. 463, 11 (2008).
* Haessler et al. (2010) S. Haessler et al., Nature Phys 6, 200 (2010).
* Mauritsson et al. (2010) J. Mauritsson et al., Phys. Rev. Lett. 105, 053001 (2010).
* Singh et al. (2010) K. P. Singh et al., Phys. Rev. Lett. 104, 023001 (2010).
* Sansone et al. (2010) G. Sansone et al., Nature 465, 763 (2010).
* Skantzakis et al. (2010) E. Skantzakis et al., Phys. Rev. Lett. 105, 043902 (2010).
* Drescher et al. (2002) M. Drescher et al., Nature 419, 803 (2002).
* Goulielmakis et al. (2010) E. Goulielmakis et al., Nature 466, 739 (2010).
* Lee et al. (2007) H. Lee, Y.-C. Cheng, and G. R. Fleming, Science 316, 1462 (2007).
* Sarovar et al. (2010) M. Sarovar et al., Nature Phys. 6, 462 (2010).
* Collini et al. (2010) E. Collini et al., Nature 463, 644 (2010).
* Harel et al. (2010) E. Harel, A. F. Fidler, and G. S. Engel, Proc. Natl. Acad. Sci. USA 107, 16444 (2010).
* Breidbach and Cederbaum (2003) J. Breidbach and L. S. Cederbaum, J. Chem. Phys. 118, 3983 (2003).
* Nest et al. (2008) M. Nest, F. Remacle, and R. D. Levine, New J. Phys. 10, 025019 (2008).
* Kuleff et al. (2010) A. Kuleff, S. Lünnemann, and L. Cederbaum, J. Phys. Chem. A 114, 8676 (2010).
* Kröner et al. (2007) D. Kröner et al., Appl. Phys. A-Mater. 88, 535 (2007).
* Muskatel et al. (2009) B. H. Muskatel, F. Remacle, and R. D. Levine, Phys. Scripta 80, 048101 (2009).
* Weinkauf et al. (1997) R. Weinkauf et al., J. Phys. Chem. A 101, 7702 (1997).
* Schlag et al. (2007) E. W. Schlag et al., Angew. Chem. Int. Ed. 46, 3196 (2007).
* Cederbaum et al. (1986) L. Cederbaum et al., Adv. Chem. Phys. 65, 115 (1986).
* Mairesse et al. (2010) Y. Mairesse et al., Phys. Rev. Lett. 104, 213601 (2010).
* Greenman et al. (2010) L. Greenman et al., Phys. Rev. A 82, 023406 (2010).
* Schlegel et al. (2007) H. B. Schlegel, S. M. Smith, and X. Li, J. Chem. Phys. 126, 244110 (2007).
* Klinkusch et al. (2009) S. Klinkusch, P. Saalfrank, and T. Klamroth, J. Chem. Phys. 131, 114304 (2009).
* Szabo and Ostlund (1996) A. Szabo and N. S. Ostlund, _Modern Quantum Chemistry_ (Dover, Mineola, NY, 1996).
* Starace (1980) A. F. Starace, Appl. Opt. 19, 4051 (1980).
* Buth et al. (2003) C. Buth, R. Santra, and L. S. Cederbaum, J. Chem. Phys. 119, 7763 (2003).
* Breuer and Petruccione (2002) H.-P. Breuer and F. Petruccione, _The Theory of Open Quantum Systems_ (Oxford University Press, 2002).
* Zholents and Fawley (2004) A. Zholents and W. Fawley, Phys. Rev. Lett. 92, 224801 (2004).
|
arxiv-papers
| 2011-01-12T10:31:27 |
2024-09-04T02:49:16.341562
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Stefan Pabst, Loren Greenman, Phay J. Ho, David A. Mazziotti, Robin\n Santra",
"submitter": "Stefan Pabst",
"url": "https://arxiv.org/abs/1101.2314"
}
|
1101.2323
|
# Semileptonic charmed $B$ meson decays in Universal Extra Dimension Model
M. Ali Paracha paracha@phys.qau.edu.pk Physics Department, Quaid-i-Azam
University, Islamabad, Pakistan Ishtiaq Ahmed ishtiaq@ncp.edu.pk National
Centre for Physics, Quaid-i-Azam University, Islambad, Pakistan M. Jamil
Aslam jamil@phys.qau.edu.pk Physics Department, Quaid-i-Azam University,
Islamabad, Pakistan.
###### Abstract
Form factors parameterizing the semileptonic decay $B_{c}\rightarrow
D_{s}^{\ast}l^{+}l^{-}$ ($l=\mu,\tau$) are calculated using the frame work of
Ward Identities. These form factors are then used to calculate the physical
observables like branching ratio and helicity fractions of final state
$D_{s}^{\ast}$ meson in these decay modes. The analysis is then extended to
the the universal extra dimension (UED) model where the dependence of above
mentioned physical variables to the compactification radius R, the only
unknown parameter in UED model, is studied. It is shown that the helicity
fractions of $D_{s}^{\ast}$ are quite sensitive to the UED model especially
when have muons as the final state lepton. Therefore, these can serve as a
useful tool to establish new physics predicted by the UED model.
## I
Introduction
Living in the LHC era, it is hoped to either verify the Standard Model (SM) or
to explore the properties of more accurate underlying theory that describes
the theory of weak scale. Flavor Changing Neutral Current (FCNC) decays of
$B$-meson are an important tool to investigate the structure of weak
interactions and also provide us a frame work to look for the physics beyond
the Standard Model (SM). This lies in the fact that FCNC decays are not
allowed at tree level in the SM and occur only at the loop level 1 ; 2 ; 2a
and makes them quite sensitive to possible small corrections that may be
result of any modification to the SM, or from the new interactions. This gives
us solid reason to study these decays both theoretically and experimentally.
Since the CLEO observations of the rare radiative $b\rightarrow s\gamma$
transition 3 , there have been intensive studies on rare semileptonic,
radiative and leptonic decays of $B_{u,d,s}$ mesons induced by FCNC
transitions of $b\rightarrow s,d$ 4 . The study will be even more complete if
one consider the similar decays of the charmed $B$ mesons $(B_{c})$.
The charmed $B_{c}$ meson is a bound state of two heavy quarks, bottom $b$ and
charm $c$, and was first observed in 1998 at Tevatron in Fermilab 8 . Because
of two heavy quarks, the $B_{c}$ mesons are rich in phenomenology compared to
the other $B$ mesons. At the Large Hadron Collider (LHC) the expected number
of events for the production of $B_{c}$ meson are about $10^{8}-10^{10}$ per
year 9 ; 10 which is a reasonable number to work on the phenomenology of the
$B_{c}$ meson. In literature, some of the possible radiative and semileptonic
exclusive decays of $B_{c}$ mesons like
$B_{c}\rightarrow\left(\rho,K^{\ast},D_{s}^{\ast},B_{u}^{\ast}\right)\gamma,B_{c}\rightarrow
l\nu\gamma$ $,B_{c}\rightarrow B_{u}^{\ast}l^{+}l^{-},B_{c}\rightarrow
D_{1}^{0}l\nu,B_{c}\rightarrow D_{s0}^{\ast}l^{+}l^{-}$ and $B_{c}\rightarrow
D_{s,d}^{\ast}l^{+}l^{-}$ have been studied using the frame work of
relativistic constituent quark model 11 , QCD Sum Rules and the Light Cone Sum
Rules 12 . The focus of the present work is the study of exclusive
$B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$ decay.
While working on the exclusive $B$-meson decays the main job is to calculate
the form factors which are the non perturbative quantities and are the scalar
functions of the square of momentum transfer. In literature the form factors
for $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$ decay were calculated using
different approaches, such as light front constituent quark models and a
relativistic quark model 11 ; 13 . In this work we calculate the form factors
for the above mentioned decay in a model independent way through Ward
identities, which was earlier applied to $B\rightarrow\rho,\gamma$ 14 ; MJAR
and $B\rightarrow K_{1}$ decays 15 . This approach enables us to make a clear
separation between the pole and non pole type contributions, the former is
known in terms of a universal function $\xi_{\perp}(q^{2})\equiv g_{+}(q^{2})$
which is introduced in the Large Energy Effective Theory (LEET) of heavy to
light transition form factors 16 . The residue of the pole is then determined
in a self consistent way in terms of $g_{+}(0)$ which will give information
about the couplings of $B_{s}^{\ast}(1^{-})$ and $B_{sA}^{\ast}(1^{+})$ with
$B_{c}D_{s}^{\ast}$ channel. The above mentioned coupling arises at lower pole
masses because the higher pole masses of $B_{c}$ meson do not contribute for
the decay $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}.$ The form factors are then
determine in terms of a known parameter $g_{+}(0)$ and the pole masses of the
particles involved, which will then be used to calculate different physical
observables like the branching ratio and the helicity fractions of
$D_{s}^{\ast}$ for these decays.
At the quark level the semileptonic decay $B_{c}\rightarrow
D_{s}^{\ast}l^{+}l^{-}$ is governed by the FCNC transition $b\rightarrow
sl^{+}l^{-},$ therefore it is an important candidate to look for physics in
and beyond the SM. Many investigations for the physics beyond the SM are now
being performed in various areas of particle physics which are expected to get
the direct or indirect evidence at high energy colliders such as LHC. During
the last couple of years there have been an increased interest in models with
extra dimensions, since they solve the hierarchy problem and they can provide
the unified framework of gravity and other interactions together with a
connection to the string theory 17 . Among them the special role plays the one
with universal extra dimensions (UED) as in this model all SM fields are
allowed to propagate in available all dimensions. The economy of UED model is
that there is only one additional parameter to that of SM which is the radius
$R$ of the compactified extra dimension. Now above the compactification scale
$1/R$ a given UED model becomes a higher dimensional field theory whose
equivalent description in four dimensions consists of SM fields and the towers
of KK modes having no partner in the SM. A simplest model of this type was
proposed by Appelquist,Cheng and Dobrescu (ACD) 18 . In this model all the
masses of the KK particles and their interactions with SM particles and also
among themselves are described in terms of the inverse of compactification
radius $R$ and the parameters of the SM 19 .
The most important property of ACD model is the conservation of parity which
implies the absence of tree level contributions of KK states to the low energy
processes taking place at scale $\mu<<1/R.$ This brings interest towards the
FCNC transitions, $b\rightarrow s$ as mentioned earlier that these transitions
occur at loop level in SM and hence the one loop contribution due to KK modes
to them could in principly be important. These processes are used to constrain
the mass and couplings of the KK states, i.e, the compactification radius
$1/R$ 19 ; 20 .
Buras et al. have computed the effective Hamiltonian of several FCNC processes
in ACD model, particularly in $b$ sector, namely $B_{s,d}$ mixing and
$b\rightarrow s$ transition such as $b\rightarrow s\gamma$ and $b\rightarrow
sl^{+}l^{-}$ decay 19 . The implications of physics with UED are examined with
data from Tevatron experiments and the bounds on the inverse of
compactification radius are found to be $1/R\geq 250-300$ GeV 21 . There
exists some studies in the literature on different $B$ to light meson decays
in ACD model, where the dependence of different physical observables like
branching ratio, forward-backward asymmetry, lepton polarization asymmetry and
the helicity fractions of final state mesons on $1/R$ is examined 21 ; 22 ; 23
.
In this work we will study the branching ratio and helicity fractions of
$D_{s}^{\ast}$ meson in $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$ decay both
in the SM and ACD model using the framework of
$B\rightarrow(K^{\ast},K_{1})l^{+}l^{-}$ decays described in refs. 22 ; 23 .
The paper is organized as follows. In Sec. II we present the effective
Hamiltonian for the decay $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}.$ Section
III contains the definitions as well as the detailed calculation of the form
factors using Ward Identities. In Sec. IV we present the basic formulas for
physical observables like decay rate and helicity fractions of $D_{s}^{\ast}$
meson where as the numerical analysis of these observables will be given in
Section V. Section VI gives the summary of the results.
## II Effective Hamiltonian and Matrix Elements
At quark level, the semileptonic decay $B_{c}\rightarrow
D_{s}^{\ast}l^{+}l^{-}$ is governed by the transition $b\rightarrow
sl^{+}l^{-}$ for which the effective Hamiltonian can be written as
$\displaystyle H_{eff}$ $\displaystyle=$
$\displaystyle-\frac{4G_{F}}{\sqrt{2}}V_{tb}V_{ts}^{\ast}\bigg{[}\sum\limits_{i=1}^{10}C_{i}(\mu)O_{i}\bigg{]},$
(1)
where $O_{i}(\mu)$ $(i=1,...,10)$ are the four quark operators and
$C_{i}(\mu)$ are the corresponding Wilson coefficients at the energy scale
$\mu$ 24 which was usually take to be the $b$-quark mass
$\left(m_{b}\right)$. The theoretical uncertainties related to the
renormalization scale can be reduced when the next to leading logarithm
corrections are included. The explicit form of the operators responsible for
the decay $B_{c}^{-}\rightarrow D_{s}^{\ast-}l^{+}l^{-}$ is
$\displaystyle O_{7}$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{16\pi^{2}}m_{b}(\bar{s}\sigma_{\mu\nu}Rb)F^{\mu\nu}$
(2) $\displaystyle O_{9}$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{16\pi^{2}}\left(\bar{s}\gamma_{\mu}Lb\right)\bar{l}\gamma^{\mu}l$
(3) $\displaystyle O_{10}$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{16\pi^{2}}\left(\bar{s}\gamma_{\mu}Lb\right)\bar{l}\gamma^{\mu}\gamma^{5}l$
(4)
with $L,R=\left(1\mp\gamma^{5}\right)/2$.
Using the effective Hamiltonian given in Eq.(1) the free quark amplitude for
$b\rightarrow sl^{+}l^{-}$ can be written as
$\displaystyle\mathcal{M(}b\rightarrow sl^{+}l^{-})$ $\displaystyle=$
$\displaystyle-\frac{G_{F}\alpha}{\sqrt{2}\pi}V_{tb}V_{ts}^{\ast}\bigg{[}{C_{9}^{eff}\left(\mu\right)(\bar{s}\gamma_{\mu}Lb)(\bar{l}\gamma^{\mu}l)+C_{10}(}\bar{s}\gamma_{\mu}Lb)(\bar{l}\gamma^{\mu}\gamma^{5}l)$
(5)
$\displaystyle-2C_{7}^{eff}\left(\mu\right)\frac{m_{b}}{q^{2}}(\bar{s}i\sigma_{\mu\nu}q^{\nu}Rb)\bar{l}\gamma^{\mu}l\bigg{]}$
where $q^{2}$ is the square of momentum transfer. Note that the operator
$O_{10}$ given in Eq.(4) can not be induced by the insertion of four quark
operators because of the absence of $Z$-boson in the effective theory.
Therefore, the Wilson coefficient $C_{10}$ does not renormalize under QCD
corrections and is independent on the energy scale $\mu.$ Additionally the
above quark level decay amplitude can get contributions from the matrix
element of four quark operators, $\sum_{i=1}^{6}\left\langle
l^{+}l^{-}s\left|O_{i}\right|b\right\rangle,$ which are usually absorbed into
the effective Wilson coefficient $C_{9}^{eff}(\mu)$ and can be written as 25 ;
26 ; 27 ; 28 ; 29 ; 30 ; 31
$C_{9}^{eff}(\mu)=C_{9}(\mu)+Y_{SD}(z,s^{\prime})+Y_{LD}(z,s^{\prime}).$
where $z=m_{c}/m_{b}$ and $s^{\prime}=q^{2}/m_{b}^{2}$. $Y_{SD}(z,s^{\prime})$
describes the short distance contributions from four-quark operators far away
from the $c\bar{c}$ resonance regions, and this can be calculated reliably in
the perturbative theory. However the long distance contribution
$Y_{LD}(z,s^{\prime})$ cannot be calculated by using the first principles of
QCD, so they are usually parameterized in the form of a phenomenological
Breit-Wigner formula making use of the vacuum saturation approximation and
quark hadron duality. Therefore, one can not calculate them reliably so we we
will neglect these long distance effects for the case of $B_{c}\rightarrow
D_{s}^{\ast}l^{+}l^{-}$. The expression for the short distance contribution
$Y_{SD}(z,s^{\prime})$ is given as
$\displaystyle Y_{SD}(z,s^{\prime})$ $\displaystyle=$ $\displaystyle
h(z,s^{\prime})(3C_{1}(\mu)+C_{2}(\mu)+3C_{3}(\mu)+C_{4}(\mu)+3C_{5}(\mu)+C_{6}(\mu))$
(6)
$\displaystyle-\frac{1}{2}h(1,s^{\prime})(4C_{3}(\mu)+4C_{4}(\mu)+3C_{5}(\mu)+C_{6}(\mu))$
$\displaystyle-\frac{1}{2}h(0,s^{\prime})(C_{3}(\mu)+3C_{4}(\mu))+{\frac{2}{9}}(3C_{3}(\mu)+C_{4}(\mu)+3C_{5}(\mu)+C_{6}(\mu)),$
with
$\displaystyle h(z,s^{\prime})$ $\displaystyle=$
$\displaystyle-{\frac{8}{9}}\mathrm{ln}z+{\frac{8}{27}}+{\frac{4}{9}}x-{\frac{2}{9}}(2+x)|1-x|^{1/2}\left\\{\begin{array}[]{l}\ln\left|\frac{\sqrt{1-x}+1}{\sqrt{1-x}-1}\right|-i\pi\quad\mathrm{for}{{\
}x\equiv 4z^{2}/s^{\prime}<1}\\\
2\arctan\frac{1}{\sqrt{x-1}}\qquad\mathrm{for}{{\ }x\equiv
4z^{2}/s^{\prime}>1}\end{array}\right.,$ (9) $\displaystyle h(0,s^{\prime})$
$\displaystyle=$
$\displaystyle{\frac{8}{27}}-{\frac{8}{9}}\mathrm{ln}{\frac{m_{b}}{\mu}}-{\frac{4}{9}}\mathrm{ln}s^{\prime}+{\frac{4}{9}}i\pi\,\,.$
(10)
Also the non factorizable effects from the charm loop brings further
corrections to the radiative transition $b\rightarrow s\gamma,$ and these can
be absorbed into the effective Wilson coefficients $C_{7}^{eff}$ which then
takes the form 32 ; 33 ; 34 ; 35 ; 36
$C_{7}^{eff}(\mu)=C_{7}(\mu)+C_{b\rightarrow s\gamma}(\mu)$
with
$\displaystyle C_{b\rightarrow s\gamma}(\mu)$ $\displaystyle=$ $\displaystyle
i\alpha_{s}\left[\frac{2}{9}\eta^{14/23}(G_{1}(x_{t})-0.1687)-0.03C_{2}(\mu)\right]$
(11) $\displaystyle G_{1}(x_{t})$ $\displaystyle=$
$\displaystyle\frac{x_{t}\left(x_{t}^{2}-5x_{t}-2\right)}{8\left(x_{t}-1\right)^{3}}+\frac{3x_{t}^{2}\ln^{2}x_{t}}{4\left(x_{t}-1\right)^{4}}$
(12)
where $\eta=\alpha_{s}(m_{W})/\alpha_{s}(\mu),$ $x_{t}=m_{t}^{2}/m_{W}^{2}$
and $C_{b\rightarrow s\gamma}$ is the absorptive part for the $b\rightarrow
sc\bar{c}\rightarrow s\gamma$ rescattering.
The new physics effects manifest themselves in rare $B$ decays in two
different ways, either through new contribution to the Wilson coefficients or
through the new operators in the effective Hamiltonian, which are absent in
the SM. Being minimal extension of SM the ACD model is the most economical one
because it has only additional parameter $R$ i.e. the radius of the
compactification leaving the operators basis same as that of the SM.
Therefore, the whole contribution from all the KK states is in the Wilson
coefficients which are now the functions of the compactification radius $R$.
At large value of $1/R$ the new states being more and more massive and will be
decoupled from the low-energy theory,therefore one can recover the SM
phenomenology.
The modified Wilson coefficients in ACD model contain the contribution from
new particles which are not present in the SM and comes as an intermediate
state in penguin and box diagrams. Thus, these coefficients can be expressed
in terms of the functions $F\left(x_{t},1/R\right)$,
$x_{t}=\frac{m_{t}^{2}}{M_{W}^{2}}$, which generalize the corresponding SM
function $F_{0}\left(x_{t}\right)$ according to:
$F\left(x_{t},1/R\right)=F_{0}\left(x_{t}\right)+\sum_{n=1}^{\infty}F_{n}\left(x_{t},x_{n}\right)$
(13)
with $x_{n}=\frac{m_{n}^{2}}{M_{W}^{2}}$ and $m_{n}=\frac{n}{R}$ 44 . The
relevant diagrams are $Z^{0}$ penguins, $\gamma$ penguins, gluon penguins,
$\gamma$ magnetic penguins, Chormomagnetic penguins and the corresponding
functions are $C\left(x_{t},1/R\right)$, $D\left(x_{t},1/R\right)$,
$E\left(x_{t},1/R\right)$, $D^{\prime}\left(x_{t},1/R\right)$ and
$E^{\prime}\left(x_{t},1/R\right)$ respectively. These functions are
calculated at next to leading order by Buras et al. 19 and can be summarized
as:
$\bullet C_{7}$
In place of $C_{7},$ one defines an effective coefficient $C_{7}^{(0)eff}$
which is renormalization scheme independent 45 :
$C_{7}^{(0)eff}(\mu_{b})=\eta^{\frac{16}{23}}C_{7}^{(0)}(\mu_{W})+\frac{8}{3}(\eta^{\frac{14}{23}}-\eta^{\frac{16}{23}})C_{8}^{(0)}(\mu_{W})+C_{2}^{(0)}(\mu_{W})\sum_{i=1}^{8}h_{i}\eta^{\alpha_{i}}$
(14)
where $\eta=\frac{\alpha_{s}(\mu_{W})}{\alpha_{s}(\mu_{b})}$, and
$C_{2}^{(0)}(\mu_{W})=1,\mbox{
}C_{7}^{(0)}(\mu_{W})=-\frac{1}{2}D^{\prime}(x_{t},\frac{1}{R}),\mbox{
}C_{8}^{(0)}(\mu_{W})=-\frac{1}{2}E^{\prime}(x_{t},\frac{1}{R});$ (15)
the superscript $(0)$ stays for leading logarithm approximation. Furthermore:
$\displaystyle\alpha_{1}$ $\displaystyle=$ $\displaystyle\frac{14}{23}\mbox{
\quad}\alpha_{2}=\frac{16}{23}\mbox{ \quad}\alpha_{3}=\frac{6}{23}\mbox{
\quad}\alpha_{4}=-\frac{12}{23}$ $\displaystyle\alpha_{5}$ $\displaystyle=$
$\displaystyle 0.4086\mbox{ \quad}\alpha_{6}=-0.4230\mbox{
\quad}\alpha_{7}=-0.8994\mbox{ \quad}\alpha_{8}=-0.1456$ $\displaystyle h_{1}$
$\displaystyle=$ $\displaystyle 2.996\mbox{ \quad}h_{2}=-1.0880\mbox{
\quad}h_{3}=-\frac{3}{7}\mbox{ \quad}h_{4}=-\frac{1}{14}$ $\displaystyle
h_{5}$ $\displaystyle=$ $\displaystyle-0.649\mbox{ \quad}h_{6}=-0.0380\mbox{
\quad}h_{7}=-0.0185\mbox{ \quad}h_{8}=-0.0057.$ (16)
The functions $D^{\prime}$ and $E^{\prime}$ are
$D_{0}^{\prime}(x_{t})=-\frac{(8x_{t}^{3}+5x_{t}^{2}-7x_{t})}{12(1-x_{t})^{3}}+\frac{x_{t}^{2}(2-3x_{t})}{2(1-x_{t})^{4}}\ln
x_{t},$ (17)
$E_{0}^{\prime}(x_{t})=-\frac{x_{t}(x_{t}^{2}-5x_{t}-2)}{4(1-x_{t})^{3}}+\frac{3x_{t}^{2}}{2(1-x_{t})^{4}}\ln
x_{t},$ (18) $\displaystyle D_{n}^{\prime}(x_{t},x_{n})$ $\displaystyle=$
$\displaystyle\frac{x_{t}(-37+44x_{t}+17x_{t}^{2}+6x_{n}^{2}(10-9x_{t}+3x_{t}^{2})-3x_{n}(21-54x_{t}+17x_{t}^{2}))}{36(x_{t}-1)^{3}}$
$\displaystyle+\frac{x_{n}(2-7x_{n}+3x_{n}^{2})}{6}\ln\frac{x_{n}}{1+x_{n}}$
$\displaystyle-\frac{(-2+x_{n}+3x_{t})(x_{t}+3x_{t}^{2}+x_{n}^{2}(3+x_{t})-x_{n})(1+(-10+x_{t})x_{t}))}{6(x_{t}-1)^{4}}\ln\frac{x_{n}+x_{t}}{1+x_{n}},$
$\displaystyle E_{n}^{\prime}(x_{t},x_{n})$ $\displaystyle=$
$\displaystyle\frac{x_{t}(-17-8x_{t}+x_{t}^{2}+3x_{n}(21-6x_{t}+x_{t}^{2})-6x_{n}^{2}(10-9x_{t}+3x_{t}^{2}))}{12(x_{t}-1)^{3}}$
(20)
$\displaystyle+-\frac{1}{2}x_{n}(1+x_{n})(-1+3x_{n})\ln\frac{x_{n}}{1+x_{n}}$
$\displaystyle+\frac{(1+x_{n})(x_{t}+3x_{t}^{2}+x_{n}^{2}(3+x_{t})-x_{n}(1+(-10+x_{t})x_{t}))}{2(x_{t}-1)^{4}}\ln\frac{x_{n}+x_{t}}{1+x_{n}}.$
Following reference 19 , one gets the expressions for the sum over $n:$
$\displaystyle\sum_{n=1}^{\infty}D_{n}^{\prime}(x_{t},x_{n})$ $\displaystyle=$
$\displaystyle-\frac{x_{t}(-37+x_{t}(44+17x_{t}))}{72(x_{t}-1)^{3}}$ (21)
$\displaystyle+\frac{\pi
M_{w}R}{2}[\int_{0}^{1}dy\frac{2y^{\frac{1}{2}}+7y^{\frac{3}{2}}+3y^{\frac{5}{2}}}{6}]\coth(\pi
M_{w}R\sqrt{y})$
$\displaystyle+\frac{(-2+x_{t})x_{t}(1+3x_{t})}{6(x_{t}-1)^{4}}J(R,-\frac{1}{2})$
$\displaystyle-\frac{1}{6(x_{t}-1)^{4}}[x_{t}(1+3x_{t})-(-2+3x_{t})(1+(-10+x_{t})x_{t})]J(R,\frac{1}{2})$
$\displaystyle+\frac{1}{6(x_{t}-1)^{4}}[(-2+3x_{t})(3+x_{t})-(1+(-10+x_{t})x_{t})]J(R,\frac{3}{2})$
$\displaystyle-\frac{(3+x_{t})}{6(x_{t}-1)^{4}}J(R,\frac{5}{2})],$
$\displaystyle\sum_{n=1}^{\infty}E_{n}^{\prime}(x_{t},x_{n})$ $\displaystyle=$
$\displaystyle-\frac{x_{t}(-17+(-8+x_{t})x_{t})}{24(x_{t}-1)^{3}}$ (22)
$\displaystyle+\frac{\pi
M_{w}R}{2}[\int_{0}^{1}dy(y^{\frac{1}{2}}+2y^{\frac{3}{2}}-3y^{\frac{5}{2}})\coth(\pi
M_{w}R\sqrt{y})]$
$\displaystyle-\frac{x_{t}(1+3x_{t})}{(x_{t}-1)^{4}}J(R,-\frac{1}{2})$
$\displaystyle+\frac{1}{(x_{t}-1)^{4}}[x_{t}(1+3x_{t})-(1+(-10+x_{t})x_{t})]J(R,\frac{1}{2})$
$\displaystyle-\frac{1}{(x_{t}-1)^{4}}[(3+x_{t})-(1+(-10+x_{t})x_{t})]J(R,\frac{3}{2})$
$\displaystyle+\frac{(3+x_{t})}{(x_{t}-1)^{4}}J(R,\frac{5}{2})],$
where
$J(R,\alpha)=\int_{0}^{1}dyy^{\alpha}[\coth(\pi
M_{w}R\sqrt{y})-x_{t}^{1+\alpha}\coth(\pi m_{t}R\sqrt{y})].$ (23)
$\bullet C_{9}$
In the ACD model and in the NDR scheme one has
$C_{9}(\mu)=P_{0}^{NDR}+\frac{Y(x_{t},\frac{1}{R})}{\sin^{2}\theta_{W}}-4Z(x_{t},\frac{1}{R})+P_{E}E(x_{t},\frac{1}{R})$
(24)
where $P_{0}^{NDR}=2.60\pm 0.25$ 12 and the last term is numerically
negligible. Besides
$\displaystyle Y(x_{t},\frac{1}{R})$ $\displaystyle=$ $\displaystyle
Y_{0}(x_{t})+\sum_{n=1}^{\infty}C_{n}(x_{t},x_{n})$ $\displaystyle
Z(x_{t},\frac{1}{R})$ $\displaystyle=$ $\displaystyle
Z_{0}(x_{t})+\sum_{n=1}^{\infty}C_{n}(x_{t},x_{n})$ (25)
with
$\displaystyle Y_{0}(x_{t})$ $\displaystyle=$
$\displaystyle\frac{x_{t}}{8}[\frac{x_{t}-4}{x_{t}-1}+\frac{3x_{t}}{(x_{t}-1)^{2}}\ln
x_{t}]$ $\displaystyle Z_{0}(x_{t})$ $\displaystyle=$
$\displaystyle\frac{18x_{t}^{4}-163x_{t}^{3}+259x_{t}^{2}-108x_{t}}{144(x_{t}-1)^{3}}$
(26)
$\displaystyle+[\frac{32x_{t}^{4}-38x_{t}^{3}+15x_{t}^{2}-18x_{t}}{72(x_{t}-1)^{4}}-\frac{1}{9}]\ln
x_{t}$
$C_{n}(x_{t},x_{n})=\frac{x_{t}}{8(x_{t}-1)^{2}}[x_{t}^{2}-8x_{t}+7+(3+3x_{t}+7x_{n}-x_{t}x_{n})\ln\frac{x_{t}+x_{n}}{1+x_{n}}]$
(27)
and
$\sum_{n=1}^{\infty}C_{n}(x_{t},x_{n})=\frac{x_{t}(7-x_{t})}{16(x_{t}-1)}-\frac{\pi
M_{w}Rx_{t}}{16(x_{t}-1)^{2}}[3(1+x_{t})J(R,-\frac{1}{2})+(x_{t}-7)J(R,\frac{1}{2})]$
(28)
$\bullet C_{10}$
$C_{10}$ is $\mu$ independent and is given by
$C_{10}=-\frac{Y(x_{t},\frac{1}{R})}{\sin^{2}\theta_{w}}.$ (29)
The normalization scale is fixed to $\mu=\mu_{b}\simeq 5$ GeV.
## III Matrix Elements and Form Factors
The exclusive $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$decay involves the
hadronic matrix elements which can be obtained by sandwiching the quark level
operators give in Eq. (5) between initial state $B_{c}$ meson and final state
$D_{s}^{\ast}$ meson. These can be parameterized in terms of form factors
which are scalar functions of the square of the four momentum
transfer($q^{2}=(p-k)^{2}).$ The non vanishing matrix elements for the process
$B_{c}\rightarrow D_{s}^{\ast}$ can be parameterized in terms of the seven
form factors as follows
$\displaystyle\left\langle
D_{s}^{\ast}(k,\varepsilon)\left|\bar{s}\gamma_{\mu}b\right|B_{c}(p)\right\rangle$
$\displaystyle=$
$\displaystyle\frac{2\epsilon_{\mu\nu\alpha\beta}}{M_{B_{c}}+M_{D_{s}^{\ast-}}}\varepsilon^{\ast\nu}p^{\alpha}k^{\beta}V(q^{2})$
(30) $\displaystyle\left\langle
D_{s}^{\ast}(k,\varepsilon)\left|\bar{s}\gamma_{\mu}\gamma_{5}b\right|B_{c}(p)\right\rangle$
$\displaystyle=$ $\displaystyle
i\left(M_{B_{c}^{-}}+M_{D_{s}^{\ast-}}\right)\varepsilon^{\ast\mu}A_{1}(q^{2})$
$\displaystyle-i\frac{(\varepsilon^{\ast}\cdot
q)}{M_{B_{c}^{-}}+M_{D_{s}^{\ast-}}}\left(p+k\right)^{\mu}A_{2}(q^{2})$
$\displaystyle-i\frac{2M_{D_{s}^{\ast-}}}{q^{2}}q^{\mu}(\varepsilon^{\ast}\cdot
q)\left[A_{3}(q^{2})-A_{0}(q^{2})\right]$
where $p$ is the momentum of $B_{c}$, $\varepsilon$ and $k$ are the
polarization vector and momentum of the final state $D_{s}^{\ast}$ meson.
Here, the form factor $A_{3}(q^{2})$ can be expressed in terms of the form
factors $A_{1}(q^{2})$ and $A_{2}(q^{2})$ as
$A_{3}(q^{2})=\frac{M_{B_{c}^{-}}+M_{D_{s}^{\ast-}}}{2M_{D_{s}^{\ast-}}}A_{1}(q^{2})-\frac{M_{B_{c}^{-}}-M_{D_{s}^{\ast-}}}{2M_{D_{s}^{\ast-}}}A_{2}(q^{2})$
(32)
with
$A_{3}(0)=A_{0}(0)$
In addition to the above form factors there are some penguin form factors,
which we can write as
$\displaystyle\left\langle
D_{s}^{\ast}(k,\varepsilon)\left|\bar{s}i\sigma_{\mu\nu}q^{\nu}b\right|B_{c}(p)\right\rangle$
$\displaystyle=$
$\displaystyle-\epsilon_{\mu\nu\alpha\beta}\varepsilon^{\ast\nu}p^{\alpha}k^{\beta}2F_{1}(q^{2})$
(33) $\displaystyle\left\langle
D_{s}^{\ast}(k,\varepsilon)\left|\bar{s}i\sigma_{\mu\nu}q^{\nu}\gamma^{5}b\right|B_{c}(p)\right\rangle$
$\displaystyle=$ $\displaystyle
i\left[\left(M_{Bc^{-}}^{2}-M_{D_{s}^{\ast-}}^{2}\right)\varepsilon_{\mu}-(\varepsilon^{\ast}\cdot
q)(p+k)_{\mu}\right]F_{2}(q^{2})$ $\displaystyle+(\varepsilon^{\ast}\cdot
q)i\left[q_{\mu}-\frac{q^{2}}{M_{Bc^{-}}^{2}-M_{D_{s}^{\ast-}}^{2}}(p+k)_{\mu}\right]F_{3}(q^{2})$
with
$F_{1}(0)=F_{2}(0)$
Now the different form factors appearing in Eqs. (30-LABEL:13b) can be related
to each other with the help of Ward identities as follows 14
$\displaystyle\left\langle
D_{s}^{\ast}(k,\varepsilon)\left|\bar{s}i\sigma_{\mu\nu}q^{\nu}b\right|B_{c}(p)\right\rangle$
$\displaystyle=$ $\displaystyle(m_{b}+m_{s})\left\langle
D_{s}^{\ast}(k,\varepsilon)\left|\bar{s}\gamma_{\mu}b\right|B_{c}(p)\right\rangle$
(35) $\displaystyle\left\langle
D_{s}^{\ast}(k,\varepsilon)\left|\bar{s}i\sigma_{\mu\nu}q^{\nu}\gamma^{5}b\right|B_{c}(p)\right\rangle$
$\displaystyle=$ $\displaystyle-(m_{b}-m_{s})\left\langle
D_{s}^{\ast}(k,\varepsilon)\left|\bar{s}\gamma_{\mu}\gamma_{5}b\right|B_{c}(p)\right\rangle$
(36) $\displaystyle+(p+k)_{\mu}\left\langle
D_{s}^{\ast}(k,\varepsilon)\left|\bar{s}\gamma_{5}b\right|B_{c}(p)\right\rangle$
By putting Eq.(30-LABEL:13b) in Eq.(35) and (36) and comparing the
coefficients of $\varepsilon_{\mu}^{\ast}$ and $q_{\mu}$ on both sides, one
can get the following relations between the form factors:
$\displaystyle F_{1}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{(m_{b}+m_{s})}{M_{B_{c}^{-}}+M_{D_{s}^{\ast-}}}V(q^{2})$
(37) $\displaystyle F_{2}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{m_{b}-m_{s}}{M_{B_{c}^{-}}+M_{D_{s}^{\ast-}}}A_{1}(q^{2})$
(38) $\displaystyle F_{3}(q^{2})$ $\displaystyle=$
$\displaystyle-(m_{b}-m_{s})\frac{2M_{D_{s}^{\ast-}}}{q^{2}}\left[A_{3}(q^{2})-A_{0}(q^{2})\right]$
(39)
The results given in Eqs. (37, 38, 39) are derived by using Ward identities
and therefore are the model independent.
The universal normalization of the above form factors at $q^{2}=0$ are
obtained by defining 14
$\displaystyle\left\langle
D_{s}^{\ast}(k,\varepsilon)\left|\bar{s}i\sigma_{\alpha\beta}b\right|B_{c}(p)\right\rangle$
$\displaystyle=$
$\displaystyle-i\epsilon_{\alpha\beta\rho\sigma}\varepsilon^{\ast\rho}\left[(p+k)^{\sigma}g_{+}+q^{\sigma}g_{-}\right]-(\varepsilon^{\ast}\cdot
q)\epsilon_{\alpha\beta\rho\sigma}(p+k)^{\rho}q^{\sigma}h$ (40)
$\displaystyle-i\left[(p+k)_{\alpha}\varepsilon_{\beta\rho\sigma\tau}\varepsilon^{\ast\rho}(p+k)^{\sigma}q^{\tau}-\alpha\leftrightarrow\beta\right]h_{1}$
Making use of the Dirac identity
$\sigma^{\mu\nu}\gamma^{5}=-\frac{i}{2}\epsilon^{\mu\nu\alpha\beta}\sigma_{\alpha\beta}$
(41)
in Eq.(40), we get
$\displaystyle\left\langle
D_{s}^{\ast}(k,\varepsilon)\left|\bar{s}i\sigma_{\mu\nu}q^{\nu}\gamma^{5}b\right|B_{c}(p)\right\rangle$
$\displaystyle=$
$\displaystyle\varepsilon_{\mu}^{\ast}\left[(M_{B_{c}^{-}}^{2}-M_{D_{s}^{\ast-}}^{2})g_{+}+q^{2}g_{-}\right]$
(42)
$\displaystyle-q\cdot\varepsilon^{\ast}\left[q^{2}(p+k)_{\mu}g_{+}-q_{\mu}g_{-}\right]$
$\displaystyle+q\cdot\varepsilon^{\ast}\left[q^{2}(p+k)_{\mu}-(M_{B_{c}^{-}}^{2}-M_{D_{s}^{\ast-}}^{2})q_{\mu}\right]h$
On comparing coefficents of $q_{\mu},\varepsilon_{\mu}^{\ast}$ and
$\epsilon_{\mu\nu\alpha\beta}$ from Eqs.(33), (LABEL:13b), (40) and (42), we
have
$\displaystyle F_{1}(q^{2})$ $\displaystyle=$
$\displaystyle\left[g_{+}(q^{2})-q^{2}h_{1}(q^{2})\right]$ (43) $\displaystyle
F_{2}(q^{2})$ $\displaystyle=$ $\displaystyle
g_{+}(q^{2})+\frac{q^{2}}{M_{B_{c}^{-}}^{2}-M_{D_{s}^{\ast-}}^{2}}g_{-}(q^{2})$
(44) $\displaystyle F_{3}(q^{2})$ $\displaystyle=$ $\displaystyle-
g_{-}(q^{2})-(M_{B_{c}^{-}}^{2}-M_{D_{s}^{\ast-}}^{2})h(q^{2})$ (45)
One can see from Eq. (43) and Eq. (44) that at $q^{2}=0,$ $F_{1}(0)=F_{2}(0).$
The form factors $V(q^{2}),A_{1}(q^{2})$ and $A_{2}(q^{2})$ can be written in
terms of $g_{+},g_{-}$ and $h$ as
$\displaystyle V(q^{2})$ $\displaystyle=$
$\displaystyle\frac{M_{B_{c}^{-}}+M_{D_{s}^{\ast-}}}{m_{b}+m_{s}}\left[g_{+}(q^{2})-q^{2}h_{1}(q^{2})\right]$
(46) $\displaystyle A_{1}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{M_{B_{c}^{-}}+M_{D_{s}^{\ast-}}}{m_{b}-m_{s}}\left[g_{+}(q^{2})+\frac{q^{2}}{M_{B_{c}^{-}}^{2}-M_{D_{s}^{\ast-}}^{2}}g_{-}(q^{2})\right]$
(47) $\displaystyle A_{2}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{M_{B_{c}^{-}}+M_{D_{s}^{\ast-}}}{m_{b}-m_{s}}\left[g_{+}(q^{2})-q^{2}h(q^{2})\right]-\frac{2M_{D_{s}^{\ast-}}}{M_{B_{c}^{-}}-M_{D_{s}^{\ast-}}}A_{0}(q^{2})$
(48)
By looking at Eq. (46) and Eq. (47) it is clear that the normalization of the
form factors $V$ and $A_{1}$ at $q^{2}=0$ is determined by a single constant
$g_{+}(0),$ where as from Eq. (48) the form factor $A_{2}$ at $q^{2}=0$ is
determined by two constants i.e. $g_{+}(0)$ and $A_{0}(0).$
### III.1 Pole Contribution
In $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$ decay, there will be a pole
contribution to $h_{1},g_{-},h$ and $A_{0}$ from
$B_{s}^{\ast}(1^{-}),B_{sA}^{\ast}(1^{+})$ and $B_{s}(0^{-})$ mesons which can
be parameterized as
$\displaystyle h_{1}|_{pole}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{g_{B_{s}^{\ast}B_{c}D_{s}^{\ast}}}{M_{B_{s}^{\ast}}^{2}}\frac{f_{T}^{B^{\ast}}}{1-q^{2}/M_{B^{\ast}}^{2}}=\frac{R_{V}}{M_{B_{s}^{\ast}}^{2}}\frac{1}{1-q^{2}/M_{B_{s}^{\ast}}^{2}}$
(49) $\displaystyle g_{-}|_{pole}$ $\displaystyle=$
$\displaystyle-\frac{g_{B_{sA}^{\ast}B_{c}D_{s}^{\ast}}}{M_{B_{sA}^{\ast}}^{2}}\frac{f_{T}^{B_{sA}^{\ast}}}{1-q^{2}/M_{B_{sA}^{\ast}}^{2}}=\frac{R_{A}^{S}}{M_{B_{sA}^{\ast}}^{2}}\frac{1}{1-q^{2}/M_{B_{sA}^{\ast}}^{2}}$
(50) $\displaystyle h|_{pole}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{f_{B_{SA}^{\ast}B_{c}D_{s}^{\ast}}}{M_{B_{sA}^{\ast}}^{2}}\frac{f_{T}^{B_{SA}^{\ast}}}{1-q^{2}/M_{B_{sA}^{\ast}}^{2}}=\frac{R_{A}^{D}}{M_{B_{sA}^{\ast}}^{2}}\frac{1}{1-q^{2}/M_{B_{sA}^{\ast}}^{2}}$
(51) $\displaystyle A_{0}(q^{2})|_{pole}$ $\displaystyle=$
$\displaystyle\frac{g_{B_{s}^{\ast}B_{c}D_{s}^{\ast}}}{M_{B_{s}^{\ast}}^{2}}f_{B_{s}}\frac{q^{2}/M_{B}^{2}}{1-q^{2}/M_{B}^{2}}=R_{0}\frac{q^{2}/M_{B_{s}}^{2}}{1-q^{2}/M_{B_{s}}^{2}}$
(52)
where the quantities $R_{V},R_{A}^{S},R_{A}^{D}$ and $R_{0}$ are related to
the coupling constants
$g_{B_{s}^{\ast}B_{c}D_{s}^{\ast}},g_{B_{sA}^{\ast}B_{c}D_{s}^{\ast}}$ and
$g_{B_{sA}^{\ast}B_{c}D_{s}^{\ast}}$, respectively. Here we would like to
mention that the above mentioned couplings aries as the lower pole mass,
because the higher pole masses of $B_{c}$ meson do not contribute for the
$B_{c}\to D_{s}^{\ast}l^{+}l^{-}$ decay. The form factors
$A_{1}(q^{2}),A_{2}(q^{2})$ and $V(q^{2})$ can be written in terms of these
quantities as
$\displaystyle V(q^{2})$ $\displaystyle=$
$\displaystyle\frac{M_{B_{c}^{-}}+M_{D_{s}^{\ast}}}{m_{b}+m_{s}}\left[g_{+}(q^{2})-\frac{R_{V}}{M_{B_{s}^{\ast}}^{2}}\frac{q^{2}}{1-q^{2}/M_{B_{s}^{\ast}}^{2}}\right]$
(53) $\displaystyle A_{1}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{M_{B_{c}^{-}}-M_{D_{s}^{\ast-}}}{m_{b}-m_{s}}\left[g_{+}(q^{2})+\frac{q^{2}}{M_{B_{c}^{-}}^{2}-M_{D_{s}^{\ast-}}^{2}}\tilde{g}_{-}(q^{2})+\frac{R_{A}^{S}}{M_{B_{sA}^{\ast}}^{2}}\frac{q^{2}}{1-q^{2}/M_{B_{sA}^{\ast}}^{2}}\right]$
(54) $\displaystyle A_{2}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{M_{B_{c}^{-}}+M_{D_{s}^{\ast-}}}{m_{b}-m_{s}}\left[g_{+}(q^{2})-\frac{R_{A}^{D}}{M_{B_{As}^{\ast}}^{2}}\frac{q^{2}}{1-q^{2}/M_{B_{sA}^{\ast}}^{2}}\right]-\frac{2M_{D_{s}^{\ast-}}}{M_{B_{c}}-M_{D_{s}^{\ast-}}}A_{0}(q^{2})$
(55)
Now, the behavior of $g_{+}(q^{2}),\tilde{g}_{-}(q^{2})$ and $A_{0}(q^{2})$ is
known from LEET and their form is 14
$\displaystyle g_{+}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{\xi_{\bot}(0)}{(1-q^{2}/M_{B}^{2})^{2}}=-\tilde{g}_{-}(q^{2})$
(56) $\displaystyle A_{0}(q^{2})$ $\displaystyle=$
$\displaystyle\left(1-\frac{M_{D_{s}^{\ast-}}^{2}}{M_{B_{c}}E_{D_{s}^{\ast-}}}\right)\xi_{\|}(0)+\frac{M_{D_{s}^{\ast-}}}{M_{B_{c}}}\xi_{\perp}(0)$
(57) $\displaystyle E_{D_{s}^{\ast}}$ $\displaystyle=$
$\displaystyle\frac{M_{B_{c}}}{2}\left(1-\frac{q^{2}}{M_{B_{c}}^{2}}+\frac{M_{D_{s}^{\ast}}^{2}}{M_{B_{c}}^{2}}\right)$
(58) $\displaystyle g_{+}(0)$ $\displaystyle=$ $\displaystyle\xi_{\bot}(0)$
(59)
The pole terms given in Eqs.(53-55) dominate near $q^{2}=M_{B_{s}^{\ast}}^{2}$
and $q^{2}=M_{B_{sA}^{\ast}}^{2}$. Just to make a remark that relations
obtained from the Ward identities can not be expected to hold for the whole
$q^{2}.$ Therefore, near $q^{2}=0$ and near the pole following parametrization
is suggested 14
$F(q^{2})=\frac{F(0)}{\left(1-q^{2}/M^{2}\right)(1-q^{2}/M^{\prime 2})}$ (60)
where $M^{2}$ is $M_{B_{s}^{\ast}}^{2}$ or $M_{B_{sA}^{\ast}}^{2}$, and
$M^{\prime}$ is the radial excitation of $M.$ The parametrization given in Eq.
(60) not only takes into account the corrections to single pole dominance
suggested by the dispersion relation approach 37 ; 38 ; 39 but also give the
correction of off-mass shell-ness of the couplings of $B_{s}^{\ast}$ and
$B_{sA}^{\ast}$ with the $B_{c}D_{s}^{\ast}$ channel.
Since $g_{+}(0)$ and $\tilde{g}_{-}(q^{2})$ have no pole at $\
q^{2}=M_{B_{s}^{\ast}}^{2},$ hence we get
$V(q^{2})(1-\frac{q^{2}}{M_{B^{\ast}}^{2}})|_{q^{2}=M_{B^{\ast}}^{2}}=-R_{V}\left(\frac{M_{B_{c}}+M_{D_{s}^{\ast}}}{m_{b}-m_{s}}\right)$
This becomes
$R_{V}\equiv-\frac{1}{2}g_{B_{s}^{\ast}B_{c}D_{s}^{\ast}}f_{B^{\ast}_{s}}=-\frac{g_{+}(0)}{1-M_{B^{\ast}}^{2}/M_{B^{\ast}}^{\prime
2}}$ (61)
and similarly
$R_{A}^{D}\equiv\frac{1}{2}f_{B_{sA}^{\ast}B_{c}D_{s}^{\ast}}f^{B^{\ast}_{sA}}_{T}=-\frac{g_{+}(0)}{1-M_{B_{sA}^{\ast}}^{2}/M_{B_{sA}^{\ast}}^{\prime
2}}$ (62)
We cannot use the parametrization given in Eq.(60) for the form factor
$A_{1}(q^{2}),$ since near $q^{2}=0,$ the behavior of $A_{1}(q^{2})$ is
$g_{+}(q^{2})\left[1-q^{2}/\left(M_{B_{c}^{-}}^{2}-M_{D_{s}^{\ast-}}^{2}\right)\right],$
therefore we can write $A_{1}(q^{2})$ as follows
$A_{1}(q^{2})=\frac{g_{+}(0)}{\left(1-q^{2}/M_{B_{sA}^{\ast}}^{2}\right)\left(1-q^{2}/M_{B_{sA}^{\ast}}^{\prime
2}\right)}\left(1-\frac{q^{2}}{M_{B_{c}^{-}}^{2}-M_{D_{s}^{\ast}}^{2}}\right)$
(63)
The only unkonown parameter in the above form factors calculation is
$g_{+}(0)$ and its value can be extracted by using the central value of
branching ratio for the decay $B_{c}^{-}\rightarrow D_{s}^{\ast-}\gamma$ 41 .
From the formula of decay rate
$\Gamma\left(B_{c}\rightarrow
D_{s}^{\ast}\gamma\right)=\frac{G_{F}^{2}\alpha}{32\pi^{4}}\left|V_{tb}V_{ts}^{\ast}\right|^{2}m_{b}^{2}M_{B_{c}}^{3}\times\left(1-\frac{M_{D_{s}^{\ast}}^{2}}{M_{B_{c}}^{2}}\right)^{3}\left|C_{7}^{eff}\right|^{2}\left|g_{+}(0)\right|^{2}$
(64)
and by putting the values of everything one can find the value of unknown
parameter $g_{+}(0)=0.32\pm 0.1$. In the forthcoming analysis we use the value
of $g_{+}(0)=0.42$ which was calculated in ref. 41 .
Using $f_{B_{c}}=0.35$ GeV we have prediction from Eq.(61) that
$g_{B_{s}^{\ast}B_{c}D_{s}^{\ast}}=10.38GeV^{-1}.$ (65)
Similarly the ratio of $S$ and $D$ wave couplings are predicted to be
$\frac{g_{B_{sA}^{\ast}B_{c}D_{s}^{\ast}}}{f_{B_{sA}^{\ast}B_{c}D_{s}^{\ast}}}=-0.42GeV^{2}$
(66)
The different values of the $F(0)$ are
$\displaystyle V(0)$ $\displaystyle=$
$\displaystyle\frac{M_{B_{c}^{-}}+M_{D_{s}^{\ast-}}}{m_{b}+m_{s}}g_{+}(0)$
(67) $\displaystyle A_{1}(0)$ $\displaystyle=$
$\displaystyle\frac{M_{B_{c}^{-}}-M_{D_{s}^{\ast-}}}{m_{b}-m_{s}}g_{+}(0)$
(68) $\displaystyle A_{2}(0)$ $\displaystyle=$
$\displaystyle\frac{M_{B_{c}^{-}}+M_{D_{s}^{\ast-}}}{m_{b}-m_{s}}g_{+}(0)-\frac{2M_{D_{s}^{\ast-}}}{M_{B_{c}^{-}}-M_{D_{s}^{\ast-}}}A_{0}(0)$
(69)
The calculation of the numerical values of $V(0)$ and $A_{1}(0)$ is quite
trivial but for the value of $A_{2}(0),$ the value of $A_{0}(0)$ has to be
known. Although LEET does not give any relationship between $\xi_{||}(0)$ and
$\xi_{\perp}(0)$, but in LCSR $\xi_{||}(0)$ and $\xi_{\perp}(0)$ are related
due to numerical coincidence 42
$\xi_{||}(0)\simeq\xi_{\perp}(0)=g_{+}(0)$ (70)
From Eq. (57) we have
$A_{0}(0)=1.12g_{+}(0)$
The value of the form factors at $q^{2}=0$ is given in Table-1
Table 1: Values of the form factors at $q^{2}=0$. $V(0)$ | $A_{1}(0)$ | $\tilde{A}_{2}(0)$ | $A_{0}(0)$ | |
---|---|---|---|---|---
$0.51\pm 0.17$ | $0.28\pm 0.08$ | $0.22\pm 0.07$ | $0.35\pm 0.11$ | |
and can be extrapolated for the other values of $q^{2}$ as follows:
$\displaystyle V(q^{2})$ $\displaystyle=$
$\displaystyle\frac{V(0)}{(1-q^{2}/M_{B_{s}^{\ast}}^{2})(1-q^{2}/M_{B_{s}^{\ast}}^{\prime
2})}$ (71) $\displaystyle A_{1}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{A_{1}(0)}{(1-q^{2}/M_{B_{sA}^{\ast}}^{2})(1-q^{2}/M_{B_{sA}^{\ast}}^{\prime
2})}$ (72) $\displaystyle A_{2}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{\tilde{A}_{2}(0)}{(1-q^{2}/M_{B_{sA}^{\ast}}^{2})(1-q^{2}/M_{B_{sA}^{\ast}}^{\prime
2})}$
$\displaystyle-\frac{2M_{D_{s}^{\ast-}}}{M_{B_{c}^{-}}-M_{D_{s}^{\ast-}}}\frac{A_{0}(0)}{(1-q^{2}/M_{B_{s}}^{2})(1-q^{2}/M_{B_{s}}^{\prime
2})}$
The behavior of form factors $V(q^{2}),$ $A_{1}(q^{2})$ and $A_{2}(q^{2})$ are
shown in Fig. 1.
(a)(b)(c) | |
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Figure 1: Form factors are plotted as a function of $q^{2}$. Solid line,
dashed line and long-dashed line correspond to $g_{+}(0)$ equal to 0.42, 0.32
and 0.22 respectively.
## IV Physical Observables for $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$
In this section we will present the calculations of the physical observables
like the decay rates and the helicity fractions of $D_{s}^{\ast}$ meson. From
Eq. (5) it is straightforward to write
$\displaystyle\mathcal{M}_{B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}}$
$\displaystyle=$
$\displaystyle-\frac{G_{F}\alpha}{2\sqrt{2}\pi}V_{tb}V_{ts}^{\ast}\left[T_{\mu}^{1}(\bar{l}\gamma^{\mu}l)+T_{\mu}^{2}\left(\bar{l}\gamma^{\mu}\gamma^{5}l\right)\right]$
(74)
where
$\displaystyle T_{\mu}^{1}$ $\displaystyle=$ $\displaystyle
f_{1}(q^{2})\epsilon_{\mu\nu\alpha\beta}\varepsilon^{\ast\nu}p^{\alpha}k^{\beta}+if_{2}(q^{2})\varepsilon_{\mu}^{\ast}+if_{3}(q^{2})(\varepsilon^{\ast}\cdot
q)P_{\mu}$ (75) $\displaystyle T_{\mu}^{2}$ $\displaystyle=$ $\displaystyle
f_{4}(q^{2})\epsilon_{\mu\nu\alpha\beta}\varepsilon^{\ast\nu}p^{\alpha}k^{\beta}+if_{5}(q^{2})\varepsilon_{\mu}^{\ast}+if_{6}(q^{2})(\varepsilon^{\ast}\cdot
q)P_{\mu}$ (76)
The functions $f_{1}$ to $f_{6}$ in Eq.(75) and Eq. (76) are known as
auxiliary functions, which contains both long distance (Form factors) and
short distance (Wilson coefficients) effects and these can be written as
$\displaystyle f_{1}(q^{2})$ $\displaystyle=$ $\displaystyle
4(m_{b}+m_{s})\frac{C_{7}^{eff}}{q^{2}}F_{1}(q^{2})+C_{9}^{eff}\frac{V(q^{2})}{M_{B_{c}}+M_{D_{s}^{\ast}}}$
$\displaystyle f_{2}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{C_{7}^{eff}}{q^{2}}4(m_{b}-m_{s})F_{2}(q^{2})\left(M_{B_{c}}^{2}-M_{D_{s}^{\ast}}^{2}\right)+C_{9}^{eff}A_{1}(q^{2})\left(M_{B_{c}}+M_{D^{\ast}}\right)$
$\displaystyle f_{3}(q^{2})$ $\displaystyle=$
$\displaystyle-\left[C_{7}^{eff}4(m_{b}-m_{s})\left(F_{2}(q^{2})+q^{2}\frac{F_{3}(q^{2})}{\left(M_{B_{c}}^{2}-M_{D_{s}^{\ast}}^{2}\right)}\right)+C_{9}^{ff}\frac{A_{2}(q^{2})}{M_{B_{c}}+M_{D_{s}^{\ast}}}\right]$
$\displaystyle f_{4}(q^{2})$ $\displaystyle=$ $\displaystyle
C_{10}\frac{V(q^{2})}{M_{B_{c}}+M_{D_{s}^{\ast}}}$ $\displaystyle
f_{5}(q^{2})$ $\displaystyle=$ $\displaystyle
C_{10}A_{1}(q^{2})\left(M_{B_{c}}+M_{D_{s}^{\ast}}\right)$ $\displaystyle
f_{6}(q^{2})$ $\displaystyle=$ $\displaystyle-
C_{10}\frac{A_{2}(q^{2})}{M_{B_{c}}+M_{D_{s}^{\ast}}}$ $\displaystyle
f_{0}(q^{2})$ $\displaystyle=$ $\displaystyle C_{10}A_{0}(q^{2})$ (77)
The next task is to calculate the decay rate and the helicity fractions of
$D_{s}^{\ast}$ meson in terms of these auxiliary functions.
### IV.1 The Differential Decay Rate of $B_{c}\rightarrow
D_{s}^{\ast}l^{+}l^{-}$
In the rest frame of $B_{c}$ meson the differential decay width of
$B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$ can be written as
$\displaystyle\frac{d\Gamma(B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-})}{dq^{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{\left(2\pi\right)^{3}}\frac{1}{32M_{B_{c}}^{3}}\int_{-u(q^{2})}^{+u(q^{2})}du\left|\mathcal{M}_{B_{c}\rightarrow
D_{s}^{\ast}l^{+}l^{-}}\right|^{2}$ (78)
where
$\displaystyle q^{2}$ $\displaystyle=$
$\displaystyle(p_{l^{+}}+p_{l^{-}})^{2}$ (79) $\displaystyle u$
$\displaystyle=$
$\displaystyle\left(p-p_{l^{-}}\right)^{2}-\left(p-p_{l^{+}}\right)^{2}$ (80)
Now the limits on $q^{2}$ and $u$ are
$\displaystyle 4m_{l}^{2}$ $\displaystyle\leq$ $\displaystyle
q^{2}\leq(M_{B_{c}}-M_{D_{s}^{\ast}})^{2}$ (81) $\displaystyle-u(q^{2})$
$\displaystyle\leq$ $\displaystyle u\leq u(q^{2})$ (82)
with
$u(q^{2})=\sqrt{\lambda\left(1-\frac{4m_{l}^{2}}{q^{2}}\right)}$ (83)
where
$\lambda\equiv\lambda(M_{B_{c}}^{2},M_{D_{s}^{\ast}}^{2},q^{2})=M_{B_{c}}^{4}+M_{D_{s}^{\ast}}^{4}+q^{4}-2M_{B_{c}}^{2}M_{D_{s}^{\ast}}^{2}-2M_{D_{s}^{\ast}}^{2}q^{2}-2q^{2}M_{B_{c}}^{2}$
The decay rate of $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$ can easily
obtained in terms of auxiliary function by integrating on $u$ (c.f. Eq. (78))
as
$\displaystyle\frac{d\Gamma(B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-})}{dq^{2}}$
$\displaystyle=$
$\displaystyle\frac{G_{F}^{2}\left|V_{tb}V_{ts}^{\ast}\right|^{2}\alpha^{2}}{2^{11}\pi^{5}3M_{B_{c}}^{3}M_{D_{s}^{\ast}}^{2}q^{2}}u(q^{2})\bigg{[}24\left|f_{0}(q^{2})\right|^{2}m_{l}^{2}M_{D_{s}^{\ast}}^{2}\lambda$
$\displaystyle+8M_{D_{s}^{\ast}}^{2}q^{2}\lambda[(2m_{l}^{2}+q^{2})\left|f_{1}(q^{2})\right|^{2}-(4m_{l}^{2}-q^{2})\left|f_{4}(q^{2})\right|^{2}]$
$\displaystyle+\lambda[(2m_{l}^{2}+q^{2})\left|f_{2}(q^{2})+(M_{B_{c}}^{2}-M_{D_{s}^{\ast}}^{2}-q^{2})f_{3}(q^{2})\right|^{2}$
$\displaystyle-(4m_{l}^{2}-q^{2})\left|f_{5}(q^{2})+(M_{B_{c}}^{2}-M_{D_{s}^{\ast}}^{2}-q^{2})f_{6}(q^{2})\right|^{2}]$
$\displaystyle+4M_{D_{s}^{\ast}}^{2}q^{2}[(2m_{l}^{2}+q^{2})\left(3\left|f_{2}(q^{2})\right|^{2}-\lambda\left|f_{3}(q^{2})\right|^{2}\right)$
$\displaystyle-(4m_{l}^{2}-q^{2})\left(3\left|f_{5}(q^{2})\right|^{2}-\lambda\left|f_{6}(q^{2})\right|^{2}\right)]\bigg{]}$
### IV.2 HELICITY FRACTIONS OF $D_{s}^{\ast}$ IN $B_{c}\rightarrow
D_{s}^{\ast}l^{+}l^{-}$
We now discuss helicity fractions of $D_{s}^{\ast}$ in $B_{c}\rightarrow
D_{s}^{\ast}l^{+}l^{-}$ which are intersting variable and are as such
independent of the uncertainities arising due to form factors and other input
parameters. The final state meson helicity fractions were already discussed in
literature for $B\rightarrow K^{\ast}\left(K_{1}\right)l^{+}l^{-}$ decays 22 ;
23 . Even for the $K^{\ast}$ vector meson, the longitudinal helicity fraction
$f_{L}$ has been measured by Babar collaboration for the decay $B\rightarrow
K^{\ast}l^{+}l^{-}(l=e,\mu)$ in two bins of momentum transfer and the results
are 46
$\displaystyle f_{L}$ $\displaystyle=$ $\displaystyle 0.77_{-0.30}^{+0.63}\pm
0.07,\ \ \ \ \ 0.1\leq q^{2}\leq 8.41GeV^{2}$ $\displaystyle f_{L}$
$\displaystyle=$ $\displaystyle 0.51_{-0.25}^{+0.22}\pm 0.08,\ \ \ \ \
q^{2}\geq 10.24GeV^{2}$
while the average value of $f_{L}$ in full $q^{2}$ range is
$f_{L}=0.63_{-0.19}^{+0.18}\pm 0.05,\ \ q^{2}\geq 0.1GeV^{2}$ (86)
The explicit expression of the helicity fractions for $B_{c}^{-}\rightarrow
D_{s}^{\ast-}l^{+}l^{-}$ decay can be written as
$\displaystyle\frac{d\Gamma_{L}(q^{2})}{dq^{2}}$ $\displaystyle=$
$\displaystyle\frac{G_{F}^{2}\left|V_{tb}V_{ts}^{\ast}\right|^{2}\alpha^{2}}{2^{11}\pi^{5}}\frac{u(q^{2})}{M_{B_{c}}^{3}}\times$
(89)
$\displaystyle\frac{1}{3}\frac{1}{q^{2}M_{D_{s}^{\ast}}^{2}}\left[\begin{array}[]{c}24\left|f_{0}(q^{2})\right|^{2}m_{l}^{2}M_{D_{s}^{\ast}}^{2}\lambda+(2m_{l}^{2}+q^{2})\left|\left(M_{B_{c}}^{2}-M_{D_{s}^{\ast}}^{2}-q^{2}\right)f_{2}(q^{2})+\lambda
f_{3}(q^{2})\right|^{2}\\\
+\left(q^{2}-4m_{l}^{2}\right)\left|\left(M_{B_{c}}^{2}-M_{D_{s}^{\ast}}^{2}-q^{2}\right)f_{5}(q^{2})+\lambda
f_{6}(q^{2})\right|^{2}\end{array}\right]$
$\displaystyle\frac{d\Gamma_{+}(q^{2})}{dq^{2}}$ $\displaystyle=$
$\displaystyle\frac{G_{F}^{2}\left|V_{tb}V_{ts}^{\ast}\right|^{2}\alpha^{2}}{2^{11}\pi^{5}}\frac{u(q^{2})}{M_{B_{c}}^{3}}\times$
(91)
$\displaystyle\frac{4}{3}\left[\left(q^{2}-4m_{l}^{2}\right)\left|f_{5}(q^{2})-\sqrt{\lambda}f_{4}(q^{2})\right|^{2}+\left(q^{2}+2m_{l}^{2}\right)\left|f_{2}(q^{2})-\sqrt{\lambda}f_{1}(q^{2})\right|^{2}\right]$
$\displaystyle\frac{d\Gamma_{-}(q^{2})}{dq^{2}}$ $\displaystyle=$
$\displaystyle\frac{G_{F}^{2}\left|V_{tb}V_{ts}^{\ast}\right|^{2}\alpha^{2}}{2^{11}\pi^{5}}\frac{u(q^{2})}{M_{B_{c}}^{3}}\times$
(92)
$\displaystyle\frac{4}{3}\left[\left(q^{2}-4m_{l}^{2}\right)\left|f_{5}(q^{2})+\sqrt{\lambda}f_{4}(q^{2})\right|^{2}+\left(q^{2}+2m_{l}^{2}\right)\left|f_{2}(q^{2})+\sqrt{\lambda}f_{1}(q^{2})\right|^{2}\right]$
where the auxiliary functions and the corresponding form factors are given in
Eq.(77) and Eqs.(71-LABEL:44). Finally the longitudinal and transverse
helicity amplitude becomes
$\displaystyle f_{L}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{d\Gamma_{L}(q^{2})/dq^{2}}{d\Gamma(q^{2})/dq^{2}}$
$\displaystyle f_{\pm}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{d\Gamma_{\pm}(q^{2})/dq^{2}}{d\Gamma(q^{2})/dq^{2}}$
$\displaystyle f_{T}(q^{2})$ $\displaystyle=$ $\displaystyle
f_{+}(q^{2})+f_{-}(q^{2})$ (93)
so that the sum of the longitudinal and transverse helicity amplitudes is
equal to one i.e. $f_{L}(q^{2})+f_{T}(q^{2})=1$ for each value of $q^{2}$22 .
## V Numerical Analysis.
In this section we present the numerical analysis of the branching ratio and
helicity fractions of $D_{s}^{\ast}$ meson in $B_{c}\rightarrow
D_{s}^{\ast}l^{+}l^{-}(l=\mu,\tau)$ both in the SM and in ACD model. One of
the main input parameters are the form factors which are non perturbative
quantities and are the major source of uncertainties. Here we calculated the
form factors using the Ward identities and their dependence on momentum
transfer $q^{2}$ is given in Section III. We have used next-to-leading order
approximation for the Wilson Coefficients at the renormalization scale
$\mu=m_{b}.$ It has already been mentioned that besides the contribution in
the $C_{9}^{eff}$, there are long distance contributions resulting from the
$c\bar{c}$ resonances like $J/\psi$ and its excited states. For the present
analysis we do not take into account these long distance effects.
The numerical results for the decay rates and helicity fractions of
$D_{s}^{\ast}$ for the decay mode $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$
both for the SM and ACD model are depicted in Figs. 2-4. Figs. 2 (a, b) shows
the differential decay rate of $B_{c}\rightarrow
D_{s}^{\ast}l^{+}l^{-}(l=\mu,\tau).$ One can see that there is a significant
enhancement in the decay rate due to KK-contribution for $1/R=300$ GeV,
whereas the value of the decay rate is shifted towards the SM at large value
of $1/R$ , both in small and large value of momentum transfer $q^{2}.$
(a)(b) | |
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Figure 2: Branching ratio for the $B\rightarrow D_{s}^{\ast}l^{+}l^{-}$
$(l=\mu,\tau)$ decays as functions of $q^{2}$ for different values of $1/R$.
Solid line correspond to SM value,dotted line is for $1/R=300$, dashed is for
$1/R=500$, long dashed line is for $1/R=700$.
In general the sensitivity on $1/R$ is usually masked by the uncertainties
which arises due to the number of sources. Among them the major one lies in
the numerical analysis of $B_{c}\to D^{*}_{s}l^{+}l^{-}$ decay originated from
the $B_{c}\to D^{*}_{s}$ transition form factors calculated in the present
approach as shown in Table I, which can bring about almost $40\%$ errors to
the differential decay rate of above mentioned decay, which showed that it is
not a very suitable tool to look for the new physics. The large uncertainties
involved in the form factors are mainly from the variations of the decay
constant of $B_{c}$ meson and also there are some uncertainties from the
strange quark mass $m_{s}$, which are expected to be very tiny on account of
the negligible role of $m_{s}$ suppressed by the much larger energy scale of
$m_{b}$. Moreover, the uncertainties of the charm quark and bottom quark mass
are at the $1\%$ level, which will not play significant role in the numerical
analysis and can be dropped out safely. It also needs to be stressed that
these hadronic uncertainties almost have no influence on the various
asymmetries including the polarization asymmetries of final state meson on
account of the serious cancelation among different polarization states and
this make them one of the best tool to look for physics beyond the SM.
Figs. 3 (a, b) shows the longitudinal and transverse helicity fractions of
$D_{s}^{\ast}$ for the decay $B_{c}\rightarrow D_{s}^{\ast}\mu^{+}\mu^{-}$
where we have used the central value of the form factors which we have
calculated in Section III. Choosing the different values of compactification
radius $1/R$, one can see from the graphs that the effect of extra dimensions
are quite significant at a particular region of $q^{2}$. These effects are
constructive for the case of transverse helicity fraction and destructive for
the case of longitudinal helicity fraction.
(a)(b) | |
---|---|---
Figure 3: Longitudinal Lepton polarization Fig.1a and Transverse Lepton
polarization Fig.2b for the $B\rightarrow D_{s}^{\ast}\mu^{+}\mu^{-}$ decays
as functions of $q^{2}$ for different values of $1/R$. Solid line correspond
to SM value,dotted line is for $1/R=300$, dashed is for $1/R=500$, long dashed
line is for $1/R=700$.
Similarly, Figs. 4 (a,b) show the helicity fraction of $D_{s}^{\ast}$ for the
decay $B_{c}\rightarrow D_{s}^{\ast}\tau^{+}\tau^{-}$ where one can see that
the effects of the extra dimensions are mild as compared to the case of
$B_{c}\rightarrow D_{s}^{\ast}\mu^{+}\mu^{-}$ . Moreover from Figs.2-4 it is
clear that each value of momentum transfer $q^{2}$ the sum of the longitudinal
and transverse helicity fractions are equal to one, i.e.
$f_{L}(q^{2})+f_{T}(q^{2})=1$.
(a)(b) | |
---|---|---
Figure 4: Longitudinal Lepton polarization Fig.1a and Transverse Lepton
polarization Fig.2b for the $B\rightarrow D_{s}^{\ast}\tau^{+}\tau^{-}$ decays
as functions of $q^{2}$ for different values of $1/R$. Solid line correspond
to SM value,dotted line is for $1/R=300$, dashed is for $1/R=500$, long dashed
line is for $1/R=700$.
## VI Conclusion:
We investigated the semileptonic decay $B_{c}\rightarrow
D_{s}^{\ast}l^{+}l^{-}$ $(l=\mu,\tau)$ using the Ward identities. The form
factors have been calculated and we found that the normalization of the form
factors in terms of a single universal constant $g_{+}(0)$. The value of
$g_{+}(0)=0.42$ is obtained from the decay $B_{c}\rightarrow
D_{s}^{\ast}\gamma$ 41 . Considering the radial excitation at lower pole
masses $M$ ( where $M=M_{B_{s}^{\ast}}$ and $M_{B_{sA}^{\ast}})$ one can
predict the coupling of $B_{s}^{\ast}$ with $B_{c}D_{s}^{\ast}$ channel as
indicated in Eq.(65) which is $g_{B_{s}^{\ast}B_{c}D_{s}^{\ast}}=10.38$
GeV${}^{-1}.$ Also we predicted the ratio of $S$ and $D$ wave couplings
$\frac{g_{B_{sA}^{\ast}B_{c}D_{s}^{\ast}}}{f_{B_{sA}^{\ast}B_{c}D_{s}^{\ast}}}=-0.42$
$GeV^{2}$ given in Eq.(66). The form factors are summarized in
Eqs.(71-LABEL:44) and their values at $q^{2}=0$ are given in Tabel-I. Using
these form factors we studied the observables, i.e. the branching ratio and
helicity fraction of $D_{s}^{\ast}$ in the decay $B_{c}\rightarrow
D_{s}^{\ast}l^{+}l^{-}$ $(l=e,\mu)$ both in SM and in ACD model, which has one
additional parameter i.e. the inverse compactification radius $1/R.$ The
effects of extra dimensions to the helicity fraction of $D_{s}^{\ast}$ is very
mild for the case when the tauon $(\tau)$ is taken as a final state lepton as
shown in fig 3, however the effects of extra dimensions are quite significant
for the case when muon ($\mu$) is taken as a final state lepton as shown in
fig 2. In near future when LHC is fully operational where more data is
available, will put a stringent constraint on compactification radius $R$ and
gives us a deep understanding of $B$ Physics.
## Acknowledgements
The authors would like to thank Profs. Riazuddin and Fayyazuddin for their
valuable guidance and helpful discussions. The authors M. A. P. and M. J. A.
would like to acknowledge the facilities provided by National Centre for
Physics during this work.
## References
* (1) S. L. Glashow, J. Iliopoulos, and L. Maiani, Phys. Rev. D2 (1970) 1285.
* (2) N. Cabbibo Phys. Rev. Lett. 10 (1963) 531; M. Kobayashi and K. Maskawa, Prog. Theor. Phys. 49 (1973) 652.
* (3) S. R. Choudhury, A. S. Cornell and Naveen Gaur, arXiv: 0911.4783 [hep-ph]
* (4) CLEO Collaboration, M. S. Alam, $et.al.,$Phys. Rev. Lett. 74 (1995) 2885
* (5) A. Ali, Int. J. Mod. Phys. A20 (2005) 5080
* (6) CDF Collaboration, F.Abe $et$ $al.,$ Phys. Rev. D 58 (1998) 112004 .
* (7) D. S. Du, Z. Wang, Phys. Rev. D39 (1989) 1342 ; C. H. Chang, Y.Q. Chen, ibid. 48 (1993) 4086 ; K. Cheung, Phys. Rev. Lett. 71 (1993) 3413 ; E. Braaten, K. Cheung, T.Yuan, Phys. Rev. D48 (1993) R5049.
* (8) Sheldon Stone, hep-ph/9709500.
* (9) A. Faessler, Th. Gutsche, M. A. Ivanov, J. G. Korner and V. E. Lyubovitskij, Eur. Phys. J. C4 (2002) 18.
* (10) K. Azizi and V. Bashiry, Phys. Rev. D76 (2007) 114007 ; T. M. Aliev and M. Savci, Phys. Lett. B434 (1998) 358 ;T. M. Aliev and M. Savci, J. Phys.G24 (1998) 2223;T. M. Aliev and M. Savci, Eur. Phys. J. C47 (2006) 413 ; T. M. Aliev and M. Savci, Phys. Lett. B480 (2000) 97; N. Ghahramany, R. Khosravi and Z. Naseri, Phys. Rev. D81 (2010) 016012; N. Ghahramany, R. Khosravi and Z. Naseri, Phys. Rev. D81 (2010) 036005; K. Azizi. F. Falahati,V. Bashiry and S. M. Zebarjad, Phys. Rev. D77 (2008)114024.
* (11) C. Q. Geng, C.W. Hwang, and C. C. Liu, Phys. Rev. D65 (2002) 094037.
* (12) A. H. S. Gilani, Riazuddin, T.A.Al-Aithan, JHEP 09 (2003) 065.
* (13) M. S. Khan, M. J. Aslam, A. H. S. Gilani and Riazuddin, Eur. Phys. J.C 49 (2007) 665-674.
* (14) M. Ali Paracha, Ishtiaq Ahmed and M. Jamil Aslam, Eur. Phys. J.C 52 (2008) 967-973.
* (15) J. Charles, A. Le Yaouanc, L. Oliver, O. Pene and J. C. Raynal, Phys. Rev. D60 (1999) 014001; M. Jamil Aslam and Riazuddin, Phys. Rev. D72 (2005) 094019; M. Jamil Aslam, Eur. Phys.J. C49 (2007) 651.
* (16) I. Antoniadis, Phys. Lett. B246 (1990) 377; K. R. Dienes, E. Dudas and T. Gherghetta, Phys. Lett. B436 (1998) 55; N. Arkani-Hamed and M. Schmaltz, Phys. Rev. D 61 (2000) 033005; N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B429 (1998) 263 ; L. Randall and R. Sundrum, Phys. Rev. Lett. 83 (1999) 3370 ; L.Randall and R.Sundrum, Phys. Rev. Lett. 83 (1999) 4690\.
* (17) T. Appelquist, H. C. Cheng and B. A. Dobrescu, Phys. Rev. D64 (2001) 035002.
* (18) A. J. Buras, M. Spranger and A.Weiler, Nucl. Phys. B660 (2003) 225; A. J. Buras, A. Poschenrieder, M. Spranger and A.Weiler, Nucl. Phys. B678 (2004) 455\.
* (19) K. Agashe, N. G. Deshpande and G. H. Wu, Phys. Lett. B514 (2001) 309\.
* (20) T. Appelquist and H. U. Yee, Phys. Rev. D67 (2003) 055002; M. V. Carlucci, P. Colangelo and F. De Fazio, Phys. Rev. D80 (2009) 055033.
* (21) P. Colangelo, F. De Fazio, R. Feerandes and T. N. Pham, Phys. Rev. D 74 (2006) 115006 [arXiv : hep-ph/0610044].
* (22) Asif Saddique, M. Jamil Aslam and Cai-Dian Lu, arXiv: 0803.0192v1 [hep-ph].
* (23) T. Goto et al., Phys. Rev. D 55 (1997) 4273; T. Goto, Y. Okada and Y. Shimizu, Phys. Rev. D 58 (1998) 094006; S. Bertolini, F. Borzynatu, A. Masiero and G. Ridolfi, Nucl. Phys. B 353 (1991) 591.
* (24) C.S. Kim, T. Morozumi, A.I. Sanda, Phys. Lett. B218 (1989) 343.
* (25) X. G. He, T. D. Nguyen and R. R. Volkas, Phys. Rev. D38 (1988) 814.
* (26) B. Grinstein, M.J. Savage, M.B. Wise, Nucl. Phys. B319 (1989) 271.
* (27) N. G. Deshpande, J. Trampetic and K. Panose, Phys. Rev. D39 (1989) 1461.
* (28) P. J. O’Donnell and H. K. K. Tung, Phys. Rev. D43 (1991) 2067.
* (29) N. Paver and Riazuddin, Phys. Rev. D45 (1992) 978.
* (30) A. Ali, T. Mannel and T. Morozumi, Phys. Lett. B273 (1991) 505.
* (31) D. Melikhov, N. Nikitin and S. Simula, Phys. Lett. B430 (1998) 332 [arXiv:hep-ph/9803343].
* (32) J. M. Soares, Nucl. Phys. B367 (1991) 575.
* (33) G. M. Asatrian and A. Ioannisian, Phys. Rev. D54 (1996) 5642 [arXiv:hep-ph/9603318].
* (34) J. M. Soares, Phys. Rev. D53 (1996) 241 [arXiv:hep-ph/9503285].
* (35) C. H. Chen and C. Q. Geng, Phys. Rev. D64 (2001) 074001 [arXiv:hep-ph/0106193].
* (36) C. A. Dominguez, N. Paver, Riazuddin, Z. Phys. C48 (1990) 55.
* (37) C. A. Dominguez, N. Paver, Riazuddin, Phys. Lett. B214 (1988) 459.
* (38) C. A. Dominguez, N. Paver, Z. Phys. C41 (1988) 217.
* (39) K. Azizi and V. Bashiry, Phys. Rev. D76 (2007) 114007.
* (40) J. Charles, A. Le Yaouanc, L. Oliver, O. Pene, J. C. Raynal, Phys. Rev. D60 (1999) 014001.
* (41) P. Colangelo, F. De Fazio, R. Ferrandes, T.N. Pham, Phys.Rev. D73 (2006) 115006; P.Colangelo, F. De Fazio, P. Santorelli and E. Scrimieri, Phys. Rev. D53 (1996) 3672; Erratum-ibid. D57 (1998) 3186.
* (42) S. Rai Choudhury , N. Gaur and N. Mahajan, Phys. Rev. D66 (2002) 054003 [arXiv: hep-ph/0203041] ; S. R. Choudhury and N. Gaur, arXiv:hep-ph/0205076 ; S. R. Choudhury and N. Gaur, arXiv:hep-ph/0207353 ; T. M. Aliev, V. Bashiry and M. Savci, Phys. Rev. D71 (2005) 035013 [arXiv:hep-ph/0411327] ; U. O. Yilmaz, B. B. Sirvanli and G. Turan, Nucl. Phys. 692 (2004) 249 [arXiv:hep-ph/0407006] ; U. O. Yilmaz, B. B. Sirvanli and G. Turan, Eur. Phys. J. C30 (2003) 197 [arXiv:hep-ph/0304100].
* (43) A. J. Buras et al. Nucl. Phys. B424 (1994) 374.
* (44) B. Aubert et al. [BABAR Collaboration], Phys. Rev. D73 (2006) 092001.
|
arxiv-papers
| 2011-01-12T11:04:36 |
2024-09-04T02:49:16.347870
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Ali Paracha, Ishtiaq Ahmed, M. Jamil Aslam",
"submitter": "Muhammad Jamil Aslam",
"url": "https://arxiv.org/abs/1101.2323"
}
|
1101.2326
|
# Nucleon form factors and structure functions from $N_{\mathrm{f}}=2$ clover
fermions
QCDSF/UKQCD Collaboration: S. Collins,a M. Göckeler,a Ph. Hägler,a T.
Hemmert,a R. Horsley,b Y. Nakamura,a,c A. Nobile,a H. Perlt,d ,e P.E.L.
Rakow,f A. Schäfer,a G. Schierholz,e A. Sternbeck,a H. Stüben,g F. Wintera and
J.M. Zanottib
a Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg,
Germany
b School of Physics, University of Edinburgh, Edinburgh EH9 3JZ, UK
c Center for Computational Sciences, University of Tsukuba, Ibaraki 305-8577,
Japan
d Institut für Theoretische Physik, Universität Leipzig, 04109 Leipzig,
Germany
e Deutsches Elektronen-Synchrotron DESY, 15738 Zeuthen, Germany
f Theoretical Physics Division, Department of Mathematical Sciences,
University of Liverpool, Liverpool L69 3BX, UK
g Konrad-Zuse-Zentrum für Informationstechnik Berlin, 14195 Berlin, Germany
Email
###### Abstract:
We give an update on our ongoing efforts to compute the nucleon’s form factors
and moments of structure functions using $N_{\mathrm{f}}=2$ flavours of non-
perturbatively improved Clover fermions. We focus on new results obtained on
gauge configurations where the pseudo-scalar meson mass is in the range of
170-270 MeV. We will compare our results with various estimates obtained from
chiral effective theories since we have some overlap with the quark mass
region where results from such theories are believed to be applicable.
## 1 Introduction
Over years significant efforts have been made to use lattice techniques to
investigate the structure of the nucleon. Of particular interest are the
Parton Distribution Functions (PDFs) and form factors. The latter encode
information about charge distribution and magnetization while the PDFs tell us
about the distribution of momentum and spin. While some of the related
observables can be determined with good accuracy by experiments (e.g. the
nucleon’s axial charge $g_{\mathrm{A}}$) other quantities are difficult to
access (like the tensor charge $g_{\mathrm{T}}$).
A precise determination of moments of nucleon PDFs and form factors on the
lattice turned out to be rather challenging. It continues to be difficult to
reach sufficient control on all systematic errors such as finite size effects,
lattice artefacts and the influence of the chiral extrapolation. Simulations
are performed in volumes of a size where some quantities exhibit significant
finite size effects. The available lattice data of the quantities of interest
show no significant discretization effects. But current simulations only probe
a small window of lattice spacings thus providing us with limited control on
the continuum extrapolation. From chiral effective theories (ChEFT) there are
indications that the quark mass dependence close to the physical pion mass is
very strong. Therefore extending lattice simulations into the region where
$m_{\pi}\leq m_{\mathrm{PS}}\lesssim 300~{}\mbox{MeV}$ has become a major goal
for recent calculations.
## 2 Simulation details
For our simulations we use Wilson glue and $N_{\mathrm{f}}=2$ degenerate
flavours of Clover fermions, where the improvement coefficient
$c_{\mathrm{SW}}$ has been determined non-perturbatively. Most of our
configurations have been generated using the BQCD implementation of the HMC
algorithm [1]. Various algorithmic improvements have been applied which
accelerate this algorithm, such as the Hasenbusch preconditioning and the use
of multiple time-scales, or reducing the time spent for matrix inversion. For
instance, chronological guess and the Schwarz Alternating Procedure (SAP) are
used to start the inversion and to precondition the fermion matrix [2].
These algorithmic improvements plus recent increase in computing resources
enabled simulations in the region of small quark masses, i.e. in a region
where the pseudo-scalar mass is smaller than 300 MeV. Fig. 1a shows the
parameter region of our simulations. When approaching physical quark masses
larger lattices are needed to stay in the region $m_{\mathrm{PS}}L\gtrsim 3$
where finite size effects are expected to be sufficiently small (see Fig. 1b).
To investigate such finite size effects we have also performed simulations
with $m_{\mathrm{PS}}L<3$.
(a)
(b)
Figure 1: The left panel shows the simulation points in the
$m_{\mathrm{PS}}^{2}$ vs. $m_{\mathrm{PS}}L$ plane. In the right panel dashed
lines show the lattice spacing and box sizes for which simulations have been
performed. In both figures the continuous lines show where
$m_{\mathrm{PS}}L=3$.
We compute the quark propagators using point sources which we (Jacobi) smear
to improve overlap with the ground state. For the three-point correlation
functions we apply standard sequential source techniques. The distance between
source and sink is about 1 fm. Throughout this paper we will ignore
contributions coming from disconnected terms. While these anyhow cancel in the
iso-vector cases, results for the iso-scalar case maybe affected by an
uncontrolled systematic error.
To set the scale we use the Sommer parameter $r_{0}/a$ which we extrapolated
to the chiral limit at each beta. While on the lattice this quantity can be
determined with small statistical errors, there is no experimental
determination. We therefore computed the dimensionless quantity
$(am_{N})(r_{0}/a)$ on the lattice and use the experimentally well known mass
of the nucleon $m_{N}$ to obtain $r_{0}=0.467\,\mbox{fm}$.
Most of the quantities considered in this paper need to be renormalised. The
renormalisation constants have been determined using the
$\mathrm{RI}^{\prime}\mathrm{-MOM}$ scheme [3], except for the vector current
renormalisation constant $Z_{V}$. Here we applied the condition that the
nucleon’s local vector current at zero momentum must be 1. If necessary, the
results are converted into $\overline{\mathrm{MS}}$ scheme using the 4- and
2-3-loop expressions of the $\beta$ function and corresponding anomalous
dimension $\gamma$, respectively.
## 3 Lowest moments of PDFs
Let us first consider the lowest moment of the polarized nucleon PDF $\langle
1\rangle_{\Delta q}$ (also known as axial coupling constant $g_{\mathrm{A}}$).
This quantity is determined from the renormalised axial vector current
$A_{\mu}^{R}=Z_{A}\,(1+b_{A}\,am_{q})A_{\mu}$, where
$am_{q}=(1/\kappa-1/\kappa^{(S)}_{c})/2$. $Z_{A}$ is known non-perturbatively
[3], for $b_{A}$ we use a tadpole improved one-loop perturbation theory
result.
(a)
(b)
Figure 2: The left panel shows $g_{\mathrm{A}}$ as a function of
$m_{\mathrm{PS}}^{2}$. The open and filled diamonds show the lattice results
before and after correction of finite size effects, respectively. The star
indicates the experimental result. The line shows a fit to the data as
described in the text. The right panel shows the relative finite size effects
determined on the lattice (symbols) and obtained from a fit to an expression
from ChEFT.
We have fitted our lattice results to an expression from ChEFT based on the
SSE formalism. Using this formalism both the quark mass dependence [4] and the
finite volume dependence [5] have been calculated. Since our results for
different lattice spacings do not exhibit clear discretization effects we
combine all our results where $m_{\mathrm{PS}}\leq 450\,\mbox{MeV}$. The fit
range has been chosen such that stable fits are obtained. Our data is not
sufficiently precise to determine all parameters. We therefore fix a few
parameters to their phenomenological value and keep only $g_{\mathrm{A}}$ in
the chiral limit, the leading $\Delta\Delta$-coupling $g_{1}$ and the SSE
coupling term $B_{9}^{r}(\lambda)$ as free fit parameters. The resulting fit
and the lattice data are shown in Fig. 2a.
In our fit we only included results for the largest lattice at a given set of
bare parameters. For some data sets we have results for different volumes. We
thus can compute the relative shift
$\delta_{g_{\mathrm{A}}}(L)=\frac{g_{\mathrm{A}}(L)-g_{\mathrm{A}}(\infty)}{g_{\mathrm{A}}(\infty)}$
(1)
both from the fit as well as from the lattice data taking the results on the
largest lattice as approximation of $g_{\mathrm{A}}(\infty)$. In Fig. 2b we
compare the relative shift for different values of the quark mass with our
lattice results at $m_{\mathrm{PS}}\simeq 270\,\mbox{MeV}$. The shift
predicted from ChEFT only slightly underestimates the relative shift computed
on the lattice.
Also after correcting for finite size effects we observe a significant
difference to the experimental value. It is interesting to notice that a much
better agreement with the experimental value is observed for the ratio
$g_{\mathrm{A}}/f_{\mathrm{PS}}$ (see Fig. 3a). In this ratio the
renormalization constant $Z_{\mathrm{A}}$ drops out.
In Fig. 3b we show our results for the nucleon tensor charge $\langle
1\rangle_{\delta q}=g_{\mathrm{T}}$. We observe only a very mild quark mass
dependence and the data reveals no systematic discretization effects. This
quantity is not well known experimentally. Our values are larger than the
phenomenological results presented in [6].
(a)
(b)
Figure 3: The left panel shows $g_{\mathrm{A}}/f_{\mathrm{PS}}$ as a function
of $m_{\mathrm{PS}}^{2}$. The right panel shows our results for
$g_{\mathrm{T}}^{\overline{\mathrm{MS}}}$ at a scale $\mu=2\,\mbox{GeV}$.
## 4 $n=2$ moments of PDFs
The lowest moment of the unpolarized PDF $\langle x\rangle_{q}=v_{2}$
corresponds to the momentum fraction carried by the quarks in the nucleon.
Lattice results from different collaborations tend to be significantly larger
than the phenomenological value. Fig. 4a and 4b show our most recent results
for the iso-vector and iso-scalar channel. In the latter case disconnected
contributions have been ignored.
Also shown are the results from a fit to results utilizing methods of
covariant Baryon Chiral Perturbation Theory (BChPT) [7]. Fits have been
performed with most parameters fixed to phenomenologically known values. The
iso-vector (iso-scalar) channel data is fitted with only 2 free parameters:
$v_{2}$ in the chiral limit and the coupling $c_{8}$ ($c_{9}$). Near the
physical light quark masses, BChPT predicts $v_{2}$ to become larger when the
quark mass becomes heavier. In our data for $m_{\mathrm{PS}}\lesssim
250\,\mbox{MeV}$ we do not see any indication for a bending down when
approaching the physical pion mass. It thus does not seem that a lack of
results at sufficiently small quark masses could explain the large discrepancy
between the phenomenological value and the lattice results. There are some
indications that part of the discrepancy can be explained by excited state
contamination [8].
(a)
(b)
Figure 4: The left and right panel show results for the second moment of the
iso-vector and iso-scalar unpolarized PDFs, respectively, as a function of
$m_{\mathrm{PS}}^{2}$. The solid lines show the fits to an expression from
ChEFT.
In Fig. 5a the results for the second moment of the polarized PDF $\langle
x\rangle_{\Delta q}=a_{1}$ is shown. Discretization effects again seem to be
absent in data. From a comparison of the results for different volumes it
seems that also finite size effects are small. Results from Heavy Baryon
Chiral Perturbation Theory (HBChPT) [9] lead to the following expression:
$a_{1}^{(u-d)}(m_{\mathrm{PS}})=C\left[1-\frac{4g_{\mathrm{A}}^{2}+1}{2(4\pi
f_{\mathrm{PS}})^{2}}m_{\mathrm{PS}}^{2}\ln\left(\frac{m_{\mathrm{PS}}^{2}}{\mu^{2}}\right)\right]+\cdots$
(2)
In Fig. 5a we plot this expression using $\mu=m_{\mathrm{N}}$ and $C$ chosen
such that it matches the phenomenological value. The bending down which we
observe in our data for $m_{\mathrm{PS}}\lesssim 0.5\,\mbox{MeV}$ is much less
than one would expect from this HBChPT result.
## 5 Electromagnetic form factors
(a)
(b)
Figure 5: The left panel shows the results for the second of the polarized
PDFs as a function of $m_{\mathrm{PS}}^{2}$. In the right panel the results
for the Dirac form factor radius $r_{1}$ are plotted. The dashed lines show
results from ChEFT as described in the text.
To compute the electromagnetic form factors one makes use of the standard
decomposition of the nucleon electromagnetic matrix elements $\langle
p^{\prime},s^{\prime}|V_{\mu}|p,s\rangle=\overline{u}\left[\gamma_{\mu}F_{1}(Q^{2})+\frac{\sigma_{\mu\nu}\,q_{\nu}}{2m_{\mathrm{N}}}F_{2}(Q^{2})\right]u$,
(in Euclidian space) where we use the local vector current $V_{\mu}$. $p$
($s$) and $p^{\prime}$ ($s^{\prime}$) denote initial and final momenta
(spins), $q=p^{\prime}-p$ the momentum transfer (with $Q^{2}=-q^{2}$). To
calculate form factor radii and the anomalous magnetic we have to parametrize
the lattice results. Here we use the ansatz
$F_{i}(Q^{2})=\frac{F_{i}(0)}{\left[1+\frac{Q^{2}}{pm_{i}^{2}}\right]^{p}}$
(3)
with $p=2$ and $p=3$ for the Dirac and Pauli form factors $F_{1}$ and $F_{2}$,
respectively. Our data is not sufficiently precise to favour a particular
parametrization (see [10] for another parametrization).
From fits to Eq. (3) we determine the form factor radii $r_{1}$ and $r_{2}$ as
well as the anomalous magnetic moment $\kappa$. The quark mass dependence of
these quantities has been calculated using the SSE formalism [11]. For $r_{1}$
the parameters are known and we therefore restrict ourselves to a comparison
of the SSE result and the lattice data (see Fig. 5b). While for
$m_{\mathrm{PS}}\gtrsim 300\,\mbox{MeV}$ the lattice results are significantly
smaller than the phenomenological value, towards smaller quark masses we
observe an increase of the radius. This is consistent with predictions from
ChEFT. For $r_{2}$ and $\kappa$ we find a similar behaviour. Since there are
no phenomenological estimates for all parameters of the SSE expressions we
perform a combined fit. The results are plotted in 6a and 6b.
(a)
(b)
Figure 6: The left panel and right panel shows the results for the Pauli
radius $r_{2}$ and the anomalous magnetic moment $\kappa$. The solid lines
show fits to an expression from ChEFT.
## 6 Summary and outlook
We have presented an update of QCDSF results on the lowest moments of
unpolarized, polarized and tensor PDFs as well as the electromagnetic form
factors. Some of our results at light quark masses with
$m_{\mathrm{PS}}\lesssim 300\,\mbox{MeV}$ confirm the expectations from ChEFT
that light quark mass effects are significant. However, this possibly does not
explain all of the observed discrepancies from phenomenological values.
## Acknowledgements
The numerical calculations have been performed on the APEmille and apeNEXT
systems at NIC/DESY (Zeuthen), the BlueGene/P at NIC/JSC (Jülich), the
BlueGene/L at EPCC (Edinburgh), the Dell PC-cluster at DESY (Zeuthen), the
QPACE systems [12] of the SFB TR-55, the SGI Altix and ICE systems at LRZ
(Munich) and HLRN (Berlin/Hannover). This work was supported in part by the
DFG (SFB TR-55) and by the European Union (grants 238353, ITN STRONGnet and
227431, HadronPhysics2, and 256594).
## References
* [1] Y. Nakamura and H. Stüben, PoS(LATTICE 2010)040.
* [2] A. Nobile, PoS(LATTICE 2010)034.
* [3] M. Göckeler et al. [QCDSF Collaboration], arXiv:1003.5756 [hep-lat].
* [4] V. Bernard et al., Nucl. Phys. A635, 121 (1998); A642, 563(E) (1998). T.R. Hemmert, M. Procura, and W. Weise, Phys. Rev. D 68, 075009 (2003).
* [5] A. Ali Khan et al. [QCDSF Collaboration], Phys. Rev. D 74 (2006) 094508.
* [6] M. Anselmino et al., Nucl. Phys. Proc. Suppl. 191 (2009) 98-107.
* [7] M. Dorati, T. A. Gail and T. R. Hemmert, Nucl. Phys. A 798 (2008) 96.
* [8] M. Göckeler et al., in preparation.
* [9] D. Arndt and M.J. Savage, Nucl. Phys. A697, 429 (2002); J.W. Chen and X. Ji, Phys. Lett. B 523, 107 (2001); W. Detmold, W. Melnitchouk and A. W. Thomas, Phys. Rev. D 66 (2002) 054501.
* [10] M. Göckeler et al. [QCDSF Collaboration], PoS(LATTICE 2007)161.
* [11] T. R. Hemmert and W. Weise, Eur. Phys. J. A 15 (2002) 487 [arXiv:hep-lat/0204005]; M. Göckeler et al. [QCDSF Collaboration], Phys. Rev. D 71 (2005) 034508 [arXiv:hep-lat/0303019].
* [12] H. Baier et al., PoS(LATTICE 2009)001.
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arxiv-papers
| 2011-01-12T11:19:52 |
2024-09-04T02:49:16.356674
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. Collins, M. G\\\"ockeler, Ph. H\\\"agler, T. Hemmert, R. Horsley, Y.\n Nakamura, A. Nobile, H. Perlt, D. Pleiter, P.E.L. Rakow, A. Sch\\\"afer, G.\n Schierholz, A. Sternbeck, H. St\\\"uben, F. Winter, J.M. Zanotti",
"submitter": "Dirk Pleiter",
"url": "https://arxiv.org/abs/1101.2326"
}
|
1101.2342
|
# A contribution to the condition number of the total least squares problem
Zhongxiao Jia 1, Bingyu Li 2,111Corresponding author.
1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084,
China
2 School of Mathematics and Statistics, Northeast Normal University,
Changchun 130024, China
jiazx@tsinghua.edu.cn (Z. Jia) mathliby@gmail.com (B. Li)
###### Abstract
This paper concerns cheaply computable formulas and bounds for the condition
number of the TLS problem. For a TLS problem with data $A$, $b$, two formulas
are derived that are simpler and more compact than the known results in the
literature. One is derived by exploiting the properties of Kronecker products
of matrices. The other is obtained by making use of the singular value
decomposition (SVD) of $[A\,\,b]$, which allows us to compute the condition
number cheaply and accurately. We present lower and upper bounds for the
condition number that involve the singular values of $[A\,\,b]$ and the last
entries of the right singular vectors of $[A\,\,b]$. We prove that they are
always sharp and can estimate the condition number accurately by no more than
four times. Furthermore, we establish a few other lower and upper bounds that
involve only a few singular values of $A$ and $[A\,\,b]$. We discuss how tight
the bounds are. These bounds are particularly useful for large scale TLS
problems since for them any formulas and bounds for the condition number
involving all the singular values of $A$ and/or $[A\ b]$ are too costly to be
computed. Numerical experiments illustrate that our bounds are sharper than a
known approximate condition number in the literature.
Keywords: total least squares, condition number, singular value decomposition.
AMS subject classification (2000): 65F35.
## 1 Introduction
For given $A\in\mathbb{R}^{m\times n}(m>n)$, $b\in\mathbb{R}^{m}$, the TLS
problem can be formulated as (see, e.g., [5, 12])
$\min\|[E\,\,r]\|_{F},\text{\quad subject to \quad}b-r\in\mathcal{R}(A+E),$
(1)
where $\|\cdot\|_{F}$ denotes the Frobenius norm of a matrix and
$\mathcal{R}(\cdot)$ denotes the range space. Suppose that
$[E_{TLS}\,\,r_{TLS}]$ solves the above problem. Then $x=x_{TLS}$ that
satisfies the equation $(A+E_{TLS})x=b-r_{TLS}$ is called the TLS solution of
(1).
The condition number measures the worst-case sensitivity of a solution of a
problem to small perturbations in the input data. Combined with backward
errors, it provides an approximate local linear upper bound on the computed
solution. Since the 1980’s, algebraic perturbation analysis for the TLS
problem has been studied extensively; see [3, 5, 9, 15, 16] and the references
therein. In recent years, asymptotic perturbation analysis and TLS condition
numbers have been studied; see, e.g., [1, 8, 18].
In the present paper, we continue our work in [8] that studied the condition
number of the TLS problem. We will derive a number of results. Firstly, we
establish two formulas that are simpler and more suitable for computational
purpose than the known results in the literature. One is derived by exploiting
the properties of Kronecker products of matrices. It improves the formulas
given in [8, 18], is independent of Kronecker products of matrices and makes
its computation convenient. The other is obtained by making use of the SVD of
$[A\,\,b]$, which can be used to compute the condition number more cheaply and
accurately than that in [1]. Secondly, we present lower and upper bounds for
the condition number that involve the singular values of $[A\,\,b]$ and the
last entries of the right singular vectors of $[A\,\,b]$. We prove that these
bounds are always sharp and can estimate the condition number accurately by no
more than four times. Finally, we focus on cheaply computable bounds of the
TLS condition number. We establish lower and upper bounds that involve only a
few singular values of $A$ and $[A\,\,b]$. We discuss how tight the bounds
are. These bounds are particularly useful for large scale TLS problems since
for them any formulas and bounds for the condition number involving all the
singular values of $A$ and/or $[A\ b]$ are too costly to be computed. So we
can compute these bounds by calculating only a few singular values of $A$
and/or $[A\ b]$ using some iterative solvers for large SVDs. In [2], an
approximate TLS condition number is presented and is applied to evaluate
conditioning of the TLS problem there. In this paper, we present numerical
experiments to demonstrate a possibly great improvement of one of our upper
bounds over the approximate condition number in [2].
The paper is organized as follows. In Section 2, we give some preliminaries
necessary. In Section 3, we present computable formulas of the TLS condition
number. The straightforward bounds on the TLS condition number are considered
in Section 4. In Section 5, we present numerical experiments to show the
tightness of bounds for the TLS condition number. We end the paper with some
concluding remarks in Section 6.
Throughout the paper, for given positive integers $m,n$, $\mathbb{R}^{n}$
denotes the space of $n$-dimensional real column vectors, $\mathbb{R}^{m\times
n}$ denotes the space of all $m\times n$ real matrices. $\|\cdot\|$ and
$\|\cdot\|_{F}$ denote 2-norm and Frobenius norm of their arguments,
respectively. Given a matrix $A$, $A(1:i,1:i)$ is a Matlab notation that
denotes the $i$th leading principal submatrix of $A$, and $\sigma_{i}(A)$
denotes the $i$th largest singular value of $A$. For a vector $a$, $a(i)$
denotes the $i$th component of $a$, and ${\rm{diag}}(a)$ is a diagonal matrix
whose diagonals are given as components of $a$. $I_{n}$ denotes the $n\times
n$ identity matrix, $O_{mn}$ denotes the $m\times n$ zero matrix, whereas $O$
denotes a zero matrix whose order is clear from the context. For matrices
$A=[a_{1},\ldots,a_{n}]=[A_{ij}]\in\mathbb{R}^{m\times n}$ and $B$, $A\otimes
B=[A_{ij}B]$ is the Kronecker product of $A$ and $B$, the linear operator
${\rm{vec}}:\mathbb{R}^{m\times n}\rightarrow\mathbb{R}^{mn}$ is defined by
${\rm{vec}}(A)=[a^{T}_{1},\ldots,a^{T}_{n}]^{T}$ for $A\in\mathbb{R}^{m\times
n}$.
## 2 Preliminaries
Throughout the paper, we let
$\hat{U}^{T}A\hat{V}={\rm{diag}}(\hat{\sigma}_{1},\ldots,\hat{\sigma}_{n})$ be
the thin SVD of $A\in\mathbb{R}^{m\times n}$, where
$\hat{\sigma}_{1}\geq\cdots\geq\hat{\sigma}_{n}$,
$\hat{U}\in\mathbb{R}^{m\times n}$, $\hat{U}^{T}\hat{U}=I_{n}$,
$\hat{V}\in\mathbb{R}^{n\times n}$, $\hat{V}^{T}\hat{V}=I_{n}$. Let
$U^{T}[A\,\,b]V={\rm{diag}}(\sigma_{1},\ldots,\sigma_{n+1})$ be the thin SVD
of $[A\,\,b]\in\mathbb{R}^{m\times(n+1)}$, where
$\sigma_{1}\geq\cdots\geq\sigma_{n+1}$,
$U=[u_{1},\ldots,u_{n+1}]\in\mathbb{R}^{m\times(n+1)}$, $U^{T}U=I_{n+1}$,
$V=[v_{1},\ldots,v_{n+1}]\in\mathbb{R}^{(n+1)\times(n+1)}$, $V^{T}V=I_{n+1}$.
The following result presents an existence and uniqueness condition for the
TLS solution [5].
###### Theorem 1
If
$\sigma_{n+1}<\hat{\sigma}_{n},$ (2)
then the TLS problem (1) has a unique solution $x_{TLS}$. Moreover,
$\displaystyle x_{TLS}$ $\displaystyle=$
$\displaystyle(A^{T}A-\sigma^{2}_{n+1}I)^{-1}A^{T}b$ (3) $\displaystyle=$
$\displaystyle-\left[\frac{v_{n+1}(1)}{v_{n+1}(n+1)},\ldots,\frac{v_{n+1}(n)}{v_{n+1}(n+1)}\right]^{T}.$
(4)
In the paper, it is always assumed that condition (2) holds. We note that, for
a given TLS problem (1), if $\sigma_{n+1}=0$, then $b\in\mathcal{R}(A)$. In
this case, the system of equations $Ax=b$ is compatible, and we can take
$[E\,\,r]=O$. As in [8, 18], in the sequel, we do not consider this trivial
case and assume that
$0<\sigma_{n+1}<\hat{\sigma}_{n}.$ (5)
We will use the following properties of the TLS problem, which are in [5]:
$\sigma^{2}_{n+1}=\frac{\|r\|^{2}}{1+\|x\|^{2}}$ (6)
and
$A^{T}r=\frac{\|r\|^{2}}{1+\|x\|^{2}}x=\sigma^{2}_{n+1}x,$ (7)
where $x=x_{TLS}$, $r=Ax-b$. By (4), it holds that
$v_{n+1}=\frac{1}{\sqrt{1+\|x\|^{2}}}\left[\begin{array}[]{c}x\\\ -1\\\
\end{array}\right]$ (8)
up to a sign $\pm 1$.
The following basic properties of the Kronecker products of matrices are
needed later and can be found in [6]:
$\displaystyle(A\otimes C)(B\otimes D)=(AB)\otimes(CD),$
$\displaystyle(A\otimes B)^{T}=A^{T}\otimes B^{T},$
where $A,B,C,D$ are matrices with appropriate sizes.
## 3 Computable formulas for the TLS condition number
Throughout the paper, we follow the definition of condition number in [4, 13].
Let $g:\mathbb{R}^{p}\longrightarrow\mathbb{R}^{q}$ be a continuous map in
normed linear spaces defined on an open set $D_{g}\subset\mathbb{R}^{p}$. For
a given $a\in D_{g}$, $a\neq 0$, with $g(a)\neq 0$, if $g$ is differentiable
at $a$, then the relative condition number of $g$ at $a$ is
$\kappa^{r}_{g}(a)=\frac{\|g^{\prime}(a)\|\|a\|}{\|g(a)\|}$ (9)
and the absolute condition number of $g$ is
$\kappa_{g}(a)=\|g^{\prime}(a)\|,$ (10)
where $g^{\prime}(a)$ denotes the derivative of $g$ at $a$. Given the TLS
problem (1), let $\tilde{A}=A+\Delta A$, $\tilde{b}=b+\Delta b$, where $\Delta
A$ and $\Delta b$ denote the perturbations in $A$ and $b$, respectively.
Consider the perturbed TLS problem
$\min\|[E\,\,r]\|_{F}\,\,\,{\text{subject
to}}\,\,\tilde{b}-r\in\mathcal{R}(\tilde{A}+E).$ (11)
In [8], we have established the following result.
###### Theorem 2
Suppose the TLS problem (1) satisfies (5). Denote by $x=x_{TLS}$ the TLS
solution, and define $r=Ax-b$, $G(x)=[x^{T}\,\,-1]\otimes I_{m}$. If
$\|[\Delta A\,\,\Delta b]\|_{F}$ is small enough, then the perturbed problem
(11) has a unique TLS solution $\tilde{x}_{TLS}$. Moreover,
$\tilde{x}_{TLS}=x_{TLS}+K~{}\left[\begin{array}[]{c}{\rm{vec}}(\Delta A)\\\
\Delta b\\\ \end{array}\right]+\mathcal{O}(\|[\Delta A\,\,\Delta
b]\|^{2}_{F}),$ (12)
where
$K=\left(A^{T}A-\sigma^{2}_{n+1}I_{n}\right)^{-1}\left(2A^{T}\frac{r}{\|r\|}\frac{r^{T}}{\|r\|}G(x)-A^{T}G(x)-[I_{n}\otimes
r^{T}\,\,O]\right).$ (13)
Denote $a={\rm{vec}}(A)$. Based on Theorem 2, in a small neighborhood of
$[a^{T},b^{T}]^{T}\in\mathbb{R}^{m(n+1)}$, we can define the function
$\begin{array}[]{ccc}g:\mathbb{R}^{m(n+1)}&\longrightarrow&\mathbb{R}^{n}\\\
\small{\left[\begin{array}[]{c}\tilde{a}\\\ \tilde{b}\\\
\end{array}\right]}&\longmapsto&\tilde{x}=(\tilde{A}^{T}\tilde{A}-\tilde{\sigma}^{2}_{n+1}I_{n})^{-1}\tilde{A}^{T}\tilde{b},\end{array}$
where $\tilde{a}=a+{\rm{vec}}(\Delta A)={\rm{vec}}(\tilde{A})$,
$\tilde{b}=b+\Delta b$, and $\tilde{x}$ is the solution to the perturbed TLS
problem (11). In particular, $g([a^{T},b^{T}]^{T})=x$. Thus, we have the
following theorem.
###### Theorem 3
Let $\kappa_{g}(A,b)$ and $\kappa^{r}_{g}(A,b)$ be the absolute and relative
condition numbers of the TLS problem, respectively. Then
$\kappa_{g}(A,b)=\|K\|,\,\,\kappa^{r}_{g}(A,b)=\frac{\|K\|\|[A\,\,b]\|_{F}}{\|x\|},$
(14)
where $K$ is defined as in (13).
Proof. By the definition of $g$ and (12), we see that $g$ is differentiable at
$[a^{T},b^{T}]^{T}$ and $g^{\prime}\left([a^{T},b^{T}]^{T}\right)=K$. Then the
assertion follows from (9) and (10). $\Box$
The dependence of Kronecker product of matrices for $K$ makes the computation
of $\kappa_{g}(A,b)$ via (14) too costly. The same are the formulas given in
[8, 18]. For a computational purpose, we will present a new formula of
$\kappa_{g}(A,b)$ that has a simpler and clearer form. To this end, we need a
lemma.
###### Lemma 1
Let
$C=A^{T}A+\sigma^{2}_{n+1}I_{n}-\frac{2\sigma^{2}_{n+1}xx^{T}}{\|x\|^{2}+1}.$
Then $C$ is positive definite.
Proof. Noticing that
$C=A^{T}A-\sigma^{2}_{n+1}I_{n}+2\sigma^{2}_{n+1}\left(I_{n}-\frac{xx^{T}}{1+\|x\|^{2}}\right),$
(15)
and that $A^{T}A-\sigma^{2}_{n+1}I_{n}$ and $I_{n}-\frac{xx^{T}}{1+\|x\|^{2}}$
are both positive definite, we complete the proof of the lemma. $\Box$
###### Theorem 4
Let
$A^{T}A+\sigma^{2}_{n+1}I_{n}-\frac{2\sigma^{2}_{n+1}xx^{T}}{\|x\|^{2}+1}=LL^{T}$
be the Cholesky factorization. Then
$\kappa_{g}(A,b)=\sqrt{\|x\|^{2}+1}\left\|(A^{T}A-\sigma^{2}_{n+1}I_{n})^{-1}L\right\|.$
(16)
Proof. Consider expression (13) of $K$. By the properties of Kronecker product
of matrices, we get
$G(x)G^{T}(x)=\left([x^{T}\,-1]\otimes
I_{m}\right)\left(\left[\begin{array}[]{c}x\\\ -1\\\ \end{array}\right]\otimes
I_{m}\right)=(\|x\|^{2}+1)I_{m},$ $[I_{n}\otimes
r^{T}\,O]G^{T}(x)=[I_{n}\otimes r^{T}\,O]\left[\begin{array}[]{c}x\otimes
I_{m}\\\ -I_{m}\\\ \end{array}\right]=(I_{n}\otimes r^{T})(x\otimes
I_{m})=xr^{T}$
and
$[I_{n}\otimes r^{T}\,O]\left[\begin{array}[]{c}I_{n}\otimes r\\\ O\\\
\end{array}\right]=(I_{n}\otimes r^{T})(I_{n}\otimes r)=\|r\|^{2}I_{n}.$
Thus, we have
$\displaystyle\left(2A^{T}\frac{r}{\|r\|}\frac{r^{T}}{\|r\|}G(x)-A^{T}G(x)-[I_{n}\otimes
r^{T}\,O]\right)$
$\displaystyle\cdot\left(2G^{T}(x)\frac{r}{\|r\|}\frac{r^{T}}{\|r\|}A-G^{T}(x)A-\left[\begin{array}[]{c}I_{n}\otimes
r\\\ O\\\ \end{array}\right]\right)$ $\displaystyle=$
$\displaystyle(\|x\|^{2}+1)A^{T}A+\|r\|^{2}I_{n}-xr^{T}A-A^{T}rx^{T}$
$\displaystyle=$
$\displaystyle(\|x\|^{2}+1)A^{T}A+\|r\|^{2}I_{n}-2\sigma^{2}_{n+1}xx^{T}.$
The last equality used $A^{T}rx^{T}=\sigma^{2}_{n+1}xx^{T}$, which follows
from (7). Denote $P=A^{T}A-\sigma^{2}_{n+1}I_{n}$. We get
$\displaystyle KK^{T}$ $\displaystyle=$ $\displaystyle
P^{-1}\left((\|x\|^{2}+1)A^{T}A+\|r\|^{2}I_{n}-2\sigma^{2}_{n+1}xx^{T}\right)P^{-1}$
(20) $\displaystyle=$
$\displaystyle(\|x\|^{2}+1)P^{-1}\left(A^{T}A+\sigma^{2}_{n+1}I_{n}-\frac{2\sigma^{2}_{n+1}xx^{T}}{\|x\|^{2}+1}\right)P^{-1}.$
(21)
In the last equality we used (6). From Theorem 3, we have
$\kappa_{g}(A,b)=(\|x\|^{2}+1)^{\frac{1}{2}}\left\|P^{-1}\left(A^{T}A+\sigma^{2}_{n+1}I_{n}-\frac{2\sigma^{2}_{n+1}xx^{T}}{\|x\|^{2}+1}\right)P^{-1}\right\|^{\frac{1}{2}}.$
Based on Lemma 1, we complete the proof. $\Box$
Compared with the formula of $\kappa_{g}(A,b)$ in Theorem 3, the formula in
Theorem 4 does not involve the Kronecker product of matrices and makes its
computation convenient. However, if $\hat{\sigma}_{n}$ and $\sigma_{n+1}$ are
close, then $A^{T}A-\sigma^{2}_{n+1}I_{n}$ becomes ill conditioned. Therefore,
it may be hard to use (16) to calculate $\kappa_{g}(A,b)$ accurately. Next we
derive a new formula that can be used to compute the condition number
accurately.
###### Theorem 5
Let $U^{T}[A\,\,b]V={\rm{diag}}(\sigma_{1},\ldots,\sigma_{n+1})$ be the SVD of
$[A\,\,b]$ with $V=[v_{1},\ldots,v_{n+1}]$. Denote $V_{11}=V(1:n,1:n)$. Then
$\kappa_{g}(A,b)=\sqrt{\|x\|^{2}+1}~{}\|V^{-T}_{11}S\|,$ (22)
where $S={\rm{diag}}([s_{1},\ldots,s_{n}])$,
$s_{i}=\frac{\sqrt{\sigma^{2}_{i}+\sigma^{2}_{n+1}}}{\sigma^{2}_{i}-\sigma^{2}_{n+1}}$,
$i=1,\ldots,n$.
Proof. Denote $P=A^{T}A-\sigma^{2}_{n+1}I_{n}$. From (21), we have
$\frac{1}{\|x\|^{2}+1}~{}KK^{T}=P^{-1}+2\sigma^{2}_{n+1}P^{-1}\left(I_{n}-\frac{xx^{T}}{1+\|x\|^{2}}\right)P^{-1}.$
(23)
Note that
$\displaystyle[A\,\,b]^{T}[A\,\,b]-\sigma^{2}_{n+1}I_{n+1}$ $\displaystyle=$
$\displaystyle\sum^{n+1}_{i=1}\sigma^{2}_{i}v_{i}v^{T}_{i}-\sigma^{2}_{n+1}\sum^{n+1}_{i=1}v_{i}v^{T}_{i}$
$\displaystyle=$
$\displaystyle\sum^{n}_{i=1}(\sigma^{2}_{i}-\sigma^{2}_{n+1})v_{i}v^{T}_{i}.$
We get
$\displaystyle P=A^{T}A-\sigma^{2}_{n+1}I_{n}$ $\displaystyle=$
$\displaystyle[I_{n}\,\,0]\sum^{n}_{i=1}(\sigma^{2}_{i}-\sigma^{2}_{n+1})v_{i}v^{T}_{i}\left[\begin{array}[]{c}I_{n}\\\
0\\\ \end{array}\right]$ (26) $\displaystyle=$
$\displaystyle[I_{n}\,\,0][v_{1},\ldots,v_{n}]\left[\begin{array}[]{ccc}\sigma^{2}_{1}-\sigma^{2}_{n+1}&&\\\
&\ddots&\\\ &&\sigma^{2}_{n}-\sigma^{2}_{n+1}\\\
\end{array}\right]\left[\begin{array}[]{c}v^{T}_{1}\\\ \vdots\\\ v^{T}_{n}\\\
\end{array}\right]\left[\begin{array}[]{c}I_{n}\\\ 0\\\ \end{array}\right]$
(35) $\displaystyle=$ $\displaystyle
V_{11}\left[\begin{array}[]{ccc}\sigma^{2}_{1}-\sigma^{2}_{n+1}&&\\\
&\ddots&\\\ &&\sigma^{2}_{n}-\sigma^{2}_{n+1}\\\
\end{array}\right]V^{T}_{11}:=V_{11}\Lambda V^{T}_{11}.$ (39)
Similarly, by (8), since
$v_{n+1}=\frac{1}{\sqrt{1+\|x\|^{2}}}\left[\begin{array}[]{c}x\\\ -1\\\
\end{array}\right],$ we have
$\displaystyle
I_{n+1}-\frac{1}{{1+\|x\|^{2}}}\left[\begin{array}[]{cc}xx^{T}&-x\\\ -x&1\\\
\end{array}\right]=I_{n+1}-v_{n+1}v^{T}_{n+1}=[v_{1},\ldots,v_{n}]\left[\begin{array}[]{c}v^{T}_{1}\\\
\vdots\\\ v^{T}_{n}\\\ \end{array}\right]$ (45)
and
$I_{n}-\frac{xx^{T}}{1+\|x\|^{2}}=V_{11}V^{T}_{11}.$ (46)
By (46), we see that $V_{11}$ is invertible. Combining (39) and (46), we have
$\displaystyle
P^{-1}+2\sigma^{2}_{n+1}P^{-1}\left(I_{n}-\frac{xx^{T}}{1+\|x\|^{2}}\right)P^{-1}$
(47) $\displaystyle=$ $\displaystyle
V^{-T}_{11}\Lambda^{-1}V^{-1}_{11}+2\sigma^{2}_{n+1}\left(V^{-T}_{11}\Lambda^{-1}V^{-1}_{11}\right)V_{11}V^{T}_{11}\left(V^{-T}_{11}\Lambda^{-1}V^{-1}_{11}\right)$
$\displaystyle=$ $\displaystyle
V^{-T}_{11}\Lambda^{-1}V^{-1}_{11}+2\sigma^{2}_{n+1}V^{-T}_{11}\Lambda^{-2}V^{-1}_{11}$
$\displaystyle=$ $\displaystyle
V^{-T}_{11}\left(\Lambda^{-1}+2\sigma^{2}_{n+1}\Lambda^{-2}\right)V^{-1}_{11}=\left(V^{-T}_{11}S\right)\left(V^{-T}_{11}S\right)^{T}.$
(48)
Then by (23) and Theorem 3 we get the desired equality. $\Box$
By Theorem 5, we can calculate $\kappa_{g}(A,b)$ by solving a linear system
with the coefficient matrix $V^{T}_{11}$. Next we show what the condition
number of $V^{T}_{11}$ is exactly.
###### Theorem 6
For $V_{11}$, we have
$\sigma_{1}(V_{11})=1,\ldots,\sigma_{n-1}(V_{11})=1,\sigma_{n}({V_{11}})=\frac{1}{\sqrt{1+\|x\|^{2}}}$
(49)
and
$\kappa(V^{T}_{11})=\frac{\sigma_{1}(V_{11})}{\sigma_{n}(V_{11})}=\sqrt{1+\|x\|^{2}}.$
(50)
Proof. By the definition of $V_{11}$ and the interlacing property [17, p.103]
for eigenvalues of symmetric matrices, we get
$\sigma_{1}(V_{11})=1,\ldots,\sigma_{n-1}(V_{11})=1.$
Noticing that
$V_{11}V^{T}_{11}x=\left(I_{n}-\frac{1}{1+\|x\|^{2}}xx^{T}\right)x=x-\frac{\|x\|^{2}}{1+\|x\|^{2}}x=\frac{1}{1+\|x\|^{2}}x,$
we know that $\frac{1}{1+\|x\|^{2}}$ is an eigenvalue of $V_{11}V^{T}_{11}$,
that is, $\sigma_{n}({V_{11}})=\frac{1}{\sqrt{1+\|x\|^{2}}}$. Thus, we have
proved (49) and (50). $\Box$
A different SVD-based closed formula for $\kappa_{g}(A,b)$ appears in [1]. It
is shown in [1] that
$\kappa_{g}(A,b)=\sqrt{\|x\|^{2}+1}\left\|\hat{D}~{}[\hat{V}^{T}\,\,O_{n,1}]~{}V~{}[D\,\,\,O_{n,1}]^{T}\right\|,$
(51)
where
$\displaystyle\hat{D}$ $\displaystyle=$
$\displaystyle{\rm{diag}}\left(\left[\frac{1}{\hat{\sigma}^{2}_{1}-\sigma^{2}_{n+1}},\ldots,\frac{1}{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}\right]\right),$
$\displaystyle D$ $\displaystyle=$
$\displaystyle{\rm{diag}}\left(\left[\sqrt{\sigma^{2}_{1}+\sigma^{2}_{n+1}},\ldots,\sqrt{\sigma^{2}_{n}+\sigma^{2}_{n+1}}\right]\right).$
Compared with (51), our (22) is simpler and more compact. Furthermore, (51)
depends on the singular values and right singular vectors of both $A$ and
$[A\,\,b]$. In contrast, (22) involves only singular values and right singular
vectors of $[A\,\,b]$. Therefore, the computational cost of the condition
number by (22) is half of that by (51). Furthermore, the following example
shows that the computed results by (22) can be more accurate than those by
(51).
A small example. We construct a TLS problem with $\hat{\sigma}_{n}$ and
$\sigma_{n+1}$ very close. We generate $A,b$ by
$[A\,\,b]={\text{\bf{generate}}}Ab\alpha(m,n,\alpha)$ (see Appendix) by taking
$m=15,n=10$, $\alpha=10^{-8}$.
$~{}\sigma_{n+1}/\sigma_{n}~{}$ | $~{}\sigma_{n+1}/\hat{\sigma}_{n}~{}$ | $~{}{\kappa}^{r}_{g}(\ref{knoneckercond})~{}$ | $~{}{\kappa}^{r}_{g}(\ref{mynewclosed})~{}$ | $~{}\kappa^{r}_{g}(\ref{Baboulin})~{}$
---|---|---|---|---
$0.608$ | $~{}1-1.49\times 10^{-15}$ | $~{}-~{}$ | $1.13\times 10^{9}$ | $3.12\times 10^{8}$
In the table, $~{}\sigma_{n+1}/\sigma_{n}~{}$ and
$~{}\sigma_{n+1}/\hat{\sigma}_{n}~{}$ denote the quotients of $\sigma_{n+1}$
over $\sigma_{n}$ and $\hat{\sigma}_{n}$, respectively.
${\kappa}^{r}_{g}(\ref{knoneckercond})$, ${\kappa}^{r}_{g}(\ref{mynewclosed})$
and $\kappa^{r}_{g}(\ref{Baboulin})$ denote the computed $\kappa^{r}_{g}(A,b)$
by calculating $\kappa_{g}(A,b)$ via (14), (22) and (51), respectively.
$\sigma_{n+1}$ and $\hat{\sigma}_{n}$ being so close makes
$A^{T}A-\sigma^{2}_{n+1}I_{n}$ numerically singular and makes
${\kappa}^{r}_{g}(\ref{knoneckercond})$ unreliable completely, so the result
of ${\kappa}^{r}_{g}(\ref{knoneckercond})$ is omitted.
We comment that ${\kappa}^{r}_{g}(\ref{mynewclosed})$ is reliable as, by the
remark in Appendix and Theorem 6, $\kappa(V^{T}_{11})=\alpha^{-1}=10^{8}$.
This means that computing $\kappa_{g}(A,b)$ via (22) amounts to solving a
moderately ill-conditioned linear system. Furthermore, the right-hand side $S$
of the system can be constructed with high accuracy since $\sigma_{n+1}$ and
$\sigma_{i}$ are not close:
$\frac{\sigma_{n+1}}{\sigma_{i}}\leq\frac{\sigma_{n+1}}{\sigma_{n}}=0.608$,
$i=1,2,\ldots,n$. In contrast, $\kappa^{r}_{g}(\ref{Baboulin})$ is inaccurate
since computing $\kappa_{g}(A,b)$ via (51) involves the diagonal matrix
$\hat{D}$ and the closeness of $\sigma_{n+1}$ (about $0.299$) and
$\hat{\sigma}_{n}$ makes its last diagonal entry both very large (about
$10^{15}$) and very inaccurate in finite precision arithmetic.
## 4 Straightforward bounds on the TLS condition number
### 4.1 Sharp lower and upper bounds based on SVD of $[A\,\,b]$
In this subsection, we further improve our result in Theorem 5 from the
viewpoint of computational cost. We will show that with the SVD
$U^{T}[A\,\,b]V={\rm{diag}}(\sigma_{1},\ldots,\sigma_{n+1})$
we are capable of estimating $\kappa_{g}(A,b)$ accurately based on the
singular values of $[A\,\,b]$ and the last row of $V$ without calculating
$\left\|V^{-T}_{11}S\right\|$, where $V_{11}=V(1:n,1:n)$,
$S={\rm{diag}}([s_{1},\ldots,s_{n}])$,
$s_{i}=\frac{\sqrt{\sigma^{2}_{i}+\sigma^{2}_{n+1}}}{\sigma^{2}_{i}-\sigma^{2}_{n+1}}$,
$i=1,\ldots,n$, as defined in Theorem 5.
From now on we denote $\alpha=\frac{1}{\sqrt{1+\|x\|^{2}}}$, which is always
smaller than one for $x\not=0$. Keep (49) in mind and note that
$s_{1}\leq s_{2}\leq\cdots\leq s_{n}.$
We then get
$s_{n}=\sigma_{n}(V^{-T}_{11})\|S\|\leq\|V^{-T}_{11}S\|\leq\|V^{-T}_{11}\|\|S\|=\alpha^{-1}s_{n}.$
Therefore, from Theorem 5 we get
$\underline{\kappa}:=\alpha^{-1}s_{n}\leq\kappa_{g}(A,b)\leq\bar{\kappa}:=\alpha^{-2}s_{n}.$
(52)
So, if $\alpha\approx 1$, that is, $V_{11}$ is nearly an orthogonal matrix,
the lower and upper bounds in (52) must be tight.
More generally, for $\alpha$ not small, say, $\frac{1}{2}<\alpha<1$, we have
$\bar{\kappa}<4s_{n}$ and $\underline{\kappa}>s_{n}$. So
$\underline{\kappa}<\bar{\kappa}<4\underline{\kappa}$. Therefore, in this
case, our lower and upper bounds on the condition number $\kappa_{g}(A,b)$ are
very tight and can estimate the condition number accurately by no more than
four times.
In the following, we only need to discuss the case that
$\alpha\leq\frac{1}{2}$. It will appear that we can establish some lower bound
$\underline{\kappa}$ and upper bound $\bar{\kappa}$ such that
$\underline{\kappa}<\bar{\kappa}<4\underline{\kappa}$ still holds. As a
result, together with the above, for any $0<\alpha<1$, we can estimate
$\kappa_{g}(A,b)$ accurately.
###### Lemma 2
$V$ can be written as
$V=\left[\begin{array}[]{cc}V_{11}&\sqrt{1-\alpha^{2}}~{}\bar{u}_{n}\\\
\sqrt{1-\alpha^{2}}~{}\bar{v}^{T}_{n}&-\alpha\\\ \end{array}\right],$
where $\bar{u}_{n}$ and $\bar{v}_{n}$ are the left and right singular vectors
associated with the smallest singular value of $V_{11}$.
Proof. Based on Theorem 6, we let
$V_{11}=\bar{U}\left[\begin{array}[]{cc}I_{n-1}&\\\ &\alpha\\\
\end{array}\right]\bar{V}^{T}$
be the SVD of $V_{11}$, where
$\bar{U}=[\bar{u}_{1},\ldots,\bar{u}_{n}]\in\mathbb{R}^{n\times n}$,
$\bar{V}=[\bar{v}_{1},\ldots,\bar{v}_{n}]\in\mathbb{R}^{n\times n}$, and
$\bar{U}^{T}\bar{U}=\bar{V}^{T}\bar{V}=I_{n}$. It is easily justified from (4)
that $|V(n+1,n+1)|=\alpha$. Without loss of generality, we assume
$V(n+1,n+1)=-\alpha$. Then, by the theorem in Section 4 of [11], we get
$\displaystyle V$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}\bar{U}&\\\ &1\\\
\end{array}\right]\left[\begin{array}[]{ccc}I_{n-1}&O_{n-1,1}&O_{n-1,1}\\\
O_{1,n-1}&\alpha&\sqrt{1-\alpha^{2}}\\\
O_{1,n-1}&\sqrt{1-\alpha^{2}}&-\alpha\\\
\end{array}\right]\left[\begin{array}[]{cc}\bar{V}^{T}&\\\ &1\\\
\end{array}\right]$ (60) $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}\bar{U}\left[\begin{array}[]{cc}I_{n-1}&\\\
&\alpha\\\
\end{array}\right]\bar{V}^{T}&~{}~{}~{}~{}\sqrt{1-\alpha^{2}}\bar{u}_{n}\\\
\sqrt{1-\alpha^{2}}\bar{v}^{T}_{n}&-\alpha\\\ \end{array}\right],$ (65)
the desired form of $V$. $\Box$
Following Lemma 2 and letting $[\beta_{1},\ldots,\beta_{n},-\alpha]$ be the
last row of $V$, we have
$\bar{v}^{T}_{n}=\frac{1}{\sqrt{1-\alpha^{2}}}[\beta_{1},\ldots,\beta_{n}].$
(66)
Noticing that ($\alpha^{-1}$, $\bar{u}_{n}$, $\bar{v}_{n}$) is the largest
singular triplet of $V^{-T}_{11}$, we denote by
$V^{-T}_{11}=[\bar{u}_{n},\bar{u}_{1}\ldots,\bar{u}_{n-1}]\left[\begin{array}[]{cccc}\alpha^{-1}&&&\\\
&1&&\\\ &&\ddots&\\\ &&&1\\\
\end{array}\right]\left[\begin{array}[]{c}\bar{v}^{T}_{n}\\\
\bar{v}^{T}_{1}\\\ \vdots\\\ \bar{v}^{T}_{n-1}\end{array}\right],$
which is the SVD of $V^{-T}_{11}$. Then, by (66) we have
$\displaystyle V^{-T}_{11}$ $\displaystyle=$
$\displaystyle\left[\alpha^{-1}\bar{u}_{n},\bar{u}_{1},\ldots,\bar{u}_{n-1}\right]\left[\begin{array}[]{ccccc}\frac{\beta_{1}}{\sqrt{1-\alpha^{2}}}&\cdots&\frac{\beta_{k}}{\sqrt{1-\alpha^{2}}}&\cdots&\frac{\beta_{n}}{\sqrt{1-\alpha^{2}}}\\\
\bar{v}_{1}(1)&\ldots&\bar{v}_{1}(k)&\cdots&\bar{v}_{1}(n)\\\
\vdots&&\vdots&&\vdots\\\
\bar{v}_{n-1}(1)&\cdots&\bar{v}_{n-1}(k)&\cdots&\bar{v}_{n-1}(n)\\\
\end{array}\right]$ (71) $\displaystyle=$
$\displaystyle\left[\frac{\alpha^{-1}\beta_{1}}{\sqrt{1-\alpha^{2}}}\bar{u}_{n}+w_{1},\ldots,\frac{\alpha^{-1}\beta_{k}}{\sqrt{1-\alpha^{2}}}\bar{u}_{n}+w_{k},\ldots,\frac{\alpha^{-1}\beta_{n}}{\sqrt{1-\alpha^{2}}}\bar{u}_{n}+w_{n}\right],$
(72)
where $\bar{v}_{i}(k)$ denotes the $k$th component of $\bar{v}_{i}$,
$w_{k}=\sum^{n-1}_{i=1}\bar{v}_{i}(k)\bar{u}_{i}$, $k=1,\ldots,n$.
###### Lemma 3
For given matrices $A_{1},A_{2}\in\mathbb{R}^{n\times n}$, if
$A^{T}_{1}A_{2}=O$, then
$\frac{1}{2}(\|A_{1}\|+\|A_{2}\|)\leq\|A_{1}+A_{2}\|.$ (73)
Proof. For an arbitrary vector $x\in\mathbb{R}^{n}$, from
$(A_{1}x)^{T}(A_{2}x)=0$ it follows that
$\|A_{1}x\|,\|A_{2}x\|\leq\|A_{1}x+A_{2}x\|$
and that
$\displaystyle\|A_{1}\|$ $\displaystyle=$
$\displaystyle{\rm{max}}_{\|x\|=1}\|A_{1}x\|\leq{\rm{max}}_{\|x\|=1}\|A_{1}x+A_{2}x\|=\|A_{1}+A_{2}\|,$
$\displaystyle\|A_{2}\|$ $\displaystyle=$
$\displaystyle{\rm{max}}_{\|x\|=1}\|A_{2}x\|\leq{\rm{max}}_{\|x\|=1}\|A_{1}x+A_{2}x\|=\|A_{1}+A_{2}\|.$
So, we get the desired inequality. $\Box$
To prove the main results of this section, we need the following two
propositions.
###### Proposition 1
Let $[\beta_{1},\ldots,\beta_{n},-\alpha]$ be the last row of $V$,
$V_{11}=V(1:n,1:n)$ and
$\bar{S}={\rm{diag}}([\bar{s}_{1},\ldots,\bar{s}_{n}])$, where
$\bar{s}_{1},\ldots,\bar{s}_{n}$ are arbitrary positive numbers and satisfy
$0<\bar{s}_{1}\leq\bar{s}_{2}\leq\cdots\leq\bar{s}_{n}$. Then
$\displaystyle\underline{c}:=\frac{1}{2}\left(\frac{\alpha^{-1}\sqrt{\beta^{2}_{1}\bar{s}^{2}_{1}+\ldots+\beta^{2}_{n}\bar{s}^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+\frac{\sqrt{1-\alpha^{2}-\beta^{2}_{n}}}{\sqrt{1-\alpha^{2}}}\bar{s}_{n}\right)$
$\displaystyle\leq$
$\displaystyle\left\|V^{-T}_{11}\bar{S}\right\|\leq\bar{c}:=\frac{\alpha^{-1}\sqrt{\beta^{2}_{1}\bar{s}^{2}_{1}+\ldots+\beta^{2}_{n}\bar{s}^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+\bar{s}_{n}.$
Proof. Following (72), we get
$\displaystyle V^{-T}_{11}\bar{S}$ $\displaystyle=$
$\displaystyle\left[\frac{\alpha^{-1}\beta_{1}\bar{s}_{1}}{\sqrt{1-\alpha^{2}}}\bar{u}_{n}+\bar{s}_{1}w_{1},\ldots,\frac{\alpha^{-1}\beta_{k}\bar{s}_{k}}{\sqrt{1-\alpha^{2}}}\bar{u}_{n}+\bar{s}_{k}w_{k},\ldots,\frac{\alpha^{-1}\beta_{n}\bar{s}_{n}}{\sqrt{1-\alpha^{2}}}\bar{u}_{n}+\bar{s}_{n}w_{n}\right].$
Define
$A_{1}=\left[\frac{\alpha^{-1}\beta_{1}\bar{s}_{1}}{\sqrt{1-\alpha^{2}}}\bar{u}_{n},\ldots,\frac{\alpha^{-1}\beta_{n}\bar{s}_{n}}{\sqrt{1-\alpha^{2}}}\bar{u}_{n}\right],\,\,A_{2}=\left[\bar{s}_{1}w_{1},\ldots,\bar{s}_{n}w_{n}\right].$
Then $V^{-T}_{11}\bar{S}=A_{1}+A_{2}$. Noticing that
$\bar{u}^{T}_{n}w_{k}=0,\,\,k=1,\ldots,n,$
we get $A^{T}_{1}A_{2}=O$. Thus, we have
$\frac{1}{2}(\|A_{1}\|+\|A_{2}\|)\leq\left\|V^{-T}_{11}\bar{S}\right\|\leq\|A_{1}\|+\|A_{2}\|,$
(74)
in which the left-hand side inequality follows from Lemma 3. Furthermore,
noticing that
$A_{1}=\frac{\alpha^{-1}}{\sqrt{1-\alpha^{2}}}\bar{u}_{n}\left[\beta_{1}\bar{s}_{1},\ldots,\beta_{n}\bar{s}_{n}\right]$
and $\|\bar{u}_{n}\|=1$, we have
$\|A_{1}\|=\frac{\alpha^{-1}}{\sqrt{1-\alpha^{2}}}\left\|\left[\beta_{1}\bar{s}_{1},\ldots,\beta_{n}\bar{s}_{n}\right]\right\|=\frac{\alpha^{-1}}{\sqrt{1-\alpha^{2}}}\sqrt{\beta^{2}_{1}\bar{s}^{2}_{1}+\ldots+\beta^{2}_{n}\bar{s}^{2}_{n}}.$
(75)
In the meantime, note that
$\|w_{n}\|=\sqrt{\sum^{n-1}_{i=1}\bar{v}^{2}_{i}(n)}=\sqrt{1-\frac{\beta^{2}_{n}}{1-\alpha^{2}}}=\frac{\sqrt{1-\alpha^{2}-\beta^{2}_{n}}}{\sqrt{1-\alpha^{2}}},$
$\|[w_{1},\ldots,w_{n}]\|=\left\|[\bar{u}_{1},\ldots,\bar{u}_{n-1}]\left[\begin{array}[]{c}\bar{v}^{T}_{1}\\\
\vdots\\\ \bar{v}^{T}_{n-1}\\\ \end{array}\right]\right\|=1,$
and
$\|\bar{S}\|=\bar{s}_{n}.$
From
$\|\bar{s}_{n}w_{n}\|\leq\|A_{2}\|\leq\left\|[w_{1},\ldots,w_{n}]\right\|\|\bar{S}\|$
we get
$\frac{\sqrt{1-\alpha^{2}-\beta^{2}_{n}}}{\sqrt{1-\alpha^{2}}}\bar{s}_{n}\leq\|A_{2}\|\leq\bar{s}_{n}.$
(76)
Combining (74), (75) and (76), we establish the desired inequality. $\Box$
###### Proposition 2
Suppose that $\alpha\leq\frac{1}{2}$. Then for $\underline{c}$ and $\bar{c}$
in Proposition 1, we have
$\underline{c}<\bar{c}<4\underline{c}.$ (77)
Proof. If $\frac{|\beta_{n}|}{\sqrt{1-\alpha^{2}}}<\frac{\sqrt{3}}{2}$, then
it is easy to verify that
$\frac{\sqrt{1-\alpha^{2}-\beta^{2}_{n}}}{\sqrt{1-\alpha^{2}}}>\frac{1}{2}$
and
$\underline{c}>\frac{1}{4}\bar{c}.$
Thus, (77) holds. If
$\frac{|\beta_{n}|}{\sqrt{1-\alpha^{2}}}\geq\frac{\sqrt{3}}{2}$, then
$\alpha^{-1}\frac{|\beta_{n}|}{\sqrt{1-\alpha^{2}}}>\frac{\sqrt{3}}{2}\alpha^{-1}>1,$
so
$\alpha^{-1}\frac{|\beta_{n}|}{\sqrt{1-\alpha^{2}}}\bar{s}_{n}>\bar{s}_{n}$,
from which and the definitions of $\bar{c}$ and $\underline{c}$ it follows
that
$\displaystyle\bar{c}$ $\displaystyle<$
$\displaystyle\frac{\alpha^{-1}\sqrt{\beta^{2}_{1}\bar{s}^{2}_{1}+\ldots+\beta^{2}_{n}\bar{s}^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+\alpha^{-1}\frac{|\beta_{n}|}{\sqrt{1-\alpha^{2}}}\bar{s}_{n}$
$\displaystyle\leq$
$\displaystyle\frac{2\alpha^{-1}\sqrt{\beta^{2}_{1}\bar{s}^{2}_{1}+\ldots+\beta^{2}_{n}\bar{s}^{2}_{n}}}{\sqrt{1-\alpha^{2}}}$
$\displaystyle\leq$
$\displaystyle\frac{2\alpha^{-1}\sqrt{\beta^{2}_{1}\bar{s}^{2}_{1}+\ldots+\beta^{2}_{n}\bar{s}^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+\frac{2\sqrt{1-\alpha^{2}-\beta^{2}_{n}}}{\sqrt{1-\alpha^{2}}}\bar{s}_{n}=4\underline{c}.$
Thus, (77) still holds. $\Box$
Now we are in a position to derive sharp bounds on $\kappa_{g}(A,b)$.
###### Theorem 7
Let $[\beta_{1},\ldots,\beta_{n},-\alpha]$ be the last row of $V$ and
$S={\rm{diag}}([s_{1},\ldots,s_{n}])$,
$s_{i}=\frac{\sqrt{\sigma^{2}_{i}+\sigma^{2}_{n+1}}}{\sigma^{2}_{i}-\sigma^{2}_{n+1}}$,
$i=1,\ldots,n$. Then
$\displaystyle\underline{\kappa}:=\frac{1}{2}\left(\frac{\alpha^{-2}\sqrt{\beta^{2}_{1}s^{2}_{1}+\ldots+\beta^{2}_{n}s^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+\frac{\sqrt{1-\alpha^{2}-\beta^{2}_{n}}}{\sqrt{1-\alpha^{2}}}\alpha^{-1}s_{n}\right)$
$\displaystyle\leq$
$\displaystyle\kappa_{g}(A,b)\leq\bar{\kappa}:=\frac{\alpha^{-2}\sqrt{\beta^{2}_{1}s^{2}_{1}+\ldots+\beta^{2}_{n}s^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+\alpha^{-1}s_{n}.$
Moreover, if $\alpha\leq\frac{1}{2}$, then
$\underline{\kappa}<\bar{\kappa}<4\underline{\kappa}.$
Proof. Noticing that $0<s_{1}\leq s_{2}\leq\cdots\leq s_{n}$ and using
Proposition 1, we have
$\displaystyle\frac{1}{2}\left(\frac{\alpha^{-1}\sqrt{\beta^{2}_{1}s^{2}_{1}+\ldots+\beta^{2}_{n}s^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+\frac{\sqrt{1-\alpha^{2}-\beta^{2}_{n}}}{\sqrt{1-\alpha^{2}}}s_{n}\right)$
$\displaystyle\leq$
$\displaystyle\left\|V^{-T}_{11}S\right\|\leq\frac{\alpha^{-1}\sqrt{\beta^{2}_{1}s^{2}_{1}+\ldots+\beta^{2}_{n}s^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+s_{n}.$
By Theorem 5, we get the first part of the theorem. Furthermore, we have the
second part of the theorem by Proposition 2. $\Box$
A small example (Continued). From Theorem 7, we have
$5.65\times 10^{8}\leq\kappa^{r}_{g}(A,b)\leq 1.13\times 10^{9}.$
The lower and upper bounds estimate
${\kappa}^{r}_{g}(\ref{mynewclosed})=1.13\times 10^{9}$ accurately, as
described in the second part of Theorem 7.
### 4.2 Lower and upper bounds based on a few of singular values of $A$ and
$[A\,\,b]$
In [10], bounds on the condition number of the Tikhonov regularization
solution are established based on a few singular values of $A$, where $A$ is
the coefficient matrix of the least squares problem under consideration. This
is particularly useful for large scale TLS problems since for them any
formulas and bounds for the condition number involving all the singular values
of $A$ and/or $[A\ b]$ are too costly to be computed. Such a bound can be
obtained by computing only a few singular values of $A$ and/or $[A\ b]$.
In the following theorem, we establish similar results for the condition
number of the TLS problem.
###### Theorem 8
We have
$\underline{\kappa}_{1}\leq\kappa_{g}(A,b)\leq\bar{\kappa}_{1},$ (78)
where
$\displaystyle\underline{\kappa}_{1}=\frac{\sqrt{1+\|x\|^{2}}\sqrt{\hat{\sigma}^{2}_{n-1}+\sigma^{2}_{n+1}}}{\hat{\sigma}^{2}_{n-1}-\sigma^{2}_{n+1}},\,\,\bar{\kappa}_{1}=\frac{\sqrt{1+\|x\|^{2}}\sqrt{\hat{\sigma}^{2}_{n}+\sigma^{2}_{n+1}}}{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}.$
(79)
Proof. Denoting
$M=(A^{T}A-\sigma^{2}_{n+1}I_{n})^{-1}\left((\|x\|^{2}+1)A^{T}A+\|r\|^{2}I_{n}\right)(A^{T}A-\sigma^{2}_{n+1}I_{n})^{-1},$
from (20) we have
$KK^{T}=M-2\sigma^{2}_{n+1}(A^{T}A-\sigma^{2}_{n+1}I_{n})^{-1}xx^{T}(A^{T}A-\sigma^{2}_{n+1}I_{n})^{-1}.$
(80)
Here and hereafter, $\lambda_{i}(M)$ denotes the $i$th largest eigenvalue of
$M$, where $M$ is an arbitrary symmetric matrix. By the Courant-Fischer
theorem [14, p.182], from (80) we get
$\lambda_{2}(M)\leq\lambda_{1}(KK^{T}).$ (81)
Furthermore, since
$2\sigma^{2}_{n+1}(A^{T}A-\sigma^{2}_{n+1}I_{n})^{-1}xx^{T}(A^{T}A-\sigma^{2}_{n+1}I_{n})^{-1}$
is nonnegative definite, the following inequality holds
$\lambda_{1}(KK^{T})\leq\lambda_{1}(M).$ (82)
Collecting (81) and (82) and based on (14), we have
$\sqrt{\lambda_{2}(M)}\leq\kappa_{g}(A,b)\leq\sqrt{\lambda_{1}(M)}.$
It is easy to verify that the set
$\left\\{\frac{(\|x\|^{2}+1)\hat{\sigma}^{2}_{j}+\|r\|^{2}}{(\hat{\sigma}^{2}_{j}-\sigma^{2}_{n+1})^{2}}\right\\}^{n}_{j=1}$
consists of all the eigenvalues of $M$. We define the function
$f(\sigma)=\frac{(\|x\|^{2}+1){\sigma}^{2}+\|r\|^{2}}{({\sigma}^{2}-\sigma^{2}_{n+1})^{2}},\,\,\sigma>\sigma_{n+1},$
and differentiate it to get
$f^{\prime}(\sigma)=\frac{-2\sigma^{3}(\|x\|^{2}+1)-2\sigma(\|x\|^{2}+1)\sigma^{2}_{n+1}-4\sigma\|r\|^{2}}{(\sigma^{2}-\sigma^{2}_{n+1})^{3}}.$
It is seen that $f^{\prime}(\sigma)<0$ and $f(\sigma)$ is decreasing in the
interval $(\sigma_{n+1},\infty)$. Thus, we get that
$\lambda_{1}(M)=\frac{(\|x\|^{2}+1)\hat{\sigma}^{2}_{n}+\|r\|^{2}}{(\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1})^{2}},\,\,\lambda_{2}(M)=\frac{(\|x\|^{2}+1)\hat{\sigma}^{2}_{n-1}+\|r\|^{2}}{(\hat{\sigma}^{2}_{n-1}-\sigma^{2}_{n+1})^{2}}$
and
$\frac{\sqrt{(\|x\|^{2}+1)\hat{\sigma}^{2}_{n-1}+\|r\|^{2}}}{\hat{\sigma}^{2}_{n-1}-\sigma^{2}_{n+1}}\leq\kappa_{g}(A,b)\leq\frac{\sqrt{(\|x\|^{2}+1)\hat{\sigma}^{2}_{n}+\|r\|^{2}}}{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}.$
Noticing that $\frac{\|r\|^{2}}{1+\|x\|^{2}}=\sigma^{2}_{n+1}$, we complete
the proof. $\Box$
Remark. In Corollary 1 of [1], the authors prove that
$\kappa_{g}(A,b)\leq\frac{\sqrt{1+\|x\|^{2}}\sqrt{{\sigma}^{2}_{1}+\sigma^{2}_{n+1}}}{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}.$
Since $\hat{\sigma}_{n}\leq\hat{\sigma}_{1},\ \hat{\sigma}_{1}\leq\sigma_{1}$,
we get
$\bar{\kappa}_{1}\leq\frac{\sqrt{1+\|x\|^{2}}\sqrt{\hat{\sigma}^{2}_{1}+\sigma^{2}_{n+1}}}{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}\leq\frac{\sqrt{1+\|x\|^{2}}\sqrt{{\sigma}^{2}_{1}+\sigma^{2}_{n+1}}}{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}.$
Therefore, our $\bar{\kappa}_{1}$ in (79) is sharper than the above upper
bound.
It is seen that the lower and upper bounds on $\kappa_{g}(A,b)$ in Theorem 8
are marginally different provided that $\hat{\sigma}_{n}$ and
$\hat{\sigma}_{n-1}$ are close. This means that in this case both bounds are
very tight. For the case that $\hat{\sigma}_{n}$ and $\hat{\sigma}_{n-1}$ are
not close, we next give a new lower bound that can be better than that in
Theorem 8.
###### Theorem 9
It holds that
$\underline{\kappa}_{2}\leq\kappa_{g}(A,b)\leq\bar{\kappa}_{1},$
where $\bar{\kappa}_{1}$ is defined as in Theorem 8 and
$\underline{\kappa}_{2}=\frac{\sqrt{1+\|x\|^{2}}}{\sqrt{{\hat{\sigma}}^{2}_{n}-\sigma^{2}_{n+1}}}.$
Moreover, when
$\hat{\sigma}_{n-1}\geq\sigma_{n+1}+\sqrt{{\hat{\sigma}}^{2}_{n}-\sigma^{2}_{n+1}}$,
we have
$\underline{\kappa}_{1}\leq\underline{\kappa}_{2}.$
Proof. Denote $P=A^{T}A-\sigma^{2}_{n+1}I_{n}$. From (23), we have
$\frac{1}{\|x\|^{2}+1}~{}KK^{T}=P^{-1}+2\sigma^{2}_{n+1}P^{-1}\left(I_{n}-\frac{xx^{T}}{1+\|x\|^{2}}\right)P^{-1}.$
Noticing the second term in the right-hand side of the above relation is
positive definite, we have
$(\|x\|^{2}+1)\lambda_{1}(P^{-1})\leq\lambda_{1}(~{}KK^{T}),$
that is,
$\frac{\|x\|^{2}+1}{{\hat{\sigma}^{2}_{n}}-\sigma^{2}_{n+1}}\leq\kappa^{2}_{g}(A,b).$
Thus, the first part of the theorem is obtained.
The second part of the theorem is proved by noting
$\frac{\sqrt{\hat{\sigma}^{2}_{n-1}+\sigma^{2}_{n+1}}}{\hat{\sigma}^{2}_{n-1}-\sigma^{2}_{n+1}}<\frac{1}{\hat{\sigma}_{n-1}-\sigma_{n+1}}\leq\frac{1}{\sqrt{{\hat{\sigma}}^{2}_{n}-\sigma^{2}_{n+1}}}$
under the assumption that
$\hat{\sigma}_{n-1}-\sigma_{n+1}\geq\sqrt{{\hat{\sigma}}^{2}_{n}-\sigma^{2}_{n+1}}$.
$\Box$
Remark 1. At first glance, the assumption in the second part of the theorem
seems not so direct but we can justify that it indeed implies that
$\hat{\sigma}_{n}$ and $\hat{\sigma}_{n-1}$ are not close. Actually, we can
verify that the second part of Theorem 9 holds under a slightly stronger but
much simpler condition that
$\hat{\sigma}_{n-1}\geq 2\hat{\sigma}_{n}.$
Remark 2. From
$\frac{\bar{\kappa}_{1}}{\underline{\kappa}_{2}}=\frac{\sqrt{\hat{\sigma}^{2}_{n}+\sigma^{2}_{n+1}}}{\sqrt{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}}=\sqrt{\frac{1+\frac{\sigma^{2}_{n+1}}{\hat{\sigma}^{2}_{n}}}{1-\frac{\sigma^{2}_{n+1}}{\hat{\sigma}^{2}_{n}}}},$
it is seen that $\frac{\bar{\kappa}_{1}}{\underline{\kappa}_{2}}>1$ provided
$\sigma_{n+1}>0$. Only for $\sigma_{n+1}=0$,
$\bar{\kappa}_{1}=\underline{\kappa}_{2}$ holds. At this time,
$b\in\mathcal{R}(A)$ and $r=0$.
We observe that the bounds on $\kappa_{g}(A,b)$ in Theorem 9 are tight when
$\frac{\sigma_{n+1}}{\hat{\sigma}_{n}}$ is small, compared with one. On the
other hand, once $\frac{\sigma_{n+1}}{\hat{\sigma}_{n}}$ is not small, these
bounds may not be tight. For this case, we will present new bounds that may
better estimate $\kappa_{g}(A,b)$.
The proof of the following theorem depends strongly on Propositions 1 and 2.
###### Theorem 10
Assume that $\alpha\leq\frac{1}{2}$. Denote
$\rho=\frac{\sigma_{n+1}}{\sigma_{n}}$. Then
$\displaystyle\underline{\kappa}_{2}:=\frac{\sqrt{1+\|x\|^{2}}}{\sqrt{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}}\leq\kappa_{g}(A,b)$
$\displaystyle<$
$\displaystyle\bar{\kappa}_{2}:=\sqrt{\frac{1+31\rho^{2}}{1-\rho^{2}}}\frac{\sqrt{1+\|x\|^{2}}}{\sqrt{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}}.$
(83)
Proof. Based on Theorem 9, it suffices to prove the right-hand side of (83).
From (23) and (47), we get
$\displaystyle\frac{1}{\|x\|^{2}+1}~{}KK^{T}$ $\displaystyle=$ $\displaystyle
P^{-1}+2\sigma^{2}_{n+1}P^{-1}\left(I_{n}-\frac{xx^{T}}{1+\|x\|^{2}}\right)P^{-1},$
(84) $\displaystyle=$ $\displaystyle
V^{-T}_{11}\Lambda^{-1}V^{-1}_{11}+2\sigma^{2}_{n+1}V^{-T}_{11}\Lambda^{-2}V^{-1}_{11}:=P^{-1}+E,$
where $P=A^{T}A-\sigma^{2}_{n+1}I_{n}$,
$\Lambda={\rm{diag}}([\sigma^{2}_{1}-\sigma^{2}_{n+1},\ldots,\sigma^{2}_{n}-\sigma^{2}_{n+1}])$.
Denote
$\displaystyle D$ $\displaystyle=$
$\displaystyle{\rm{diag}}([d_{1},\ldots,d_{n}]),\,d_{i}=\frac{\sigma_{n+1}}{\sigma^{2}_{i}-\sigma^{2}_{n+1}},i=1,\ldots,n,$
$\displaystyle T$ $\displaystyle=$
$\displaystyle{\rm{diag}}([t_{1},\ldots,t_{n}]),\,t_{i}=\frac{1}{\sqrt{\sigma^{2}_{i}-\sigma^{2}_{n+1}}},i=1,\ldots,n.$
Then $P^{-1}=\left(V^{-T}_{11}T\right)\left(TV^{-1}_{11}\right)$ and
$E=2\left(V^{-T}_{11}D\right)\left(DV^{-1}_{11}\right)$.
Note that $0<d_{1}\leq d_{2}\leq\cdots\leq d_{n}$ and $0<t_{1}\leq
t_{2}\leq\cdots\leq t_{n}$. Applying Proposition 1, we get
$\displaystyle\frac{1}{2}\left(\frac{\alpha^{-1}\sqrt{\beta^{2}_{1}d^{2}_{1}+\ldots+\beta^{2}_{n}d^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+\frac{\sqrt{1-\alpha^{2}-\beta^{2}_{n}}}{\sqrt{1-\alpha^{2}}}d_{n}\right)$
(85) $\displaystyle\leq$
$\displaystyle\left\|V^{-T}_{11}D\right\|\leq\frac{\alpha^{-1}\sqrt{\beta^{2}_{1}d^{2}_{1}+\ldots+\beta^{2}_{n}d^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+d_{n}$
and
$\displaystyle\frac{1}{2}\left(\frac{\alpha^{-1}\sqrt{\beta^{2}_{1}t^{2}_{1}+\ldots+\beta^{2}_{n}t^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+\frac{\sqrt{1-\alpha^{2}-\beta^{2}_{n}}}{\sqrt{1-\alpha^{2}}}t_{n}\right)$
$\displaystyle\leq$
$\displaystyle\left\|V^{-T}_{11}T\right\|\leq\frac{\alpha^{-1}\sqrt{\beta^{2}_{1}t^{2}_{1}+\ldots+\beta^{2}_{n}t^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+t_{n},$
respectively, where $[\beta_{1},\ldots,\beta_{n},-\alpha]$ denotes the last
row of $V$ as before. Define
$k_{n}=\frac{d_{n}}{t_{n}}=\frac{\sigma_{n+1}}{\sqrt{\sigma^{2}_{n}-\sigma^{2}_{n+1}}}$.
Then
$\frac{d_{1}}{t_{1}}=\frac{\sigma_{n+1}}{\sqrt{\sigma^{2}_{1}-\sigma^{2}_{n+1}}}\leq
k_{n}\,,\ldots,\frac{d_{n-1}}{t_{n-1}}=\frac{\sigma_{n+1}}{\sqrt{\sigma^{2}_{n-1}-\sigma^{2}_{n+1}}}\leq
k_{n}.$
Thus, by (85) we have
$\displaystyle\frac{1}{\sqrt{2}}\|E\|^{\frac{1}{2}}=\left\|V^{-T}_{11}D\right\|\leq
k_{n}\left(\frac{\alpha^{-1}\sqrt{\beta^{2}_{1}t^{2}_{1}+\ldots+\beta^{2}_{n}t^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+t_{n}\right).$
(86)
Note that for the lower and upper bounds on $\left\|V^{-T}_{11}T\right\|$
above, by Proposition 2 it holds that
$\displaystyle\frac{\alpha^{-1}\sqrt{\beta^{2}_{1}t^{2}_{1}+\ldots+\beta^{2}_{n}t^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+t_{n}$
$\displaystyle<$ $\displaystyle
2\left(\frac{\alpha^{-1}\sqrt{\beta^{2}_{1}t^{2}_{1}+\ldots+\beta^{2}_{n}t^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+\frac{\sqrt{1-\alpha^{2}-\beta^{2}_{n}}}{\sqrt{1-\alpha^{2}}}t_{n}\right)$
(87) $\displaystyle<$ $\displaystyle 4\left\|V^{-T}_{11}T\right\|.$
Based on (86) and (87), we derive that
$\frac{1}{\sqrt{2}}\|E\|^{\frac{1}{2}}<4k_{n}\left\|V^{-T}_{11}T\right\|=4k_{n}\|P^{-1}\|^{\frac{1}{2}}$
and that
$\|E\|<32k^{2}_{n}\|P^{-1}\|.$ (88)
Combining (88) and (84), we establish that
$\displaystyle\kappa_{g}(A,b)=\|K\|=\|KK^{T}\|^{\frac{1}{2}}$ $\displaystyle<$
$\displaystyle\sqrt{1+32k^{2}_{n}}\sqrt{1+\|x\|^{2}}\|P^{-1}\|^{\frac{1}{2}}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{1+31\rho^{2}}{1-\rho^{2}}}\frac{\sqrt{1+\|x\|^{2}}}{\sqrt{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}}.$
So, the proof of the theorem is completed. $\Box$
Remark. It is clear that the bounds in Theorem 10 are tight when
$\rho=\frac{\sigma_{n+1}}{\sigma_{n}}$ is small, compared with one. The result
in this theorem is of particular importance in the case that
$\frac{\sigma_{n+1}}{\hat{\sigma}_{n}}$ is close to one. Recall that the lower
and upper bounds in Theorem 9 differ considerably when
$\frac{\sigma_{n+1}}{\hat{\sigma}_{n}}$ is close to one. Theorem 10 tells us
that, if only $\frac{\sigma_{n+1}}{\sigma_{n}}$ is not so close to one,
$\kappa_{g}(A,b)$ should be close to the lower bound.
The improvement of $\bar{\kappa}_{2}$ to $\bar{\kappa}_{1}$ can be illustrated
as follows. For $\frac{\sigma_{n+1}}{\sigma_{n}}$ small, i.e.,
${\sigma_{n+1}}$ and ${\sigma_{n}}$ not close, as an upper bound of
$\kappa^{r}_{g}(A,b)$,
$\displaystyle\bar{\kappa}^{r}_{2}:=\frac{\bar{\kappa}_{2}}{\|x\|}\|[A\,\,b]\|_{F}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{1+31\rho^{2}}{1-\rho^{2}}}\frac{\sqrt{1+\|x\|^{2}}}{\|x\|}\frac{\|[A\,\,b]\|_{F}}{\sqrt{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}}$
$\displaystyle\approx$
$\displaystyle\sqrt{\frac{1+31\rho^{2}}{1-\rho^{2}}}\frac{\|[A\,\,b]\|_{F}}{\sqrt{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}}$
is a moderate multiple of
$\frac{1}{\sqrt{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}}$. In contrast,
$\displaystyle\bar{\kappa}^{r}_{1}:=\frac{\bar{\kappa}_{1}}{\|x\|}\|[A\,\,b]\|_{F}$
$\displaystyle=$
$\displaystyle\frac{\sqrt{1+\|x\|^{2}}}{\|x\|}\frac{\sqrt{\hat{\sigma}^{2}_{n}+\sigma^{2}_{n+1}}}{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}\|[A\,\,b]\|_{F}$
$\displaystyle\approx$
$\displaystyle\frac{\sqrt{\hat{\sigma}^{2}_{n}+\sigma^{2}_{n+1}}}{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}\|[A\,\,b]\|_{F}$
is a moderate multiple of $\frac{1}{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}$.
The improvement of $\bar{\kappa}^{r}_{2}$ over $\bar{\kappa}^{r}_{1}$ becomes
significant as ${\sigma_{n+1}}$ and ${\hat{\sigma}_{n}}$ are close. Similarly,
$\bar{\kappa}^{r}_{2}$ also improves the approximate condition number used in
[2]:
$\bar{\kappa}^{r}_{\cite[cite]{[\@@bibref{}{BjorckHeggernesMatstoms:2000}{}{}]}}:=\frac{\hat{\sigma}_{1}}{\hat{\sigma}_{n}-\sigma_{n+1}}=\frac{\hat{\sigma}_{1}(\hat{\sigma}_{n}+\sigma_{n+1})}{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}.$
We will further illustrate the improvement by numerical experiments to be
presented in Section 5.
## 5 Numerical experiments
We present numerical experiments to illustrate how tight the bounds in
Theorems 9 and 10 are, and to compare the bounds with the related result in
[2]. For a given TLS problem, the TLS solution is computed by (4). All
experiments were run using Matlab 7.8.0 with the machine precision
$\epsilon_{\rm mach}=2.22\times 10^{-16}$ under the Microsoft Windows XP
operating system.
Example 1. In this example, the TLS problem comes from [7]. Specifically, an
$m\times(m-2\omega)$ convolution matrix $\bar{T}$ is constructed to have the
first column
$t_{i,1}=\left\\{\begin{array}[]{ll}\frac{1}{\sqrt{2\pi\alpha^{2}}{\rm{exp}}\left[\frac{-(\omega-i+1)^{2}}{2\alpha^{2}}\right]}&\hbox{\,\,\,$i=1,2,\ldots,2\omega+1$,}\\\
0&\hbox{\,\,\,otherwise,}\end{array}\right.$
and the first row
$t_{1,j}=\left\\{\begin{array}[]{ll}t_{1,1}&\hbox{\,\,\,if $j=1$,}\\\
0&\hbox{\,\,\,otherwise,}\end{array}\right.$
where $\alpha=1.25$ and $\omega=8$. A Toeplitz matrix $A$ and a right-hand
side vector $b$ are constructed as $A=\bar{T}+E$ and $b=\bar{g}+e$, where
$\bar{g}=[1,\ldots,1]^{T}$, $E$ is a random Toeplitz matrix with the same
structure as $\bar{T}$ and $e$ is a random vector. The entries in $E$ and $e$
are generated randomly from a normal distribution with mean zero and variance
one, and scaled so that
$\|e\|=\gamma\|\bar{g}\|,\,\,\,\|E\|=\gamma\|\bar{T}\|,\,\,\gamma=0.001.$
Table 1: $m$ | $\sigma_{n+1}/\sigma_{n}$ | $\sigma_{n+1}/\hat{\sigma}_{n}$ | $\kappa^{r}_{g}(A,b)$ | $\underline{\kappa}^{r}_{2}$ | $\bar{\kappa}^{r}_{2}$ | $\bar{\kappa}^{r}_{1}$ | $\bar{\kappa}^{r}_{\cite[cite]{[\@@bibref{}{BjorckHeggernesMatstoms:2000}{}{}]}}$
---|---|---|---|---|---|---|---
$~{}100~{}$ | $0.981$ | $~{}1-7.85\times 10^{-9}$ | $7.70\times 10^{7}$ | $7.04\times 10^{7}$ | $2.01\times 10^{9}$ | $7.94\times 10^{11}$ | $1.03\times 10^{11}$
$~{}300~{}$ | $0.995$ | $~{}1-2.05\times 10^{-8}$ | $1.40\times 10^{8}$ | $1.26\times 10^{8}$ | $6.90\times 10^{9}$ | $8.83\times 10^{11}$ | $6.54\times 10^{10}$
$~{}500~{}$ | $0.998$ | $~{}1-5.66\times 10^{-8}$ | $9.01\times 10^{7}$ | $7.89\times 10^{7}$ | $6.56\times 10^{9}$ | $3.32\times 10^{11}$ | $1.90\times 10^{10}$
In the table,
$\underline{\kappa}^{r}_{2}=\frac{\underline{\kappa}_{2}}{\|x\|}\|[A\,\,b]\|_{F},\,\,\bar{\kappa}^{r}_{2}=\frac{\bar{\kappa}_{2}}{\|x\|}\|[A\,\,b]\|_{F},\,\,\bar{\kappa}^{r}_{1}=\frac{\bar{\kappa}_{1}}{\|x\|}\|[A\,\,b]\|_{F},$
see Theorems 10 and 9, respectively. We calculate the approximate condition
number used in [2]:
$\bar{\kappa}^{r}_{\cite[cite]{[\@@bibref{}{BjorckHeggernesMatstoms:2000}{}{}]}}=\frac{\hat{\sigma}_{1}}{\hat{\sigma}_{n}-\sigma_{n+1}}.$
As indicated by the table, all the given TLS problems are similar in that
$\sigma_{n+1}$ and $\hat{\sigma}_{n}$ are close but $\sigma_{n+1}$ and
$\sigma_{n}$ are not so close. As estimates of $\kappa^{r}_{g}(A,b)$, the
lower bounds $\underline{\kappa}^{r}_{2}$ are very accurate, and the upper
bounds $\bar{\kappa}^{r}_{2}$ improve the corresponding $\bar{\kappa}^{r}_{1}$
and
$\bar{\kappa}^{r}_{\cite[cite]{[\@@bibref{}{BjorckHeggernesMatstoms:2000}{}{}]}}$
significantly by one or two orders.
Example 2. In this example, the TLS problems are generated by the function
described in Appendix. For given $m,n$ and $\alpha$, $A$ and $b$ are generated
by
$[A\,\,b]={\text{generate}}Ab\alpha(m,n,\alpha).$
A different $\alpha$ gives rise to a different TLS problem with different
properties. As $\alpha$ becomes smaller, $\sigma_{n+1}$ and $\hat{\sigma}_{n}$
become closer, so that the TLS problem becomes worse conditioned. For each of
the TLS problems, we calculate the same quantities as those in Example 1 and
list them in Table 2 in which the first set of data is for $(m,n)=(500,350)$
and the second set is for $(m,n)=(1000,750)$.
Table 2: $\alpha$ | $\sigma_{n+1}/\sigma_{n}$ | $\sigma_{n+1}/\hat{\sigma}_{n}$ | $\kappa^{r}_{g}(A,b)$ | $\underline{\kappa}^{r}_{2}$ | $\bar{\kappa}^{r}_{2}$ | $\bar{\kappa}^{r}_{1}$ | $\bar{\kappa}^{r}_{\cite[cite]{[\@@bibref{}{BjorckHeggernesMatstoms:2000}{}{}]}}$
---|---|---|---|---|---|---|---
$10^{-2}$ | $0.953$ | $1-3.05\times 10^{-4}$ | $2.55\times 10^{4}$ | $8.98\times 10^{3}$ | $1.60\times 10^{5}$ | $5.14\times 10^{5}$ | $6.29\times 10^{5}$
$10^{-3}$ | $0.980$ | $1-3.16\times 10^{-6}$ | $2.01\times 10^{5}~{}$ | $8.75\times 10^{4}$ | $2.42\times 10^{6}$ | $4.92\times 10^{7}$ | $6.03\times 10^{7}$
$10^{-5}$ | $0.953$ | $1-2.77\times 10^{-10}$ | $1.97\times 10^{7}$ | $9.78\times 10^{6}$ | $1.74\times 10^{8}$ | $5.87\times 10^{11}$ | $7.20\times 10^{11}$
$10^{-7}$ | $0.966$ | $1-1.80\times 10^{-14}$ | $3.28\times 10^{9}$ | $1.12\times 10^{9}$ | $2.38\times 10^{10}$ | $8.38\times 10^{15}$ | $1.02\times 10^{16}$
$10^{-2}$ | $0.983$ | $1-2.78\times 10^{-4}$ | $6.76\times 10^{4}$ | $1.65\times 10^{4}$ | $4.97\times 10^{5}$ | $9.90\times 10^{5}$ | $1.21\times 10^{6}$
$10^{-3}$ | $0.978$ | $1-1.95\times 10^{-6}$ | $6.70\times 10^{5}$ | $1.93\times 10^{5}$ | $5.09\times 10^{6}$ | $1.38\times 10^{8}$ | $1.69\times 10^{8}$
$10^{-5}$ | $0.968$ | $1-3.01\times 10^{-10}$ | $4.33\times 10^{7}$ | $1.60\times 10^{7}$ | $3.52\times 10^{8}$ | $9.24\times 10^{11}$ | $1.13\times 10^{12}$
$10^{-7}$ | $0.993$ | $1-3.82\times 10^{-14}$ | $1.13\times 10^{10}$ | $1.44\times 10^{9}$ | $7.02\times 10^{10}$ | $7.38\times 10^{15}$ | $9.03\times 10^{15}$
We can see from the table that, for $\alpha=10^{-2}$ in which
$\hat{\sigma}_{n}$ and $\sigma_{n+1}$ are not very close,
$\bar{\kappa}^{r}_{1}$ and
$\bar{\kappa}^{r}_{\cite[cite]{[\@@bibref{}{BjorckHeggernesMatstoms:2000}{}{}]}}$
are very tight and they estimate $\kappa^{r}_{g}(A,b)$ quite accurately; for
$\alpha\leq 10^{-3}$, $\hat{\sigma}_{n}$ and $\sigma_{n+1}$ become closer with
decreasing $\alpha$, $\bar{\kappa}^{r}_{1}$ and
$\bar{\kappa}^{r}_{\cite[cite]{[\@@bibref{}{BjorckHeggernesMatstoms:2000}{}{}]}}$
estimate $\kappa^{r}_{g}(A,b)$ increasingly more poorly. In contrast, however,
for all the cases, since ${\sigma}_{n}$ and $\sigma_{n+1}$ are not so close,
$\underline{\kappa}^{r}_{2}$ and $\bar{\kappa}^{r}_{2}$ estimate
$\kappa^{r}_{g}(A,b)$ accurately. Particularly, for $\alpha\leq 10^{-5}$,
$\bar{\kappa}^{r}_{2}$ improves $\bar{\kappa}^{r}_{1}$ and
$\bar{\kappa}^{r}_{\cite[cite]{[\@@bibref{}{BjorckHeggernesMatstoms:2000}{}{}]}}$
very considerably by several orders.
## 6 Concluding Remarks
In the paper, we have mainly studied the condition number of the TLS problem
and its lower and upper bounds that can be numerically computed cheaply. For
the TLS condition number, we have derived a new closed formula. For a
computational purpose, we can use it to compute the condition number more
accurately. We have derived a few bounds, which are quite sharp and can be
calculated cheaply. We have confirmed our results numerically and demonstrated
the tightness of our bounds by numerical experiments.
ACKNOWLEDGEMENTS
The work was partially supported by National Basic Research Program of China
2011CB302400 and National Science Foundation of China (No. 11071140) and
Specialized Research Fund for the Doctoral Program of Higher Education (No.
20070200009)
## Appendix A Codes for generating tested TLS problems
The following codes produce an $m\times(n+1)$ matrix $[A\,\,b]$, which has the
SVD $[A\,\,b]=U\Sigma V^{T}$ with $V(n+1,n+1)=-\alpha$, where $0<\alpha<1$.
$\displaystyle[A\,\,b]={\text{{\bf{generate}}}}{\bf{Ab}}{\bf{\alpha}}(m,n,\alpha)$
$\displaystyle\%~{}m,n:{\text{two given positive integers with }}m\geq n$
$\displaystyle\%~{}\alpha:{\text{a given positive number with $0<\alpha<1$}}$
$\displaystyle\text{Generate}~{}\tilde{V};~{}~{}\text{\% a random orthogonal
matrix }~{}\text{of order}~{}n$ $\displaystyle
V=\text{\bf{generateV}}(n,\tilde{V},\alpha);$ $\displaystyle
B=\text{rand}(m,n+1);~{}~{}\text{\% the Matlab function rand(~{})}$
$\displaystyle[U,\Sigma,\hat{V}]=\text{svd}(B,0);~{}~{}\text{\% the Matlab
function svd(~{})}$ $\displaystyle[A\,\,b]=U*\Sigma*V^{T}$
The subfunction ${\bf{generateV}}()$ is shown as follows. It is used to
produce an $(n+1)\times(n+1)$ orthogonal matrix $V$ with $V(n+1,n+1)=-\alpha$,
where $0<\alpha<1$. The idea of construction comes from Lemma 2.
$\displaystyle[V]={\text{{\bf{generate}}}}{\bf{V}}(n,\tilde{V},\alpha)$
$\displaystyle\%~{}n:{\text{a given positive integer}}$
$\displaystyle\%~{}\tilde{V}:{\text{a given orthogonal matrix of order $n$}}$
$\displaystyle\%~{}\alpha:{\text{a given positive number with $0<\alpha<1$}}$
$\displaystyle\text{partition}~{}\tilde{V}=[\tilde{v}_{1},\ldots,\tilde{v}_{n}];$
$\displaystyle\text{generate}~{}U=[u_{1},\ldots,u_{n}];~{}~{}~{}{\text{\% a
random orthogonal matrix of order $n$}}$ $\displaystyle
V_{11}=[u_{1},\ldots,u_{n-1}][\tilde{v}_{1},\ldots,\tilde{v}_{n-1}]^{T}+\alpha
u_{n}\tilde{v}^{T}_{n};$ $\displaystyle
V=\left[\begin{array}[]{cc}V_{11}&\sqrt{1-\alpha^{2}}u_{n}\\\
\sqrt{1-\alpha^{2}}\tilde{v}^{T}_{n}&-\alpha\\\ \end{array}\right]$ (91)
Remark. Lemma 4.3 in [5] gives
$\frac{|\hat{u}^{T}_{n}b|}{2(\hat{\sigma}_{n}-\sigma_{n+1})}\leq\|x\|\leq\frac{\|b\|}{\hat{\sigma}_{n}-\sigma_{n+1}}.$
Equivalently, it holds that
$\frac{|\hat{u}^{T}_{n}b|}{2\|x\|}\leq\hat{\sigma}_{n}-\sigma_{n+1}\leq\frac{\|b\|}{\|x\|},$
(92)
where it is supposed that $x\neq 0$. Note that $V(n+1,n+1)=-\alpha$ and
$\alpha=\frac{1}{\sqrt{1+\|x\|^{2}}}$. From (92) we see that a small $\alpha$
implies that $\hat{\sigma}_{n}$ and $\sigma_{n+1}$ are close in some sense.
## References
* [1] M. Baboulin, S. Gratton, A contribution to the conditioning of the total least squares problem, arXiv:1012.5484v1.
* [2] Å. Björck, P. Heggernes, P. Matstoms, Methods for large scale total least squares problems, SIAM J. Matrix Anal. Appl., 22 (2000) 413–429.
* [3] R. D. Fierro, J. R. Bunch, Perturbation theory for orthogonal projection methods with applications to least squares and total least squares, Linear Algebra Appl., 234 (1996) 71–96.
* [4] I. Gohberg, I. Koltracht, Mixed, componentwise, and structured condition numbers, SIAM J. Matrix. Anal. Appl., 14 (1993) 688–704.
* [5] G. H. Golub,C. F. Van Loan, An analysis of the total least squares problem, SIAM J. Numer. Anal., 17 (1980) 883–893.
* [6] A. Graham, Kronecker Products and Matrix Calculus with Application, Wiley, New York, 1981.
* [7] J. Kamm and J. G. Nagy, A total least squares method for Toeplitz system of equations, BIT, 38 (1998) 560–582.
* [8] B. Li, Z. Jia, Some results on condition numbers of the scaled total least squares problem, Linear Algebra Appl. (2010), doi:10.1016/j.laa.2010.07.022.
* [9] X. Liu, On the solvability and perturbation analysis for total least squares problem, Acta Mathematicae Applicatae Sinica, 19 (1996) 253–262 (in Chinese).
* [10] A. N. Malyshev,A unified theory of conditioning for linear least squares and Tikhonov regularization solutions, SIAM J. Matrix. Anal. Appl., 24 (2003) 1186–1196.
* [11] C. C. Paige, M. A. Saunders, Towards a generalized singular value decomposition, SIAM J. Numer. Anal., 18 (1981) 398–405.
* [12] C. C. Paige, Z. Strako$\check{s}$, Scaled total least squares fundamentals, Numer. Math., 91 (2002) 117–146.
* [13] J. R. Rice, A theory of condition, SIAM J. Numer. Anal., 3 (1966) 287–310.
* [14] Roger A. Horn, Charles R. Johnson, Matrix Analysis, Cambridge University Press, New York, 1985.
* [15] M. Wei, The analysis for the total least squares problem with more than one solution, SIAM J. Matrix. Anal. Appl., 13 (1992) 746–763.
* [16] M. Wei, On the perturbation of the LS and TLS problems, Mathematica Numerica Sinica, 20 (1998) 267–278 (in Chinese).
* [17] J. H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University Press, London, 1965.
* [18] L. Zhou, L. Lin, Y. Wei, S. Qiao, Perturbation analysis and condition numbers of Scaled Total Least Squares problems. Numer. Algor., 51 (2009) 381–399.
|
arxiv-papers
| 2011-01-12T12:42:18 |
2024-09-04T02:49:16.362561
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhongxiao Jia and Bingyu Li",
"submitter": "Li",
"url": "https://arxiv.org/abs/1101.2342"
}
|
1101.2363
|
In this article we review the theory of anafunctors introduced by Makkai
and Bartels, and show that given a subcanonical site $S$, one can form a
bicategorical localisation of various 2-categories of internal categories or
groupoids at weak equivalences using anafunctors as 1-arrows. This unifies
a number of proofs throughout the literature, using the fewest assumptions
possible on $S$.
§ INTRODUCTION
It is a well-known classical result of category theory that a functor
is an equivalence (that is, in the 2-category of categories) if and
only if it is fully faithful and essentially surjective.
This fact is equivalent to the axiom of choice. It is therefore
not true if one is working with categories internal to a category
$S$ which doesn't satisfy the (external) axiom of choice. This is may fail
even in a category very much like the category of sets, such as a
well-pointed boolean topos, or even the category of sets in constructive
foundations. As internal
categories are the objects of a 2-category $\Cat(S)$ we can talk about
internal equivalences, and even fully faithful functors. In the case
$S$ has a singleton
pretopology $J$ (i.e. covering families
consist of single maps) we can define an analogue of essentially
surjective functors. Internal functors which are fully faithful
and essentially surjective are called weak equivalences in the
literature, going back to [16]. We shall call them $J$-equivalences for
clarity. We can recover the classical result mentioned above if we localise the
2-category $\Cat(S)$ at the class $W_J$ of $J$-equivalences.
We are not just interested in localising $\Cat(S)$, but various
full sub-2-categories $C \into \Cat(S)$ which arise in the
study of presentable stacks, for example algebraic, topological, differentiable,
etc. stacks. As such it is necessary to ask for a compatibility
condition between the pretopology on $S$ and the sub-2-category we are
interested in. We call this condition existence of base change for
covers of the pretoplogy, and demand that for any cover $p\colon U\to X_0$
(in $S$) of the object of objects of $X\in C$, there is a fully faithful
functor in $C$ with object component $p$.
Let $S$ be a category with singleton pretopology $J$ and let $C$ be a
full sub-2-category of $\Cat(S)$ which admits base change along arrows in
$J$. Then $C$ admits a calculus of fractions for the $J$-equivalences.
Pronk gives us the appropriate notion of a calculus of fractions for a
2-category in [53] as a generalisation of the usual construction
for categories [29]. In her construction, 1-arrows are spans and 2-arrows are
equivalence classes of bicategorical spans of spans. This construction,
while canonical, can be a little unwieldy so we look for a simpler construction
of the localisation.
We find this in the notion of anafunctor, introduced by Makkai
for plain small categories [41] (Kelly described them briefly
in [34] but did not develop the concept further). In his setting an anafunctor
is a span of functors such that the left (or source) leg is a surjective-on-objects,
fully faithful functor.[Anafunctors were so named by Makkai, on the
suggestion of Pavlovic, after profunctors,
in analogy with the pair of terms anaphase/prophase from biology. For more on the
relationship between anafunctors and profunctors, see below.]
For a general category $S$ with a subcanonical
singleton pretopology $J$ [6], the analogon is a span with left leg a
fully faithful functor with object component a cover. Composition of
anafunctors is given by composition of spans in the usual way, and there
are 2-arrows between anafunctors (a certain sort of span of spans) that
give us a bicategory $\Cat_\ana(S,J)$ with objects internal categories
and 1-arrows anafunctors. We can also define the full sub-bicategory
$C_\ana(J) \into \Cat_\ana(S,J)$ analogous to $C$, and there is a strict
inclusion 2-functor $C \into C_\ana(J)$. This gives us our second main
Let $S$ be a category with subcanonical singleton pretopology $J$ and let
$C$ be a full sub-2-category of $\Cat(S)$ which admits base change
along arrows in $J$, Then $C \into C_\ana(J)$ is a localisation
of $C$ at the class of $J$-equivalences.
So far we haven't mentioned the issue of size, which usually is important
when constructing localisations. If the site $(S,J)$ is locally small,
then $C$ is locally small, in the sense that the hom-categories are
small. This also implies that $C_\ana(J)$ and hence any
$C[W_J^{-1}]$ has locally small hom-categories i.e. has only
a set of 2-arrows between any pair of 1-arrows. To prove that the
localisation is locally essentially small (that is, hom-categories are equivalent
to small categories), we need to assume a size restriction axiom on the
pretopology $J$, called WISC (Weakly Initial Sets of Covers).
WISC can be seen as an extremely weak choice principle, weaker than the existence
of enough projectives, and states that for every object $A$ of $S$, there is
a set of $J$-covers of $A$ which is cofinal in all $J$-covers of $A$. It is
automatically satisfied if the pretopology is specified as an assignment of
a set of covers to each object.
Let $S$ be a category with subcanonical singleton pretopology $J$ satisfying
WISC, and let $C$ be a full sub-2-category of $\Cat(S)$ which admits
base change along arrows in $J$. Then any localisation of $C$ at the
class of $J$-equivalences is locally essentially small.
Since a singleton pretopology can be conveniently defined as a certain wide
subcategory, this is not a vacuous statement for large sites, such as $\Top$
or $\Grp(E)$ (group objects in a topos $E$). In fact WISC is independent
of the Zermelo-Fraenkel axioms (without Choice) [13, 54].
It is thus possible to have the theorem fail for the topos $S = \Set_{\neg AC}$
with surjections as covers.
Since there have been many very closely related approaches to localisation
of 2-categories of internal categories and groupoids, we give a brief sketch
in the following section. Sections 3 to 6 of this article then give necessary background and notation
on sites, internal categories, anafunctors and bicategories of fractions respectively.
Section 7 contains our main results, while section 8 shows examples from
the literature that are covered by the theorems from section 7. A short appendix
detailing superextensive sites is included, as this material does not appear to
be well-known (they were discussed in the recent [58], Example 11.12).
This article started out based on the first chapter of the author's PhD thesis,
which only dealt with groupoids in the site of topological spaces
and open covers.
Many thanks are due to Michael Murray, Mathai Varghese and Jim Stasheff,
supervisors to the author. The patrons of the $n$-Category Café and $n$Lab,
especially Mike Shulman and Toby Bartels, provided helpful input and feedback.
Steve Lack suggested a number of improvements, and the referee asked for
a complete rewrite of this article, which
has greatly improved the theorems, proofs, and hopefully also the exposition. Any delays
in publication are due entirely to the author.
§ ANAFUNCTORS IN CONTEXT
The theme of giving 2-categories of internal categories or groupoids more
equivalences has been approached in several different ways over the decades.
We sketch a few of them, without necessarily finding the original references, to
give an idea of how widely the results of this paper apply. We give some
more detailed examples of this applicability in section 8.
Perhaps the oldest related construction is the distributors of Bénabou, also
known as modules or profunctors [9] (see [31] for
a detailed treatment of internal profunctors, as the original article is difficult to source).
Bénabou pointed out [12], after a preprint of this article was
released, that in the case of the category $\Set$ (and more generally in a finitely
complete site with reflexive coequalisers that are stable under pullback,
see [42]), the bicategory
of small (resp. internal) categories with representable profunctors as 1-arrows is
equivalent to the bicategory of small categories with anafunctors as 1-arrows.
In fact this was discussed by Baez and Makkai [8],
where the latter pointed out that representable profunctors correspond to
saturated anafunctors in his setting. The author's preference for anafunctors
lies in the fact they can be defined with weaker assumptions on the site $(S,J)$,
and in fact in the sequel [56], do not require the 2-category to have objects
which are internal categories. In a sense this is analogous to [60],
where the formal bicategorical approach to profunctors between objects of
a bicategory is given, albeit still requiring more colimits to exist than anafunctors do.
Bénabou has pointed out in private communication that he has
an unpublished distributor-like construction that does not rely on existence of
reflexive coequalisers; the author has not seen any details of this and is
curious to see how it compares to anafunctors.
Related to this is the original work of Bunge and Paré [16],
where they consider functors between indexed categories associated to
internal categories, that is, the externalisation of an internal category and stack
completions thereof. This was
one motivation for considering weak equivalences in the first place, in that a pair
of internal categories have equivalent stack completed externalisations if and only if they are
connected by a span of internal functors which are weak equivalences.
Another approach is constructing bicategories of fractions à la Pronk [53].
This has been followed by a number of authors, usually followed up by an explicit
construction of a localisation simplifying the canonical one. Our work here sits at the
more general end of this spectrum, as others have tailored their constructions to
take advantage of the structure of the site they are interested in. For example,
butterflies (originally called papillons) have been used for the category of
groups [50, 3, 4],
abelian categories [15] and semiabelian categories [1, 42].
These are similar to the meromorphisms of [52], introduced in
the context of the site of smooth manifolds; though these only use a 1-categorical
approach to localisation.
Vitale [62], after first showing that the
2-category of groupoids in a regular category has a bicategory
of fractions, then shows that for protomodular regular categories one can
generalise the pullback congruences of Bénabou in [11] to
discuss bicategorical localisation. This approach can be applied to internal
categories, as long as one restricts to invertible 2-arrows. Similarly, [42]
give a construction of what they call fractors between internal
groupoids in a Mal'tsev category, and show that in an efficiently regular category
(e.g. a Barr-exact category) fractors are 1-arrows in a localisation of the 2-category of
internal groupoids. The proof also works for internal categories if one considers
only invertible 2-arrows.
Other authors, in dealing with internal groupoids, have adopted the
approach pioneered by Hilsum and Skandalis
[30], which has gone by various names including
Hilsum-Skandalis morphisms, Morita morphisms, bimodules, bibundles, right principal bibundles
and so on. All of these are very closely related to saturated anafunctors, but
in fact no published definition of a saturated anafunctor in a site other than $\Set$
([41]) has appeared, except in the guise of internal profunctors
(e.g. [31], section B2.7). Note also that this approach has only been applied to internal
groupoids. The review [37] covers the case of Lie groupoids, and in
particular orbifolds, while [48] treats bimodules between groupoids in the
category of affine schemes, but from the point of view of Hopf algebroids.
The link between localisation at weak equivalences and presentable stacks is considered
in (of course) [53], as well as more recently in [18], [57],
in the cases of topological and algebraic stacks respectively, and for example
[61] in the case of differentiable stacks.
A third approach is by considering a model category structure on the 1-category
of internal categories. This is considered in [32] for categories
in a topos, and in [28] for categories in a
finitely complete subcanonical site $(S,J)$.
In the latter case the authors show when it is possible to construct a Quillen model category
structure on $\Cat(S)$ where the weak equivalences are the weak equivalences
from this paper. Sufficient conditions on $S$ include being a topos with nno, being
locally finitely presentable or being finitely complete regular Mal'tsev – and additionally
having enough $J$-projective objects. If one is willing to consider other model-category-like
structures, then these assumptions can be dropped. The proof from [28]
can be adapted to show that for a finitely complete site $(S,J)$, the category of groupoids
with source and target maps restricted to be $J$-covers has the structure of a category of fibrant objects,
with the same weak equivalences. We note that [20] gives a Quillen
model structure for the category of orbifolds, which are there defined to be proper topological
groupoids with discrete hom-spaces.
In a similar vein, one could consider a localisation using hammock localisation
<cit.> of a category of internal categories,
which puts one squarely in the realm of $(\infty,1)$-categories. Alternatively, one could
work with the $(\infty,1)$-category arising from a 2-category of internal categories, functors
and natural isomorphisms and consider a localisation of this as given in, say
[38]. However, to deal with general 2-categories of internal categories in this way,
one needs to pass to $(\infty,2)$-categories to handle the non-invertible 2-arrows. The
theory here is not so well-developed, however, and one could see the results
of the current paper as giving
toy examples with which one could work. This is one motivation for making sure the
results shown in this paper apply to not just 2-categories of groupoids. Another is extending
the theory of presentable stacks from stacks of groupoids to stacks of categories [55].
§ SITES
The idea of surjectivity is a necessary ingredient when talking about
equivalences of categories—in the guise of just essential surjectivity—but it doesn't
generalise in a straightforward way from the category $\Set$. The necessary properties
of the class of surjective maps are encoded in the definition of a Grothendieck pretopology,
in particular a singleton pretopology. This section gathers definitions and
notations for later use.
A Grothendieck pretopology (or simply pretopology) on a category $S
$ is a collection $J$ of families
\[
\{ (U_i \to A)_{i\in I} \}_{A\in \Obj(S)}
\]
of morphisms for each object $A \in S$ satisfying the following properties
* $(A' \stackrel{\sim}{\to} A)$ is in $J$ for every isomorphism $A'\simeq A$.
* Given a map $B \to A$, for every $(U_i \to A)_{i\in I}$ in $J$ the pullbacks
$B \times_A A_i$ exist and $(B \times_A A_i \to B)_{i\in I}$ is in $J$.
* For every $(U_i \to A)_{i\in I}$ in $J$ and for a collection $(V_k^i \to
U_i)_{k\in K_i}$ from $J$ for each $i \in I$, the family of composites
\[
(V_k^i \to A)_{k\in K_i,i\in I}
\]
are in $J$.
Families in $J$ are called covering families. We call a category $S$ equipped with a
pretopology $J$ a site, denoted $(S,J)$ (note that often one sees
a site defined as a category equipped with a Grothendieck topology).
The pretopology $J$ is called a singleton pretopology if
every covering family consists of a single arrow $(U \to A)$. In this case a covering
family is called a cover and we call $(S,J)$ a unary site.
Very often, one sees the definition of a pretopology as being an assignment of a set
covering families to each object. We do not require this, as
one can define a singleton pretopology as a subcategory with certain properties,
and there is not necessarily then
a set of covers for each object. One example is the category of groups with surjective
homomorphisms as covers. This distinction will be important later.
One thing we will require is that sites come with specified pullbacks of
covering families. If one does not mind applying the axiom of choice (resp. axiom of choice
for classes) then any small site (resp. large site) can be so equipped. But often sites that
arise in practice have more or less canonical choices for pullbacks, such as the category
of ZF-sets.
The prototypical example is the pretopology $\mathcal{O}$ on $\Top$, where a covering family is an open cover.
The class of numerable open covers (i.e. those that admit a subordinate
partition of unity [25]) also forms a pretopology on $\Top$. Much of traditional bundle theory
is carried out using this site; for example the Milnor classifying space classifies
bundles which are locally trivial over numerable covers.
A covering family $(U_i \to A)_{i\in I} $ is called effective if $A$ is the colimit
of the following diagram: the objects are the $U_i$ and the pullbacks $U_i \times_A
U_j$, and the arrows are the projections
\[
U_i \leftarrow U_i \times_A U_j \to U_j.
\]
If the covering family consists of a single arrow $(U \to A)$, this is the same as saying
$U \to A$ is a regular epimorphism.
A site is called subcanonical if every covering family is effective.
On $\Top$, the usual pretopology $\mathcal{O}$ of opens, the pretopology of numerable covers
and that of open surjections are subcanonical.
In a regular category, the class of regular epimorphisms forms a subcanonical singleton
In fact we can make the following definition.
For a category $S$, the largest class of regular epimorphisms of which all pullbacks exist,
and which is stable under pullback, is called the canonical singleton pretopology
and denoted $\can$.
This is a to be contrasted to the canonical topology on a category, which consists
of covering sieves rather than covers. The canonical singleton pretopology is the largest
subcanonical singleton pretopology on a category.
Let $(S,J)$ be a site. An arrow $P \to A$ in $S$ is called a $J$-epimorphism
if there is a covering family $(U_i \to A)_{i\in I}$ and a lift
\[
\xymatrix{
& P \ar[d] \\
U_i \ar@{-->}[ur] \ar[r] & A
\]
for every $i \in I$. A $J$-epimorphism is called universal if its pullback
along an arbitrary map exists. We denote the singleton pretopology of universal $J$-epimorphisms
by $J_{un}$.
This definition of $J$-epimorphism is equivalent to the definition in III.7.5 in
[40]. The
dotted maps in the above definition are called local sections, after the case of
the usual open cover pretopology on $\Top$. If $J$ is a singleton pretopology,
it is clear that $J \subset J_{un}$.
The universal $\mathcal{O}$-epimorphisms for the pretopology $\mathcal{O}$ of
open covers on $\Diff$ form $Subm$, the pretopology of surjective submersions.
In a finitely complete category the universal $triv$-epimorphisms are the split
epimorphisms, where $triv$ is the trivial pretopology where all covering
families consist of a single isomorphism. In $\Set$ with the axiom of
choice there are all the epimorphisms.
Note that for a finitely complete site $(S,J)$, $J_{un}$ contains $triv_{un}$,
hence all the split epimirphisms.
Although we will not assume that all sites we consider are finitely complete,
results similar to ours have, and so in that case we can say a little more,
given stronger properties on the pretopology.
A singleton pretopology $J$ is called saturated if whenever the composite
$A \stackrel{h}{\to} B \stackrel{g}{\to} C$ is in $J$, then $g\in J$.
The concept of a saturated pretopology was introduced by Bénabou under the name
calibration [10]. It follows from the definition that a saturated singleton
pretopology contains the split epimorphisms (take $h$ to be a section of the epimorphism $g$).
The canonical singleton pretopology $\can$ in a regular category (e.g. a topos) is saturated.
Given a pretopology $J$ on a finitely complete category, $J_{un}$ is saturated.
Sometimes a pretopology $J$ contains a smaller pretopology that still has enough
covers to compute the same $J$-epimorphisms.
If $J$ and $K$ are two singleton pretopologies with $J \subset K$, such that
$K \subset J_{un}$, then $J$ is said to be cofinal in $K$.
Clearly $J$ is cofinal in $J_{un}$ for any singleton pretopology $J$.
If $J$ is cofinal in $K$, then $J_{un} = K_{un}$.
We have the following lemma, which is essentially proved in [31], C2.1.6.
If a pretopology $J$ is subcanonical, then so any pretopology in which it is cofinal.
In particular, $J$ subcanonical implies $J_{un}$ subcanonical.
As mentioned earlier, one may be given a singleton pretopology such that each
object has more than a set's worth of covers. If such a pretopology contains a
cofinal pretopology with set-many covers for each object, then we can pass to
the smaller pretopology and recover the same results (in a way that
will be made precise later). In fact, we can get away
with something weaker: one could ask only that the category of all covers of an
object (see definition <ref> below) has a set of weakly initial
objects, and such set may not form a pretopology.
This is the content of the axiom WISC below. We first give some more
precise definitions.
A category $C$ has a weakly initial set $\mathcal{I}$ of objects if for every
object $A$ of $C$ there is an arrow $O\to A$ from some object $O\in \mathcal{I}$.
For example the large category $\Fields$ of fields has a weakly initial set, consisting of
the prime fields $\{\mathbb{Q},\mathbb{F}_p|p\textrm{ prime}\}$.
To contrast, the category of sets with surjections for arrows doesn't have a
weakly initial set of objects. Every small category has a weakly initial set, namely its set of objects.
We pause only to remark that the statement of the adjoint functor theorem can be
expressed in terms of weakly initial sets.
Let $(S,J)$ be a site. For any object $A$, the category of covers of $A$,
denoted $J/A$ has as objects the covering families $(U_i \to A)_{i\in I}$ and as
morphisms $(U_i \to A)_{i\in I} \to (V_j \to A)_{j\in J}$ tuples consisting of a function
$r\colon I\to J$ and arrows $U_i \to V_{r(i)}$ in $S/A$.
When $J$ is a singleton pretopology this is simply a full subcategory of $S/A$.
We now define the axiom WISC (Weakly Initial Set of Covers), due independently to Mike
Shulman and Thomas Streicher.
A site $(S,J)$ is said to satisfy WISC if for every object $A$ of $S$, the
category $J/A$ has a weakly initial set of objects.
A site satisfying WISC is in some sense constrained by a small amount of data for each object.
Any small site satisfies WISC, for example, the usual site of finite-dimensional smooth
manifolds and open covers. Any pretopology $J$ containing a cofinal pretopology $K$
such that $K/A$ is small for every object $A$ satisfies WISC.
Any regular category (for example a topos) with enough projectives, equipped with the canonical
singleton pretopology, satisfies WISC. In the case of $\Set$ `enough projectives' is the Presentation
Axiom (PAx), studied, for instance, by Aczel [2] in the context of constructive set theory.
$(\Top,\mathcal{O})$ satisfies WISC, using AC in $\Set$.
Choice may be more than is necessary here; it would be interesting to see if weaker
choice principles in the site $(\Set,surjections)$ are enough to prove WISC for
$(\Top,\mathcal{O})$ or other concrete sites.
If $(S,J)$ satisfies WISC, then so does $(S,J_{un})$.
It is instructive to consider an example where WISC fails in a non-artificial
way. The category of sets and surjections with all arrows covers clearly
doesn't satisfy WISC, but is contrived and not a `useful' sort of category.
For the moment, assume the existence
of a Grothendieck universe $\mathbb{U}$ with cardinality $\lambda$,
and let $\mathrm{Set}_\mathbb{U}$ refer to the category of
$\mathbb{U}$-small sets. Clearly we can define WISC relative to $\mathbb{U}$,
call it WISC${}_\mathbb{U}$.
Let $G$ be a $\mathbb{U}$-large group and $\mathbf{B}G$
the $\mathbb{U}$-large groupoid with one object associated to $G$. The boolean
topos $\mathrm{Set}_\mathbb{U}^{\mathbf{B}G}$ of $\mathbb{U}$-small $G$-sets is
a unary site with the class $epi$ of epimorphisms for covers. One could consider
this topos as being an exotic sort of forcing construction.
If $G$ has at least $\lambda$-many conjugacy classes of subgroups,
then $(\mathrm{Set}_\mathbb{U}^{\mathbf{B}G},epi)$ does not satisfy
Alternatively, one could work in foundations where it is legitimate to discuss
a proper class-sized group, and then consider the topos of sets with an action
by this group. If there is a proper class of conjugacy classes of subgroups, then
this topos with its canonical singleton pretopology will fail to satisfy WISC. Simple
examples of such groups are $\mathbb{Z}^\mathbb{U}$ (given a universe $\mathbb{U}$) and
$\mathbb{Z}^K$ (for some proper class $K$).
Recently, [13] (relative to a large cardinal axiom) and [54] (with no large cardinals)
have shown that the category of sets may fail to satisfy WISC. The models constructed
in [33] are also conjectured to not satisfy WISC.
Perhaps of independent interest is a form of WISC with a bound: the weakly
initial set for each category $J/A$ has cardinality less than some cardinal $\kappa$
(call this WISC${}_\kappa$).
Then one could consider, for example, sites where each
object has a weakly initial finite or countable set of covers. Note that the condition
`enough projectives' is the case $\kappa = 2$.
§ INTERNAL CATEGORIES
Internal categories were introduced in [27], starting
with differentiable and topological categories (i.e. internal to $\Diff$ and
$\Top$ respectively). We collect here the necessary definitions, terminology and
notation. For a thorough recent account, see [7] or the encyclopedic
Fix a category $S$, referred to as the ambient category.
An internal category $X$ in a category $S$ is a diagram
\[
X_1 \times_{X_0} X_1 \times_{X_0} X_1\rightrightarrows X_1 \times_{X_0}
X_1 \xrightarrow{m} X_1 \stackrel{s,t}{\st} X_0 \xrightarrow{e} X_1
\]
in $S$ such that the multiplication $m$ is associative (we demand the
limits in the
diagram exist), the unit map $e$
is a two-sided unit for $m$ and $s$ and $t$ are the usual source and
target. An internal groupoid is an internal category with an
\[
(-)^{-1}\colon X_1 \to X_1
\]
satisfying the usual diagrams for an inverse.
Since multiplication is associative, there is a well-defined map
$X_1 \times_{X_0} X_1 \times_{X_0} X_1 \to X_1$, which will also be denoted by
$m$. The pullback in the diagram in definition <ref> is
\[
\xymatrix{
X_1 \times_{X_0} X_1 \ar[r] \ar[d] & X_1 \ar[d]^-{s}\\
X_1 \ar[r]_-{t} & X_0\;.
\]
and the double pullback is the limit of
$X_1 \stackrel{t}{\rightarrow} X_0 \stackrel{s}{\leftarrow} X_1
\stackrel{t}{\rightarrow} X_0 \stackrel{s}{\leftarrow}X_0$.
These, and pullbacks like these (where source is pulled back along target), will
occur often. If confusion can arise, the maps in question will be explicity
written, as in $X_1 \times_{s,X_0,t} X_1$. One usually sees the requirement that
$S$ is finitely complete in order to define internal categories. This is not
strictly necessary, and not true in the well-studied case of $S = \Diff$, the
category of smooth manifolds.
Often an internal category will be denoted $X_1 \st X_0$, the arrows $m,s,t,e$
(and $(-)^{-1}$) will be referred to as structure maps and $X_1$ and
$X_0$ called the object of arrows and the object of objects respectively. For
example, if $S = \Top$, we have the space of arrows and the space of objects,
for $S = \Grp$ we have the group of arrows and so on.
If $X \to Y$ is an arrow in $S$ admitting iterated kernel pairs, there is an
internal groupoid $\check{C}(X)$ with
$\check{C}(X)_0 = X$, $\check{C}(X)_1 = X \times_Y X$,
source and target are projection on first and second factor, and the
multiplication is projecting out the middle factor in $X \times_Y X \times_Y X$.
This groupoid is called the Čech groupoid of the map $X \to Y$. The origin of
the name is that in $\Top$, for maps of the form $\coprod_I U_i \to Y$ (arising
from an open cover), the Čech groupoid $\check{C}(\coprod_I U_i)$ appears in
the definition of Čech cohomology.
Let $S$ be a category with binary products. For each object $A \in S$ there is an internal groupoid
$\disc(A)$ which has $\disc(A)_1 = \disc(A)_0 = A$ and all structure maps equal
to $id_A$. Such a category is called discrete.
There is also an internal groupoid $\codisc(A)$ with
\[
\codisc(A)_0 = A,\
\codisc(A)_1 = A \times A
\]
and where source and target are projections on the
first and second factor respectively. Such a groupoid is called
Given internal categories $X$ and $Y$ in $S$, an internal functor
is a pair of maps
\[
f_0\colon X_0 \to Y_0 \quad\textrm{and}\quad f_1\colon X_1 \to Y_1
\]
called the object and arrow component respectively. Both components are
required to commute with all the structure maps.
If $A\to C$ and $B\to C$ are maps admitting iterated kernel pairs, and $A \to B$
is a map over $C$, there is a functor $\check{C}(A) \to \check{C}(B)$.
If $(S,J)$ is a subcanonical unary site, and $U \to A$ is a cover,
a functor $\check{C}(U) \to \disc(B)$
gives a unique arrow $A\to B$. This follows immediately from the fact $A$ is the
colimit of the diagram underlying $\check{C}(U)$.
Given internal categories $X,Y$ and internal functors $f,g\colon X \to Y$, an
internal natural transformation (or simply transformation)
\[
a\colon f \Rightarrow g
\]
is a map $a\colon X_0 \to Y_1$ such that $s \circ a = f_0,\ t\circ a = g_0$ and
the following diagram commutes
\begin{equation}\label{diag:naturality}
\xymatrix{
X_1 \ar[r]^-{(g_1,a\circ s)} \ar[d]_{(a \circ t,f_1)} &
Y_1 \times_{Y_0} Y_1 \ar[d]^{m} \\
Y_1 \times_{Y_0} Y_1 \ar[r]^-{m} & Y_1
\end{equation}
expressing the naturality of $a$.
Internal categories (resp. groupoids), functors and transformations
in a locally small category $S$ form a
locally small 2-category $\Cat(S)$ (resp. $\Gpd(S)$) [27]. There
is clearly an inclusion 2-functor $\Gpd(S) \to \Cat(S)$.
Also, $\disc$ and $\codisc$, described in example <ref>, are
2-functors $S \to \Gpd(S)$, whose underlying functors
are left and right adjoint to the functor
\[
\Obj\colon\Cat(S)_{\leq 1} \to S,\qquad (X_1\st X_0)\mapsto X_0.
\]
Here $\Cat(S)_{\leq 1}$ is the 1-category underlying the 2-category
$\Cat(S)$. Hence for an internal category $X$ in $S$, there are functors
$\disc(X_0) \to X$ and $X \to \codisc(X_0)$, the arrow component of the latter
being $(s,t):X_1\to X_0^2$.
We say a natural transformation is a natural isomorphism if it has an
inverse with respect to vertical composition. Clearly there is no distinction
between natural transformations and natural isomorphisms when the codomain of
the functors is an internal groupoid.
We can reformulate the naturality diagram (<ref>) in the case
that $a$ is a natural isomorphism. Denote by $-a$ the inverse of $a$.
Then the diagram (<ref>) commutes if and only if the diagram
\begin{equation}\label{naturality}
\xymatrix{
X_0 \times_{X_0} X_1 \times_{X_0} X_0
\ar[rr]^{-a\times f_1 \times a}
\ar[d]_{\simeq} &&Y_1 \times_{Y_0} Y_1 \times_{Y_0} Y_1 \ar[d]^m \\
X_1 \ar[rr]_{g_1} && Y_1
\end{equation}
commutes, a fact we will use several times.
If $X$ is a category in $S$, $A$ is an object of $S$ and $f,g:X \to \codisc(A)$
are functors, there is a unique natural isomorphism
$f\stackrel{\sim}{\Rightarrow} g$.
An internal or strong equivalence of internal categories is an
equivalence in the 2-category of internal categories. That is, an internal functor $f
\colon X\to Y$ such that there is a functor $f'\colon Y\to X$ and natural isomorphisms
$f\circ f' \Rightarrow \id_Y$, $f'\circ f \Rightarrow \id_X$.
For an internal category $X$ and a map $p:M\to X_0$ in $S$ the base change of $X$ along
$p$ is any category $X[M]$ with object of objects $M$ and object of arrows given by the pullback
\[
\xymatrix{
M^2 \times_{X_0^2} X_1 \ar[r] \ar[d] & X_1 \ar[d]^{(s,t)} \\
M^2 \ar[r]_{p^2} & X_0^2
\]
If $C\subset \Cat(S)$ denotes a full sub-2-category and if the base change
along any map in a given class $K$ of maps exists in $C$ for all objects
of $C$, then we say $C$ admits base change along maps in $K$,
or simply admits base change for $K$.
In all that follows, `category' will mean object of $C$ and similarly for
`functor' and `natural transformation/isomorphism'.
The strict pullback of internal categories
\[
\xymatrix{
X \times_Y Z \ar[r] \ar[d] & Z \ar[d] \\
X \ar[r] & Y
\]
when it exists, is the internal category with objects $X_0 \times_{Y_0} Z_0$, arrows
$X_1 \times_{Y_1} Z_1$, and all structure maps given componentwise by those of
$X$ and $Z$. Often we will be able to prove that certain pullbacks exist because of
conditions on various component maps in $S$. We do not assume that all strict
pullbacks of internal categories exists in our chosen $C$.
It follows immediately from definition <ref> that given maps $N\to M$ and $M\to X_0$,
there is a canonical isomorphism
\begin{equation}\label{induced_cat_1}
X[M][N] \simeq X[N].
\end{equation}
with object component the identity map, when these base changes exist.
If we agree to follow the convention that $M \times_N N = M$ is the pullback along
the identity arrow $\id_N$, then $X[X_0] = X$. This also simplifies other results of this
paper, so will be adopted from now on.
One consequence of this assumption is that the iterated fibre product
\[
M\times_M M \times_M \ldots \times_M M,
\]
bracketed in any order, is equal to $M$. We cannot, however, equate two
bracketings of a general iterated fibred product; they are only canonically isomorphic.
Let $Y\to X$ be a functor in $S$ and $j_0\colon U \to X_0$ a map.
If the base change along $j_0$ exists, the following square is a strict pullback
\[
\xymatrix{
Y[Y_0\times_{X_0}U] \ar[r] \ar[d] & X[U] \ar[d]^j \\
Y \ar[r] & X
\]
assuming it exists.
Since base change along $j_0$ exists, we know that we have the functor
$Y[Y_0\times_{X_0}U] \to Y$, we just need to show it is a strict pullback of $j$. On
the level of objects this is clear, and on the level of arrows, we have
\begin{align*}
(Y_0\times_{X_0}U)^2 \times_{Y_0^2}Y_1 &\simeq U^2\times_{X_0^2} Y_1\\
&\simeq (U^2\times_{X_0^2}X_1) \times_{X_1}Y_1 \\
&\simeq X[U]_1\times_{X_1}Y_1
\end{align*}
so the square is a pullback.
We are interested in 2-categories $C$ which admits base change for a
given pretopology $J$ on $S$, which we shall cover in more detail in section <ref>.
Equivalences in $\Cat$—assuming the axiom of choice—are precisely the fully
faithful, essentially surjective functors. For internal categories, however, this is not
the case. In addition, we need to make use of a pretopology to make the `surjective'
part of `essentially surjective' meaningful.
Let $(S,J)$ be a unary site. An internal functor $f:X \to Y$ in $S$ is called
* fully faithful if
\[
\xymatrix{
X_1 \ar[r]^{f_1} \ar[d]_{(s,t)} & Y_1 \ar[d]^{(s,t)}\\
X_0 \times X_0 \ar[r]_{f_0 \times f_0} & Y_0 \times Y_0
\]
is a pullback diagram;
* $J$-locally split if there is a $J$-cover $U\to Y_0$ and a diagram
\[
\xymatrix{
Y[U] \ar[d]_{\bar f} \ar@/^.5pc/[dr]_{\ }="s1"^{u}& \\
X\ar[r]_{f}^(.33){\ }="t1"&Y
\ar@{=>}"s1";"t1"
\]
commuting up to a natural isomorphism;
* a $J$-equivalence if it is fully faithful and $J$-locally split.
The class of $J$-equivalences will be denoted $W_J$. If mention of $J$ is
suppressed, they will be called weak equivalences.
There is another defintion of full faithfulness for internal categories, namely
that of a functor $f\colon Z\to Y$ being representably fully faithful.
This means that for all categories $Z$, the functor
\[
f_\ast\colon \Cat(S)(Z,X) \to \Cat(S)(Z,Y)
\]
is fully faithful. It is a well-known result that these two notions coincide, so we
shall use either characterisation as needed.
If $f:X \to Y$ is a fully faithful functor such that $f_0$ is in $J$, then $f$ is $J$-locally split.
That is, the canonical functor $X[U] \to X$ is a $J$-equivalence whenever the
base change exists. Also, we do not require that $J$ is subcanonical. We record
here a useful lemma.
Given a fully faithful functor $f\colon X \to Y$ in $C$ and a natural
isomorphism $f \Rightarrow g$, the functor $g$ is also fully faithful. In
particular, an internal equivalence is fully faithful.
This is a simple application of the definition of representable full faithfulness
and the fact that the result is true in $\Cat$.
The first definition of weak equivalence of internal categories along the lines we are
considering appeared in [16] for $S$
a regular category, and $J$ the class of regular epimorphisms (i.e. $\can$), in the
context of stacks and indexed categories. This was later generalised in
[28] to more general finitely complete sites to discuss
model structures on the category of internal categories. Both work only
with saturated singleton pretopologies.
Note that when $S$ is finitely complete, the object
$X_1^{iso} \into X_1$ of isomorphisms of a category $X$ can be constructed as a finite limit
[16], and in the case when $X$ is a groupoid we have $X_1^{iso} \simeq X_1$.
[16, 28]
For a finitely complete unary site $(S,J)$ with $J$ saturated, a functor $f$ is called
essentially $J$-surjective if the arrow labelled $\circledast$ below is in $J$.
\[
\xymatrix{
&\ar[dl] X_0 \times_{Y_0} Y_1^{iso} \ar@/^1pc/[ddr]^\circledast \ar[d]&\\
X_0 \ar[d]_{f_0} & \ar[dl]^s Y_1^{iso} \ar[dr]_t &\\
Y_0 && Y_0
\]
A functor is called a Bunge-Paré $J$-equivalence if it is
fully faithful and essentially $J$-surjective. Denote the class of such
maps by $W_J^{BP}$.
Definition <ref> is equivalent to the one in [16, 28]
in the sites they consider but seems more appropriate for sites without all finite
limits. Also, definition <ref> makes sense in 2-categories other than $\Cat(S)$
or sub-2-categories thereof.
Let $(S,J)$ be a finitely complete unary site with $J$ saturated.
Then a functor is a $J$-equivalence if and only if it is a Bunge-Paré
Let $f\colon X \to Y$ be a Bunge-Paré $J$-equivalence, and consider the
$J$-cover given by the map $U := X_0 \times_{Y_0} Y_1^{iso} \to Y_0$. Denote by
$\iota\colon U\to Y_1^{iso}$ the projection on the second factor, by $-\iota$ the composite
of $\iota$ with the inversion map $(-)^{-1}$ and by $s_0\colon U\to X_0$
the projection on the first factor.
The arrow $s_0$ will be the object component of a functor $s\colon Y[U] \to X$,
we need to define the arrow component $s_1$. Consider the composite
\begin{align*}
Y[U]_1 \simeq U\times_{Y_0} Y_1 \times_{Y_0} U \xrightarrow{(s,\iota)\times\id\times(-\iota,s)}
(X_0 \times_{Y_0} Y_1^{iso}) \times_{Y_0} Y_1 \times_{Y_0} ( Y_1^{iso} \times_{Y_0} X_0) \\
\hookrightarrow X_0 \times_{Y_0} Y_3 \times_{Y_0} X_0 \xrightarrow{\id\times m\times\id}
X_0 \times_{Y_0} Y_1 \times_{Y_0} X_0 \simeq X_1
\end{align*}
where the last isomorphism arises from $f$ being fully faithful. It is clear that this
commutes with source and target, because these are given by projection on the first and last
factor at each step. To see that it respects identities and composition, one can use generalised
elements and the fact that the $\iota$ component will cancel with the
$-\iota = (-)^{-1}\circ \iota$ component.
We define the natural isomorphism $f\circ s \Rightarrow j$ (here $j\colon Y[U] \to Y$
is the canonical functor) to have component $\iota$ as denoted above.
Notice that the composite $f_1\circ s_1$ is just
\[
Y[U]_1 \simeq U \times_{Y_0} Y_1 \times_{Y_0} U \xrightarrow{\iota\times\id
\times -\iota} Y_1^{iso} \times_{Y_0} Y_1 \times_{Y_0} Y_1^{iso} \hookrightarrow
Y_3 \xrightarrow{m} Y_1.
\]
Since the arrow component of $Y[U] \to Y$ is $U \times_{Y_0} Y_1 \times_{Y_0} U
\xrightarrow{\pr_2} Y_1$, $\iota$ is indeed a natural isomorphism using the diagram
(<ref>). Thus a Bunge-Paré $J$-equivalence is a $J$-equivalence.
In the other direction, given a $J$-equivalence $f\colon X\to Y$, we have a
$J$-cover $j\colon U\to Y_0$ and a map $(\overline{f},a)\colon U \to X_0 \times Y_1^{iso}$
such that $j = (t\circ pr_2)\circ(\overline{f},a)$. Since $J$ is saturated, $(t\circ pr_2)\in J$
and hence $f$ is a Buge-Paré $J$-equivalence.
We can thus use definition <ref> as we like, and it will still
refer to the same sorts of weak equivalences that appear in the literature.
§ ANAFUNCTORS
We now let $J$ be a subcanonical singleton pretopology on the ambient
category $S$. In this section we assume that $C\into \Cat(S)$ admits base change along
arrows in the given pretopology $J$. This is a slight generalisation of what is
considered in [6], where only $C = \Cat(S)$ is considered.
[41, 6]
An anafunctor in $(S,J)$ from a category $X$ to a category $Y$ consists of a
$J$-cover $(U \to X_0)$ and an internal functor
\[
f\colon X[U] \to Y.
\]
Since $X[U]$ is an object of $C$, an anafunctor is a span in $C$, and
can be denoted
\[
(U,f)\colon X \gento Y.
\]
For an internal functor $f\colon X \to Y$ in $S$, define the anafunctor $(X_0,f)\colon
X \gento Y$ as the following span
\[
X \xleftarrow{=} X[X_0] \xrightarrow{f} Y.
\]
We will blur the distinction between these two descriptions. If $f=id\colon X \to X$,
then $(X_0,id)$ will be denoted simply by $id_X$.
If $U \to A$ is a cover in $(S,J)$ and $\mathbf{B}G$ is a groupoid with one object in
$S$ (i.e. a group in $S$), an anafunctor $(U,g)\colon\disc(A) \gento \mathbf{B}G$ is the
same thing as a Čech cocycle.
[41, 6]
Let $(S,J)$ be a site and let
\[
(U,f),(V,g)\colon X \gento Y
\]
be anafunctors in $S$. A transformation
\[
\alpha\colon (U,f) \Rightarrow (V,g)
\]
from $(U,f)$ to $(V,g)$ is a natural transformation
\[
\xymatrix{
& \ar[dl] X[U\times_{X_0}V] \ar[dr] & \\
X[U] \ar[dr]_f & \stackrel{\alpha}{\Rightarrow} & X[V] \ar[dl]^g\\
& Y &
\]
If $\alpha$ is a natural isomorphism, then $\alpha$ will be called an
isotransformation. In that case we say $(U,f)$ is
isomorphic to $(V,g)$. Clearly all transformations between anafunctors between
internal groupoids are isotransformations.
Given functors $f,g\colon X \to Y$ between categories in $S$, and a natural
transformation $a\colon f \Rightarrow g$, there is a transformation $a\colon (X_0,f)
\Rightarrow (X_0,g)$ of anafunctors, given by the component $X_0\times_{X_0}X_0
= X_0 \xrightarrow{a} Y_1$.
If $(U,g),(V,h)\colon \disc(A) \gento \mathbf{B}G$ are two Čech cocycles, a
transformation between them is a coboundary on the cover $U\times_A V\to A$.
Let $(U,f)\colon X \gento Y$ be an anafunctor in $S$. There is an isotransformation
$1_{(U,f)}\colon (U,f) \Rightarrow (U,f)$ called the identity transformation,
given by the natural transformation with component
\begin{equation}\label{id_transf_component}
U \times_{X_0} U \simeq (U \times U) \times_{X_0^2} X_0 \xrightarrow{id_U^2
\times e} X[U]_1 \xrightarrow{f_1} Y_1
\end{equation}
[41]
Given anafunctors $(U,f)\colon X\to Y$ and $(V,f\circ k)\colon X \to Y$ where $k
\colon V\to U$ is a cover (over $X_0$), a renaming transformation
\[
(U,f)\Rightarrow(V,f\circ k)
\]
is an isotransformation with component
\[
1_{(U,f)}\circ (k\times \id):V\times_{X_0} U \to U\times_{X_0} U \to Y_1.
\]
(We also call its inverse for vertical composition a renaming transformation.)
If $k$ is an isomorphism, then it will itself be referred to as a
renaming isomorphism.
We define (following [6]) the composition of anafunctors as follows. Let
\[
(U,f)\colon X \gento Y \quad \textrm{and} \quad (V,g)\colon Y \gento Z
\]
be anafunctors in the site $(S,J)$. Their composite $(V,g)\circ(U,f)$ is the composite
span defined in the usual way. It is again a span in $C$:
\[
\xymatrix{
&& \ar[dl] X[U\times_{Y_0}V] \ar[dr]^{f^V} & \\
&\ar[dl]X[U] \ar[dr]_f & & Y[V] \ar[dl] \ar[dr]^g\\
X&& Y &&Z
\]
The square is a pullback by lemma <ref>
(which exists because $V\to
Y_0$ is a cover), and the resulting span is an anafunctor because $V \to Y_0$,
hence $U\times_{Y_0}V\to X_0$, are covers, and using the isomorphism (<ref>). We will
sometimes denote the composite by $(U\times_{Y_0}V,g\circ f^V)$.
Here we are using the fact we have specified pullbacks of covers in $S$. Without this
we would not end up with a bicategory (see theorem <ref>),
but what [41] calls an
anabicategory. This is similar to a bicategory, but composition
and other structural maps are only anafunctors, not functors.
Consider the special case when $V = Y_0$, so that $(Y_0,g)$ is just an ordinary
functor. Then there is a renaming transformation (the identity transformation!)
$(Y_0,g)\circ(U,f) \Rightarrow (U,g\circ f)$, using the equality $U \times_{Y_0} Y_0= U$
(by remark <ref>).
If we let $g=\id_Y$, then we see that $(Y_0,\id_Y)$ is a strict unit on the left for
anafunctor composition. Similarly, considering $(V,g)\circ(Y_0,\id)$,
we see that $(Y_0,\id_Y)$ is a two-sided strict unit for anafunctor composition. In
fact, we have also proved
Given two functors $f\colon X\to Y$, $g\colon Y \to Z$ in $S$, their composition as
anafunctors is equal to their composition as functors:
\[
(Y_0,g)\circ(X_0,f) = (X_0,g\circ f).
\]
As a concrete and relevant example of a renaming transformation we can consider
the triple composition of anafunctors
\begin{align*}
(U,f)\colon & X \gento Y,\\
(V,g)\colon & Y \gento Z,\\
(W,h)\colon & Z \gento A.
\end{align*}
The two possibilities of composing these are
\[
\left((U\times_{Y_0} V)\times_{Z_0}W,h\circ(gf^V)^W\right)\quad \text{and}\quad \left(U
\times_{Y_0} (V\times_{Z_0} W),h\circ g^W\circ f^{V\times_{Z_0}W}\right).
\]
The unique isomorphism $(U\times_{Y_0} V)\times_{Z_0}W \simeq U\times_{Y_0} (V
\times_{Z_0} W)$ commuting with the various projections is a
renaming isomorphism. The isotransformation arising from this renaming
transformation is called the associator.
A simple but useful criterion for describing isotransformations where one of the
anafunctors involved is a functor is as follows.
An anafunctor $(V,g)\colon X \gento Y$ is isomorphic to a functor $(X_0,f)\colon X \gento Y$ if
and only if there is a natural isomorphism
\[
\xymatrix{
& \ar[dl] X[V] \ar[dr]^g \\
X \ar@/_1.5pc/[rr]_(.6){f}& \stackrel{\sim}{\Rightarrow} & Y
\]
Just as there is a vertical composition of natural transformations between internal functors,
there is a vertical composition of transformations between internal anafunctors [6].
This is where the subcanonicity of $J$ will be used in order to construct a map
locally over some cover. Consider the following diagram
\[
\xymatrix{
&& \ar[dl] X[U\times_{X_0} V\times_{X_0} W] \ar[dr]\\
& \ar[dl] X[U\times_{X_0} V] \ar[dr] &
& \ar[dl] X[V\times_{X_0} W] \ar[dr]\\
X[U] \ar[drr]_f & \stackrel{a}{\Rightarrow} & X[V] \ar[d]^g& \stackrel{b}
{\Rightarrow} & X[W] \ar[dll]^h \\
\]
We can form a natural transformation between the leftmost and the
rightmost composites as functors in $S$. This will have as its component the arrow
\[
\widetilde{ba}\colon U\times_{X_0} V\times_{X_0} W \xrightarrow{\id\times \Delta
\times \id} U\times_{X_0}V\times_{X_0}V\times_{X_0} W \xrightarrow{a\times b}
Y_1\times_{Y_0} Y_1 \xrightarrow{m} Y_1
\]
in $S$. Notice that the Čech groupoid of the cover
\begin{equation}\label{iterated_cover}
U\times_{X_0} V\times_{X_0} W \to U \times_{X_0} W
\end{equation}
\[
U\times_{X_0} V\times_{X_0} V\times_{X_0} W \st U\times_{X_0} V\times_{X_0} W,
\]
with source and target arising from the two projections $V\times_{X_0} V \to V$.
Denote this pair of parallel arrows
by $s,t\colon UV^2W \st UVW$ for brevity. In [6], section 2.2.3, we find the
commuting diagram
\begin{equation}\label{tobys_diag}
\xymatrix{
UV^2W \ar[r]^t \ar[d]_s & UVW \ar[d]^{\widetilde{ba}}\\
UVW \ar[r]_{\widetilde{ba}} & Y_1
\end{equation}
(this can be checked by using generalised elements) and so we have a functor
\[
\check{C}(U\times_{X_0} V\times_{X_0} W) \to \disc(Y_1).
\]
Our pretopology $J$ is assumed to
be subcanonical, so example <ref> gives us a unique arrow
$ba\colon U\times_{X_0} W \to Y_1$, which is the data for the composite of $a$ and $b$.
In the special case that $U\times_{X_0} V\times_{X_0} W \to U \times_{X_0} W$ is
split (e.g. is an isomorphism), the composite transformation has
\[
U \times_{X_0} W\to U\times_{X_0} V\times_{X_0} W \xrightarrow{\widetilde{ba}} Y_1
\]
as its component arrow. In particular, this is the case if one of $a$ or $b$ is a
renaming transformation.
Let $(U,f):X\gento Y$ be an anafunctor and $U'' \xrightarrow{j'} U' \xrightarrow{j} U$
successive refinements of $U \to X_0$ (e.g isomorphisms). Let $(U',f_{U'})$ and
$(U'',f_{U''})$ denote the composites of $f$ with $X[U'] \to X[U]$ and $X[U''] \to X[U]$
respectively. The arrow
\[
U \times_{X_0} U'' \xrightarrow{\id_U\times j\circ j'} U \times_{X_0} U \to Y_1
\]
is the component for the composition of the isotransformations $(U,f)
\Rightarrow(U',f_{U'}),\Rightarrow(U'',f_{U''})$ described in example
<ref>. Thus we can see that the composite of renaming
transformations associated to isomorphisms $\phi_1,\phi_2$ is simply the renaming
transformation associated to their composite $\phi_1\circ \phi_2$.
This can be used to show that the associator satisfies the necessary coherence conditions.
If $a\colon f\Rightarrow g,\ b\colon g\Rightarrow h$ are natural transformations
between functors $f,g,h\colon X\to Y$ in $S$, their composite as transformations
between anafunctors
\[
(X_0,f),(X_0,g),(X_0,h)\colon X\gento Y.
\]
is just their composite as natural transformations. This uses the equality
\[
X_0\times_{X_0} X_0\times_{X_0} X_0= X_0\times_{X_0} X_0 = X_0,
\]
which is due to our choice in remark <ref> of canonical
Even though we don't have pseudoinverses for weak equivalences of internal
categories, one might guess that the local splitting guaranteed to exist by definition
is actually more than just a splitting of sorts. This is in fact the case, if we use anafunctors.
Let $f\colon X \to Y$ be a $J$-equivalence in $S$. There is an anafunctor
\[
(U,\bar{f})\colon Y \gento X
\]
and isotransformations
\begin{align*}
\iota\colon (X_0,f)\circ (U,\bar{f}) & \Rightarrow id_Y\\
\epsilon\colon(U,\bar{f})\circ (X_0,f) & \Rightarrow id_X
\end{align*}
We have the anafunctor $(U,\bar{f})$ by definition as $f$ is $J$-locally split.
Since the anafunctors $\id_X,\ \id_Y$ are actually
functors, we can use lemma <ref>. Using the special case of
anafunctor composition when the second is a functor, this tells us that $\iota$ will be
given by a natural isomorphism
\[
\xymatrix{
& X \ar[dr]^{f}_(0.2){\ }="s" & \\
Y[U] \ar[rr]^{\ }="t" \ar[ur]^{\bar{f}} && Y
\ar@{=>}"s";"t"
\]
with component $\iota\colon U \to Y_1$. Notice that the composite $f_1\circ \bar{f}_1$ is just
\[
Y[U]_1 \simeq U \times_{Y_0} Y_1 \times_{Y_0} U \xrightarrow{\iota\times\id
\times -\iota} Y_1 \times_{Y_0} Y_1 \times_{Y_0} Y_1 \hookrightarrow
Y_3 \xrightarrow{m} Y_1.
\]
Since the arrow component of $Y[U] \to Y$ is $U \times_{Y_0} Y_1 \times_{Y_0} U
\xrightarrow{\pr_2} Y_1$, $\iota$ is indeed a natural isomorphism using the diagram
The other isotransformation $\epsilon$ is between $(X_0\times_{Y_0} U,\bar{f}\circ \pr_2)$ and
$(X_0,\id_X)$, and is given by the component
\[
\epsilon\colon X_0 \times_{X_0} X_0\times_{Y_0} U = X_0\times_{Y_0} U
\xrightarrow{\id\times (\bar{f}_0,\iota)} X_0\times_{Y_0} (X_0\times_{Y_0} Y_1) \simeq X_0^2
\times_{Y_0^2} Y_1 \simeq X_1
\]
The diagram
\[
\xymatrix{
(X_0\times_{Y_0^2} U)^2 \times_{X_0^2} X_1 \ar[d]_\simeq \ar[rr]^{\pr_2}
& &X_1 \ar[dd]^\simeq\\
U \times_{Y_0} X_1 \times_{Y_0}U \ar[d]_{-\iota\times f\times\iota} & \\
(X_0 \times_{Y_0} Y_1) \times_{Y_0} Y_1 \times_{Y_0} (Y_1 \times_{Y_0} X_0)
\ar[rr]_(.6){\id\times m \times \id}
&& X_0\times_{Y_0} Y_1 \times_{Y_0} X_0
\]
commutes (a fact which can be checked using generalised elements), and using (<ref>)
we see that $\epsilon$ is natural.
The first half of the following theorem is proposition 12 in [6], and the
second half follows because all the constructions of categories involved in dealing
with anafunctors outlined above are still objects of $C$.
[6]
For a site $(S,J)$ where $J$ is a subcanonical singleton pretopology, internal
categories, anafunctors and transformations form a bicategory $\Cat_\ana(S,J)$. If
we restrict attention to a full sub-2-category $C$ which admits base change for
arrows in $J$, we have an analogous full sub-bicategory $C_\ana(J)$.
In fact the bicategory $C_{ana}(J)$ fails to be a strict 2-category
only in the sense that the associator
is given by the non-identity isotransformation from lemma <ref>.
All the other structure is strict.
There is a strict 2-functor $C_\ana(J) \to \Cat_\ana(S,J)$ which is an inclusion on
objects and fully faithful in the strictest sense, namely
being the identity functor on hom-categories. The following is the main
result of this section, and allows us to relate anafunctors to the localisations
considered in the next section.
There is a strict, identity-on-objects 2-functor
\[
\alpha_J\colon C \to C_\ana(J)
\]
sending $J$-equivalences to equivalences, and commuting with the respective
inclusions into $\Cat(S)$ and $\Cat_\ana(S,J)$.
We define $\alpha_J$ to be the identity on objects, and as described in examples
<ref>, <ref> on 1-arrows and 2-arrows (i.e. functors
and transformations). We need first to show that this gives a functor $C(X,Y)
\to C_\ana(J)(X,Y)$. This is precisely the content of example
<ref>. Since the identity 1-cell on a category $X$ in
$C_\ana(J)$ is the image of the identity functor on $S$ in $C$, $\alpha_J$
respects identity 1-cells. Also, lemma <ref> tells us that $
\alpha_J$ respects composition. That $\alpha_J$ sends $J$-equivalences to
equivalences is the content of lemma <ref>.
The 2-category $C$ is locally small (i.e. enriched in small categories)
if $S$ itself is locally small (i.e. enriched in sets),
but a priori the collection of anafunctors $X\gento Y$ do not constitute
a set for $S$ a large category.
Let $(S,J)$ be a locally small, subcanonical unary site satisfying
WISC and let $C$ admit base change along arrows in $J$. Then
$C_\ana(J)$ is locally essentially small.
Given an object $A$ of $S$, let $I(A)$ be a weakly initial set for $J/A$. Consider the locally full
sub-2-category of $C_\ana(J)$ with the same objects, and arrows those
anafunctors $(U,f):X \gento Y$ such that $U \to X_0$ is in $I(X_0)$. Every
anafunctor is then isomorphic, by example
<ref>, to one in this sub-2-category. The collection of anafunctors $(U,f):X \gento Y$
for a fixed $U$ forms a set, by local smallness of $C$, and similarly the collection
of transformations between a pair of anafunctors forms a set by local smallness of $S$.
Examples of locally small sites $(S,J)$ where $C_\ana(J)$ is not known to be locally
essentially small are the category of sets from the model of ZF used in [13],
the model of ZF constructed in [54] and the topos from
proposition <ref>. We note that local essential
smallness of $C_\ana(J)$ seems to be a condition just slightly weaker
than WISC.
§ LOCALISING BICATEGORIES AT A CLASS OF 1-CELLS
Ultimately we are interesting in inverting all $J$-equivalences in $C$ and so
need to discuss what it means to add the formal pseudoinverses to a class of 1-cells
in a 2-category – a process known as localisation. This was done in
[53] for the more general case of a class of 1-cells in a bicategory, where
the resulting bicategory is constructed and its universal properties examined.
The application in loc. cit. is to show the
equivalence of various bicategories of stacks to localisations of 2-categories of
smooth, topological and algebraic groupoids. The results of this article can be seen
as one-half of a generalisation of these results to more general sites.
[53]
Let $B$ be a bicategory and $W \subset B_1$ a class of 1-cells. A localisation
of $B$ with respect to $W$ is a bicategory $B[W^{-1}]$ and a weak 2-functor
\[
U \colon B \to B[W^{-1}]
\]
such that $U$ sends elements of $W$ to equivalences, and is universal with this
property i.e. precomposition with $U$ gives an equivalence of bicategories
\[
U^* \colon Hom(B[W^{-1}],D) \to Hom_W(B,D),
\]
where $Hom_W$ denotes the sub-bicategory of weak 2-functors that send elements
of $W$ to equivalences (call these $W$-inverting, abusing notation slightly).
The universal property means that $W$-inverting weak 2-functors $F\colon B \to D$
factor, up to an equivalence, through $B[W^{-1}]$, inducing an essentially unique
weak 2-functor $\widetilde{F}\colon B[W^{-1}] \to D$.
[53]
Let $B$ be a bicategory with a class $W$ of 1-cells. $W$ is said to admit
a right calculus of fractions if it satisfies the following conditions
$W$ contains all equivalences
a) $W$ is closed under composition
b) If $a\in W$ and there is an isomorphism $a \stackrel{\sim}{\Rightarrow} b$
then $b\in W$
For all $w\colon A' \to A,\ f\colon C \to A$ with $w\in W$ there exists a
2-commutative square
\[
\xymatrix{
P \ar[dd]^v \ar[rr]^g && A'\ar[dd]^w_{\ }="s" \\
\\
C \ar[rr]^{f}="t" & & A
\ar@{=>}_{\simeq} "s"; "t"
\]
with $v\in W$.
If $\alpha\colon w \circ f \Rightarrow w \circ g$ is a 2-arrow and $w\in W$
there is a 1-cell $v \in W$ and a 2-arrow $\beta\colon f\circ v \Rightarrow g \circ v$ such
that $\alpha\circ v = w \circ \beta$. Moreover: when $\alpha$ is an isomorphism, we
require $\beta$ to be an isomorphism too; when $v'$ and $\beta'$ form another such
pair, there exist 1-cells $u,\,u'$ such that $v\circ u$ and $v'\circ u'$ are in $W$, and
an isomorphism $\epsilon\colon v\circ u \Rightarrow v' \circ u'$ such that the following
diagram commutes:
\begin{equation}\label{2cf4.diag}
\xymatrix{
f \circ v \circ u \ar@{=>}[rr]^{\beta\circ u}
\ar@{=>}[dd]_{f\circ \epsilon}^\simeq &&
g\circ v \circ u \ar@{=>}[dd]^{g\circ \epsilon}_\simeq \\
\\
f\circ v' \circ u' \ar@{=>}[rr]_{\beta'\circ u'} && g\circ v' \circ u'
\end{equation}
For a bicategory $B$ with a calculus of right fractions, [53]
constructs a localisation of $B$ as a bicategory of fractions; the 1-arrows are
spans and the 2-arrows are equivalence classes of bicategorical spans-of-spans diagrams.
From now on we shall refer to a calculus of right fractions as simply a calculus
of fractions, and the resulting localisation constructed by Pronk as a
bicategory of fractions. Since $B[W^{-1}]$ is defined only up to equivalence, it
is of great interest to know when a bicategory $D$, in which elements of $W$ are
sent to equivalences by a 2-functor $B \to D$, is equivalent to
$B[W^{-1}]$. In particular, one might be interested in finding such an
equivalent bicategory with a simpler description than that which appears in
[53]
A weak 2-functor $F:B \to D$ which sends elements of $W$ to equivalences induces
an equivalence of bicategories
\[
\widetilde{F} \colon B[W^{-1}] \xrightarrow{\sim} D
\]
if the following conditions hold
$F$ is essentially surjective,
For every 1-cell $f \in D_1$ there are 1-cells $w \in W$ and $g\in B_1$
such that
$Fg \stackrel{\sim}{\Rightarrow} f \circ Fw$,
$F$ is locally fully faithful.
Thanks are due to Matthieu Dupont for pointing out (in personal communication)
that proposition <ref> actually only holds in the one direction, not
in both, as claimed in loc. cit.
The following is useful in showing a weak 2-functor sends weak equivalences to
equivalences, because this condition only needs to be checked on a class that is in
some sense cofinal in the weak equivalences.
Let $V \subset W$ be two classes of 1-cells in a bicategory $B$ such that for all
$w\in W$, there exists $v\in V$ and $s\in W$ and an invertible 2-cell
\[
\xymatrix{
&& a \ar[dd]^w \\
& & \\
b \ar[rr]_v^{\ }="t1" \ar[uurr]^s_{\ }="s1" && c\; .
\ar@{=>}"s1";"t1"^{\simeq}
\]
Then a weak 2-functor $F\colon B \to D$ that sends elements of $V$ to equivalences
also sends elements of $W$ to equivalences.
In the following the coherence arrows will be present, but unlabelled. It is
enough to prove that if in a bicategory $D$ with a class of maps $M$ (in our case
$M=F(W)$) such that for all $w\in M$ there is an equivalence $v$ and an
isomorphism $\alpha$,
\[
\xymatrix{
&& a \ar[dd]^w \\
& & \\
b \ar[rr]_v^{\ }="t1" \ar[uurr]^s_{\ }="s1" && c
\ar@{=>}"s1";"t1"^{\simeq}_\alpha
\]
where $s\in M$, then all elements of $M$ are also equivalences.
Let $\bar v$ be a pseudoinverse
for $v$ and let $j = s \circ \bar v$. Then there is sequence of isomorphisms
\[
w\circ j \Rightarrow (w\circ s)\circ \bar v \Rightarrow v \circ \bar v
\Rightarrow I.
\]
Since $s\in M$, there is an equivalence $u$, $t\in M$ and an isomorphism
$\beta$ giving the following diagram
\[
\xymatrix{
d \ar[dd]_{t} \ar[rr]^{u}_{\ }="s2" && a \ar[dd]^w \\
& & \\
b \ar[rr]_v^{\ }="t1" \ar[uurr]^s="t2"_{\ }="s1" && c \; .
\ar@{=>}"s1";"t1"^\alpha
\ar@{=>}"s2";"t2"_\beta
\]
Let $\bar u$ be a pseudoinverse of $u$. We know from the first part of the proof
that we have a pseudosection $k = t\circ \bar u$ of $s$, with an isomorphism
$s \circ k \Rightarrow I$. We then have the following sequence of isomorphisms:
\[
j\circ w
= (s\circ \bar v) \circ w
\Rightarrow ((s\circ \bar v) \circ w) \circ (s \circ k)
\Rightarrow s \circ ((\bar v \circ v) \circ (t\circ \bar u))
\Rightarrow (s\circ t) \circ u
\Rightarrow \bar u \circ u
\Rightarrow I.
\]
Thus all elements of $M$ are equivalences.
§ 2-CATEGORIES OF INTERNAL CATEGORIES ADMIT BICATEGORIES OF FRACTIONS
In this section we prove the result that $C\into \Cat(S)$ admits a calculus of
fractions for the $J$-equivalences, where $J$ is a
singleton pretopology on $S$.
The following is the first main theorem of the paper, and subsumes a number of other,
similar theorems throughout the literature (see section <ref> for details).
Let $S$ be a category with a singleton pretopology $J$. Assume the full sub-2-category
$C \into \Cat(S)$ admits base change along maps in $J$. Then $C$ admits a
right calculus of fractions for the class $W_J$ of $J$-equivalences.
We show the conditions of definition <ref> hold.
2CF1. An internal equivalence is clearly $J$-locally split. Lemma
<ref> gives us the rest.
a) That the composition of fully faithful functors is again fully faithful is trivial.
Consider the composition $g\circ f$ of two $J$-locally split functors,
\[
\xymatrix{
Y[U] \ar[d] \ar@/^.5pc/[dr]_{\ }="s1"^{u}&Z[V] \ar[d]\ar@/^.5pc/[dr]_(.5){\ }="s2"^{v}& \\
X\ar[r]_{f}^(.33){\ }="t1"&Y \ar[r]_{g}^(.33){\ }="t2" & Z
\ar@{=>}"s1";"t1"
\ar@{=>}"s2";"t2"
\]
By lemma <ref> the functor $u$ pulls back to a functor
$Z[U\times_{Y_0}V] \to Z[V]$. The composite $Z[U\times_{Y_0}V] \to Z$
is fully faithful with object component in $J$, hence $g\circ f$ is $J$-locally split.
b) Lemma <ref> tells us that fully faithful functors
are closed under isomorphism, so we just need to show $J$-locally split functors are closed
under isomorphism.
Let $w,f\colon X\to Y$ be functors and $a\colon w \Rightarrow f$ be a natural
isomorphism. First, let $w$ be $J$-locally split. It is immediate from the diagram
\[
\xymatrix{
Y[U] \ar[dd] \ar@/^.7pc/[ddrr]_{\ }="s1"^{u} \\
\\
X\ar@/^1pc/[rr]^{w}="t1"_{\ }="s2" \ar@/_1pc/[rr]_{f}^{\ }="t2"
\ar@{=>}"s1";"t1"
\ar@{=>}"s2";"t2"^{a}
\]
that $f$ is also $J$-locally split.
2CF3. Let $w\colon X\to Y$ be a $J$-equivalence, and let $f\colon Z\to Y$ be a functor. From the definition of $J$-locally split, we have the diagram
\[
\xymatrix{
Y[U] \ar[d] \ar@/^.5pc/[dr]_{\ }="s1"^{u}& \\
X\ar[r]_{w}^(.33){\ }="t1"&Y
\ar@{=>}"s1";"t1"
\]
We can use lemma <ref> to pull $u$ back along $f$ to get
a 2-commuting diagram
\[
\xymatrix{
& Z[U\times_{Y_0} Z_0] \ar[dr]^{v} \ar[dl] \\
Y[U] \ar[d] \ar@/^.5pc/[dr]_{\ }="s1"^{u}& &Z \ar[dl]^f\\
X\ar[r]_{w}^(.33){\ }="t1"&Y
\ar@{=>}"s1";"t1"
\]
with $v\in W_J$ as required.
Since $J$-equivalences are representably fully faithful, given
\[
\xymatrix{
&Y \ar[dr]^w \\
X \ar[ur]^f \ar[dr]_g & \Downarrow a & Z\\
& Y \ar[ur]_w
\]
where $w\in W_J$, there is a unique $a'\colon f
\Rightarrow g$ such that
\[
\raisebox{36pt}{
\xymatrix{
&Y \ar[dr]^w \\
X \ar[ur]^f \ar[dr]_g & \Downarrow a & Z\\
& Y \ar[ur]_w
\equals
\raisebox{36pt}{
\xymatrix{
X \ar@/^1.5pc/[rr]^f \ar@/_1.5pc/[rr]_g&\Downarrow a'& Y \ar[r]^w & Z
\,.
\]
The existence of $a'$ is the first half of 2CF4, where $v=\id_X$. Note that if $a$ is
an isomorphism, so if $a'$, since $w$ is representably fully faithful.
Given $v'\colon W\to X \in W_J$ such that
there is a transformation
\[
\xymatrix{
&X \ar[dr]^f \\
W \ar[ur]^{v'} \ar[dr]_{v'} & \Downarrow b & Y\\
& X \ar[ur]_g
\]
\begin{align}\label{antiwhisker_eqn}
\raisebox{36pt}{
\xymatrix{
&X \ar[dr]^f \\
W \ar[ur]^{v'} \ar[dr]_{v'} & \Downarrow b & Y \ar[r]^w & Z\\
& X \ar[ur]_g
\equals &
\raisebox{36pt}{
\xymatrix{
&&Y \ar[dr]^w \\
W \ar[r]^{v'} &X \ar[ur]^f \ar[dr]_g & \Downarrow a & Z\\
&& Y \ar[ur]_w
} \nonumber \\
\equals &
\raisebox{36pt}{
\xymatrix{
W\ar[r]^{v'}&X \ar@/^1.5pc/[rr]^f \ar@/_1.5pc/[rr]_g &\Downarrow a'&
Y \ar[r]^w & Z
}\, ,
\end{align}
then uniqueness of $a'$, together with equation (<ref>) gives us
\[
\raisebox{36pt}{
\xymatrix{
&X \ar[dr]^f \\
W \ar[ur]^{v'} \ar[dr]_{v'} & \Downarrow b & Y \\
& X \ar[ur]_g
\equals
\raisebox{36pt}{
\xymatrix{
W\ar[r]^{v'}&X \ar@/^1.5pc/[rr]^f \ar@/_1.5pc/[rr]_g
&\Downarrow a'
& Y
}\, .
\]
This is precisely the diagram (<ref>) with $v=\id_X$, $u=v'$, $u'=\id_W$ and
$\epsilon$ the identity 2-arrow.
Hence 2CF4 holds.
The proof of theorem <ref> is written using only the language of
2-categories, so can be generalised from $C$ to other 2-categories. This approach
will be taken up in [56].
The second main result of the paper is that we want to know when this
bicategory of fractions is equivalent to a bicategory of anafunctors, as the latter
bicategory has a much simpler construction.
Let $(S,J)$ be a subcanonical unary site and let the full sub-2-category
$C\into \Cat(S)$ admit base change along arrows in $J$.
Then there is an equivalence of bicategories
\[
C_\ana(J) \simeq C[W_J^{-1}]
\]
under $C$.
Let us show the conditions in proposition <ref> hold. To begin with, the 2-functor
$\alpha_J\colon C \to C_{ana}(J)$ sends $J$-equivalences to equivalences by
proposition <ref>.
EF1. $\alpha_J$ is the identity on 0-cells, and hence surjective on objects.
EF2. This is equivalent to showing that for any anafunctor $(U,f)\colon X\gento Y$
there are functors $w,g$ such that $w$ is in $W_J$ and
\[
(U,f) \stackrel{\sim}{\Rightarrow} \alpha_J(g)\circ\alpha_J(w)^{-1}
\]
where $\alpha_J(w)^{-1}$ is some pseudoinverse for $\alpha_J(w)$.
Let $w$ be the functor $X[U] \to X$ and let $g=f\colon X[U] \to Y$. First, note that
\[
\xymatrix{
& \ar[dl] X[U] \ar[dr]^= &\\
X && X[U]
\]
is a pseudoinverse for
\[
\alpha_J(w)
\equals
\left(\raisebox{24pt}{
\xymatrix{
& \ar[dl]_{=} X[U][U] \ar[dr] &\\
X[U] && X
\]
Then the composition $ \alpha_J(f)\circ\alpha_J(w)^{-1}$ is
\[
\xymatrix{
& \ar[dl] X[U\times_U U \times_U U]\ar[dr]\\
X && Y\; ,
\]
which is just $(U,f)$ (recall we have the equality $U\times_U U \times_U U = U$
by remark <ref>).
EF3. If $a\colon(X_0,f)\Rightarrow(X_0,g)$ is a transformation of anafunctors for
functors $f,g\colon X\to Y$, it is given by a natural transformation
\[
f \Rightarrow g\colon X = X[X_0 \times_{X_0} X_0] \to Y.
\]
Hence we get a unique natural
transformation $a\colon f\Rightarrow g$ such that $a$ is the image of $a'$ under
We now give a series of results following from this theorem, using basic properties of
pretopologies from section <ref>.
When $J$ and $K$ are two subcanonical singleton pretopologies on $S$ such that
$J_{un}=K_{un}$, for example $J$ cofinal in $K$, there is an equivalence of bicategories
\[
C_\ana(J) \simeq C_\ana(K).
\]
The class of maps in $\Top$ of the form $\coprod U_i \to X$ for an
open cover $\{U_i\}$ of $X$ form a singleton pretopology. This is because $\mathcal{O}$
is a superextensive pretopology (see the appendix).
Given a site with a superextensive pretopology $J$,
we have the following result which is useful when $J$ is not a singleton
pretopology (the singleton pretopology $\amalg J$ is defined analogously
to the case of $\Top$, details are in the appendix).
Let $(S,J)$ be a superextensive site where $J$ is a subcanonical pretopology. Then
\[
C[W_{J_{un}}^{-1}] \simeq C_\ana(\amalg J).
\]
This essentially follows by lemma <ref>.
Obviously this can be combined with previous results, for example if $K$ is cofinal in $\amalg J$,
for $J$ a non-singleton pretopology, $K$-anafunctors localise $C$ at the
class of $J_{un}$-equivalences.
Finally, given WISC we have a bound on the size of the hom-categories, up to equivalence.
Let $(S,J)$ be a subcanonical unary site satisfying WISC with $S$ locally small
and let $C\into \Cat(S)$ admit base change along arrows in $J$.
Then any localisation $C[W_J^{-1}]$ is locally essentially small.
Recall that this localisation can be chosen such that the class of objects is the same as the class
of objects of $C$, and so it is not necessary to consider additional set-theoretic mechanisms for dealing with
large (2-)categories here.
We note that the issue of size of localisations is not touched on in [53]. even
though such issues are commonly addressed in localisation of 1-categories. If we have
a specified bound on the hom-sets of $S$ and also know that some WISC${}_\kappa$ holds,
then we can put specific bounds on the size of the hom-categories of the localisation.
This is important if examining fine size requirements or implications for localisation
theorems such as these, for example higher versions of locally presentable categories.
§ EXAMPLES
The simplest example is when we take the trivial singleton pretopology $triv$, where
covering families are just single isomorphisms: $triv$-equivalences are
internal equivalences and, up to equivalence, localisation at $W_{triv}$ does nothing. It is
worth pointing out that if we localise at $W_{triv_{un}}$, which is equivalent to
considering anafunctors with source leg having a split epimorphism for its
object component, then by corollary <ref> this is equivalent
to localising at $W_{triv}$, so $C_{ana}(triv_{un}) \simeq C_{ana}(triv)\simeq C$.
The first non-trivial case is that of a regular category with
the canonical singleton pretopology $\can$. This is the setting of [16].
Recall that $W_J^{BP}$ is the class of Bunge-Paré $J$-equivalences (definition
<ref>). For now, let $C$ denote either $\Cat(S)$ or $\Gpd(S)$.
Let $(S,J)$ be a finitely complete unary site with $J$ saturated. Then we have
\[
C[(W_J^{BP})^{-1}] \simeq C[W_J^{-1}]
\]
This is merely a restatement of the fact Bunge-Paré $J$-equivalences
and ordinary $J$-equivalences coincide in this case.
The canonical singleton pretopology $\can$ on a finitely complete category $S$
is saturated. Hence $W_{\can}^{BP} = W_{\can}$ for this site, and
\[
C[(W_{\can}^{BP})^{-1}] \simeq C[W_{\can}^{-1}]\simeq C_\ana(\can)
\]
We can combine this corollary with corollary <ref> so that the
localisation of either $\Cat(S)$ or $\Gpd(S)$ at the Bunge-Paré weak equivalences
can be calculated using $J$-anafunctors for $J$ cofinal in $\can$. We note that $\can$
does not satisfy WISC in general (see proposition <ref> and the comments
following), so the localisation might not be locally essentially small.
The previous corollaries deal with the case when we are interested in the 2-categories
consisting of all of the internal categories or groupoids in a site. However, for many applications
of internal categories/groupoids it is not sufficient to take all of $\Cat(S)$ or $\Gpd(S)$.
One widely used example is that of Lie groupoids, which are groupoids internal to the
category of (finite-dimensional) smooth manifolds such that source and target maps
are submersions (more on these below). Other examples are used in the theory
of algebraic stacks, namely
groupoids internal to schemes or algebraic spaces. Other types of such presentable
stacks use groupoids internal to some site with specified conditions on the source and target
maps. Although it is not covered explicitly in the literature, it is possible to consider presentable
stacks of categories, and this will be taken up in future work [55].
We thus need to furnish examples of sub-2-categories $C$, specified by restricting
the sort of maps that are allowed for source and target, that admit base change
along some class of arrows. The following lemma gives a sufficiency condition for this
to be so.
Let $\Cat^\mathcal{M}(S)$ be defined as the full sub-2-category of $\Cat(S)$ with
objects those categories such that the source and target maps belong to a singleton pretopology
$\mathcal{M}$. Then $\Cat^\mathcal{M}(S)$ admits base change along arrows in $\mathcal{M}$,
as does the corresponding 2-category $\Gpd^\mathcal{M}(S)$ of groupoids.
Let $X$ be an object of $\Cat^\mathcal{M}(S)$ and $f\colon M\to X_0 \in \mathcal{M}$. In the
following diagram, all the squares are pullbacks and all arrows are in $\mathcal{M}$.
\[
\SelectTips {cm}{}%
\xymatrix{
X[M]_1 \ar[d] \ar[r] \ar @/_2.4pc/ [dd]_{s'} \ar @/^1pc/[rr]^{t'} & X_1\times_{X_0} M \ar[r] \ar[d] & M \ar[d] \\
M\times_{X_0} X_1 \ar[d] \ar[r] & X_1 \ar[r] \ar[d] & X_0 \\
M \ar[r] & X_0
\]
The maps marked $s',t'$ are the source and target maps for the base change along
$f$, so $X[M]$ is in $\Cat^\mathcal{M}(S)$. The same argument holds for groupoids verbatim.
In practice one often only wants base change along a subclass of $\mathcal{M}$,
such as the class of open covers sitting inside the class of open maps in $\Top$.
We can then apply theoerems <ref> and <ref>
to the 2-categories $\Cat^\mathcal{M}(S)$
and $\Gpd^\mathcal{M}(S)$ with the classes of $\mathcal{M}$-equivalences, and indeed
to sub-2-categories of these, as we shall in the examples below.
We shall focus of a few concrete cases to show how the results of this paper subsume
similar results in the literature proved for specific sites.
The category of smooth manifolds is not finitely complete so the localisation results in this
section so far do not apply to it. There are two ways around this. The first is to expand the category
of manifolds to a category of smooth spaces which is finitely complete
(or even cartesian closed). In that
case all the results one has for finitely complete sites can be applied. The other
is to take careful note of which finite limits are actually needed, and show that all constructions
work in the original category of manifolds. There is then a hybrid approach, which is to
work in the expanded category, but point out which results/constructions actually fall inside
the original category of manifolds. Here we shall take the second approach. First, let us pin down
some definitions.
Let $\Diff$ be the category of smooth, finite-dimensional manifolds. A Lie category is
a category internal to $\Diff$ where the source and target maps are submersions (and hence
the required pullbacks exist). A Lie groupoid is a Lie category which is a groupoid.
A proper Lie groupoid is one where the map $(s,t)\colon X_1 \to X_0 \times X_0$ is
proper. An étale Lie groupoid is one where the source and target maps are local
By lemma <ref> the 2-categories of Lie categories,
Lie groupoids and proper Lie groupoids admit base change along any of the following
classes of maps: open covers ($\amalg\mathcal{O}$), surjective local diffeomorphisms ($\acute{e}t$),
surjective submersions ($Subm$).
The 2-categories of étale Lie groupoids and proper étale Lie groupoids admit base change
along arrows in $\acute{e}t$ and $Subm$. We should note that
we have $\amalg\mathcal{O}$ cofinal in $\acute{e}t$, which is cofinal in
We can thus apply the main results of this paper to the sites $(\Diff,\mathcal{O})$,
$(\Diff,\amalg\mathcal{O})$, $(\Diff,\acute{e}t)$
and $(\Diff,Subm)$ and the 2-categories of Lie categories, Lie groupoids, proper Lie goupoids
and so on. However, the definition of weak equivalence we have here, involving $J$-locally
split functors, is not one that apppears in the Lie groupoid literature, which is actually Bunge-Paré
$Subm$-equivalence. However, we have the following result:
A functor $f\colon X\to Y$ between Lie categories is a $Subm$-equivalence if and only if
it is a Bunge-Paré
Before we prove this, we need a lemma proved by Ehresmann.
[26]
For any Lie category $X$, the subset of invertible arrows, $X_1^{iso} \into X_1$ is an open
Hence there is a Lie groupoid $X^{iso}$ and an identity-on-objects functor $X^{iso} \to X$
which is universal for functors from Lie groupoids. In particular, a natural isomorphism between
functors with codomain $X$ is given by a component map that factors through $X_1^{iso}$, and
the induced source and target maps $X_1^{iso} \to X_0$ are submersions.
(proposition <ref>)
Full faithfulness is the same for both definitions, so we just need to show that $f$ is $Subm$-locally
split if and only if it is essentially $Subm$-surjective. We first show the forward implication.
The special case of a $\amalg\mathcal{O}$-equivalence between Lie groupoids is a small
generalisation of the proof of proposition 5.5 in [46], which states than
an internal equivalence of Lie groupoids is a Bunge-Paré $Subm$-equivalence. Since $\amalg\mathcal{O}$ is
cofinal in $Subm$, a $Subm$-equivalence is a $\amalg\mathcal{O}$-equivalence, hence a
Bunge-Paré $Subm$-equivalence.
For the case when $X$ and $Y$ are Lie categories, we use the fact that we can define
$X_0\times_{Y_0}Y_1^{iso}$ and that the local sections constructed in Moerdijk-Mrčun's proof
factor through this manifold to set up the proof as in the groupoid case.
For the reverse implication, the construction in the first half of the proof of proposition
<ref> goes through verbatim, as all the pullbacks used
involve submersions.
The need to localise the category of Lie groupoids at $W_{Subm}$
was perhaps first noted in [52],
where it was noted that something other than the standard construction of a category of fractions
was needed. However Pradines lacked the necessary 2-categorical localisation results.
Pronk considered the sub-2-category of étale Lie groupoids, also localised at $W_{Subm}$, in order
to relate these groupoids to differentiable étendues [53]. Lerman discusses the 2-category
of orbifolds qua stacks [37] and argues that it should be a localisation
of the 2-category of proper étale
Lie groupoids (again at $W_{Subm}$). These three cases use different constructions of the 2-categorical
localisation: Pradines used what he called meromorphisms, which are equivalence classes of
butterfly-like diagrams and are related to Hilsum-Skandalis morphisms, Pronk introduces the techniques
outlined in this paper, and Lerman uses Hilsum-Skandalis morphisms, also known as right principal
Interestingly, [19] considers this localisation of the 2-category of Lie groupoids then
considers a further localisation, not given by the results of this paper.[In fact this is the only
2-categorical localisation result involving internal categories or groupoids known to the author to
not be covered by theorem <ref> or its sequel [56].]
Colman in essence shows that the full
sub-2-category of topologically discrete groupoids, i.e. ordinary small groupoids, is a localisation at
those internal functors which induce an equivalence on fundamental groupoids.
Our next example is that of topological groupoids, which correspond to various flavours of stacks on
the category $\Top$. The idea of weak equivalences of topological groupoids predates the
case of Lie groupoids, and [52] credits it to Haefliger, van Est,
and [30]. In particular the first two were ultimately interested in
defining the fundamental group of a foliation, that is to say, of the topological groupoid
associated to a foliation, considered up to weak eqivalence.
However more recent examples have focussed on topological stacks, or variants thereon.
In particular, in parallel with the algebraic and differentiable cases, the topological stacks
for which there is a good theory correspond to those topological groupoids with conditions
on their source and target maps. Aside from étale topological groupoids (which were considered
by [53] in relation to étendues), the real advances here have come from
work of Noohi, starting with [49], who axiomatised the
concept of local fibration and asked that the source and target maps of topological
groupoids are local fibrations.
A singleton pretopology $LF$ in $\Top$ is called a class of local fibrations if the following
conditions hold:[We have packaged the conditions in a way slightly different to
[49], but the definition is in fact identical.]
* $LF$ contains the open embeddings
* $LF$ is stable under coproducts, in the sense that $\coprod_{i\in I} X_i \to Y$ is in $LF$
if each $X_i\to Y$ is in $LF$
* $LF$ is local on the target for the open cover pretopology.
That is, if the pullback of a map $f\colon X\to Y$ along an open cover of $Y$ is in $LF$, then $f$
is in $LF$.
Conditions 1. and 2. tell us that $\amalg\mathcal{O} \subset LF$, and that $LF$ is $\amalg J$
for some superextensive pretopology $J$ containing the open embeddings as singleton
`covering' families (beware the misleading terminology here: covering families are not
assumed to be jointly surjective). Note that $LF$ will not be subcanonical,
by condition 1. As an example, given any of the following pretopologies $K$:
* Serre fibrations,
* Hurewicz fibrations,
* open maps,
* split maps,
* projections out of a cartesian product,
* isomorphisms;
one can define a class of local fibrations by choosing those maps which are in $K$ on pulling
back to an open cover of the codomain. Such maps are then called local $K$.
As an example of the usefulness of this concept, the topological stacks corresponding to
topological groupoids with local Hurewicz fibrations as source and target have a nicely behaved
homotopy theory. The case of étale groupoids corresponds to the last named class of maps,
which give us local isomorphisms, i.e. étale maps.
We can then apply lemma <ref> and theorem <ref>
to the 2-category $\Grp^{LF}(\Top)$ to localise at the class $W_{\amalg \mathcal{O}}$ (as
$\amalg \mathcal{O} \subset LF$), or any other singleton pretopology contained in $LF$,
using anafunctors whenever this pretopology is subcanonical. Note that if $C$ satisfies WISC,
so will the corresponding $LF$, although this is probably not necessary to consider in the presence
of full AC.
A slightly different approach is taken in [18], where the author introduces a new
pretopology on the category $CGH$ of compactly generated Hausdorff spaces. We give a definition
equivalent to the one in loc cit.
A (not necessarily open) cover $\{V_i\into X\}_{i\in I}$ is called a $\mathcal{CG}$-cover if for any map
$K\to X$ from a compact space $K$, there is a finite open cover $\{U_j \into K\}$ which refines
the cover $\{V_i\times_X K\to K\}_{i\in I}$. $\mathcal{CG}$-covers form a pretopology $\mathcal{CG}$
on $CGH$.
Compactly generated stacks then correspond to groupoids in $CGH$ such that source and
target maps are in the pretopology $\mathcal{CG}_{un}$. Again, we can localise
$\Gpd^{\mathcal{CG}}(CGH)$ at $W_{\mathcal{CG}_{un}}$ using lemma
<ref> and theorem <ref>, and anafunctors
can be again pressed into service.
We now arrive at the more involved case of algebraic stacks (cf. the
continually growing [59]
for the extent of the theory of algebraic stacks), which were the first presentable
stacks to be defined. There are some subtleties about the site of definition for algebraic stacks,
and powerful representability theorems, but we can restrict to three main cases: groupoids
in the category of affine schemes $\Aff = \Ring^{op}$; groupoids in the category $\Sch$ of schemes; and
groupoids in the category $\AlgSp$ of algebraic spaces. Algebraic spaces reduce to
algebraic stacks on $\Sch$ represented by groupoids with trivial automorphism groups, and the category of
schemes is a subcategory of $Sh(\Aff)$, so we shall just consider the case when our ambient category
is $\Aff$. In any case, all the special properties of classes of maps in all three sites are ultimately
defined in terms of properties of ring homomorphisms.
Note that groupoids in $\Aff$ are exactly the same thing as cogroupoid objects in $\Ring$, which are more
commonly known as Hopf algebroids.
Despite the possibly unfamiliar language used by algebraic geometry, algebraic stacks reduce
to the following semiformal definition. We fix three singleton pretopologies on our site $\Aff$: $J$, $E$
and $D$ such that $E$ and $D$ are local on the target for the pretopology $J$. An algebraic
stack then is a stack on $\Aff$ for the pretopology $J$ which `corresponds' to a groupoid $X$ in $\Aff$
such that source and target maps belong to $E$
and $(s,t)\colon X_1 \to X_0^2$ belongs to $D$. We recover the algebraic stacks by localising the 2-category
of such groupoids at $W_E$ (this claim of course needs substantiating, something we will not do here
for reasons of space, referring rather to [53, 57] and the forthcoming [55]).
In practice, $D$ can be something like closed maps (to recover Hausdorff-like conditions) or all maps,
and $E$ consists of either smooth or étale maps, corresponding to Artin and Deligne-Mumford stacks
respectively. $J$ is then something like the étale topology (or rather, the singleton pretopology associated
to it, as the étale topology is superextensive), and we can apply lemma <ref>
to see that base change exists along $J$, along with the fact that asking for $(s,t) \in D$ is automatically
stable under forming the base change. In practice, a variety of combinations of $J,E$ and $D$ are used,
as well as passing from $\Aff$ to $\Sch$ and $\AlgSp$, so there are various compatibilities to check in order
to know one can apply theorem <ref>.
A final application we shall consider is when our ambient category consists of algebraic objects. As
mentioned in section 2, a number of authors have considered localising groupoids in
Mal'tsev, or Barr-exact, or protomodular, or semi-abelian categories, which are hallmarks of categories
of algebraic objects rather than spatial ones, as we have been considering so far.
In the case of groupoids in $\Grp$ (which, as in any Mal'tsev category, coincide with the internal categories)
it is a well-known result that they can be described using crossed modules.
A crossed module (in $\Grp$) is a homomorphism $t\colon G\to H$ together with a homomorphism
$\alpha\colon H\to \Aut(G)$ such that $t$ is $H$-equivariant (using the conjugation action of $H$ on itself),
and such that the composition $\alpha\circ t\colon G\to\Aut(G)$ is the action of $G$ on itself by conjugation.
A crossed module is often denoted, when no confusion will arise, by $(G\to H)$.
A morphism $(G \to H) \to (K\to L)$ of crossed modules is a pair of maps $G\to K$ and $H\to L$
making the obvious square commute, and commuting with all the action maps.
Similar definitions hold for groups internal to cartesian closed categories, and even just finite-product
categories if one replaces $H\to \Aut(G)$ with its transpose $H\times G\to G$. Ultimately of course there
is a definition for crossed modules in semiabelian categories (e.g. [1]), but we shall consider just groups.
There is a natural definition of 2-arrow between maps of crossed modules, but the specifics are not
important for the present purposes, so we refer to <cit.> for details.
The 2-categories of groupoids internal to $\Grp$ and crossed modules are equivalent, so we shall just
work with the terminology of the latter.
Given the result that crossed modules correspond to pointed, connected homotopy 2-types,
it is natural to ask if all maps of such arise from maps between crossed modules. The answer is,
perhaps unsurprisingly, no, as one needs maps which only weakly preserve the group structure.
One can either write down the definition of some generalised form of map (<cit.>),
or localise the 2-category of crossed modules ([51] considers a model structure on the
category of crossed modules). To localise the 2-category of crossed modules we can consider
the singleton pretopology $epi$ on $\Grp$ consisting of the epimorphisms, and localise $\Gpd(\Grp)$ at $W_{epi}$.
There are potentially interesting sub-2-categories of crossed modules that one might want to consider,
for example, the one corresponding to nilpotent pointed connected 2-types. These are crossed
modules $t\colon G \to H$ where the cokernel of $t$ is a nilpotent group and the (canonical) action of $\coker t$
on $\ker t$ is nilpotent. The correspondence between such crossed modules and the corresponding internal
groupoids is a nice exercise, as well as seeing that this 2-category admits base change for the pretopology $epi$.
§ SUPEREXTENSIVE SITES
The usual sites of topological spaces, manifolds and schemes all share a common property:
one can (generally) take coproducts of covering families and end up with a cover.
In this appendix we gather some results that generalise this fact, none of which are especially
deep, but help provide examples of bicategories of anafunctors. Another reference for superextensive
sites is [58].
[17]
A finitary (resp. infinitary) extensive category is a category
with finite (resp. small) coproducts such that the following condition holds: let $I$ be a
a finite set (resp. any set), then, given a collection of commuting diagrams
\[
\xymatrix{
X_i \ar[r] \ar[d] &Z \ar[d] \\
A_i \ar[r] & \coprod_{i\in I} A_i\;,
\]
one for each $i\in I$, the squares are all pullbacks if and only if the collection $\{X_i
\to Z\}_{i\in I}$ forms a coproduct diagram.
In such a category there is a strict initial object: given a map $A \to 0$ with $0$
initial, we have $A \simeq 0$.
$\Top$ is infinitary extensive. $\Ring^{op}$, the category of affine schemes,
is finitary extensive.
In $\Top$ we can take an open cover $\{U_i\}_I$ of a space $X$ and replace it with
the single map $\coprod_I U_i \to X$, and work just as before using this new sort of
cover, using the fact $\Top$ is extensive. The sort of sites that mimic this behaviour
are called superextensive.
A superextensive site is an extensive category $S$ equipped with a
pretopology $J$ containing the families
\[
(U_i \to \coprod_I U_i)_{i\in I}
\]
and such that all covering families are bounded; this means that for a finitely
extensive site, the families are finite, and for an infinitary site, the families are small.
The pretopology in this instance will also be called superextensive.
Given an extensive category $S$, the extensive pretopology has as covering
families the bounded collections $(U_i \to \coprod_I U_i)_{i\in I}$. The pretopology on
any superextensive site contains the extensive pretopology.
The category $\Top$ with its usual pretopology of open covers is a superextensive
An elementary topos with the coherent pretopology is finitary superextensive, and a Grothendieck
topos with the canonical pretopology is infinitary superextensive.
Given a superextensive site $(S,J)$, one can form the class $\amalg J$ of arrows of the
form $\coprod_I U_i \to A$ for covering families $\{U_i \to A\}_{i\in I}$ in $J$ (more precisely,
all arrows isomorphic in $S/A$ to such arrows).
The class $\amalg J$ is a singleton pretopology, and is subcanonical if and only if $J$
Since isomorphisms are covers for $J$ they are covers for $\amalg J$. The pullback
of a $\amalg J$-cover $\coprod_I U_i \to A$ along $B \to A$ is a $\amalg J$-cover as
coproducts and pullbacks commute by definition of an extensive category. Now for
the third condition we use the fact that in an extensive category a map
\[
f\colon B \to \coprod_I A_i
\]
implies that $B\simeq \coprod_I B_i$ and $f=\coprod_i f_i$. Given $\amalg J$-covers
$\coprod_I U_i \to A$ and $\coprod_J V_j \to (\coprod_I U_i)$, we see that
$\coprod_J V_j \simeq \coprod_I W_i$ for some objects $W_i$. By the previous point, the pullback
\[
\coprod_I U_k \times_{\coprod_I U_{i'}} W_i
\]
is a $\amalg J$-cover of $U_i$, and hence
$(U_k \times_{\coprod_I U_{i'}} W_i \to U_k)_{i\in I}$
is a $J$-covering family for each $k\in I$. Thus
\[
(U_k \times_{\coprod_I U_{i'}} W_i \to A)_{i,k\in I}
\]
is a $J$-covering family, and so
\[
\coprod_J V_j \simeq \coprod_{k\in I} \left( \coprod_{i\in I} U_k \times_{\coprod_I U_{i'}}
W_i\right) \to A
\]
is a $\amalg J$-cover.
The map $\coprod_I U_i \to A$ is the coequaliser of $\coprod_{I\times I} U_i \times_A
U_j \st \coprod_I U_i$ if and only if $A$ is the colimit of the diagram in definition
<ref>. Hence $(\coprod_I U_i \to A)$ is effective if and only if $
(U_i \to A)_{i\in I}$ is effective
Notice that the original superextensive pretopology $J$ is generated by the union of
$\amalg J$ and the extensive pretopology.
One reason we are interested in superextensive sites is the following.
In a superextensive site $(S,J)$, we have $J_{un} = (\amalg J)_{un}$.
This means we can replace the singleton pretopology $J_{un}$ (e.g. local-section-admitting
maps of topological spaces) with the singleton pretopology $\amalg J$ (e.g. disjoint unions of open covers)
when defining anafunctors. This makes for much smaller pretopologies in practice.
One class of extensive categories which are of particular interest is those that also
have finite/small limits. These are called lextensive. For example, $\Top$ is
infinitary lextensive, as is a Grothendieck topos. In contrast, an elementary topos is
in general only finitary lextensive. We end with a lemma about WISC.
If $(S,J)$ is a superextensive site, $(S,J)$ satisfies WISC if and only if $(S,\amalg J)$
One reason for why superextensive sites are so useful is the following result from
Let $(S,J)$ be a superextensive site, and $F$ a stack for the extensive topology on $S$.
Then the associated stack $\widetilde{F}$ on the site $(S,\amalg J)$ is also the associated
stack for the site $(S,J)$.
As a corollary, since every weak 2-functor $F\colon S\to \Gpd$ for extensive $S$
represented by an internal groupoid is automatically a stack for the extensive topology,
we see that we only need to stackify $F$ with respect to a singleton pretopology on $S$.
This will be applied in [55].
[1]
O. Abbad, S. Mantovani, G. Metere, and E.M. Vitale, Butterflies are
fractions of weak equivalences, preprint (2010). Available from
[2]
P. Aczel, The type theoretic interpretation of constructive set
theory, Logic Colloquium '77, Stud. Logic Foundations Math., vol. 96,
North-Holland, 1978, pp 55–66.
[3]
E. Aldrovandi, B. Noohi, Butterflies I: Morphisms of 2-group stacks,
Adv. Math., 221, issue 3 (2009), pp 687–773, [arXiv:0808.3627].
[4]
E. Aldrovandi, B. Noohi, Butterflies II: Torsors for 2-group stacks,
Adv. Math., 225, issue 2 (2010), pp 922–976, [arXiv:0909.3350].
[6]
T. Bartels, Higher gauge theory I: 2-Bundles, Ph.D. thesis, University
of California Riverside, 2006, [arXiv:math.CT/0410328].
[7]
J. Baez and A. Lauda, Higher dimensional algebra V: 2-groups, Theory
and Application of Categories 12 (2004), no. 14, pp 423–491.
[8]
J. Baez and M. Makkai, Correspondence on the category theory mailing
list, January 1997, available at <http://www.mta.ca/ cat-dist/catlist/1999/anafunctors>.
[9]
J. Bénabou, Les distributeurs, rapport 33,
Université Catholique de Louvain, Institut de Mathématique Pure et
Appliquée, 1973.
[10]
J. Bénabou, Théories relatives à un corpus, C. R. Acad. Sci. Paris
Sér. A-B 281 (1975), no. 20, Ai, A831–A834.
[11]
J. Bénabou, Some remarks on 2-categorical algebra, Bulletin de la
Société Mathématique de Belgique 41 (1989), pp 127–194.
[12]
J. Bénabou, Anafunctors versus distributors, email to Michael Shulman,
posted on the category theory mailing list 22 January 2011, available from
[13]
B. van den Berg, Predicative toposes, preprint (2012), [arXiv:1207.0959].
[15]
M. Breckes, Abelian metamorphosis of anafunctors in butterflies,
preprint (2009).
[16]
M. Bunge and R. Paré, Stacks and equivalence of indexed categories,
Cah. Topol. Géom. Différ. 20 (1979), no. 4,
pp 373–399.
[17]
A. Carboni, S. Lack, and R. F. C. Walters, Introduction to
extensive and distributive categories, J. Pure Appl. Algebra 84
(1993), pp 145–158.
[18]
D. Carchedi, Compactly generated stacks: a cartesian-closed theory of
topological stacks, Adv. Math. 229 (2012), no. 6, pp 3339–33397, [arXiv:0907.3925]
[19]
H. Colman, On the homotopy 1-type of Lie groupoids, Appl. Categ.
Structures 19, Issue 1 (2010), pp 393–423, [arXiv:math/0612257].
[20]
H. Colman and C. Costoya, A Quillen model structure for orbifolds,
preprint (2009). Available from <http://faculty.ccc.edu/hcolman/>.
[22]
W. G. Dwyer and D. M. Kan, Simplicial localizations of categories, J. Pure
Appl. Algebra 17 (1980), no. 3, pp 267–284.
[25]
A. Dold, Partitions of unity in the theory of fibrations, Ann. Math.
78 (1963), no. 2, pp 223–255.
[26]
C. Ehresmann, Catégories topologiques et catégories
différentiables, Colloque Géom. Diff. Globale (Bruxelles, 1958), Centre
Belge Rech. Math., Louvain, 1959, pp 137–150.
[27]
C. Ehresmann, Catégories structurées, Annales de l'Ecole Normale et
Superieure 80 (1963), pp 349–426.
[28]
T. Everaert, R.W. Kieboom, and T. van der Linden, Model structures for
homotopy of internal categories, Theory and Application of Categories 15 (2005), no. 3, pp 66–94.
[29]
P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory,
Springer-Verlag, 1967.
[30]
M. Hilsum and G. Skandalis, Morphismes-orientés d`epsaces de feuilles
et fonctorialitè en théorie de Kasparov (d'après une conjecture
d'A. Connes), Ann. Sci. École Norm. Sup. 20 (1987),
pp 325–390.
[31]
P. Johnstone, Sketches of an elephant, a topos theory compendium, Oxford
Logic Guides, vol. 43 and 44, The Clarendon Press Oxford University Press,
[32]
A. Joyal and M. Tierney, Strong stacks and classifying spaces,
Category theory (Como, 1990), Lecture Notes in Math., vol. 1488, Springer, 1991,
pp 213–236.
[33]
A. Karaglia, Embedding posets into cardinals with $DC_{\kappa}$, preprint
(2012), [arXiv:1212.4396].
[34]
G. M. Kelly, Complete functors in homology. I. Chain maps and endomorphisms
Proc. Cambridge Philos. Soc. 60 (1964), pp 721–735.
[37]
E. Lerman, Orbifolds as stacks?, L'Enseign Math. (2) 56 (2010),
no. 3-4, pp 315–363, [arXiv:0806.4160].
[38]
J. Lurie, Higher Topos Theory, Annals of Mathematics Studies 170,
Princeton University Press, 2009. Available from <http://www.math.harvard.edu/ lurie/>.
[40]
S. MacLane and I. Moerdijk, Sheaves in geometry and logic,
Springer-Verlag, 1992.
[41]
M. Makkai, Avoiding the axiom of choice in general category theory, J. Pure Appl. Algebra
108 (1996), pp 109–173. Available from <http://www.math.mcgill.ca/makkai/>.
[42]
S. Mantovani, G. Metere and E. M. Vitale,
Profunctors in Mal'cev categories and fractions of functors, preprint (2012).
Available from <http://perso.uclouvain.be/enrico.vitale/research.html>.
[46]
I. Moerdijk and J. Mrčun, Introduction to foliations and lie
groupoids, Cambridge studies in advanced mathematics, vol. 91, Cambridge
University Press, 2003.
[48]
J. Mrčun, The Hopf algebroids of functions on étale groupoids and their
principal Morita equivalence,
J. Pure Appl. Algebra 160 (2001), no. 2-3, pp 249–262.
[49]
B. Noohi, Foundations of topological stacks I, preprint (2005),
[50]
B. Noohi, On weak maps between 2-groups, preprint (2005), [arXiv:math/0506313].
[51]
B. Noohi, Notes on 2-groupoids, 2-groups and crossed-modules, preprint (2005)
[52]
J. Pradines, Morphisms between spaces of leaves viewed as fractions,
Cah. Topol. Géom. Différ. Catég. 30 (1989),
no. 3, pp 229–246, [arXiv:0803.4209].
[53]
D. Pronk, Etendues and stacks as bicategories of fractions, Compositio
Math. 102 (1996), no. 3, pp 243–303.
[54]
D. M. Roberts, Con(ZF+ $\neg$WISC), preprint, (2013).
[55]
D. M. Roberts, All presentable stacks are stacks of anafunctors, forthcoming (A).
[56]
D. M. Roberts, Strict 2-sites, $J$-spans and localisations, forthcoming (B).
[57]
D. Schäppi, A characterization of categories of coherent sheaves of certain algebraic stacks,
preprint (2012), [arXiv:1206.2764].
[58]
M. Shulman, Exact completions and small sheaves, Theory and Application of
Categories, 27 (2012), no. 7, pp 97–173.
[59]
The Stacks project authors, Stacks project, <http://stacks.math.columbia.edu>.
[60]
R. Street, Fibrations in bicategories,
Cah. Topol. Géom. Différ. Catég. 21 (1980),
pp 111–160.
[61]
J.-L. Tu, P. Xu and C. Laurent-Gengoux
Twisted K-theory of differentiable stacks,
Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 6, pp 841–910, [arXiv:math/0306138].
[62]
E. M. Vitale, Bipullbacks and calculus of fractions, Cah. Topol.
Géom. Différ. Catég. 51 (2010), no. 2, pp 83–113.
Available from <http://perso.uclouvain.be/enrico.vitale/>.
|
arxiv-papers
| 2011-01-12T13:56:33 |
2024-09-04T02:49:16.375698
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "David M. Roberts",
"submitter": "David Roberts",
"url": "https://arxiv.org/abs/1101.2363"
}
|
1101.2437
|
# New Physics effects on decay $B_{s}\to\gamma\gamma$ in Technicolor Model
Qin XiuMei snowlotusqin@yahoo.com.cn Wujun Huo whuo@ictp.it Xiaofang Yang
Department of Physics Department, Southeast University, Nanjing, Jiangsu
211189, China
###### Abstract
In this paper we calculate the contributions to the branching ratio of
$B_{s}\to\gamma\gamma$ from the charged Pseudo-Goldstone bosons appeared in
one generation Technicolor model. We find that the theoretical values of the
branching ratio, $BR(B_{s}\to\gamma\gamma)$, including the contributions of
PGBs, $P^{\pm}$ and $P^{\pm}_{8}$ , are much different from the $SM$
prediction. The new physics effects can be enhance 2-3 levels to $SM$ result.
It is shown that the decay $B_{s}\to\gamma\gamma$ can give the test the new
physics signals from the technicolor model.
## I introduction
As is well known, the rare radiative decays of $B$ mesons is in particular
sensitive to contributions from those new physics beyond the standard
model(SM). Both inclusive and exclusive processes, such as the decays
$B_{s}\to X\gamma$, $B_{s}\to\gamma\gamma$ and $B\to X_{s}\gamma$ have been
received some attention in the literature[1-14]. In this paper, we will
present our results in Technicolor theories.
The one generation Technicolor model (OGTM)[15-16]is the simplest and most
frequently studied model which contained the parameters are less than SM. Same
as other models, the OGTM has its defects such as the S parameter large and
positive[17]. But we can relax the constraints on the OGTM form the $S$
parameter by introducing three additional parameters $(V,W,X)$[18]. The basic
idea of the OGTM is: we introduce a new set of asymptotically free gauge
interactions and the Technicolor force act on Technifermions. The Technicolor
interaction at $1Tev$ become strong and cause a spontaneous breaking of the
global flavor symmetry $SU(8)_{L}\times SU(8)_{R}\times U(1)_{Y}$. The result
is $8^{2}-1=63$ massless Goldstone bosons. Three of the these objects replace
the Higgs field and induce a mass of $W^{\pm}$ and $Z^{0}$ gauge bosons. And
at the new strong interaction other Goldstone bosons acquire masses. As for
the $B_{s}\to\gamma\gamma$, only the charged color single and color octets
have contributions. The gauge couplings of the PGBs are determined by their
quantum numbers. In Table 1 we listed the relevant couplings[19] needed in our
calculation, where the $V_{ud}$ is the corresponding element of $Kobayashi-
Maskawa$ matrix . The Goldstone boson decay constant $F_{\pi}$[20] should be
$F_{\pi}=v/2=123GeV$, which corresponds to the vacuum expectation of an
elementary Higgs field .
$P^{+}P^{-}\gamma_{\mu}$ | $-ie(p_{+}-p_{-})_{\mu}$
---|---
$P^{+}_{8a}P^{-}_{8b}\gamma_{\mu}$ | $-ie(p_{+}-p_{-})_{\mu}\delta_{ab}$
$P^{+}\;u\;d$ | $i\frac{V_{ud}}{2F_{\pi}}\sqrt{\frac{2}{3}}[M_{u}(1-\gamma_{5})-M_{d}(1+\gamma_{5})]$
$P^{+}_{8a}\;u\;d$ | $i\frac{V_{ud}}{2F_{\pi}}\lambda_{a}[M_{u}(1-\gamma_{5})-M_{d}(1+\gamma_{5})]$
$P^{+}_{8a}P^{-}_{8b}g_{c\mu}$ | $-gf_{abc}(p_{a}-p_{b})_{\mu}$
Table 1: The relevant gauge couplings and Effective Yukawa couplings for the
OGTM.
At the LO in QCD the effective Hamiltonian is
${\cal
H}_{eff}=\frac{-4G_{F}}{\sqrt{2}}V_{tb}V_{ts}^{*}\displaystyle{\sum_{i=1}^{8}}C_{i}(M_{W}^{-})O_{i}(M_{W}^{-}).$
(1)
Where, as usual, $G_{F}$ denotes the Fermi coupling constant and
$V_{tb}V_{ts}^{*}$ indicates the Cabibbo-Kobayashi-Maskawa matrix element.And
the current-current, QCD penguin, electromagnetic and chromomagnetic dipole
operators are of the form
$\displaystyle O_{1}$ $\displaystyle=$
$\displaystyle(\overline{c}_{L\beta}\gamma^{\mu}b_{L\alpha})(\overline{s}_{L\alpha}\gamma_{\mu}c_{L\beta})\;$
(2) $\displaystyle O_{2}$ $\displaystyle=$
$\displaystyle(\overline{c}_{L\alpha}\gamma^{\mu}b_{L\alpha})(\overline{s}_{L\beta}\gamma_{\mu}c_{L\beta})\;$
(3) $\displaystyle O_{3}$ $\displaystyle=$
$\displaystyle(\overline{s}_{L\alpha}\gamma^{\mu}b_{L\alpha})\sum_{q=u,d,s,c,b}(\overline{q}_{L\beta}\gamma_{\mu}q_{L\beta})\;$
(4) $\displaystyle O_{4}$ $\displaystyle=$
$\displaystyle(\overline{s}_{L\alpha}\gamma^{\mu}b_{L\beta})\sum_{q=u,d,s,c,b}(\overline{q}_{L\beta}\gamma_{\mu}q_{L\alpha})\;$
(5) $\displaystyle O_{5}$ $\displaystyle=$
$\displaystyle(\overline{s}_{L\alpha}\gamma^{\mu}b_{L\alpha})\sum_{q=u,d,s,c,b}(\overline{q}_{R\beta}\gamma_{\mu}q_{R\beta})\;$
(6) $\displaystyle O_{6}$ $\displaystyle=$
$\displaystyle(\overline{s}_{L\alpha}\gamma^{\mu}b_{L\beta})\sum_{q=u,d,s,c,b}(\overline{q}_{R\beta}\gamma_{\mu}q_{R\alpha})\;$
(7) $\displaystyle O_{7}$ $\displaystyle=$
$\displaystyle(e/16\pi^{2})m_{b}\overline{s}_{L}\sigma^{\mu\nu}b_{R}F_{\mu\nu}\;$
(8) $\displaystyle O_{8}$ $\displaystyle=$
$\displaystyle(g/16\pi^{2})m_{b}\overline{s}_{L}\sigma^{\mu\nu}T^{a}b_{R}G_{\mu\nu}^{a}\;$
(9)
where $\alpha$ and $\beta$ are color indices, $\alpha=1,...,8$ labels SU(3)c
generators, e and $g$ refer to the electromagnetic and strong coupling
constants, while $F_{\mu\nu}$ and $G^{a}_{\mu\nu}$ denote the QED and QCD
field strength tensors, respectively.
The Feynman diagrams that contribute to the matrix element as the following
Figure 1: Examples of Feynman diagrams that contribute to the matrix element.
Figure 2: The Feynman diagrams that contribute to the Wilson coefficients
C7,C8.
In Fig.2 the shot-dash lines represent the charged PGBs $P^{\pm}$ and
$P^{\pm}_{8}$ of OGTM. We at first integrate out the top quark and the weak
$W$ bosons at $\mu=M_{W}$ scale, generating an effective five-quark theory and
run the effective field theory down to b-quark scale to give the leading log
QCD corrections by using the renormalization group equation. The Wilson
coefficients are process independent and the coefficients $C_{i}(\mu)$ of 8
operators are calculated from the Fig.2.The Wilson coefficients are read[21]
$\displaystyle C_{i}(M_{W})=0,\;\;i=1,3,4,5,6,\;\;\;C_{2}(M_{W})=1,$ (10)
$\displaystyle
C_{7}(M_{W})=-A(\delta)+\frac{B(x)}{3\sqrt{2}G_{F}F_{\pi}^{2}}+\frac{8B(y)}{3\sqrt{2}G_{F}F_{\pi}^{2}}$
(11) $\displaystyle
C_{8}(M_{W})=-C(\delta)+\frac{D(x)}{3\sqrt{2}G_{F}F_{\pi}^{2}}+\frac{8D(y)+E(y)}{3\sqrt{2}G_{F}F_{\pi}^{2}}$
(12)
with $\delta=M_{W}^{2}/m_{t}^{2}$, $x=(m(P^{\pm})/m_{t})^{2}$ and
$y=(m(P^{\pm}_{8})/m_{t})^{2}$.From the $Eq(11),(12)$ , we can see the
situation of the color-octet charged PGBs is more complicate than that of the
color-singlet charged PGBs ,because of the involvement of the color
interactions. where
$\displaystyle A(\delta)$ $\displaystyle=$
$\displaystyle\frac{\frac{1}{3}+\frac{5}{24}\delta-\frac{7}{24}\delta^{2}}{(1-\delta)^{3}}+\frac{\frac{3}{4}\delta-\frac{1}{2}\delta^{2}}{(1-\delta)^{4}}\log[\delta]$
(13) $\displaystyle B(y)$ $\displaystyle=$
$\displaystyle\frac{-\frac{11}{36}+\frac{53}{72}y-\frac{25}{72}y^{2}}{(1-y)^{3}}$
(14) $\displaystyle+$
$\displaystyle\frac{-\frac{1}{4}y+\frac{2}{3}y^{2}-\frac{1}{3}y^{3}}{(1-y)^{4}}\log[y]$
$\displaystyle C(\delta)$ $\displaystyle=$
$\displaystyle\frac{\frac{1}{8}-\frac{5}{8}\delta-\frac{1}{4}\delta^{2}}{(1-\delta)^{3}}-\frac{\frac{3}{4}\delta^{2}}{(1-\delta)^{4}}\log[\delta]$
(15) $\displaystyle D(y)$ $\displaystyle=$
$\displaystyle\frac{-\frac{5}{24}+\frac{19}{24}y-\frac{5}{6}y^{2}}{(1-y)^{3}}$
(16) $\displaystyle+$
$\displaystyle\frac{\frac{1}{4}y^{2}-\frac{1}{2}y^{3}}{(1-y)^{4}}\log[y]$
$\displaystyle E(y)$ $\displaystyle=$
$\displaystyle\frac{\frac{3}{2}-\frac{15}{8}y-\frac{15}{8}y^{2}}{(1-y)^{3}}+\frac{\frac{9}{4}y-\frac{9}{2}y^{2}}{(1-y)^{4}}\log[y]$
(17)
By caculate the graphs of the exchanged $W$ boson in the SM we gained the
function $A$ and $C$;And by caculate the graphs of the exchanged color-singlet
and color-octet charged PGBs in OGTM we gained the function $B$, $D$ and $E$.
when $\delta<1$, $x,y>>1$, the OGTM contribution $B$, $D$ and $E$ have always
a relative minus sign with the SM contribution $A$ and $C$. As a result, the
OGTM contribution always destructively interferes with the SM contribution.
The leading-order results for the Wilson coefficients of all operators
entering the effective Hamiltonian in Eq.(1) can be written in an analytic
form. They are
$\displaystyle C_{7}^{eff}(m_{b})$ $\displaystyle=$
$\displaystyle\eta^{16/23}C_{7}(M_{W})+\frac{8}{3}(\eta^{14/23}-\eta^{16/23})\times$
(18) $\displaystyle
C_{8}(M_{W})+C_{2}(M_{W})\displaystyle\sum_{i=1}^{8}h_{i}\eta^{a_{i}}.$
With $\eta=\alpha_{s}(M_{W})/\alpha_{s}(m_{b})$,
$\displaystyle h_{i}$ $\displaystyle=$
$\displaystyle(\frac{626126}{272277},-\frac{56281}{51730},-\frac{3}{7},-\frac{1}{14},-0.6494,$
(19) $\displaystyle-0.0380,-0.0186,-0.0057).$ $\displaystyle a_{i}$
$\displaystyle=$
$\displaystyle(\frac{14}{23},\frac{16}{23},\frac{6}{23},-\frac{12}{23},$ (20)
$\displaystyle 0.4086,-0.4230,-0.8994,0.1456).$
To calculate $B_{s}\to\gamma\gamma$ , one may follow a perturbative QCD
approach which includes a proof of factorization, showing that soft gluon
effects can be factorized into $B_{s}$ meson wave function; and a systematic
way of resumming large logarithms due to hard gluons with energies between
1Gev and $m_{b}$. In order to calculate the matrix element of Eq(1) for the
$B_{s}\to\gamma\gamma$ , we can work in the weak binding approximation and
assume that both the $b$ and the $s$ quarks are at rest in the $B_{s}$ meson,
and the $b$ quarks carries most of the meson energy, and its four velocity can
be treated as equal to that of $B_{s}$. Hence one may write $b$ quark momentum
as $p_{b}=m_{b}v$ where is the common four velocity of $b$ and $B_{s}$. We
have
$\displaystyle p_{b}\cdot k_{1}$ $\displaystyle=$ $\displaystyle m_{b}v\cdot
k_{1}={1\over 2}m_{b}m_{B_{s}}=p_{b}\cdot k_{2},$ $\displaystyle p_{s}\cdot
k_{1}$ $\displaystyle=$ $\displaystyle(p-k_{1}-k_{2})\cdot k_{1}=$ (21)
$\displaystyle-{1\over 2}m_{B_{s}}(m_{B_{s}}-m_{b})=p_{s}\cdot k_{2},$
We compute the amplitude of $B_{s}\to\gamma\gamma$ using the following
relations
$\displaystyle\left\langle
0|\bar{s}\gamma_{\mu}\gamma_{5}b|B_{s}(P)\right\rangle$ $\displaystyle=$
$\displaystyle-if_{B_{s}}P_{\mu},$ $\displaystyle\left\langle
0|\bar{s}\gamma_{5}b|B_{s}(P)\right\rangle$ $\displaystyle=$ $\displaystyle
if_{B_{s}}M_{B},$ (22)
where $f_{B_{s}}$ is the $B_{s}$ meson decay constant which is about $200$ MeV
.
The total amplitude is now separated into a CP-even and a CP-odd part
$T(B_{s}\to\gamma\gamma)=M^{+}F_{\mu\nu}F^{\mu\nu}+iM^{-}F_{\mu\nu}\tilde{F}^{\mu\nu}.$
(23)
We find that
$\displaystyle M^{+}$ $\displaystyle=$ $\displaystyle{-4{\sqrt{2}}\alpha
G_{F}\over 9\pi}f_{B_{s}}m_{b_{s}}V_{ts}^{*}V_{tb}\times$ (24)
$\displaystyle\left(\frac{m_{b}}{m_{B_{s}}}BK(m_{b}^{2})+{3C_{7}\over
8\bar{\Lambda}}\right).$
with $B=-(3C_{6}+C_{5})/4$, $\bar{\Lambda}=m_{B_{s}}-m_{b}$, and
$\displaystyle M^{-}$ $\displaystyle=$ $\displaystyle{4{\sqrt{2}}\alpha
G_{F}\over 9\pi}f_{B_{s}}m_{b_{s}}V_{ts}^{*}V_{tb}\times$ (25)
$\displaystyle\left(\sum_{q}A_{q}J(m_{q}^{2})+\frac{m_{b}}{m_{B_{s}}}BL(m_{b}^{2})+{3C_{7}\over
8\bar{\Lambda}}\right).$
where
$\displaystyle A_{u}$ $\displaystyle=$
$\displaystyle(C_{3}-C_{5})N_{c}+(C_{4}-C_{6})$ $\displaystyle A_{d}$
$\displaystyle=$ $\displaystyle{1\over
4}\left[(C_{3}-C_{5})N_{c}+(C_{4}-C_{6})\right]$ $\displaystyle A_{c}$
$\displaystyle=$ $\displaystyle(C_{1}+C_{3}-C_{5})N_{c}+(C_{2}+C_{4}-C_{6})$
$\displaystyle A_{s}$ $\displaystyle=$ $\displaystyle{1\over
4}\left[(C_{3}+C_{4}-C_{5})N_{c}+(C_{3}+C_{4}-C_{6})\right]$ (26)
$\displaystyle A_{s}$ $\displaystyle=$ $\displaystyle{1\over
4}\left[(C_{3}+C_{4}-C_{5})N_{c}+(C_{3}+C_{4}-C_{6})\right].$ (27)
The functions $J(m^{2})$, $K(m^{2})$ and $L(m^{2})$ are defined by
$\displaystyle J(m^{2})$ $\displaystyle=$ $\displaystyle I_{11}(m^{2}),$
$\displaystyle K(m^{2})$ $\displaystyle=$ $\displaystyle
4(I_{11}(m^{2})-I_{00}(m^{2})),$ $\displaystyle L(m^{2})$ $\displaystyle=$
$\displaystyle I_{00}(m^{2}),$ (28)
with
$I_{pq}(m^{2})=\int_{0}^{1}{dx}\int_{0}^{1-x}{dy}\frac{x^{p}y^{q}}{m^{2}-2xyk_{1}\cdot
k_{2}-i\varepsilon}$ (29)
The decay width for $B_{s}\to\gamma\gamma$ is simply
$\Gamma(B_{s}\to\gamma\gamma)={m_{B_{s}}^{3}\over
16\pi}({|M^{+}|}^{2}+{|M^{-}|}^{2}).$ (30)
In SM, with $C_{2}=C_{2}(M_{W})=1$ , and the other Wilson coefficients are
zero, we find $\Gamma(B_{s}\to\gamma\gamma)=1.3\times 10^{-10}\ {\rm eV}$
which amounts to a branching ratio $Br(B_{s}\to\gamma\gamma)=3.5\times
10^{-7}$, for the given $\Gamma^{total}_{B_{s}}=4\times 10^{-4}\ {\rm eV}$. In
numerical calculations we use the corresponding input parameters
$M_{W}=80.22\;GeV$, $\alpha_{s}(m_{Z})=0.117$, $m_{c}=1.5\;GeV$,
$m_{b}=4.8\;GeV$ and $|V_{tb}V_{ts}^{*}|^{2}/|V_{cb}|^{2}=0.95$ ,
respectively. The present experimental limit[22] on the decay
$B_{s}\to\gamma\gamma$ is
$\displaystyle{\rm Br}(B_{s}\to\gamma\gamma)\leq 8.6\times 10^{-6},$ (31)
which is far from the theoretical results. So, we can not put the constraint
to the masses of PGBs. The constraints of the masses of $P^{\pm}$ and
$P^{\pm}_{8}$ can be from the decay[24] $B\to s\gamma$ :
$m_{P^{\pm}_{8}}>400$GeV.
Figure 3: the $Br(B_{s}\to\gamma\gamma)$ about the mass of $P_{8}^{\pm}$ under
different values of $m_{P^{\pm}}$. Figure 4: the $Br(B_{s}\to\gamma\gamma)$
about the mass of $P^{\pm}$ under different values of $P_{8}^{\pm}$.
Fig.3(4) denotes the $Br(B_{s}\to\gamma\gamma)$ about the mass of
$P_{8}^{\pm}$ ($P^{\pm}$) under different values of $m_{P^{\pm}}$
($P_{8}^{\pm}$). From Fig.3 and 4, we find the the curves are much different
from the the SM one. It can be enhanced about 1-2 levels to the SM prediction
in the reasonable region of the masses of PGBs. This gives the strong new
physics signals from the Technicolor Model. The branching ratio of
$B_{s}\to\gamma\gamma$ decrease along with the mass of $P_{8}^{\pm}$ and
$P^{\pm}$ reduce. This is from the decoupling theorem that for heavy enough
nonstandard boson. When $m(P^{\pm})$ and $m(P^{\pm}_{8})$ have large values,
the contributions from OGTM is small.From the $Eq(16),(17),(18)$ ,we can see
the functions $B$, $D$ and $E$ go to zero, as $x$, $y\to\infty$.The branching
ratio in the Fig.(3) is changed much faster than that in the Fig.(4).This is
because the contribution to $B_{s}\to\gamma\gamma$ from the color octet
$P_{8}^{\pm}$ is large when compared with the contribution from color singlet
$P^{\pm}$.
As a conclusion, the size of contribution to the rare decay of
$B_{s}\to\gamma\gamma$ from the PGBs strongly depends on the values of the
masses of the charged PGBs. This is quite different from the SM case. By the
comparison of the theoretical prediction with the current data one can derived
out the the contributions of the PGBs: $P^{\pm}$ and $P^{\pm}_{8}$ to
$B_{s}\to\gamma\gamma$ and give the new physics signals of new physics.
## References
* (1)
* (2) A N Mitra. Phys. Lett., 2000, B473: 297-304
* (3) Junjie Cao,Zhenjun Xiao,Gongru Lu. Phys. Rev., 2001, D64: 014012
* (4) Z.J.Xiao, C.D.L, W.J.Huo.Phys. Rev., 2003, D67: 094021
* (5) L. Reina, G. Ricciardi, A. Soni. Phys. Rev., 1997, D56: 5805
* (6) G. Hiller, E.O. Iltan. Phys. Lett., 1997, B409: 425
* (7) C.-H. Chang, G.-L. Lin, Y.-P. Yao. Phys. Lett., 1997, B415: 395-401
* (8) M.R. Ahmady, E. Kou. hep-ph/9708347
* (9) G. Hiller, E.O. Iltan, Mod.Phys.Lett., 1997.A12: 2837-2846
* (10) S. Choudlhury, J. Ellis. hep-ph/9804300
* (11) T.M. Aliev, G. Turan.Phys. Rev., 1993, D48: 1176
* (12) T.M. Aliev, G. Hiller, E.O. Iltan. Nucl.Phys.,1998, B515: 321-341
* (13) T.M. Aliev, E.O. Iltan. Phys. Rev., 1998, D58: 095014
* (14) S. Bertolini, J. Matias. Phys. Rev., 1998, D57: 4197-4204
* (15) P. Singer, D.-X. Zhang.Phys. Rev., 1997, D56: 4274
* (16) S. Dimopoulos. Nucl.Phys.,1980, B168: 69
* (17) E. Farhi, L. Susskind.Phys. Rev., 1979, D20: 3404
* (18) M.E. PeskinT. Takeuchi. Phys.Rev. Lett., 1990, 65: 964
* (19) I. Maksymyk, C.P. Burgess. Phys. Rev., 1994, D50: 529
* (20) Z.J.Xiao, L.D.Wan, J.M.Yang, etal. Phys. Rev., 1994, D49: 5949
* (21) E.Eichten,I. Hinchliffe,K. Lane,etal. Phys. Rev., 1986, D34: 1547
* (22) C.D. Lu ,Z.J. Xiao. Phys. Rev., 1996, D53: 2529
* (23) J.Wicht, I. Adachi, H. Aihara, etal. Phys.Rev. Lett., 2008, 100:121801
|
arxiv-papers
| 2011-01-12T20:18:57 |
2024-09-04T02:49:16.393518
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Qin XiuMei, Wujun Huo, Xiaofang Yang",
"submitter": "Wujun Huo Dr",
"url": "https://arxiv.org/abs/1101.2437"
}
|
1101.2471
|
# CATEGORY OF FUZZY HYPER BCK-ALGEBRAS
J.DONGHO ****Department of Mathematics, University of Yaounde, BP 812,
Cameroon josephdongho@yahoo.fr
###### Abstract.
In this paper we first define the category of fuzzy hyper BCK-algebras. After
that we show that the category of hyper BCK-algebras has equalizers,
coequalizers, products. It is a consequence that this category is complete and
hence has pullbacks.
## 1\. Introduction
The study of hyperstructure was initiated in 1934 by F. Marty at 8th congress
of Scandinavian Mathematiciens. Y.B. Jun et al. applied the hyperstructures to
BCK-algebras, and introduces the notion of hyper BCK-algebra. Now we follow
[1,2,3,4] and introduce the category of fuzzy hyperBCK-algebra and obtain some
result, as mentioned in the abstarct.
## 2\. Preliminaries
We now review some basic definitions that are very useful in the paper.
###### Definition 1.
$[\ref{3}]$ Let $H$ be an non empty set.
A hyperoperation $*$ on $H$ is a mapping of $H\times H$ family of non-empty
subsets of $H$ $\mathcal{P}^{*}(H)$
###### Definition 2.
Let $*$ be an hyperoperation on $H$ and $O$ a constant element of $H$ An
hyperorder on $H$ is subset $<$ of
$\mathcal{P}^{*}(H)\times\mathcal{P}^{*}(H)$ define by:
for all $x,y\in H,x<y$ iff $O\in x*y$ and for every $A,B\subseteq H,A<B$ iff
$\forall a\in A,\exists b\in B$ such that $a<b.$
###### Definition 3.
If $*$ is hyperoperation on $H$.
For all $A,B\subseteq H,A*B:=\underset{a\in A,b\in B}{\bigcup}a*b$
###### Definition 4.
$[1]$ By hyper BCK-algebra we mean a non empty set $H$ endowed with a hyper-
operation $*$ and a constant $O$ satisfying the following axioms.
* (HK1)
$(x*z)*(y*z)<(x*y)$
* (HK2)
$(x*y)*z=(x*z)*y$
* (HK3)
$x*H<\\{x\\}$
###### Definition 5.
A fuzzy hyper BCK-algebra is a pair $(\mathbf{H};\mu_{H})$ where
$\mathbf{H}=(H;*;O)$ is hyper BCK-algebra and $\mu_{H}:H\longrightarrow[0,1]$
is a map satisfy the following property:
$\inf(\mu_{H}(x*y))\geq\min(\mu_{H}(x),\mu_{H}(y))$
for all $x,y\in H.$
###### Example 1.
$[\ref{5}]$ Let $n\in\mathbb{N}^{*}.$ Define the hyperoperation $*$ on
$H=[n,+\infty)$ as follows:
$x*y=\left\\{\begin{array}[]{lllc}[n,x]&\texttt{iff}\quad x<y\\\
(n,y]&\texttt{iff}\quad x>y\neq n\\\ \\{x\\}&\texttt{iff}\quad
y=n\end{array}\right.$
for all $x,y\in H$. To show that $(H,*,n)$ is hyper BCK-algebra, it suffice to
show axiom $HK3.$ For all $x\in H,x*H=\underset{t\in H}{\bigcup}x*t.$ For all
$x\in H$ then $x*x\subseteq x*H.$ And then $n\in[n,x]*\\{x\\}$
## 3\. The category of fuzzyhyper BCK-algebras
###### Lemma 1.
Let $(\mathbf{H};\mu_{H})$ be a fuzzy hyper BCK-algebra.
For all $x\in H,\mu_{H}(O)\geq\mu_{H}(x)$
Proof. For all $x\in H,x<x;$ then $O\in x*x$.
$\begin{array}[]{ccll}O\in
x*x&\texttt{imply}&\mu_{H}(O)\geq\inf(\mu_{H}(x*y))\geq\min(\mu_{H}(x),\mu_{H}(y))\\\
&\texttt{i.e}&\mu_{H}(O)\geq\min(\mu_{H}(x),\mu_{H}(y))=\mu_{H}(x)\\\
&\texttt{i.e}&\mu_{H}(O)\geq\mu_{H}(x).\end{array}$
###### Definition 6.
Let $(\mathbf{H};\mu_{H})$ be a fuzzy hyperBCK-algebra. $\mu_{H}$ is called a
fuzzy map.
###### Lemma 2.
Let $(\mathbf{H};\mu_{H})$ a fuzzy hyper BCK-algebra.The following properties
are trues:
* i)
If for all $x,y\in H,x<y$ imply $\mu_{H}(x)\leq\mu_{H}(y)$
then for all $x\in H,\mu_{H}(x)=\mu_{H}(O)$
* ii)
If $\mu_{H}(O)=0$ then $\mu_{H}(x)=0$
Proof.
* i)
For all $x\in H$, $x*H<\\{x\\}$ then $x*O<x$.
$O<x\Rightarrow\mu_{H}(O)\leq\mu_{H}(x)$.
Then $\mu_{H}(x)\leq\mu_{H}(O)$ and $\mu_{H}(O)\leq\mu_{H}(x)$ for all $x\in
H$.
i.e $\mu_{H}(x)=\mu_{H}(O)$ for all $x\in H$.
* ii)
$\mu_{H}(O)=O\Rightarrow\mu_{H}(O)\leq\mu_{H}(x),$ for all $x\in H.$
Then $\mu_{H}(x)=\mu_{H}(O)$ for all $x\in H$.
###### Definition 7.
Let $(\mathbf{H};\mu_{H})$ and $(\mathbf{F},\mu_{F})$ two fuzzy hyperBCK-
algebras. An homomorphism from $(\mathbf{H},\mu_{H})$ to
$(\mathbf{F},\mu_{F})$ is an homomorphism
$f:\mathbf{H}\longrightarrow\mathbf{F}$ of hyper BCK-algebra such that for all
$x\in H$, $\mu_{F}(f(x))\geq\mu_{H}(x)$
###### Proposition 1.
Let $(\mathbf{H},\mu_{H})$ an hyperBCK-algebra. Let
$\mathbf{G},\mathbf{F}\subset H$ two hyperBCK-sualgebras of $\mathbf{H}$. If
there exist $\alpha\in]0,1[$ such that
$\mu_{H}(G^{*})\subset[0,\alpha[$ and $\mu_{H}(F)\subseteq]\alpha,1]$. Then
any homorphism of hyper BCK-algebra $f:G\longrightarrow F$ is homomorphism of
fuzzy hyperBCK-algebra.
Proof. Suppose that there is $\alpha\in]0,1]$ such that
$\mu_{H}(G^{*})\subset[0,\alpha[$ and $\mu_{H}(F)\subseteq]\alpha,1[.$
Let $f:G\longrightarrow F$ an homomorphism of hyper BCK-algebra.
For all $x\in G^{*},f(x)\in F.$ And $\mu_{F}(f(x))>\alpha>\mu_{H}(x).$
Then $\mu_{F}(f(x))>\mu_{H}(x)$ for all $x\in G^{*}$
$f(O)=O$ then $\mu_{F}(f(O))=\mu_{F}(O)=\mu_{H}(O)$
i.e $\mu_{F}(f(O))=\mu_{H}(O).$ therefore, for all $x\in x\in
G,\mu_{F}(x)\geq\mu_{H}(x)$
###### Example 2.
$[\ref{1}]$ Define the hyper operation $"*"$ on $H=[1;+\infty]$ as follow.
$x*y=\left\\{\begin{array}[]{ccccll}&[1,x]&&\texttt{if}&\quad x\leq y\\\
&(1,y]&&\texttt{if}&\quad x>y\neq 1\\\ &\\{x\\}&&\texttt{if}&\quad
y=1\end{array}\right.$
For all $x,y\in H,(\mathbf{H},*,1)$ is hyperBCK-algebra. Define the fuzzy
structure $\mu_{H}$ on $H$ by:
$\begin{array}[]{lllcc}\mu_{H}:&H&\longrightarrow&[0,1]&\\\
&x&\mapsto&\frac{1}{x}&\end{array}$
We show that $(\mathbf{H},\mu_{H})$ is a fuzzy hyper BCK-algebra.
Let $x,y\in H.$
1. (i)
If $x\leq y$, then $x*y=[1,x]$; i.e for all $t\in x*y,1\leq t\leq x\leq y$
and so $\frac{1}{y}\leq\frac{1}{x}\leq\frac{1}{t}.$ So,
$\mu_{H}(t)\geq\frac{1}{y}=\min\\{\frac{1}{y},\frac{1}{x}\\}=\min\\{\mu_{H}(x),\mu_{H}(y)\\}$.
Then $\inf\\{x*y\\}\geq\min\\{\mu_{H}(x),\mu_{H}(y)\\}$
2. (ii)
If $x>y\neq 1$ then $x*y=(1,y]$. For all $t\in H\cap
x*y,\frac{1}{x}\leq\frac{1}{y}\leq\frac{1}{t}\leq 1.$ therefore,
$\mu_{H}(t)=\frac{1}{t}\geq\frac{1}{x}=\min\\{\mu_{H}(x),\mu_{H}(y)\\}$ for
all $t\in x*y$. Then
$\in\\{\mu_{H}(x*y)\\}\geq\min\\{\mu_{H}(x),\mu_{H}(y)\\}.$
3. (iii)
If $y=1,x*y=\\{x\\}$, hence $\mu_{H}(x*y)=\\{\mu_{H}(x)\\}=\\{\frac{1}{x}\\}$.
$y=1$imply $y\leq x$ and $\frac{1}{x}\leq\frac{1}{y}$ for all $x\in H$; i.e;
$\min\\{\mu_{H}(x),\mu_{H}(y)\\}=\frac{1}{x}$. Then
$\mu_{H}(x*y)=\\{\frac{1}{x}\\}.$
Thus $\inf\\{\mu_{H}(x*y)\\}=\frac{1}{x}\geq\min\\{(\mu_{H}(x),\mu_{H}(y))\\}$
###### Proposition 2.
The fuzzy hyperBCK-algebras and homomorphisms of fuzzy hyperBCK-algebras form
a category.
Proof. The proof is easy.
###### Notes 1.
In the following we let $\mathcal{H}$ the category of hyperBCK-algebras;
$\mathbb{F}_{\mathcal{H}}$ the category of fuzzy hyperBCK-algebras;
$\mathbb{H}$ the fuzzy hyper BCK-algebra $(\mathbf{H},\mu_{H})$
For any fuzzy hyper BCK-algebra $\mathbb{H}$, we associate for all
$\alpha\in[0,1]$ the set $H_{\alpha}:=\\{x\in H,\mu_{H}(x)\geq\alpha\\}$
###### Lemma 3.
Let $\mathbb{H}$ a fuzzy hyper BCK-algebra. For all $\alpha\in[0,1],O\in
H_{\alpha}$ and for all $x,y\in H,x*y\subseteq H_{\alpha}$
Proof. By lemma 1, for all $x\in H,\mu_{H}(x)\leq\mu_{H}(0).$
Then for all $x\in H_{\alpha},\mu_{H}(O)\geq\mu_{H}(x)>\alpha$ i.e $O\in
H_{\alpha}$.
Let $x,y\in H_{\alpha}$;
for all $t\in x*y,$
$\mu_{H}(t)\geq\inf\\{\mu_{H}(x*y)\geq\min\\{\mu_{H}(x),\mu_{H}(y)\\}\\}\geq\alpha$
then $t\in H_{\alpha}$. therefore, $x*y\subseteq H_{\alpha}$
###### Definition 8.
Let $(H,*,O)$ be an hyper BCK-algebra. An hyper BCK-subalgebra of $H$ is a non
empty subset $S$ of $H$ such that $O\in S$ and $S$ is hyper BCK-algebra with
respect to the hyper operation $"*"$ on $H$
###### Proposition 3.
Let $(H,*,O)$ be an hyper BCK-algebra. A non empty subset $S$ of $H$ is hyper
BCK-subalgebra of $H$ iff for all $x,y\in S,x*y\in S$
Proof. The proof is easy.
###### Definition 9.
A fuzzy hyper BCK-subalgebra of $\mathbb{H}$ is an hyper BCK-subalgebra $S$ of
$\mathbf{H}$ with the restriction $\mu_{S}$ of $\mu_{H}$ on $S.$
###### Proposition 4.
For all $\alpha\in[0,1],$ $(H_{\alpha},\mu_{H})$ is fuzzy hyper BCK-subalgebra
of $\mathbb{H}$
Proof. By lemma 3, $H_{\alpha}$ is hyper BCK-subalgebra of $\mathbf{H}$
and $\inf\\{\mu_{H}(x*y)\\}\geq\min\\{\mu_{H}(x),\mu_{H}(y)\\}$
###### Definition 10.
Let $\mathbb{H}$ by an fuzzy hyper BCK-algebra. The fuzzy-hyperBCK-subalgebra
$\mathbf{H}_{\alpha}:=(H_{\alpha};\mu_{H})$ is calling hyper $\alpha$-cut of
$\mathbb{H}$
###### Proposition 5.
Let $\mathbb{H}$ be fuzzy hyper BCK-algebra. A hyper BCK-subalgebra $S$ of
$\mathbf{H}$ is fuzzy hyper BCK-subalgebra iff $S$ is hyper $\alpha$-cut of
$H.$
Proof. By prosition 4, any hyper $\alpha$-cut is fuzzy hyper BCK-subalgebra.
Conversely, let $S$ be fuzzy hyper BCK-subalgebra of $\mathbb{H}$. Then
$\mu_{H}(S)$ is subset of $[0,1].$
If $0\in\mu_{H}(S),$ then $S=H_{0}=\mathbb{H}.$
If $0<\inf(\mu_{H}(S)),$ then $S=H_{\inf(\mu_{H}(S))}.$
###### Proposition 6.
Let $\mathbb{H}$ and $\mathbb{F}$ be two fuzzy hyper BCK algebras. An
$\mathcal{H}$-morphism $f:H\longrightarrow F$ is
$\mathbb{F}_{\mathcal{H}}$-morphism iff for all
$\alpha\in[0,1],f(H_{\alpha})\subseteq F_{\alpha}.$
Proof. Suppose that $f(H_{\alpha})\subseteq F_{\alpha}$ for all
$\alpha\in[0,1]$ Let $x\in[0,1]$ we need $\mu_{H}(x)\leq\mu_{F}(f(x))$. Let
$\alpha=\mu_{H}(x);\quad x\in H_{\alpha}$ and $f(x)\in f(H_{\alpha})\subseteq
F_{\alpha}$. Then $\mu_{F}(f(x))>\alpha=\mu_{H}(x).$ whence for all $x\in
H,\mu_{F}(f(x))\geq\mu_{H}(x)$.
Conversely, suppose that $f:\mathbb{H}\longrightarrow\mathbb{F}$ is
$\mathbb{F}_{\mathcal{H}}$-morphism.
For all $x\in H_{\alpha}$ for some $\alpha\in[0,1]$,
$\mu_{F}(f(x))\geq\mu_{H}(x)\geq\alpha$i.e; $f(x)\in[0,1]$. Then
$f(H_{\alpha})\subseteq F_{\alpha}$ for all $\alpha\in[0,1]$.
###### Proposition 7.
A $\mathbb{F}_{\mathcal{H}}$-morphism $f:\mathbb{H}\longrightarrow\mathbb{F}$
is $\mathbb{F}_{\mathcal{H}}$-iso iff it is both $\mathcal{H}$-iso and
$\mu_{H}=\mu_{F}f.$
Proof. Suppose that $f$ is $\mathcal{H}$-iso and $\mu_{H}=\mu_{F}f$. there is
$g\in Hom_{\mathcal{H}}(\mathbf{F},\mathbf{H})$; $g\circ f=Id_{H}$ and $g\circ
g=Id_{F}.$
Then, for all $x\in F,\mu_{H}(g(x))=\mu_{F}(f(g(x)))\mu_{F}(x).$
And then, $g\in Hom_{\mathbb{F}_{\mathcal{H}}}(\mathbb{F},\mathbb{H}).$
Conversely, Suppose that $f$ is $\mathbb{F}_{\mathcal{H}}$-iso.
There is $g\in Hom_{\mathbb{F}_{\mathcal{H}}}(\mathbb{F},\mathbb{H});g\circ
f=Id_{F}$ and $f\circ g=Id_{H}.$
Since $f\in
Hom_{\mathbb{F}_{\mathcal{H}}}(\mathbb{F},\mathbb{H}),\mu_{H}\leq\mu_{F}f.$
Since $g\in
Hom_{\mathbb{F}_{\mathcal{H}}}(\mathbb{H},\mathbb{H}),\mu_{F}\leq\mu_{H}g.$
$x\in H$ imply $f(x)\in F$. Then
$\mu_{F}(f(x))\leq\mu_{H}(g(f(x)))=\mu_{H}(x)$ i.e; $\mu_{F}f\leq\mu_{H}.$
therefore, $\mu_{F}f=\mu_{H}$
###### Proposition 8.
Let $f\in Hom_{\mathbb{F}_{\mathcal{H}}}(\mathbb{F},\mathbb{H}).$
$f$ is $\mathbb{H}_{\mathcal{H}}$-mono iff $f$ is $\mathcal{H}$-mono
Proof. Suppose that $f$ is $\mathbb{H}_{\mathcal{H}}$-mono.
For all $h,g\in Hom_{\mathcal{H}}(\mathbf{K},\mathbf{H})$, such that $fh=fg,$
we define $\mu_{K}=\min(\mu_{H}(h(x);\mu_{H}(g(x))))$ for all $x\in K.$
1. a)
We show that $(K;\mu_{K})$ is fuzzy hyper BCK-algebra.
$\begin{array}[]{lllccccllll}\inf(\mu_{K}(x*y))&=&\inf\\{\mu_{H}(h(x*y);\mu_{H}(g(x*y)))\\}\\\
&=&\inf\\{\mu_{H}(h(x)*h(y);\mu_{H}(g(x)*g(y)))\\}\\\
&=&\min\\{\inf\\{\mu_{H}(h(x)*h(y)\\};\inf\\{\mu_{H}(g(x)*g(y)))\\}\\}\\\
&\geq&\min\\{\min\\{\mu_{H}(h(x);\mu_{H}(h(y)\\};\min\\{\mu_{H}(g(x);g(y))\\}\\}\\\
&\geq&\min\\{\min\\{\mu_{K}(x);\mu_{K}(y)\\}\\}\\\
&\geq&\min\\{\mu_{K}(x);\mu_{K}(y)\\}\par\par\end{array}$
Then, for all $x,y\in
K,\inf(\mu_{K}(x*y))\geq\min\\{\mu_{K}(x),\mu_{K}(y)\\}$. therefore,
$(K;\mu_{K})$ is fuzzy hyper BCK-algebra.
2. b)
We show that $h$ and $g$ are $\mathbb{F}_{\mathcal{H}}$-homomorphism.
For all $x\in K,$ $\mu_{K}(x)=\min\\{\mu_{H}(h(x),\mu_{H}(g(x)))\\}.$
Then $\mu_{K}(x)\leq\mu_{H}(g(x))$ and $\mu_{K}(x)\leq\mu_{H}(h(x))$.
therefore, $h$ and $g$ are $\mathbb{F}_{\mathcal{}}H$-morphism.
Since $f$ is $\mathbb{F}_{\mathcal{H}}$-mono and $h,g\in
Hom_{\mathbb{F}_{\mathcal{H}}}(\mathbb{F},\mathbb{H}),$ $fh=fg$ imply $f=g$
Conversely, if $f$ is $\mathbb{F}_{\mathcal{H}}$-mono, it is
$\mathcal{H}$-mono.
###### Lemma 4.
The pair $\mathbb{O}=(\\{O\\},\mu_{o})$ where
$\begin{array}[]{lcccll}\mu_{o}:&\\{o\\}&\longrightarrow&[0,1]\\\
&o&\mapsto&0\end{array}$
is fuzzy hyper BCK-algebra
Proof. Easy
###### Lemma 5.
$\mathbb{O}$ is final objet of $\mathbb{F}_{\mathcal{H}}$
###### Proposition 9.
The category $\mathbb{F}_{\mathcal{H}}$ has products.
Proof. Let $(\mathbb{H}_{i};\mu_{H_{i}})_{i\in I}$ a family of fuzzy hyper
BCK-algebras.
Denote $\mathbf{H}=\underset{i\in I}{\prod}H_{i}$ the $\mathcal{H}$-product of
$(H_{i})_{i\in I}$ with the projection morphisms
$p_{i}:\mathbf{H}\longrightarrow H_{i}$. Consider the following map
$\mu_{H}:H\longrightarrow[0,1]$ define by:
$\mu_{H}(x)=\underset{i\in I}{\bigwedge}\mu_{H_{i}}p_{i}(x)$
for all $x\in H$
* a)
We show that the pair $(\mathbf{H};\mu_{H})$ is fuzzy hyper BCK-algebra.
For all $x,y\in H,p_{i}(x*y)=p_{i}(x)*p_{i}(y)$ for all $i\mathbb{N}I.$
Then
$\begin{array}[]{lcllc}\inf(\mu_{H_{i}}p_{i}(x*y))&=&\inf(\mu_{H_{i}}(p_{i}(x)*p_{i}(y))\\\
&\geq&\min\\{\mu_{H_{i}}(p_{i}(x));\mu_{H_{i}}(p_{i}(y))\\}\end{array}$
for all $i\in I$.
Then,
$\begin{array}[]{lllcc}\inf(\underset{i\in
I}{\bigwedge}\mu_{H_{i}}p_{i}(x*y))&\geq&\underset{i\in
I}{\bigwedge}\inf\\{\mu_{H_{i}}(p_{i}(x)*p_{i}(y)\\}\\\ &\geq&\underset{i\in
I}{\bigwedge}\min\\{\mu_{H_{i}}(p_{i}(x));\mu_{H_{i}}(p_{i}(y))\\}\\\
&\geq&\min\\{\underset{i\in
I}{\bigwedge}\mu_{H_{i}}p_{i}(x);\mu_{H_{i}}p_{i}(y)\\}\\\
&\geq&\min\\{\mu_{H}(x),\mu_{H}(y)\\}\end{array}.$
* b)
For all $i\in I,x\in H;\mu_{H_{i}}p_{i}(x)\geq(\underset{i\in
I}{\bigwedge}\mu_{H_{i}}p_{i})(x)$.
Then each $p_{i}$ is $\mathbb{F}_{\mathcal{H}}$-morphism.
* c)
If $q_{j}:\mathbb{F}\longrightarrow\mathbb{H}_{j}$ is family of
$\mathbb{F}_{\mathcal{H}}$-morphism, there is unique $\mathcal{H}$-morphism
$\varphi:\mathbf{F}\longrightarrow\mathbf{H}$ such that the following diagram
commute.
$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{j}}$$\textstyle{H_{j}}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{q_{j}}$
i.e $p_{j}\varphi=q_{j}$ for all $j\in I$
For all $x\in F$, $\mu_{F}(x)\leq\mu_{H_{i}}q_{i}(x)$.
$\begin{array}[]{lllcc}\texttt{Then}&\mu_{F}(x)&\leq\mu_{H_{i}}p_{i}\varphi(x)\quad\texttt{for
all }\quad x\in F,i\in I\\\ \texttt{i.e}&\mu_{F}(x)&\leq\underset{i\in
I}{\bigwedge}\mu_{H_{i}}p_{i}\varphi(x)\\\ &&\leq(\underset{i\in
I}{\bigwedge}\mu_{H_{i}}p_{i})\varphi(x)\\\
&&\leq\mu_{H}(\varphi(x))\quad\texttt{for all}\quad x\in F.\\\
&\texttt{Then}&\mu_{F}\leq\mu_{H}\varphi\leq\end{array}$
Then $\varphi$ is $\mathbb{F}_{\mathcal{H}}$-morphism.
###### Proposition 10.
$\mathbb{F}_{\mathcal{H}}$ have equalizers.
Proof. Let $f,g\in
Hom_{\mathbb{F}_{\mathcal{H}}}(\mathbb{H},\mathbb{F}),K:=\\{x\in
H,f(x)=g(x)\\}$.
It is prove in [1] that $K$ is hyper BCK-subalgebra of $H$. It is clear that
$(K,\mu_{H})$ is fuzzy hyper BCK-algebra. Let $i:K\longrightarrow H$ the
inclusion map. $i\in Hom_{\mathbb{F}_{\mathcal{H}}}(\mathbb{K},\mathbb{F}).$
For all $x\in K,fi(x)=f(x)=g(x)=gi(x).$
Let $h\in Hom_{\mathbb{F}_{\mathcal{H}}}(\mathbb{L},\mathbb{F})$ such that
$fh=gh,$ for all $x\in L,f(h(x))=g(h(x))$. Then $Imh\subseteq L$. Define
$\delta:L\longrightarrow K$ by $\delta(x)=h(x)$ for all $x\in L.$ $\delta\in
Hom_{\mathbb{F}_{\mathcal{H}}}(\mathbb{\mathbb{H}},\mathbb{F})$ and
$i\delta=h.$ So, the following diagram commute.
|
---|---
$\textstyle{\mathbb{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\mathbb{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{\mathbb{F}}$$\textstyle{\mathbb{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\scriptstyle{h}$
Since $i$ is monic, $\delta$ is unique $\mathbb{F}_{\mathcal{H}}$-morphism
such that the above diagram commute.
therefore, $\mathbb{F}_{\mathcal{H}}$ have equalizers.
###### Proposition 11.
$\mathbb{F}_{\mathcal{H}}$ is complet.
Proof. By proposition 9, each family of objets of $\mathbb{F}_{\mathcal{H}}$
has product.
By proposition 10, each pair of parallel arrows has an equalizer. Then
$\mathbb{F}_{\mathcal{H}}$ is complet.
###### Corollary 1.
$\mathbb{F}_{\mathcal{H}}$ has pulbacks
Proof. By propositions 9 and 10, $\mathbb{F}_{\mathcal{H}}$ has equalizers and
products. therefore, $\mathbb{F}_{\mathcal{H}}$ has pulbacks.
###### Proposition 12.
$\mathcal{\mathbb{F}_{H}}$ have coequalizers
Proof. Let $f,g\in Hom_{\mathcal{\mathbb{F}_{H}}}(\mathbb{H},\mathbb{K})$. Let
$\sum_{fg}=\\{\theta,\theta\quad\texttt{regular congruence relation
on}\quad\mathbf{K}\quad\texttt{such that}\quad f(a)\theta g(a)\forall a\in
H\\}$
$\sum_{fg}\neq\phi$ because $K\times K\in\sum_{fg}$
Let $\rho=\underset{\theta\in\sum_{fg}}{\bigcap}\theta$. Then, $\rho$ is
regular congruence relation.
Define on $K/\rho$ the following hyper operation
$[x]_{\rho}*[y]_{\rho}=[x*y]_{{\rho}}.$
$(K/\rho;*;[0]_{\rho})$ is an objet of $\mathcal{H}$ (see $[\ref{1}]$).
Define on $K/\rho$ the following map
$\begin{array}[]{lllcc}\mu_{K/\rho}:&K/\rho&\longrightarrow&[0,1]\\\
{}&\mu_{K/\rho}([x]_{\rho})&\longmapsto&\underset{a\in[x]_{\rho}}{\bigvee}\mu_{K}(a)\end{array}$
* a)
We show that $(K/\rho,\mu_{K/\rho})$ is objet of $\mathbb{F}_{\mathcal{H}}$.
If $x,y\in K$ such that $[x]_{\rho}=[y]_{\rho}$.
Then
$\underset{a\in[x]_{\rho}}{\bigvee}\mu_{K}(a)=\underset{a\in[y]_{\theta}}{\bigvee}\mu_{K}(a)$
$\forall x\in K,\,\mu_{K}(x)\leq\underset{a\in[x]_{\rho}}{\bigvee}\mu_{K}(a)$.
Then
$\mu_{K}\leq\mu_{K/\rho}([x]_{\rho})=\mu_{K/\rho}(\pi(x))$
Then, the canonical projection $\pi$ is an $\mathcal{\mathbb{F}_{H}}-morphism$
Since for all $x\in H,f(x)\rho g(x)$, then $[f(x)]_{\rho}=[g(x)]_{\rho}$.
therefore, $(\pi\circ f)(x)=(\pi\circ g)(x).$
Then, $\pi\circ f=\pi\circ g.$
* b)
Universal property of coequalizer.
Let $\varphi:\mathbb{K}\longrightarrow\mathbb{L}$ and
$\mathbb{F}_{\mathcal{H}}$-morphism such that $\varphi\circ f=\varphi\circ g$.
Define the following mapping.
$\begin{array}[]{lllcc}\psi:&K/\rho&\longrightarrow&L\\\
{}&[x]_{\rho}&\longmapsto&\varphi(x)\end{array}$
* c)
We prove that $\psi$ is well define.
If $[x]_{\rho}=[y]_{\rho}$ then, for all $a\in H,\varphi(f(a))=\varphi(g(a))$
imply $f(a)R_{\varphi}g(a)$ because $R_{\varphi}$ is regular congruence on
$K$. Then $R_{\varphi}\in\sum_{f,g}.$ The minimality of $\rho$ on $\sum_{f,g}$
imply $\rho\subseteq R_{\varphi}$.
therefore, $[x]_{\rho}=[y]_{\rho}$ imply $x\rho y$.
Then $xR_{\varphi}y$. i.e $\varphi(x)=\varphi(y)$
And then, $\psi([x]_{\rho})=\psi([y]_{\rho})$ therefore, $\psi$ is well
define.
For all $x\in
K,\mu_{L}(\psi(\pi)(x))=\mu_{L}(\varphi(x))\geq\mu_{K}(x),\forall x\in K$ then
for all $a\in[x]_{\rho}$. $\mu_{L}(\varphi(a))\geq\mu_{K}(a).$ By the
minimality of $\rho$, $[a]_{\rho}=[x]_{\rho}$ imply $a\rho x$ then
$aR_{\varphi}x$ i.e $\varphi(a)=\varphi(x)$ (because $\rho\subseteq
R_{\varphi}$). Then
$\underset{a\in[x]_{\rho}}{\bigvee}\tilde{L}(\varphi(a))=\underset{a\in[x]_{\rho}}{\bigvee}\tilde{L}(\varphi(x))=\tilde{L}(\varphi(x))$
therefore
$\tilde{L}(\varphi(x))\geq\underset{a\in[x]_{\rho}}{\bigvee}(a)=\tilde{K}/\rho([x]_{\rho})$
i.e $\tilde{L}(\psi([x]_{\rho}))\geq\tilde{k}/\rho([x]_{\rho})\forall x\in H$.
It is clean that
$\psi(\pi(x))=\psi([x]_{\rho})=\varphi(x),\forall x\in H$
i.e $\psi\circ\pi=\varphi$
This prove the commutativity of the following diagram:
$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$.$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\varphi}$$\textstyle{K/\rho\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{L}$
The unicity of $\psi$ is thus to the fat that $\pi$ is epimorphism.
Then, $\mathcal{F}_{\mathcal{H}}$ have coequalizer.
## ACKNOWLEDGEMENTS
## References
* [1] H. Harizavi,J. Macdonald AND A. Borzooei _Category of bck-algebras_ :Scientiae Mathematicae Japonicae Online, e-2006, 529-537.
* [2] R. A. Borzooei H.Harizavi._Regular Congruence Relations on hyper BCK-algebras_ : vol.61, No.1(2005),83-97
* [3] Eun Hwan ROH, B. Davvaz And Kyung Ho Kim _T-fuzzy subhypernea-rings of hypernear-ring_ , Scientiae Mathematicae Japonicae Online, e-2005,19-29 .
* [4] Carol L. Walker, _Category of fuzzy sets_.
* [5] R.A. Borzoei and M.M. Zahedi, _Positive implicative hyperK-ideals_ , Scientiae Mathematicae Japonicae Online, Vol. 4,(2001), 381-389.
* [6] C. LELE and M. Salissou, _Discussiones Mathematicae, general Algebra and Application_ 26: 111-135(2006).
|
arxiv-papers
| 2011-01-13T01:09:39 |
2024-09-04T02:49:16.399475
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Joseph Dongho",
"submitter": "Joseph Dongho",
"url": "https://arxiv.org/abs/1101.2471"
}
|
1101.2513
|
# Pattern fluctuations in transitional plane Couette Flow
Joran Rolland, Paul Manneville
Laboratoire d’Hydrodynamique de l’École Polytechnique, 91128 Palaiseau, France
###### Abstract
In wide enough systems, plane Couette flow, the flow established between two
parallel plates translating in opposite directions, displays alternatively
turbulent and laminar oblique bands in a given range of Reynolds numbers $R$.
We show that in periodic domains that contain a few bands, for given values of
$R$ and size, the orientation and the wavelength of this pattern can fluctuate
in time. A procedure is defined to detect well-oriented episodes and to
determine the statistics of their lifetimes. The latter turn out to be
distributed according to exponentially decreasing laws. This statistics is
interpreted in terms of an activated process described by a Langevin equation
whose deterministic part is a standard Landau model for two interacting
complex amplitudes whereas the noise arises from the turbulent background.
## 1 Introduction
The main features of the transition to turbulence are well understood in
systems prone to a linear instability like convection where chaos emerges at
the end of an instability cascade. A much wilder transition is observed in
wall-bounded shear flows for which the laminar and turbulent regimes are both
possible states at intermediate values of the Reynolds number $R$, the natural
control parameter, whereas no linear instability mechanism is effective. A
direct transition can take then place via the coexistence of laminar and
turbulent domains in physical space. Two emblematic cases are the pipe flow
and plane Couette flow (PCF), the simple shear flow developing between two
parallel plates translating in opposite directions. Both of them are stable
against infinitesimal perturbations for all values of $R$ and become turbulent
only provided sufficiently strong perturbations are present. In both cases,
strong hysteresis is observed and, upon decreasing $R$, the turbulent state
can be maintained down to a value $R_{\rm g}$. Above $R_{\rm g}$, turbulence
remains localised in space, in the form of turbulent puffs in pipe flow and
turbulent patches in PCF. A striking property of PCF or counter-rotating
cylindrical Couette flow (CCF) is the spatial organisation of turbulence in
alternatively turbulent and laminar oblique bands that takes place in large
enough systems in a specific range of Reynolds numbers [1, Ch.7]. This regime
was studied in depth at Saclay by Prigent et al. [2]. It can be obtained by
decreasing the Reynolds number continuously from featureless turbulence below
$R_{\rm t}$, the Reynolds number above which the flow is uniformly turbulent,
or triggered from laminar flow by finite amplitude perturbations above $R_{\rm
g}$, the Reynolds number below which laminar flow is expected to prevail in
the long time limit. A similar situation is observed in pipe flow but things
are complicated by the global downstream advection so that the existence of a
threshold $R_{\rm t}$ above which turbulence is uniform is still a debated
matter. In contrast for PCF, the pattern is essentially time-independent and
can be characterised by two wavelengths $\lambda_{x}$ and $\lambda_{z}$ in the
streamwise and spanwise direction, $x$ and $z$ respectively,111In the case of
CCF, the pattern is time-independent in a frame that rotates at the mean
angular velocity and the axial (azimuthal) direction corresponds to the
spanwise (streamwise) direction. or equivalently by a wavevector
$\mathbf{k}=(k_{x},k_{z})$ with $k_{x,z}=2\pi/\lambda_{x,z}$. From symmetry
considerations, two orientations are possible, corresponding to two possible
combinations $(k_{x},\pm k_{z})$. Whereas a single orientation is present
sufficiently far from $R_{\rm t}$ so that either mode $(k_{x},+k_{z})$ or mode
$(k_{x},-k_{z})$ is selected, patches of one or the other orientation have
been reported to fluctuate in space and time when $R$ approaches $R_{\rm t}$
from below [2, Figs. 2 & 3]. The main features of the bifurcation diagram
could then be accounted for at a phenomenological level by an approach in
terms of Ginzburg–Landau equations subjected to random noise featuring the
small-scale turbulent background.
This patterning was reproduced by Duguet et al. [3] using fully resolved
numerical simulations in an extended system of size comparable with that of
the Saclay apparatus but the computational load was so heavy that a
statistical study of the upper transitional range was inconceivable. Earlier,
Barkley & Tuckerman [4] also succeeded in obtaining the bands by means of
fully resolved simulations with less computational burden but using narrow
elongated domains aligned with the pattern’s wavevector. By construction, the
fluctuating domain regime could not be obtained, whereas a re-entrant
featureless turbulence regime, called ‘intermittent’ was obtained closer to
$R_{\rm t}$.
In our previous work on this problem, we first showed that full numerical
resolution was not necessary to obtain realistic patterning but that a good
account of the long range streamwise correlation of velocity fluctuations was
essential [5]. This next incited us to consider reduced-resolution simulations
in systems of sizes sufficient to contain at least an elementary cell
$(\lambda_{x},\lambda_{z})$ of the pattern [6], thus avoiding the orientation
constraint inherent in the Barkley–Tuckerman approach. Here, we expand our
previous work to focus on pattern fluctuations in the upper part of the PCF’s
bifurcation diagram when $R$ approaches $R_{\rm t}$ from below, taking the
best possible use of the inescapable resolution lowering to perform long
duration simulations, so as to obtain meaningful statistics about the dynamics
of this regime.
Systems considered in our numerical experiment, to be described in §2, produce
patterns with a few wavelengths. In the neighbourhood of $R_{\rm t}$,
fluctuations manifest themselves as orientation changes in time instead of the
spatiotemporal evolution of well-ordered patches. It turns out that episodes
of well-formed pattern between two orientation changes can be identified
reliably, so that the lifetimes of such episodes can be measured and their
average determined as a function of $R$. The Langevin approach initiated by
Prigent et al. in [2] was resumed in [6] as providing an appropriate framework
to interpret our numerical results. Orientation fluctuations were taken into
account but their detailed statistical properties left aside, which are the
subject of the present paper.
In the context of pattern formation, the Langevin/Fokker–Planck approach has a
long history, dating back to the 1970’s when it was applied to convecting
systems [7]. Noise of thermal origin is however extremely weak so that the
region of parameter space where the system is sensitive to this noise is
exceedingly narrow [8] and nontrivial effects can be observed only in very
specific conditions [9]. When applying the approach to the description of the
bifurcation from featureless turbulence to pattern in shear flows, Prigent et
al. [2] implicitly took for granted that the noise intensity was an adjustable
parameter linked to the turbulent background at $R>R_{\rm t}$. Here we extend
the analysis started in [6] within this conceptual framework, the subject of
§3, and analyse simulation results presented in §2.4 in the light of this
theory. We conclude in §4 by discussing how well this approach is suited to
describe mode competition and intermittent re-entrance of featureless
turbulence [4, 6] and, more generally, how the noisy temporal dynamics of
coherent modes can hint at the spatio-temporal nature of transitional wall-
bounded flows and explain the exponentially decreasing probability
distributions of residence times or decay times often observed in this field
[12].
## 2 Conditions of the numerical experiment
### 2.1 Numerical procedure
Direct numerical simulation (DNS) of the incompressible Navier–Stokes
equations in the geometry of PCF are performed using Gibson’s open source code
ChannelFlow [10] that assumes no-slip boundary conditions at the plates
driving the flow and in-plane periodic boundary conditions. The parallel
plates producing the shear are placed at a distance $2h$ from each other in
the wall-normal direction $y$, they move at speeds $\pm U$ in the streamwise
direction $x$, $z$ labelling the spanwise direction. The length unit is $h$,
the velocity unit $U$, the time unit $h/U$, and the Reynolds number
$R=Uh/\nu$, where $\nu$ is the kinematic viscosity of the fluid. The problem
is completely specified when the in-plane dimensions $L_{x}$ and $L_{z}$ of
the set-up are chosen. The perturbation to the laminar flow
$\mathbf{U}=y\,\mathbf{\hat{x}}$ is noted $\mathbf{u}$, so that
$\mathbf{u}^{2}$ is the local Euclidian distance to the base flow squared.
Periodic in-plane boundary conditions allow the definition of the wave-vectors
$k_{x,z}=2\pi n_{x,z}/L_{x,z}$, where the wavenumbers $n_{x,z}$ are integers.
Without loss of generality, we can assume $n_{x}\geq 0$.
The resolution of the simulation is fixed by the number $N_{y}$ of Chebyshev
polynomials used to represent the wall-normal dependence, and the numbers
$N_{x,z}$ of collocation points used to evaluate the nonlinear terms in
pseudo-spectral scheme of integration of the Navier–Stokes equations. The
number of Fourier modes involved in the simulation is then $2N_{x,z}/3$, owing
to the 3/2-rule applied to de-aliase the velocity field. The computational
load necessary to obtain meaningful results in sufficiently wide domains with
fully resolved simulations is unrealistically heavy. Accordingly, we take
advantage of our previous work devoted to the validation of systematic under-
resolution as a modelling strategy [5]. In that work, we showed that
qualitatively excellent and quantitatively acceptable results could be
obtained by taking $N_{y}=15$ and $N_{x,z}=8L_{x,z}/3$. The price to be paid
for the resolution lowering was apparently just a downward shift of the range
$[R_{\rm g},\,R_{\rm t}]$ in which the bands are obtained, but everything else
was preserved, including wavelengths. Of course, as far as resolution is
concerned, the finest is the best on a strictly quantitative basis but we do
not expect that the observed trends and our conclusions be sensitive to our
rules to fix $N_{y}$ and $N_{x,z}$.
### 2.2 Orientation fluctuations.
In this article we consider domains able to contain pattern with one or two
elementary cells, i.e. $L_{x,z}=|n_{x,z}|\lambda_{x,z}$ where $n_{x}=1$ or 2
and $n_{z}=\pm 1$ or $\pm 2$. According to [2], in PCF wavelength
$\lambda_{x}$ is found to be approximately equal to $110$ over the whole range
$[R_{\rm g},\,R_{\rm t}]$, while wavelength $\lambda_{z}$ varies as a function
of $R$ in the range $[40,\,85]$ These observations serve us to fix the size of
the systems that we are going to consider below. As shown in [6], the
specificity of such systems is to convert the spatio-temporal evolution of
fluctuating domains observed in the neighbourhood of $R_{\rm t}$ into the
temporal evolution of coherent patterns characterised by the amplitudes of the
corresponding fundamental Fourier modes; possible orientation changes are
associated with changes of sign of the spanwise wavenumbers. Close enough to
$R_{\rm t}$, there is also some probability that featureless turbulence, the
state that prevails for $R>R_{\rm t}$, be observed transiently, which is akin
to the intermittent regime identified in [4]. In contrast, in the lowest part
of the transitional range, close to $R_{\rm g}$, the orientation remain frozen
as expected for well-formed steady oblique bands. We first illustrate this
phenomenon using snapshots of $\mathbf{u}^{2}$ in Figures 1 and 2.
Figure 1: Snapshots of $\mathbf{u}^{2}$ in the plane $y=-0.57$ for $R=315$ in
a system of size $L_{x}\times L_{z}=128\times 84$. From left to right: one
band pure state with each of the two possible orientations ($n_{x}=1$,
$n_{z}=+1$) or ($n_{x}=1$, $n_{z}=-1$), two band pure state ($n_{x}=1$,
$n_{z}=+2$), and mixed or defective pattern. Deep blue corresponds to laminar
flow.
Figure 2: Snapshots of $\mathbf{u}^{2}$ in the plane $y=-0.57$ for $R=290$ in
a system of size $L_{x}\times L_{z}=170\times 48$. From left to right:
($n_{x}=1$, $n_{z}=-1$), ($n_{x}=1$, $n_{z}=+1$), and ($n_{x}=2$, $n_{z}=+1$).
The left and centre panels of Fig. 1 display well-oriented patterns or ‘pure
states’ showing the organised cohabitation of laminar and turbulent flow; an
example of defective pattern or ‘mixed state’ without much spatial
organisation is shown in the right panel. (Orientation defects between well-
oriented domains require wider systems to be clearly identified as such.)
Figure 2 similarly displays snapshots of $\mathbf{u}^{2}$ obtained in a
narrower but longer system.
Typically, during long-lasting simulations at given $L_{x,z}$ and $R$, the
flow displays a pure pattern for some time, then experience a brief defective
stage, and next recovers a pure state, possibly with different orientation
or/and wavelength, and so on. The spatial organisation of the pattern is
detected via the Fourier transform of the perturbation velocity field
$\hat{\mathbf{u}}$. It turns out that most of the information about the
modulation is encoded in the amplitude of the dominant wavenumber [2, 11, 6].
We consider time series of
$m^{2}(n_{x},n_{z},t)=\frac{1}{2}\int_{-1}^{+1}|\hat{u}_{x}(n_{x},y,n_{z},t)|^{2}\,{\rm
d}y\,,$ (1)
which thus characterises a flow pattern with wavelengths
$(\lambda_{x},\lambda_{z})=(L_{x}/n_{x},L_{z}/|n_{z}|)$ and orientation given
by the sign of $n_{z}$. In the present study, we focus on the amplitude of the
turbulence modulation in the flow and not on its phase, i.e. on the position
of the pattern in the system, which was shown to be a random function of time
[6].
An example of such time series is displayed in Fig. 3.
Figure 3: Time series of $m^{2}(t)$ for several wave numbers $n_{z}=\pm 1,\pm
2$ for $R=315$ in a system of size $L_{x}\times L_{z}=128\times 84$.
A pure pattern stage corresponds to a single $m(n_{x},n_{z})$ fluctuating
around a non zero value, the other $m(n_{x}^{\prime},n_{z}^{\prime})$
remaining negligible. For instance the pattern keeps wave-number $n_{z}=+2$
from $t=3\,10^{3}$ to $t=10^{4}$. The defective stage corresponds to
$m(n_{x},n_{z})$ decaying to zero while another one
$m(n_{x}^{\prime},n_{z}^{\prime})$ grows. Wavenumbers
$(n_{x}^{\prime},n_{z}^{\prime})$ may be different from $(n_{x},n_{z})$, in
which case there is an effective change of the orientation if
$|n_{z}^{\prime}|=|n_{z}|$ or a change of wavelength (sometimes combined with
orientation changes) if $|n_{z}^{\prime}|\neq|n_{z}|$. In Fig. 3, a change of
orientation takes place at time $t=4\,10^{4}$ ($n_{z}=-1\to+1$), a change of
wavelength at time $t=1.7\,10^{4}$ ($n_{z}=-2\to+1$), the pattern with
$n_{z}=+1$ growing back from a defective stage at time $t=4.3\,10^{4}$. Most
of the time there is no ambiguity about the value of $n$ involved so that we
shall use simplified notations, i.e. just $m$ or $m_{\pm}$ instead of
$m(n_{x},\pm n_{z})$, as often as possible.
Except very close to $R_{\rm t}$, pure state intermissions last long and
defective episodes are short, so that series of lifetime $T_{i}$ of well-
oriented lapses can be defined from recording simulations of duration
sufficient to make reliable statistics.
### 2.3 Lifetime computations
Orientation and wavelength fluctuations are best characterised by lifetimes
distributions. Beforehand, we have to define a systematic method to detect the
beginning and the end of pure pattern episodes from the $m^{2}$ time series.
This is done by using two thresholds: one, $s_{1}$, for the start of a pure
pattern episode and the other, $s_{2}$, for its termination, see Fig. 4 (top).
Figure 4: Time series of $m^{2}$ for $n_{z}=\pm 1$ at $R=330$ (top) and
$R=345$ (bottom). The two horizontal lines in the top panel locate threshold
$s_{1}$ (full line) and $s_{2}$ (dashed line). Close to $R_{\rm t}$, bottom
panel, orientation fluctuations are short-lived and much smaller, rendering
the detection of well-oriented episodes more difficult.
The fast growth of $m^{2}$ makes it easy to choose $s_{1}$ and the results are
not much sensitive to its exact value. In contrast, detecting the decay is
more problematic. This will be discussed in detail after the presentation of a
typical result obtained by assuming that the difficulty has been properly
resolved.
For practical reasons, we use a byproduct of the cumulated probability density
function (PDF) $Q$:
$Q(T)=\frac{\\#\\{T_{i}\geq T\\}}{\\#\\{T_{i}\\}}=1-\frac{\\#\\{T_{i}\leq
T\\}}{\\#\\{T_{i}\\}}\,.$
Empirical distributions obtained in an experiment with $L_{x}=110$ and
$L_{z}=32$ for $R=330$ are displayed in Fig. 5. They
Figure 5: Logarithm of $Q$ (right) for $L_{x}\times L_{z}=128\times 84$,
$R=315$, computed with $s_{1}=0.001$, $s_{2}=0.005$, for both wave numbers
$|n_{z}|=1$ and $|n_{z}|=2$. We have about 40 events for $|n_{z}|=1$ and about
20 events for $|n_{z}|=2$.
are obtained from the time series, a small part of which is shown in Fig. 4,
distinguishing $n_{z}=\pm 1$ from $n_{z}=\pm 2$. Since, for symmetry reasons,
the two orientations are supposed to have identical distributions, we sum over
the $\pm$ in each case. The semi-logarithmic coordinates used to represent
$Q(T)$ suggest exponentially decreasing variations, which makes orientation
changes look like deriving from a Poisson process. Assuming that they are
indeed in the form $\exp(-T/\langle T\rangle)$, we can obtain the mean
lifetime $\langle T\rangle$ from the plain arithmetic average of the liftetime
series. $\langle T\rangle$ can also be obtained by fitting the empirical
cumulated distribution against an exponential law or its logarithm against a
linear law. In addition to raw data, Fig. 5 displays the second kind of fits
for $|n_{z}|=1$ and $2$. These three different estimates are close to each
other provided that the lifetime series comprise sufficiently large numbers of
events. An average of these three values will be used to define the mean
lifetime and the corresponding unbiased standard deviation will give an
estimate of the “error” for each lifetime series.
Let us now come to the problem of the sensitivity of $Q(T)$ to the value of
the thresholds $s_{1}$ and $s_{2}$ used to determine the lifetimes of the pure
pattern episodes. In Figure 6 (left) the mean lifetime $\langle T\rangle$
displays a clear plateau as a functions of $s_{1}$. The width of this plateau
does not depend on $s_{2}$ though its value depends on it.
Figure 6: Mean lifetimes functions of $s_{1}$ given $s_{2}$ (left) and of
$s_{2}$ given $s_{1}$ (right). $R=315$ and system size $L_{x}\times
L_{z}=128\times 84$, $|n_{z}|=1$.
The existence of this plateau is easily seen to be related to the fast growth
of $m$ when the pattern sets in: $m$ always goes through most of the values
corresponding to the plateau in a very short time. In practice, for
$10^{-3}\leq s_{1}\leq 1.5\,10^{-3}$ the very same episodes are detected
whatever the precise value of $s_{1}$. That the plateau value still depends on
$s_{2}$ just expresses that the duration of the detected episodes are modified
in the same way due to changes in the detection of their termination. Of
course, when $s_{1}$ is taken too large, some less-well ordered episodes
escape detection or are detected too late, which artificially decreases the
mean. On the other hand, if $s_{1}$ is taken too small, the “signal” gets lost
in the “noise”: a large number of brief noisy excursions are detected as
relevant ordered episodes, again decreasing the mean.
The variation of the mean lifetime with $s_{2}$ is completely different as
seen in Fig. 6 (right). Here, $\langle T\rangle$ varies roughly linearly with
$s_{2}$ in a wide interval above the noise level ($\sim 3\,10^{-4}$, see Fig.
4):
$\langle T\rangle(s_{2})\simeq a(1-bs_{2})\,.$
Coefficient $b=1000\pm 100$ does not vary significantly over the cases that we
have considered. This dependence fully explains the change of plateau value in
plots of $\langle T\rangle$ as a function of $s_{1}$. Coefficient $a$,
corresponding to $\langle T\rangle$ extrapolated toward $s_{2}=0$ however
still depend on $R$ and the geometry. Henceforth, we define this extrapolated
value as the relevant average lifetime $\langle T\rangle$, which will be
supported by the theoretical considerations to be developed in the next
section.
The observed dependence of $\langle T\rangle$ on $s_{2}$ can be explained by
the fact that the decay of a pure pattern is much more gradual than its
growth, which causes significant differences when the duration of an episode
is measured, leading to a decrease of $\langle T\rangle$ as $s_{2}$ increases
since the termination of the episode is detected earlier. A second reason why
the mean lifetime increases as $s_{2}$ decreases arises from the fact that
some excursions are not counted as decay events. In physical space, this
corresponds to an irregular and slow disorganisation of turbulence,
contrasting with the fast installation of the pattern. In fact $\langle
T\rangle$ cannot be obtained otherwise than by extrapolation of threshold
$s_{2}$ to zero, as will be discussed in §3.3.
### 2.4 DNS results
The two systems sizes, $L_{x}\times L_{z}=128\times 64$ and $110\times 32$,
already considered in our previous work [5, 6] are studied here over the whole
range of Reynolds numbers where the pattern exists at the chosen numerical
resolution, $R\in[R_{\rm g},\,R_{\rm t}]=[275,345]$. Orientation fluctuations
are systematically found close enough to $R_{\rm t}$, see Cases 1 & 2 below.
In addition, wavelength fluctuations can take place when the size of the
system is too far away from resonating with the pattern’s elementary cell
$\lambda_{x}^{\rm opt}\times\lambda_{z}^{\rm opt}$, where ‘opt’ means
‘optimal’, in a sense to be defined below in §3.1. Orientation and wavelength
fluctuations are observed at $R=315$ for $L_{x}=128$, $L_{z}=84$ and $90$, and
for $L_{x}=110$, $L_{z}=84$, meaning that both $|n_{z}|=1$ and $|n_{z}|=2$ are
competitive for $L_{z}=84$ or $L_{z}=90$. In contrast, lifetimes of single
mode patterns are extremely long for $L_{z}<84$ and $L_{z}>90$, meaning that
$L_{z}<84$ is optimal for $|n_{z}|=1$ and $L_{z}>90$ is optimal pour
$|n_{z}|=2$. Orientation and wavelength fluctuations are similarly present in
several other circumstances, at lower Reynolds number $R=272$ and $R=275$ for
$L_{x}\times L_{z}=110\times 32$, as well as at $R=290$ for $L_{z}=48$ and
$L_{x}=80$ or at $R=330$ for $L_{x}=90$, $140$, and $150$.
#### Case 1: $L_{x}=128$, $L_{z}=84$, $R=315$, wavelength fluctuations.
Several experiments under the same protocol have been performed, using
different initial conditions. Integration times ranged from $5\,10^{4}$ to
$10^{5}$ $h/U$. A large enough ensemble of lifetimes has been sampled, both
for $|n_{z}|=1$ and $|n_{z}|=2$, allowing us to compute the corresponding
order parameters $M$ – the conditional time averages of $m(t)$ as defined in
(1) – with sufficient accuracy. Snapshots corresponding to this aspect ratio
are displayed in Fig. 1, a typical part of the corresponding time series is
shown in Fig. 3. For $|n_{z}|=1$ and $|n_{z}|=2$, we obtain $M_{1}=0.033\pm
0.001$ and $M_{2}=0.038\pm 0.001$, respectively. From the lifetime
distributions in Fig 5, we get $\tau_{1}=8100\pm 200$ and $\tau_{2}=3800\pm
100$. The fact that $M_{1}<M_{2}$ is not surprising and is understood in term
of optimal wavelength (§3.1, $\lambda^{\rm opt}_{z}\simeq 39$ at $R=315$ [6]).
The reason why one has $\tau_{1}>\tau_{2}$ is however not clear.
#### Case 2: $L_{x}=110$, $L_{z}=32$, variable $R$, orientation fluctuations.
A thorough account of the behaviour of $M$ and the re-entrance featureless
turbulence has been given in [6]. Here, lifetimes are computed for Reynolds
number ranging from $R=325$ to $R=340$. Below $R=325$, the lifetimes are so
long that a small number of events is observed despite the length of time
series used ($>2\,10^{5}$), which forbids the determination of $\tau$ as a
meaningful average (Fig. 10). Above $R=340$, a clear separation of time scales
is lacking, which now forbids the definition of lifetimes of individual
events, compare the two panels in Fig. 4.
Figure 7 displays the variation of the average lifetime $\tau$ with $R$,
showing that it increases by a factor of 10 as $R$ decreases from $R=340$,
which is somewhat below $R_{\rm t}=355$, down to $R=325$ below which it is too
long to be measured reliably. “Error bars” suggested by up and down triangles
in Fig. 7 correspond to the unbiased standard deviation of the three estimates
for $\tau$ mentioned earlier.
Figure 7: Mean lifetime $\tau$ as a function of $R$ for $L_{x}\times
L_{z}=110\times 32$ (log scale).
## 3 Conceptual framework and application to DNS results
### 3.1 The Landau–Langevin model
Prigent et al. proposed to consider the turbulent bands as resulting from a
conventional pattern formation problem described at lowest order, from
symmetry arguments, by two coupled cubic Ginzburg–Landau equations, one for
each band orientation, further subjected to noise featuring the turbulent
background above $R_{\rm t}$. The slowly varying part of the velocity field
component away from the laminar profile can be written as
$u_{x}=A_{+}(\tilde{x},\tilde{z},\tilde{t})e^{ik_{x}^{\rm c}x+ik_{z}^{\rm
c}z}+A_{-}(\tilde{x},\tilde{z},\tilde{t})e^{ik_{x}^{\rm c}x-k_{z}^{\rm
c}z}+cc\,,$
where $A_{\pm}\in\mathbb{C}$ are the amplitude fields accounting for the two
modulation waves, and $\tilde{x}$, $\tilde{z}$ and $\tilde{t}$ are slow
variables [1]. Then, following this approach, we assume
$\tau_{0}\partial_{\tilde{t}}A_{\pm}=(\epsilon+\xi_{x}^{2}\partial_{\tilde{x}\tilde{x}}^{2}+\xi_{z}^{2}\partial_{\tilde{z}\tilde{z}}^{2})A_{\pm}-g_{1}|A_{\pm}|^{2}A_{\pm}-g_{2}|A_{\mp}|^{2}A_{\pm}+\alpha\zeta_{\pm}\,,$
(2)
the quantity $\epsilon=(R_{\rm t}-R)/R_{\rm t}$ measures the relative distance
to the threshold222The existence of a well defined threshold in this system is
attested by the behaviour of the turbulent fraction and spatially averaged
kinetic energy which display a marked change of slope at $R_{\rm t}$[6]
$R_{\rm t}$, $\tau_{0}$ is the ‘natural’ time scale for pattern formation,
$\xi_{x,z}$ are streamwise and spanwise correlation lengths, $g_{1}$ and
$g_{2}$ are the self-coupling and cross-coupling nonlinear coefficients, and
$\alpha$ the strength of the noise $\zeta_{\pm}$ supposed to be a centred
Gaussian process with unit variance. The strength $\alpha$ of the noise is
expected to grow smoothly with $R$, regardless of the existence of the pattern
since the local intensity of the turbulence is empirically not directly
correlated to the amplitude and phase of the modulation $A_{\pm}$. The tilde
variables describe the long-wave modulations to an ideal pattern with critical
wavelengths $\lambda_{x,z}^{\rm c}$ to which correspond critical wavevectors
$k_{x,z}^{\rm c}=2\pi/\lambda_{x,z}^{\rm c}$, the term critical referring to
the most unstable wave vector near $R=R_{\rm t}$. The systems that we consider
have periodic boundary conditions placed at distances $L_{x,z}$. Fourier
analysis then leads to characterise the pattern by wavevectors ${\bf
k}=(k_{x},k_{z})$, with $k_{x,z}=2\pi n_{x,z}/L_{x,z}$. It is assumed that the
wave numbers obtained during a given experiment will be the integers that will
be as close as possible of $n_{x,z}^{\rm c}=L_{x,z}/\lambda_{x,z}^{\rm c}$.
Furthermore, our systems can accommodate a small number of cells of size
$(\lambda_{x},\lambda_{z})$ so their modes are well isolated [1, Ch.4].
Assuming that a single pair $(n_{x},\pm n_{z})$ is involved, the partial
differential equation (2) is turned into an ordinary differential equation for
$A(n_{x},\pm n_{z})$ simply denoted $A_{\pm}\equiv A_{\pm}^{\rm
r}+iA_{\pm}^{\rm i}$, close enough to $R_{\rm t}$[6]:
$\tau_{0}\mbox{$\frac{\rm d}{{\rm
d}t}$}A_{\pm}=\tilde{\epsilon}A_{\pm}-g_{1}|A_{\pm}|^{2}A_{\pm}-g_{2}|A_{\mp}|^{2}A_{\pm}+\alpha\zeta_{\pm}\,,$
(3)
where $\tilde{\epsilon}=\epsilon-\xi_{x}^{2}\delta k_{x}^{2}-\xi_{z}^{2}\delta
k_{z}^{2}$ controlling the linear stability of these modes, is evaluated for
$\delta k_{x,z}=k_{x,z}-k_{x,z}^{\rm c}$ with the relevant $k_{x,z}=2\pi
n_{x,z}/L_{x,z}$, as well as the nonlinear coefficients $g_{1,2}$
($\in\mathbb{R}$ because the pattern does not drift, at least in the absence
of noise). Coefficient $\alpha$ is the effective strength of the noise
affecting the mode that we consider. Equation (3) can be written as deriving
from a potential:
$\tau_{0}\mbox{$\frac{\rm d}{{\rm d}t}$}A_{\pm}^{\rm
r,i}=-\frac{\partial\mathcal{V}}{\partial A_{\pm}^{\rm
r,i}}+\alpha\zeta_{\pm}\,,$
with
$\mathcal{V}=-\mbox{$\frac{1}{2}$}\tilde{\epsilon}\left(|A_{+}|^{2}+|A_{-}|^{2}\right)+\mbox{$\frac{1}{4}$}g_{1}\left(|A_{+}|^{4}+|A_{-}|^{4}\right)+\mbox{$\frac{1}{2}$}g_{2}|A_{+}|^{2}|A_{-}|^{2}\,.$
(4)
Usually, when making use of phenomenological equations such as (2), one relies
on values of critical wavevectors $k^{\rm c}$ that are computed once for all
from some linear stability theory and further introduced in the perturbation
expansions solving the nonlinear wavelength selection problem beyond the
threshold [13]. Here the theory is not developed enough to have such a
definition and such an evaluation of nonlinearly selected ‘optimal’
wavevectors far enough from threshold. Accordingly, in (3) we introduce values
of $\tilde{\epsilon}$ that do not make reference to some explicit computation
involving measured values of $\epsilon$ and $\xi_{x,z}$ but values that are
just estimates consistent with the empirically determined optimal wavelengths.
In the same way, we keep the cubic Landau expressions (3), neglecting higher
order terms that would introduce too many little-constrained parameters,
without deeper insight into the problem.
The stable fixed points of the deterministic part of (3) were shown to
correspond to the permanent state of the pattern and the additive noise term
seen to account for fluctuations quite well by solving the corresponding
Fokker–Planck equation [6]. The stationary probability distribution for the
moduli $|A_{\pm}|=A_{\pm}^{\rm m}$ was obtained in the form:
$\Pi(A_{+}^{\rm m},A_{-}^{\rm m})=Z^{-1}A_{+}^{\rm m}A_{-}^{\rm
m}\exp(-2\mathcal{V}/\alpha^{2})\,,\qquad Z=\int A_{+}^{\rm m}A_{-}^{\rm
m}\exp(-2\mathcal{V}/\alpha^{2})\,{\rm d}A_{+}^{\rm m}{\rm d}A_{-}^{\rm m}\,.$
(5)
The time behaviour of $A_{\pm}^{\rm m}$ is easily discussed by considering the
shape of $\mathcal{V}$ within the stochastic process framework. Two limiting
cases can be identified, depending on whether $\tilde{\epsilon}$ is
$\mathcal{O}(1)$ or $\ll 1$. In the first case, excursions from the
neighbourhood of the minima of $\mathcal{V}$ are rare; the lifetime of an
ordered episode can be defined as the average time necessary for the system to
go from the neighbourhood of a minimum to the potential’s saddle. It is
expected to increase with the height of the potential barrier, i.e. as
parameter $\tilde{\epsilon}$ grows, and to fall off as $\alpha$ increases. The
lack of symmetry between the growth and the decay of a pattern has then a
clear explanation when $\tilde{\epsilon}$ is large: The growth corresponds to
the system falling from the neighbourhood of the saddle into one of the wells;
even in the presence of noise, this evolution is fast and mostly
deterministic. In contrast, the decay corresponds to the system slowly
climbing toward the saddle against the deterministic flow, driven by the sole
effect of noise. In the opposite limit, when $\tilde{\epsilon}$ approaches
zero, the definition of a lifetime no longer makes sense since the
characteristic times for growth and decay become of the same order of
magnitude.
### 3.2 Orientation lifetimes from the model
Orientation changes and associated lifetimes are analysed in terms of first
passage time and escape from metastable states [14]. The distribution of
lifetimes is anticipated to be Poissonian as expected from a jump process
controlled by an activation “energy”. In a simplified one-dimensional version
of potential $\cal V$ [14, Ch.11,§2,6–7], if the well is deep enough, within a
parabolic approximation the mean escape time, the average time necessary to go
from a well to another, is given by
$\tau/\tau_{0}=\frac{2\pi}{\sqrt{\mathcal{V}_{\rm
w}^{\prime\prime}\,|\mathcal{V}_{\rm
s}^{\prime\prime}|}}\exp\left(2\frac{\mathcal{V}_{\rm s}-\mathcal{V}_{\rm
w}}{\alpha^{2}}\right)\,,$ (6)
where ‘w’ stands for ‘well’ and ‘s’ for ‘saddle’; $\mathcal{V}_{\rm w,s}$ are
the values of the potentials at the corresponding points and $\mathcal{V}_{\rm
w,s}^{\prime\prime}$ the values of the second order derivatives of the
potential with respect to the variable at these points. The derivation of this
formula shows that $\tau$ is dominated by the time spent around the saddle. In
our two dimensional system with potential (4), at lowest order in $\alpha$ the
coordinates of the well and saddle points are:
$(A_{\pm}^{\rm w},A_{\mp}^{\rm
w})=\left(\sqrt{\tilde{\epsilon}/g_{1}},\,0\right)\qquad\mbox{and}\qquad(A_{+}^{\rm
s},A_{-}^{\rm
s})=\left(\sqrt{\tilde{\epsilon}/(g_{1}+g_{2})},\,\sqrt{\epsilon/(g_{1}+g_{2})}\right)$
and the corresponding values of the potential:
$\mathcal{V}_{\rm
w}=-\tilde{\epsilon}^{2}/4g_{1}\qquad\mbox{and}\qquad\mathcal{V}_{\rm
s}=-\tilde{\epsilon}^{2}/2(g_{1}+g_{2})\,.$
The second derivatives have to be replaced by the eigenvalues of the Hessian
matrix of $\cal V$ computed at these points:
$H_{\rm w}=\left(\begin{matrix}2\tilde{\epsilon}&0\\\
0&\tilde{\epsilon}(g_{2}/g_{1}-1)\end{matrix}\right)\qquad\mbox{and}\qquad
H_{\rm
s}=\frac{2\tilde{\epsilon}}{g_{2}+g_{1}}\left(\begin{matrix}g_{1}&g_{2}\\\
g_{2}&g_{1}\end{matrix}\right)\,.$
At point ‘w’, $H_{\rm w}$ is diagonal and the eigen-direction pointing to
point ‘s’ has eigen-value $\tilde{\epsilon}(g_{2}/g_{1}-1)$. At point ‘s’,
$H_{\rm s}$ is diagonal in the basis $\\{(1,1),(1,-1)\\}$ and has eigenvalues
$\\{2\tilde{\epsilon},\,2\tilde{\epsilon}(g_{1}-g_{2})/(g_{1}+g_{2})\\}$. The
unstable eigen-direction correspond to the second one which is negative
($g_{2}>g_{1}$). Inserting these values in (6), we obtain:
$\tau/\tau_{0}=\frac{2\pi\sqrt{2}}{\tilde{\epsilon}}\frac{\sqrt{g_{1}(g_{1}+g_{2})}}{g_{2}-g_{1}}\exp\left(\frac{\tilde{\epsilon}^{2}}{2\alpha^{2}g_{1}}\,\frac{g_{2}-g_{1}}{g_{1}+g_{2}}\right)\,.$
(7)
In its exponential factor, this formula points out an “energy” scale
$\tilde{\epsilon}^{2}/g_{1}$ to be compared to the characteristic noise energy
$\alpha^{2}$ which play the role of the Boltzmann energy in thermal problems.
It also shows that, especially when $g_{2}$ is larger but comparable to
$g_{1}$, the noise energy has to remain small enough because the parabolic
approximation which underlies the formula assumes sufficiently deep wells. The
main difference between the one and two dimensions cases are in the shape of
the “energy landscape”, corrections are therefore expected to be
multiplicative and not depend on the value of $\tilde{\epsilon}$.
### 3.3 Simulation of the model
For the deterministic part of equation (3) we use a simple first-order
implicit Euler algorithm, while the additive noise $\alpha\zeta(t)$ is treated
as a Gaussian random variable with standard deviation $\alpha\sqrt{dt}$ at
each time step. The model is integrated over a range of $\tilde{\epsilon}$,
given $g_{1}$, $g_{2}$ (several values), and $\alpha$ (assumed constant). The
time series of $|A_{\pm}|^{2}$ displayed in Fig. 8 are indeed reminiscent of
those obtained by numerical integration of Navier–Stokes equations (Figs. 3
and 4): pure states at the bottom of the wells correspond to $|A_{-}|^{2}$
fluctuating around 0 and $|A_{+}|^{2}$ away from 0, or the reverse.
Figure 8: Typical time series from model (3), for a set of parameters
corresponding to the Navier–Stokes DNS, $\tilde{\epsilon}=0.05$, $g_{1}=60$,
$g_{2}=120$, $\alpha=0.002$ [6].
Large excursions can lead to a change of the dominant orientation. These
excursions are more likely to occur when $\tilde{\epsilon}$ is decreased.
Lifetimes have been computed in the same way as for Fig. 6. The dependence of
the mean lifetimes on thresholds $s_{1,2}$ is displayed in Figure 9.
Figure 9: Mean lifetime $\tau$, extracted from the model as a function of
$s_{1}$, for $s_{2}=0.0163$, $0.0193$, and $0.0225$ (left) and as a function
of $s_{2}$ for several $s_{1}$ ranging from $0.0784$ to $0.1296$ (right).
$\tilde{\epsilon}=0.075$, $g_{1}=1$, $g_{2}=2$, $\alpha=0.002$.
A neat plateau is obtained for $s_{1}\in[0.075,0.115]$ for different values of
$s_{2}$ (Fig. 9, left), which corresponds to the trajectory getting away from
the saddle. Extremes values of $s_{1}$ lead to bad estimates of $\tau$ for the
same reasons as stated before. As to threshold $s_{2}$, an orientation change
has taken place when the trajectory goes beyond the saddle, while a pure state
corresponds to one amplitude large and the other at the noise level. We have
thus to detect the change from large to small for one or the other amplitude.
It is extremely difficult to detect the precise passage at the saddle, since
it is dominated by the time spent in that region, contribution from the two
sides of the saddle point having the same weight. On the contrary the passage
from one state to another leaves no doubt as to its definition. Therefore, we
prefer to compute the mean first passage time from one well to another, which
is obtained in our simulation by the extrapolation at $s_{2}=0$. Approximating
the curves in Fig. 9 (right) by linear functions $\tau=a(1-bs_{2})$, one finds
that the slope $b$ depends on $\tilde{\epsilon}$ and $s_{1}$ only weakly; the
value of $\tau$ retained is then the one given by the extrapolation $s_{2}=0$,
i.e. coefficient $a$. Improving the definition of $\tau$ with approximations
better than the linear one has not been found necessary. The general
expression of the mean first passage time gives no hint as to the quantitative
behaviour on the distance to the second well $s_{2}$, although it shows the
same qualitative behaviour as seen in Figures 6 (right) and 9 (right). In
Figure 10 (semilog coordinates), the lifetime $\tau$ measured in this way is
compared to the asymptotic expression from the theory (7) as a function of
$\tilde{\epsilon}$.
Figure 10: $\log(\tau)$ as a function of $\tilde{\epsilon}$ for $g_{1}=60$,
$g_{2}=120$, $\tilde{\alpha}=0.002$; the model was integrated over $10^{5}$
time units.
It can be seen that the asymptotic formula is not valid for the smallest
values of $\tilde{\epsilon}$, when the wells are not longer deep enough for
the approximation to be valid, similarly to what is found in the DNS close to
$R_{\rm t}$. The values given by this formula for small values of
$\tilde{\epsilon}$, especially around and below the minimum it predicts at
$\tilde{\epsilon}=\alpha\sqrt{g_{1}(g_{2}+g_{1})/(g_{2}-g_{1})}$, cannot be
trusted. For large $\tilde{\epsilon}$, the lifetime computed from the
simulation saturates because it becomes of the order of the total integration
time so that only a few events smaller than this total time can be recorded.
Accordingly the long time tail of the distribution is badly sampled with an
under-representation of lifetimes larger than the average expected from the
theory. In Fig. 10, the numerical and the asymptotic estimates of the mean
lifetimes are seen to differ by a constant of order unity, which is attributed
to the one-dimensional character of the approximation.
### 3.4 Generalisation
This approach can be extended to wavelength fluctuations. When the size
$(L_{x},L_{z})$ of the system is such that it ‘hesitates’ between two pairs of
modes $(n_{x},\pm n_{z})$ and $(n^{\prime}_{x},\pm n^{\prime}_{z})$, we
introduce two supplementary amplitudes $A(n^{\prime}_{x},\pm n^{\prime}_{z})$
that we denote $B_{\pm}$ for short and, extending notations straightforwardly
with primes for quantities related to $B_{\pm}$, we arrive at:
$\displaystyle\tau_{0}\mbox{$\frac{\rm d}{{\rm
d}t}$}A_{\pm}=\tilde{\epsilon}A_{\pm}-g_{1}|A_{\pm}|^{2}A_{\pm}-g_{2}|A_{\pm}|^{2}A_{\pm}+g_{3}(|B_{\pm}|^{2}+|B_{\mp}|^{2})A_{\pm}+\alpha\zeta_{\pm}\,,$
(8) $\displaystyle\tau_{0}\mbox{$\frac{\rm d}{{\rm
d}t}$}B_{\pm}=\tilde{\epsilon}^{\prime}B_{\pm}-g^{\prime}_{1}|B_{\pm}|^{2}B_{\pm}-g^{\prime}_{2}|B_{\pm}|^{2}B_{\pm}+g^{\prime}_{3}(|A_{\pm}|^{2}+|A_{\mp}|^{2})B_{\pm}+\alpha^{\prime}\zeta^{\prime}_{\pm}\,,$
(9)
where $\tilde{\epsilon}$ and $\tilde{\epsilon}^{\prime}$ as well as the
nonlinear coupling constants $g_{1,2,3}$, $g^{\prime}_{1,2,3}$ and even the
effective noise intensities $\alpha$, $\alpha^{\prime}$ may differ since they
relate to pure patterns with different $\delta k_{x,z}=k_{x,z}-k_{x,z}^{\rm
c}$. A first guess would be to assume the primed and non-primed variables
equal, which would bring us immediately back to the previous approach with an
effective potential, wells, saddles, and potential barriers, leading to
estimates for the different lifetimes involved.
It is not clear how the case of turbulence re-entrance (the intermittent
regime of [4]) would fit this framework but it is well described by a PDF with
three peaks [6] corresponding to a probability potential with three wells and
thus hopefully amenable to a similar treatment with a similar output.
These generalisations have not been worked out in detail numerically since
they introduces a discouragingly large number of parameters to be fitted
against the experiments and from which we would learn little, owing to their
phenomenological basis. Only the case involving a single pair of modes was
examined in §3.3 above, mostly in order to validate the procedure followed to
determine lifetimes in §2.3.
## 4 Summary and conclusion
In this paper, numerical simulations of the Navier–Stokes equation in plane
Couette flow configuration have been performed in a range of Reynolds numbers
where the transition to turbulence happens in the form of oblique bands.
Systems with sizes fitting a few elementary cells
$\lambda_{x}\times\lambda_{z}$ of the pattern have been considered. These
sizes are much larger than the minimal flow unit which allows the reduction of
the transition problem to a temporal process familiar to chaos theory [12].
Accordingly, the considered systems are able to display the first
manifestations of a genuinely spatiotemporal dynamics via patterning.
Following the patterns in time, we showed that they experience orientation and
wavelength fluctuations in the upper part of the range of transitional
Reynolds numbers $[R_{\rm g},R_{\rm t}]$. A systematic procedure to detect the
start and the termination of well-oriented episodes was defined, leading to
the observation of exponentially decreasing distributions for their lifetimes
(Fig. 5).
A consistent interpretation scheme was then provided by adapting the noisy
Ginzburg–Landau model proposed in [2] to our case, transforming the original
stochastic PDE into a Landau–Langevin stochastic ODE. Besides supporting the
procedure used to determine lifetimes, the approach directly leads to the
determination of probability distributions for the patterned states from the
shape of the potential obtained by solving the corresponding Fokker–Planck
equation, as already suggested in [2, Fig. 19]. The variation of the patterns’
mean lifetimes is thus linked to the relative distance to threshold and noise
intensity through an asymptotic formula involving the “energy” barrier between
wells corresponding to the different well-oriented states in competition.
Ingredients in the relative distance to threshold $\tilde{\epsilon}$ which is
a function of both the Reynolds number and the optimal wavelength, are amply
sufficient to explain most of the dependance of the mean lifetimes as
functions of $R$, $L_{x,z}$, and the spontaneous appearance of defects
separating patches of well-oriented patterns close enough to $R_{\rm t}$ in
larger aspect-ratio systems as illustrated in Fig. 11 and seen in the
experiments [2].
Figure 11: Orientation defect spontaneously appearing in the flow for $R=340$
in a domain of size $L_{x}\times L_{z}=660\times 48$.
In order to explain the occurrence of exponentially decreasing lifetime
distributions, the theory of dynamical systems appeals to the sensitivity to
initial conditions of trajectories visiting a homoclinic tangle [12]. Here,
the modelling that fits well our observations implies that exponential
distributions arise from some jump random process [14]. As soon as the size of
the system is much larger that the minimal flow unit (for which the temporal
behaviour inherent in low dimensional dynamical systems is relevant), a
spatiotemporal perspective becomes in order, and the jumps in question can
easily be associated to the local chaotic dynamics of pieces of streaks and
streamwise vortices involved in the self sustaining process of turbulence
[15]. This local chaotic dynamics would then be responsible for the wandering
of the global system through some “energy” landscape with wells and saddles.
With system sizes of the order of the elementary pattern cell
$\lambda_{x}\times\lambda_{z}$, this wandering amounts to orientation and/or
wavelength changes. Extending these views to larger systems would then explain
the statistical properties of fluctuating laminar-turbulent patches observed
in the upper transitional range close enough to $R_{\rm t}$ [2].
Whereas the origin of the noise introduced in the description is
understandable from chaos at the local (microscopic) scale, it remains however
to understand why the coexistence of laminar and turbulent flow takes the form
of oblique bands at the global (macroscopic) scale, i.e. to justify the
Ginzburg–Landau approach from the first principles rather than taking it as an
educated phenomenological guess.
## References
* [1] For an introductory review see, e.g. P. Manneville, Instabilities, Chaos and Turbulence, 2nd edition (Imperial College Press, 2010).
* [2] A. Prigent, G. Grégoire, H. Chaté, O. Dauchot, Long-wavelength modulation of turbulent shear flow, Physica D 174 100–113 (2003).
* [3] Y. Duguet, P. Schlatter, D.S. Henningson Formation of turbulent patterns near the onset of transition in plane Couette flow, J. Fluid Mech. 650, 119–129 (2010).
* [4] D. Barkley, L. Tuckerman, Computational study of turbulent laminar patterns in Couette flow, Phys. Rev. Lett. 94 014502 (2005).
* [5] P. Manneville, J. Rolland, On modelling transitional turbulent flows using under-resolved direct numerical simulations, Theor. Comput. Fluid Dyn. in press, DOI : 10.1007/s00162-010-0215-5.
* [6] J. Rolland, P. Manneville, Ginzburg–Landau description of laminar-turbulent oblique bands in transitional plane Couette flow, Eur. Phys. J. B, submitted.
* [7] R. Graham, Hydrodynamics fluctuations near the convection instability, Phys. Rev. A, 10, 1762-1784, (1974).
* [8] P. Hohenberg, J. Swift, Effects of additive noise at the onset of Rayleigh-Bénard convection, Phys. Rev A, 46, 4773–4785 (1992).
* [9] M. Scherer, G. Ahler, F. Hörner, I. Rehberg, Deviation from linear theory for fluctuations below the super-critical primary bifurcation to electroconvection, Phys. Rev. Lett. 85 3754–3760 (2000).
* [10] J. Gibson, http://www.channelflow.org/.
* [11] D. Barkley, O. Dauchot, L. Tuckerman, Statistical analysis of the transition to turbulent-laminar banded patterns in plane Couette flow, Journal of Physics, Conference Series 137 (2008) 012029.
* [12] B. Eckhardt, H. Faisst, A. Schmiegel, T.M. Schneider, Dynamical systems and the transition to turbulence in linearly stable shear flows, Phil. Trans. R. Soc. A 366 1297–1315 (2008).
* [13] M.C. Cross, P.C. Hohenberg, Rev. Mod. Phys 65 (1993) 851–1112
* [14] N.G Van Kampen, Stochastic processes in physics and chemistry, North-Holland (1990).
* [15] F. Waleffe, On a self-sustaining process in shear flows, Phys. Fluids 9 883–900 (1997).
|
arxiv-papers
| 2011-01-13T09:36:24 |
2024-09-04T02:49:16.406874
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Joran Rolland, Paul Manneville",
"submitter": "Joran Rolland",
"url": "https://arxiv.org/abs/1101.2513"
}
|
1101.2616
|
CECS-PHY-10/15
Accelerating black hole in 2+1 dimensions
and 3+1 black (st)ring
Marco Astorino***marco.astorino@gmail.com
Instituto de Física,
Pontificia Universidad Católica de Valparaíso
and
Centro de Estudios Científicos (CECS), Valdivia,
Chile
Abstract
A C-metric type solution for general relativity with cosmological constant is
presented in 2+1 dimensions. It is interpreted as a three-dimensional black
hole accelerated by a strut. Positive values of the cosmological constant are
admissible too.
Some embeddings of this metric in the 3+1 space-time are considered:
accelerating BTZ black string and a black ring where the gravitational force
is sustained by the acceleration.
## 1 Introduction
Beside the fact that general relativity in three space-time dimensions is
trivial, because the lack of dynamical degrees of freedom, still it admits
black holes solutions. In fact the static and rotating black hole in 2+1
dimension is very well known as remarkably discovered by Bañados, Teitelboim
and Zanelli (BTZ) in [1]. Also well known is the charged and the electro-
rotating flavours [2]. Taking inspiration from the four dimensional C-metric
(for references see [3], [4] and [5]), we are here interested to apply
acceleration to the three dimensional black hole (section 2). This is done
with the motivation that such acceleration, and the physical object that
produce it, could play an important role in 3+1 general relativity solutions
with non-trivial topology, as shown in section 3. Our supposition is based on
analogy with the five dimensional case, as explained in [6]: to forge a black
ring one has to bend a string built with the cross product of a line and a
Schwarzschild black hole, and then balance the gravitational self attraction
which makes the string self collide. If one would do so in four dimensions,
the natural candidate to start with is the cross product of a line and the
three dimensional black hole, thus the cosmological constant is necessary.
Cosmological constant is also prominent in introducing a length scale to
obtain different scales for the radius and the thickness of the ring. In [6]
the absence of black (st)rings in vacuum 4D general relativity is attributed
to the lack of such a scale.
## 2 Accelerating black hole in 2+1 dimensions
We begin considering the Einstein-Hilbert action for standard general
relativity, with generic cosmological constant $\Lambda$, in three
dimensions111We have set, for convenience, the gravitational constant $G=1/8$:
$I[g_{\mu\nu}]=\frac{1}{2\pi}\int d^{3}x\sqrt{-g}(R-2\Lambda)$ (2.1)
Extremization of the action with respect the metric $g_{\mu\nu}$ yields the
Einstein field equations:
$R_{\mu\nu}-\frac{1}{2}R\ g_{\mu\nu}+\Lambda g_{\mu\nu}=0$ (2.2)
Being inspired by the four dimensional C-metric [5], we start proposing a
similar ansatz but in $2+1$ dimensions:
$ds^{2}=\frac{1}{(1+\alpha
r\cos\theta)^{2}}\left[-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+\frac{r^{2}d\theta^{2}}{g(\theta)}\right]$
(2.3)
a general solution for (2.2) in terms of the unknown functions $f(r)$ and
$g(\theta)$ can be find:
$f(r)=c_{0}+c_{1}\ r+c_{2}\ r^{2}\ \ ,\qquad
g(\theta)=\frac{c_{0}\alpha^{2}\cos^{2}(\theta)-c_{1}\alpha\cos(\theta)+c_{2}+\Lambda}{\alpha^{2}(\cos^{2}(\theta)-1)}$
(2.4)
Where $c_{0},c_{1},c_{2}$ are arbitrary integration constant. This metric
describes locally a constant curvature space-time: $R^{\mu\nu}_{\ \
\rho\sigma}=\Lambda\ (\delta^{\mu}_{\ \rho}\delta^{\nu}_{\
\sigma}-\delta^{\mu}_{\ \sigma}\delta^{\nu}_{\ \rho})$. In order to have a
significant $\alpha\rightarrow 0$ limit we select a particular metric
choosing, from (2.4), the following integration constants:
$c_{0}=1-m\quad,\qquad c_{1}=0\quad,\qquad c_{2}=\alpha^{2}(m-1)-\Lambda$
After rescaling the $\theta$ coordinate one obtains the solution:
$\displaystyle ds^{2}=\frac{1}{\left[1+\alpha
r\cos\left(\theta\sqrt{1-m}\right)\right]^{2}}\bigg{\\{}$
$\displaystyle-\Big{[}1-m+r^{2}\big{[}\alpha^{2}(m-1)-\Lambda\big{]}\Big{]}dt^{2}$
(2.5)
$\displaystyle+\frac{dr^{2}}{1-m+r^{2}\big{[}\alpha^{2}(m-1)-\Lambda\big{]}}+r^{2}d\theta^{2}\bigg{\\}}$
The coordinates $(t,r,\theta)$ are chosen to be polar, so their range is
$-\infty<t<\infty$, $r\geq 0$, $-\pi\leq\theta\leq\pi$; since $\theta$ is an
angular coordinate, the points $\theta=-\pi$ and $\theta=\pi$ can be
considered identified. Other choises for the coordinate ranges and
identification are possible and give rise to different spacetime geometries;
our choice is motivated by the will to model an accelerating 2+1 black hole.
In fact when the parameter $\alpha$ is imposed to be null one has:
$ds^{2}=-\big{(}1-m-\Lambda\ r^{2}\big{)}dt^{2}+\frac{dr^{2}}{1-m-\Lambda\
r^{2}}+r^{2}d\theta^{2}$ (2.6)
which is precisely the static BTZ black hole metric in the gauge where the
ground state is the (A)dS space-time. We prefer this form of the metric
respect to the one of [1] because when the mass222Computed respect to the
(A)dS background, as done for instance in [7] $m\rightarrow 0$ one does
recover from (2.6), not just an asymptotically (A)dS space-time as in [1], but
exactly the standard vacuum (A)dS space-time. Similarly this is what happens
in 3+1 (or higher) dimension, for instance to the Schwarzschild-(A)dS black
hole. The fact that in this gauge when $m\in[0,1)$ naked singularity occurs
(while black holes for $m\in[1,\infty)$) it’s a typical feature of Chern-
Simons gravity or odd-dimensional Lovelock theories, such is three-dimensional
general relativity.333Recently $m\in[0,1)$ states of the black hole spectrum
were physically dignified as topological defects in [8], representing
particles in $AdS_{3}$ (see also [9]) .
The metric (2.5) describes, for $m\geq 1$ an accelerating black hole whose
event horizon is located at:
$r_{h}=\sqrt{\frac{m-1}{\alpha^{2}(m-1)-\Lambda}}$
A physical interpretation to the $\alpha$ parameter can be given, following
the argument of [3] and [4]. Usually, as for the 3+1 C-metric, one considers
the weak field limit of the solution (2.5), that is $m\rightarrow 0$:
$ds^{2}=\frac{1}{\big{[}1+\alpha
r\cos(\theta)\big{]}^{2}}\left\\{-\big{[}1-r^{2}(\alpha^{2}+\Lambda)\big{]}dt^{2}+\frac{dr^{2}}{1-r^{2}(\alpha^{2}+\Lambda)}+r^{2}d\theta^{2}\right\\}$
(2.7)
The weak field limit is usefull because, in this case, black holes can be
considered as test particles and cease to deform the spacetime and inertial
frames around them.444A metric similar to (2.7) is studied by [10] in the case
of null cosmological constant. The 3D timelike worldlines $x^{\mu}(\lambda)$
of an observer with $r=\bar{r}=constant$ and $\theta=0$ can be obtained by the
property $u_{\mu}u^{\mu}=-1$ of the 3-velocity defined by
$u^{\mu}=dx^{\mu}/d\lambda$:
$x^{\mu}(\lambda)=\left[\frac{1+\alpha\bar{r}\cos(\theta)}{\sqrt{1-\bar{r}^{2}(\alpha^{2}+\Lambda)}}\lambda,\bar{r},0\right]\
\ \ ;$
where $\lambda$ is the proper time of the observer. Then the magnitude $a$ of
the 3-acceleration, $a^{\mu}=(\nabla_{\nu}u^{\mu})u^{\nu}$, for this kind of
observer results
$|a|=\sqrt{a_{\mu}a^{\mu}}\ \Big{|}_{\bar{r}=0}=\alpha$ (2.8)
Since $a_{\mu}u^{\mu}=0$, the value $|a|$ is also the magnitude of the
2-acceleration in the rest frame of the observer. From eq. (2.8) we achieve
the conclusion that the origin of the metric (2.7), $\bar{r}=0$ is being
accelerated with an uniform acceleration whose value is precisely given by the
constant $\alpha$, so (2.7) is nothing but the accelerating (A)dS space-time.
Outside of the weak field limit similar results can also be obtained:
$a_{\mu}a^{\mu}\ |_{\bar{r}=0}=\alpha^{2}\ (1-m)$ for $m\neq 1$, while
$a_{\mu}a^{\mu}\ |_{\bar{r}=0}=-\Lambda$ for $m=1$.
Thus now we can interpret the BTZ metric (2.6) as the non-accelerating limit
of the (2.5). Thanks to this limit the parameter $m$ which appears in (2.5)
can be interpreted as the mass parameter of our solution. Of course in case of
non null acceleration this does not coincide with the mass of the metric
(2.5), but is somehow related to the mass. Actually is not clear how to
compute energy in this class of accelerating space-time, neither in standard
3+1 dimensions, because of the non trivial asymptotic. So usually in the
literature (see [5]) the mass value is estimated by thermodynamical argument
as follows: The first law of black hole thermodynamics states that
$dM=TdS=\frac{k}{8\pi G}dA$; while the surface gravity $k$ is defined by
$k^{2}=-\frac{1}{2}\nabla^{\mu}\chi^{\nu}\nabla_{\mu}\chi_{\nu}$ where
$\chi^{\mu}$ is the killing vector $\partial_{t}$:
$k=r_{h}\left[(m-1)\alpha^{2}-\Lambda\right]\bigg{|}_{m=1+\frac{\Lambda
r_{h}^{2}}{r_{h}^{2}\alpha^{2}-1}}=\frac{\Lambda
r_{h}}{r_{h}^{2}\alpha^{2}-1}$
The area of the accelerating black hole horizon $A$ is given by:
$A=\int_{-\pi}^{\pi}\sqrt{g_{\theta\theta}}\>\bigg{|}_{\\!\\!\begin{array}[]{l}\scriptscriptstyle
r=r_{h}\\\\[-3.61664pt] \scriptscriptstyle t=\hbox{\tiny
const.}\end{array}}\\!\\!\\!\\!\ d\theta\ =\
\frac{4}{\sqrt{\Lambda}}\arctan\left[\sqrt{\frac{\alpha r_{h}-1}{\alpha
r_{h}+1}}\tanh\left(\frac{\pi}{2}\sqrt{\frac{\Lambda
r_{h}^{2}}{r_{h}^{2}\alpha^{2}-1}}\right)\right]$ (2.9)
So the mass estimation is, up to a integration constant:
$M=\int
dM=\int\frac{2\sqrt{\Lambda}}{\pi(\alpha^{2}r_{h}^{2}-1)^{3/2}}\frac{\alpha
r_{h}\sinh\left(\pi\sqrt{\frac{\Lambda
r_{h}^{2}}{\alpha^{2}r_{h}^{2}-1}}\right)-\pi\sqrt{\frac{\Lambda
r_{h}^{2}}{\alpha^{2}r_{h}^{2}-1}}}{1+\alpha
r_{h}\cosh\left(\pi\sqrt{\frac{\Lambda
r_{h}^{2}}{\alpha^{2}r_{h}^{2}-1}}\right)}dr_{h}$
For small values of the acceleration parameter $\alpha<<1$, thus, in practice
we restrict only to the $\Lambda<0$ sector, this integral can be evaluated:
$M=-\Lambda r_{h}^{2}+2\alpha\left[\frac{r_{h}}{\pi^{2}}\cos\left(\pi
r_{h}\sqrt{\Lambda}\right)+\frac{\Lambda\pi^{2}r_{h}^{2}-1}{\sqrt{\Lambda}\pi^{3}}\sin\left(\pi
r_{h}\sqrt{\Lambda}\right)\right]+O(\alpha^{2})$
Evidently in case of null acceleration this results is coherent with the BTZ
one: $M[r_{h}]=m[r_{h}]+k_{0}$. Note that the integration constant $k_{0}$ is
background dependent, $k_{0}=0$ for the AdS background.
The main difference of this three dimensional accelerating black hole,
compared with the four dimensional C-metric, is the absence of a pure
acceleration horizon. In fact the effect of the acceleration merges with the
cosmological constant pressure to give an unique event horizon. This happens
because also the standard three dimensional event horizon is just that of an
accelerated observer, until one identifies $\theta$ as an angular coordinate
[1]. The peculiar combination between $\Lambda$ and $\alpha$ has the
remarkable consequence that also positive values for the cosmological constant
are admissible in order to get a black hole configuration. This is something
unexpected since even a no-go theorem (see [11]) is found to justify the lack
of a 2+1 black hole, in standard general relativity, for positive cosmological
constant. In our case the no-go theorem is circumvented in one of the
hypothesis of regularity of the horizon: the horizon of the accelerating
solution (2.5) is continuous but not smooth as required in [11]. It is worth
to note that the Riemannian curvature tensor remains locally constant
everywhere, but in fact in $\theta=\pm\pi$ the metric is not differentiable
because of an angular singularity, a common feature which characterises this
kind of accelerating black holes. Here the singularity is not a conical one
due to a standard deficit angle as the four dimensional case, this may appear
as another difference. But it is just because otherwise in one dimension less
there would be no room for a strut or a string.
Figure 1: Polar plot ($\bar{r},\theta$) of $\bar{r}=$constant radial curves
embedded in the plane $\mathbb{E}^{2}$. When $\bar{r}=r_{h}$ they illustrate
horizon deformations for various values of the mass: $m=0$, $0<m<1$, $m=1$ and
$m>1$.
In figure 1 are polar plotted the shapes of the horizon, for various values of
the mass parameter $m$, as embedded in the two dimensional euclidean plane
$\mathbb{E}^{2}$. Note that the sharp vertex occurring for $0<m<1$ and $m>1$
is an artifact of the embedding, in the real curved space-time there are no
sharp vertexes. In fact the two vectors normal to the horizon in $\theta=\pi$
and $\theta=-\pi$, denoted by $n^{\mu}_{(\pm)}$, are parallels:
$n^{\mu}_{(+)}\ n_{(-)\mu}=1$. Nevertheless the angular singularity in the
conformal factor persist as pointed out by the undifferenciability of the
metric in $\theta=\pm\pi$ (for $m\neq 0,1$). It is usually associated to the
presence of a semi-infinite cosmic string, or a strut, which is pulling the
black hole along the $\theta=\pi$ axis. In this picture the tension of the
string $\tau$ is responsible for the acceleration of the 2+1 dimensional black
hole. Thanks to the Israel junction condition it is possible to compute the
strength of the strut’s force, geometrically due by a jump in the extrinsic
curvature $\mathcal{K}_{\mu\nu}$:
$\big{[}\mathcal{K}_{\mu\nu}\big{]}^{\theta=\pi}_{\theta=-\pi}-h_{\mu\nu}\big{[}\mathcal{K}\big{]}^{\theta=\pi}_{\theta=-\pi}=-\pi
h_{\mu\nu}\tau$ (2.10)
where $h_{\mu\nu}$ is the induced metric on the $\theta=\pm\pi$ surface,
$\mathcal{K}=\mathcal{K}_{\mu\nu}h^{\mu\nu}$ and the tension is
$\tau=-2\alpha\ \sqrt{m-1}\ \sinh(\pi\sqrt{m-1}).$
The negativity of $\tau$ indicate that we are dealing with a strut that is
pushing the black hole, rather than a pulling string. As can be read from the
metric (2.5) or perceived from figure 1, the strut tension $\tau$ vanishes for
$m=0,1$, where the metric acquire regularity. The metric (2.5) is supported by
the surface stress-energy tensor that can be extract from (2.10) and, for sake
of precision, it should be better added in the equation of motion (2.2).
The casual structure and the Carter-Penrose diagram is similar to the BTZ one,
although now the casual singularity in $r=0$ has an acceleration $\alpha$.
A rotating and accelerating metric can be obtained by an improper boost in the
($t,\theta$) plane of (2.5) as explained in [2]. Since in three dimensions
there is no room for rotating around the strut axis as in four dimensions, the
only way to pursue rotation is have the strut rotate with the black hole. As
expected that metric reduces to the (2.5) when the rotation is null, while it
reduces to the rotating BTZ metric when the acceleration parameter vanish.
Since the metric (2.5) remains finite for $r\rightarrow\infty$ one may also
think to extend the spacetime also in the negative $r$ sector in order to
reach the conformal infinity for $r=-1/(\alpha\cosh(\theta\sqrt{m-1}))$. This
can be done in general for $\Lambda>0$. While for $\Lambda<0$, depending also
on the reciprocal values of the parameters $\Lambda,m$ and the angular
direction $\theta$, one may encounter another killing horizon corresponding to
the negative root of the $f(r)$ function before the conformal infinity. But
here we are not interested in that. Instead we are more interested on the
possible embeddings of this metric in the $3+1$ dimensional space time, as
presented in the next section.
## 3 Embeddings in 3+1
### 3.1 Accelerating BTZ black string
The most direct embedding in 3+1 dimensions of the 2+1 solution (2.5) is
obtained in the spirit of [12]. A string like object can be written by a
warped product of the three dimensional metric and a line element $dz$:
$\displaystyle ds^{2}=\frac{\cos^{2}(z)}{\left[1+\alpha
r\cos\left(\theta\sqrt{1-m}\right)\right]^{2}}\bigg{\\{}$
$\displaystyle-\left[1-m+r^{2}\left(\alpha^{2}(m-1)-\frac{\Lambda}{3}\right)\right]dt^{2}+$
(3.1)
$\displaystyle+\frac{dr^{2}}{1-m+r^{2}\left(\alpha^{2}(m-1)-\frac{\Lambda}{3}\right)}+r^{2}d\theta^{2}\bigg{\\}}\
+\ \frac{3}{\Lambda}\ dz^{2}$
To preserve the correct metric signature $(-,+,+,+)$ the z coordinate have to
be rotated to the imaginary plane when dealing with negative cosmological
constant: $z\rightsquigarrow iz$. The standard BTZ black string of [12] is
precisely recovered from (3.1) in the limit of $\alpha=0$ (of course in that
case just negative cosmological constant can be considered). But the periodic
dependence on the $z$ coordinate, for positive $\Lambda$ (and $\alpha\neq 0$),
suggests that in this case the string singularity can be though of closed
type. Thus the horizon’s topology has a toroidal geometric structure, but with
a couple of points in $z=\pm\pi/2$ where the horizon throat shrinks to
zero555Alternatively is possible to achieve the spherical topology, for
$\Lambda>0$, considering $-\pi/2<z<\pi$; the casual singularity would be a
$\pi$-length segment along the $z$-axes.. Anyway many other features are
shared between the static and accelerating black string. For instance the
space-time curvature still remains trivially constant as the 2+1-dimensional
case: $R^{\mu\nu}_{\ \ \rho\sigma}=\Lambda/3\ (\delta^{\mu}_{\
\rho}\delta^{\nu}_{\ \sigma}-\delta^{\mu}_{\ \sigma}\delta^{\nu}_{\ \rho})$.
In this case the acceleration is provided by a two space dimensions membrane,
which results a semi-infinite plane, at least for negative $\Lambda$.
### 3.2 Black Ring
Our main interest is in topological non-trivial solution, so we consider an
ansatz with toroidal base manifold, finally. Not the one of constant curvature
obtained by identification of a flat rectangle edges, but rather a doughnut
embedded in the full 3+1 space-time, whose metric and curvature is giving,
thinking $r$ and $t$ constant, as follows:
$ds^{2}=r^{2}d\theta^{2}+\left[R_{0}+r\cos\left(\theta\sqrt{1-m}\right)\right]^{2}d\phi^{2}\
,\qquad R^{\theta\phi}_{\ \
\theta\phi}=\frac{(1-m)\cos\left(\theta\sqrt{1-m}\right)}{r\left[R_{0}+r\cos\left(\theta\sqrt{1-m}\right)\right]}$
When $m\neq 0\neq 1$, but still $\theta=\pm\pi$ identified, the circular
section of the torus is deformed, respect to the smooth $m=0,m=1$ cases, in a
drop or a cardioid shaped section (see figure 1). Using this base manifold
with generic $m$ and the acceleration conformal factor of the previous section
2 one can obtain a ring-like solution of Einstein equations (2.2):
$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle\frac{1}{\left[1+\alpha
r\cos\left(\theta\sqrt{1-m}\right)\right]^{2}}\bigg{\\{}-\left[1-m+r^{2}\left[\alpha^{2}(m-1)-\frac{\Lambda}{3}\right]\right]dt^{2}+$
$\displaystyle+$
$\displaystyle\frac{dr^{2}}{1-m+r^{2}\left[\alpha^{2}(m-1)-\frac{\Lambda}{3}\right]}+r^{2}d\theta^{2}+\big{[}R_{0}+r\cos(\theta\sqrt{1-m})\big{]}^{2}d\phi^{2}\bigg{\\}}$
where the bigger radius of the torus $R_{0}$ is related to the acceleration
and mass parameter in this way:
$R_{0}=\frac{\alpha(m-1)}{\alpha^{2}(m-1)-\Lambda/3}=\alpha r_{0}^{2}$
It is evident that the acceleration plays a fundamental role in the
topological structure of the solution: when the acceleration parameter
decreases also the radius of the torus $R_{0}$ shrinks until it vanishes for
$\alpha=0$; in the latter case local spherical symmetry in the base manifold
is achieved (while globally one has a portion of $(A)dS_{4}$). So, from a
physical point of view, the acceleration sustains and balances the
gravitational attraction of the ring singularity, providing an equilibrium
configuration (whose stability is unclear).
Figure 2: “Accelerating” black ring horizon embedded in $E^{3}$
Considering a $\phi$-constant slice of (3.2) one has exactly the accelerating
black hole in 2+1 of section 2, modulo the dimensional rescaling of the
cosmological constant.
The topology of the horizon, located at $r_{0}$, can be confirmed with the
help of the Gauss-Bonnet theorem. It is simple to compute the Euler
characteristic when the mass parameter gives smooth 2-surface for constant
time and radius, that is $m=0$ and $m=1$. For different values of $m$ one has
to take into account the correction to the Gauss-Bonnet theorem due to the
sharp edge, which is given by the Gibbons-Hawking term. This involves the jump
on the trace of the extrinsic curvature $\mathcal{\bar{K}}$ and the induced
metric $\bar{h}$ on the circular sharp edge. Consider the surface
$\mathcal{S}$ described by the two dimensional metric obtained by (3.2) fixing
$r=\bar{r}=$const and $t=$const, whose embedding in the three dimensional
euclidean space $E^{3}$ is portrayed in figure 2:
$d\bar{s}^{2}=\frac{1}{\left[1+\alpha\bar{r}\cos\left(\theta\sqrt{1-m}\right)\right]^{2}}\left\\{\bar{r}^{2}d\theta^{2}+\big{[}R_{0}+\bar{r}\cos(\theta\sqrt{1-m})\big{]}^{2}d\phi^{2}\right\\}$
Its Euler characteristic is null:
$\displaystyle\chi(\mathcal{S})$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\left(\int_{\mathcal{S}}\sqrt{\bar{g}}\ \bar{R}\
d\theta\ d\phi\ +2\int_{-\pi}^{\pi}\left[\sqrt{\bar{h}}\
\bar{\mathcal{K}}\right]^{\theta=\pi}_{\theta=-\pi}\ d\phi\right)=$
$\displaystyle=$ $\displaystyle\frac{\alpha
R_{0}-1}{4\pi}\sqrt{1-m}\int_{-\pi}^{\pi}d\phi\int_{-\pi}^{\pi}\frac{\cos(\theta\sqrt{1-m})+\bar{r}\alpha}{[1+\alpha\bar{r}\cos(\theta\sqrt{1-m})]^{2}}d\theta$
$\displaystyle-$ $\displaystyle\frac{\alpha
R_{0}-1}{2\pi}\int_{-\pi}^{\pi}\frac{2\sin(\pi\sqrt{1-m})}{1+\alpha\bar{r}\cos(\pi\sqrt{1-m})}\
d\phi\ \ =\ 0$
so, since $\chi(\mathcal{S})=2-2g$, the genus of $\mathcal{S}$ is 1: toroidal
topology $S^{1}\times S^{1}$. Observe that the irregularity on the horizon can
be cast also in the external part of the torus, as happens in the 5D static
ring [13].
The causal singularity is located along $r=0$, forming a circle in the $\phi$
direction. It’s a naked singularity with a cosmological horizon for
$m\in(0,1)$ while a black hole type for $m>1$. Coordinate $r$ is not a
standard polar radius but rather the distance from the ring singularity. Of
course in order to have a proper black ring torus horizon the parameters of
the metric are somewhat constrained to assure that the ring radius is larger
than the ring thickness, $R_{0}\geq r_{0}$: $0\leq\Lambda\leq
3(m-1)\alpha^{2}$. Note that in the null cosmological constant case we can
have just a plump horn ring with $R_{0}=r_{0}$, as expected in [6], because a
lack of the lengh scale furnished by $\Lambda$. It’s not anymore possible tune
the acceleration parameter $\alpha$ to make one radius arbitrary larger than
the other one. While in the other limiting case, that is
$\Lambda=3(m-1)\alpha^{2}$, $R_{0}$ grows to infinity. To have a well behaved
coordinate’s set and an Hausdorff manifold, the range of $r$ have to be
restricted when $|\theta|>\pi/2$ to $r\geq-R_{0}/\cos(\theta\sqrt{1-m})$. That
fact is maybe clearer in toroidal coordinates (3.3) but, on the other hand,
this coordinates patch makes the physical interpretation of the metric (3.2)
more opaque. Anyway to get the the metric (3.2) in the usual ring coordinate
just rename $y=-1/\alpha r$, $x=\cos(\theta\sqrt{1-m})$ and rescale time:
$\displaystyle ds^{2}=\frac{1}{\alpha^{2}(y-x)^{2}}\bigg{\\{}$
$\displaystyle-$
$\displaystyle\left[(y^{2}-1)(1-m)-\frac{\Lambda}{3\alpha^{2}}\right]dt^{2}+\frac{dy^{2}}{(y^{2}-1)(1-m)-\frac{\Lambda}{3\alpha^{2}}}+$
(3.3) $\displaystyle+$
$\displaystyle\frac{dx^{2}}{(1-x^{2})(1-m)}+\left(\alpha R_{0}\
y-x\right)^{2}d\phi^{2}\bigg{\\}}$
This form of the solution may be of some utility for those people interested
in finding a (A)dS ring five or in higher dimensions.
Overall this metric (3.2) or (3.3) is again locally (A)dS again despite the
fact that the base manifold is not of constant curvature. However note that
this behaviour is conceptually different from the topology of the accelerating
black string (3.1), where the toroidal topology is archived by means of the
identification on the fourth coordinate, when $\Lambda>0$. That identification
forced a topology change in the whole universe, which is not here the case.
## 4 Comments and Conclusions
In this paper a C-metric type solution for 2+1 general relativity with
cosmological constant is presented. It is analysed following the standard
four-dimensional techniques: in the weak field approximation the extra
parameter $\alpha$ is found to be the acceleration of an observer in the
origin of coordinates. When $\alpha$ parameter vanishes the usual static BTZ
black hole is recovered. Thus the metric (2.5) is interpreted as an
accelerating three-dimensional black hole, for certain range of the mass
parameters: $m>1$. The acceleration is provided by a one-space-dimensional
semi-infinite strut whose tension is proportional to the jump in the extrinsic
curvature. For $0\leq m<1$ the same solution represents an accelerating naked
singularity with a cosmological horizon pulled by a string. A remarkable fact
is that black hole configurations are admissible even for positive
cosmological constant (whenever is smaller than a certain amount of
acceleration: $\Lambda<\alpha^{2}(m-1)$). This because the pushing effect of
the strut arithmetically adds to the acceleration provided by the cosmological
constant.
Moreover a couple of embeddings of this metric in the 3+1 dimension are
considered. First an accelerating (by a two space dimension membrane) black
string, whose horizon topology depends on the sign of the cosmological
constant: cylindrical or toroidal for negative or positive $\Lambda$
respectively. Again the standard BTZ black string is retrieved when
$\alpha=0$. Lastly it is proposed a ring singularity covered by a toroidal
horizon where the gravitational force is balanced by the acceleration supplied
by a disk (or puncured plane in case of external irregularity) that, in fact,
sustains the ring. Each $\phi$-constant slice represents an accelerating black
hole of one dimension less. Up to the author knowledge this is, though not
regular, the first analytical black ring solution in four dimensions and also
the first not asymptotically flat in any dimensions.
For future perspective would be very interesting counterweight the
gravitational attraction, instead of the acceleration only, by a smoother
centrifugal effect due to the angular momentum. This may provide a regular
horizon such as been done in five dimension by Emparan and Reall passing from
an irregular static ring in [13] to a regular rotating one in [14]. The case
of null acceleration implies only negative cosmological constant, so the
Hawking topology censorship theorem can be avoided. Nevertheless the presence
of the cosmological constant preclude the use of standard generating solutions
techniques.
## Acknowledgements
I would like to thank Eloy Ayon Beato, Fabrizio Canfora, Fiorenza de Micheli,
Dietmar Klemm, Julio Oliva, Souya Ray, David Tempo, Steve Willison and Jorge
Zanelli for fruitful discussions. I’m deeply indebted to Hideki Maeda for his
continuous encouragement, suggestions and comments, without his help this work
just wouldn’t have risen.
This work has been partially funded by the Fondecyt grant 1100755 and by the
Conicyt grant “Southern Theoretical Physics Laboratory” ACT-91. The Centro de
Estudios Científicos (CECS) is funded by the Chilean Government through the
Centers of Excellence Base Financing Program of Conicyt. CECS is also
supported by a group of private companies which at present includes
Antofagasta Minerals, Arauco, Empresas CMPC, Indura, Naviera Ultragas and
Telefónica del Sur.
## References
* [1] M. Banados, M. Henneaux, C. Teitelboim and J. Zanelli, “Geometry of the (2+1) black hole,” Phys. Rev. D 48 (1993) 1506 [arXiv:gr-qc/9302012]
* [2] C. Martinez, C. Teitelboim and J. Zanelli, “Charged rotating black hole in three spacetime dimensions,” Phys. Rev. D 61, 104013 (2000) [arXiv:hep-th/9912259].
* [3] J. Podolsky and J. B. Griffiths, “Uniformly accelerating black holes in a de Sitter universe,” Phys. Rev. D 63 (2001) 024006 [arXiv:gr-qc/0010109].
J. Podolsky, “Accelerating black holes in anti-de Sitter universe,” Czech. J.
Phys. 52 (2002) 1 [arXiv:gr-qc/0202033].
* [4] O. J. C. Dias and J. P. S. Lemos, “Pair of accelerated black holes in anti-de Sitter background: The AdS C-metric,” Phys. Rev. D 67 (2003) 064001 [arXiv:hep-th/0210065].
* [5] J. B. Griffiths, P. Krtous and J. Podolsky, “Interpreting the C-metric,” Class. Quant. Grav. 23 (2006) 6745 [arXiv:gr-qc/0609056].
* [6] R. Emparan and H. S. Reall, “Black Holes in Higher Dimensions,” Living Rev. Rel. 11 (2008) 6 [arXiv:0801.3471 [hep-th]]
* [7] S. Deser and B. Tekin, “Energy in generic higher curvature gravity theories,” Phys. Rev. D 67 (2003) 084009 [arXiv:hep-th/0212292].
* [8] O. Miskovic and J. Zanelli, “On the negative spectrum of the 2+1 black hole,” Phys. Rev. D 79 (2009) 105011 [arXiv:0904.0475 [hep-th]].
* [9] A. R. Steif,“Time-Symmetric Initial Data for Multi-Body Solutions in Three Dimensions,” Phys. Rev. D 53 (1996) 5527 [arXiv:gr-qc/9511053].
* [10] M. M. Anber, “AdS4/CFT3+Gravity for Accelerating Conical Singularities,” JHEP 0811 (2008) 026 [arXiv:0809.2789 [hep-th]]
* [11] D. Ida, “No black hole theorem in three-dimensional gravity,” Phys. Rev. Lett. 85 (2000) 3758 [arXiv:gr-qc/0005129].
* [12] R. Emparan, G. T. Horowitz and R. C. Myers, “Exact description of black holes on branes. II: Comparison with BTZ black holes and black strings,” JHEP 0001 (2000) 021 [arXiv:hep-th/9912135].
* [13] R. Emparan and H. S. Reall, “Generalized Weyl solutions,” Phys. Rev. D 65 (2002) 084025 [arXiv:hep-th/0110258].
* [14] R. Emparan and H. S. Reall, “A rotating black ring in five dimensions,” Phys. Rev. Lett. 88 (2002) 101101 [arXiv:hep-th/0110260].
|
arxiv-papers
| 2011-01-13T17:36:22 |
2024-09-04T02:49:16.418108
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Marco Astorino",
"submitter": "Marco Astorino",
"url": "https://arxiv.org/abs/1101.2616"
}
|
1101.2618
|
UNIVERSITÀ DEGLI STUDI DELL’INSUBRIA
Dipartimento di scienze MM. FF. NN. Como
Anno accademico 2008 - 2009
Sessione di laurea del 30 Settembre 2009
Tesi di Matematica:
Potenziali di Evans
su varietà paraboliche
Autore:
Daniele Valtorta, matricola 617528
e-mail: danielevaltorta@gmail.com
Relatore:
Alberto Giulio Setti
Co-relatore:
Stefano Pigola
### 0.1 Riassunto
La tesi si occupa di una particolare caratterizzazione delle varietà
paraboliche, in particolare una varietà Riemanniana $R$ è parabolica se e solo
se ammette potenziali di Evans, funzioni armoniche proprie definite fuori da
un compatto.
Il principale riferimento bibliografico è il libro di Sario e Nakai [SN],
libro nel quale gli autori dimostrano l’esistenza di potenziali di Evans su
superfici riemanniane. La generalizzazione al caso di varietà di dimensione
qualsiasi presenta alcune differenze tecniche non irrilevanti, ad esempio
leggermente diverse sono le tecniche utilizzate nel paragrafo 4.4.3 relativo
al principio dell’energia. Si ringrazia il professor Wolfhard Hansen
(University of Bielefeld) per i suggerimenti gentilmente forniti riguardo alla
teoria del potenziale, che gioca un ruolo fondamentale in queste
dimostrazioni.
La tesi si articola in una parte introduttiva in cui vengono ripresi alcuni
concetti di geometria Riemanniana e di teoria delle funzioni armoniche.
Speciale attenzione è stata posta sulla soluzione del problema di Dirichlet su
varietà, in particolare è stata trovata una dimostrazione geometrica della
solubilità del problema di Dirichlet per domini particolari, dimostrazione
riportata nella sezione 1.9. In seguito vengono definite e studiate l’algebra
di Royden $\mathbb{M}(R)$ e la compattificazione di Royden $R^{*}$ di una
varietà Riemanniana $R$, strumento essenziale per la dimostrazione di
esistenza dei potenziali di Evans. Essendo $R^{*}$ uno spazio topologico non
primo numerabile, un capitolo della tesi (il capitolo 2) è dedicato allo
studio di filtri e ultrafiltri, concetti utili a descrivere questa topologia e
soprattutto a fornire esempi espliciti di elementi in $R^{*}$ non banali (vedi
paragrafo 3.3.2).
Importanti strumenti tecnici introdotti nella tesi sono le formule di Green e
il principio di Dirichlet, relazioni che sono state dimostrate con ipotesi
poco restrittive sulla regolarità degli insiemi e delle funzioni in gioco
(vedi paragrafo 3.2.4). Anche il principio del massimo ha un ruolo essenziale,
sia nelle sue forme “standard”, sia nelle sue versioni più sofisticate
presentate nel paragrafo 3.3.5.
Nel capitolo 4 viene introdotta la capacità di un insieme, e questo concetto è
utilizzato per la caratterizzazione delle varietà paraboliche, seguendo il
percorso tracciato nella monografia [G2]. Sono introdotti anche il diametro
transfinito e la costante di Tchebycheff, e viene dimostrato che il diametro
transfinito di ogni compatto contenuto nel bordo irregolare di una varietà
parabolica è infinito. Questa dimostrazione è basata fortemente sulla teoria
del potenziale, teoria che tratta delle proprietà di misure di Borel regolari,
della loro energia e delle proprietà dei relativi potenziali di Green. La non
finitezza del diametro transfinito è essenziale per la costruzione di funzioni
armoniche che tendono a infinito sul bordo irregolare di una varietà
iperbolica, e grazie al legame tra queste varietà e le varietà paraboliche
esposto nel paragrafo 4.4.6, otteniamo la dimostrazione (costruttiva)
dell’esistenza dei potenziali di Evans, riportata nel teorema 4.76.
In tutte queste dimostrazioni la linearità è essenziale, in particolare si
sfrutta il fatto che la somma di due funzioni armoniche è armonica. Un
possibile sviluppo futuro di questa tesi è cercare di caratterizzare le
varietà $p$-paraboliche con l’esistenza di potenziali di Evans $p$-armonici,
che richiederebbe di adattare le tecniche utilizzate in questo lavoro al caso
non lineare.
###### Contents
1. 0.1 Riassunto
2. 1 Richiami di matematica
1. 1.1 Geometria Riemanniana
1. 1.1.1 Gradiente e laplaciano di una funzione
2. 1.1.2 Punti critici, valori critici e teorema di Sard
3. 1.1.3 Coordinate polari geodetiche e modelli
4. 1.1.4 Integrazione su varietà e formula di Green
5. 1.1.5 Coordinate di Fermi e applicazioni
6. 1.1.6 Esaustioni regolari
7. 1.1.7 Insiemi chiusi e funzioni lisce
2. 1.2 Funzioni assolutamente continue
3. 1.3 Spazi L2 di forme
4. 1.4 Derivazione sotto al segno d’integrale
5. 1.5 Convoluzioni
6. 1.6 Duali di spazi di Banach
7. 1.7 Funzioni armoniche
1. 1.7.1 Principio del massimo
2. 1.7.2 Stime sul gradiente
3. 1.7.3 Disuguaglianza, funzione e principio di Harnack
4. 1.7.4 Funzioni di Green
5. 1.7.5 Singolarità di funzioni armoniche
6. 1.7.6 Principio di Dirichlet
7. 1.7.7 Funzioni super e subarmoniche
8. 1.8 Algebre di Banach e caratteri
9. 1.9 Problema di Dirichlet
1. 1.9.1 Metodo di Perron
2. 1.9.2 Domini con bordo liscio
3. 1.9.3 Altri domini regolari
4. 1.9.4 Regolarità sul bordo
3. 2 Ultrafiltri e funzionali lineari moltiplicativi
1. 2.1 Filtri, ultrafiltri e proprietà
2. 2.2 Applicazioni: caratteri sulle successioni limitate
4. 3 Algebra e compattificazione di Royden
1. 3.1 Funzioni di Tonelli
1. 3.1.1 Definizioni e proprietà fondamentali
2. 3.1.2 Operazioni con le funzioni di Tonelli
2. 3.2 Algebra di Royden
1. 3.2.1 Definizione
2. 3.2.2 Topologie sull’algebra di Royden
3. 3.2.3 Densità di funzioni lisce
4. 3.2.4 Formule di Green e principio di Dirichlet
5. 3.2.5 Ideali dell’algebra di Royden
3. 3.3 Compattificazione di Royden
1. 3.3.1 Definizione
2. 3.3.2 Esempi non banali di caratteri
3. 3.3.3 Caratterizzazione del bordo
4. 3.3.4 Bordo armonico e decomposizione
5. 3.3.5 Principio del massimo
5. 4 Varietà paraboliche e iperboliche
1. 4.1 Capacità
2. 4.2 Bordo Armonico
3. 4.3 Funzioni di Green
1. 4.3.1 Funzioni di Green sulla compattificazione di Royden
4. 4.4 Potenziali di Evans
1. 4.4.1 Diametro transfinito
2. 4.4.2 Stime per il diametro transfinito
3. 4.4.3 Pricipio dell’energia
4. 4.4.4 Il diametro transfinito è infinito
5. 4.4.5 Funzioni armoniche che tendono a infinito sul bordo di R
6. 4.4.6 Varietà iperboliche irregolari
7. 4.4.7 Potenziali di Evans su varietà paraboliche
6. A Glossario
7. Bibliography
## Chapter 1 Richiami di matematica
### 1.1 Geometria Riemanniana
Lo scopo di questo capitolo è passare brevemente in rassegna alcune
definizioni e risultati che riguardano le varietà Riemanniane. Nel farlo
considereremo solo funzioni lisce e varietà $C^{\infty}$.
Verranno dati per scontati i concetti di varietà differenziale e di metrica
riemanniana, come referenza su questi argomenti consigliamo [C1], [P1] e [G1].
Ricordiamo che una varietà riemanniana $M$ di dimensione $m$ può essere dotata
di una metrica (cioè una funzione $g:T(M)\times T(M)\to\mathbb{R}$ simmetrica
definita positiva e liscia), e indicheremo con $g_{ij}$ la matrice definita da
$g_{ij}(p)=g(\frac{\partial}{\partial x^{i}},\frac{\partial}{\partial
x^{j}})$, con $\left|g\right|$ il suo determinante e con $g^{ij}$ la sua
inversa.
#### 1.1.1 Gradiente e laplaciano di una funzione
Data una varietà Riemanniana $(M,g)$, è possibile associare a ogni funzione
differenziabile $f:M\to\mathbb{R}$ il suo differenziale $df:T(M)\to
T(\mathbb{R})$, che in carte locali assume la forma
$\displaystyle df|_{x}(v)=\left.\frac{\partial\tilde{f}}{\partial
x^{i}}\right|_{x}v^{i}$
dove $\tilde{f}$ è la rappresentazione locale della funzione $f$ rispetto a
una qualsiasi carta, e $v^{i}$ le componenti del vettore $v$ nella stessa
carta. Il gradiente della funzione $f$ è il duale del suo differenziale, nel
senso che $\nabla f$ è l’unico elemento di $T(M)$ tale che per ogni campo
vettoriale $V\in T(M)$:
$\displaystyle g_{ij}(\nabla f)^{i}v^{j}=\left\langle\nabla
f\middle|V\right\rangle=D_{V}(f)\equiv V(f)=\sum_{i=1}^{m}v^{i}\frac{\partial
f}{\partial x^{i}}$
da questa relazione risulta che il gradiente in coordinate locali assume la
forma
$\displaystyle(\nabla f)^{i}=g^{ij}\frac{\partial f}{\partial x^{j}}$ (1.1)
Oltre al gradiente, per una funzione reale possiamo definire anche l’operatore
di Laplace-Beltrami (o laplaciano) $\Delta:C^{m+2}(M,\mathbb{R})\to
C^{m}(M,\mathbb{R})$. La forma locale di questo operatore (che è l’unico
aspetto di interesse in questa tesi) è la seguente:
$\displaystyle\Delta f=\frac{1}{\sqrt{g}}\frac{\partial}{\partial
x^{i}}\left(g^{ij}\sqrt{g}\frac{\partial f}{\partial x^{j}}\right)$ (1.2)
dove si sottointende la somma degli indici ripetuti (convenzione di Einstein).
È utile definire anche l’operatore divergenza. Dato un campo vettoriale $V$
liscio, la sua divergenza è una funzione reale e in coordinate locali questa
funzione è descritta da:
$\displaystyle div(V)=\frac{1}{\sqrt{g}}\frac{\partial}{\partial
x^{i}}\left(\sqrt{g}V^{i}\right)$
dalla definizione osserviamo subito che
$\displaystyle\Delta f=div(\nabla f)$
e per un campo vettoriale $V$ e una funzione $f$ qualsiasi vale che:
$\displaystyle div(fV)=\left\langle\nabla f\middle|V\right\rangle+f\Delta V$
A questo punto ha senso parlare di funzioni armoniche, che per definizione
sono le funzioni per le quali $\Delta f=0$.
Per approfondimenti sul laplaciano consigliamo il libro [P1] (nell’esercizio
10 cap. 2.8 pag 57 si trova la maggior parte delle informazioni necessarie per
questo lavoro).
#### 1.1.2 Punti critici, valori critici e teorema di Sard
In questa sezione riportiamo brevemente le definizioni di punto critico,
valore critico e il teorema di Sard. Per approfondimenti e chiarimenti
rimandiamo a [H3]. Tutte le definizioni e i teoremi sono riportati con la
generalità sufficiente agli scopi di questo lavoro, in modo da non appesantire
la tesi con dettagli eccessivi, quindi in particolare tutte le varietà
differenziali saranno varietà lisce.
La prima domanda che ci poniamo è se le funzioni $f:M\to\mathbb{R}$ possono
essere utili per definire in qualche senso una sottovarietà di $M$.
###### Definizione 1.1.
Data una varietà differenziale $M$ e una funzione $f\in C^{1}(M,\mathbb{R})$,
si dice che $p\in M$ è un punto critico se $df|_{p}=0$ (o equivalentemente
$\nabla f|_{p}=0$). Si dice invece che $x\in\mathbb{R}$ è un valore critico
per $f$ se $f^{-1}(x)$ contiene almeno un punto critico.
Al contrario si dice che $p\in M$ è un valore regolare per $f$ se non è un
punto critico, cioè se $df|_{p}\neq 0$, mentre $x\in\mathbb{R}$ è un valore
regolare per $f$ se $f^{-1}(p)$ non contiene punti critici, cioè contiene solo
punti regolari 111nel caso banale $f^{-1}(x)=\emptyset$, si dice che $x$ è un
valore regolare per $f$.
Indichiamo $\mathcal{C}(f)\subset\mathbb{R}$ l’insieme dei valori critici di
$f:M\to\mathbb{R}$.
###### Definizione 1.2.
Data una funzione $f:M\to\mathbb{R}$, per ogni valore $c\in\mathbb{R}$,
definiamo insieme di sottolivello di $f$ rispetto a $c$ l’insieme
$f^{-1}(-\infty,c]$, insieme di sopralivello di $f$ rispetto a $c$ l’insieme
$f^{-1}[c,\infty)$ e insieme di livello di $f$ rispetto a $c$ l’insieme
$f^{-1}(c)$.
Osserviamo subito che il bordo degli insiemi di sotto/sopra livello è
contenuto nel relativo insieme di livello, cioè
$\partial(f^{-1}(-\infty,c])\subset f^{-1}(c)$, e che se $c$ è un valore
regolare per $f$, questa inclusione si trasforma in uguaglianza, cioè
$\partial(f^{-1}(-\infty,c])=f^{-1}(c)$.
Ha senso chiedersi quando gli insiemi di sottolivello e di livello sono
sottovarietà di $M$.
###### Proposizione 1.3.
Se $c\in\mathbb{R}$ è un valore regolare di $f\in C^{r}(M,\mathbb{R})$
222$r\geq 1$, allora $f^{-1}(c)$ è una sottovarietà $m-1$ dimensionale di $M$
di regolarità almeno $C^{r}$, e quindi $f^{-1}(-\infty,c]$ è una sottovarietà
con bordo.
Questa proposizione corrisponde al teorema 3.2 cap. 1.3 pag 22 di [H3], e
rimandiamo a questo libro per la dimostrazione.
Ora possiamo chiederci “quanti” siano i valori regolari di una funzione
indipendentemente dalla funzione data. La risposta è contenuta nel teorema di
Sard. Prima di enunciarlo definiamo gli insiemi di misura nulla su una
varietà. Consigliamo il paragrafo 3.1 pag 68 di [H3] per approfondimenti.
###### Definizione 1.4.
Un insieme $A\subset\mathbb{R}^{m}$ si dice essere di misura nulla se e solo
se per ogni $\epsilon>0$ esiste un’insieme al più numerabile di cubi $C_{n}$
in $\mathbb{R}^{m}$ 333cioè di insiemi della forma
$\prod_{i=1}^{m}[a_{i},b_{i}]$ che hanno misura
$\lambda(\prod_{i=1}^{m}[a_{i},b_{i}])=\prod_{i=1}^{m}(b_{i}-a_{i})$ tali che
1. 1.
$A\subset\bigcup_{n}C_{n}$
2. 2.
$\sum_{n=1}^{\infty}\lambda(C_{n})<\epsilon$
Ricordiamo che questa definizione è equivalente alla richiesta che la misura
di Lebesgue di $A$ sia nulla.
Valgono le seguenti proprietà
###### Proposizione 1.5.
Per gli insiemi di misura nulla vale che
1. 1.
L’unione numerabile di insiemi di misura nulla ha misura nulla
2. 2.
Gli insiemi aperti non vuoti non hanno misura nulla, e il complementare di
insiemi di misura nulla è denso
3. 3.
I sottoinsiemi di insiemi di misura nulla hanno misura nulla
4. 4.
L’immagine attraverso una funzione localmente Lipschitziana di un insieme di
misura nulla ha misura nulla
Grazie alla proprietà (1) ha senso definire
###### Definizione 1.6.
Un insieme $A\subset M$ si dice essere di misura nulla se per ogni carta
locale $(U,\phi)$ di $M$, l’insieme $\phi(A)$ ha misura nulla. Osserviamo che
per paracompattezza di $M$, esiste sempre un atlante al più numerabile
$\\{(U_{n},\phi_{n})\\}$ di $M$ 444anzi se la dimensione di $M$ è $m$, esiste
sempre un atlante formato da al più $m+1$ carte, vedi problema 2.8 pag 21 di
[M3], quindi dato che $A=\cup_{n}(A\cap U_{n})$, ha senso la definizione di
insieme di misura nulla e non dipende dall’atlante scelto.
Il teorema di Sard (o meglio una sua versione non molto generale) garantisce
che
###### Teorema 1.7 (Teorema di Sard).
Data $M$ varietà differenziale $m-$dimensionale, sia $r\geq\max\\{1,m-1\\}$.
Allora se $f\in C^{r}(M,\mathbb{R})$, l’insieme dei valori critici
$\mathcal{C}(f)$ ha misura nulla in $\mathbb{R}$, quindi il suo complementare
è denso.
#### 1.1.3 Coordinate polari geodetiche e modelli
Un sistema di coordinate che utilizzeremo spesso sulle varietà sono le
coordinate polari geodetiche. In questa sezione ci occupiamo di dare una breve
carrellata sulle coordinate polari e sulle varietà modello, ovvero varietà
sfericamente simmetriche. Come referenze per questa sezione consigliamo il
capitolo 3 di [C1] e il capitolo 3 di [G2]. In tutta la sezione lavoreremo con
una varietà riemannana $R$ di dimensione $m$.
Fissato un punto $p\in R$, esiste sempre un intorno normale $U$ di $p$, un
intorno cioè dove la mappa esponenziale $\exp:TR\to R$ è un diffeomorfismo.
Visto che $TR$ è isomorfo a $\mathbb{R}^{m}$, possiamo definire su
$TR\setminus\\{0\\}$ le classiche coordinate polari $m$-dimensionali 555ogni
punto può essere individuato dalla distanza dall’origine e da una coordinata
su $S^{m-1}$ che rappresenta la direzione del punto e grazie alla mappa
esponenziale possiamo portare queste coordinate sulla varietà. Sull’aperto
$U\setminus\\{p\\}$ chiamiamo le coordinate definite da
$\displaystyle\phi(q)=\\{r(\exp^{-1}(q)),\theta(\exp^{-1}(q))\\}=(r,\theta)$
coordinate polari geodetiche. In questo sistema di coordinate la metrica
assume la forma:
$\displaystyle g=\begin{bmatrix}1&0\\\ 0&A(q)\end{bmatrix}$ (1.3)
oppure utilizzando una notazione più famigliare alla geometria riemanniana:
$\displaystyle ds^{2}=dr^{2}+A(r,\theta)_{ij}d\theta^{i}d\theta^{j}$
dove $A(q)$ è una matrice definita positiva 666vedi equazione 3.1 di [G2]
oppure pagina 136 di [P1]. In coordinate polari inoltre l’operatore laplaciano
assume la forma:
$\displaystyle\Delta(f)=\frac{1}{\sqrt{\left|A\right|}}\frac{\partial}{\partial
r}\left(\sqrt{\left|A\right|}\frac{\partial f}{\partial
r}\right)+\Delta_{S}(f)=\frac{\partial^{2}f}{\partial
r^{2}}+\frac{1}{2}\frac{\partial\log(\left|A\right|)}{\partial
r}\frac{\partial f}{\partial r}+\Delta_{S}(f)$ (1.4)
Dove $\Delta_{S}$ indica il laplaciano sulla sottovarietà $r=$costante 777vedi
equazione 3.4 di [G2].
Si nota quindi che per le funzioni “radiali”, ossia quelle funzioni che
dipendono solo da $r$, il laplaciano assume una forma abbastanza semplificata.
Grazie alla semplice forma di $g$ in coordinate polari anche il gradiente di
una funzione radiale è molto semplice da calcolare, infatti grazie alla 1.1
notiamo che per le funzioni radiali $f$:
$\displaystyle(\nabla f)^{1}=\frac{\partial f}{\partial r}$ (1.5)
mentre tutte le altre componenti del gradiente sono nulle.
Queste coordinate risultano particolarmente adatte per descrivere le varietà
modello. Queste varietà sono varietà sfericamente simmetriche rispetto a
rotazioni attorno a un punto fisso.
###### Definizione 1.8.
Definiamo $R$ una varietà con polo $o$, se la mappa esponenziale $\exp|_{o}$ è
un diffeomorfismo globale tra $\mathbb{R}^{m}$ e $R$. Inoltre se la metrica di
questa varietà rispetto alle coordinate polari geodetiche in $o$ è della forma
$\displaystyle ds^{2}=dr^{2}+\sigma(r)^{2}d\theta^{2}$
dove $d\theta^{2}$ è la metrica euclidea standard di $S^{m-1}$, definiamo
questa una varietà modello
La definizione trova la sua giustificazione in questa osservazione. Se
consideriamo una rotazione $\rho:S^{m-1}\to S^{m-1}$, possiamo definire una
funzione
$\displaystyle\psi_{\rho}:R\to R\ \
\psi_{\rho}(r,\theta)=\psi_{\rho}(r,\rho(\theta))$
questa funzione (che possiamo considerare a tutti gli effetti una rotazione su
$R$), ha la proprietà di tenere la metrica invariata, cioè
$\displaystyle\psi_{\rho}^{*}(g)=g$
dove $\psi_{\rho}^{*}$ è l’operatore di push-forward, definito da:
$\displaystyle\psi_{\rho}^{*}(g)(v_{1},v_{2})=g(d\psi_{\rho}(v_{1}),d\psi_{\rho}(v_{2}))$
#### 1.1.4 Integrazione su varietà e formula di Green
In questa sezione riportiamo brevemente la definizione di integrazione su
varietà riemanniane e la formula di Green per domini regolari. Rimandiamo al
testo [M2] per approfondimenti.
Ai nostri scopi interessa solo ricordare brevemente la definizione di
integrale di una funzione su una varietà e di una 1-forma su un bordo. Data
una funzione reale $f:R\to\mathbb{R}$, per definire il suo integrale
utilizziamo l’integrale di Lebesgue su $\mathbb{R}^{m}$. Non introduciamo la
teoria dell’operatore duale di Hodge, ma ne utilizziamo comunque il simbolo
($\ast$) definendolo quando necessario.
###### Definizione 1.9.
Sia $f:R\to\mathbb{R}$ una funzione continua a supporto compatto
$supp(f)\Subset U$ con $(U,\phi)$ carta locale. Allora definiamo
$\displaystyle\int_{R}fdV=\int_{\phi(U)}\tilde{f}(x)\sqrt{\left|g\right|}dx^{1}\cdots
dx^{m}=\int_{\phi(U)}f\phi^{-1}(x)\sqrt{\left|g\right|}dx^{1}\cdots dx^{m}$
Osserviamo che la presenza di $\sqrt{\left|g\right|}$ garantisce che la
definizione non dipenda dalla carta locale scelta. Data una funzione
$f:R\to\mathbb{R}$ senza ulteriori condizioni sul supporto, definiamo:
$\displaystyle\int_{R}fdV=\sum_{n=1}^{\infty}\int_{R}f\lambda_{n}dV$
dove $\\{\lambda_{n}\\}$ è una partizione dell’unità di $R$ subordinata a un
ricoprimento di carte locali qualsiasi.
###### Definizione 1.10.
Date due funzioni $f,h:R\to\mathbb{R}$, e un dominio regolare $\Omega\subset
R$ 888con regolare si intende un dominio aperto relativamente compatto con
$\partial\Omega$ bordo liscio (almeno a pezzi), definiamo
$\displaystyle\int_{\partial\Omega}h\ast
df=\int_{\phi(\partial\Omega)}h(x)g^{im}(x)\frac{\partial f}{\partial
x^{i}}(x)\sqrt{\left|g\right|}dx^{1}\cdots dx^{m-1}$
quando $f$ ha supporto contenuto in una carta $(U,\phi)$ regolare per
$\partial\Omega$, cioè una carta in cui
$\phi(\partial\Omega)\subset\\{x^{m}=0\\}$. Se $f$ non ha queste
caratteristiche, l’integrale si ottiene come sopra grazie alle partizioni
dell’unità.
Osserviamo che la notazione $\ast df$ indica il duale di Hodge della forma
$df$. In questa tesi però questo operatore verrà usato solo in casi simili a
quello appena descritto, quindi per brevità tralasciamo la sua definizione e
lo utilizziamo solo per comodità di scrittura.
Grazie a quanto appena descritto possiamo enunciare la prima identità di Green
per varietà Riemanniane:
###### Proposizione 1.11 (Prima identità di Green).
Dato un dominio regolare $\Omega\subset R$, e due funzioni lisce
$f,h:R\to\mathbb{R}$, si ha che:
$\displaystyle\int_{\Omega}\left\langle\nabla f\middle|\nabla h\right\rangle
dV+\int_{\Omega}f\Delta hdV=\int_{\partial\Omega}f\ast dh$
In seguito rilasseremo le ipotesi sulla regolarità di $f$ in questa identità.
La dimostrazione di questa identità è una facile conseguenza del teorema di
Stokes (consigliamo come referenza il capitolo 7 di [M2]).
#### 1.1.5 Coordinate di Fermi e applicazioni
In questo paragrafo descriveremo un sistema di coordinate locali intorno a una
sottovarietà regolare particolarmente utile in quanto per molti aspetti molto
simile alle coordinate euclidee di $\mathbb{R}^{n}$. Il riferimento principale
in questa sezione è il libro [G1], in particolare il II capitolo, dal quale
riporteremo alcuni risultati senza dimostrazione.
Data una varietà riemanniana $R$ di dimensione $m$ e una sua sottovarietà
regolare $S$ di dimensione $s$, cerchiamo un sistema di coordinate locali
$(x_{1},\cdots,x_{m})$ in un intorno $U$ di $x_{0}\in S$ tali che ogni punto
$p\in U\cap S$ abbia $x^{m-s+1}(p)=\cdots=x^{m}(p)=0$ e tali che la metrica in
queste coordinate assuma la forma:
$\displaystyle g(p)=\begin{bmatrix}A(p)&0\\\ 0&B(p)\end{bmatrix}$
dove $A$ è una matrice $(m-s)\times(m-s)$ e $B$ è una matrice $s\times s$. In
realtà per gli scopi di questo lavoro siamo interessati solo al caso $s=m-1$,
quindi per comodità di notazione tratteremo solo questo caso, anche se i
risultati di questo paragrafo possono facilmente essere estesi a sottovarietà
di dimensioni qualsiasi.
Per comodità, indicheremo con $p,\ q$ i punti sulla varietà riemanniana, con
$(x^{1},\cdots,x^{m-1})\equiv\vec{x}$
le prime $m-1$ funzioni coordinate sulla varietà e con $y$ l’ultima
coordinata.
###### Proposizione 1.12.
Data una sottovarietà $S$ di codimensione $1$ in $R$, per ogni punto $p_{0}\in
S$, esiste un suo intorno $U$ e delle coordinate locali tali che:
1. 1.
$y(U\cap S)=0$
2. 2.
la metrica assume la forma
$\displaystyle g(p)=\begin{bmatrix}A(p)&0\\\ 0&1\end{bmatrix}$
dove $A$ è una matrice $(m-1)\times(m-1)$. Inoltre è possibile scegliere le
coordinate in modo che su $S$ le funzioni $x^{1}\cdots x^{m-1}$ siano
coordinate qualsiasi relative alla sottovarietà $S$.
###### Proof.
Il sistema di coordinate cercato prende il nome di coordinate di Fermi per la
sottovarietà $S$. L’esistenza di queste coordinate è dimostrata in [G1]. In
particolare a pagina 17 si trova la definizione di coordinate di Fermi, e
grazie al lemma 2.3 di pagina 18 e al corollario 2.14 di pagina 31 999il lemma
di Gauss generalizzato, si dimostra la proprietà (2). ∎
Introduciamo ora gli intorni geodeticamente convessi su una varietà
riemanniana.
###### Definizione 1.13.
Un insieme aperto $U\subset R$ si dice geodeticamente convesso se e solo se
per ogni coppia di punti $(p,q)\subset U\times U$ la geodetica che minimizza
la distanza tra questi due punti esiste unica e la sua traccia è contenuta in
$U$.
L’esempio più classico di intorno geodeticamente convesso sono le bolle in
$\mathbb{R}^{n}$. L’esistenza di intorni geodeticamente convessi è garantita
da questa proposizione, di cui non riportiamo la dimostrazione.
###### Proposizione 1.14.
Dato un aperto $W\in R$ e $x\in W$, esiste sempre un intorno aperto
geodeticamente convesso $U$ tale che $x\in U\subset W$.
###### Proof.
La dimostrazione di questa proposizione può essere trovata su [C2],
proposizione 10.5.4 pagina 334. ∎
Ora siamo pronti per dimostrare un lemma molto tecnico che servirà in seguito
per dimostrare una proprietà dei potenziali di Green.
###### Proposizione 1.15.
Dato $K$ insieme aperto relativamente compatto in $R$ con bordo liscio,
definiamo per ogni $q\in R$:
$\displaystyle\pi(q)\ \ \ t.c.\ \ \ d(\pi(q),q)=\inf\\{d(q,p)\ t.c.\ p\in
K\\}$
Dimostriamo che per ogni $p_{0}\in\partial K$, esiste un intorno $V$ di
$p_{0}$ dove $\pi$ è una funzione continua ben definita, e per ogni
$\epsilon>0$, esiste un intorno $U_{\epsilon}$ di $p_{0}$ per il quale per
ogni $p\in K$ e per ogni $q\in U_{\epsilon}$:
$\displaystyle d(p,q)\leq(1+\epsilon)d(p,\pi(q))$
###### Proof.
Consideriamo un intorno $V_{1}$ di $p_{0}$ dove siano definite le coordinate
di Fermi per la sottovarietà regolare $\partial K$, e in particolare
consideriamo un sistema di coordinate per cui la metrica nel punto $p_{0}$
assume la forma euclidea standard, cioè
$\displaystyle g_{ij}(p_{0})=\delta_{ij}$
Definiamo $K_{1}=K\cap\overline{V}_{1}$ e $K_{2}\equiv K\cap V_{1}^{C}$.
L’insieme definito da
$\displaystyle A=\\{p\in R\ t.c.\ d(p,K_{1})<d(p,K_{2})$
è un’insieme aperto per la continuità della funzione distanza, non vuoto
perché $p_{0}\in A$. Se $p\in A$, sicuramente il punto (o i punti) $\pi(p)$
sono da cercare sono nell’insieme $K_{1}$. Consideriamo un cilindro $V$
rispetto all’ultima coordinata 101010quindi un insieme della forma
$B\times(-\epsilon,\epsilon)$ con $B$ aperto contenente la proiezione $p_{0}$
contenuto nell’aperto $A\cap V_{1}$. Data la particolare forma delle
coordinate di Fermi, è facile dimostrare che se $q\in V$, allora il punto
$\pi(q)$ è il punto di coordinate
$\displaystyle y(\pi(q))=0,\ \ x^{i}(\pi(q))=x^{i}(q)\ \ \forall i=1\cdots
m-1$
Infatti supponiamo per assurdo che $y(\pi(q))\neq 0$ 111111dove l’indice $1$
può essere sostituito da qualsiasi altro indice, e sia $\gamma$ la geodetica
che unisce $\gamma(0)=\pi(q)$ e $\gamma(1)=q$. La curva:
$\displaystyle\tilde{\gamma}(t)=(x^{1}(q),\cdots,x^{m-1}(q),y(\gamma(t)))$
è una curva che unisce i punti $(x^{1}(q),\cdots,m^{m-1}(q),0)$ con $q$, e la
sua lunghezza è
$\displaystyle
L(\tilde{\gamma})=\int_{0}^{1}g_{ij}\dot{\tilde{\gamma}}^{i}(t),\dot{\tilde{\gamma}}^{j}(t)dt=\int_{0}^{1}(\dot{\gamma}^{m}(t))^{2}dt<\int_{0}^{1}g_{ij}\dot{\gamma}^{i}(t),\dot{\gamma}^{j}(t)dt=L(\gamma)$
Per dimostrare la seconda parte della proposizione, ragioniamo sulle prime
$m-1$ righe e colonne della matrice $g_{ij}$, cioè sulla matrice $A_{ij}$ per
$i,j=1\cdots m-1$. Possiamo sempre scrivere che per ogni punto di $V$:
$\displaystyle
A_{ij}(p)=A_{ij}(p_{0})+\alpha_{ij}(p)=\delta_{ij}+\alpha_{ij}(p)$
Per continuità della metrica, per ogni $\epsilon>0$, esiste un intorno
$U^{\prime}_{\epsilon}(p_{0})$ contenuto in $V$ per il quale il modulo di
tutti gli autovalori di $\alpha_{ij}(p)$ è minore di $\epsilon$. Sia
$U^{\prime\prime}_{\epsilon}(p_{0})$ un cilindro rispetto all’ultima
coordinata della varietà contenuto in $U^{\prime}_{\epsilon}$, e
$U_{\epsilon}$ un intorno di $p_{0}$ geodeticamente convesso contenuto in
$U^{\prime\prime}_{\epsilon}$. Siano $p\in U_{\epsilon}\cap K$ e $q\in
U_{\epsilon}$. Se $q\in K$, non c’è niente da dimostrare in quanto $q=\pi(q)$,
quindi consideriamo solo il caso $q\not\in K$.
Sia $\gamma$ una geodetica normalizzata (cioè con
$g_{ij}(\gamma(t))\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)=1$) che congiunge $p$
a $q$. Questa geodetica è necessariamente contenuta in $U_{\epsilon}$. Se
chiamiamo $l=d(p,q)$, sappiamo che
$\displaystyle
l=\int_{0}^{l}\sqrt{g_{ij}|_{\gamma(t)}\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)}dt=\int_{0}^{l}\sqrt{A_{ij}|_{\gamma(t)}\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)+\left|\dot{\gamma}^{m}(t)\right|^{2}}dt$
Sia $\eta(t)$ una curva che unisce i punti $p$ e $\pi(q)$ descritta da:
$\displaystyle\eta(t)=\left(\gamma^{1}(t),\cdots,\gamma^{m-1}(t),\frac{y(p)}{y(p)-y(q)}(\gamma^{m}(t)-y(q))\right)$
questa curva è contenuta nell’insieme $U^{\prime\prime}_{\epsilon}$ e anche se
non è necessariamente una geodetica, vale che:
$\displaystyle
d(\pi(q),p)\leq\int_{0}^{l}\sqrt{g_{ij}|_{\eta(t)}\dot{\eta}^{i}(t)\dot{\eta}^{j}(t)}dt=$
$\displaystyle=\int_{0}^{l}\sqrt{A_{ij}|_{\eta(t)}\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)+\kappa^{2}\left|\dot{\gamma}(t)\right|^{2}}dt\leq\int_{0}^{l}\sqrt{A_{ij}|_{\eta(t)}\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)+\left|\dot{\gamma}(t)\right|^{2}}dt$
dove $\kappa=\left|\frac{y(p)}{y(p)-y(q)}\right|<1$ grazie al fatto che $p\in
K$, quindi $y(p)\leq 0$, e $q\not\in K$, quindi $y(q)>0$.
Consideriamo inoltre che, pochè $\gamma$ è normalizzata:
$\displaystyle A_{ij}|_{\gamma(t)}\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)\leq
1$
Se indichiamo
$\left\|\dot{\gamma}(t)\right\|_{m-1}=\sum_{i,j\leq
m-1}\delta_{ij}\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)$
ricaviamo che per ogni $t$:
$\displaystyle\delta_{ij}\dot{\gamma}^{i}(t)\dot{\gamma}^{i}(t)+\alpha|_{\gamma(t)}\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)\leq
1$ $\displaystyle\left\|\dot{\gamma}(t)\right\|_{m-1}\leq
1-\alpha|_{\gamma(t)}\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)\leq
1+\epsilon\left\|\dot{\gamma}(t)\right\|_{m-1}$
$\displaystyle\left\|\dot{\gamma}(t)\right\|_{m-1}\leq\frac{1}{1-\epsilon}$
dove abbiamo utilizzato il fatto che il modulo di tutti gli autovalori di
tutte le matrici $\alpha_{ij}$ in $U^{\prime}_{\epsilon}$ è minore di
$\epsilon$. Osserviamo che questa stima è indipendente dalla scelta di $p$ e
$q$.
Grazie alla definizione delle matrici $\alpha_{ij}$ abbiamo anche che:
$\displaystyle
A_{ij}|_{\eta(t)}\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)=[A_{ij}|_{\gamma(t)}-\alpha_{ij}|_{\gamma(t)}+\alpha_{ij}|_{\eta(t)}]\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)$
quindi:
$\displaystyle[A_{ij}|_{\eta(t)}-A_{ij}|_{\gamma(t)}]\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)=[-\alpha_{ij}|_{\gamma(t)}+\alpha_{ij}|_{\eta(t)}]\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)$
$\displaystyle\left|[A_{ij}|_{\eta(t)}-A_{ij}|_{\gamma(t)}]\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)\right|\leq
2\epsilon\left\|\dot{\gamma}(t)\right\|_{m-1}\leq\frac{2\epsilon}{1-\epsilon}$
Grazie a quest’ultima disuguaglianza e ricordando che
$\sqrt{a+b}\leq\sqrt{a}+\sqrt{\left|b\right|}$ per ogni coppia di numeri reali
tali che $a\geq 0,\ a+b\geq 0$, possiamo concludere che:
$\displaystyle
d(\pi(q),p)\leq\int_{0}^{l}\sqrt{A_{ij}|_{\eta(t)}\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)+\left|\dot{\gamma}(t)\right|^{2}}dt=$
$\displaystyle=\int_{0}^{l}\sqrt{A_{ij}|_{\gamma(t)}\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)+\left|\dot{\gamma}(t)\right|^{2}+[A_{ij}|_{\eta(t)}-A_{ij}|_{\gamma(t)}]\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)}dt\leq$
$\displaystyle\leq\int_{0}^{l}\sqrt{A_{ij}|_{\gamma(t)}\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)+\left|\dot{\gamma}(t)\right|^{2}}dt+\int_{0}^{l}\sqrt{\frac{2\epsilon}{1-\epsilon}}dt\leq
l\left(1+\sqrt{\frac{2\epsilon}{1-\epsilon}}\right)$
ricordiamo che $l=d(p,q)$. La tesi segue dal fatto che
$\displaystyle f(\epsilon)=\sqrt{\frac{2\epsilon}{1-\epsilon}}$
è una funzione continua per $0\leq\epsilon<1$ e $f(0)=0$. Osserviamo che
$\epsilon$ non è stata scelta in funzione di $p$ e $q$, ma solo in funzione
dell’intorno $U_{\epsilon}$, quindi la stima vale per ogni coppia di punti $p$
e $q$ in $U_{\epsilon}(p_{0})$. ∎
Osserviamo che questa proposizione vale anche in un altro caso, nel caso in
cui $K$ sia una sottovarietà di $R$ con bordo regolare. La dimostrazione
appena conclusa non tiene conto della possibilità che $K$ abbia un bordo
regolare, ma può essere adattata facilmente. Basta considerare che se
$\pi(q)\in\partial K$ (bordo inteso come bordo della sottovarietà regolare),
allora la geodetica che unisce $p$ e $\pi(q)$ è di certo più corta della
geodetica che unisce $p$ alla proiezione di $q$ sul piano contenente la parte
di bordo considerata. Riportiamo quindi la proposizione lasciando i dettagli
della dimostrazione al lettore.
###### Proposizione 1.16.
Data $K$ sottovarietà regolare di $R$ possibilmente con bordo liscio,
definiamo per ogni $q\in R$:
$\displaystyle\pi(q)\ \ \ t.c.\ \ \ d(\pi(q),q)=\inf\\{d(q,p)\ t.c.\ p\in
K\\}$
Per ogni $p_{0}\in K$, esiste un intorno $V$ di $p_{0}$ dove $\pi$ è una
funzione continua ben definita, e per ogni $\epsilon>0$, esiste un intorno
$U_{\epsilon}$ di $p_{0}$ per il quale per ogni $p\in K$ e per ogni $q\in
U_{\epsilon}$:
$\displaystyle d(p,q)\leq(1+\epsilon)d(p,\pi(q))$
#### 1.1.6 Esaustioni regolari
In questa sezione ci occupiamo di dimostrare l’esistenza di esaustioni
regolari per una qualsiasi varietà differenziale liscia connessa $M$ 121212in
realtà come si può dedurre dalle dimostrazioni, un’esaustione esiste per ogni
varietà $M$, e per ogni varietà esiste un’esaustione regolare quanto lo è la
varietà. Per prima cosa dimostriamo l’esistenza di esaustioni non regolari,
poi sfruttiamo la densità delle funzioni lisce nelle funzioni continue su una
varietà per ottenere l’esistenza di esaustioni regolari. Come risulta evidente
dalle definizioni, ha senso procurarsi un’esaustione di $M$ solo nel caso $M$
non compatta.
Oltre che ad esaustioni regolari, avremo bisogno anche di esaustioni che si
comportano bene in relazione ad una sottovarietà di $R$. In particolare,
vogliamo dimostrare che per ogni sottovarietà regolare con bordo $S\subset R$
($dim(S)=dim(R)$), esiste un’esausione regolare $K_{n}$ tale che $C=\partial
K_{n}\cap S$ è ancora una sottovarietà regolare di codimensione $1$ con bordo.
Ovviamente cominciamo con le definizioni del caso.
###### Definizione 1.17.
Data una varietà differenziale non compatta $M$, diciamo che $\\{K_{n}\\}$ è
un’esaustione di $M$ se $K_{n}$ sono insiemi aperti connessi relativamente
compatti tali che
1. 1.
$\overline{K_{n}}\Subset K_{n+1}$
2. 2.
$M=\bigcup_{n}K_{n}$
Spesso nella definzione si confondono $K_{n}$ e $\overline{K_{n}}$.
Diciamo che questa esaustione è regolare se i bordi $\partial K_{n}$ sono
$C^{\infty}$, o equivalentemente è una sottovarietà di $M$.
Dimostriamo per prima cosa l’esistenza di un’esaustione.
###### Proposizione 1.18.
Ogni varietà differenziale connessa $M$ ammette un’esaustione $K_{n}$.
###### Proof.
Per ogni punto $p\in M$ consideriamo un intorno aperto connesso relativamente
compatto $U(p)$ 131313che esiste poiché $M$ è localmente euclidea. Al variare
di $p\in M$, $U(p)$ è un ricoprimento aperto di $M$, quindi poiché $M$ è II
numerabile, esiste un sottoricoprimento numerabile $U(p_{n})\equiv U_{n}$.
Costruiamo per induzione l’esaustione $K_{n}$.
Per prima cosa rinumeriamo gli insiemi $U_{n}$ in modo che per ogni $N$,
$S_{N}\equiv\bigcup_{n=1}^{N}U_{n}$ sia un insieme connesso. Questo è
sicuramente vero se $N=1$. Supponiamo che sia vero per $N$. Dato che
$M=\cup_{n}U_{n}$, se non esistesse nessun $\bar{n}\geq N$ tale che
$\displaystyle S_{N}\cap U_{\bar{n}}\neq\emptyset$
allora $M$ sarebbe disconnessa perché
$S_{N}\bigcap\left(\bigcup_{n=N+1}^{\infty}U_{n}\right)=\emptyset$.
Rinominando $\bar{n}=N+1$, si ottiene che $S_{N+1}$ è l’unione di insiemi
connessi a intersezione non vuota, quindi è connesso.
Consideriamo ora $K_{1}=U_{1}$, e costruiamo per induzione una successione a
valori interi strettamente crescente $k(n)$ tale che
1. 1.
$k(1)=1$
2. 2.
per ogni $n$,
$\bigcup_{j=1}^{k(n)}\overline{U_{j}}\Subset\bigcup_{j=1}^{k(n+1)}U_{j}$
Se definiamo $K_{n}\equiv\bigcup_{j=1}^{k(n)}U_{j}$, il gioco è fatto.
Questo è possibile perché
$\overline{K_{n}}=\bigcup_{j=1}^{k(n)}\overline{U_{j}}$ è un insieme compatto
(unione finita di compatti), ricoperto dagli aperti $U_{j}$, quindi esiste un
sottoricoprimento finito di $U_{j}$ che ricopre $K_{n}$. Se chiamiamo $k(n+1)$
l’indice massimo di questo sottoricoprimento, la successione $k(n)$ ha le
proprietà desiderate. ∎
Grazie all’esistenza di questa esaustione, possiamo costruire una funzione di
esaustione $f:M\to\mathbb{R}$ in questo modo:
###### Proposizione 1.19.
Per ogni varietà $M$, esiste una funzione continua positiva propria
$f:M\to\mathbb{R}$ 141414ricordiamo che una funzione è detta propria se la
controimmagine di un qualsiasi compatto è compatta.
###### Proof.
Definiamo $f(\overline{K_{1}})=1$, e $f(\partial K_{n})=n$. Per ogni insieme
$\overline{K_{n+1}}\setminus K_{n}$ ($n=1,2,\cdots$) estendiamo $f$ a una
funzione continua $n\leq f\leq n+1$ grazie al lemma di Urysohn 151515vedi
teorema 4.1 pag 146 di [D]. Ricordiamo che ogni spazio metrizzabile (e quindi
ogni varietà o ogni suo sottoinsieme) è normale. In questo modo otteniamo una
funzione continua $f:M\to\mathbb{R}$ tale che $\overline{K_{n}}\Subset
f^{-1}(-\infty,n]\Subset K_{n+1}$, il che garantisce che $f$ sia propria. ∎
Osserviamo che ci sono altri modi di definire una funzione continua positiva
propria su $M$. Ad esempio se $M$ è riemanniana geodeticamente completa (e per
ogni varietà è possibile definire una metrica $g$ che la renda completa), la
funzione distanza da un punto soddisfa queste caratteristiche.
Trovata questa funzione, siamo in grado di dimostrare che
###### Proposizione 1.20.
Ogni varietà liscia $M$ ammette un’esaustione regolare $C_{n}$.
###### Proof.
Data un’esaustione $K_{n}$ e definita $f$ come nella proposizione precedente,
sia $h\in C^{\infty}(M,\mathbb{R})$ tale che $\left\|h-f\right\|_{\infty}<1/4$
161616questo è possibile per densità delle funzioni lisce nello spazio delle
funzioni continue con la norma uniforme, vedi teorema 2.6 pag. 49 di [H3],
oppure è possibile dimostrare questo fatto adattando la dimostrazione della
proposizione 3.27 a pagina 3.27. Per ogni $n$ consideriamo l’insieme
$\Delta_{n}=(n+1/4,n+3/4)$ aperto in $\mathbb{R}$. Grazie al teorema di Sard,
sappiamo che esiste un numero reale $x_{n}\in\Delta_{n}$ che sia valore
regolare di $h$. Grazie alle proposizioni di questa sezione, questo garantisce
che l’insieme $A_{n}\equiv h^{-1}(-\infty,x_{n})$ sia un aperto di $M$ con
bordo $\partial A_{n}=h^{-1}(x_{n})$ sottovarietà regolare di $M$.
Per definizione di $h$, si verifica facilmente che $K_{n}\subset A_{n}\subset
K_{n+1}$, quindi $A_{n}$ è relativamente compatto, però non è detto che
$A_{n}$ sia connesso. Per questo motivo definiamo $C_{n}$ la componente
connessa di $A_{n}$ che contiene $K_{n}$. Poiché $\partial C_{n}$ è una
componente connessa di $\partial A_{n}$, mantiene le sue proprietà di
regolarità. Infine, dato che per ogni $n$ $K_{n}\subset C_{n}\subset K_{n+1}$,
vale che
1. 1.
$\overline{C}_{n}\Subset C_{n+1}$
2. 2.
$M=\bigcup_{n}C_{n}$
∎
Oltre che ad esaustioni regolari, avremo bisogno anche di esaustioni che si
comportano bene in relazione ad una sottovarietà di $R$. In particolare,
vogliamo dimostrare che per ogni sottovarietà regolare con bordo $S\subset R$
($dim(S)=dim(R)$), esiste un’esausione regolare $K_{n}$ tale che $C=\partial
K_{n}\cap S$ è ancora una sottovarietà regolare di codimensione $1$ con bordo.
A questo scopo, assumiamo che $S$ sia chiusa ma non compatta (se $S$ compatta,
$S\subset K_{n}^{\circ}$ definitivamente). Dato che $C^{\circ}=\partial
K_{n}\cap S^{\circ}$ è di certo una sottovarietà regolare, resta da dimostrare
solo che anche il bordo $\partial C=\partial K_{n}\cap\partial S$ è regolare.
Riportiamo ora un risultato riguardo alla regolarità dell’intersezione tra due
sottovarietà regolari.
###### Definizione 1.21.
Data $R$ varietà riemanniana e $S,M$ sue sottovarietà regolari, diciamo che
$M$ è trasversale rispetto a $S$ se e solo se per ogni $x\in S\cap M$:
$\displaystyle T_{x}(M)+T_{x}(S)=T_{x}(R)$
###### Proposizione 1.22.
Data $R$ varietà riemanniana e date $S,M$ sue sottovarietà regolari senza
bordo, se $S$ ed $M$ sono trasversali, allora $S\cap M$ è ancora una
sottovarietà regolare di $R$ con:
$\displaystyle codim(S\cap M)=codim(S)+codim(M)$
dove $codim(S)$ indica la codimensione della sottovarietà, cioè
$codim(S)\equiv dim(R)-dim(S)$
###### Proof.
La dimostrazione di questo risultato ed alcuni approfondimenti sulla teoria
della trasversalità possono essere trovati su [GP]. In particolare questo
teorema è il teorema pag. 30 del cap. 1. ∎
Con questo risultato possiamo dimostrare che:
###### Proposizione 1.23.
Data una sottovarietà regolare $S\subset R$ della stessa dimensione di $R$,
chiusa e con bordo regolare $\partial S$, esiste un’esaustione regolare
$K_{n}$ tale che $\partial K_{n}\cap S$ è una sottovarietà regolare di
codimensione $1$ con bordo.
###### Proof.
Come affermato sopra, è sufficiente dimostrare che $\partial K_{n}\cap\partial
S$ sia una sottovarietà regolare di codimensione $2$ (è il bordo della
sottovarietà $\partial K_{m}\cap S$).
Consideriamo una funzione di esaustione liscia positiva $h$ come nella
proposizione 1.20. La funzione $\bar{h}\equiv h|_{\partial S}$ è anch’essa una
funzione liscia propria e positiva. Sia $\\{A_{n},\phi_{n}\\}$ una successione
di aperti relativamente compatti di $R$ su cui siano definite le coordinate di
Fermi $\phi_{n}$ relative a $\partial S$ tali che
$\displaystyle\partial S\subset\bigcup_{n}A_{n}$
Chiamiamo le funzioni coordinate di Fermi $(x_{1},\cdots,x_{m-1},y)$ e
definiamo delle funzioni $f_{n}$ in modo che $f_{n}:A_{n}\to\mathbb{R}$,
$f_{n}|_{\partial S}=\bar{h}$ e che rispetto alle carte $\phi_{n}$ si abbia:
$\displaystyle\tilde{f_{n}}(x,y)\equiv
f_{n}\circ\phi_{n}^{-1}(x,y)=f_{n}(x,0)=\bar{h}(x)$
Cioè definiamo $f_{n}$ in modo che siano costanti sulle geodetiche
perpendicolari a $\partial S$.
Consideriamo ora il ricoprimento di aperti di $R$ dato da $\\{A_{n},(\partial
S)^{C}\\}$, e sia $\\{\lambda_{n},\lambda\\}$ una partizione dell’unità
subordinata a questo ricoprimento. Definiamo:
$\displaystyle F(p)\equiv\sum_{n}f_{n}(p)\lambda_{n}(p)+h(p)\lambda(p)$
Osserviamo subito che $F\in C^{\infty}(R,\mathbb{R}^{+})$. Inoltre vale che
per ogni punto $p\in\partial S$, $\nabla(F)|_{p}\in T_{p}(S)$. Infatti, sia
$p\in A_{n}\cap S$, indichiamo le sue coordinate rispetto alla carta
$\phi_{n}$ come $p=(\bar{x},0)$ 171717$\bar{x}$ rappresenta il vettore delle
prime $m-1$ coordinate :
$\displaystyle\left.\frac{\partial F}{\partial
y}\right|_{p}=\frac{\partial}{\partial t}F(\bar{x},t)=\frac{\partial}{\partial
t}\left(\sum_{n}f_{n}(\bar{x},t)\lambda_{n}(\bar{x},t)\right)=$
$\displaystyle=\frac{\partial}{\partial
t}\left(\sum_{n}h(\bar{x})\lambda_{n}(\bar{x},t)\right)=h(\bar{x})\frac{\partial}{\partial
t}\sum_{n}\lambda_{n}(\bar{x},t)=0$
dove abbiamo sfruttato il fatto che in un intorno di $\partial S$, la funzione
$\lambda=0$, quindi $\sum_{n}\lambda_{n}(\bar{x},t)=1$. Questo garantische che
l’ultima componente del differenziale sia nulla, e data la particolare forma
della metrica, anche l’ultima componente di $\nabla(F)|_{p}$ è nulla. Dato che
in questa carta l’insieme $\partial S$ è il piano avente ultima coordinata
nulla, questo dimostra che $\nabla(F)|_{p}\in T_{p}(\partial S)$.
Consideriamo ora un valore regolare $c\in\mathbb{R}$ di $F$ 181818grazie al
teorema di Sard, i valori regolari sono densi in $\mathbb{R}$. Allora
l’insieme $F^{-1}(c)$ è una sottovarietà regolare $F_{c}$ di $R$. Ricordiamo
che per queste sottovarietà:
$\displaystyle T_{p}(F_{c})=\\{\nabla(F)|_{p}\\}^{\perp}$
Dato che $\nabla(F)|_{p}\in T_{p}(\partial S)$:
$\displaystyle T_{p}(F_{c})+T_{p}(\partial
S)\supset\\{\nabla(F)|_{p}\\}^{\perp}+\\{\nabla(F)|_{p}\\}=T_{p}(R)$
quindi le sottovarietà $F_{c}$ e $\partial S$ sono trasversali, e la loro
intersezione è regolare.
Se verifichiamo anche che la funzione $F$ è una funzione propria, allora
possiamo scegliere una qualsiasi successione $c_{n}\nearrow\infty$ di valori
regolari di $F$ e definire
$\displaystyle K_{n}\equiv F^{-1}[0,c_{n}]$
per ottenere la tesi.
Per dimostrare che $F$ è propria, dimostriamo che per ogni $x\in R^{+}$,
esiste un compatto $C_{x}$ tale che
$\displaystyle p\not\in C_{x}\ \Rightarrow\ F(p)>x$
Sia $K=h^{-1}[0,x]$, insieme compatto per il fatto che $h$ è propria. Per
locale finitezza delle partizioni dell’unità, esiste un numero finito di
indici $n$ per il quale i supporti di $\lambda_{n}$ intersecano l’insieme $K$.
Indichiamo con $I_{x}$ l’insieme finito di questi indici e consideriamo
l’insieme compatto:
$\displaystyle C_{x}=K\bigcup_{n\in I_{x}}\overline{A_{n}}$
Supponiamo che $p\not\in C_{x}$. Allora $h(p)>x$, e anche
$f_{n}(p)=h(\pi(p))>x$, dove $\pi(p)$ indica la proiezione di $p$ su $\partial
S$, cioè il punto che nelle coordinate di Fermi è caratterizzato da:
$\displaystyle\bar{x}(\pi(p))=\bar{x}(p),\ \ y(\pi(p))=0$
Infatti, $p\not\in C_{x}\ \Rightarrow\ \pi(p)\not\in K$. Supponiamo per
assurdo il contrario. Dato che deve esistere $\bar{n}$ per cui $p\in
A_{\bar{n}}$, e dato che $\pi(p)\in A_{\bar{n}}$ per costruzione delle
coordinate di Fermi, allora se $\pi(p)\in K$ necessariamente $\bar{n}\in
I_{x}$, assurdo poiché abbiamo assunto $p\not\in C_{x}$.
Ricordando la definizione di $F$, è immediato verificare che $F(p)>x$. ∎
Ripetendo questa costruzione per una successione di sottovarietà $\\{S_{n}\\}$
con bordi disgiunti possiamo ottenere:
###### Proposizione 1.24.
Sia $\\{S_{n}\\}$ una successione di sottovarietà di $R$ della sua stessa
dimensione con bordi $\partial S_{n}$ lisci disgiunti tra loro. Allora esiste
un’esaustione regolare $K_{m}$ tale che per ogni $n$, $m$ l’insieme $\partial
K_{m}\cap S_{n}$ è una sottovarietà di $R$ $m-1$ dimensionale con bordo
liscio.
###### Proof.
L’idea della dimostrazione è la stessa della dimostrazione precedente.
Scegliamo per ogni sottovarietà $\partial S_{n}$ un insieme numerabile di suoi
intorni $\\{I_{n}^{m}\\}_{m=1}^{\infty}$ che ammettono coordinate di Fermi,
restringendoli in modo che questi intorni siano disgiunti dagli altri bordi
$\partial S_{k}$ (con $k\neq n$). Fissiamo inoltre una partizione dell’unità
subordinata al ricoprimento di aperti
$\left\\{\\{I_{n}^{m}\\}_{n,m=1}^{\infty};\ \bigcap_{n}\partial
S_{n}^{C}\right\\}$
Ripetendo una costruzione del tutto analoga alla precedente otteniamo la tesi.
∎
#### 1.1.7 Insiemi chiusi e funzioni lisce
Lo scopo di questo paragrafo è ottenere l’osservazione 1.27, che sarà utile
nel seguito del lavoro.
###### Lemma 1.25.
Dato un insieme chiuso $C\subset\mathbb{R}^{n}$, esiste una funzione
$f:\mathbb{R}^{n}\to\mathbb{R}$ liscia positiva tale che $C=f^{-1}(0)$.
###### Proof.
Grazie al fatto che $\mathbb{R}^{n}$ è uno spazio metrico, è facile trovare
una funzione $g:\mathbb{R}^{n}\to\mathbb{R}$ continua per cui $C=g^{-1}(0)$,
basta considerare infatti la funzione
$\displaystyle g(x)=d(x,C)\equiv\inf_{y\in C}d(x,y)$
ha le caratteristiche cercate. Consideriamo ora gli insiemi
$\displaystyle C_{\epsilon}\equiv\\{x\in\mathbb{R}^{n}\ t.c.\
d(x,C)\leq\epsilon\\}$
data la continuità della funzione $d(\cdot,C)$, tutti questi insiemi sono
insiemi chiusi, quindi esistono funzioni
$g_{\epsilon}:\mathbb{R}^{n}\to\mathbb{R}$ tali che
$g_{\epsilon}^{-1}(0)=C_{\epsilon}$. Notiamo che per ogni $y\in C$,
$B_{\epsilon}(y)\subset C_{\epsilon}$, quindi in particolare
$g(B_{\epsilon(y)})=0$ per ogni $y\in C$, e che
$\displaystyle
C=\bigcap_{\epsilon>0}C_{\epsilon}=\bigcap_{n=1}^{\infty}C_{1/n}$
Grazie al lemma 1.40, esiste un nucleo di convoluzione $\Theta_{\epsilon}$ con
supporto compatto contenuto in $B_{\epsilon}(0)$, e grazie al lemma 1.43, la
funzione
$\displaystyle g^{\prime}_{\epsilon}\equiv g_{\epsilon}\ast\Theta_{\epsilon}$
è una funzione liscia da $\mathbb{R}^{n}$ a $\mathbb{R}$ per la quale
$g^{\prime}_{\epsilon}(y)=0$ per ogni $y\in C$ e $g^{\prime}_{\epsilon}(x)>0$
per ogni $x\in\mathbb{R}^{n}$ tale che $d(x,C)>\epsilon$.
Definiamo la funzione
$\displaystyle
f(x)\equiv\sum_{n=1}^{\infty}\frac{g^{\prime}_{1/n}(x)}{A_{n}}\frac{1}{2^{n}}$
dove
$A_{n}=\max_{i=1}^{n}\left\|D^{i}g^{\prime}_{1/n}\right\|_{\infty,B_{n}(0)}$.
Grazie a questa definizione è facile verificare che la serie che definisce
$f(x)$ converge localmente uniformemente in $\mathbb{R}^{n}$ assieme alla
serie delle derivate di qualunque ordine, quindi la funzione $f$ è una
funzione liscia. Ovviamente $f$ è positiva, e dato che per ogni $n$,
$g^{\prime}_{1/n}(y)=0$ per ogni $y\in C$, anche $f(y)=0$ per ogni $y\in C$.
Se $y\not\in C$, grazie al fatto che $C$ è chiuso, esiste $\bar{n}$
sufficientemente grande per cui $y\not\in C_{1/\bar{n}}$, quindi dalle
considerazioni precedenti abbiamo che $g^{\prime}_{1/\bar{n}}(y)>0$, quindi
anche $f(y)>0$. Da cui la tesi. ∎
###### Osservazione 1.26.
Utilizzando le partizioni dell’unità, è possibile estendere questo lemma a una
varietà differenziale $M$ qualsiasi 191919se vogliamo $f$ liscia, la varietà
$M$ deve avere una struttura $C^{\infty}$.
###### Proof.
Sia $M$ una varietà differenziale qualsiasi e sia $\lambda_{n}$ una partizione
dell’unità di $M$ subordinata a un ricoprimento di carte locali
$(U_{n},\phi_{n})$. Sia $f_{n}$ la funzione descritta nel lemma precedente
relativa all’insieme chiuso in $\mathbb{R}^{m}$
$\phi_{n}(C\cap\leavevmode\nobreak\ supp(\lambda_{n}))$. Allora la funzione
$\displaystyle f(p)\equiv\sum_{n}f_{n}\circ\phi_{n}(p)\cdot\lambda_{n}(p)$
ha le caratteristiche cercate. Infatti dato che la somma è localmente finita,
$f$ è una funzione liscia. La positività di $f$ è ovvia conseguenza della
positività delle funzioni $f_{n}$ e $\lambda_{n}$; se $p\in C$, allora
$f_{n}(\phi_{n}(p))\cdot\lambda_{n}(p)=0$, mentre se $p\not\in C$, esiste
$\bar{n}$ tale che $\lambda_{\bar{n}}(p)>0$ 202020quindi in particolare $p\in
supp(\lambda_{\bar{n}})$ e quindi
$f_{n}(\phi_{n}(p))\cdot\lambda_{\bar{n}}(p)>0$. ∎
###### Osservazione 1.27.
Grazie alla precedente osservazione e al teorema di Sard, possiamo concludere
che per ogni insieme $C$ chiuso in una varietà $R$, esiste una successione di
insiemi aperti con bordo liscio $A_{n}\subset A_{n-1}$ in $R$ tali che
$C=\cap_{n}A_{n}$. Sia infatti $f$ una funzione con le caratteristiche appena
descritte. Consideriamo una successione di valori regolari $a_{n}$ di $f$ tali
che $a_{n}\searrow 0$, allora è facile verificare che gli insiemi
$\displaystyle A_{n}=f^{-1}[0,a_{n})$
soddisfano le proprietà richieste. Osserviamo inoltre che se $C$ è compatto,
gli insiemi possono essere scelti relativamente compatti (lasciamo al lettore
i facili dettagli di questa dimostrazione).
###### Osservazione 1.28.
Sia $\Omega$ un aperto di $R$. Allora esiste una successione di aperti $A_{n}$
con bordo liscio tali che:
$\displaystyle A_{n}\subset\Omega\ \ \ A_{n}\subset A_{n+1}\ \ \
\bigcup_{n}A_{n}=\Omega$
Se $\Omega$ è relativamente compatto, anche gli insiemi $A_{n}$ lo sono.
###### Proof.
Sia $f$ una funzione liscia positiva tale che $f^{-1}(0)=\Omega^{C}$, e sia
$\epsilon_{n}\searrow 0$ una successione di valori regolari per la funzione
$f$. È facile verificare che gli insiemi
$\displaystyle A_{n}\equiv f^{-1}(\epsilon_{n},\infty)$
hanno le caratteristiche desiderate. ∎
### 1.2 Funzioni assolutamente continue
In questa sezione riportiamo alcuni risultati sulle funzioni assolutamente
continue senza dimostrazione. Come referenza consigliamo il paragrafo 5.4 di
[R1].
###### Definizione 1.29.
Una funzione $f:U\to\mathbb{R}$ (dove $U\subset\mathbb{R}$ è un aperto) è
detta assolutamente continua se per ogni $\epsilon>0$, esiste $\delta>0$ tale
che per ogni scelta di $\\{x_{i},x_{i}^{\prime}\\}\subset U$ 212121in questa
definizione è equivalente chiedere che $i$ vari su un insieme finito o
numerabile di indici tali che $x_{i}^{\prime}>x_{i}>x_{i-1}^{\prime}$ e tali
che $\sum_{i}(x_{i}^{\prime}-x_{i})<\delta$, allora:
$\displaystyle\sum_{i}\left|f(x_{i}^{\prime})-f(x_{i})\right|<\epsilon$
###### Proposizione 1.30.
Se una funzione $f$ è assolutamente continua, allora la sua derivata esiste
finita quasi ovunque ed è Lebesgue integrabile sui compatti. Inoltre $f$ è
l’integrale della sua derivata se e solo se è assolutamente continua. Tutte le
funzioni di Lipschitz sono assolutamente continue.
###### Proposizione 1.31.
Date due funzioni assolutamente continue limitate $f$ e $g$, il loro prodotto
è assolutamente continuo.
###### Proof.
La dimostrazione segue facilmente dalla disuguaglianza:
$\displaystyle\left|f(x)g(x)-f(y)g(y)\right|\leq\left|f(x)\cdot(g(x)-g(y))\right|+\left|g(y)\cdot(f(x)-f(y))\right|\leq$
$\displaystyle\leq\max\\{\left\|f\right\|_{\infty},\left\|g\right\|_{\infty}\\}\cdot(\left|f(x)-f(y)\right|+\left|g(x)-g(y)\right|)$
∎
### 1.3 Spazi $\mathcal{L}^{2}$ di forme
In teoria dell’intergrazione sono famosi gli spazi $L^{p}$, spazi di funzioni
misurabili il cui modulo elevato alla $p$ è integrabile sullo spazio. Per
un’introduzione su questa teoria consigliamo [R4] (capitolo 3). I risultati
che ci interessano sono comunque riportati in questa breve rassegna:
###### Definizione 1.32.
Dato uno spazio di misura $(X,\mu)$ con $\mu$ misura positiva, definiamo per
$1\leq p<\infty$ $L^{p}(X,\mu)$ come lo spazio delle funzioni
$f:X\to\mathbb{R}$ (o anche $f:X\to\mathbb{C}$) misurabili 222222rispetto alla
misura $\mu$ e all’algebra di Borel su $\mathbb{R}$ tali che
$\displaystyle\left\|f\right\|_{p}^{p}\equiv\int_{X}\left|f\right|^{p}d\mu<\infty$
Se introduciamo la relazione di equivalenza $f\sim g$ se $f=g$ quasi ovunque,
lo spazio quoziente $L^{p}/\sim$ con la norma $\left\|\cdot\right\|_{p}$,
$L^{p}$ è uno spazio di Banach, e per $p=2$ uno spazio di Hilbert.
###### Proposizione 1.33.
Se $f_{n}\to f$ rispetto a una qualsiasi norma $\left\|\cdot\right\|_{p}$,
allora esiste una sottosuccesione $f_{n_{k}}$ che converge puntualmente quasi
ovunque a $f$. Questo implica anche che se $f_{n}$ converge uniformemente a
$f$ e $f_{n}$ converge in norma $p$ ad $h$, allora $f=h$ quasi ovunque.
###### Proposizione 1.34.
Data una varietà Riemanniana $R$, gli spazi $L^{p}(R)$ sono separabili per
$1\leq p<\infty$.
###### Proof.
Grazie al teorema 3.14 pag 68 di [R4], sappiamo che l’insieme delle funzioni
continue a supporto compatto in $R$ (che indicheremo $C_{C}(R)$ è denso nello
spazio $L^{p}(R)$. Sia $K_{n}$ un’esaustione di $R$, allora:
$C_{C}(R)=\bigcup_{n}C(K_{n})$
dato che $C(K_{n})$ dotato della norma del sup è separabile, e dato che la
norma del sup è più forte della norma $L^{p}$ su insiemi compatti, allora
$C(K_{n})$ è separabile anche nella norma $L^{p}$. Siano $D_{n}$ insiemi
numerabili densi in $C(K_{n})$, e sia $D=\cup_{n}D_{n}$. Allora $D$ è
numerabile e denso in $L^{p}(R)$, infatti data $f\in L^{p}(R)$ e dato
$\epsilon>0$, esiste $g\in C_{C}(R)$ tale che
$\left\|g-f\right\|_{p}<\epsilon$. Sia $K_{n}$ tale che $supp(g)\subset
K_{n}$, allora esiste $h\in D_{n}\subset D$ tale che
$\left\|g-h\right\|_{p}<\epsilon$, quindi:
$\displaystyle\left\|f-h\right\|_{p}\leq\left\|f-g\right\|_{p}+\left\|g-h\right\|_{p}\leq
2\epsilon$
data l’arbitrarietà di $\epsilon$, si ottiene la tesi. ∎
Una facile generalizzazione di questi spazi sono gli spazi $L^{p}$ per forme
su varietà Riemanniane. La teoria di questi spazi è del tutto analoga agli
spazi $L^{p}$ di funzioni, quindi anche in questo caso non riporteremo tutte
le dimostrazioni. Un riferimento su questo argomento può essere [S1], capitolo
7 232323in realtà questo libro si occupa solo di superfici riemanniane, ma in
questo frangente non c’è nessuna sostanziale differenza con una varietà di
dimensione generica.
###### Definizione 1.35.
Data una varietà Riemanniana $(R,g)$, definiamo lo spazio $\mathcal{L}^{2}(R)$
come lo spazio delle 1-forme a quadrato integrabile su $R$. Sia cioè
$\alpha\in T^{*}(R)$ una 1-forma (in coordinate locali
$\alpha=\sum_{i=1}^{m}\alpha_{i}(x)dx^{i}$), $\alpha\in\mathcal{L}^{2}(R)$ se
e solo se $\alpha_{i}$ sono tutte misurabili e
$\int_{R}\left|\alpha\right|^{2}dV<\infty$
Ricordiamo che l’integrale su una varietà è definito tramite partizioni
dell’unità. Sia $\\{\lambda_{n}\\}$ una partizione dell’unità di $R$
subordinata a un ricoprimento di aperti coordinati. Per definizione
$\displaystyle\int_{R}fdV=\sum_{n=1}^{\infty}\int_{supp(\lambda_{n})}\lambda_{n}\cdot
f\
dV=\sum_{n=1}^{\infty}\int_{\phi_{n}(supp(\lambda_{n}))}\tilde{\lambda}_{n}(x)\tilde{f}(x)\
\sqrt{\left|g\right|}dx^{1}\dots dx^{m}$
Ricoriamo anche che
$\left|\tilde{\alpha}(x)\right|^{2}=g_{ij}(x)\tilde{\alpha}^{i}(x)\tilde{\alpha}^{j}(x)$,
dove le funzioni con la tilde indicano le rappresentazioni in coordinate delle
relative funzioni.
Anche per questo spazio valgono gli analoghi delle proposizioni 1.33 e 1.34:
###### Proposizione 1.36.
Lo spazio $\mathcal{L}^{2}(R)$ è uno spazio di Hilbert, e se
$\alpha_{n}\to\alpha$ in norma, allora esiste una sottosuccesione
$\alpha_{n_{k}}$ che converge puntualmente quasi ovunque ad $\alpha$. Inoltre
$\mathcal{L}^{2}(R)$ è separabile.
Anche se in questa tesi non toccheremo l’argomento, osserviamo che
$\left|\alpha\right|^{2}dV=\alpha\ast\alpha$
dove $\ast$ indica l’operatore duale di Hodge.
Osserviamo che l’integrale di Dirichlet di una funzione
$\displaystyle\int_{R}\left|\nabla
f\right|^{2}dV=\int_{R}\left|df\right|^{2}dV$
è la norma nello spazio $\mathcal{L}^{2}(R)$ della 1-forma $df$.
### 1.4 Derivazione sotto al segno d’integrale
In questa sezione ripotiamo due lemmi che consentono sotto certe ipotesi di
scambiare derivata e integrale. Il secondo lemma è una generalizzazione del
primo, anche se più difficile da dimostrare.
###### Lemma 1.37 (Derivazione sotto al segno di integrale).
Sia $U\subset\mathbb{R}$ un aperto e $(\Omega,\mu)$ uno spazio di misura.
Supponiamo che $f:U\times\Omega\to\mathbb{R}$ abbia le proprietà:
1. 1.
$f(x,\omega)$ è Lebesgue-integrabile per ogni $x\in U$
2. 2.
quasi ovunque rispetto a $\mu$ la funzione $\frac{\partial
f(x,\omega)}{\partial x}$ esiste per ogni $x\in U$
3. 3.
esiste una funzione $g\in L^{1}(\Omega,\mu)$ tale che $\left|\frac{\partial
f(x,\omega)}{\partial x}\right|\leq g(\omega)$ per ogni $x\in U$
Allora vale che:
$\displaystyle\frac{d}{dx}\int_{\Omega}f(x,\omega)d\mu=\int_{\Omega}\frac{\partial}{\partial
x}f(x,\omega)d\mu$
###### Proof.
La dimostrazione è una semplice applicazione del teorema di convergenza
dominata (vedi teorema 1.34 pag. 26 di [R4]). ∎
Prima di enunciare il secondo lemma, riportiamo una proposizione che sarà
utile per la sua dimostrazione
###### Proposizione 1.38.
Se $f\in L^{1}(X,\mu)$, dove $(X,\mu)$ è uno spazio di misura positiva
qualsiasi, allora per ogni $\epsilon>0$ esiste $\delta>0$ tale che
$\displaystyle\mu(E)<\delta\Rightarrow\int_{E}\left|f\right|d\mu<\epsilon$
dove $E$ è un qualsiasi sottoinsieme misurabile di $(X,\mu)$.
###### Proof.
Osserviamo che questo è il testo dell’esercizio 12 pag 32 di [R4]. Supponiamo
per assurdo che esista $\epsilon>0$ tale che per ogni $n\in\mathbb{N}$ esiste
$E_{n}$ con $\mu(E_{n})<1/n$ tale che
$\int_{E_{n}}\left|f\right|\geq\epsilon$. Consideriamo la successione
$\left|f\right|\chi_{E_{n}}$. Questa successione converge in misura a $0$,
quindi per il teorema di convergenza dominata
$\int_{E_{n}}\left|f\right|d\mu=\int_{X}f\chi_{E_{n}}d\mu\to 0$, assurdo. ∎
###### Lemma 1.39 (Derivazione sotto al segno di integrale).
Sia $U\subset\mathbb{R}$ un aperto e $(\Omega,\mu)$ uno spazio di misura.
Supponiamo che $f:U\times\Omega\to\mathbb{R}$ abbia le proprietà:
1. 1.
$f(x,\omega)$ è misurabile su $U\times\Omega$, e per quasi ogni $x\in U$ è
integrabile su $\Omega$
2. 2.
quasi ovunque rispetto a $\mu$ la funzione $f(x,\omega)$ è assolutamente
continua in $x$
3. 3.
$\displaystyle\int_{U}dx\int_{\Omega}d\mu\left|\frac{\partial f}{\partial
x}(x,\omega)\right|<\infty$
Allora $F(x)\equiv\int_{\Omega}f(x,\omega)d\mu$ è assolutamente continua in
$x$ e per quasi ogni $x$:
$\displaystyle\frac{d}{dx}\int_{\Omega}f(x,\omega)d\mu=\int_{\Omega}\frac{\partial}{\partial
x}f(x,\omega)d\mu$
###### Proof.
Per prima cosa dimostriamo che $F(x)$ è assolutamente continua. Grazie alla
proposizione precedente (la 1.38), per ogni $\epsilon>0$ esiste $\delta>0$ per
cui se $\lambda(E)<\delta$ 242424$\lambda$ indica la misura di Lebesgue su
$\mathbb{R}$, allora
$\displaystyle\int_{E}dx\int_{\Omega}d\mu\left|\frac{\partial f}{\partial
x}(x,\omega)\right|<\epsilon$
Siano $\\{a^{i},b^{i}\\}$ tali che $b_{i}>a_{i}>b_{i-1}$ e
$\sum_{i}(b_{i}-a_{i})<\delta$, e chiamiamo $E=\cup_{i}(a^{i},b^{i})$. Allora
$\displaystyle\sum_{i}\left|F(b^{i})-F(a^{i})\right|=\sum_{i}\left|\int_{\Omega}f(b^{i},\omega)-f(a^{i},\omega)d\mu\right|=$
$\displaystyle=\sum_{i}\left|\int_{\Omega}\int_{a^{i}}^{b^{i}}\frac{\partial
f}{\partial x}(x,\omega)\ dx\
d\mu\right|\leq\int_{E}dx\int_{\Omega}d\mu\left|\frac{\partial f}{\partial
x}(x,\omega)\right|<\epsilon$
quindi $F$ è assolutamente continua in $x$. Questo significa che esiste quasi
ovunque $F^{\prime}(x)\equiv\partial F(x)/\partial x$ e che
$F(b)-F(a)=\int_{a}^{b}F^{\prime}(t)dt$ ogni volta che $[a,b]\subset U$. Ma
$\displaystyle F(b)-F(a)=\int_{\Omega}f(b,\omega)-f(a,\omega)\
d\mu=\int_{\Omega}\int_{a}^{b}\frac{\partial f}{\partial x}(x,\omega)\ dx\
d\mu=$ $\displaystyle=\int_{a}^{b}\int_{\Omega}\frac{\partial f}{\partial
x}(x,\omega)\ d\mu\ dx$
dove l’ultimo passaggio è giustificato dal teorema di Fubini (vedi teorema 7.8
pag 140 di [R4]). Ma allora si ha che per ogni $[a,b]\subset U$:
$\displaystyle\int_{a}^{b}\left(F^{\prime}(x)-\int_{\Omega}\frac{\partial
f}{\partial x}(x,\omega)\ d\mu\right)dx=0$
quindi grazie a una proprietà nota degli integrali 252525vedi teorema 1.39 (b)
pag 29 di [R4]
$\displaystyle F^{\prime}(x)=\int_{\Omega}\frac{\partial f}{\partial
x}(x,\omega)\ d\mu$
dove l’uguaglianza è intesa quasi ovunque in $x$ in ogni componente connessa
dell’insieme $U$, quindi in tutto $U$. ∎
### 1.5 Convoluzioni
In questa sezione introdurremo la convoluzione tra funzioni reali (non nella
forma più generale possibile, ma nella forma utile ai nostri scopi), e
esploreremo alcune tecniche di regolarizzazione di funzioni tramite
convoluzione. Per prima cosa dimostriamo l’esistenza dei nuclei di
convoluzione.
###### Lemma 1.40.
Per ogni $\alpha>0$ esiste una funzione positiva $\Theta_{\alpha}\in
C^{\infty}(\mathbb{R}^{m},\mathbb{R})$ con
$supp(\Theta_{\alpha})\Subset\overline{B_{\alpha}(0)}$ e
$\int_{\mbox{\footnotesize{$\mathbb{R}$}}^{m}}\Theta_{\alpha}(x)dx=1$.
Chiamiamo queste funzioni nuclei di convoluzione.
###### Proof.
È sufficiente trovare una funzione con le caratteristiche descritte per
$\alpha=1$. Infatti è facile verificare che la funzione
$\Theta_{\alpha}(x)\equiv\frac{1}{\alpha^{m}}\Theta\left(\frac{x}{\alpha}\right)$
verifica tutte le richieste.
Per trovare la funzione $\Theta_{1}$ basta considerare la funzione
$\displaystyle\tilde{\Theta}_{1}(x)\equiv\begin{cases}exp\left(\frac{1}{\left\|x\right\|^{2}-1}\right)&se\left\|x\right\|\leq
1\\\ 0&se\left\|x\right\|\geq 1\end{cases}$
$\displaystyle\Theta_{1}(x)\equiv\frac{\tilde{\Theta}_{1}(x)}{\int_{\mbox{\footnotesize{$\mathbb{R}$}}^{m}}\tilde{\Theta}_{1}(x)dx}$
oppure è possibile sfruttare l’esistenza delle partizioni dell’unità. Trovata
una funzione a supporto compatto $0\leq\lambda(x)\leq 1$ (non identicamente
nulla) con le tecniche descritte qui sopra la si può traslare e scalare in
modo da ottenere la funzione desiderata. ∎
Passiamo a definire la convoluzione, operazione che si rivelerà molto utile
per approssimare funzioni abbastanza generiche con una funzioni lisce. La
dimostrazione di queste (e altre) proprietà può essere trovata su [R4] al
§7.13, oppure su [F1] al §8.2.
###### Proposizione 1.41.
Disuguaglianza di Young: Date $f\in L_{1}(\mathbb{R}^{m})$ e $g\in
L_{p}(\mathbb{R}^{m})$ ($1\leq p\leq\infty$), si definisce:
$f*g(x)=\int_{\mbox{\footnotesize{$\mathbb{R}$}}}\ f(x-y)g(y)dy$
Questa definizione ha senso solo quasi ovunque e vale che:
$\left\|f*g\right\|_{p}\leq\left\|f\right\|_{1}\cdot\left\|g\right\|_{p}$
(1.6)
###### Proposizione 1.42.
Date $f\in L_{p}(\mathbb{R}^{m})$ e $g\in L_{q}(\mathbb{R}^{m})$ con
$\frac{1}{p}+\frac{1}{q}=1,\ 1\leq p\leq\infty$, ha senso definire $\forall
x$:
$f*g(x)=\int_{\mbox{\footnotesize{$\mathbb{R}$}}}\ f(x-y)g(y)dy$
e vale che $f*g$ è uniformemente continua su $\mathbb{R}^{m}$ e:
$\left\|f*g\right\|_{\infty}\leq\left\|f\right\|_{p}\cdot\left\|g\right\|_{q}$
Osserviamo subito che $\ast$ è un’operazione commutativa, cioè $f\ast g=g\ast
f$. La convoluzione può essere utilizzata per regolarizzare una funzione, cioè
per trovare una funzione liscia vicina a piacere alla funzione data. I
dettagli sono nei lemmi seguenti.
Per comodità di notazione, ricordiamo brevemente la definizione di
multiindice. Un vettore di numeri interi non negativi $\vec{k}=(k_{1},\cdots
k_{m})$ è detto multiindice. La sua lunghezza è per definizione
$\left|\vec{k}\right|=\sum_{i=1}^{m}k_{i}$, inoltre con il simbolo
$D^{\vec{k}}f$ intendiamo:
$\displaystyle
D^{\vec{k}}f\equiv\frac{\partial^{\left|\vec{k}\right|}f}{\partial
x_{1}^{k_{1}}\cdots\partial x_{m}^{k_{m}}}$
###### Lemma 1.43.
Sia $\Theta\in C^{r}(\mathbb{R}^{m},\mathbb{R})$ con
$supp(\Theta)\Subset\overline{B_{\alpha}(0)}$ e $\int\Theta(x)dx=1$ con $0\leq
r\leq\infty$, sia $f\in L^{1}(\mathbb{R}^{m})$. Allora si ha che:
1. 1.
$(\Theta\ast f)\in C^{r}(\mathbb{R}^{m},\mathbb{R})$
2. 2.
Per ogni $\left|\vec{k}\right|\leq r$, $D^{\vec{k}}(\Theta\ast
f)|_{x}=(D^{\vec{k}}(\Theta)\ast f)|_{x}$
Inoltre se $f\in C^{s}(\mathbb{R}^{m},\mathbb{R})$ ($0\leq s\leq\infty$) con
$supp(f)=C$, allora si ha anche che:
1. 1.
$(\Theta\ast f)\in C^{s}(\mathbb{R}^{m},\mathbb{R})$
2. 2.
Per ogni $\left|\vec{k}\right|\leq s$, $D^{\vec{k}}(\Theta\ast
f)|_{x}=(\Theta\ast(D^{\vec{k}}f))|_{x}$
3. 3.
$supp(\Theta\ast f)\subset C+\alpha=\\{x\in\mathbb{R}^{m}\ t.c.\
d(x,C)\leq\alpha\\}$
###### Proof.
La dimostrazione è una semplice applicazione del lemma 1.37. ∎
Nel seguito avremo bisogno di una versione più raffinata di questo lemma, in
particolare con richieste meno stringenti sulla regolarità di $f$. A questo
scopo dimostriamo che:
###### Lemma 1.44.
Sia $f\in C(\mathbb{R}^{m})$ con
$supp(f)=K\Subset\prod_{i=1}^{m}(a_{i},b_{i})\equiv U$. Sia inoltre
$f(\bar{x}^{1},\cdots,x^{i},\cdots,\bar{x}^{m})$ assolutamente continua
rispetto a $x^{i}$ quasi ovunque rispetto a
$(\bar{x}^{1},\cdots,\bar{x}^{i-1},\bar{x}^{i+1},\cdots,\bar{x}^{m})$, con
$\partial f/\partial x^{i}\in L^{1}(\mathbb{R}^{m})$. Allora se $\Theta$ è un
nucleo di convoluzione:
$\displaystyle\frac{\partial}{\partial x^{i}}(\Theta\ast
f)=\Theta\ast\frac{\partial f}{\partial x^{i}}$
###### Proof.
Questo lemma è conseguenza del lemma 1.39, ma per completezza riportiamo anche
una dimostrazione più “elementare”.
Per comodità di notazione indicheremo
$\displaystyle\frac{\partial}{\partial x^{i}}\equiv\partial_{i}$
Osserviamo che quasi ovunque rispetto alle $\bar{x}$,
$f(\bar{x},x^{i})=\int_{-\infty}^{x^{i}}\partial_{i}f(\bar{x},t)dt$.
Consideriamo $f^{\prime}_{n}$ successione di funzioni continue a supporto
compatto $f^{\prime}_{n}:\mathbb{R}^{m}\to\mathbb{R}$ tali che
$\left\|f^{\prime}_{n}-\partial_{i}f\right\|_{L^{1}}\to 0$ 262626questa
successione esiste per densità di $C_{C}(\mathbb{R}^{m})$ in
$L^{1}(\mathbb{R}^{m})$, vedi teorema 3.14 pag 68 di [R4]. Definiamo
$\displaystyle
f_{n}(\bar{x},x^{i})=\int_{-\infty}^{x^{i}}f^{\prime}_{n}(\bar{x},t)dt$
in questo modo la successione $f_{n}$ converge in norma $L_{1}$ a $f$,
infatti:
$\displaystyle\left\|f_{n}-f\right\|_{L^{1}}\leq\int_{\mbox{\footnotesize{$\mathbb{R}$}}^{m}}\int_{\infty}^{x^{i}}\left|f^{\prime}_{n}(\bar{x},x^{i})-\partial_{i}f(\bar{x},x^{i})\right|dt\
d\bar{x}dx^{i}\leq$
$\displaystyle\leq(b_{i}-a_{i})\left\|f^{\prime}_{n}-\partial_{i}f\right\|_{L^{1}}\to
0$
Grazie ai lemmi precedenti osserviamo che
$\displaystyle\partial_{i}(\Theta\ast f_{n})=(\partial_{i}\Theta)\ast
f_{n}\to(\partial_{i}\Theta)\ast f=\partial_{i}(\Theta\ast f)$
e anche
$\displaystyle\partial_{i}(\Theta\ast
f_{n})=\Theta\ast(\partial_{i}f_{n})=\Theta\ast
f^{\prime}_{n}=\to\Theta\ast(\partial_{i}f)$
dove la convergenza nelle ultime due uguaglianze è intesa nel senso di
$L^{1}$. Per unicità del limite, $\partial_{i}(\Theta\ast
f)=\Theta\ast(\partial_{i}f)$ nel senso di $L^{1}$, quindi quasi ovunque. Ma
le funzioni $\Theta\ast(\partial_{i}f)$ e $\partial_{i}(\Theta\ast f)$ sono
funzioni lisce, quindi uguaglianza quasi ovunque implica uguaglianza ovunque.
∎
Ora analizziamo le proprietà di approssimazione della convoluzione. A questo
scopo utilizzeremo la notazione
$\displaystyle\left\|f\right\|_{\infty}\equiv\max_{x\in\mathbb{R}^{m}}\left|f(x)\right|$
###### Lemma 1.45.
Sia $f\in C^{r}(\mathbb{R}^{m},\mathbb{R})$ con $supp(f)=K$ compatto e sia
$\vec{k}$ un multiindice di lunghezza $\left|\vec{k}\right|\leq r$. Dato
$\epsilon>0$, esiste $\alpha>0$ tale che per ogni $\Theta$ con supporto in
$\overline{B_{\alpha}(0)}$ e $\int\Theta(x)dx=1$ si ha che
$\left\|D^{\vec{k}}(\Theta\ast f)-D^{\vec{k}}(f)\right\|_{\infty}<\epsilon$.
###### Proof.
Per uniforme continuità di $D^{\vec{k}}(f)$ su $K$, vale che dato
$\epsilon>0$, esiste $\alpha>0$ tale che
$\left\|x-y\right\|\leq\alpha\Rightarrow\left|D^{\vec{k}}f|_{y}-D^{\vec{k}}f|_{x}\right|<\epsilon$.
Dato che:
$\displaystyle D^{\vec{k}}(\Theta\ast
f)|_{x}-D^{\vec{k}}(f)|_{x}=\int_{B_{\alpha}(0)}{\Theta(y)}(D^{\vec{k}}(f)|_{x-y}-D^{\vec{k}}(f)|_{x})dy$
si ottiene:
$\displaystyle\left|D^{\vec{k}}(\Theta\ast
f)|_{x}-D^{\vec{k}}(f)|_{x}\right|\leq$
$\displaystyle\leq\int_{B_{\alpha}(0)}\left|{\Theta(y)}\right|\left|(D^{\vec{k}}(f)|_{x-y}-D^{\vec{k}}(f)|_{x})\right|dy\leq\epsilon\int_{B_{\alpha}(0)}\left|{\Theta(y)}\right|=\epsilon$
∎
Questo lemma garantisce che ogni funzione a supporto compatto può essere
approssimata uniformemente a ogni suo ordine di regolarità con una funzione
liscia.
### 1.6 Duali di spazi di Banach
In questa sezione ricordiamo brevemente alcuni risultati riguardo agli spazi
duali degli spazi di Banach, in particolare la definizione di topologia
debole-* e il teorema di Banach-Alaoglu. Per approfondimenti rimandiamo ai
capitoli 3 e 4 di [R2].
###### Definizione 1.46.
Dato uno spazio di Banach (reale) $(X,\left\|\cdot\right\|)$, si definisce
$X^{*}$ il suo spazio duale, cioè:
$\displaystyle X^{*}=\\{\phi:X\to\mathbb{R}\ \ t.c.\ \phi\ lineare\ e\
continuo\\}$
Possiamo rendere questo spazio uno spazio di Banach con la norma:
$\displaystyle\left\|\phi\right\|^{*}\equiv\sup_{x\neq
0}\frac{\left|\phi(x)\right|}{\left\|x\right\|}=\sup_{\left\|x\right\|\leq
1}\left|\phi(x)\right|=\sup_{\left\|x\right\|=1}\left|\phi(x)\right|$
Notiamo che per un funzionale lineare, essere continuo, essere continuo nel
punto 0 ed essere limitato (cioè avere $\left\|\phi\right\|^{*}<\infty$) sono
proprietà equivalenti. Sullo spazio $X^{*}$ però è possibile definire anche
una topologia vettoriale più debole della topologia indotta da questa norma,
la topologia debole-*.
###### Definizione 1.47.
Sullo spazio $X^{*}$ definiamo $\tau^{*}$ la topologia debole-*, una topologia
invariante per traslazioni tale che una base di intorni del punto $0$ è
costituita dagli insiemi
$\displaystyle V(0,\epsilon,x_{1},\cdots,x_{n})=\\{\phi\ t.c.\
\left|\phi(x_{i})\right|<\epsilon\ \forall\ 1\leq i\leq n\\}$
al variare di $\epsilon$ e di $x_{1},\cdots,x_{n}\in X$ 272727al variare degli
elementi e del numero degli elementi, che può essere un numero finito
qualsiasi. Questa topologia rende $X^{*}$ uno spazio vettoriale topologico.
Ricordiamo che per gli spazi di Hilbert reali esiste un’isometria lineare tra
lo spazio $H$ e il suo duale $H^{*}$, isometria data dal teorema di
rappresentazione di Riesz (vedi ad esempio teorema 3.4 di [C3]).
Per la topologia debole-* vale un teorema molto famoso, il teorema di Banach
Alaoglu (vedi teorema 3.15 pag 68 di [R2]). In questa rassegna riportiamo una
versione del teorema adatta ai nostri scopi
###### Teorema 1.48 (Teorema di Banach-Alaoglu).
Nello spazio $(X^{*},\tau^{*})$, l’insieme
$B^{*}=\\{\left\|\phi\right\|^{*}\leq 1\\}$
è compatto. Inoltre se $X$ è separabile, allora $B^{*}$ è anche
sequenzialmente compatto.
###### Proof.
La compattezza di $B^{*}$ è dimostrata nel teorema 3.15 pag 68 di [R2], per
quanto riguarda la compattezza per successioni, il teorema 3.17 a pagina 70 di
[R2] garantisce che se $X$ è separabile, allora gli insiemi compatti di
$X^{*}$ sono metrizzabili, quindi anche sequenzialmente compatti. ∎
### 1.7 Funzioni armoniche
In questa sezione riportiamo alcuni risultati riguardo alle funzioni armoniche
su varietà riemanniane. In tutta la sezione $R$ sarà una varietà Riemanniana
liscia senza bordo di dimensione $m$.
#### 1.7.1 Principio del massimo
In questo paragrafo riportiamo alcuni risultati sugli operatori ellittici, in
particolare il principio del massimo.
###### Definizione 1.49.
Un operatore differenziale lineare del secondo ordine
$D:C^{2}(\Omega,\mathbb{R})\to C(\Omega,\mathbb{R})$
dove $\Omega$ è un aperto in $\mathbb{R}^{m}$ è detto ellittico se ha la forma
$\displaystyle
D(f)=a^{ij}(x)\partial_{i}\partial_{j}f+b^{i}(x)\partial_{i}f+c(x)f$
dove le funzioni $a$ e $b$ sono lisce in $\Omega$ e la matrice $a^{ij}$ è
definita positiva (quindi ha tutti gli autovalori strettamente maggiori di
$0$).
L’operatore $D$ è detto strettamente ellittico se esiste un numero positivo
$\lambda>0$ tale che per ogni vettore $v$ e per ogni $x\in\Omega$
$\displaystyle a^{ij}(x)v_{i}v_{j}\geq\lambda\sum_{i}\left|v_{i}\right|^{2}$
o equivalentemente se l’autovalore minimo della matrice $a^{ij}$ è limitato
dal basso sull’insieme $\Omega$.
L’operatore $D$ è detto uniformemente ellittico su $\Omega$ se il rapporto tra
l’autovalore massimo e l’autovalore minimo di $a^{ij}(x)$ è limitato
(indipendentemente da $x$).
La definizione di operatore ellittico può essere estesa facilmente a operatori
su varietà Riemanniane chiedendo semplicemente che ogni loro rappresentazione
locale abbia le caratteristiche descritte sopra.
La teoria di questi operatori viene sviluppata in maniera esaustiva su [GT],
testo dal quale estraiamo solo i risultati che serviranno in seguito per
questa tesi.
Una proprietà interessante di questi operatori è che se i coefficienti
$a^{ij},b^{i},c$ sono funzioni lisce e una funzione $C^{2}$ soddisfa $Du=f$
con $f\in C^{\infty}(\Omega)$, allora automaticamente la funzione $u\in
C^{\infty}(\Omega)$. Anzi si può dimostrare che questo continua a valere anche
se $Du=f$ solo nel senso delle distrubuzioni.
Un’altra proprietà che sfrutteremo molto in questo lavoro è il principio del
massimo, che permette di controllare il valore di una funzione con i suoi
valori al bordo dell’insieme di definizione.
###### Proposizione 1.50 (Principio del massimo).
Sia $D$ un operatore ellittico in un dominio relativamente compatto $\Omega$.
Se la funzione $u:\Omega\to\mathbb{R}$ soddisfa:
1. 1.
$u\in C^{2}(\Omega)\cap C^{0}(\overline{\Omega})$
2. 2.
$Du=0$
allora il massimo e il minimo di $u$ su $\overline{\Omega}$ sono raggiunti sul
bordo $\partial\Omega$. Cioè:
$\displaystyle\sup_{x\in\Omega}u(x)=\sup_{x\in\partial\Omega}u(x)\ \ \
\inf_{x\in\Omega}u(x)=\inf_{x\in\partial\Omega}u(x)$
Inoltre se $u$ non è continua su $\overline{\Omega}$, la conclusione può
essere sostituita da:
$\displaystyle\sup_{x\in\Omega}u(x)=\limsup_{x\to\partial\Omega}u(x)\ \ \
\inf_{x\in\Omega}u(x)=\liminf_{x\to\partial\Omega}u(x)$
dove con $\limsup_{x\to\partial\Omega}u(x)$ intendiamo il limite di
$\sup_{x\in K_{n}^{C}}u(x)$ quando $K_{n}$ è un’esaustione di $\Omega$.
Rimandiamo al teorema 3.1 pagina 31 di [GT] per la dimostrazione di questo
teorema.
Vale un principio simile anche se $\Omega$ non è relativamente compatto,
infatti:
###### Proposizione 1.51.
Sia $D$ un operatore ellittico in un dominio $\Omega$, e sia $u$ una funzione
$u\in C^{2}(\Omega)\cap C^{0}(\overline{\Omega})$. Allora
$\displaystyle\sup_{x\in\Omega}u(x)\leq\limsup_{x\to\infty}u(x)\ \ \
\inf_{x\in\Omega}u(x)\geq\liminf_{x\to\infty}u(x)$
dove con $\limsup_{x\to\infty}u(x)$ intendiamo il limite di $\sup_{x\in
K_{n}^{C}}u(x)$ quando $K_{n}$ è un’esaustione di $\Omega$.
###### Proof.
Supponiamo per assurdo che
$\sup_{x\in\Omega}u(x)>\limsup_{x\to\infty}u(x)\equiv L$, e consideriamo
l’insieme $A=u^{-1}(L,\infty)$, aperto non vuoto per ipotesi. Dato che
$\limsup_{x\to\infty}u(x)=L$, $A$ è relativamente compatto, e applicando il
precedente principio a questo insieme si ottiene che $u\equiv L$ in $A$,
assurdo. ∎
Una forma leggermente più forte del principio del massimo è la seguente:
###### Proposizione 1.52 (Principio del massimo).
Sia $D$ un operatore uniformemente ellittico con $c=0$. Se una funzione $u$
soddisfa $Du=0$ assume il suo massimo in un punto interno a $\Omega$, allora è
constante
Rimandiamo al teorema 3.5 pagina 34 di [GT] per la dimostrazione di questo
teorema.
Il principio del massimo può essere espresso anche in forme più sofisticate,
alcune delle quali verranno esposte nel seguito della tesi.
Osserviamo subito che questi principi possono essere applicati nel caso
dell’operatore laplaciano su varietà Riemanniane. Infatti la rappresentazione
locale del laplaciano (vedi 1.2) dimostra che questo è un operatore ellittico
282828ricordiamo che per definizione di metrica la matrice $g^{ij}$ è definita
postiva sulla varietà $R$, quindi vale il corollario:
###### Proposizione 1.53.
Sia $u$ una funzione armonica in $\Omega$ dominio relativamente compatto in
$R$ e continua fino al bordo. Allora il massimo e il minimo della funzione
sono assunti sul bordo $\partial\Omega$.
Inoltre è facile verificare che su ogni insieme compatto $K\Subset R$
l’operatore $\Delta$ è uniformemente ellittico, quindi:
###### Proposizione 1.54.
Sia $u$ una funzione armonica in $\Omega$ dominio relativamente compatto in
$R$. Se $u$ assume il suo massimo in un punto interno a $\Omega$, allora $u$ è
costante su $\Omega$.
Ovviamente qualunque funzione armonica su tutta la varietà $R$ è armonica su
ogni dominio relativamente compatto, quindi per una funzione di questo tipo
che non sia costante vale che:
$\displaystyle\inf_{x\in\partial\Omega}u(x)<u(p)<\sup_{x\in\partial\Omega}u(x)$
per ogni punto $p\in\Omega$.
#### 1.7.2 Stime sul gradiente
Uno strumento importantissimo nello studio delle funzioni armoniche sono le
stime sul gradiente, cioé stime sul modulo del gradiente di una funzione
armonica positiva. Riportiamo solo il risultato, la cui dimostrazione può
essere trovata su [SY] 292929teorema 3.1 pagina 17
###### Proposizione 1.55.
Sia $R$ una varietà Riemanniana completa con $dim(R)\equiv m\geq 2$, e sia
$B_{2r}(x_{0})$ la bolla geodetica di raggio $2r$ centrata in $x_{0}$.
Supponiamo che $u$ sia una funzione armonica positiva su $B_{2r}(x_{0})$, e
sia $Ric(M)\geq-(m-1)K$ su $B_{2r}$ 303030essendo la curvatura di Ricci una
funzione continua su $R$, su ogni insieme compatto assume un minimo finito,
quindi la costante $K$ si può sempre trovare dove $K\geq 0$ è una costante.
Allora:
$\displaystyle\frac{\left|\nabla u\right|}{u}\leq C_{m}\frac{1+r\sqrt{K}}{r}$
sull’insieme $B_{r}(x_{0})$, dove $C_{m}$ è una costante che dipende solo
dalla dimensione $m$ della varietà.
Osserviamo che sul testo [SY], il teorema è enunciato in una forma diversa, si
richiede infatti che il limite inferiore sulla curvatura valga su tutta la
varietà $R$. Dalla dimostrazione però è evidente che questa ipotesi può essere
rilassata.
#### 1.7.3 Disuguaglianza, funzione e principio di Harnack
Un’altra proprietà che vale per le funzioni armoniche su varietà è la
disuguaglianza di Harnack:
###### Proposizione 1.56 (Disuguaglianza di Harnack).
Dato un dominio $\Omega$ e un insieme $K\Subset\Omega$, esiste una costante
$\Lambda$ 313131chiamata costante di Harnack che dipende solo da $\Omega$ e
$K$ tale che per ogni funzione armonica positiva $u$ su $\Omega$ vale che:
$\displaystyle\sup_{x\in K}u(x)\leq\Lambda\inf_{x\in K}u(x)$
Questa disuguaglianza è una conseguenza del corollario 8.21 pagina 189 di
[GT].
Grazie a questa disuguaglianza siamo in grado di provare il principio di
Harnack, che riguarda successioni di funzioni armoniche positive su varietà
###### Proposizione 1.57 (Principio di Harnack).
Sia $u_{m}$ una successione crescente di funzioni armoniche positive su
$\Omega$ dominio in $R$. Allora o $u_{m}$ diverge localmente uniformemente, o
converge localmente uniformemente in $\Omega$.
Se la successione di funzioni $u_{m}$ è uniformemente limitata (non
necessariamente positiva o crescente), allora esiste una sua sottosuccessione
che converge localmente uniformemente su $\Omega$.
Questo principio di può trovare su [ABR] pagina 49 323232in realtà il testo
[ABR] tratta il caso del laplaciano standard in $\mathbb{R}^{m}$, ma per molte
dimostrazioni le tecniche usate si basano su proprietà (come la disuguaglianza
di Harnack) che valgono anche nel caso di laplaciano su varietà, quindi
possono essere facilmente estese.
Questo principio è molto significativo anche perché vale che
###### Proposizione 1.58.
Se una successione di funzioni armoniche $u_{n}$ converge localmente
uniformemente a una funzione $u$, allora $u$ è armonica.
Oltre alla disuguaglianza di Harnack e alla relativa costante, possiamo
definire una funzione di Harnack in questo modo:
###### Definizione 1.59.
Su una varietà Riemanniana $R$ dato un aperto $\Omega$, possiamo definire una
funzione $k:\Omega\times\Omega\to[1,\infty)$ nel seguente modo:
$\displaystyle k(x,y)\equiv\sup\\{c\ t.c.\ c^{-1}u(x)\leq u(y)\leq cu(x)\ \
\forall\ u\in HP(\Omega)\\}$
dove $HP(\Omega)$ è l’insieme delle funzioni armoniche positive su $\Omega$.
Chiamiamo questa funzione funzione di Harnack.
Grazie all’esistenza della costante di Harnack $\Lambda(K,\Omega)$, sappiamo
che la funzione $k$ è ben definita su $\Omega$, infatti dati due punti
$x,y\in\Omega$, se consideriamo un compatto $K$ che li contiene, abbiamo che:
$\displaystyle k(x,y)\leq\Lambda(K)$
però vale anche un’altra importante proprietà di questa funzione:
###### Proposizione 1.60.
Per ogni $\Omega$ dominio in $R$, e per ogni $x\in\Omega$, vale che:
$\displaystyle\lim_{y\to x,\ y\in\Omega}k(x,y)=1$
###### Proof.
Questa dimostrazione è un’applicazione delle stime sul gradiente 1.55.
Consideriamo una bolla geodetica $B_{2r}(x)\subset\Omega$, e sia $K$ la
costante descritta nella proposizione 1.55. Sia $y\in B_{r}(x)$ tale che
$d(x,y)\equiv d$, sia $\gamma:[0,d]\to R$ la geodetica che unisce $x$ e $y$ e
per una qualsiasi funzione $u$ armonica positiva su $\Omega$ definiamo la
funzione
$\displaystyle\phi:[0,d]\to R\ \ \ \ \ \phi(t)\equiv\log\circ
u\circ\gamma(t)=\log(u(\gamma(t)))$
Sappiamo che
$\displaystyle\phi(d)-\phi(0)=\log\left(\frac{u(y)}{u(x)}\right)$
e grazie alle stime sul gradiente possiamo osservare che:
$\displaystyle\left|\phi(d)-\phi(0)\right|=\left|\int_{0}^{d}\frac{d\phi}{dt}(s)ds\right|\leq\int_{0}^{d}\left|\frac{d\phi}{dt}(s)\right|ds=\int_{0}^{d}\left|\left\langle\nabla\log(u)\middle|\frac{d\gamma}{dt}\right\rangle\right|ds=$
$\displaystyle=\int_{0}^{d}\frac{\left|\nabla u\right|}{u}ds\leq
C_{m}\frac{1+r\sqrt{K}}{r}d$
Questo significa che per qualunque funzione armonica positiva $u$:
$\displaystyle\frac{u(y)}{u(x)}\leq\exp\left(C_{m}\frac{1+r\sqrt{K}}{r}d(x,y)\right)$
e anche:
$\displaystyle\frac{u(x)}{u(y)}\leq\exp\left(C_{m}\frac{1+r\sqrt{K}}{r}d(x,y)\right)$
quindi questa quantità tende a $1$ se $d(x,y)\to 0$. ∎
#### 1.7.4 Funzioni di Green
###### Definizione 1.61.
Una funzione di Green per un insieme aperto $\Omega\subset R$ rispetto a un
punto $p$ è una funzione $G\in C^{\infty}(\Omega\times\Omega\setminus
D;\mathbb{R})$, dove $D=\\{(p,p)\ t.c.\ p\in\Omega\\}$ è la diagonale di
$\Omega\times\Omega$, tale che:
1. 1.
$G$ è strettamente positiva su $\Omega$
2. 2.
$G$ è simmetrica, cioè $G(p,q)=G(q,p)$
3. 3.
Fissato $p\in\Omega$, la funzione $G(p,q)$ è armonica rispetto a $q$
sull’insieme $\Omega\setminus\\{p\\}$ e superarmonica su tutto $\Omega$.
4. 4.
$G$ soddisfa la condizione di Dirichlet al bordo, cioè per ogni $p\in\Omega$,
$G(p,q)=0$ per ogni $q\in\partial\Omega$ 333333il valore di $G$ sul bordo è
inteso come il limite per $q_{n}\to q$ dove $q_{n}\in\Omega$
5. 5.
$G$ è soluzione fondamentale dell’operatore $\Delta$, cioé per ogni funzione
liscia $f$ a supporto compatto in $\Omega$:
$\displaystyle\Delta_{x}\int_{\Omega}G(x,y)f(y)dy=\int_{\Omega}G(x,y)\Delta_{y}(f)(y)dy=-f(x)$
questo significa che nel senso delle distribuzioni
$\Delta_{y}G(x,y)=-\delta_{x}$
6. 6.
Il flusso di $G_{\Omega}(\ast,p)$ attraverso il bordo di un’insieme regolare
$K\Subset\Omega$ con $p\not\in\partial K$ vale:
$\displaystyle\int_{\partial K}\ast dG(\cdot,p)=\begin{cases}-1&se\ p\in K\\\
0&se\ p\not\in K\end{cases}$
7. 7.
La funzione $G$ ha un comportamento asintotico della forma:
$\displaystyle G(x,y)\sim C(m)\begin{cases}-log(d(x,y))&m=2\\\
d(x,y)^{m-2}&m\geq 3\end{cases}$
quando $d(x,y)\to 0$. La costante $C(m)$ dipende solo dalla dimensione della
varietà e può essere determinata sfruttando la condizione (5).
Segnaliamo che per domini $\Omega$ con bordo liscio esiste unica la funzione
di Green associata a questo dominio. Un riferimento per questa proposizione è
[PSR] a pagina 165.
###### Proposizione 1.62.
Dato un insieme aperto relativamente compatto $\Omega$ con bordo liscio,
esiste unica la funzione di Green $G_{\Omega}$
Osserviamo che grazie al principio del massimo possiamo dimostrare che
###### Proposizione 1.63.
Sia $\Omega$ un dominio relativamente compatto dal bordo liscio in $R$ e sia
$G(\cdot,p)$ la funzione di Green relativa a $\Omega$. Se $K$ è un dominio
relativamente compatto tale che
$\displaystyle p\in K\Subset\Omega$
Allora la funzione $G(\cdot,p)$ rispetto all’insieme
$\overline{\Omega}\setminus K$ assume il suo massimo su $\partial K$. Inoltre
se $p\in K\Subset K^{\prime}\Subset\Omega$, $G$ assume il suo massimo rispetto
a $\overline{K^{\prime}}\setminus K$ su $\partial K$.
###### Proof.
La dimostrazione segue dal principio del massimo applicato all’insieme
$\Omega\setminus K$. Essendo $G(\cdot,p)|_{\partial\Omega}=0$ per definizione,
ed essendo $\Delta$ un operatore uniformemente ellittico su
$\overline{\Omega}\setminus K$ 343434grazie alla compattezza di questo
insieme, vale il principio 1.52, e quindi
$\displaystyle G(\cdot,p)|_{\Omega\setminus\overline{K}}<\max_{x\in\partial
K}G(x,p)$ (1.7)
Questo dimostra anche che sull’insieme compatto $\partial
K^{\prime}\Subset\Omega\setminus\overline{K}$ il massimo è strettamente minore
del massimo di $G$ su $\partial K$. ∎
Se $R$ non è compatta, ha senso chiedersi se esiste una funzione con proprietà
simili a quelle descritte definita su tutta la varietà. La risposta a questa
domanda è legata alla parabolicità della varietà $R$ che introdurremo in
seguito, e si trova nella sezione 4.3
#### 1.7.5 Singolarità di funzioni armoniche
Grazie al principio del massimo e alle funzioni di Green siamo in grado di
caratterizzare le singolarità delle funzioni armoniche positive.
###### Proposizione 1.64.
Dato un dominio aperto relativamente compatto $\Omega\subset R$ con bordo
liscio e una funzione
$v\in H(\Omega\setminus\\{x_{0}\\})\cap
C^{0}(\overline{\Omega}\setminus\\{x_{0}\\})$
positiva 353535in realtà è sufficiente che sia limitata dal basso, detta
$G_{0}$ la funzione di Green $G_{\Omega}(\cdot,x_{0})$, se
$\displaystyle\lim_{d(x,x_{0})\to 0}\frac{v(x)}{G_{0}(x)}=0$
allora la funzione $v$ è estendibile a una funzione armonica su tutta
$\Omega$.
###### Proof.
Sia $\Omega^{\prime}$ un dominio aperto relativamente compatto con bordo
liscio tale che
$\displaystyle
x_{0}\in\Omega^{\prime}\subset\overline{\Omega^{\prime}}\Subset\Omega$
Sia $[v]$ la soluzione del problema di Dirichlet su $\Omega^{\prime}$ con
valore al bordo $v|_{\partial\Omega^{\prime}}$. L’obiettivo della
dimostrazione è mostrare che $v=[v]$ su
$\Omega_{0}\equiv\Omega^{\prime}\setminus\\{x_{0}\\}$. A questo scopo
chiamiamo $\delta=v-[v]$ la funzione differenza. Evidentemente
$\displaystyle\delta\in H(\Omega_{0})\cap
C^{0}({\overline{\Omega}^{\prime}\setminus\\{x_{0}\\}})\ \
\delta|_{\partial\Omega^{\prime}}=0$
Consideriamo la funzione $\phi(x)=\frac{\delta(x)}{G_{0}(x)}$ definita su
$\Omega_{0}$. Questa funzione è identicamente nulla sul bordo di
$\Omega^{\prime}$ e per $x$ che tende a $x_{0}$ il suo limite vale $0$
363636grazie al fatto che $G_{0}$ tende a infinito in $x_{0}$, $[v]$ è
limitata e questa proprietà vale per il rapporto $v/G_{0}$. Inoltre questa
funzione soddisfa:
$\displaystyle\nabla\left(\frac{\delta}{G_{0}}\right)=\frac{\nabla\delta}{G_{0}}-\frac{\delta\nabla
G_{0}}{G_{0}^{2}}$
$\displaystyle\Delta(\phi)=div(\nabla(\phi))=div\left(\frac{\nabla\delta}{G_{0}}-\frac{\delta\nabla
G_{0}}{G_{0}^{2}}\right)=$
$\displaystyle=\frac{\Delta\delta}{G_{0}}-2\frac{\left\langle\nabla\delta\middle|\nabla
G_{0}\right\rangle}{G_{0}^{2}}-\delta\frac{\Delta
G_{0}}{G_{0}^{2}}+2\frac{\delta\left\|\nabla
G_{0}\right\|^{2}}{G_{0}^{3}}=-2\frac{\left\langle\nabla\delta\middle|\nabla
G_{0}\right\rangle}{G_{0}^{2}}+2\frac{\delta\left\|\nabla
G_{0}\right\|^{2}}{G_{0}^{3}}=$
$\displaystyle-\frac{2}{G_{0}}\left(\left\langle\frac{\nabla\delta}{G_{0}}\middle|\nabla
G_{0}\right\rangle-\left\langle\frac{\delta\nabla
G_{0}}{G_{0}^{2}}\middle|\nabla
G_{0}\right\rangle\right)=-2\left\langle\nabla\phi\middle|\frac{\nabla
G_{0}}{G_{0}}\right\rangle$
quindi:
$\displaystyle\Delta(\phi)+2\left\langle\nabla\phi\middle|\frac{\nabla
G_{0}}{G_{0}}\right\rangle=0$
Dato che l’operatore
$\displaystyle
D(\cdot)=\Delta(\cdot)+2\left\langle\nabla\cdot\middle|\frac{\nabla
G_{0}}{G_{0}}\right\rangle$
è un’operatore ellittico su $\Omega_{0}$ (i coefficienti sono funzioni lisce e
$\Delta$ è ellittico), allora grazie al principio del massimo 1.50 otteniamo
che $\phi\equiv 0$ su tutto l’insieme $\Omega_{0}$, da cui la tesi. ∎
Grazie a questa proposizione siamo in grado di dimostrare il corollario:
###### Proposizione 1.65.
Dato un dominio aperto relativamente compatto $\Omega\subset R$ con bordo
liscio e una funzione
$v\in H(\Omega\setminus\\{x_{0}\\})\cap
C^{0}(\overline{\Omega}\setminus\\{x_{0}\\})$
positiva 373737in realtà è sufficiente che sia limitata dal basso, detta
$G_{0}$ la funzione di Green $G_{\Omega}(\cdot,x_{0})$, se
$\displaystyle\lim_{d(x,x_{0})\to 0}\frac{v(x)}{G_{0}(x)}=c$
allora esiste una funzione armonica $\delta$ su $\Omega$ continua fino al
bordo tale che
$\displaystyle v(x)=\delta(x)+cG_{0}(x)$
###### Proof.
Basta applicare la proposizione precedente alla funzione
$v(x)-cG_{0}(x)$
∎
#### 1.7.6 Principio di Dirichlet
In questo paragrafo ci occupiamo di introdurre il principio di Dirichlet nella
sua forma più standard. Questo principio afferma che le funzioni armoniche
sono le funzioni che hanno integrale di Dirichlet minimo in una certa famiglia
di funzioni. È possibile rilassare le ipotesi di regolarità su queste
funzioni, come riportato nella sezione 3.2.4.
Data una funzione su una varietà $f:R\to\mathbb{R}$ e un insieme misurabile
$\Omega$, possiamo definire il suo integrale di Dirichlet come:
$\displaystyle D_{\Omega}(f)\equiv\int_{\Omega}\left|\nabla f\right|^{2}dV$
e in maniera simile definiamo anche
$\displaystyle D_{\Omega}(f,h)\equiv\int_{\Omega}\left\langle\nabla
f\middle|\nabla g\right\rangle dV$
Il principio di Dirichlet 383838o meglio una versione del principio, in
seguito dimostreremo versioni con ipotesi meno restrittive sulla regolarità
delle funzioni in gioco afferma che:
###### Proposizione 1.66 (Principio di Dirichlet).
Dato un dominio regolare 393939un insieme aperto relativamente compatto con
bordo liscio a tratti $\Omega$ e una funzione continua
$h:\partial\Omega\to\mathbb{R}$, per ogni funzione
$f:\overline{\Omega}\to\mathbb{R}$ liscia tale che $f=h$ su $\partial\Omega$:
$\displaystyle D_{\Omega}(f)=D_{\Omega}(u)+D_{\Omega}(f-u)$
dove $u$ è l’unica soluzione del problema di Dirichlet che ha $h$ come valore
al bordo 404040cioè $u$ è l’unica funzione continua in $\overline{\Omega}$,
tale che $\Delta u=0$ in $\Omega$ e $u=h$ su $\partial\Omega$. In particolare
$u$ ha integrale di Dirichlet minimo tra tutte le funzioni $f$.
La dimostrazione di questo principio si può trovare su [S2], sezione 7.1
pagina 173.
#### 1.7.7 Funzioni super e subarmoniche
Oltre alle funzioni armoniche, è possibile definire altre due categorie di
funzioni legate all’armonicità: le funzioni superarmoniche e le funzioni
subarmoniche. Prima di definire queste funzioni, ricordiamo alcune definizioni
preliminari.
###### Definizione 1.67.
Una funzione $f:R\to\mathbb{R}\cup\\{+\infty\\}$ si dice semicontinua
inferiormente se vale una delle seguenti proprietà equivalenti:
1. 1.
per ogni $x\in R$, $f(x)\leq\liminf_{y\to x}f(y)$
2. 2.
per ogni $a\in R$, $\\{x\in\mathbb{R}\ t.c.\ f(x)>a\\}$ è un’insieme aperto
3. 3.
per ogni $a\in R$, $\\{x\in\mathbb{R}\ t.c.\ f(x)\leq a\\}$ è un’insieme
chiuso
$f$ si dice semicontinua superiormente se e solo se $-f$ è semicontinua
inferiormente, o equivalentemente se e solo se
$f:R\to\mathbb{R}\cup\\{-\infty\\}$ e:
1. 1.
per ogni $x\in R$, $f(x)\geq\limsup_{y\to x}f(y)$
2. 2.
per ogni $a\in R$, $\\{x\in\mathbb{R}\ t.c.\ f(x)<a\\}$ è un’insieme aperto
3. 3.
per ogni $a\in R$, $\\{x\in\mathbb{R}\ t.c.\ f(x)\geq a\\}$ è un’insieme
chiuso
Ricordiamo che:
###### Proposizione 1.68.
Una successione crescente di funzioni $f_{n}$ continue converge a una funzione
semicontinua inferiormente. Una successione decrescente di funzioni continue
converge a una funzione semicontinua superiormente. Il minimo e il massimo tra
due (o tra un numero finito) di funzioni semicontinue inferiormente o
superiormente è ancora una funzione semicontinua inferiormente o
superiormente.
Il seguente lemma sarà utile per confrontare funzioni sub e superarmoniche. La
sua formulazione può essere data con ipotesi meno restrittive, ma per gli
scopi della tesi è sufficiente assumere di lavorare su spazi metrici. Questo
lemma è tratto del lemma 4.3 pag 171 di [D].
###### Lemma 1.69.
Sia $X$ uno spazio metrico, e siano $G:X\to\mathbb{R}$ semicontinua
superiormente, $g:X\to\mathbb{R}$ semicontinua inferiormente. Se $G(x)<g(x)$
per ogni $x\in X$, allora esiste una funzione continua $\phi:X\to\mathbb{R}$
tale che per ogni $x\in X$:
$\displaystyle G(x)<\phi(x)<g(x)$
###### Proof.
Costruiamo la funzione $\phi$ grazie alle partizioni dell’unità. Per ogni
numero razionale positivo $r>0$, sia
$\displaystyle U_{r}=\\{x\in X\ t.c.\ G(x)<r\\}\cap\\{x\in X\ t.c.\ g(x)>r\\}$
per semicontinuità tutti questi insiemi sono aperti, inoltre evidentemente
formano un ricoprimento dell’insieme $X$. Dato che $X$ è uno spazio metrico,
ogni ricoprimento aperto ammette una partizione dell’unità subordinata a tale
ricoprimento. Siano $\lambda_{r}$ le funzioni di questa partizione, definiamo:
$\displaystyle\phi(x)=\sum_{r}r\lambda_{r}$
Dato che la somma è localmente finita, la funzione $\phi:X\to\mathbb{R}$ è
continua su $X$. Inoltre, dato che per ogni $r$ $supp(\lambda_{r})\subset
U_{r}$, si ha che per ogni $x\in X$:
$\displaystyle
G(x)=G(x)\sum_{r}\lambda_{r}<\phi(x)<g(x)\sum_{n}\lambda_{r}=g(x)$
∎
Passiamo ora a trattare le funzioni sub e superarmoniche.
Intuitivamente, una funzione superarmonica è una funzione che confrontata con
una funzione armonica è maggiore di questa funzione. Se la funzione $v$ è
continua, si dice che è superarmonica se e solo se per ogni compatto con bordo
liscio $K$, la funzione armonica determinata da $v|_{\partial K}$ è minore
della funzione $v$ su tutto $K$. È però possibile rilassare l’ipotesi sulla
continuità di $v$. Comunque vogliamo che abbia senso confrontare la funzione
$v$ con una funzione armonica $f$ su $K$ tale che $f|_{\partial K}\leq
v_{\partial K}$. Se vogliamo che $f\leq v$ su tutto $K$, è necessario chiedere
che la funzione $v$ sia semicontinua inferiormente. Quindi definiamo:
###### Definizione 1.70.
Una funzione $v:R\to\mathbb{R}\cup\\{+\infty\\}$ si dice superarmonica se e
solo se è una funzione semicontinua inferiormente e se per ogni insieme
compatto $K\Subset R$ con bordo liscio e per ogni funzione $f$ armonica in
$K^{\circ}$, continua su $K$ e tale che $f|_{\partial K}\leq v|_{\partial K}$,
allora $f\leq v$ su tutto l’insieme $K$.
Una funzione $v:R\to\mathbb{R}\cup\\{-\infty\\}$ si dice subarmonica se e solo
se $-v$ è superarmonica, quindi se e solo se è una funzione semicontinua
superiormente e se per ogni insieme compatto $K\Subset R$ con bordo liscio e
per ogni funzione $f$ armonica in $K^{\circ}$, continua su $K$ e tale che
$f|_{\partial K}\geq v|_{\partial K}$, allora $f\geq v$ su tutto l’insieme
$K$.
Osserviamo subito che la richiesta che $K$ abbia bordo liscio può essere
rilassata.
###### Proposizione 1.71.
Sia $\Omega$ un dominio aperto relativamente compatto in $R$. Se $v$ è
superarmonica su $\overline{\Omega}$ e se $f\in
H(\Omega)\cap(C(\overline{\Omega}))$ è tale che:
$v|_{\partial\Omega}\geq f|_{\partial\Omega}$
allora $v\geq f$ su tutto l’insieme $\Omega$.
###### Proof.
Sia $\epsilon>0$. Data la semicontinuità inferiore di $v-f$, l’insieme
$\Omega_{\epsilon}\equiv(v-f)^{-1}(-\epsilon,\infty)=\\{x\ t.c.\
v(x)>f(x)-\epsilon\\}$
è aperto, e per ipotesi contiene $\partial\Omega$. Grazie all’osservazione
1.28, esiste una successione di aperti relativamente compatti con bordo liscio
$K_{n}$ tali che
$\displaystyle K_{n}\Subset\Omega\ \ \ K_{n}\Subset K_{n+1}\ \ \
\bigcup_{n}K_{n}=\Omega$
È facile verificare che $\Omega_{\epsilon}^{C}\subset K_{n}$ definitivamente.
Infatti $\Omega_{\epsilon}^{C}$ è un compatto di $R$ ed è ricoperto dagli
aperti $\\{K_{n}\\}$.
Dato che $f\in H(\overline{K}_{n})$ e dato che definitivamente in $n$,
$v|_{\partial K_{n}}<f|_{\partial K_{n}}-\epsilon$
poiché il bordo di $K_{n}$ è liscio, per la definizione di superarmonicità si
ha che definitivamente in $n$:
$\displaystyle v|_{K_{n}}\geq f|_{K_{n}}-\epsilon\ \ \Rightarrow\ \
v|_{\Omega}\geq f|_{\Omega}-\epsilon$
per l’arbitrarietà di $\epsilon$ si ottiene la tesi. ∎
Le funzioni sub e superarmoniche si possono confrontare tra loro, in
particolare si ha che:
###### Proposizione 1.72.
Sia $K$ un compatto con bordo liscio, e siano $u$ subarmonica su $K$ e $v$
superarmonica su $K$, allora se $u|_{\partial K}\leq v|_{\partial K}$ la
disuguaglianza vale su tutto l’insieme $K$.
###### Proof.
Grazie al lemma 1.69, sappiamo che per ogni $\epsilon>0$ esiste una funzione
continua $\phi:\partial K\to\mathbb{R}$ tale che
$\displaystyle u|_{\partial K}<\phi_{\epsilon}<v|_{\partial K}+\epsilon$
Sia $\Phi_{\epsilon}$ la soluzione del problema di Dirichlet su $K$ con
condizioni al bordo $\phi_{\epsilon}$. Allora per definizione di
superarmonicità, sappiamo che:
$\displaystyle u\leq\Phi_{\epsilon}\leq v+\epsilon$
su tutto l’insime $K$. Data l’arbitrarietà di $\epsilon$, otteniamo la tesi. ∎
###### Osservazione 1.73.
Con una tecnica analoga a quella utilizzata per la dimostrazione della
proposizione 1.71, si può dimostrare che 1.72 vale anche se si toglie
l’ipotesi di liscezza del bordo di $K$.
Osserviamo che condizione necessaria e sufficiente affinché una funzione $v$
sia armonica su $R$ è che sia contemporaneamente sub e superarmonica.
Una proprietà elementare delle funzioni sub e superarmoniche è che:
###### Proposizione 1.74.
Il minimo in una famiglia finita di funzioni superarmoniche è superarmonico, e
il massimo in una famiglia finita di funzioni subarmoniche è subarmonico.
Lo spazio delle funzioni sub e superarmoniche è un cono in uno spazio
vettoriale, più precisamente:
###### Proposizione 1.75.
Combinazioni lineari a coefficienti positivi di funzioni superarmoniche sono
superarmoniche.
###### Proof.
Se $f$ è superarmonica, è ovvio che per ogni $t\geq 0$, anche $tf$ è una
funzione superarmonica. Resta da dimostrare che la somma mantiene la proprietà
di superarmonicità. È facile dimostrare che la somma di una funzione armonica
e una superarmonica è superarmonica, e analogamente la somma di una funzione
armonica e una subarmonica è subarmonica. Consideriamo ora due funzioni
$v_{1}$ e $v_{2}$ entrambe superarmoniche. Sia $K$ un compatto con bordo
liscio in $R$ e sia $u\in H(K^{\circ})\cap C(K)$ tale che:
$\displaystyle u|_{\partial K}\leq(v_{1}+v_{2})|_{\partial K}=v_{1}|_{\partial
K}+v_{2}|_{\partial K}$
dato che la funzione $u-v_{1}$ è subarmonica mentre $v_{2}$ è superarmonica,
grazie alla proposizione 1.72 sappiamo che $u-v_{1}\leq v_{2}$ su tutto
l’insieme $K$, da cui la tesi. ∎
Evidentemente vale un’affermazione analoga per le funzioni subarmoniche.
Riportiamo ora due proposizioni che saranno utili in seguito per dimostrare la
superarmonicità dei potenziali di Green.
###### Proposizione 1.76.
Data una funzione $f:R\times R\to\mathbb{R}$ continua per cui per ogni $y\in
R$, $f(\cdot,y)$ è superarmonica in $R$, e data una misura di Borel positiva a
supporto compatto $K$ con $\mu(K)=\mu(R)<\infty$, la funzione
$\displaystyle F(x)\equiv\int_{R}f(x,y)d\mu(y)$
è una funzione superarmonica.
###### Proof.
Grazie al teorema di convergenza dominata, è facile dimostrare che la funzione
$F$ è continua su $R$, quindi in particolare semicontinua inferiormente.
Consideriamo ora una successione con indice $n$ di partizioni di $K$
costituite da insiemi $E_{k}^{(n)}$ tali che il diametro di ogni $E_{k}^{(n)}$
sia minore di $1/n$, e per ogni $n$ e $k$ sia $y_{k}^{(n)}$ un punto qualsiasi
dell’insieme $E_{k}^{(n)}$. La successione di funzioni
$\displaystyle
F^{n}(x)=\int_{K}f(x,y_{k}^{(n)})\chi(E_{k}^{(n)})(y)d\mu(y)=\sum_{k}f(x,y_{k}^{(n)})\mu(E_{k}^{(n)})$
è una successione di funzioni superarmoniche grazie alla proposizione 1.75,
inoltre converge localmente uniformemente alla funzione $F(x)$. Sia infatti
$C$ un qualsiasi insieme compatto in $R$. L’insieme $C\times K$ è compatto in
$R\times R$, e quindi la funzione $f(x,y)$ è uniformemente continua su questo
insieme. Questo significa che per ogni $\epsilon>0$, esiste $\delta>0$ tale
che uniformemente in $x$ si ha:
$\displaystyle d(y_{1},y_{2})<\delta\ \Rightarrow\
\left|f(x,y_{1})-f(x,y_{2})\right|<\epsilon$
Allora per ogni $\epsilon>0$, se scegliamo $n$ in modo che $1/n\leq\delta$,
otteniamo:
$\displaystyle\left|F(x)-F^{n}(x)\right|\leq\int_{K}\left|f(x,y)-\sum_{k}f(x,y_{k}^{(n)})\chi(E_{k}^{(n)})(y)\right|d\mu(y)\leq\mu(K)\epsilon$
Ora osserviamo che una successione di funzioni superarmoniche che converge
localmente uniformemente ha limite superarmonico.
Infatti sia $C$ un compatto con bordo liscio in $R$ e $u$ una funzione
armonica sulla parte interna di $C$ continua fino al bordo tale che
$\displaystyle u|_{\partial C}(x)\leq F|_{\partial C}(x)$
Allora per ogni $\epsilon>0$, esiste un $N$ tale che per ogni $n\geq N$:
$\displaystyle F^{n}|_{\partial C}\geq F|_{\partial C}-\epsilon\geq
u|_{\partial C}-\epsilon$
Per superarmonicità di $F_{n}$, vale che su tutto $C$:
$\displaystyle F^{n}(x)\geq u(x)-\epsilon$
Passando al limite su $n$ e grazie all’arbitrarietà di $\epsilon$, otteniamo
la tesi. ∎
###### Proposizione 1.77.
Una successione crescente di funzioni continue superarmoniche ha limite
superarmonico.
###### Proof.
Sia $f_{n}(x)$ una successione crescente di funzioni continue superarmoniche,
e sia $f(x)$ il suo limite 414141automaticamente $f$ è una funzione
semicontinua inferiormente. Per dimostrare la superarmonicità di $f$,
consideriamo $K$ un compatto con bordo liscio in $R$ e una funzione $u$
armonica sull’interno di $K$ continua fino al bordo con $u|_{\partial K}\leq
f|_{\partial K}$. Consideriamo la successione di funzioni
$\displaystyle g_{n}(x)=\min\\{f_{n}|_{\partial K},u|_{\partial K}\\}$
questa successione è una successione di funzioni continue, crescenti, e ha
come limite la funzione continua $u|_{\partial K}$. Grazie al teorema di Dini
(riportato dopo questa dimostrazione, teorema 1.78) la convergenza è uniforme,
quindi per ogni $\epsilon>0$, esiste $n$ tale che
$\displaystyle g_{n}|_{\partial K}\geq u|_{\partial K}-\epsilon\ \Rightarrow\
f_{n}|_{\partial K}\geq u|_{\partial K}-\epsilon$
data la superarmonicità di $f_{n}$, si ha che
$\displaystyle f_{n}(x)\geq u(x)-\epsilon$
per ogni $x\in K$. Per monotonia si ottiene che per ogni $x\in K$:
$\displaystyle f(x)\geq u(x)-\epsilon$
data l’arbitrarietà di $\epsilon$, si ottiene la tesi. ∎
Riportiamo ora il teorema di Dini. Per ulteriori approfondimenti su questo
teorema rimandiamo al testo [L] (il teorema di Dini è il teorema 1.3 pag 381).
###### Teorema 1.78 (Teorema di Dini).
Sia $X$ uno spazio topologico compatto, e sia $f_{n}$ una successione
crescente di funzioni $f_{n}:X\to\mathbb{R}$ continue. Se $f_{n}$ converge
puntualmente a una funzione $f:X\to\mathbb{R}$ continua, allora la convergenza
è uniforme.
###### Proof.
Fissato $\epsilon>0$, consideriamo $X_{n}$ gli insiemi aperti
$\displaystyle X_{n}\equiv\\{x\in X\ t.c.\ f(x)-f_{n}(x)<\epsilon\\}$
Dato che $f_{n}$ è una successione crescente, $X_{n}\subset X_{n+1}$, e vista
la convergenza puntuale di $f$, sappiamo che $X=\cup_{n}X_{n}$. Per
compattezza di $X$, esiste un indice $\bar{n}$ per cui $X=X_{\bar{n}}$, cioè
per ogni $x\in X$ e per ogni $n\geq\bar{n}$ si ha che
$\displaystyle f_{n}(x)>f(x)-\epsilon$
quindi la convergenza è uniforme. ∎
Concludiamo questo paragrafo con una proposizione che sarà spesso utilizzata
nel seguito.
###### Proposizione 1.79.
Dato $K$ compatto con bordo liscio in $R$, se $f$ è armonica in $R\setminus
K$, costante sull’insieme $K$, minore o uguale alla costante fuori da $K$ e
continua su $R$, allora è superarmonica.
###### Proof.
Senza perdita di generalità, supponiamo che $f\equiv 1$ sull’insieme $K$.
Osserviamo che se esiste $v$ superarmonica su $R$ tale che $v|_{R\setminus
K}=f|_{R\setminus K}$, questa proposizione è conseguenza del fatto che il
minimo tra funzioni superarmoniche è superarmonico. Non è necessario però
richiedere che esista una tale funzione.
Sia $C$ un insieme compatto con bordo liscio in $R$, e $u$ una funzione
armonica sulla parte interna di $C$ continua fino al bordo la cui restrizione
al bordo sia minore o uguale alla restrizione di $f$. Se $C\subset R\setminus
K$ o $C\subset K$ non c’è niente da dimostrare. Negli altri casi, sia
$C_{1}\equiv C\cap K$ e $C_{2}\equiv C\cap K^{C}$. Grazie al principio del
massimo, sappiamo che $u\leq 1$ su tutto l’insieme $C$, quindi in particolare
su $C_{1}$. Sempre con il principio del massimo (la forma descritta in 1.53),
confrontando $u$ e $f$ sull’insieme $\overline{C_{2}}$ abbiamo la tesi. ∎
### 1.8 Algebre di Banach e caratteri
Questa sezione è dedicata a una breve rassegna sulle algebre di Banach e
alcuni risultati che saranno utili nello svolgimento della tesi. Per
approfondimenti sull’argomento consigliamo il testo [R2], in particolare il
capitolo 10.
Iniziamo con il ricordare alcune definizioni di base.
###### Definizione 1.80.
Un’algebra associativa è uno spazio vettoriale su un campo $\mathbb{K}$ (che
d’ora in avanti noi assumeremo sempre essere $\mathbb{R}$) con un’operazione
di moltiplicazione associativa e distributiva rispetto alla somma. In simboli,
uno spazio vettoriale $V$ è un’algebra associativa se è definita una funzione
$\cdot:V\times V\to V$ tale che:
1. 1.
$(x\cdot y)\cdot z=x\cdot(y\cdot z)$
2. 2.
$(x+y)\cdot z=x\cdot z+y\cdot z$
3. 3.
$z\cdot(x+y)=z\cdot z+z\cdot y$
4. 4.
$a(x\cdot y)=(ax)\cdot y=x\cdot(ay)$
Per ogni $x,y,z\in V$ e per ogni $a\in\mathbb{K}$.
Nel seguito il simbolo di moltiplicazione $\cdot$ sarà sottointeso quando
questo non causerà confusione.
Un’algebra di Banach è semplicemente un’algebra associativa dotata di una
norma compatibile con le operazioni di somma e moltiplicazione e che sia
completa rispetto a questa norma.
###### Definizione 1.81.
Un’algebra di Banach è un’algebra associativa su cui è definita un’operazione
$\left\|\cdot\right\|:V\to\mathbb{R}^{+}$ tale che $\left\|x\cdot
y\right\|\leq\left\|x\right\|\left\|y\right\|$ e che lo spazio normato
$(V,\left\|\cdot\right\|)$ sia completo (o di Banach).
Si dice che l’algebra di Banach $A$ sia dotata di unità se esiste un elemento
$e$ tale che per ogni $x\in A$, $ex=xe=x$ e anche $\left\|e\right\|=1$. È
facile dimostrare che se esiste, l’elemento $e$ è unico. D’ora in avanti ci
occuperemo solo di algebre di Banach con unità sul campo dei numeri reali.
##### Elementi invertibili
Nelle algebre di Banach con unità (che indicheremo con $A$ in tutta la
sezione), ha senso parlare di “elementi invertibili”. Si dice invertibile un
elemento $x$ se esiste $x^{-1}$ tale che $xx^{-1}=x^{-1}x=e$. Grazie
all’associatività è facile dimostrare che gli elementi invertibili sono chiusi
rispetto alla moltiplicazione, e ovviamente $e$ è un elemento invertibile, il
cui inverso è sè stesso. È facile dimostrare che una categoria particolare di
elementi di $A$ è sempre invertibile:
###### Proposizione 1.82.
Se $\left\|x\right\|<1$, allora $(e-x)$ è invertibile in $A$.
###### Proof.
Lo scopo di questa dimostrazione è provare che
$\displaystyle(e-x)^{-1}=\sum_{i=0}^{\infty}x^{i}$
dove per convenzione $x^{0}=e$. Per prima cosa, grazie al fatto che
$\left\|x\right\|<1$, la serie converge totalmente, quindi converge essendo
$A$ uno spazio di Banach. Inoltre:
$\displaystyle\left(\sum_{i=0}^{\infty}x^{i}\right)\cdot(e-x)=\left(\lim_{n\to\infty}\sum_{i=0}^{n}x^{i}\right)\cdot(e-x)=\lim_{n\to\infty}\left[\left(\sum_{i=0}^{n}x^{i}\right)\cdot(e-x)\right]=$
$\displaystyle=\lim_{n\to\infty}\left[\sum_{i=0}^{n}x^{i}-\sum_{i=1}^{n+1}x^{i})\right]=e-\lim_{n\to\infty}x^{n+1}=e$
e un ragionamento del tutto analogo vale per
$(e-x)\cdot\left(\sum_{i=0}^{\infty}x^{i}\right)$. ∎
##### Caratteri
Introduciamo ora una classe particolare di funzionali lineari sulle algebre di
Banach, i caratteri, o funzionali moltiplicativi. Anche in questa sezione
riporteremo solo i risultati che interessano agli scopi della tesi, e per
ulteriori approfondimenti rimandiamo a [R2].
###### Definizione 1.83.
Un funzionale lineare $\phi:A\to\mathbb{R}$ si dice moltiplicativo se conserva
la moltiplicazione, cioè se $\forall x,y\in A$
$\displaystyle\phi(x\cdot y)=\phi(x)\phi(y)$
Le proprietà che risultano evidenti dalla definizione sono che:
1. 1.
$\phi(e)=1$
2. 2.
per ogni elemento invertibile $x$, $\phi(x)\neq 0$
3. 3.
$\phi(x^{-1})=\left(\phi(x)\right)^{-1}$
Grazie all’ultima proprietà e alla proposizione 1.82 possiamo dimostrare che
qualunque carattere è necessariamente continuo e ha norma 1 424242ricordiamo
che la norma di un funzionale può essere definita da
$\left\|\phi\right\|\equiv\sup_{\left\|x\right\|\leq
1}(\left|\phi(x)\right|)$, un funzionale è continuo se e solo se ha norma
finita.
###### Proposizione 1.84.
Ogni carattere $\phi$ ha norma 1.
###### Proof.
Dal fatto che $\phi(e)=1$ si ricava facilmente che $\left\|\phi\right\|\geq
1$. Consideriamo ora un qualunque $x$ con norma minore o uguale a 1, e un
qualunque numero reale $\lambda$ con $\left|\lambda\right|>1$. Dalla
proposizione 1.82 segue che $e-\lambda^{-1}x$ è un elemento invertibile,
quindi
$\displaystyle\phi(e-\lambda^{-1}x)\neq 0\Longleftrightarrow
1-\frac{\phi(x)}{\lambda}\neq 0\Longleftrightarrow\phi(x)\neq\lambda$
Questa considerazione è valida per qualunque numero $\left|\lambda\right|>1$,
e quindi si ricava che $\sup_{\left\|x\right\|\leq
1}\\{\left|\phi(x)\right|\\}\leq 1$, da cui la tesi. ∎
È utile osservare che:
###### Proposizione 1.85.
L’insieme $\mathcal{C}$ dei funzionali lineari moltiplicativi è compatto
rispetto alla topologia debole-*.
###### Proof.
Grazie al teorema 1.48, è sufficiente dimostare che $\mathcal{C}$ è chiuso
nella topologia debole-*434343ricordiamo che questo insieme è limitato in
norma.. A questo scopo, consideriamo $\phi\in\overline{\mathcal{C}}$, e
verifichiamo se $\phi(xy)=\phi(x)\phi(y)$. Dato che
$\phi\in\overline{\mathcal{C}}$, per ogni $\epsilon>0$, esiste
$\tilde{\phi}\in\mathcal{C}$ tale che:
$\displaystyle\tilde{\phi}\in V(\phi,\epsilon,x,y,xy)\equiv\\{\psi\in A^{*}\
t.c.\ \left|\psi(t)-\phi(t)\right|<\epsilon\ \ t=x,y,xy\\}$
quindi per ogni $\epsilon>0$:
$\displaystyle\left|\phi(xy)-\phi(x)\phi(y)\right|\leq$
$\displaystyle\leq\left|\phi(xy)-\tilde{\phi}(xy)\right|+\left|\tilde{\phi}(xy)-\tilde{\phi}(x)\tilde{\phi}(y)\right|+\left|\tilde{\phi}(x)\tilde{\phi}(y)-\phi(x)\phi(y)\right|\leq$
$\displaystyle\leq
2\epsilon+\left|\tilde{\phi}(x)\tilde{\phi}(y)-\tilde{\phi}(x)\phi(y)\right|+\left|\tilde{\phi}(x)\phi(y)-\phi(x)\phi(y)\right|\leq$
$\displaystyle\leq
2\epsilon+(\left|\tilde{\phi}(x)\right|+\left|\phi(y)\right|)\epsilon\leq
2\epsilon+(\left|\phi(x)\right|+\left|\phi(y)\right|+\epsilon)\epsilon$
Data l’arbitrarietà di $\epsilon$, si ottiene la tesi. ∎
### 1.9 Problema di Dirichlet
Lo scopo di questa sezione è illustrare alcuni risultati per risolvere il
problema di Dirichlet su domini limitati su varietà riemanniane. Daremo la
definizione di dominio regolare per il problema di Dirichlet e utilizzeremo il
metodo di Perron e le barriere per caratterizzare questi domini, in seguito
dimostreremo che alcuni domini particolari sulle varietà sono regolari, in
particolare i domini limitati con bordo liscio e i domini della forma
$\Omega\setminus K$, dove $\Omega$ è un dominio limitato con bordo liscio e
$K$ una sottovarietà di codimensione $1$ con bordo liscio contenuta in
$\Omega$. Il lemma 1.89 (che citeremo senza dimostrazione) sarà lo strumento
principale per i nostri scopi, in quanto lega la solubilità del problema di
Dirichlet per un operatore ellittico abbastanza generico al problema di
Dirichlet del più noto e studiato laplaciano standard in $\mathbb{R}^{n}$.
In tutta la sezione $\Omega$ indicherà un dominio (cioè un insieme aperto
connesso relativamente compatto) in $R$ varietà riemanniana o in
$\mathbb{R}^{n}$, mentre $L:C^{2}(R,\mathbb{R})\to C(R,\mathbb{R})$ indicherà
un operatore differenziale del II ordine strettamente ellittico (il laplaciano
su varietà riemanniane soddisfa queste ipotesi come illustrato nella sezione
1.7.1). Inoltre considereremo solo funzioni sub e superarmoniche continue fino
alla chiusura dell’insieme di definizione.
Prima di cominciare diamo una definizione preliminare.
###### Definizione 1.86.
Un dominio $\Omega\subset R$ ha bordo $C^{k}$ con $0\leq k\leq\infty$ se e
solo se per ogni punto $p\in\partial\Omega$, esiste un intorno $V(p)$ e una
funzione $f\in C^{k}(V,\mathbb{R})$ tale che:
$\displaystyle\Omega\cap V=f^{-1}(\infty,0)\ \ \ \partial\Omega\cap
V=f^{-1}(0)$
Inoltre $0$ deve essere un valore regolare di $f$.
Osserviamo che non è sufficiente per un dominio in $R$ avere il bordo
costutuito da sottovarietà regolari per essere considerato di bordo liscio. Ad
esempio consideriamo l’insieme $A=B(0,2)\setminus\\{(x,0)\ -1\leq x\leq
1\\}\subset\mathbb{R}^{2}$. Il bordo di questo dominio è costituito dal bordo
della bolla e dal segmento $\\{(x,0)\ -1\leq x\leq 1\\}$, e sebbene per ogni
punto del segmento $A=\\{(x,0)\ -1<x<1\\}$ esiste un intorno $V$ e una
funzione $f:V\to\mathbb{R}$ tale che $A\cap V=f^{-1}(0)$, non è possibile fare
in modo che $\Omega\cap V=f^{-1}(\infty,0)$.
#### 1.9.1 Metodo di Perron
Per prima cosa definiamo il problema di Dirichlet.
###### Definizione 1.87.
Dati $\Omega\in R$ e $L$ con le caratteristiche descritte appena sopra,
$\phi:\partial\Omega\to\mathbb{R}$ funzione continua, diciamo che
$\Phi:\overline{\Omega}\to\mathbb{R}$ è soluzione del problema di Dirichlet
se:
$\displaystyle\Phi\in C^{2}(\Omega)\cap C(\overline{\Omega})\ \ \ e\ \ \
L(\Phi)=0\ \ su\ \ \Omega\ \ \ e\ \ \ \Phi|_{\partial\Omega}=\phi$ (1.8)
La solubilità del problema di Dirichlet è fortemente legata alla regolarità
del dominio $\Omega$. Ad esempio, se
$\Omega=B(0,1)\setminus\\{0\\}\subset\mathbb{R}^{n}$ 444444$B(0,1)$ indica la
bolla di raggio $1$ in $\mathbb{R}^{n}$, non esistono soluzioni del problema
di Dirichlet “classico” (cioè con $L=\Delta$) uguali a $0$ sul bordo di
$B(0,1)$ e diverse da $0$ ma limitate nell’origine (vedi esercizi 15 e 16 cap
3 pag 57 di [ABR], e per un altro esempio vedi es 25 cap 2 pag 54 dello stesso
libro).
###### Definizione 1.88.
Un dominio $\Omega\in R$ si dice essere regolare rispetto al problema di
Dirichlet se il più generico di questi problemi ha un’unica soluzione.
Il problema di Dirichlet per domini in $\mathbb{R}^{n}$ rispetto all’operatore
laplaciano standard è un argomento molto studiato in matematica e con molti
risultati nella letteratura. Il seguente lemma (che riportiamo senza
dimostrazione) lega la solubilità del problema di Dirichlet per un operatore
ellittico abbastanza generico a quella del laplaciano standard, il che
semplifica molto il problema.
###### Lemma 1.89.
Dato un dominio $\Omega\in\mathbb{R}^{n}$ con
$\Omega\subset\overline{\Omega}\Subset\Omega_{1}$ e dato $L$ operatore
ellittico della forma:
$\displaystyle L=a^{ij}(x)\partial_{i}\partial_{j}+b^{i}(x)\partial_{i}$
con coefficienti $a^{ij}$ e $b^{i}$ localmente lipschitziani in $\Omega_{1}$ e
per il quale esiste una funzione di Green su $\Omega_{1}$, allora il generico
problema di Dirichlet su $\Omega$ è risolubile se e solo se lo è anche il
generico problema di Dirichlet legato all’operatore laplaciano.
###### Proof.
Questo lemma è il corollario al teorema 36.3 in [H2] (ultimo risultato
presente nell’articolo). ∎
Se consideriamo $\Omega\subset R$ dominio contenuto in una carta locale, il
laplaciano sulla varietà $R$ soddisfa tutte le ipotesi del teorema. Questo
implica in particolare che:
###### Proposizione 1.90.
Per ogni dominio $\Omega\subset R$ e per ogni $p\in\Omega$, esiste un intorno
aperto $V(p)\subset\overline{V(p)}\Subset\Omega$ tale che il generico problema
di Dirichlet rispetto al laplaciano su varietà è risolubile su $V(p)$.
###### Proof.
La dimostrazione è un semplice corollario del teorema precedente. Per il
laplaciano standard, il problema di Dirichlet su ogni bolla
$B(x_{0},r)\subset\mathbb{R}^{n}$ è risolubile (vedi ad esempio teorema 1.17
pag 13 di [ABR]), quindi se consideriamo un qualsiasi insieme $A$ aperto
intorno di $p$ la cui chiusura è contenuta in $\Omega\cap U$ dove $(U,\phi)$ è
una carta locale di $R$ e tale che $\phi(A)=B(x_{0},r)$, il problema di
Dirichlet relativo al laplaciano su varietà è risolubile su $A$. ∎
Per caratterizzare alcuni dei domini su cui è sempre possibile risolvere il
problema di Dirichlet, utilizzeremo il metodo di Perron.
###### Definizione 1.91.
Dati $\Omega$, $L$, $\phi$ come sopra, indichiamo con $P[\phi]$ la funzione
$P[\phi]:\overline{\Omega}\to\mathbb{R}$ definita da
$\displaystyle P[\phi](x)=\sup_{u\in S_{\phi}}u(x)$
dove $S_{\phi}$ è l’insieme delle funzioni subarmoniche
$u:\overline{\Omega}\to\mathbb{R}$ continue su $\overline{\Omega}$ tali che
$\displaystyle u|_{\partial\Omega}\leq\phi$
Osserviamo che per la compattezza di $\partial\Omega$, la funzione $\phi$ ha
minimo $m$ e massimo $M$ finiti, e dato che le tutte le funzioni costanti
soddisfano $\Delta_{R}(c)=0$, l’insieme $S_{\phi}$ non è vuoto in quanto
contiene la funzione costante uguale a $m$, e tutte le funzioni in $S_{\phi}$,
in quanto funzioni subarmoniche sono limitate da $M$, quindi $P[\phi](x)\leq
M$ per ogni $x\in\overline{\Omega}$.
Il metodo di Perron è illustrato ad esempio nel paragrafo 11.3 pag 226 di
[ABR] o nel paragrafo 2.8 pag 23 di [GT] per risolvere il problema di
Dirichlet per il laplaciano standard su domini qualsiasi. I cardini essenziali
di questo metodo però sono il principio del massimo e la possibilità di
risolvere il problema di Dirichlet sulla bolla. Come abbiamo visto, questi
principi valgono anche per il laplaciano su varietà, non è quindi difficile
immaginare che anche per questo problema il metodo di Perron fornisca una
soluzione adeguata.
Verifichiamo ora sotto quali condizioni $P[\phi]$ risolve 1.8. Per prima cosa
dimostriamo che indipendentemente dal dominio $\Omega$, $P[\phi]$ è una
funzione armonica.
###### Proposizione 1.92.
$P[\phi]$ è armonica sull’insieme $\Omega$.
###### Proof.
Per dimostrare che la funzione $P[\phi]$ è armonica, dimostriamo che per ogni
punto $p\in\Omega$, esiste un intorno $V=V(p)$ su cui la funzione è armonica.
Consideriamo a questo scopo $V(p)$ un aperto con chiusura contenuta in
$\Omega\cap U$, dove $U$ è un intorno coordinato di $R$ tale che la
rappresentazione in carte locali di $V(p)$ sia una bolla. Grazie a 1.89,
sappiamo che è possibile risolvere il problema di Dirichlet relativo a
$\Delta_{R}$ in carte locali su $V(p)$. Per ogni funzione continua
$v:\Omega\to\mathbb{R}$, possiamo definire il suo sollevamento armonico come
la funzione data da:
$\displaystyle\bar{v}(x)=\begin{cases}v(x)&se\ x\in V^{C}\\\ D[v](x)&se\
x\in\overline{V}\end{cases}$
dove $D[v]$ è la soluzione del problema di Dirichlet relativo a $\Delta_{R}$
su $V(p)$ con $v|_{\partial V}$ come condizione al bordo. Osserviamo che
questa funzione è continua su $\Omega$ e se $v$ è subarmonica, allora
$\bar{v}$ mantiene questa proprietà in quanto massimo di due funzioni
subarmoniche.
Sia ora $u_{k}$ una successione di funzioni in $S_{\phi}$ tali che
$u_{k}(p)\to P[\phi](p)$. Consideriamo $\bar{u}_{k}$ la successione dei
sollevamenti armonici di queste funzioni relativamente all’intorno $V(p)$.
Dato che $\bar{u}_{k}(p)\geq u_{k}(p)$ e che $\bar{u}_{k}\in S_{\phi}$, si ha
che $\bar{u}_{k}(p)\to P[\phi](p)$. Data l’uniforme limitatezza delle funzioni
$\bar{u}_{k}$, per il principio di Harnack 1.57 esiste una sottosuccessione
che per comodità continueremo a indicare con lo stesso indice che converge
localmente uniformemente su $V(p)$ a una funzione armonica. Sia
$u=\lim_{k}\bar{u}_{k}$, vogliamo dimostrare che $u|_{V(p)}=P[\phi]|_{V(p)}$.
È facile osservare che $u\leq P[\phi]$ su tutto $\Omega$, infatti ogni
funzione $\bar{u}_{k}\in S_{\phi}$. Inoltre come osservato in precedenza
$u(p)=P[\phi](p)$. Supponiamo per assurdo che esista $q\in V(p)$ tale che
$u(q)<P[\phi](q)$. Allora per definizione per ogni $\epsilon>0$, esiste una
funzione $\tilde{w}\in S_{\phi}$ tale che
$\displaystyle u(q)<\tilde{w}(q)\leq P[\phi](q)\ \ \
\tilde{w}(q)>P[\phi](q)-\epsilon$
Se definiamo $w_{k}$ come il sollevamento armonico della funzione
$\max\\{\tilde{w},u_{k}\\}$ rispetto all’insieme $V(p)$, otteniamo come prima
una successione di funzioni armoniche limitate che, a patto di passare a una
sottosuccessione, converge localmente uniformemente a una funzione $w$
armonica su $V(p)$. Poiché tutte le funzioni $w_{k}\in S_{\phi}$, sappiamo che
$w\leq P[\phi]$, e quindi in particolare $w(p)\leq P[\phi](p)$. Inoltre per
costruzione $u_{k}\leq w_{k}$, quindi passando al limite otteniamo che $u\leq
w$ sull’insieme $V(p)$. Ma dato che $P[\phi](p)=u(p)\leq w(p)\leq P[\phi](p)$,
abbiamo che $w(p)=u(p)$, cioè la funzione armonica $u-w$ assume il suo massimo
in $V(p)$ in un punto interno all’insieme, quindi per il principio del massimo
1.52 $u-w=0$. Dato che per costruzione $w(q)>P[\phi](q)-\epsilon$, abbiamo
che:
$\displaystyle u(q)>P[\phi](q)-\epsilon$
e l’assurdo segue dall’arbitrarietà di $\epsilon$. ∎
Resta da verificare se la funzione $P[\phi]$ è continua su $\overline{\Omega}$
e se $P[\phi]|_{\partial\Omega}=\phi$. Prima di dare condizioni per verificare
queste proprietà, sottolineamo che il problema di Dirichlet è risolvibile se e
solo se valgono queste proprietà per $P[\phi]$.
###### Proposizione 1.93.
Il problema di Dirichlet 1.8 è risolubile se e solo se $P[\phi]$ risulta
essere continua su $\overline{\Omega}$ e $P[\phi]|_{\partial\Omega}=\phi$.
###### Proof.
Se le condizioni su $P[\phi]$ sono verificate, automaticamente $P[\phi]$ è
l’unica soluzione del problema di Dirichlet. Al contrario, supponiamo che
esista $\Phi$ soluzione del problema. Allora $\Phi\in S_{\phi}$, anzi
$\Phi=P[\phi]$. Infatti la funzione $\Phi$ è armonica, quindi maggiora tutte
le funzioni in $S_{\phi}$. ∎
Come accennato in precedenza, non sempre il problema di Dirichlet è
risolubile. Data l’ultima equivalenza questo implica che sebbene $P[\phi]$ sia
sempre una funzione armonica, non sempre soddisfa tutte le condizioni del
problema di Dirichlet. Un modo per verificare quando il problema è risolubile
è il criterio delle barriere.
###### Definizione 1.94.
Dato un punto $p\in\partial\Omega$, diciamo che una funzione
$S:\overline{\Omega}\to\mathbb{R}$ è una barriera per il punto $p$ se $S$ è
una funzione superarmonica in $\Omega$, continua in $\overline{\Omega}$ tale
che
$\displaystyle S(p)=0\ \ \ S|_{\overline{\Omega}\setminus\\{p\\}}>0$
Se la funzione $\beta$ ha queste caratteristiche ma è definita solamente su
$\overline{\Omega}\cap V(p)$, dove $V(p)$ è un intorno qualsiasi del punto
$p$, diciamo che $\beta$ è una barriera locale.
Per prima cosa osserviamo che l’esistenza di una barriera “globale” è
equivalente all’esistenza di una barriera locale, infatti:
###### Proposizione 1.95.
Se esiste una barriera locale $\beta$ per un punto $p\in\partial\Omega$,
allora esiste anche una barriera globale per $p$.
###### Proof.
La dimostrazione è relativamente facile. Sia $V(p)$ un intorno aperto tale che
$\beta$ sia definita su $\overline{\Omega}\cap V(p)$. Sia
$W\subset\overline{W}\subset V$ un secondo intorno di $p$. Per continuità
$\beta$ assume minimo $m$ strettamente positivo sull’insieme $W^{C}$.
Consideriamo la funzione $S$ definita da:
$\displaystyle S(q)=\begin{cases}\min\\{\beta(q),m\\}&se\
q\in\overline{\Omega}\cap W\\\ m&se\ q\in\overline{\Omega}\cap
W^{C}\end{cases}$
dato che $S$ è il minimo tra funzioni superarmoniche, è ancora una funzione
superarmonica. È facile dimostrare che anche le proprietà richieste a una
barriera sono verificate. ∎
L’utilità del concetto di barriera è contenuta nella seguente proposizione:
###### Proposizione 1.96.
Sia $p\in\partial\Omega$. Se esiste una barriera per $p$ rispetto a $\Omega$,
allora
$\displaystyle P[\phi](p)=\lim_{x\to p}P[\phi](x)=\phi(p)$
###### Proof.
Poiché $\phi$ è continua sull’insieme $\partial\Omega$, per ogni $\epsilon>0$
esiste un intorno $U_{\epsilon}(p)$ tale che per ogni $x\in
U_{\epsilon}\cap\partial\Omega$:
$\displaystyle\phi(p)-\epsilon<\phi(x)<\phi(p)+\epsilon$
Dato che $S$ è una funzione strettamente positiva su $\partial\Omega\setminus
U_{\epsilon}$, esiste una costante $c>0$ tale che
$\displaystyle\phi(p)-\epsilon-cS(x)<\phi(x)<\phi(p)+\epsilon+cS(x)$
per ogni $x\in\partial\Omega\setminus U_{\epsilon}$. Data la positività di
$S$, questa relazione vale su tutto l’insieme $\partial\Omega$. Dato che $S$ è
superarmonica, la funzione $\phi(p)-\epsilon-cS(x)\in S_{\phi}$, quindi
$\displaystyle\phi(p)-\epsilon-cS(x)\in S_{\phi}\leq P[\phi](x)$ (1.9)
ora consideriamo una funzione $u\in S_{\phi}$. Vale che:
$\displaystyle u|_{\partial\Omega}\leq\phi<\phi(p)+\epsilon+cS(x)\ \
\Rightarrow\ \ [u-cS]|_{\partial\Omega}\leq\phi(p)+\epsilon$
Data la subarmonicità della funzione continua $[u-cS]$, questa relazione è
valida su tutto l’insieme $\overline{\Omega}$, e data l’arbitrarietà di $u\in
S_{\phi}$ otteniamo che:
$\displaystyle P[\phi](x)\leq\phi(p)+\epsilon+cS(x)$ (1.10)
Passando al limite per $x\to p$ nelle relazioni 1.9 e 1.10, otteniamo che per
ogni $\epsilon>0$:
$\displaystyle\phi(p)-\epsilon\leq\liminf_{x\to p}P[\phi](x)\leq\limsup_{x\to
p}P[\phi](x)\leq\phi(p)+\epsilon$
data l’arbitrarietà di $\epsilon>0$, si ottiene la tesi. ∎
Il criterio delle barriere è molto utile perchè l’esistenza delle barriere è
condizione necessaria e sufficiente per risolvere il problema di Dirichlet,
infatti:
###### Proposizione 1.97.
Il generico problema di Dirichlet su $\Omega\subset R$ è risolubile se e solo
se per ogni $p\in\partial\Omega$ esiste una barriera relativa a $p$.
###### Proof.
Grazie alla proposizione precedente, se ogni punto del bordo di $\Omega$ ha
una barriera, $P[\phi]$ soddisfa tutte le condizioni del problema di
Dirichlet, quindi esiste unica la soluzione di tale problema. Supponiamo al
contrario che il problema sia sempre risolubile. Allora fissato
$p\in\partial\Omega$, sia $\phi_{p}$ una funzione continua sul bordo tale che
$\phi_{p}(p)=0$, e $\phi_{p}(x)>0$ per ogni $x\neq p$ (ad esempio,
$\phi_{p}(x)=d(x,p)$). Sia $\Phi_{p}$ la soluzione del relativo problema di
Dirichlet. Questa funzione è una barriera per $p$, infatti è una funzione
armonica (quindi anche superarmonica) in $\Omega$, continua in
$\overline{\Omega}$ e uguale a $\phi_{p}$ su $\partial\Omega$, e grazie al
principio del massimo, $\Phi_{p}(x)>0$ per ogni $x\in\Omega$. ∎
Come corollario immediato di questo teorema, dimostriamo che
###### Proposizione 1.98.
Se $\Omega$ e $\Omega^{\prime}$ sono domini regolari rispetto a $\Delta_{R}$,
allora anche la loro intersezione
$\Omega^{\prime\prime}=\Omega\cap\Omega^{\prime}$ è un dominio regolare.
###### Proof.
La regolarità dei domini è equivalente all’esistenza di barriere per ogni
punto del bordo. Dato che
$\displaystyle\partial(\Omega\cap\Omega^{\prime})\subset\partial\Omega\cup\partial\Omega^{\prime}$
ogni punto del bordo di $\Omega^{\prime\prime}$ appartiene ad almeno uno dei
due bordi. Senza perdita di generalità, consideriamo
$p\in(\partial\Omega^{\prime\prime}\cap\partial\Omega)$. Per regolarità di
$\Omega$, esiste una barriera $B$ per $p$ rispetto a $\Omega$. La funzione $B$
è superarmonica in $\Omega$, quindi automaticamente anche in
$\Omega^{\prime\prime}$, strettamente positiva su
$\overline{\Omega}\setminus\\{p\\}$, quindi anche su
$\overline{\Omega^{\prime\prime}}\setminus\\{p\\}$, e ovviamente
$B|_{\overline{\Omega^{\prime\prime}}}$ è una funzione continua. Questo
dimostra che ogni punto di $\partial{\Omega^{\prime\prime}}$ possiede una
barriera, $\Omega^{\prime\prime}$ è regolare. ∎
Riassumendo, un dominio $\Omega\subset R$ è regolare rispetto al laplaciano
sulla varietà $R$ se e solo se per ogni punto del bordo di $\Omega$ esiste una
barriera, barriera che come abbiamo visto è un concetto locale. Se il dominio
$\Omega$ è contenuto in una carta locale, allora grazie al lemma 1.89,
possiamo utilizzare tutti i risultati noti per il laplaciano standard e
concludere che ogni insieme regolare per il laplaciano standard è regolare
anche per il laplaciano su varietà (e viceversa). Utilizzando assieme queste
due tecniche, dimostreremo che i domini con bordo liscio in $R$ sono regolari,
ma anche i domini della forma $\Omega=A\setminus K$, dove $A$ è un dominio con
bordo liscio, e $K$ una sottovarietà con bordo regolare di $R$ di codimensione
$1$ contenuta in $A$.
#### 1.9.2 Domini con bordo liscio
Dalla letteratura sul laplaciano standard in $\mathbb{R}^{n}$, sappiamo che
una condizione geometrica semplice per garantire la regolarità di un dominio è
la condizione della bolla esterna.
###### Proposizione 1.99 (Condizione della bolla esterna).
Dato un dominio $\Omega\subset\mathbb{R}^{n}$, $\Omega$ è regolare per il
laplaciano standard se per ogni punto di $\Omega$ esiste una bolla
$B(x(p),\epsilon(p))$ tale che
$\displaystyle\overline{B(x,\epsilon)}\cap\overline{\Omega}=\\{p\\}$
cioè una bolla esterna al dominio $\Omega$ la cui chiusura interseca il bordo
di $\Omega$ solo nel punto $p$.
###### Proof.
Grazie all’omogeneità del dominio $\mathbb{R}^{n}$ e dell’operatore $\Delta$,
e soprattutto grazie al grande dettaglio con cui questo problema è stato
studiato, la trattazione dell’argomento è particolarmente semplice. Per
dimostrare questa proposizione, dimostriamo che per ogni punto $p$ che
soddisfa la condizione della bolla esterna, esiste una barriera $B_{p}$. Ad
esempio possiamo considerare la funzione:
$\displaystyle
B_{p}(y)=\begin{cases}\log(\left\|x(p)-y\right\|)-\log(\epsilon(p))&se\ n=2\\\
-\frac{1}{\left\|x(p)-y\right\|^{n-2}}+\frac{1}{\epsilon(p)}&se\ n\geq
3\end{cases}$
È facile dimostrare che questa funzione soddisfa tutte le proprietà richieste
a una barriera per $\Omega$ (vedi ad esempio teorema 11.13 pag 229 di [ABR]).
∎
Come corollario di questa condizione, è facile dimostrare che
###### Proposizione 1.100.
Qualunque dominio $\Omega\subset\mathbb{R}^{n}$ con bordo $C^{2}$ è regolare
per il problema di Dirichlet.
###### Proof.
È sufficiente dimostrare che ogni punto del bordo di questi domini soddisfa la
condizione della bolla esterna. Per i dettagli vedi corollario 11.13 di [ABR].
∎
La condizione della bolla esterna implica in particolare che ogni anello in
$\mathbb{R}^{n}$, cioè ogni insieme della forma
$\displaystyle A(x_{0},r_{1},r_{2})\equiv
B(x_{0},r_{2})\setminus\overline{B(x_{0},r_{1})}=\\{x\in\mathbb{R}^{n}\ t.c.\
r_{1}<\left\|x-x_{0}\right\|<r_{2}\\}$
è un dominio regolare per il laplaciano standard, e grazie al lemma 1.89, ogni
anello in un insieme coordinato 454545cioè ogni insieme
$A\subset\overline{A}\Subset U\subset R$ tale che $(U,\psi)$ sia un intorno
coordinato e $\psi(A)$ sia un anello in $\psi(U)$ è regolare rispetto al
laplaciano su $R$.
Questo ci permette di dimostrare che
###### Proposizione 1.101.
Ogni dominio $\Omega\subset R$ con bordo liscio 464646è sufficiente con bordo
$C^{2}$ è un dominio regolare per $\Delta_{R}$.
###### Proof.
Dimostriamo che ogni punto del bordo possiede una barriera locale. Sia a
questo scopo $p\in\partial\Omega$ e $(U,\psi)$ un intorno coordinato di $p$
tale che
$\displaystyle U\cap\Omega=\psi^{-1}\\{(x_{1},\cdots,x_{m-1},0)\\}$
cioè sia $\psi$ una carta che manda $U\cap\Omega$ in un semispazio chiuso di
$\mathbb{R}^{n}$. È facile immaginare (vedi disegno 1.1) che esiste un anello
$A(x_{0},r_{1},r_{2})$ in $\psi(U)$ tale che
$\overline{B(x_{0},r_{1})}\cap\psi(\Omega)=\\{\psi(p)\\}$.
Figure 1.1: L’area in grigio è l’area di definizione della barriera locale
$\beta$
Visto che questo insieme è regolare per il laplaciano $\Delta_{R}$, esiste una
funzione $\beta$ armonica su $A$, continua su $\overline{A}$, tale che
$\beta|_{\partial B(x_{0},r_{1})}=0$, $\beta|_{\partial B(x_{0},r_{2})}=1$.
Grazie al principio del massimo, è facile dimostrare che questa funzione è una
barriera locale per il punto $p$. ∎
#### 1.9.3 Altri domini regolari
Lo scopo di questa sezione è dimostrare che ogni dominio della forma
$R\supset\Omega=A\setminus K$ dove $A$ è un dominio con bordo liscio e $K$ una
sottovarietà regolare di codimensione $1$ con bordo regolare contenuta in $A$,
è un dominio regolare per $\Delta_{R}$. Cercheremo di tenere la dimostrazione
al livello più elementare e “geometrico” possibile. Il lemma 1.89 sarà
essenziale per ottenere questo risultato.
Cominciamo con alcuni risultati preliminari sul laplaciano standard in
$\mathbb{R}^{n}$. Nel seguito chiameremo armoniche le funzioni armoniche nel
senso di $\mathbb{R}^{n}$ e considereremo quindi domini in $\mathbb{R}^{n}$
($n\geq 2$), indicando con $B(\bar{x},r)$ la bolla euclidea aperta di centro
$\bar{x}$ e raggio $r$, e con $D(\bar{x},r)$ la bolla aperta $n-1$
dimensionale centrata in $\bar{x}$ di raggio $r$, cioè l’insieme
$\displaystyle D(\bar{x},r)=\\{x\in\mathbb{R}^{n}\ t.c.\
\left\|\bar{x}-x\right\|<r\ e\
x_{m}=\bar{x}_{m}\\}=B(\bar{x},r)\cap\\{x_{m}=\bar{x}_{m}\\}$
###### Proposizione 1.102.
Sull’insieme $\Omega\equiv
B(\bar{x},r_{2})\setminus\overline{D(\bar{x},r_{1})}$ con $r_{2}>r_{1}$,
esiste una funzione $f:\Omega\to\mathbb{R}$ tale che
$\displaystyle f\in H(\Omega)\cap C(\overline{\Omega})=H(\Omega)\cap
C(\overline{B(\bar{x},r)})\ \ \ f|_{\partial B(\bar{x},r)}=0,\ \ \
f|_{\overline{D(\bar{x},r)}}=1$
###### Proof.
Vista l’invarianza per riscalamento di $\mathbb{R}^{n}$ e dell’operatore
laplaciano standard, possiamo assumere senza perdita di generalità $r\equiv
r_{1}<r_{2}=1$. In tutta la dimostrazione, $B=B(\bar{x},1)$, $D=D(\bar{x},r)$.
Grazie all’osservazione 1.27, esiste una successione di aperti relativamente
compatti con bordo liscio $A_{n}$ tali che $A_{n}\subset A_{n-1}$ e
$\overline{D}=\cap_{n}A_{n}$. Definitivamente in $n$, vale che
$\overline{A_{n}}\subset B(0,1)$. Assumiamo senza perdita di generalità che
$\overline{A_{1}}\subset B$. Per ogni $n$, l’insieme
$\Omega_{n}=B\setminus\overline{A_{n}}$ è un insieme regolare per il
laplaciano (ha bordo liscio), quindi per ogni $n$ esiste una funzione:
$\displaystyle f_{n}\in C(\overline{B},\mathbb{R})\ \ \ f_{n}\in
H(\Omega_{n})\ \ \ f_{n}|_{\partial B}=0\ \ \ f_{n}|_{\overline{A_{n}}}=1$
Grazie al principio del massimo e al fatto che $A_{n}\subset A_{n-1}$, è
facile verificare che questa successione è una successione decrescente, e che
per ogni $n$, $0\leq f_{n}\leq 1$. Grazie al principio di Harnack (vedi 1.57),
la successione $f_{n}$ converge localmente uniformemente su $\Omega$ a una
funzione $f$ armonica su $\Omega$. Dimostriamo che questa funzione $f$ è
continua e che soddisfa le richieste sui valori al bordo.
Per prima cosa consideriamo $\bar{y}\in D$. Per definizione di $D$, esiste un
raggio $r(\bar{y})$ sufficientemente piccolo in modo che
$D(\bar{y},r(\bar{y}))\subset D$ e $B(\bar{y},r(\bar{y}))\subset B$.
Consideriamo la semisfera:
$\displaystyle
B^{+}(\bar{y},r(\bar{y}))=B(\bar{y},r(\bar{y}))\cap\\{x\in\mathbb{R}^{n}\
t.c.\ x_{m}\geq 0\\}$
Grazie alla proposizione 1.98, questo insieme è un’insieme regolare per il
laplaciano. Consideriamo una qualsiasi funzione continua $h:\partial
B^{+}\to\mathbb{R}$ tale che $h|_{\partial B^{+}}=0$, $h(\bar{y})=1$ e $0\leq
h\leq 1$, e sia $u$ la soluzione del relativo problema di Dirichlet (cioè
$u|_{\partial B^{+}}=h$). Grazie al principio del massimo, è facile concludere
che per ogni funzione $f_{n}$, $f_{n}\geq u$ su $B^{+}\cap A_{n}$, quindi
anche su tutta $B^{+}$. Passando al limite otteniamo che
$\displaystyle f(x)\geq u(x)$
Questo implica che la funzione $f$ non può essere identicamente nulla e che:
$\displaystyle\liminf_{x\to\bar{y}}f(x)\geq\lim_{x\to\bar{y}}u(x)=1$
L’arbitrarietà di $\bar{y}\in D$ garantisce che
$\displaystyle\liminf_{x\to\bar{y}}f(x)\geq 1$
per ogni $\bar{y}\in D$.
Dato che per costruzione $\limsup_{x\to\bar{y}}f(x)\leq 1$, otteniamo che per
ogni punto $\bar{y}\in D$, $f$ è continua e $f|_{D}=1$. Rimangono da
considerare i punti sull’insieme $E\equiv\overline{D}\setminus D$. Sia a
questo scopo $\tilde{y}\in E$. Per costruzione di $f$:
$\displaystyle
0\leq\liminf_{x\to\tilde{y}}f(x)\leq\limsup_{x\to\tilde{y}}f(x)=1$
Consideriamo un numero $0<\lambda<1$. Ricordiamo che dato un insieme $I$, si
dice
$\displaystyle\lambda I=\\{x\in\mathbb{R}^{n}\ t.c.\ x/\lambda\in
I\\}=\\{\lambda x\ t.c.\ x\in I\\}$
Grazie all’invarianza per traslazioni di $\mathbb{R}^{n}$ e dell’operatore
laplaciano standard, possiamo ridefinire per comodità il sistema di
riferimento in modo che $\tilde{y}=0$.
Scegliamo $\lambda$ in modo che $\partial(\lambda B)=\lambda(\partial
B)\subset B\setminus\Omega$ (vedi figura 1.2).
Figure 1.2: Le linee piene rappresentano il bordo degli insiemi $m-1$
dimensionali, le linee tratteggiate il bordo degli insiemi $m$ dimensionali.
Il colore rosso indica gli insiemi moltiplicati per $\lambda$.
La funzione $f$ è armonica su $\Omega$, quindi assume un valore minimo su
$\lambda(\partial B)$. Grazie al principio del massimo 1.52, questo valore è
strettamente positivo, diciamo $\epsilon>0$. Consideriamo la funzione
$\displaystyle f_{\lambda}:\lambda B\to\mathbb{R}\ \ \
f_{\lambda}(x)=f(\lambda^{-1}x)$
Osserviamo subito che
$\liminf_{x\to\tilde{y}}f(x)=\liminf_{x\to\tilde{y}}f_{\lambda}(x)$
(ricordiamo che abbiamo scelto un riferimento dove $\tilde{y}=0$). Definiamo
inoltre
$\displaystyle f_{\epsilon}(x)=\frac{f_{\lambda}(x)+\epsilon}{1+\epsilon}$
Questa funzione è armonica su $\lambda\Omega$, quindi essendo $\lambda<1$ è
armonica su $\lambda B\setminus\overline{D}$. Inoltre vale che
$f_{\epsilon}|_{\partial(\lambda B)}=\frac{\epsilon}{1+\epsilon}<\epsilon$, e
$f_{\epsilon}\leq 1$ ovunque definita. Dato che la successione $f_{n}$
converge a $f$ monotonamente dall’alto, per ogni $n$ si ha che:
$\displaystyle f_{\epsilon}|_{\partial(\lambda B)}<f_{n}|_{\partial(\lambda
B)}$
Inoltre visto che $f_{\epsilon}\leq 1$ ovunque definita, grazie al principio
del massimo otteniamo che per ogni $n$:
$\displaystyle f_{n}>f_{\epsilon}\ \ \Rightarrow\ \ f\geq f_{\epsilon}$
Questo significa che
$\displaystyle\liminf_{x\to\tilde{y}=0}f(x)\geq\liminf_{x\to\tilde{y}=0}f|_{\epsilon}(x)=\frac{\epsilon+\liminf_{x\to\tilde{y}=0}f(x)}{1+\epsilon}$
se definiamo $\liminf_{x\to\tilde{y}=0}f(x)\equiv L$, otteniamo che:
$\displaystyle L+\epsilon L\geq L+\epsilon\ \ \Rightarrow\ \ L\geq 1$
Quindi:
$\displaystyle
1\leq\liminf_{x\to\tilde{y}}f(x)\leq\limsup_{x\to\tilde{y}}f(y)=1$
ripetendo la costruzione per ogni $\tilde{y}\in E$, otteniamo la tesi. ∎
Osserviamo che a patto di riscalare la funzione $f$, è possibile ottenere una
funzione tale che:
$\displaystyle f\in H(\Omega)\cap
C(\overline{\Omega})=C(\overline{B(\bar{x},r)})\ \ \ f|_{\partial
B(\bar{x},r)}=a,\ \ \ f|_{\overline{D(\bar{x},r)}}=b$
per qualsiasi valore fissato di $a$ e $b\in\mathbb{R}$.
Passiamo ora a dimostrare che ogni dominio $\Omega$ della forma descritta qui
sopra è regolare rispetto al laplaciano standard.
###### Proposizione 1.103.
L’insieme $\Omega\equiv B(\bar{x},r_{2})\setminus\overline{D(\bar{x},r_{1})}$
con $r_{2}>r_{1}$ è regolare per il laplaciano standard, cioè per ogni
funzione $\phi:\partial\Omega\to\mathbb{R}$ continua, esiste una soluzione del
relativo problema di Dirichlet.
###### Proof.
Come sopra, possiamo assumere $\bar{x}=0$ e $r\equiv r_{1}<r_{2}=1$. Iniziamo
con il considerare solo funzioni $\phi$ identicamente nulle su $\partial B$ e
$0\leq\phi\leq 1$ ovunque su $\partial\Omega$, e dimostriamo che $P[\phi]$
risolve il relativo problema di Dirichlet.
Sappiamo che $0\leq P[\phi]\leq 1$ su tutto $\overline{\Omega}$. Inoltre
grazie all’esistenza di barriere per ogni punto di $\partial B$, sappiamo che
$P[\phi]$ è continua su un intorno di $\partial B$ (a priori disgiunto da
$\overline{D}$) e $P[\phi]|_{\partial B}=0$. Sia $y\in\overline{D}$. Per
continuità, dato $\epsilon>0$, esiste $\delta>0$ tale che
$\displaystyle\phi(y)-\epsilon<\phi|_{B(y,\delta)\cap\overline{D}}<\phi(y)+\epsilon$
Consideriamo ora un insieme
$\Omega_{y}=B(x,\rho_{2})\setminus\overline{D(x,\rho_{1})}$, dove $x$,
$\rho_{1}$, $\rho_{2}$ sono scelti in modo che $B(x,\rho_{2})\subset
B(y,\delta)$, $\overline{D}(x,\rho_{1})\subset D$,
$y\in\overline{D}(x,\rho_{1})$. Ad esempio, se $y\in D$ è sufficiente
scegliere $x=y$ e $\rho_{1}<\rho_{2}<\delta$, mentre se
$y\in\overline{D}\setminus D$ (cioè se $y$ sta sul bordo del disco $D$),
allora si può scegliere $D(x,\rho_{1})$ in modo che sia un disco interno e
tangente a $D$ nel punto $y$, sufficientemente piccolo in modo che
$\overline{D(x,\rho_{1})}\subset B(y,\delta)$.
Consideriamo ora due funzioni, $f_{1}$ e $f_{2}$, definite in questo modo:
$\displaystyle f_{1}|_{\partial B(x,\rho_{2})}=0\ \ \
f_{1}|_{\overline{D}(x,\rho_{1})}=\phi(y)-\epsilon\ \ \ f\in
H(B(x,\rho_{2})\setminus\overline{D(x,\rho_{1}}))$ $\displaystyle
f_{2}|_{\partial B(x,\rho_{2})}=1\ \ \
f_{1}|_{\overline{D}(x,\rho_{1})}=\phi(y)+\epsilon\ \ \ f\in
H(B(x,\rho_{2})\setminus\overline{D(x,\rho_{1}}))$
estendiamo le due funzioni a costanti su $\overline{B}\setminus
B(x,\rho_{2})$, in particolare $f_{1}=0$ e $f_{2}=1$ fuori dall’insieme
$B(x,\rho_{2})$. Entrambe le funzioni risultano continue su $\overline{B}$,
$f_{1}$ è subarmonica su $\Omega$ (in quanto massimo tra due funzioni
armoniche), mentre $f_{2}$ è superarmonica su $\Omega$ (in quanto minimo tra
due funzioni armoniche). Inoltre per costruzione
$f_{1}|_{\partial\Omega}\leq\phi$ e $f_{2}|_{\partial\Omega}\geq\phi$.
Per definizione di $P[\phi]$, otteniamo quindi che:
$\displaystyle P[\phi]\geq f_{1}$
quindi vale che:
$\displaystyle\phi(y)-\epsilon=\lim_{x\to y}f_{1}(x)\leq\liminf_{x\to
y}P[\phi](x)$
Inoltre, qualunque funzione $u\in S_{\phi}$ 474747vedi 1.91 per il principio
del massimo è minore della funzione $f_{2}$, quindi anche $P[\phi]$ conserva
questa proprietà. Da questo ricaviamo che:
$\displaystyle\phi(y)+\epsilon=\lim_{x\to y}f_{2}(x)\geq\limsup_{x\to
y}P[\phi](x)$
data l’arbitrarietà di $\epsilon$ e del punto $y$, possiamo concludere che
$P[\phi]$ risolve il problema di Dirichlet considerato.
Osserviamo che con il procedimento appena descritto permette di costruire una
barriera per ogni punto di $\overline{D}$. Indatti fissato
$\bar{y}\in\overline{D}$, se scegliamo
$\displaystyle\phi_{\bar{y}}(x)=\begin{cases}1-d(x,\bar{y})&se\
x\in\overline{D}\\\ 0&se\ x\in\partial B\end{cases}$
grazie a quanto appena dimostrato la funzione $P[\phi]$ è armonica in
$\Omega$, continua su $\overline{B}$, $P[\phi_{y}](y)=1$ e $P[\phi_{y}](x)<1$
per ogni $x\neq y$. Quindi la funzione $1-P[\phi_{y}]$ è una barriera per il
punto $y$.
Dato che i punti di $\partial B$ sono regolari (hanno tutti una barriera
locale), abbiamo dimostrato che il dominio $\Omega$ è regolare per l’operatore
laplaciano standard. ∎
Ricordiamo che grazie al lemma 1.89, i domini appena considerati sono regolari
anche per l’operatore $\Delta_{R}$ (in carte locali).
Passiamo ora a dimostrare la proposizione principale di questo paragrafo:
###### Proposizione 1.104.
Ogni dominio su una varietà riemanniana $R$ della forma $\Omega=A\setminus K$
dove $A$ è un dominio con bordo liscio e $K$ una sottovarietà regolare di
codimensione $1$ con bordo regolare contenuta in $A$, è un dominio regolare
per $\Delta_{R}$.
###### Proof.
Dimostriamo che per ogni punto di $\partial\Omega$ è un punto regolare per il
problema di Dirichlet.
Se $p\in\partial A$, $p$ è regolare grazie alla proposizione 1.101, quindi
possiamo limitarci a considerare il caso $p\in K$. D’ora in avanti indicheremo
con $\partial K$ il bordo di $K$ inteso come bordo della sottovarietà, non
come bordo topologico, e $K^{\circ}$ sarà la parte interna sempre intesa nel
senso di sottovarietà.
Se $p\in K^{\circ}$, consideriamo $(V(p),\psi)$ un aperto (nella topologia di
$R$) coordinato intorno di $p$ tale che
$\displaystyle V\cap K=\psi^{-1}\\{(x_{1},\cdots,x_{m-1},0)\\},\ \ \
\psi(p)=0$
ad esempio possiamo considerare le coordinate di Fermi su un intorno di $p$
(vedi proposizione 1.12). Sia $r_{2}$ tale che
$\overline{B(0,r_{2})}\subset\psi(V)$ e sia $r_{1}<r_{2}$. Allora l’insieme
$\displaystyle S=B(0,r_{2})\setminus\overline{D(0,r_{1})}$
è un’insieme regolare per l’operatore $\Delta_{R}$, quindi esiste la soluzione
del problema di Dirichlet
$\displaystyle\Delta_{R}(\Phi)=0\ su\ S,\ \ \ \Phi\in C(\overline{S}),\ \ \
\Phi(x)=d(x,0)\ \ \forall x\in\partial S$
È facile verificare che la funzione $\Phi$ è una barriera locale per il punto
$p$.
Se $p\in\partial K$, la dimostrazione della sua regolarità è molto simile.
Consideriamo $(V,\psi)$ un’aperto con coordinate di Fermi tale che:
$\displaystyle V\cap K=\psi^{-1}\\{(x_{1},\cdots,x_{m-1},0)\ t.c.\ x_{m-1}\leq
0\\}\ \ \ \psi(p)=0$
cioè un sistema di coordinate che rappresenta $K$ come un semipiano in
$\mathbb{R}^{n}$. Sia $r_{2}$ abbastanza piccolo da verificare
$B(0,2r_{2})\subset\psi(V)$. Se $r_{1}<r_{2}$, e
$\bar{x}=(0,\cdots,0,-r_{1},0)$, l’insieme
$\displaystyle S=B(\bar{x},r_{2})\setminus D(\bar{x},r_{1})$
è un’insieme regolare per l’operatore $\Delta_{R}$, quindi esiste la soluzione
del problema di Dirichlet:
$\displaystyle\Delta_{R}(\Phi)=0\ su\ S,\ \ \ \Phi\in C(\overline{S}),\ \ \
\Phi(x)=d(x,0)\ \ \forall x\in\partial S$
Come nel caso precedente, è facile verificare che la funzione $\Phi$ è una
barriera locale per il punto $p$. ∎
#### 1.9.4 Regolarità sul bordo
In questo paragrafo riportiamo un lemma che garantisce sotto certe condizioni
la regolarità delle soluzioni di particolari problemi di Dirichlet fino al
bordo del dominio. Il lemma è basato sul teorema 17.3 pag 28 di [F2], di cui
riportiamo una versione semplificata senza dimostrazione.
###### Teorema 1.105.
Sia $\Omega$ un dominio con bordo liscio, e $L$ un operatore uniformemente
ellittico su $\overline{\Omega}$ con coefficienti $a,b,c$ lisci. Se $u$ è
soluzione del problema di Dirichlet generalizzato
$\displaystyle Lu=f\ in\ \Omega\ \ \ \ u|_{\partial\Omega}=0$
con $f\in C^{\infty}(\overline{\Omega})$, allora $u\in
C^{\infty}(\overline{\Omega})$.
Ricordiamo che, grazie alla teoria sviluppata in [F2], sotto le ipotesi citate
il problema di Dirichlet generalizzato ha sempre un’unica soluzione.
Osserviamo che considerando un dominio relativamente compatto con bordo liscio
$\Omega\subset R$, il laplaciano su varietà soddisfa tutte le condizioni del
teorema.
Grazie a questo teorema siamo in grado di dimostrare che:
###### Lemma 1.106.
Siano $L$ e $\Omega$ come nel teorema precedente, sia $u\in C^{2}(\Omega)\cap
C(\overline{\Omega})$ con $Lu=0$. Se esiste una funzione $h\in
C^{\infty}(\overline{\Omega})$ tale che
$u|_{\partial\Omega}=h|_{\partial\Omega}$
allora $u\in C^{\infty}(\overline{\Omega})$.
###### Proof.
La funzione $u-h$ soddisfa il problema di Dirichlet generalizzato:
$\displaystyle L(u-h)=L(u)-L(h)=-L(h)\in C^{\infty}(\overline{\Omega})\ \ \ \
\ (u-h)|_{\partial\Omega}=0$
grazie al teorema precedente, la funzione $(u-h)\in
C^{\infty}(\overline{\Omega})$, quindi per differenza anche la funzione $u\in
C^{\infty}(\overline{\Omega})$. ∎
Nella tesi, spesso utilizzeremo questo lemma applicato a domini $\Omega$ con
bordo costituito da 2 componenti connesse $\partial\Omega_{0}$ e
$\partial\Omega_{1}$ 484848ad esempio un anello in $\mathbb{R}^{2}$ ha queste
caratteristiche e funzioni $u$ costanti su ogni componente 494949ad esempio
$u|_{\partial\Omega_{0}}=0$ e $u|_{\partial\Omega_{1}}=1$. In questo caso
sappiamo che la funzione $h$ esiste, infatti:
###### Proposizione 1.107.
Sia $\Omega$ un dominio con bordo costituito da due componenti connesse
$\partial\Omega_{0}$ e $\partial\Omega_{1}$. Allora esiste una funzione
$h\in\overline{\Omega}$ tale che $h|_{\partial\Omega_{0}}=0$ e
$h|_{\partial\Omega_{1}}=1$.
###### Proof.
Siano $V_{0}$ e $V_{1}$ due intorni disgiunti di $\partial\Omega_{0}$ e
$\partial\Omega_{1}$ 505050gli insiemi $\partial\Omega_{0}$ e
$\partial\Omega_{1}$ sono chiusi e disgiunti in $R$ spazio metrico, e sia
$\\{\lambda_{0},\lambda_{1},\lambda\\}$ uan partizione dell’unità subordinata
al ricoprimento aperto $\\{V_{0},V_{1},(\partial\Omega)^{C}\\}$. Chiamiamo
$f_{0}$ e $f_{1}$ due funzioni lisce con le caratteristiche descritte in 1.25,
in particolare:
$\partial\Omega_{0}=f_{0}^{-1}(0)\ \ \ \partial\Omega_{1}=f_{1}^{-1}(0)$
È facile verificare che la funzione:
$\displaystyle h\equiv\lambda_{0}\cdot f_{0}+\lambda_{1}\cdot(1-f_{1})$
è una funzione liscia con $h|_{\partial\Omega_{0}}=0\ \
h|_{\partial\Omega_{1}}=1$. ∎
Questo dimostra ad esempio che tutte le funzioni
$u\in H(\Omega),\ \ u|_{\partial\Omega_{0}}=0\ \ u|_{\partial\Omega_{1}}=1$
sono funzioni $u\in C^{\infty}(\overline{\Omega})$.
## Chapter 2 Ultrafiltri e funzionali lineari moltiplicativi
In questo capitolo introduciamo gli ultrafilti. L’introduzione è tratta da
[D]. Il risultato principale riguarda l’esistenza di funzionali lineari
moltiplicativi sull’algebra di $l_{\infty}(\mathbb{N})$, ed è tratto
dall’articolo [N].
Lo scopo di questo capitolo è di descrivere un modo alternativo alle
successioni per caratterizzare le topologie che non sono sequenziali, nelle
quali cioè la chiusura per successioni non coincide con la chiusura
topologica. Un esempio di questi spazi topologici è lo spazio $\mathbb{M}(R)$
con la topologia $B$ descritto nella definizione 3.13. Inoltre gli ultrafiltri
possono essere utilizzati per definire dei caratteri sullo spazio
$l_{\infty}(\mathbb{N})$ (vedi paragrafo 2.2), e questo risultato può essere
utilizzato per dare un esempio non banale di carattere sull’algebra di Royden.
A questo scopo rimandiamo alla sezione 3.3.2.
### 2.1 Filtri, ultrafiltri e proprietà
Per prima cosa introduciamo la definizione di filtro.
###### Definizione 2.1.
Sia $(X,\tau)$ uno spazio topologico. Un filtro in questo spazio è una
collezione $\mathcal{A}=\\{A_{\alpha}\ t.c.\ \alpha\in I\\}$ di sottoinsiemi
di $X$ tale che:
1. 1.
$\forall\alpha\in I\ A_{\alpha}\neq\emptyset$
2. 2.
$\forall\alpha,\ \beta\in I$ $\exists\gamma\in I\ t.c.\ A_{\gamma}\subset
A_{\alpha}\cap A_{\beta}$
Osserviamo dalla definizione che ogni coppia di insiemi in $\mathcal{A}$ ha
intersezione non vuota, e per induzione qualunque collezione finita di insiemi
di $\mathcal{A}$ ha intersezione non vuota. Gli ultrafiltri generalizzano la
nozione di convergenza di successioni e sono utili negli spazi non primo
numerabili (come ad esempio l’algebra di Royden). A questo scopo definiamo:
###### Definizione 2.2.
Dato un ultrafiltro $\mathcal{A}$ nello spazio $(X,\tau)$ diciamo che
$\mathcal{A}$ converge a $x_{0}$ e scriviamo che $\mathcal{A}\to x_{0}$ se e
solo se $\forall U(x_{0})$ intorno aperto in $\tau$, esiste $\alpha$ tale che
$\mathcal{A}_{\alpha}\subset U$.
Diciamo invece che $\mathcal{A}$ si accumula in $x_{0}$ e scriviamo
$\mathcal{A}\triangleright x_{0}$ se e solo se $\forall U(x_{0})$,
$\forall\alpha$, $A_{\alpha}\cap U\neq\emptyset$
Osserviamo subito che $\mathcal{A}\to
x_{0}\Rightarrow\mathcal{A}\triangleright x_{0}$, e che negli spazi di
Hausdorff $\mathcal{A}$ non si accumula in nessun altro punto 111questo è il
teorema 3.2 a pag. 214 di [D]. Questo è evidente dal fatto che ogni coppia di
insiemi in $\mathcal{A}$ ha intersezione non vuota. Inoltre dalla definizione
è evidente che $\mathcal{A}\triangleright x_{0}$ se e solo se
$x_{0}\in\bigcap_{\alpha}\overline{A_{\alpha}}$.
Un esempio banale di filtro convergente è l’insieme degli intorni di un punto.
Dato $x_{0}\in X$, indichiamo con $\mathcal{U}(x_{0})$ il filtro dei suoi
intorni. È evidente che $\mathcal{U}(x_{0})\to x_{0}$.
È possibile definire in alcuni casi operazioni di unione e intersezione tra
filtri:
###### Definizione 2.3.
Dati due filtri $\mathcal{A}$ e $\mathcal{B}$ su $X$ definiamo i filtri:
1. 1.
$\mathcal{A}\cup\mathcal{B}=\\{A_{\alpha}\cup B_{\beta}\ t.c.\ \alpha\in I,\
\beta\in Y\\}$
2. 2.
se $A_{\alpha}\cap B_{\beta}\neq\emptyset$ per ogni scelta di $\alpha$ e
$\beta$, allora definiamo
$\mathcal{A}\cap\mathcal{B}=\\{A_{\alpha}\cap B_{\beta}\ t.c.\ \alpha\in I,\
\beta\in Y\\}$
La facile verifica che questi insiemi sono filtri è lasciata al lettore.
###### Definizione 2.4.
Dati due filtri $\mathcal{A}=\\{A_{\alpha}\ t.c.\ \alpha\in I\\}$ e
$\mathcal{B}=\\{B_{\beta}\ t.c.\ \beta\in Y\\}$, diciamo che $\mathcal{B}$ è
subordinato ad $\mathcal{A}$ e scriviamo $\mathcal{B}\vdash\mathcal{A}$ se e
solo se $\forall\alpha\ \exists\beta\ t.c.\ B_{\beta}\subset A_{\alpha}$
È facile verificare che la relazione di subordinazione è una relazione di
preordine, ma non di ordine parziale. Infatti è una relazione riflessiva e
transitiva, ma non è vero che $\mathcal{A}\vdash\mathcal{B}\ \wedge\
\mathcal{B}\vdash\mathcal{A}\Rightarrow\mathcal{A}=\mathcal{B}$. Un
controesempio si può trovare considerando $X=\mathbb{N}$,
$\mathcal{A}=\\{[2n,\infty)\ n\in\mathbb{N}\\}$ e $\mathcal{B}=\\{[n,\infty)\
n\in\mathbb{N}\\}$.
Per la relazione di subordinazione valgono le seguenti proprietà:
###### Proposizione 2.5.
1. 1.
$\mathcal{A}\subset\mathcal{B}\Rightarrow\mathcal{B}\vdash\mathcal{A}$
2. 2.
Se $\mathcal{B}\vdash\mathcal{A}$, allora ogni elemento di $\mathcal{B}$ ha
intersezione non vuota con ogni elemento di $\mathcal{A}$
3. 3.
$\mathcal{A}\to x_{0}$ se e solo se $\mathcal{A}\vdash\mathcal{U}(x_{0})$
4. 4.
Se $\mathcal{B}\vdash\mathcal{A}$, allora $\mathcal{A}\to
x_{0}\Rightarrow\mathcal{B}\to x_{0}$ e anche $\mathcal{B}\triangleright
x_{0}\Rightarrow\mathcal{A}\triangleright x_{0}$
###### Proof.
La dimostrazione di questi punti è lasciata al lettore. ∎
Per le successioni negli spazi topologici vale che una successione converge a
un punto $p$ se e solo se da ogni sua sottosuccessione si può estrarre una
sottosottosuccessione convergente a $p$, e una successione si accumula in un
punto se e solo se esiste una sua sottosuccessione convergente a quel punto.
Valgono risultati analogi per i filtri:
###### Proposizione 2.6.
Un filtro $\mathcal{A}$ converge a $x_{0}$ se e solo se per ogni
$\mathcal{B}\vdash\mathcal{A}$, esiste $\mathcal{C}\vdash\mathcal{B}$ tale che
$\mathcal{C}\to x_{0}$
###### Proof.
Dalla proposizione precedente risulta ovvia una delle due implicazioni. Per
dimostrare l’altra, supponiamo che $\mathcal{A}$ non converga a $x_{0}$.
Quindi esiste $U(x_{0})$ intorno di $x_{0}$ tale che $\forall\alpha,\
A_{\alpha}\cap U^{C}\neq\emptyset$. La collezione
$\mathcal{B}=\\{A_{\alpha}\cap U^{C}\ t.c.\ \alpha\in I\\}$ è un filtro
subordinato a $\mathcal{A}$ e per il quale $x_{0}$ non è punto di
accumulazione. Allora grazie alla proposizione precedente risulta che ogni
filtro subordinato a $\mathcal{B}$ non può convergere a $x_{0}$. ∎
###### Proposizione 2.7.
$\mathcal{A}\triangleright x_{0}$ se e solo se esiste
$\mathcal{B}\vdash\mathcal{A}$ tale che $\mathcal{B}\to x_{0}$
###### Proof.
Se esiste $\mathcal{B}\vdash\mathcal{A}$ tale che $\mathcal{B}\to x_{0}$, è
chiaro che $\mathcal{A}\triangleright x_{0}$. Supponiamo ora che
$\mathcal{A}\triangleright x_{0}$. La collezione
$\mathcal{B}=\mathcal{A}\cap\mathcal{U}(x_{0})$ è un filtro dato che
$\forall\alpha\ \forall U(x_{0}),\ A_{\alpha}\cap U\neq\emptyset$. Inoltre è
chiaro che $\mathcal{B}\vdash\mathcal{U}(x_{0})$. Allora per le proprietà
della relazione di subordinazione $\mathcal{B}\to x_{0}$. ∎
I filtri si possono definire anche tramite funzioni.
###### Proposizione 2.8.
Dato un filtro $\mathcal{A}$ su $X$ e una funzione $f:X\to Y$, la collezione
$f(\mathcal{A})=\\{f(A_{\alpha}\ t.c.\ A_{\alpha}\in\mathcal{A}\\}$ è un
filtro su $Y$
###### Proof.
Il fatto che ogni elemento di $f(\mathcal{A})$ sia non vuoto è ovvio. La
seconda proprietà caratterizzante i filtri segue facilmente dalla
considerazione che $f(A\cap B)\subset f(A)\cap f(B)$ ∎
Passiamo ora a definire gli ultrafiltri, o filtri massimali, e a dimostrarne
l’esistenza.
###### Definizione 2.9.
Un filtro $\mathcal{M}$ è detto massimale (o ultrafiltro) se non ha filtri
propriamente subordinati, cioè se
$\displaystyle\mathcal{A}\vdash\mathcal{M}\Rightarrow\mathcal{M}\vdash\mathcal{A}$
Non tutti gli autori condividono questa definizione. Su alcuni testi si
definisce ultrafiltro un filtro che non è contenuto propriamente in nessun
altro filtro. Anche la definizione di subordinazione è diversa. Si definisce
$\mathcal{B}$ subordinato a $\mathcal{A}$ se $\mathcal{A}\subset\mathcal{B}$.
In questo modo la relazione di subordinazione diventa una relazione d’ordine
parziale. Per gli scopi di questa tesi però non c’è differenza tra le due
definizioni, quindi adotteremo quella meno restrittiva e più generale di [D].
Passiamo ora a caratterizzare gli ultrafiltri e a dimostrarne l’esistenza.
###### Proposizione 2.10.
Un filtro $\mathcal{M}=\\{M_{\alpha}\ t.c.\ \alpha\in I\\}$ è un ultrafiltro
se e solo se per ogni $S\subset X$, uno dei due insiemi $S$ o $S^{C}$ contiene
un elemento di $\mathcal{M}$.
###### Proof.
Dato un ultrafiltro $\mathcal{M}$ e un insieme $S$ è chiaro che non può
accadere che sia $S$ che $S^{C}$ contengano un elemento di $\mathcal{M}$,
altrimenti questi due elementi avrebbero necessariamente intersezione vuota.
Assumiamo allora che $\forall\alpha,\ M_{\alpha}$ non sia contenuto in $S$,
allora $\forall\alpha M_{\alpha}\cap S^{C}\neq\emptyset$. Questo implica che
l’insieme $\mathcal{M}_{1}=\mathcal{M}\cap S^{C}=\\{A_{\alpha}\cap S^{C}\
t.c.\ \alpha\in I\\}$ sia un filtro tale che
$\mathcal{M}_{1}\vdash\mathcal{M}$. Ma allora per ipotesi
$\mathcal{M}\vdash\mathcal{M}_{1}$, quindi per ogni $\alpha$ esiste $\gamma$
tale che $A_{\gamma}\subset A_{\alpha}\cap S^{C}\subset S^{C}$.
L’altra implicazione si dimostra considerando un filtro
$\mathcal{M}_{1}\vdash\mathcal{M}$. L’ipotesi assicura che per ogni
$A^{1}_{\beta}\in\mathcal{M}_{1}$ esiste un
$\mathcal{A}_{\alpha}\in\mathcal{M}$ tale che $\mathcal{A}_{\alpha}\subset
A^{1}_{\beta}$ oppure $\mathcal{A}_{\alpha}\subset(A^{1}_{\beta})^{C}$. La
seconda possibilità è esclusa dal fatto che
$\mathcal{M}_{1}\vdash\mathcal{M}$, quindi $\mathcal{M}\vdash\mathcal{M}_{1}$,
cioè $\mathcal{M}$ è massimale. ∎
Grazie a questa caratterizzazione è facile dimostrare che dato $x_{0}\in X$,
l’insieme $\mathcal{M}=\\{A\subset X\ t.c.\ x_{0}\in A\\}$ è un ultrafiltro,
anche se è abbastanza banale e poco pratico. Per i nostri scopi è necessario
dimostrare che ogni filtro è contenuto in un ultrafiltro, e per fare questo è
necessario assumere il lemma di Zorn, che ricordiamo:
###### Teorema 2.11 (Lemma di Zorn).
Ogni insieme preordinato in cui ogni catena contiene un limite superiore
possiede almeno un elemento massimale
Solitamente il lemma di Zorn è enunciato per insiemi parzialmente ordinati e
non semplicemente preordinati. Si può dimostrare però che queste due versioni
del teorema sono equivalenti (a questo scopo rimandiamo a [M1], esercizio
1.16). Osserviamo che negli insiemi preordinati (con una relazione d’ordine
$\geq$) un elemento $m$ è detto massimale se $a\geq m\Rightarrow m\geq a$.
###### Teorema 2.12.
Dato un insieme $X$ e un filtro $\mathcal{A}$ su questo insieme, esiste un
filtro massimale (ultrafiltro) $\mathcal{M}$ tale che
$\mathcal{M}\vdash\mathcal{A}$.
###### Proof.
Chiamiamo $\hat{B}$ la famiglia dei filtri per i quali
$\mathcal{B}\vdash\mathcal{A}$, e definiamo una relazione di preordine $\geq$
su questo insieme in questo modo:
$\displaystyle\mathcal{B}_{2}\geq\mathcal{B}_{1}\Longleftrightarrow\mathcal{B}_{2}\vdash\mathcal{B}_{1}$
Ricordiamo che dalla proposizione 2.5,
$\mathcal{B}_{1}\subset\mathcal{B}_{2}\Rightarrow\mathcal{B}_{2}\vdash\mathcal{B}_{1}$,
quindi anche $\mathcal{B}_{2}\geq\mathcal{B}_{1}$. Consideriamo una catena
$\mathcal{B}_{i}$ di elementi di $\hat{B}$ 222ricordiamo che con catena in un
insieme preordinato si intende un suo sottoinsieme tale che per ogni sua
coppia di elementi $a$ e $b$, $a\geq b$ oppure $b\geq a$, dove è possibile che
entrambe le affermazioni siano vere. L’insieme
$\tilde{\mathcal{B}}=\bigcup_{i}\mathcal{B}_{i}$ è un elemento massimale per
questa catena. Per prima cosa dimostriamo che è un filtro. Ovviamente ogni suo
elemento è non vuoto, inotlre dati due suoi elementi
$A_{1}\in\tilde{\mathcal{B}}$ e $A_{2}\in\tilde{\mathcal{B}}$, esistono
$\mathcal{B}_{i_{1}}$ e $\mathcal{B}_{i_{2}}$ filtri nella catena che li
contengono. Supponiamo che $\mathcal{B}_{i_{1}}\vdash\mathcal{B}_{i_{2}}$,
allora esiste $A_{3}\in\mathcal{B}_{i_{1}}$ tale che $A_{3}\subset A_{2}$, e
quindi poiché $\mathcal{B}_{i_{1}}$ è un filtro, esiste anche
$E\in\mathcal{B}_{i_{1}}$ tale che $E\subset A_{1}\cap A_{3}\subset A_{1}\cap
A_{2}$. È poi evidente che
$\tilde{\mathcal{B}}\geq\mathcal{B}_{i}\geq\mathcal{A}$ per ogni $i$, quindi
$\tilde{\mathcal{B}}$ è un limite superiore per la catena. Applicando il lemma
di Zorn otteniamo che esiste almeno un elemento massimale $\mathcal{M}$ in
$\hat{B}$. $\mathcal{M}$ è un filtro massimale perché se
$\mathcal{C}\vdash\mathcal{M}$, per transitività
$\mathcal{C}\vdash\mathcal{A}$, quindi $\mathcal{C}\in\hat{B}$ e quindi per
massimalità $\mathcal{M}\geq\mathcal{C}$, cioè $\mathcal{M}\vdash\mathcal{C}$.
∎
Osserviamo ora due proprietà degli ultrafiltri che ci saranno utili nel
seguito:
###### Proposizione 2.13.
Dato un ultrafiltro $\mathcal{M}$ su $X$, $\mathcal{M}\triangleright x_{0}$ se
e solo se $\mathcal{M}\to x_{0}$
###### Proof.
Questa proposizione è un corollario della definizione di ultrafiltro e della
proposizione 2.7 ∎
###### Proposizione 2.14.
Data una qualunque funzione $f:X\to Y$ insiemi qualsiasi, se $\mathcal{M}$ è
un ultrafiltro su $X$, allora $f(\mathcal{M})=\\{f(M_{\alpha})\ t.c.\
M_{\alpha}\in\mathcal{M}\\}$ è un ultrafiltro in $Y$.
###### Proof.
$f(\mathcal{M})$ è un filtro grazie alla proposizione 2.8, quindi rimane da
dimostrare solo la sua massimalità. Utilizziamo la caratterizzazione 2.10 dei
filtri massimali. Sia $S\in Y$. Visto che $f^{-1}(S^{C})=(f^{-1}(S))^{C}$,
esiste un elemento $M_{\alpha}$ contenuto o in $f^{-1}(S)$ o nel suo
complementare. $M_{\alpha}\subset f^{-1}(S)\Rightarrow f(M_{\alpha})\subset S$
e lo stesso vale per $S^{C}$, quindi dato ogni sottoinsieme di $Y$, esiste un
elemento di $f(\mathcal{M})$ contenuto o in esso o nel suo complementare. ∎
Il tipo di ultrafiltri massimali che ci interessano sono gli ultrafiltri non
costanti, ovvero gli ultrafiltri che non contengono nessun elemento finito
###### Definizione 2.15.
Un ultrafiltro $\mathcal{M}$ è detto non costante se e solo se non contiene
nessun insieme di cardinalità finita.
Per dimostrare l’esistenza di ultrafiltri non costanti sull’insieme dei numeri
naturali costruiamo un filtro con queste caratteristiche e lo estendiamo a
massimale grazie al teorema 2.12 333dato che il teorema fa uso del lemma di
Zorn, questa dimostrazione non è costruttiva. Non si possono cioè trovare
esempi descrivibili facilmente di ultrafiltri massimali non costanti con
questa tecnica..
###### Osservazione 2.16.
La collezione $\mathcal{F}$ di sottoinsiemi a complementare finito è un filtro
sull’insieme $\mathbb{N}$.
###### Proof.
La dimostrazione è immediata ∎
Indichiamo con $\mathcal{M}_{\mathcal{F}}$ un ultrafiltro subordinato al
filtro $\mathcal{F}$. Questo ultrafiltro necessariamente non conterrà nessun
insieme a cardinalità finita. Sia per assurdo $S\in\mathcal{M}_{\mathcal{F}}$
insieme a cardinalità finita. $S^{C}\in\mathcal{F}$ per definizione. Ma visto
che $\mathcal{M}_{\mathcal{F}}\vdash\mathcal{F}$, esiste un elemento di
$\mathcal{M}_{\mathcal{F}}$ contenuto in $S^{C}$, ma allora questo elemento ha
intersezione vuota con $S$, assurdo.
Prima di concludere questa sezione riportiamo la relazione tra compattezza di
un insieme e filtri su quell’insieme.
###### Proposizione 2.17.
Per un insieme $K\subset(X,\tau)$ spazio topologico le seguenti affermazioni
sono equivalenti:
1. 1.
Ogni ricoprimento di aperti di $M$ ha un sottoricoprimento finito
2. 2.
Per ogni famiglia di chiusi $C_{i}$ tale che $\bigcap_{i}C_{i}=\emptyset$,
esiste un numero finito di indici $i_{1},\dots,i_{n}$ tali che
$\bigcap_{k=1}^{n}C_{i_{k}}=\emptyset$
3. 3.
Ogni filtro su $K$ ha un punto di accumulazione
4. 4.
Ogni ultrafiltro in $K$ è convergente
Se una qualsiasi delle proprietà precedenti è valida, l’insieme $K$ è detto
compatto.
###### Proof.
(1) e (2) sono equivalenti grazie alle leggi di De Morgan, (3)$\Rightarrow$(4)
grazie alla proposizione 2.13, (4)$\Rightarrow$(3) grazie al teorema 2.12 e
alla proposizione 2.5. (2)$\Rightarrow$(3) poiché dato un filtro
$\mathcal{A}$, ogni collezione finita di suoi elementi $A_{\alpha_{i}}$ ha
intersezione non vuota, quindi anche
$\bigcap_{i=1}^{n}\overline{A_{\alpha_{i}}}\neq\emptyset$. Grazie a (2) allora
$\bigcap_{\alpha}\overline{A_{\alpha}}\neq\emptyset$, quindi l’insieme dei
punti di accumulazione di $\mathcal{A}$ è non vuoto. (3)$\Rightarrow$(2)
perché data una qualsiasi collezione di chiusi $C_{\alpha}$ tale che ogni sua
sottocollezione finita abbia intersezione non vuota, la collezione dei chiusi
e di tutte le possibili intersezioni finite è un filtro, che per (2) ha un
punto di accumulazione, quindi
$\bigcap_{\alpha}\overline{C_{\alpha}}=\bigcap_{\alpha}C_{\alpha}\neq\emptyset$.
∎
Vale anche la seguente caratterizzazione della continuità delle funzioni
tramite ultrafiltri:
###### Proposizione 2.18.
Una funzione $f:X\to Y$ spazi topologici è continua in $x_{0}$ se e solo se
per ogni filtro $\mathcal{A}\to x_{0}$, $f(\mathcal{A})\to f(x_{0})$
###### Proof.
Supponiamo che $f$ sia continua in $x_{0}$. Chiamando $\mathcal{U}(x)$ il
filtro degli intorni aperti di $x_{0}$, dalla definizione di continuità si ha
che per ogni intorno aperto $W$ di $f(x_{0})$, esiste un aperto $U$ in $X$
tale che $f(U)\subset W$. Quindi per definizione $f(\mathcal{U}(x_{0}))\to
f(x_{0})$. Ora, se consideriamo un filtro $\mathcal{A}\to x_{0}$ qualsiasi,
dalle proposizioni precedenti risulta che
$\mathcal{A}\vdash\mathcal{U}(x_{0})$, quindi anche $f(\mathcal{A})\vdash
f(\mathcal{U}(x_{0}))$, quindi necessariamente $f(\mathcal{A})\to f(x_{0})$.
Per dimostrare l’implicazione inversa, dato che $f(\mathcal{U}(x_{0}))\to
f(x_{0})$, per definizione per ogni intorno di $W(f(x_{0}))$, esiste
$U(x_{0})$ tale che $f(U(x_{0}))\subset W$, quindi $f$ è continua. ∎
Per una successione in $\mathbb{R}$ possono accadere due cose: o la
successione è limitata, quindi tutta contenuta in un compatto, oppure no. Se
la successione è limitata e converge, allora converge a un punto al finito di
$\mathbb{R}$, altrimenti una successione illimitata può “convergere”
all’infinito. Per filtri e ultrafiltri /che giocano il ruolo delle successioni
“convergenti” in senso generalizzato), vale una caratterizzazione simile.
###### Definizione 2.19.
Un filtro $\mathcal{A}$ si dice avere supporto in un insieme $K\subset X$ se
esiste $\alpha$ tale che $A_{\alpha}\subset K$
Da questa definizione si ricava immediatamente che
###### Proposizione 2.20.
Se $\mathcal{A}$ ha supporto in $K$, allora $\mathcal{A}\triangleright
x_{0}\Rightarrow x_{0}\in\overline{K}$ (quindi anche $\mathcal{A}\to
x_{0}\Rightarrow x_{0}\in\overline{K}$)
###### Proof.
La dimostrazione è una conseguenza quasi immediata della definizione.
Supponiamo per assurdo che $x_{0}\not\in\overline{K}$. Allora esiste $U$
intorno di $x_{0}$ separato da $\overline{K}$. Ma allora $A_{\alpha}\cap
U=\emptyset$ (dove $\alpha$ è l’indice tale che $A_{\alpha}\subset K$), quindi
$x_{0}$ non può essere punto di accumulazione per $A$. ∎
###### Proposizione 2.21.
Se $\mathcal{A}$ ha supporto in $K$, allora $\mathcal{B}\equiv\mathcal{A}\cap
K$ è un filtro subordinato ad $\mathcal{A}$. Inoltre se $\mathcal{A}$ è un
ultrafiltro, anche $\mathcal{B}$ è un ultrafiltro.
###### Proof.
Visto che esiste $\alpha$ tale che $A_{\alpha}\subset K$, allora ogni insieme
del filtro $\mathcal{A}$ ha intersezione non vuota con $K$, quindi
$\mathcal{B}\equiv\mathcal{A}\cap K$ ben definisce un filtro, ovviamente
subordinato ad $\mathcal{A}$. Il fatto che $\mathcal{B}$ sia un ultrafiltro se
$\mathcal{A}$ lo è è una facile conseguenza della caratterizzazione 2.10 ∎
Un filtro si dice avere supporto compatto se esiste un insieme compatto che è
un suo supporto. In uno spazio localmente compatto, ogni filtro convergente ha
supporto compatto (basta considerare un intorno compatto del punto al quale il
filtro converge). Se il filtro $\mathcal{A}$ non ha supporto compatto, vuol
dire che per ogni compatto $K$ e per ogni indice $\alpha$, $A_{\alpha}\cap
K^{C}\neq\emptyset$, quindi per ogni compatto $K$ il filtro
$\mathcal{B}=\mathcal{A}\cap K^{C}$ è un filtro subordinato ad $\mathcal{A}$,
e se $\mathcal{A}$ è un ultrafiltro, anche $\mathcal{B}$ lo è.
### 2.2 Applicazioni: caratteri sulle successioni limitate
Come applicazione delle due sezioni precedenti, e soprattutto come esempio per
il seguito, presentiamo l’esistenza di particolari caratteri sullo spazio
$l_{\infty}(\mathbb{N})$, cioè lo spazio di Banach delle successioni limitate
a valori reali dotato della norma del sup. Questo spazio diventa un’algebra di
Banach se definiamo la moltiplicazione tra successioni punto per punto, cioè
date due successioni $x(n)$ e $y(n)$, definiamo $(x\cdot y)(n)\equiv x(n)\cdot
y(n)$ 444lasciamo le dovute verifiche al lettore. È facile osservare che la
successione costante uguale a 1 è l’unità di ques’algebra.
Tutti i funzionali $\phi_{n}:l_{\infty}(\mathbb{N})\to\mathbb{R}$ definiti da
$\phi_{n}(x)=x(n)$ sono caratteri. Cerchiamo però di definire altri
funzionali, legati solo al comportamento di ogni successione all’infinito.
Questa sezione è tratta dall’articolo [N].
Per prima cosa definiamo un’operazione di limite su $l_{\infty}(\mathbb{N})$:
###### Definizione 2.22.
Un’operazione di limite sullo spazio $l_{\infty}(\mathbb{N})$ è un funzionale
lineare $\phi:l_{\infty}(\mathbb{N})\to\mathbb{R}$ tale che per ogni
successione
$\displaystyle\liminf_{n}x(n)\leq\phi(x)\leq\limsup_{n}x(n)$
Un’operazione di limite è quindi un funzionale lineare che vale $\lim_{n}x(n)$
se questo limite esiste. Grazie al teorema di Hahn-Banach è possibile
dimostrare l’esistenza di alcune operazioni di limite (vedi ad esempio [N],
oppure l’esercizio 4 cap.3 pag.85 di [R2]). È anche possibile dimostrare
l’esistenza di operazioni di limite moltiplicative su
$l_{\infty}(\mathbb{N})$, e a questo scopo utilizzeremo la teoria sviluppata
sugli ultrafiltri non costanti. Prima della proposizione ricordiamo che:
###### Osservazione 2.23.
Ricordiamo la definizione di somma di insiemi. Dati due sottoinsiemi $A$ e $B$
di uno spazio vettoriale $V$, $A+B=\\{a+b\ \ t.c.\ \ a\in A,\ b\in B\\}$.
Dalla definizione è chiaro che $(f+g)(A)\subset f(A)+g(A)$. In modo analogo
alla somma di insiemi si può definire il loro prodotto, e vale ancora che
$(f\cdot g)(A)\subset f(A)\cdot g(A)$.
###### Proposizione 2.24.
Sull’insieme $l_{\infty}(\mathbb{N})$ esistono operazioni di limite
moltiplicative.
###### Proof.
Fissiamo un ultrafiltro non costante $\mathcal{M}$ sull’insieme $\mathbb{N}$
555l’esistenza di questo tipo di ultrafiltro è stata dimostrata
nell’osservazione 2.16. Una qualunque successione $x:\mathbb{N}\to\mathbb{R}$
permette di definire un ultrafiltro $x(\mathcal{M})$ su $\mathbb{R}$, anzi
visto che la successione è limitata, l’ultrafiltro sarà contenuto nell’insieme
compatto $[-\left\|x\right\|,\left\|x\right\|]$, quindi grazie a 2.17 sarà
convergente. Definiamo
$\displaystyle\phi_{\mathcal{M}}(x)\equiv\lim_{\mathcal{M}}x\equiv\lim
x(\mathcal{M})$
Per prima cosa osserviamo che se sostituiamo $\mathcal{M}$ con il filtro
$\mathcal{F}$ degli insiemi a complementare finito, il limite non è sempre
definito ma coincide con il limite di una successione in senso classico. Resta
da dimostrare che $\phi_{\mathcal{M}}$ è lineare e moltiplicativo. A questo
scopo consideriamo due successioni $x,\ y:\mathbb{N}\to\mathbb{R}$, e
dimostriamo che
$\phi_{\mathcal{M}}(x+y)=\phi_{\mathcal{M}}(x)+\phi_{\mathcal{M}}(y)$. Sia
$\phi_{\mathcal{M}}(x)=s$ e $\phi_{\mathcal{M}}(y)=t$. Dalla definizione di
limite di ultrafiltro otteniamo che per ogni $\epsilon>0$, esistono $\alpha$ e
$\beta$ tali che
$\displaystyle x(M_{\alpha})\subset B_{\epsilon}(s)\ \wedge\
y(M_{\beta})\subset B_{\epsilon}(t)$
dove $M_{\alpha},\ M_{\beta}\in\mathcal{M}$ e $B_{\epsilon}(s)$ indica
l’insieme aperto $(s-\epsilon,s+\epsilon)$. Dato che $\mathcal{M}$ è un
filtro, esiste $\gamma$ tale che $M_{\gamma}\subset M_{\alpha}\cap M_{\beta}$.
Questo significa che per ogni $\epsilon>0$
$\displaystyle(x+y)(M_{\gamma})\subset x(M_{\gamma})+y(M_{\gamma})\subset
B_{\epsilon}(s)+B_{\epsilon}(t)=B_{2\epsilon}(s+t)$
dove abbiamo usato l’osservazione 2.23. Questo dimostra che
$\lim_{\mathcal{M}}(x+y)=\lim_{\mathcal{M}}x+\lim_{\mathcal{M}}y$
Consideriamo ora un qualunque numero $a\in\mathbb{R}$ e una successione $x\in
l_{\infty}(\mathbb{N})$. La dimostrazione del fatto che
$\phi_{\mathcal{M}}(ax)=a\phi_{\mathcal{M}}(x)$ è del tutto analoga a quella
appena mostrata, quindi lasciamo i dettagli al lettore 666questo completa la
dimostrazione che $\phi_{\mathcal{M}}$ è lineare..
La dimostrazione che $\phi_{\mathcal{M}}$ è moltiplicativo è anch’essa analoga
a questa, l’unico punto un po’ più difficile è capire come caratterizzare
l’insieme $B_{\epsilon}(s)\cdot B_{\epsilon}(t)$. Osserviamo che
$\displaystyle\left|a-s\right|<\epsilon\ \wedge\ \left|b-t\right|<\epsilon\
\Longrightarrow$ $\displaystyle\Longrightarrow\left|ab-st\right|\leq\left|ab-
at\right|+\left|at-
st\right|\leq\epsilon(\left|a\right|+t)\leq\epsilon(\left|s\right|+\left|t\right|+\epsilon)$
quindi $B_{\epsilon}(s)\cdot B_{\epsilon}(t)\subset
B_{[\epsilon(\left|s\right|+\left|t\right|+\epsilon)]}(st)$. Dato che
$\epsilon(\left|s\right|+\left|t\right|+\epsilon)$ diventa piccolo a piacere
al diminuire di $\epsilon$, seguendo un ragionamento del tutto analogo al
precedente, si ottiene che per ogni $\epsilon>0$, esiste $\gamma$ tale che
$\displaystyle(x\cdot y)(M_{\gamma})\subset x(M_{\gamma})\cdot
y(M_{\gamma})\subset B_{\epsilon}(s+t)$
da cui per ogni ultrafiltro $\mathcal{M}$, $\phi_{\mathcal{M}}$ è lineare e
moltiplicativo (quindi anche continuo) su $l_{\infty}(\mathbb{N})$. Resta da
dimostrare solo che è un’operazione di limite. A questo scopo procediamo per
assurdo: sia $\phi_{\mathcal{M}}(x)>\limsup_{n}x(n)$, e scegliamo $p$ in modo
che $\phi_{\mathcal{M}}(x)>p>\limsup_{n}x(n)$. L’insieme $(p,\infty)$ è un
intorno di $\phi_{\mathcal{M}}(x)$, quindi contiene un elemento di
$x(\mathcal{M})$, quindi esiste un elemento di $\mathcal{M}$ contenuto in
$x^{-1}(p,\infty)$. Dato che $p>\limsup_{n}(x)$, l’insieme $x^{-1}(p,\infty)$
ha cardinalità finita, quindi per definizione di ultrafiltro non costante è
impossibile che $x^{-1}(p,\infty)$ contenga qualunque elemento di
$\mathcal{M}$, da cui $\phi_{\mathcal{M}}(x)\leq\limsup_{n}x(n)$ $\forall x,\
\forall\mathcal{M}$ ultrafiltro non costante.
Osservando che $\liminf(x)=-\limsup(-x)$, si ottiene immediatamente anche
l’altra disuguaglianza. ∎
Quello che abbiamo ottenuto è un’operazione $\phi_{\mathcal{M}}$ che è
lineare, moltilicativa, definita su ogni successione limitata e che coincide
con il limite standard quando questo è definito. Il limite standard, oltre che
ad essere lineare e moltiplicativo, è anche invariante per traslazioni, nel
senso che $\lim_{n}x(n)=\lim_{n}x(n+1)$ quando è definito. Possiamo chiederci
se vale una proprietà simile per $\phi_{\mathcal{M}}$.
###### Osservazione 2.25.
Nessuna operazione di limite lineare e moltiplicativa è anche invariante per
traslazioni
###### Proof.
Dimostriamo questa affermazione per assurdo. Consideriamo la successione
limitata $x:\mathbb{N}\to\mathbb{R}$, $x(n)\equiv(-1)^{n}$, e sia $\phi$
un’operazione di limite lineare e moltiplicativa. Dato che
$\phi(x)^{2}=\phi(x\cdot x)=\phi(1)=1$, $\phi(x)=\pm 1$. Per linearità
$\phi(x(n))+\phi(x(n+1))=\phi(x(n)+x(n+1))=0$. Se $\phi(x(n))=\phi(x(n+1))$,
allora dall’ultima uguaglianza si otterrebbe $\phi(x)=0$, assurdo. ∎
Ricordiamo che esistono operazioni di limite lineari invarianti per
traslazione (che evidentemente non possono essere moltiplicative). La loro
esistenza è dimostrata ad esempio su [R2] nell’esercizio 4 cap.3 pag.85.
## Chapter 3 Algebra e compattificazione di Royden
Questo capitolo segue le orme del capitolo III di [SN] e dell’articolo [CSL].
Lo scopo è quello di introdurre l’algebra di Royden su varietà Riemanniane (di
dimesione finita qualsiasi) e quindi la compattificazione di Royden delle
varietà.
### 3.1 Funzioni di Tonelli
#### 3.1.1 Definizioni e proprietà fondamentali
Per prima cosa definiamo un’insieme particolare di funzioni, le funzioni di
Tonelli. Lo scopo della sezione è dimostrare che queste funzioni su una
varietà constituiscono un’algebra di Banach rispetto a una particolare norma.
Iniziamo con il definire queste funzioni su rettangoli in $\mathbb{R}^{m}$
###### Definizione 3.1.
Una funzione $f:\prod_{i=1}^{m}(a_{i},b_{i})\to\mathbb{R}$ si dice di Tonelli
se soddisfa:
1. 1.
$f$ è continua su $\prod_{i=1}^{m}(a_{i},b_{i})$
2. 2.
per ogni $i$,
$f^{i}(z)=f(\bar{x}_{1},\dots,\bar{x}_{i-1},z,\bar{x}_{i+1},\dots,\bar{x}_{m})$
è assolutamente continua rispetto a $z$ per quasi ogni valore di
$(\bar{x}_{1},\dots,\bar{x}_{m})$ 111rimandiamo alla sezione 1.2 per la
definizione e alcune proprietà delle funzioni assolutamente continue
3. 3.
Per ogni $i=1,\cdots,m$, $\frac{\partial f}{\partial x_{i}}$ è a quadrato
integrabile su ogni sottoinsieme compatto di $\prod_{i=1}^{m}(a_{i},b_{i})$,
cioè $\frac{\partial f}{\partial x_{i}}\in
L^{2}_{loc}(\prod_{i=1}^{m}(a_{i},b_{i}))$
Notiamo che tutte le funzioni
$C^{\infty}(\prod_{i=1}^{m}(a_{i},b_{i}),\mathbb{R})$ sono ovviamente funzioni
di Tonelli. L’insieme delle funzioni di Tonelli è ovviamente uno spazio
vettoriale, e in un certo senso è l’insieme pià piccolo di funzioni continue
per cui ha senso parlare di integrale di Dirichlet. A questo proposito
dimostriamo che:
###### Proposizione 3.2.
Data una funzione $f:M\to\mathbb{R}$ che sia di Tonelli su un insieme compatto
$K\Subset U$, dove $U=\phi^{-1}(\prod_{i=1}^{m}(a_{i},b_{i}))$ è un insieme
coordinato, allora detto
$\displaystyle
D_{K}(f)=\int_{K}\left|\nabla(f)\right|^{2}dV=\int_{\phi(K)}g^{ij}\frac{\partial
f}{\partial x^{i}}\frac{\partial f}{\partial
x^{j}}\sqrt{\left|g\right|}dx^{1}\dots dx^{n}$ (3.1)
si ha che $D_{K}(f)$ è finito.
###### Proof.
Dalla definizione sappiamo che
$\int_{\phi(K)}\sum_{i=1}^{m}\left|\frac{\partial f}{\partial
x^{i}}\right|^{2}dx^{1}\dots dx^{n}<\infty$. Iniziamo la dimostrazione col
notare che per ogni punto $p$ e ogni m-upla di numeri $(y_{1},\dots,y_{m})$,
esiste una costante $c_{p}$ (indipendente dai numeri $y_{i}$) tale che:
$\displaystyle
c_{p}^{-1}\sum_{i=1}^{m}(y_{i})^{2}\leq\sum_{i,j=1}^{m}g^{ij}y_{i}y_{j}\leq
c_{p}\sum_{i=1}^{m}(y_{i})^{2}$ (3.2)
questo è vero grazie al fatto che $\sum_{i,j=1}^{m}g^{ij}y_{i}y_{j}$ definisce
una norma su $\mathbb{R}^{m}$, e tutte le norme negli spazi finito-
dimensionali sono equivalenti. Inoltre questa costante $c_{p}$ dipende con
continuità da $p$ 222le funzioni $g^{ij}$ dipendono con continuità da $p$,
quindi su ogni compatto $K\Subset M$, la funzione $c_{p}$ ha un massimo, che
indicheremo semplicemente con $c$. Da queste considerazioni risulta che:
$\displaystyle g^{ij}\frac{\partial f}{\partial x^{i}}\frac{\partial
f}{\partial x^{j}}\leq\sum_{i=1}^{m}c\left|\frac{\partial f}{\partial
x^{i}}\right|^{2}$ (3.3)
dove gli indici ripetuti si intendono sommati. A questo punto, dato che su
ogni compatto anche $\sqrt{\left|g\right|}$ assume massimo finito (che
indicheremo $G$), si ottiene facilmente che:
$\displaystyle D_{K}(f)=\int_{\phi(K)}g^{ij}\frac{\partial f}{\partial
x^{i}}\frac{\partial f}{\partial x^{j}}\sqrt{\left|g\right|}dx^{1}\dots
dx^{n}\leq cG\sum_{i=1}^{m}\int_{\phi(K)}\left|\frac{\partial f}{\partial
x^{i}}\right|^{2}dx^{1}\dots dx^{m}<\infty$
∎
La definzione di funzione di Tonelli può essere estesa in maniera naturale
anche a funzioni $f:M\to\mathbb{R}$
###### Definizione 3.3.
Una funzione $f:M\to\mathbb{R}$ si dice di Tonelli se è di Tonelli in ogni
insieme parametrizzabile come rettangolo.
La proposizione precedente assicura che su ogni compatto coordinato (contenuto
in un rettangolo coordinato per la precisione), l’integrale di Dirichlet di
una funzione di Tonelli è finito. La stessa cosa vale anche per un qualsiasi
insieme compatto.
###### Proposizione 3.4.
Per ogni compatto $K\Subset M$, $D_{K}(f)<\infty$ se $f$ è una funzione di
Tonelli
###### Proof.
La dimostrazione segue tecniche standard della geometria differenziale. Per
ogni punto $p$ della varietà $M$ è possibile trovare un intorno compatto
coordinato che descritto in carte locali abbia la forma
$\prod_{i=1}^{m}[a_{i},b_{i}]$. Su ciascuno di questi insiemi l’integrale di
Dirichlet di $f$ è finito grazie alla proposizione precedente. Dato che
l’insieme $K$ può essere ricoperto da un numero finito di questi insiemi,
anche l’integrale di Dirichlet su $K$ sarà finito. ∎
#### 3.1.2 Operazioni con le funzioni di Tonelli
Oltre che somma e moltiplicazione per un numero reale, le funzioni di Tonelli
ammettono anche altre operazioni tra loro. In particolare il valore assoluto
di una funzione di Tonelli è di Tonelli, e quindi anche massimo e minimo tra
due funzioni di Tonelli sono funzioni di Tonelli. In questo paragrafo ci
occuperemo di queste operazioni.
###### Proposizione 3.5.
Se $f$ è di Tonelli, anche $\left|f\right|$ è di Tonelli. Inoltre su ogni
insieme misurabile $S\subset M$, $D_{S}(\left|f\right|)\leq D_{S}(f)$.
###### Proof.
Consideriamo una funzione $f:\prod_{i=1}^{m}(a_{i},b_{i})\to\mathbb{R}$ di
Tonelli. Ovviamente $\left|f\right|$ è una funzione continua. L’assoluta
continuità delle funzioni $\left|f\right|^{i}$ è garantita dal fatto che
$\left|\left|f\right|^{i}(z_{1})-\left|f\right|^{i}(z_{2})\right|\leq\left|f^{i}(z_{1})-f^{i}(z_{2})\right|$
333vedi definizione 1.29. Rimane da verificare solo la maggiorazione degli
integrali di Dirichlet. Consideriamo come primo caso
$K\Subset\prod_{i=1}^{m}(a_{i},b_{i})$.
Sia $E$ un insieme di misura nulla tale che tutte le derivate
$\partial\left|f\right|/\partial x^{i}$ e $\partial f/\partial x^{i}$ esistano
su $L\equiv E^{C}=\prod_{i=1}^{m}(a_{i},b_{i})\setminus E$. Siano ora
$L_{+}=f^{-1}(0,\infty)\cap L$, $L_{-}=f^{-1}(-\infty,0)\cap L$,
$L_{0}=f^{-1}(0)\cap L$. Questi insiemi sono ovviamente misurabili.
Sull’insieme $L_{+}$, $\partial\left|f\right|/\partial x^{i}=\partial
f/\partial x^{i}$, infatti per ogni punto $x\in L_{+}$, grazie al teorema di
permanenza del segno esiste un intorno di $x$ sul quale $\left|f\right|=f$. In
modo analogo, su $L_{-}$, $\partial\left|f\right|\partial x^{i}=-\partial
f/\partial x^{i}$.
Su $L_{0}$ invece $\partial\left|f\right|\partial x^{i}=0$. Infatti per
definizione:
$\displaystyle\frac{\partial\left|f\right|}{\partial x^{i}}=\lim_{h\to
0}\frac{\left|f(\bar{x}^{1},\dots,\bar{x}^{i-1},x+h,\bar{x}^{i+1},\dots\bar{x}^{m})\right|-0}{h}$
dove abbiamo sfruttato il fatto che $x\in L_{0}$. Data la definizione di $L$,
questo limite esiste necessariamente. Visto che scegliendo $h>0$ il limite è
$\geq 0$ e scegliendo $h<0$ succede il contrario, questo limite è
necessariamente nullo.
Consideriamo ora un compatto $K\Subset\prod_{i=1}^{m}(a_{i},b_{i})$. Per
additività degli integrali:
$\displaystyle\int_{K}\left|\nabla\left|f\right|\right|^{2}dV=\int_{K\cap
L_{+}}\left|\nabla\left|f\right|\right|^{2}dV+\int_{K\cap
L_{-}}\left|\nabla\left|f\right|\right|^{2}dV+\int_{K\cap
L_{0}}\left|\nabla\left|f\right|\right|^{2}dV$
Visto che in coordinate
$\left|\nabla f\right|^{2}=g^{ij}\frac{\partial f}{\partial
x^{i}}\frac{\partial f}{\partial x^{j}}$
si ottiene che
$\displaystyle\int_{K\cap
L_{+}}\left|\nabla\left|f\right|\right|^{2}dV=\int_{K\cap L_{+}}\left|\nabla
f\right|^{2}dV$ $\displaystyle\int_{K\cap
L_{-}}\left|\nabla\left|f\right|\right|^{2}dV=\int_{K\cap L_{-}}\left|\nabla
f\right|^{2}dV$ $\displaystyle\int_{K\cap
L_{0}}\left|\nabla\left|f\right|\right|^{2}dV=0$
da cui si ottiene $D_{K}(\left|f\right|)\leq D_{K}(f)$.
Per un insieme $S\subset M$ qualsiasi, basta applicare la definizione di
integrale con le partizioni dell’unità. A questo scopo sia $\\{\lambda_{n}\\}$
una partizione dell’unità di $M$ subordinata a un ricoprimento di aperti
coordinati. Per definizione
$\displaystyle\int_{M}\left|\nabla
f\right|^{2}dV=\sum_{n=1}^{\infty}\int_{supp(\lambda_{n})}\lambda_{n}\cdot\left|\nabla
f\right|^{2}dV$
con un ragionamento analogo a quello illustrato sopra, si ottiene
$D_{S}(\left|f\right|)\leq D_{S}(f)$ per ogni $S\subset M$ misurabile. ∎
Grazie a questa proposizione, è immediato verificare che
###### Proposizione 3.6.
Date due funzioni di Tonelli $f$ e $g$, anche $\max\\{f,g\\}$ e
$\min\\{f,g\\}$ sono funzioni di Tonelli
###### Proof.
La dimostrazione è immediata se si considera che
$\displaystyle\max\\{f,g\\}=\frac{1}{2}(f+g)+\frac{1}{2}\left|f-g\right|\ \ \
\min\\{f,g\\}=\frac{1}{2}(f+g)-\frac{1}{2}\left|f-g\right|$
∎
### 3.2 Algebra di Royden
In questa sezione ci occupiamo dell’algebra di Royden su una varietà
Riemanniana $R$ e delle sue proprietà.
#### 3.2.1 Definizione
###### Definizione 3.7.
Data una varietà riemanniana $R$, definiamo l’algebra di Royden
$\mathbb{M}(R)$ l’insieme delle funzioni $f:R\to\mathbb{R}$ tali che:
1. 1.
$f$ è una funzione continua e limitata su $R$
2. 2.
$f$ è una funzione di Tonelli
3. 3.
$D_{R}(f)=\int_{R}\left|\nabla f\right|^{2}dV<\infty$
Notiamo subito che $\mathbb{M}(R)$ è un’algebra commutativa di funzioni, cioè
un insieme chiuso rispetto a somma, prodotto e prodotto per uno scalare.
###### Proposizione 3.8.
$\mathbb{M}(R)$ è un’algebra commutativa di funzioni
###### Proof.
La verifica che $\mathbb{M}(R)$ è chiusa rispetto a somma e prodotto per uno
scalare è ovvia. Rimane solo da verificare che date due funzioni $f,\
g\in\mathbb{M}(R)$, anche il loro prodotto appartiene all’algebra.
Per prima cosa, se $f$ e $h$ sono continue e limitate, $f\cdot h$ è continua e
limitata da $\left\|f\right\|_{\infty}\cdot\left\|h\right\|_{\infty}$ 444dove
$\left\|f\right\|_{\infty}$ è la norma del sup. Se
$f(\bar{x}^{1},\dots,\bar{x}^{i-1},x,\bar{x}^{i+1},\dots,\bar{x}^{m})$ e
$h(\bar{x}^{1},\dots,\bar{x}^{i-1},x,\bar{x}^{i+1},\dots,\bar{x}^{m})$ sono
entrambe assolutamente continue rispetto a $x$ (eventualità che si verifica
quasi ovunque rispetto alle variabili barrate), allora la proposizione 1.31
garantisce che anche il prodotto sia assolutamente continuo, quindi esiste
quasi ovunque la derivata del prodotto e vale che:
$\displaystyle\frac{\partial(fh)}{\partial x^{i}}=\frac{\partial f}{\partial
x^{i}}h+f\frac{\partial h}{\partial x^{i}}$
dove questa uguaglianza si intende quasi ovunque, quindi praticamente sempre
dal punto di vista integrale.
Mancano da verificare le proprietà sull’integrale di Dirichlet. Iniziamo con
il verificare che dato un compatto $K$ contenuto in un rettangolo coordinato,
$D_{K}(fh)<\infty$. Questa verifica segue dalla catena di disuguaglianze:
$\displaystyle D_{K}(fh)=\int_{\phi(K)}g^{ij}\left(\frac{\partial f}{\partial
x^{i}}h+f\frac{\partial h}{\partial x^{i}}\right)\left(\frac{\partial
f}{\partial x^{j}}h+f\frac{\partial h}{\partial
x^{j}}\right)\sqrt{\left|g\right|}dx^{1}\dots dx^{m}=$
$\displaystyle=\int_{\phi(K)}g^{ij}\frac{\partial f}{\partial
x^{i}}\frac{\partial f}{\partial
x^{j}}h^{2}dV+\int_{\phi(K)}g^{ij}\frac{\partial h}{\partial
x^{i}}\frac{\partial h}{\partial
x^{j}}f^{2}dV+2\int_{\phi(K)}g^{ij}\frac{\partial f}{\partial
x^{i}}\frac{\partial h}{\partial x^{j}}fhdV$
dove $dV=\sqrt{\left|g\right|}dx^{1}\dots dx^{m}$ per semplicità di notazione.
Considerando che la forma quadratica $g^{ij}$ è sempre definita positiva,
l’argomento dei primi due integrali è sempre positivo, quindi il loro modulo
può facilmente essere maggiorato da
$\displaystyle\left|\int_{\phi(K)}g^{ij}\frac{\partial f}{\partial
x^{i}}\frac{\partial f}{\partial
x^{j}}h^{2}dV+\int_{\phi(K)}g^{ij}\frac{\partial h}{\partial
x^{i}}\frac{\partial h}{\partial x^{j}}f^{2}dV\right|\leq$
$\displaystyle\leq\left\|h\right\|_{\infty}^{2}D_{K}(f)+\left\|f\right\|_{\infty}^{2}D_{K}(h)$
Per l’ultimo integrale, applichiamo due volte la disuguaglianza di Schwartz
(vedi ad esempio paragrafo 10.8 pag 210 di [R1]) in modo da ottenere:
$\displaystyle\left|\int_{\phi(K)}g^{ij}\frac{\partial f}{\partial
x^{i}}\frac{\partial h}{\partial
x^{j}}fhdV\right|\leq\left\|f\right\|_{\infty}\left\|h\right\|_{\infty}\int_{\phi(K)}\left|g^{ij}\frac{\partial
f}{\partial x^{i}}\frac{\partial h}{\partial x^{j}}\right|dV\leq$
$\displaystyle\leq\left\|f\right\|_{\infty}\left\|h\right\|_{\infty}\int_{\phi(K)}\left|g^{ij}\frac{\partial
f}{\partial x^{i}}\frac{\partial f}{\partial
x^{j}}\right|^{1/2}\left|g^{ij}\frac{\partial h}{\partial x^{i}}\frac{\partial
h}{\partial x^{j}}\right|^{1/2}dV\leq$
$\displaystyle\leq\left\|f\right\|_{\infty}\left\|h\right\|_{\infty}\left(\int_{\phi(K)}g^{ij}\frac{\partial
f}{\partial x^{i}}\frac{\partial f}{\partial
x^{j}}dV\right)^{1/2}\left(\int_{\phi(K)}g^{ij}\frac{\partial h}{\partial
x^{i}}\frac{\partial h}{\partial x^{j}}dV\right)^{1/2}=$
$\displaystyle=\left\|f\right\|_{\infty}\left\|h\right\|_{\infty}D_{K}(f)^{1/2}D_{K}(g)^{1/2}$
dove dalla prima alla seconda riga si sfrutta il fatto che $g^{ij}$ è definita
positiva, quindi $g^{ij}x_{i}y_{j}$ è in prodotto scalare tra $x$ e $y$
555quindi possiamo applicare $\left|\left\langle
x\middle|y\right\rangle\right|\leq\left\|x\right\|\left\|y\right\|$, e nella
terza riga sfruttiamo la disuguaglianza di Schwartz per integrali.
Riassumendo, con queste disuguaglianze otteniamo che
$\displaystyle
D_{K}(fh)\leq\left\|h\right\|_{\infty}^{2}D_{K}(f)+2\left\|f\right\|_{\infty}\left\|h\right\|_{\infty}(D_{K}(f)D_{K}(h))^{1/2}+\left\|f\right\|_{\infty}^{2}D_{K}(h)=$
(3.4)
$\displaystyle=\left(\left\|h\right\|_{\infty}D_{K}(f)^{1/2}+\left\|f\right\|_{\infty}D_{K}(h)^{1/2}\right)^{2}$
Sfruttanto le partizioni dell’unità, con un argomento del tutto analogo a
quello utilizzato nella dimostrazione di 3.5, si ottiene la stessa
disuguaglianza anche per gli integrali di Dirichlet estesi a tutta $R$,
quindi:
$\displaystyle
D_{R}(fh)\leq\left(\left\|h\right\|_{\infty}D_{R}(f)^{1/2}+\left\|f\right\|_{\infty}D_{R}(h)^{1/2}\right)^{2}$
(3.5)
∎
L’algebra $\mathbb{M}(R)$ è quindi un’algebra dotata di unità (la funzione
costante uguale a 1). Ha senso chiedersi quali siano i suoi elementi
invertibili. È chiaro che l’inversa di una funzione $f$ è necessariamente la
funzione $f^{-1}=1/f$, che esiste ed è continua solo se $f\neq 0$ ovunque.
Però questa funzione è limitata solo se $\inf\left|f\right|>0$. Questo
suggerisce che
###### Proposizione 3.9.
Data $f\in\mathbb{M}(R)$, $f^{-1}\in\mathbb{M}(R)$ se e solo se
$\inf\left|f\right|>0$.
###### Proof.
Supponiamo che $\inf\left|f\right|>0$. Allora $f^{-1}$ esiste ed è continua e
limitata. Inoltre se $f$ è derivabile in $x$, lo è anche $f^{-1}$, con
$\displaystyle\left.\frac{\partial f^{-1}}{\partial
x^{i}}\right|_{f(x)}=\left.-\frac{1}{f^{2}}\frac{\partial f}{\partial
x^{i}}\right|_{x}$
Dato che $\left|f\right|\geq\inf\left|f\right|>0$, si ottiene che
$\displaystyle D_{S}(f^{-1})\leq\frac{1}{(\inf\left|f\right|)^{4}}D_{S}(f)$
per ogni sottoinsieme $S$ misurabile di $R$.
Supponiamo invece che $\inf\left|f\right|=0$. Se esiste $x$ tale che $f(x)=0$,
$f^{-1}$ non è definita, ma anche se la funzione non assume mai il valore $0$,
$\inf\left|f\right|=0\Rightarrow f^{-1}$ non limitata. ∎
#### 3.2.2 Topologie sull’algebra di Royden
In questa sezione ci occupiamo di definire alcune topologie sull’algebra di
Royden $\mathbb{M}(R)$ e di caratterizzarne alcune proprietà, in particolare
la completezza. Con una di queste topologie, $\mathbb{M}(R)$ diventa
un’algebra di Banach.
La prima topologia che introduciamo è $\tau_{C}$, la topologia della
convergenza uniforme sui compatti.
###### Definizione 3.10.
Data l’algebra $\mathbb{M}(R)$, definiamo una base per una topologia
$\tau_{C}$ gli insiemi della forma
$\displaystyle V(f,\epsilon,K)=\\{h\in\mathbb{M}(R)\ t.c.\ \
\left\|f-h\right\|_{\infty,K}\equiv\max_{p\in
K}\\{\left|f(p)-h(p)\right|<\epsilon\\}\\}$
dove $K\Subset R$ qualsiasi.
Lasciamo al lettore la facile verifica che questa è una base per una
topologia. Si nota subito che $\tau_{C}$ è primo numerabile, infatti fissata
$f$, e fissata un’esaustione $K_{n}$ di compatti in $R$, una base di intorni è
ad esempio
$\displaystyle V_{n}(f)=V\left(f,\frac{1}{n},K_{n}\right)$
Seguendo la traccia di [R2] (1.38 (c) a pagina 29), si ottiene che questa
topologia è metrizzabile. Se $R$ è compatta, questa topologia è anche
normabile, in caso contrario no poiché non è localmente limitata (vedi teorema
1.9 pag. 9 di [R2]). Una metrica per $\tau_{C}$ può essere ad esempio definita
da
$\displaystyle
d(f,h)=\max_{n}\frac{1}{n}\frac{\left\|f-h\right\|_{n}}{1+\left\|f-h\right\|_{n}}\
\ \ \left\|f-h\right\|_{n}=\max_{p\in K_{n}}\left|f(p)-h(p)\right|$
Sia nel caso $R$ compatta che nel caso $R$ non compatta questa metrica non è
completa. L’idea è che l’algebra di Royden contiene solo funzioni derivabili
in qualche senso, e una topologia che non chiede alcun controllo sulle
derivate non può essere completa.
###### Osservazione 3.11.
La topologia $\tau_{C}$ non è completa su $\mathbb{M}(R)$.
###### Proof.
Le funzioni lisce limitate a supporto compatto appartengono all’algebra di
Royden. Questo insieme però è denso nell’insieme delle funzioni continue su
$R$ rispetto a $\tau_{C}$, quindi come sottospazio $\mathbb{M}(R)$ non può
essere chiuso, quindi $(\mathbb{M}(R),\tau_{c})$ non può essere uno spazio
completo. ∎
La topologia $\tau_{C}$ essendo metrizzabile può essere caratterizzata dal
comportamento delle sue successioni convergenti
###### Proposizione 3.12.
Rispetto alla topologia $\tau_{C}$, $f_{n}\to f$ se e solo se $f_{n}$ converge
localmente uniformemente ad $f$, cioè se e solo se
$\displaystyle\forall K\Subset R\ \
\lim_{n}\max_{K}\left|f_{n}(p)-f(p)\right|=0$
In questo caso scriviamo che
$\displaystyle f=C-\lim_{n}f_{n}$ (3.6)
Un altro modo di definire una topologia vettoriale su $\mathbb{M}(R)$ è il
seguente:
###### Definizione 3.13.
Fissata $K_{n}$ un’esaustione di $R$ 666ricordiamo che esaustione significa
che $K_{n}$ sono compatti, $K_{n}\Subset K_{n+1}^{\circ}$ e $\cup_{n}K_{n}=R$,
definiamo una base per la topologia $\tau_{B}$ di intorni di $0$ come:
$\displaystyle V(0,E_{n})=\\{f:R\to\mathbb{R}\ t.c.\ \max_{p\in
K_{n}}\left|f(p)\right|<E_{n}\\}$ (3.7)
dove $E_{n}$ è una qualunque successione di numeri strettamente positivi tali
che
$\lim_{n\to\infty}E_{n}=\infty$
La topologia su $R$ è univocamente determinata dalla sua invarianza per
traslazioni.
Visto che si verifica facilmente che
$\displaystyle V(0,E_{n})\cap
V(0,^{\prime}E_{n})=V(0,\min\\{E_{n},E^{\prime}_{n}\\})$
e che il minino di due successioni strettamente positive e tendenti a infinito
mantiene queste proprietà, la definizione di base è ben posta.
Osserviamo subito che la definizione non dipende dalla scelta di $K_{n}$ (cioè
al variare di $K_{n}$, tutte le topologie generate sono equivalenti tra loro),
e nella definizione si può fare l’ulteriore richiesta che $E_{n}$ sia una
successione crescente senza perdere di generalità.
È inoltre possibile descrivere in maniera alternativa questa topologia. A
questo scopo, data una funzione $e:R\to\mathbb{R}$, diciamo che
$\displaystyle\lim_{p\to\infty}e(p)=+\infty\ \ \Longleftrightarrow\ \ \forall
N\in\mathbb{N},\ \exists K\Subset R\ t.c.\ e(p)>N\ \forall p\not\in K$
Con questa notazione la topologia $\tau_{B}$ può essere descritta da una base
di intorni fatta così:
$\displaystyle V(0,e)=\\{f:R\to\mathbb{R}\ t.c.\ \left|f(p)\right|<e(p)\\}$
Al variare di $e$ tra le funzioni continue positive che tendono a infinito,
questa base di intorni definisce la stessa topologia definita dagli intorni in
3.7.
Un altro modo differente per definire questa topologia è il seguente. Dato
$N\in\mathbb{N}$, definiamo l’insieme
$\displaystyle X_{N}\equiv\\{f\in\mathbb{M}(R)\ t.c.\
\left\|f\right\|_{\infty}\leq N\\}$
Dotiamo $X_{N}$ 777che ovviamente non è uno spazio vettoriale della topologia
indotta da $(\mathbb{M}(R),\tau_{C})$. Diciamo che un insieme $V$ è aperto in
$\tau_{B}$ se e solo se $\forall N$ l’intersezione $V\cap X_{N}$ è aperto
nella topologia $(X_{N},\tau_{C})$. Lasciamo al lettore la verifica che questa
topologia è una topologia vettoriale.
###### Proposizione 3.14.
La topologia generata da questi aperti è la topologia $\tau_{B}$ definita
sopra.
###### Proof.
Consideriamo $V$ aperto intorno di $0$ secondo la vecchia definizione. Allora
esiste una successione $E_{n}\to\infty$ tale che
$\displaystyle V(0,E_{n})\subset V$
L’intersezione $V(0,E_{n})\cap X_{N}$ è l’insieme delle funzioni tali che:
$\displaystyle V(0,E_{n})\cap X_{N}=\\{f\in X_{N}\ t.c.\
\left\|f\right\|_{\infty,\ K_{n}}\leq E_{n}\ \forall n\\}$
Consideriamo $E_{n}$ successione crescente. Dato che $E_{n}\to\infty$, esiste
$\bar{n}$ tale che $E_{n}\geq N$ per ogni $n\geq\bar{n}$, questo significa che
$\displaystyle\left\|f\right\|_{\infty}\leq N\ \Rightarrow\
\left\|f\right\|_{\infty,\ K_{\bar{n}}^{C}}\leq E_{n}$
per ogni $n\geq\bar{n}$. Quindi possiamo scrivere:
$\displaystyle V(0,E_{n})\cap X_{N}=\\{f\in X_{N}\ t.c.\
\left\|f\right\|_{\infty,\ K_{n}}\leq E_{n}\ \forall n\leq\bar{n}\\}\supset$
$\displaystyle\supset\\{f\in X_{N}\ t.c.\ \left\|f\right\|_{\infty,\
K_{\bar{n}}}\leq E_{1}\\}$
e questo è per definizione un insieme della topologia $(X_{N},\tau_{C})$.
Sia ora un insieme aperto intorno di $0$ secondo la nuova definizione, cioè
tale che per ogni $N$:
$\displaystyle V\cap X_{N}\in\tau_{C}$
Allora per $N=1$, esiste un insieme $K_{1}$ e un numero $E_{1}$ tale che
$\displaystyle\\{f\ t.c.\ \left\|f\right\|_{\infty,\ K_{1}}\leq
E_{1}\\}\subset V\cap X_{1}\subset V$
e lo stesso per ogni valore di $N$. Questo significa che esiste una
successione $E_{N}$ tale che:
$\displaystyle\\{f\ t.c.\ \left\|f\right\|_{\infty,\ K_{N}}\leq E_{N}\ \forall
N\\}\subset V$
Supponiamo per assurdo che $E_{N}\not\to\infty$, quindi $E_{N}$ limitata da
$M$. Questo è impossibile perchè altrimenti l’insieme:
$\displaystyle\\{f\ t.c.\ \left\|f\right\|_{\infty,\ K_{N}}\leq E_{N}\ \forall
N\\}\cap X_{M}$
non sarebbe aperto nella topologia di $(X_{M},\tau_{C})$. ∎
La topologia $\tau_{B}$ ha lo svantaggio di non essere I numerabile (se $R$
non è compatta), quindi non è una topologia metrizzabile e soprattutto non può
essere descritta in maniera completa dal comportamento delle sue successioni
convergenti.
###### Osservazione 3.15.
$\tau_{B}$ non è I numerabile, cioè fissato un punto in $\mathbb{M}(R)$, non
esiste una base numerabile di intorni del punto.
###### Proof.
Supponiamo per assurdo che esista una base numerabile $V_{k}$ di intorni di
$0$. Questi intorni avranno la forma
$\displaystyle V_{k}\equiv V(0,E^{(k)}_{n})$
dove per ogni $k$ la successione $\\{E^{(k)}_{n}\\}$ è strettamente positiva e
divergente. Costruiamo per induzione una successione strettamente crescente
$k_{m}$ di interi tali che
$\displaystyle\forall n\geq k_{m},\ \ E^{(m)}_{n}\geq m$
e definiamo la successione $B_{m}$ strettamente positiva e tendente a infinito
come:
$\displaystyle k_{m}\leq n<k_{m+1}\ \Rightarrow\ B_{n}=m/2$
i termini $B_{1},\cdots,B_{k_{1}-1}$ possono essere assegnati casualmente.
Dalla definizione, otteniamo che per nessun valore di $k$ $B_{n}\leq
E^{(k)}_{n}\ \forall n$, quindi non esiste $k$ per cui
$V(0,E^{(k)}_{n})\subset V(0,B_{n})$, che contraddice il fatto che
$\\{V(0,E^{(k)}_{n})\\}$ sia una base di intorni di $0$. ∎
Le successioni convergenti in questa topologia sono caratterizzate da
###### Proposizione 3.16.
Una successione $f_{n}$ converge a $f$ rispetto alla topologia $\tau_{B}$ se e
solo se:
1. 1.
esiste $M$ tale che $\left\|f_{n}\right\|_{\infty}\leq M\ \forall n$
2. 2.
$f=C-\lim_{n}f_{n}$
###### Proof.
Dimostriamo prima che se valgono (1) e (2), allora la successione converge
rispetto a $\tau_{B}$. Vista l’invarianza per traslazioni della topologia, è
sufficiente verificare questa condizione con $f=0$. Consideriamo un qualsiasi
intorno aperto di $0$ della forma $V(0,E_{n})$. Dato che $E_{n}\to\infty$,
esiste un numero $N_{1}$ tale che
$E_{n}>M\ \forall n\geq N_{1}$
Inoltre poiché $f_{n}$ converge localmente uniformemente a $0$, esiste un
numero $N_{2}$ tale che
$\displaystyle\forall n\geq N_{2},\ \max_{p\in
K_{N-1}}\\{\left|f_{n}(p)\right|\\}\leq\min_{1\leq i\leq N-1}\\{E_{i}\\}$
quindi se $N=\max\\{N_{1},N_{2}\\}$, $f_{n}\in V\ \forall n\geq N$, cioè
$f_{n}$ converge a $f$ rispetto a $\tau_{B}$.
Per l’implicazione inversa, dato che $\tau_{C}\subset\tau_{B}$, la convergenza
nella topologia $B$ implica la convergenza locale uniforme. Inoltre supponiamo
per assurdo che $\left\|f_{n}\right\|_{\infty}\to\infty$, quindi esiste
$\\{x_{n}\\}\subset R$ tale che $\left|f_{n}(x_{n})\right|\to\infty$. Dato che
$f_{n}$ converge localmente uniformemente a $0$, necessariamente
$x_{n}\to\infty$ 888cioè $x_{n}$ abbandona definitivamente ogni compatto.
Fissata un’esaustione $K_{n}$ di $R$, per ogni $n$ consideriamo
$\displaystyle E_{n}=\min_{x_{i}\in K_{n}}\\{\left|f_{i}(x_{i})\right|\\}/2$
Non è difficile verificare che $E_{n}\to\infty$, e che quindi a meno di un
numero finito di termini 999che in questo ragionamento sono ininfluenti,
$E_{n}>0$.
In questo modo la successione $\\{f_{n}\\}$ non è contenuta definitivamente
nell’aperto $V(0,E_{n})$, quindi non può convergere a $0$. ∎
Indipendentemente dal fatto che $\tau_{B}$ non è primo numerabile, ha senso
chiedersi se le successioni bastano a descrivere la topologia di questo spazio
101010spazi topologici di questo genere si chiamano spazi di Frechet-Urysohn.
Visti gli scopi della tesi, ci limitiamo a rimandare a [A] capitolo I pag 13
per ulteriori approfondimenti. La risposta a questa domanda è negativa, anzi
qualunque topologia generi la convergenza $B$ è destinata a non essere una
topologia “strana”. Infatti in questi spazi la chiusura per successioni non è
idempotente, cioè indicando $[A]_{seq}$ la chiusura per successioni di $A$,
cioè l’insieme di tutti i possibili limiti di successioni in $A$, si ha che in
generale $[[A]_{seq}]_{seq}\neq[A]_{seq}$, quindi necessariamente
$[A]_{seq}\neq\overline{A}$.
###### Proposizione 3.17.
Data una successione $f_{n}\in\mathbb{M}(R)$, se definiamo
$\displaystyle f=B-\lim_{n}f_{n}\Longleftrightarrow f=C-\lim_{n}f_{n}\ \wedge\
\left(\exists M\ t.c.\ \left\|f_{n}\right\|_{\infty}\leq M\right)$
allora se $R$ non è compatta, qualunque topologia generi questa convergenza
non è di Frechet-Urysohn, quindi non è descrivibile con il comportamento delle
sue successioni convergenti.
###### Proof.
La dimostrazione è ispirata all’esercizio 9 cap 3 pag 87 di [R2]. Consideriamo
una successione discreta $\\{x_{n}\\}\subset\mathbb{R}$, $x_{n}\to\infty$, e
sia $\\{U_{n}\\}$ una collezione di intorni disgiunti di $x_{n}$. Siano
$g_{n}\in\mathbb{M}(R)$ funzioni tali che $supp(g_{n})\subset U_{n}$, $0\leq
g_{n}\leq 1$ e $g_{n}(x_{n})=1$. Definiamo $f_{n,m}\in\mathbb{M}(R)$ come
$f_{n,m}=g_{n}+ng_{m}$, e consideriamo
$\displaystyle A=\\{f_{n,m}\ n,m\in\mathbb{N}\\}$
Allora $0\not\in[A]_{seq}$, ma $0\in[[A]_{seq}]_{seq}$. Infatti qualunque
successione $\\{f_{n(k),m(k)}\\}_{k=1}^{\infty}$ in $A$ che converge nel senso
$B$ a $0$ necessariamente converge localmente uniformemente a $0$, quindi
$n_{k}\to\infty$, ma allora anche
$\left\|f_{n(k),m(k)}\right\|_{\infty}=n_{k}\to\infty$, il che significa che
questa successione non può convergere a $0$.
D’altro canto è facile verificare che tutte le funzioni $g_{n}\in[A]_{seq}$,
basta considerare che la successione $f_{n(k),m(k)}\to g_{n}$ se $n(k)=n$,
$m(k)\to\infty$. Ma dato che $g_{n}\to 0$ rispetto alla convergenza $B$, la
tesi è dimostrata. ∎
Questa proposizione impone di trattare la convergenza $B$ con particolare
attenzione, perché non può essere descritta semplicemente dalle sue
successioni (come accade per gli spazi metrici).
Un’altra topologia che possiamo definire su $R$ è la classica topologia della
convergenza uniforme ovunque:
###### Definizione 3.18.
La norma del sup è una norma sullo spazio $\mathbb{M}(R)$. Definiamo la
topologia descritta da questa norma $\tau_{U}$. Diciamo che
$\displaystyle f=U-\lim_{n}f_{n}$ (3.8)
se $f_{n}$ converge ad $f$ rispetto a questa norma, quindi se
$\displaystyle\lim_{n}\left\|f_{n}-f\right\|_{\infty}=\lim_{n}\sup_{R}\left|f_{n}(p)-f(p)\right|=0$
###### Osservazione 3.19.
È facile verificare che $\tau_{C}\subset\tau_{B}\subset\tau_{U}$. Le tre
topologie coincidono nel caso $R$ compatta, mentre l’inclusione è stretta se
$R$ non è compatta.
Anche in questo caso vale un’osservazione molto simile a 3.11 (e la
dimostrazione è del tutto analoga, basta sostituire lo spazio delle funzioni
continue con lo spazio delle funzioni continue limitate).
###### Osservazione 3.20.
La norma dell’estremo superiore non rende $\mathbb{M}(R)$ uno spazio completo.
Dalla definizione di $\mathbb{M}(R)$, non è difficile immaginare che per
rendere questa algebra un’algebra di Banach bisogna in qualche senso tenere in
considerazione il comportamento delle derivate delle funzioni. A questo scopo
introduciamo il concetto di $D-\lim$:
###### Definizione 3.21.
Data una successione di funzioni $f_{n}\in\mathbb{M}(R)$, diciamo che
$\displaystyle D-\lim_{n}f_{n}=f$ (3.9)
se e solo se $D_{R}(f_{n}-f)\to 0$
Ricordiamo che l’integrale di Dirichlet è molto legato al concetto di norma
nello spazio $\mathcal{L}^{2}(R)$, lo spazio di Hilbert delle 1-forme a
quadrato integrabili su $R$ 111111a questo scopo rimandiamo alla sezione 1.3.
Questi concetti di convergenza possono essere mischiati fra loro:
###### Definizione 3.22.
Data una successione $f_{n}\in\mathbb{M}(R)$, diciamo che
$\displaystyle f=QD-\lim_{n}f_{n}$ (3.10)
se e solo se $f=D-\lim_{n}f_{n}$ e anche $f=Q-\lim_{n}f_{n}$, dove $Q$
rappresenta una convergenza qualsiasi tra $C$, $B$, $U$.
I due concetti di convergenza che saranno più usati in questo lavoro sono la
$BD$-convergenza e la $UD$-convergenza. La seconda convergenza è generata da
una norma, che indicheremo $\left\|\cdot\right\|_{R}$, norma che rende
$\mathbb{M}(R)$ un’algebra di Banach.
###### Teorema 3.23.
Sull’algebra $\mathbb{M}(R)$ definiamo la norma
$\displaystyle\left\|f\right\|_{R}\equiv\left\|f\right\|_{\infty}+D_{R}(f)^{1/2}$
questa norma genera la convergenza $UD$, e rispetto a questa norma
$\mathbb{M}(R)$ è un’algebra di Banach commutativa con unità.
###### Proof.
Il fatto che $\left\|\cdot\right\|_{R}$ sia una norma è facile conseguenza del
fatto che $\left\|\cdot\right\|_{\infty}$ è una norma, e $D_{R}(\cdot)^{1/2}$
è una seminorma.
Ovviamente $\left\|1\right\|=1$, e grazie alla relazione (3.5) si ottiene:
$\displaystyle\left\|fh\right\|_{R}=\left\|fh\right\|_{\infty}+D_{R}(fh)^{1/2}\leq\left\|f\right\|_{\infty}\left\|h\right\|_{\infty}+\left\|h\right\|_{\infty}D_{R}(f)^{1/2}+\left\|f\right\|_{\infty}D_{R}(h)^{1/2}\leq$
$\displaystyle\leq(\left\|f\right\|_{\infty}+D_{R}(f)^{1/2})(\left\|h\right\|_{\infty}+D_{R}(h)^{1/2})=\left\|f\right\|_{R}\left\|h\right\|_{R}$
questo rende $(\mathbb{M}(R),\left\|\cdot\right\|_{R})$ un’algebra normata.
Rimane da verificare la completezza.
Sia a questo scopo $\\{f_{n}\\}\subset\mathbb{M}(R)$ una successione di
Cauchy. Allora $\\{f_{n}\\}$ è di Cauchy uniforme, quindi esiste una funzione
continua limitata sulla varietà $R$ tale che
$\left\|f_{n}-f\right\|_{\infty}\to 0$. La parte più complicata è dimostrare
che questa funzione è di Tonelli e che $D_{R}(f_{n}-f)\to 0$.
Ricordiamo che $\mathcal{L}^{2}(R)$, lo spazio delle 1-forme su $R$ normato
con l’integrale del modulo quadro della forma, è uno spazio di Hilbert (vedi
sezione 1.3). L’ipotesi che $f_{n}$ sia di Cauchy rispetto a
$\left\|\cdot\right\|_{R}$ implica che $df_{n}$ sia una successione di Cauchy
nello spazio $\mathcal{L}^{2}(R)$, quindi esiste una 1-forma
$\alpha\in\mathcal{L}^{2}(r)$ tale che $df_{n}\to\alpha$. Se dimostriamo che
$f$ è di Tonelli e $df=\alpha$, abbiamo la tesi. Osserviamo che queste due
affermazioni hanno carattere locale, quindi fissiamo un qualunque aperto
coordinato $(U,\phi)$ con l’accortezza che $\phi$ sia definita in un intorno
di $\overline{U}$ e tale che $\phi(U)=\prod_{i=1}^{m}(a_{i},b_{i})$ e
dimostriamo l’uguaglianza $df=\alpha$ in questa carta locale.
A questo scopo indichiamo con $\alpha_{i}(x)$ le componenti locali di
$\alpha$, cioè:
$\displaystyle\alpha=\alpha_{i}(x)dx^{i}$
mentre per semplicità di notazione continuiamo a indicare con $f_{n}(x)$ e
$f(x)$ le rappresentazioni locali di $f_{n}$ ed $f$ rispettivamente.
Dividiamo la dimostrazione in due parti: l’idea e i conti. L’idea è dimostrare
$f=f^{(i)}$, dove $f^{(i)}$ sono funzioni per definizione assolutamente
continue e la cui derivata parziale rispetto a $x^{i}$ è proprio $\alpha_{i}$.
A questo scopo mostriamo che le funzioni $f_{n}$, che convergono uniformemente
a $f$, convergono in norma $L^{2}(\phi(U))$ a $f^{(i)}$ (per tutti gli indici
$1\leq i\leq m$). Quindi $f=f^{(i)}$ quasi ovunque (vedi proposizione 1.33). E
questo conclude la dimostrazione.
Ora passiamo a conti. Definiamo le funzioni $f^{(i)}$ come:
$\displaystyle f^{(i)}(x^{1},\cdots,x^{i},\cdots,x^{m})\equiv
f(x^{1},\cdots,c_{i},\cdots,x^{m})+\int_{c_{i}}^{x^{i}}\alpha_{i}(x^{1},\cdots,t,\cdots,x^{m})dt$
(3.11)
dove $c_{i}=(a_{i}+b_{i})/2$ 121212in realtà per la dimostrazione va bene
qualsiasi $c_{i}\in(a_{i},b_{i})$. Osserviamo che $\partial f_{n}/\partial
x^{i}(x)$ converge in norma $L^{2}$ a $\alpha_{i}(x)$ quando $n$ tende a
infinito. Infatti
$\displaystyle\int_{\phi(U)}\left(\frac{\partial f_{n}}{\partial
x^{i}}(x)-\alpha_{i}(x)\right)^{2}dx^{1}\cdots
dx^{m}\leq\int_{\phi(U)}\sum_{i=1}^{m}\left(\frac{\partial f_{n}}{\partial
x^{i}}(x)-\alpha_{i}(x)\right)^{2}dx^{1}\cdots dx^{m}\leq$
$\displaystyle\leq\int_{\phi(U)}Cg^{ij}\left(\frac{\partial f_{n}}{\partial
x^{i}}(x)-\alpha_{i}(x)\right)\left(\frac{\partial f_{n}}{\partial
x^{j}}(x)-\alpha_{j}(x)\right)dx^{1}\cdots dx^{m}\leq$
$\displaystyle\leq\frac{C}{k}\int_{\phi(U)}g^{ij}\left(\frac{\partial
f_{n}}{\partial x^{i}}(x)-\alpha_{i}(x)\right)\left(\frac{\partial
f_{n}}{\partial x^{j}}(x)-\alpha_{j}(x)\right)\
\sqrt{\left|g\right|}dx^{1}\cdots dx^{m}=$
$\displaystyle=\frac{C}{k}\left\|df_{n}-\alpha\right\|_{\mathcal{L}^{2}(R)}\to
0$
dove per passare dalla prima alla seconda riga abbiamo sfruttato la relazione
3.2, scegliendo come $C$ il massimo dei $c_{p}$ al variare di
$p\in\phi(\overline{U})$, mentre $k$ è il minimo della funzione
$\sqrt{\left|g\right|}$ sempre sullo stesso insieme compatto 131313per questo
motivo è importante scegliere $\phi$ definita su un intorno di $\overline{U}$.
Fissato $i$, tutte le funzioni
$f_{n}(\bar{x}^{i},\cdots,x^{i},\cdots,\bar{x}^{m})$ sono contemporaneamente
assolutamente continue rispetto a $x^{i}$, quasi ovunque rispetto a
$(\bar{x}^{1},\cdots,\bar{x}^{i-1},\bar{x}^{i+1},\cdots,\bar{x}^{m})$
141414ogni funzione $f_{n}$ è assolutamente continua a meno di un insieme di
misura nulla di $\bar{x}$, ma visto che l’unione numerabile di insiemi di
misura nulla ha misura nulla, tutte le funzioni sono assolutamente continue
contemporaneamente sullo stesso insieme con complementare di misura nulla,
quindi vale che
$\displaystyle
f_{n}(\bar{x}^{i},\cdots,x^{i},\cdots,\bar{x}^{m})=f_{n}(\bar{x}^{i},\cdots,c_{i},\cdots,\bar{x}^{m})+\int_{c_{i}}^{x^{i}}\frac{\partial
f_{n}(\bar{x}^{i},\cdots,t,\cdots,\bar{x}^{m})}{\partial t}dt$ (3.12)
Confrontando questa relazione con la relazione 3.11 otteniamo:
$\displaystyle\left|f_{n}(\bar{x}^{i},\cdots,x_{i},\cdots,\bar{x}^{m})-f^{(i)}(\bar{x}^{i},\cdots,x_{i},\cdots,\bar{x}^{m})\right|^{2}=$
$\displaystyle=\left|f_{n}(\bar{x}^{i},\cdots,c_{i},\cdots,\bar{x}^{m})-f(\bar{x}^{i},\cdots,c_{i},\cdots,\bar{x}^{m})+\right.$
$\displaystyle\left.+\int_{c_{i}}^{x^{i}}\frac{\partial
f_{n}(\bar{x}^{i},\cdots,t,\cdots,\bar{x}^{m})}{\partial
t}dt-\int_{c_{i}}^{x^{i}}\alpha_{i}(\bar{x}^{i},\cdots,t,\cdots,\bar{x}^{m})dt\right|^{2}\leq$
$\displaystyle\leq
2\left|f_{n}(\bar{x}^{i},\cdots,c_{i},\cdots,\bar{x}^{m})-f(\bar{x}^{i},\cdots,c_{i},\cdots,\bar{x}^{m})\right|^{2}+$
$\displaystyle+2\left|\int_{c_{i}}^{x^{i}}\frac{\partial
f_{n}(\bar{x}^{i},\cdots,t,\cdots,\bar{x}^{m})}{\partial
t}dt-\int_{c_{i}}^{x^{i}}\alpha_{i}(\bar{x}^{i},\cdots,t,\cdots,\bar{x}^{m})dt\right|^{2}$
per semplicità di notazione chiamiamo $2\left|A_{n}\right|^{2}$ la prima riga
dopo l’ultimo segno di disuguaglianza, e la seconda $2\left|B_{n}\right|^{2}$.
Grazie alle proprietà dell’integrale e alla disuguaglianza di Schwartz
otteniamo:
$\displaystyle\left|B_{n}\right|^{2}\leq\left(\int_{c_{i}}^{x^{i}}\left|\frac{\partial
f_{n}(\bar{x}^{i},\cdots,t,\cdots,\bar{x}^{m})}{\partial
t}-\alpha_{i}(\bar{x}^{i},\cdots,t,\cdots,\bar{x}^{m})\right|dt\right)^{2}\leq$
$\displaystyle\leq\left(b_{i}-a_{i}\right)\int_{a_{i}}^{b^{i}}\left|\frac{\partial
f_{n}(\bar{x}^{i},\cdots,t,\cdots,\bar{x}^{m})}{\partial
t}-\alpha_{i}(\bar{x}^{i},\cdots,t,\cdots,\bar{x}^{m})\right|^{2}dt$
e questa relazione vale quasi ovunque rispetto a
$(\bar{x}^{1},\cdots,\bar{x}^{i-1},\bar{x}^{i+1},\cdots,\bar{x}^{m})$.
Grazie a queste disuguaglianze ora siamo in grado di dimostrare che la
successione $f_{n}$ (o meglio la successione delle rappresentazioni locali di
$f_{n}$) converge in norma $L^{2}(\phi(U))$ alla funzione $f^{(i)}$ (con
$1\leq i\leq m$ qualsiasi). Infatti:
$\displaystyle\int_{\phi(U)}\left|f_{m}(x^{1},\cdots,x^{m})-f^{(i)}(x^{1},\cdots,x^{m})\right|^{2}dx^{1}\cdots
dx^{m}\leq$ $\displaystyle\leq 2\left|A_{n}\right|^{2}Vol(\phi(U))+$
$\displaystyle+2(b_{i}-a_{i})\int_{\phi(U)}\left\\{\int_{a_{i}}^{b^{i}}\left|\frac{\partial
f_{n}(\bar{x}^{i},\cdots,t,\cdots,\bar{x}^{m})}{\partial
t}-\alpha_{i}(\bar{x}^{i},\cdots,t,\cdots,\bar{x}^{m})\right|^{2}dt\right\\}dx=$
$\displaystyle=2\left|A_{n}\right|^{2}Vol(\phi(U))+2(b_{i}-a_{i})^{2}\left\|\frac{\partial
f_{n}}{\partial x^{i}}-\alpha_{i}\right\|^{2}_{L^{2}(\phi(U))}$
Sappiamo che $A_{n}$ tende a 0 (poichè $f_{n}$ converge uniformemente a $f$),
e anche $\left\|\frac{\partial f_{n}}{\partial
x^{i}}-\alpha_{i}\right\|^{2}_{L^{2}(\phi(U))}$ tende a zero come dimostrato
prima.
∎
Modificando leggermente questa dimostrazione, si può ottenere che:
###### Proposizione 3.24.
Se $f_{n}$ è una successione di funzioni in $\mathbb{M}(R)$ tale che
$\displaystyle f=C-\lim_{n}f_{n}$
dove $f$ è una funzione (continua) limitata, e se esiste $K<\infty$ tale che
per ogni $n$:
$\displaystyle D_{R}(f_{n})\leq K<\infty$
allora $f\in\mathbb{M}(R)$, $D_{R}(f)\leq K$ e inoltre esiste una
sottosuccessione $n_{k}$ tale che per ogni $g\in\mathbb{M}(R)$:
$\displaystyle D_{R}(f-f_{n_{k}},g)\to 0$
###### Proof.
Grazie al teorema 1.48, esiste una sottosuccessione di $f_{n}$ che
continueremo a indicare nello stesso modo tale che $df_{n}$ converge
debolmente a $\alpha\in\mathcal{L}^{2}(R)$.
Consideriamo un rettangolo coordinato $T$ qualsiasi e sia $\phi\in
C^{\infty}_{0}(T)$ lo spazio delle funzioni lisce a supporto compatto in $T$.
Grazie a un’integrazione per parti otteniamo che:
$\displaystyle\int_{T}f_{n}\frac{\partial\phi}{\partial
x^{i}}dV=-\int_{T}\phi\frac{\partial f_{n}}{\partial x^{i}}dV$
Ora poiché $df_{n}\to\alpha$ debolmente, si ha che:
$\displaystyle\int_{T}\phi\frac{\partial f_{n}}{\partial
x^{i}}dV\to\int_{T}\phi\alpha_{i}dV$
e dato che $f=C-\lim_{n}f_{n}$ si ha che:
$\displaystyle\int_{T}f_{n}\frac{\partial\phi}{\partial
x^{i}}dV\to\int_{T}f\frac{\partial\phi}{\partial x^{i}}dV$
Da queste consideraziono otteniamo che:
$\displaystyle\int_{T}f\frac{\partial\phi}{\partial
x^{i}}dV=\int_{T}\phi\alpha_{i}dV$
Questo significa che le derivate distribuzionali di $f$ rispetto a $x^{i}$
coincidono con $\alpha_{i}$, e dato che $\alpha\in\mathcal{L}^{2}(R)$, grazie
al lemma 3.25 (che riportiamo alla fine di questa proposizione), $f$ è di
Tonelli con $df=\alpha$ nel senso standard e quindi $D_{R}(f)<\infty$, il che
dimostra che $f\in\mathbb{M}(R)$.
Il fatto che
$\displaystyle D_{R}(f-f_{n},g)\to 0$
è conseguenza diretta del fatto che $df_{n}$ converge debolmente nel senso di
$\mathcal{L}^{2}(R)$ a $df$. Inoltre dalla teoria della convergenza debole,
sappiamo che
$\left\|df\right\|_{L_{2}}\leq\limsup_{n}\left\|df_{n}\right\|_{L_{2}}$
quindi ad esempio
$\displaystyle\left\|df_{n}\right\|_{L_{2}}\equiv D_{R}(f_{n})\leq k\ \
\forall n\ \Rightarrow\ \left\|df\right\|_{L_{2}}\equiv D_{R}(f)\leq k$
∎
###### Lemma 3.25.
Sia $f:T\to\mathbb{R}$ una funzione $f\in L^{2}(T)$ 151515osserviamo che tutte
le funzioni continue sono a quadrato integrabile su insiemi relativamente
compatti di misura finita, quindi sui rettangoli coordinati $T$ rispetto alla
forma volume tale che per tutti gli indici $i=1,\cdots,m$:
$\displaystyle\frac{\partial f}{\partial x_{i}}\in L^{2}(T)$
dove le derivate sono intese in senso distrubuzionale. Allora $f$ è
assolutamente continua rispetto a quasi tutti i segmenti paralleli agli assi
coordinati in $T$, e le sue derivate standard coincidono quasi ovunque con le
derivate distribuzionali. Quindi se $f$ è continua, è una funzione di Tonelli.
###### Proof.
Dato che questo teorema riguarda la teoria delle distribuzioni e gli spazi di
Sobolev, che sono argomenti marginali in questa tesi, per la dimostrazione
rimandiamo il lettore a [Z] (teorema 2.1.4 pag. 44), oppure a [MZ] (teorema
1.41 pag. 22). ∎
###### Osservazione 3.26.
Nella dimostrazione, il fatto che $f$ sia limitata è utile esclusivamente per
dimostrare che $f\in\mathbb{M}(R)$. Quindi se $f$ non è limitata, valgono
tutte le conclusioni del teorema a meno dell’appartenenza all’algebra di
Royden.
#### 3.2.3 Densità di funzioni lisce
In questo paragrafo dimostreremo la densità delle funzioni lisce nello spazio
$\mathbb{M}(R)$, e utilizzeremo questo risultato per dimostrare formule di
Green generalizzate.
###### Proposizione 3.27.
Sia $f$ una funzione di Tonelli su $R$. Per ogni $\epsilon>0$, esiste
$f_{\epsilon}\in C^{\infty}(R,\mathbb{R})$ tale che
$\left\|f_{\epsilon}-f\right\|_{R}<\epsilon$ 161616in questa proposizione non
è richiesto che $f$ sia limitata, e che il suo integrale di Dirichlet
$D_{R}(f)<\infty$, quindi potrebbe non avere senso $\left\|f\right\|_{R}$.
Inoltre se $f$ ha supporto contenuto in un aperto $U\subset R$, anche
$f_{\epsilon}$ può essere scelta con supporto contenuto in $U$.
###### Proof.
Come spesso accade in questi casi, la dimostrazione si divide in due parti,
una “locale” e una “globale”. Nella prima parte dimostreremo il risultato per
funzioni $f$ a supporto compatto in una carta locale, poi generalizzeremo il
risultato utilizzando le partizioni dell’unità.
Sia quindi $f$ come nelle ipotesi e anche a supporto compatto in un intorno
coordinato $(U,\phi)$ di $R$, e chiamiamo $\tilde{f}$ la rappresentazione di
$f$ in questa carta locale. Consideriamo una successione
$\Theta_{n}:\mathbb{R}^{m}\to\mathbb{R}$ di nuclei di convoluzione con
supporto contenuto in $B_{1/n}(0)$. Grazie al lemma 1.45 la successione
$\tilde{f}_{n}\equiv\Theta_{n}\ast\tilde{f}$ (che ha supporto definitivamente
contenuto in $\phi(U)$) converge nella norma del sup a $\tilde{f}$. Inoltre
grazie al lemma 1.44, si ha che:
$\displaystyle\frac{\partial\tilde{f}_{n}}{\partial
x^{i}}=\Theta_{n}\ast\frac{\partial\tilde{f}}{\partial x^{i}}$
Passiamo ora a considerare $D_{R}(f_{n}-f)$. Grazie al fatto che esiste un
insieme compatto $K$ tale che $supp(\tilde{f}_{n})\subset K\Subset U$
definitivamente rispetto a $n$, e grazie alla relazione 3.3, otteniamo che:
$\displaystyle g^{ij}\left(\frac{\partial\tilde{f}_{n}}{\partial
x^{i}}(x)-\frac{\partial\tilde{f}}{\partial
x^{i}}(x)\right)\left(\frac{\partial\tilde{f}_{n}}{\partial
x^{j}}(x)-\frac{\partial\tilde{f}}{\partial x^{j}}(x)\right)\leq
c\sum_{i=1}^{m}\left(\frac{\partial\tilde{f}_{n}}{\partial
x^{i}}(x)-\frac{\partial\tilde{f}}{\partial x^{i}}(x)\right)^{2}=$
$\displaystyle=c\sum_{i=1}^{m}\left(\int_{B_{n}}\Theta_{n}(y)\left(\frac{\partial\tilde{f}}{\partial
x^{i}}(x-y)-\frac{\partial\tilde{f}}{\partial
x^{i}}(x)\right)dV(y)\right)^{2}\leq$ $\displaystyle\leq
c\sum_{i=1}^{m}\int_{B_{n}}\Theta_{n}(y)\left(\frac{\partial\tilde{f}}{\partial
x^{i}}(x-y)-\frac{\partial\tilde{f}}{\partial x^{i}}(x)\right)^{2}dV(y)$
dove l’ultimo passaggio è giustificato dalla disuguaglianza di Jensen
171717infatti $\int\Theta_{n}(y)dV(y)=1$. Per riferimenti sulla disuguaglianza
di Jensen, vedi teorema 3.3 pag 61 di [R4].
Queste considerazioni (assieme al teorema di Fubuni 181818vedi teorema 7.8 pag
140 di [R4]) permettono di concludere:
$\displaystyle
D_{R}(f_{n}-f)=\int_{K}g^{ij}\left(\frac{\partial\tilde{f}_{n}}{\partial
x^{i}}(x)-\frac{\partial\tilde{f}}{\partial
x^{i}}(x)\right)\left(\frac{\partial\tilde{f}_{n}}{\partial
x^{j}}(x)-\frac{\partial\tilde{f}}{\partial x^{j}}(x)\right)dV(x)\leq$
$\displaystyle\leq
c^{\prime}\int_{K}dV(x)\sum_{i=1}^{m}\int_{B_{n}}dV(y)\Theta_{n}(y)\left(\frac{\partial\tilde{f}}{\partial
x^{i}}(x-y)-\frac{\partial\tilde{f}}{\partial x^{i}}(x)\right)^{2}=$
$\displaystyle=c^{\prime}\sum_{i=1}^{m}\int_{B_{n}}dV(y)\Theta_{n}(y)\int_{K}dV(x)\left(\frac{\partial\tilde{f}}{\partial
x^{i}}(x-y)-\frac{\partial\tilde{f}}{\partial x^{i}}(x)\right)^{2}$
dove $c^{\prime}=\max_{x\in K}\\{c\sqrt{\left|g\right|(x)}\\}$.
Chiamiamo
$\displaystyle h(y)\equiv\int_{K}dV(x)\left(\frac{\partial\tilde{f}}{\partial
x^{i}}(x-y)-\frac{\partial\tilde{f}}{\partial x^{i}}(x)\right)^{2}$
Grazie alla continuità dell’operatore traslazione in $L^{2}(R)$ 191919vedi
teorema 9.5 pag 183 di [R2], e dato che $f$ è di Tonelli, possiamo concludere
che:
$\displaystyle\lim_{y\to 0}h(y)=0$
Per definizione questo significa che
$\displaystyle\lim_{n\to\infty}\max_{y\in B_{n}(0)}\left|h(y)\right|=0$
E quindi anche:
$\displaystyle\lim_{n\to\infty}\left|\int_{B_{n}}dV(y)\Theta_{n}(y)h(y)\right|\leq\lim_{n\to\infty}\max_{y\in
B_{n}(0)}\left|h(y)\right|\int_{B_{n}}dV(y)\Theta_{n}(y)=0$
Questo conclude la dimostrazione che $D_{R}(f_{n}-f)\to 0$.
Consideriamo ora una funzione $f$ di Tonelli qualsiasi, $\epsilon>0$, e sia
$\lambda_{n}:R\to\mathbb{R}$ una partizione dell’unità liscia subordinata a
aperti coordinati e a supporto compatto. Allora per ogni $n$,
$f\cdot\lambda_{n}$ è una funzione del tipo descritto sopra. Scegliamo per
ogni $n$ una funzione liscia $f_{n}$ 202020se $supp(f)\subset U$, allora
$supp(f\cdot\lambda_{n})\Subset U$, quindi possiamo scegliere
$supp(f_{n})\Subset U$ tale che
$\left\|f_{n}-f\cdot\lambda_{n}\right\|_{R}<\epsilon/2^{n}$. Allora la
funzione $f_{\epsilon}\equiv\sum_{n=1}^{\infty}f_{n}$ soddisfa le richieste
fatte.
$f_{\epsilon}$ è una funzione liscia per locale finitezza della partizione
dell’unità $\lambda_{n}$, inoltre:
$\displaystyle\left\|f_{\epsilon}-f\right\|_{R}=\left\|\sum_{n=1}^{\infty}(f_{n}-f\cdot\lambda_{n})\right\|_{R}\leq\sum_{n=1}^{\infty}\left\|f_{n}-f\cdot\lambda_{n}\right\|_{R}<\epsilon$
Inoltre se $supp(f_{n})\Subset U$ per ogni $n$, allora anche $supp(f)\subset
U$. Questo significa che se $f$ ha supporto compatto, visto che ogni compatto
in $R$ è contenuto in un aperto relativamente compatto, allora $f_{\epsilon}$
può essere scelta a supporto compatto. ∎
Vale anche una proposizione più forte rispetto a questa. Possiamo infatti
chiedere che la funzione $f_{\epsilon}$ sia uguale ad $f$ su un insieme
chiuso, ovviamente a patto di rilassare le ipotesi di regolarità sulla
funzione $f_{\epsilon}$.
###### Proposizione 3.28.
Sia $f$ una funzione di Tonelli su $R$ e $A\subset R$ aperto con bordo
$\partial A$ regolare. Per ogni $\epsilon>0$, esiste $f_{\epsilon}\in
C^{\infty}(A,\mathbb{R})$ di Tonelli su tutta la varietà tale che
$\left\|f_{\epsilon}-f\right\|_{R}<\epsilon$ e $f_{\epsilon}=f$ sull’insieme
$A^{C}$.
###### Proof.
Riportiamo solo il filo conduttore della dimostrazione, lasciando alcuni
dettagli al lettore. Questa dimostrazione è ispirata dal lemma 2.8 pagina 50
di [H3].
Sia $K_{n}$ un ricoprimento di aperti relativamente compatti localmente finiti
in $A$ 212121con localmente finito si intende un ricoprimento tale che per
ogni $p\in A$ esiste un intorno che interseca solo un numero finito di insiemi
$K_{n}$, e sia $\epsilon_{n}$ una successione di numeri positivi che tende a 0
tale che $\epsilon_{n}\leq\epsilon$. Lo scopo è riuscire a creare una funzione
$h_{\epsilon}\in C^{\infty}(A,\mathbb{R})$ tale che
$\displaystyle\left\|h_{\epsilon}-f\right\|_{K_{n}}\equiv\left\|h_{\epsilon}-f\right\|_{\infty,\
K_{n}}+D_{K_{n}}(h_{\epsilon}-f)^{1/2}<\epsilon_{n}$ (3.13)
In questo modo se estendiamo la definizione di $h_{\epsilon}$ come
$\displaystyle f_{\epsilon}(p)=\begin{cases}h_{\epsilon}(p)&p\in A\\\
f(p)&p\in A^{C}\end{cases}$
questa funzione risulta di Tonelli su tutta $R$. Infatti $f_{\epsilon}$ è
ovviamente continua e di Tonelli su tutti i punti di $R$ tranne che sul bordo
$\partial A$. Consideriamo $p\in\partial A$, e consideriamo una successione
$p_{k}\in A$, $p_{k}\to p$. Visto che $p_{k}$ converge al bordo di $A$,
necessariamente $p_{k}\in K_{n(k)}$ dove $n(k)$ tende a infinito quando $k$
tende a infinito. Questo significa che
$\displaystyle\left|f_{\epsilon}(p_{k})-f(p)\right|\leq\left|f_{\epsilon}(p_{k})-f(p_{k})\right|+\left|f(p_{k})-f(p)\right|\leq\epsilon_{n(k)}+\left|f(p_{k})-f(p)\right|$
se $k$ tende a infinito, $\epsilon_{n(k)}$ tende a zero per ipotesi, mentre il
secondo termine tende a zero per continuità di $f$. Questo prova che
$f_{\epsilon}$ è continua.
Consideriamo un punto $p\in\partial A$ e una parametrizzazione locale
$\phi(q)=(x^{1},\cdots x^{m})$ attorno a $p$ tale che $\partial
A=\\{x^{m}=0\\}$. Allora la funzione $f_{\epsilon}$ è di certo assolutamente
contiua quasi ovunque nell’insieme $x^{m}\neq 0$, quindi quasi ovunque.
L’integrale di Dirichlet di $f_{\epsilon}$ è finito su ogni compatto che ha
intersezione vuota con il bordo perché $f_{\epsilon}$ su questo insieme è una
funzione liscia o una funzione di Tonelli per ipotesi. Se il compatto
interseca il bordo di $A$, allora poiché $D_{R}(f-f_{\epsilon})$ è finito,
necessariamente l’integrale di Dirichlet di $f_{\epsilon}$ su questo compatto
è finita.
Resta da dimostrare che è possibile scegliere una funzione $h_{\epsilon}$ che
soddisfi le condizioni 3.13. La tecnica che utilizzeremo è un’adattamento
della dimostrazione della proposizione 3.27.
Scegliamo una partizione dell’unità $\\{\lambda_{n}\\}$ di $A$ subordinata al
ricoprimento $K_{n}$. Per ogni $n$, definiamo
$\alpha_{m}\equiv\frac{\epsilon_{m}}{\\#n\ t.c.\ supp(\lambda_{n})\cap
K_{m}\neq\emptyset}$ $\delta_{n}\equiv\min\\{\alpha_{m}\ t.c.\ K_{m}\cap
supp(\lambda_{n})\neq\emptyset\\}$
Per locale finitezza del ricoprimento $\\{K_{m}\\}$ tutte queste quantità sono
ben definite e strettamente positive.
Ora grazie alle tecniche esposte nella dimostrazione della proposizione 3.27,
scegliamo per ogni $n$ una funzione liscia $f_{n}$ con supporto in $K_{n}$
tale che:
$\displaystyle\left\|f_{n}-f\lambda_{n}\right\|_{R}<\delta_{n}$
e definiamo $h_{\epsilon}\equiv\sum_{n}f_{n}$. Per locale finitezza della
partizione dell’unità questa serie è ben definita e $h_{\epsilon}\in
C^{\infty}(A,\mathbb{R})$, inoltre:
$\displaystyle\left\|h_{\epsilon}-f\right\|_{K_{m}}=\left\|\sum_{n}(f_{n}-f\lambda_{n})\right\|_{K_{m}}<\epsilon_{m}$
da cui la tesi. ∎
#### 3.2.4 Formule di Green e principio di Dirichlet
Ricordando la notazione definita nella sezione 1.1.4, grazie ai risultati di
densità appena descritti possiamo dimostrare che:
###### Proposizione 3.29.
Se $f$ è di Tonelli con $D_{R}(f)<\infty$, e $u\in H(\Omega)\cap
C^{\infty}(\overline{\Omega})$ 222222$H(\Omega)$ è l’insieme delle funzioni
armoniche su $\Omega$. $C^{\infty}(\overline{\Omega})$ è l’insieme delle
funzioni che possono essere estese a funzioni lisce in un intorno di
$\overline{\Omega}$, dove $\Omega$ è un dominio regolare, abbiamo che:
$\displaystyle D_{\Omega}(f,u)\equiv\int_{\Omega}\left\langle\nabla
f\middle|\nabla u\right\rangle dV=\int_{\partial\Omega}f\ast du$
###### Proof.
Consideriamo una successione $f_{n}\in C^{\infty}(R,\mathbb{R})$ che converge
in norma $\left\|\cdot\right\|_{R}$ a $f$. Allora grazie alla proposizione
1.11, abbiamo che per ogni $m$:
$\displaystyle D_{\Omega}(f_{m},u)\equiv\int_{\Omega}\left\langle\nabla
f_{m}\middle|\nabla u\right\rangle dV=\int_{\partial\Omega}f_{m}\ast du$
Poiché $D_{R}(f_{m}-f)\to 0$ e anche $\left\|f_{m}-f\right\|_{\infty}\to 0$,
si ha che:
$\displaystyle\left|D_{\Omega}(f_{m},u)-D_{\Omega}(f,u)\right|\equiv\left|\int_{\Omega}\left\langle\nabla(f_{m}-f)\middle|\nabla
u\right\rangle dV\right|\leq D_{\Omega}(u)D_{\Omega}(f_{m}-f)\to 0$
$\displaystyle\left|\int_{\partial\Omega}f_{m}\ast
du-\int_{\partial\Omega}f_{m}\ast
du\right|\leq\int_{\partial\Omega}\left|f_{m}-f\right|\ast du\to 0$
da cui la tesi. ∎
Possiamo anche rilassare le ipotesi su $u$, e ottenere:
###### Proposizione 3.30.
Sia $\Omega\subset R$ un dominio regolare, $f$ una funzione di Tonelli con
$D_{R}(f)<\infty$, e $u\in H(\Omega)$ con $D_{\Omega}(u)<\infty$. Siano
inoltre $\gamma_{1}$ e $\gamma_{2}$ due insiemi connessi disgiunti tali che
$\gamma_{1}\cup\gamma_{2}=\partial\Omega$. Se $f=0$ su $\gamma_{1}$, e $u\in
C^{\infty}(\gamma_{2})$, allora:
$\displaystyle D_{\Omega}(f,u)\equiv\int_{\Omega}\left\langle\nabla
f\middle|\nabla u\right\rangle dV=\int_{\gamma_{2}}f\ast du$ (3.14)
###### Proof.
Per cominciare dimostriamo questa proposizione in un caso particolare. Sia $g$
una funzione di Tonelli con integrale di Dirichlet finito su $\Omega$, $g\geq
0$ su $\Omega$, $g|_{\gamma_{1}}=0$ e $g|_{\gamma_{2}}\geq\delta>0$.
Consideriamo ora una successione di insiemi $\Omega_{n}$ con bordo liscio tali
che $\Omega_{n}\subset\Omega$, $\partial\Omega_{n}=\gamma_{2}\cup\beta_{n}$ e
$\cup_{n}\Omega_{n}=\Omega$ 232323è possibile costruire una successione con
tali caratteristiche ad esempio considerando una funzione liscia
$h:\overline{\Omega}\to\mathbb{R}$ tale che $h(\gamma_{1})=0$,
$h(\gamma_{2})=1$ e $0<h(x)<1$ per ogni $x\in\Omega$. Scegliendo una
successione decrescente $r_{n}\searrow 0$ di valori regolari per $h$, gli
insiemi $\Omega_{n}\equiv h^{-1}(r_{n},1)$ hanno le caratteristiche cercate e
definiamo le funzioni
$\displaystyle g_{c}(x)\equiv\max\\{g(x)-c,0\\}$
per $0<c<\delta$. Per costruzione, queste funzioni coincidono con $g(x)-c$ su
un intorno di $\gamma_{2}$ e sono tutte nulle in un intorno di $\gamma_{1}$,
quindi fissato $c$, $g_{c}$ si annulla identicamente su $\beta_{n}$
definitivamente in $n$.
Grazie a queste considerazioni, e grazie alla proposizione 3.29, sappiamo che
esiste $n$ per cui:
$\displaystyle
D_{\Omega}(g_{c},u)=D_{\Omega_{n}}(g_{c},u)=\int_{\beta_{n}}g_{c}\ast
du+\int_{\gamma_{2}}g_{c}\ast du=\int_{\gamma_{2}}g\ast
du-c\int_{\gamma_{2}}\ast du$
Facendo tendere $c$ a $0$, la seconda parte della disuguaglianza tende a
$\int_{\gamma_{2}}g\ast du$, mentre la prima tende a $D_{\Omega}(g,u)$.
Infatti se chiamiamo $A_{c}=\\{x\in\Omega\ t.c.\ g(x)\leq\leavevmode\nobreak\
c\\}$
$\displaystyle\left|D_{\Omega}(g_{c},u)-D_{\Omega}(g,u)\right|=\left|\int_{\Omega}\left\langle\nabla(g_{c}-g)\middle|\nabla
u\right\rangle dV\right|=\left|\int_{A_{c}}\left\langle\nabla g\middle|\nabla
u\right\rangle dV\right|\leq$ $\displaystyle\leq\int_{A_{c}}\left|\nabla
g\right|^{2}dV\int_{A_{c}}\left|\nabla u\right|^{2}dV\to 0$
e dato che entrambi gli integrali di Dirichlet estesi a tutta $\Omega$ sono
finiti, se l’insieme la misura dell’insieme di integrazione tende a zero,
anche l’integrale tende a 0.
Questo dimostra la tesi su $g|_{\gamma_{2}}\geq\delta<0$. Se consideriamo una
funzione $f\geq 0$ qualsiasi, la tesi si ottiene per linearità, infatti basta
scegliere una funzione $g$ con le caratteristiche descritte sopra e applicare
quanto appena per ottenere:
$\displaystyle
D_{\Omega}(f,u)=D_{\Omega}((f+g)-g,u)=D_{\Omega}(f+g,u)-D_{\Omega}(g,u)=$
$\displaystyle=\int_{\gamma_{2}}(f+g)\ast du-\int_{\gamma_{2}}g\ast du$
Se la funzione $f$ non è positiva, la tesi si ottiene applicando il
ragionamento appena esposto alle funzioni $f^{+}$ e $f^{-}$, in particolare:
$\displaystyle
D_{\Omega}(f,u)=D_{\Omega}(f^{+}-f^{-},u)=\int_{\gamma_{2}}(f^{+}-f^{-})\ast
du=\int_{\gamma_{2}}f\ast du$
∎
Ora siamo in grado di dimostrare una generalizzazione del principio di
Dirichlet esposto in 1.66.
###### Proposizione 3.31.
Sia $f:R\to\mathbb{R}$ una funzione di Tonelli con $D_{R}(f)<\infty$, $u\in
H(\Omega)\cap C(\overline{\Omega})$, dove $\Omega$ è un dominio regolare in
$R$ varietà riemanniana. Se $f|_{\partial\Omega}=u|_{\partial\Omega}$, allora
$D_{\Omega}(u)<\infty$ e:
$\displaystyle D_{\Omega}(f)=D_{\Omega}(u)+D_{\Omega}(f-u)$
###### Proof.
La tesi di questa proposizione è esattamente identica alla tesi della
proposizione 1.66, la differenza è nelle richieste di regolarità su $f$, che
in questo caso sono molto meno stringenti. Come negli esempi sopra,
dimostreremo questo teorema approssimando $f$ con funzioni lisce.
Sia $f_{m}$ una successione di funzioni tali che
$\left\|f_{m}-f\right\|_{R}\to 0$, e siano $u_{m}$ le soluzioni del problema
di Dirichlet su $\Omega$ con $f_{m}|_{\partial\Omega}$ come dato al bordo. La
proposizione 1.66 afferma che:
$\displaystyle D_{\Omega}(f_{m})=D_{\Omega}(u_{m})+D_{\Omega}(f_{m}-u_{m})$
Grazie al principio del massimo otteniamo anche che
$\left\|u_{m}-u\right\|_{\infty}\leq\left\|f_{m}|_{\partial\Omega}-f|_{\partial\Omega}\right\|_{\infty}\to
0$
quindi $u=U-\lim_{m}u_{m}$. Inoltre:
$\displaystyle
D_{\Omega}(u_{m}-u_{k})=D_{\Omega}(f_{m}-f_{k})-D_{\Omega}(f_{m}-f_{k}+u_{k}-u_{m})\leq
D_{\Omega}(f_{m}-f_{k})$
e dato che $f_{m}\to f$ e che:
$\displaystyle D_{\Omega}(f_{m}-f_{k})^{1/2}\leq
D_{\Omega}(f_{m}-f)^{1/2}+D_{\Omega}(f_{k}-f)^{1/2}$
allora la successione $\\{u_{m}\\}$ è di Cauchy rispetto alla norma
$\left\|\cdot\right\|_{R}$, quindi per completezza di $\mathbb{M}(R)$, $u_{m}$
converge e necessariamente converge a $u$. Questo implica che
$D_{\Omega}(u)<\infty$. Osserviamo che:
$\displaystyle D_{\Omega}(f)=\int_{\Omega}\left|\nabla
f\right|^{2}dV=\int_{\Omega}\left|\nabla(f-u)+u\right|^{2}dV=$
$\displaystyle=\int_{\Omega}\left|\nabla(f-u)\right|^{2}dV+\int_{\Omega}\left|\nabla
u\right|^{2}dV+2\int_{\Omega}\left\langle\nabla(f-u)\middle|\nabla
u\right\rangle dV=$
$\displaystyle=D_{\Omega}((f-u))+D_{\Omega}(u)+2D_{\Omega}((f-u),u)$
Applicando il lemma 3.30 con $\gamma_{1}=\partial\Omega$ otteniamo che:
$\displaystyle
D_{\Omega}((f-u),u)=\int_{\Omega}\left\langle\nabla(f-u)\middle|\nabla
u\right\rangle dV=0$
dato che per ipotesi $(f-u)|_{\partial\Omega}=0$. ∎
In realtà le ipotesi sulla regolarità del bordo di $\Omega$ non sono
necessarie, infatti vale la proposizione
###### Proposizione 3.32.
Sia $f:R\to\mathbb{R}$ una funzione di Tonelli con $D_{R}(f)<\infty$,
$u\in\leavevmode\nobreak\ H(\Omega)\cap C(\overline{\Omega})$, con $\Omega$ è
un aperto relativamente compatto in $R$ varietà riemanniana. Se
$f|_{\partial\Omega}=u|_{\partial\Omega}$, allora $D_{\Omega}(u)<\infty$ e:
$\displaystyle D_{\Omega}(f)=D_{\Omega}(u)+D_{\Omega}(f-u)$
###### Proof.
Per comodità, estendiamo l’insieme di definizione di $u$ a tutto $R$, ponendo
$u|_{\Omega^{C}}=f|_{\Omega^{C}}$. In questo modo $u$ ha tutte le proprietà di
una funzione in $\mathbb{M}(R)$, tranne il fatto di avere integrale di
Dirichlet finito 242424proprietà che seguirà dalla dimostrazione.
Sia $\Omega_{n}$ un’esaustione regolare per $\Omega$, definiamo le funzioni:
$\displaystyle u_{n}:R\to\mathbb{R}\ \ u_{n}\in H(\Omega_{n})\cap C(R)\ \ \ \
u_{n}|_{\partial\Omega_{n}}=f|_{\partial\Omega_{n}}\ \ \ \
u_{n}|_{\Omega_{n}^{C}}=f|_{\Omega_{n}^{C}}$
Dalla proposizione precedente è evidente $\forall n$
$D_{\Omega}(u_{n})<\infty$, quindi $u_{n}\in\mathbb{M}(R)$, inoltre per ogni
$m>n$ vale che:
$\displaystyle
D_{\Omega_{m}}(u_{n})=D_{\Omega_{m}}(u_{m})+D_{\Omega_{m}}(u_{n}-u_{m})$
quindi considerando che
$u_{n}|_{\Omega_{m}^{C}}=u_{n}|_{\Omega_{m}^{C}}=f|_{\Omega_{m}^{C}}$, si ha
che $D_{\Omega_{m}^{C}}(u_{n})=D_{\Omega_{m}^{C}}(u_{m})$ e
$D_{\Omega_{m}^{C}}(u_{n}-u_{m})=0$, quindi:
$\displaystyle\underbrace{D_{\Omega_{m}}(u_{n})+D_{\Omega_{m}^{C}}(u_{n})}_{D_{R}(u_{n})}=\underbrace{D_{\Omega_{m}}(u_{m})+D_{\Omega_{m}^{C}}(u_{m})}_{D_{R}(u_{m})}+\underbrace{D_{\Omega_{m}}(u_{n}-u_{m})+D_{\Omega_{m}^{C}}(u_{n}-u_{m})}_{D_{R}(u_{n}-u_{m})}$
Grazie a un ragionamento simile a quello riportato nella dimostrazione del
teorema 3.53, si ha che la successione $\\{u_{n}\\}$ è $D$-cauchy. Inoltre
grazie al principio del massimo si può dimostrare che $u_{n}$ converge
uniformemente a $u$ su $\Omega$, infatti
$\displaystyle\left|u(x)-u_{n}(x)\right|\begin{cases}=0&\text{se
}x\in\Omega^{C}\\\ =\left|f(x)-u(x)\right|&\text{se
}x\in\Omega\setminus\Omega_{n}\\\
\leq\max_{x\in\partial\Omega_{n}}\\{\left|f(x)-u(x)\right|\\}&\text{se
}x\in\Omega_{n}\end{cases}$
dove l’ultima riga segue dal principio del massimo. Da questa relazione
ricaviamo la stima:
$\displaystyle\left\|u-u_{n}\right\|_{\infty}=\max_{x\in\Omega\setminus\Omega_{n}}\\{\left|u(x)-f(x)\right|\\}$
dato che $\Omega$ è compatto e $f|_{\partial\Omega}=u|_{\partial\Omega}$, la
parte sinistra dell’ultima uguaglianza tende a $0$ quando $n$ tende a
infinito, da cui $u$ è il limite uniforme di $u_{n}$.
Dato che la successione $\\{u_{n}\\}$ è di Cauchy rispetto alla metrica $CD$,
converge in questa metrica, e per unicità del limite converge a $u$, in
particolare:
$\displaystyle\lim_{n\to\infty}D_{R}(u-u_{n})=0$
Per dimostrare la seconda parte della proposizione, applichiamo ancora una
volta la proposizione precedente, ottenendo che:
$\displaystyle
D_{\Omega_{n}}(f)=D_{\Omega_{n}}(u_{n})+D_{\Omega_{n}}(f-u_{n})$
e quindi anche
$\displaystyle D_{\Omega}(f)=D_{\Omega}(u_{n})+D_{\Omega}(f-u_{n})$
La tesi si ottiene passando al limite per $n$ che tende a infinito. ∎
#### 3.2.5 Ideali dell’algebra di Royden
In questo paragrafo parleremo di alcuni ideali dell’algebra di Royden.
Ricordiamo la definizione di ideale di un’algebra (commutativa)
###### Definizione 3.33.
Data un’algebra $A$, un suo sottoinsieme $I\subset A$ è detto ideale se:
1. 1.
$I$ è un sottospazio vettoriale di $A$
2. 2.
per ogni $x\in A$ e $y\in I$, $xy\in I$
Un esempio di ideale su $\mathbb{M}(R)$ è l’insieme delle funzioni che si
annullano in un punto $p$. Questo ideale è per altro anche il nucleo del
carattere $\tau(p)$. Gli ideali a cui saremo interessati in questa sezione
però sono: $\mathbb{M}_{0}(R)$ e $\mathbb{M}_{\Delta}(R)$, dove:
$\displaystyle\mathbb{M}_{0}(R)\equiv\\{f\in\mathbb{M}(R)\ t.c.\ supp(f)\
compatto\\}$
mentre $\mathbb{M}_{\Delta}(R)$ è l’insieme di tutte le funzioni che sono
$BD-$limiti di successioni di funzioni in $\mathbb{M}_{0}(R)$ 252525quindi
$M_{\Delta}(R)$ è la chiusura sequenziale dell’insieme $M_{0}(R)$. È
abbastanza facile verificare che $\mathbb{M}_{0}(R)$ è un ideale di
$\mathbb{M}(R)$, meno banale è la verifica che anche $\mathbb{M}_{\Delta}(R)$
è un ideale.
###### Proposizione 3.34.
$\mathbb{M}_{\Delta}(R)$ è un ideale di $\mathbb{M}(R)$.
###### Proof.
Per dimostrare questa affermazione consideriamo una funzione
$f\in\mathbb{M}_{\Delta}(R)$ e $h\in\mathbb{M}(R)$. Allora esiste una
successione $f_{n}\in\mathbb{M}_{0}(R)$ tale che
$f=BD-\lim_{n}f_{n}$
Ovviamente la successione $\\{hf_{n}\\}\subset\mathbb{M}_{0}(R)$. Dimostriamo
che la successione $hf_{n}$ converge nella topologia $BD$ a $hf$, in questo
modo otteniamo la tesi.
Poiché $h$ è continua e limitata, la successione $hf_{n}$ converge localmente
uniformemente a $hf$ ed è uniformemente limitata (quindi
$hf=B-\lim_{n}hf_{n}$). Resta da dimostrare che $hf=D-\lim_{n}hf_{n}$. A
questo scopo osserviamo che:
$\displaystyle D_{R}(hf-hf_{n})=\int_{R}g^{ij}\left(\frac{\partial(hf-
hf_{n})}{\partial x^{i}}\right)\left(\frac{\partial(hf-hf_{n})}{\partial
x^{j}}\right)dV=$ $\displaystyle=\int_{R}g^{ij}\left(\frac{\partial
h}{\partial x^{i}}f+h\frac{\partial f}{\partial x^{i}}-\frac{\partial
h}{\partial x^{i}}f_{n}-h\frac{\partial f_{n}}{\partial
x^{i}}\right)\left(\frac{\partial h}{\partial x^{j}}f+h\frac{\partial
f}{\partial x^{j}}-\frac{\partial h}{\partial x^{j}}f_{n}-h\frac{\partial
f_{n}}{\partial x^{j}}\right)dV=$
$\displaystyle\int_{R}g^{ij}\left(\frac{\partial h}{\partial
x^{i}}(f-f_{n})+h\left(\frac{\partial f}{\partial x^{i}}-\frac{\partial
f_{n}}{\partial x^{i}}\right)\right)\left(\frac{\partial h}{\partial
x^{j}}(f-f_{n})+h\left(\frac{\partial f}{\partial x^{j}}-\frac{\partial
f_{n}}{\partial x^{j}}\right)\right)dV\leq$ $\displaystyle\leq
2\int_{R}g^{ij}\frac{\partial h}{\partial x^{i}}\frac{\partial h}{\partial
x^{j}}(f-f_{m})^{2}dV+2\int_{R}g^{ij}\left(\frac{\partial f}{\partial
x^{i}}-\frac{\partial f_{n}}{\partial x^{i}}\right)\left(\frac{\partial
f}{\partial x^{j}}-\frac{\partial f_{n}}{\partial x^{j}}\right)h^{2}dV$
dove l’ultima disuguaglianza segue dal fatto che per ogni norma
$\left\|A+B\right\|^{2}\leq
2\left(\left\|A\right\|^{2}+\left\|B\right\|^{2}\right)$
applicando questa disuguaglianza alla norma
$\left\|A\right\|=g^{ij}A_{i}A_{j}$ si ottiene il risultato.
Ora consideriamo che per ogni $K\Subset R$:
$\displaystyle\int_{R}g^{ij}\frac{\partial h}{\partial x^{i}}\frac{\partial
h}{\partial x^{j}}(f-f_{m})^{2}dV=$
$\displaystyle=\int_{K}g^{ij}\frac{\partial h}{\partial x^{i}}\frac{\partial
h}{\partial x^{j}}(f-f_{m})^{2}dV+\int_{R\setminus K}g^{ij}\frac{\partial
h}{\partial x^{i}}\frac{\partial h}{\partial x^{j}}(f-f_{m})^{2}dV$
e poiché $f_{n}$ converge localmente uniformemente a $f$, il primo integrale
converge a $0$ quando $m$ tende a infinito, mentre il secondo è limitato in
modulo da:
$\displaystyle\left|\int_{R\setminus K}g^{ij}\frac{\partial h}{\partial
x^{i}}\frac{\partial h}{\partial x^{j}}(f-f_{m})^{2}dV\right|\leq
ND_{R\setminus K}(h)$
dove $N$ è tale che $\left|f_{n}(x)-f(x)\right|^{2}\leq N\ \forall n\ \forall
x$ 262626poiché $f$ è limitata e $f_{n}$ è uniformemente limitata, questo
numero esiste.
Inoltre abbiamo che:
$\displaystyle\left|\int_{R}g^{ij}\left(\frac{\partial f}{\partial
x^{i}}-\frac{\partial f_{n}}{\partial x^{i}}\right)\left(\frac{\partial
f}{\partial x^{j}}-\frac{\partial f_{n}}{\partial
x^{j}}\right)h^{2}dV\right|\leq$ $\displaystyle\leq
M\int_{R}g^{ij}\left(\frac{\partial f}{\partial x^{i}}-\frac{\partial
f_{n}}{\partial x^{i}}\right)\left(\frac{\partial f}{\partial
x^{j}}-\frac{\partial f_{n}}{\partial x^{j}}\right)dV=MD_{R}(f_{n}-f)$
dove $M=\left\|g\right\|_{\infty}^{2}$. Dato che $D_{R}(f_{n}-f)\to 0$,
otteniamo che:
$\displaystyle\limsup_{n\to\infty}D_{R}(hf-hf_{n})\leq 2ND_{R\setminus K}(h)$
data l’arbitrarietà di $K$ e dato che $D_{R}(h)<\infty$, otteniamo che:
$\displaystyle\lim_{n\to\infty}D_{R}(hf-hf_{n})=0$
Quindi $BD-\lim_{n}gf_{n}=gf$, il che dimostra che
$gf\in\mathbb{M}_{\Delta}(R)$. ∎
Vale una proposizione simile a 3.24 per lo spazio $\mathbb{M}_{\Delta}(R)$:
###### Proposizione 3.35.
Sia $f_{n}$ una successione in $\mathbb{M}_{\Delta}(R)$ tale che
$\displaystyle f=C-\lim_{n}f_{n}$
con $f$ funzione (continua) limitata e
$\displaystyle D_{R}(f_{n})\leq K$
dove $K$ non dipende da $n$. Allora $f\in\mathbb{M}_{\Delta}(R)$.
###### Proof.
Grazie al teorema 3.24, $f\in\mathbb{M}(R)$ e esiste una sottosuccessione
$f_{n_{k}}$ (che per comodità continueremo a indicare con $f_{n}$) tale che
per ogni $g\in\mathbb{M}(R)$
$\displaystyle D_{R}(f-f_{n};g)\to 0$
Sia $R_{n}$ un’esaustione regolare di $R$, e definiamo le funzioni $\phi_{n}$
in modo che:
1. 1.
$\phi_{n}=0$ su $R\setminus\overline{R_{2}}$
2. 2.
$\phi_{n}=f_{n}$ su $R_{1}$
3. 3.
$\phi_{n}\in H(R_{2}\setminus\overline{R_{1}})$
e sia $\phi$ definita in maniera analoga per $f$.
Dato che per ogni $n$, $\phi_{n}\in\mathbb{M}_{0}(R)$, al posto che $f$ e
$f_{n}$, nella dimostrazione possiamo considerare le funzioni $f-\phi$ e
$f_{n}-\phi_{n}$, cioè possiamo assumere per ipotesi che $f_{n}=f=0$
sull’insieme $R_{1}$.
Sia $u_{m}$ una successione di funzioni definite da:
1. 1.
$u_{m}=0$ su $R_{1}$
2. 2.
$u_{m}=f$ su $R_{m}\setminus R_{1}$
3. 3.
$u_{m}\in H(R_{m}\setminus\overline{R_{1}})$
Chiaramente $u_{m}\in\mathbb{M}(R)$, e per il principio del massimo
$\left\|u_{m}\right\|_{\infty}\leq\left\|f\right\|_{\infty}$. Poiché
$u_{m}=u_{n}=f$ su $\partial R_{p}$ e $u_{p}$ è armonica su $R_{p}$, grazie al
principio di Dirichlet (vedi 3.31) otteniamo che se $p>m$:
$\displaystyle D_{R}(u_{p}-u_{m})=D_{R}(u_{m})-D_{R}(u_{p})$
questo dimostra che la successione $D_{R}(u_{m})$ è decrescente al crescere di
$m$ 272727infatti se $p>m$, $D_{R}(u_{m})-D_{R}(u_{p})\geq 0$, e ovviamente
limitata dal basso da $0$, quindi è convergente, e quindi se $m$ è
sufficientemente grande e $p>m$, $D_{R}(u_{m}-u_{p})$ è piccolo a piacere,
cioè $\\{u_{m}\\}$ è $D$-Cauchy.
Grazie al principio 1.57, esiste una sottosuccessione di $u_{m}$ (che
continueremo a indicare nello stesso modo) tale che:
$\displaystyle u=B-\lim_{n}u_{n}\ \ \Rightarrow\ \ u=BD-\lim_{n}u_{n}$
allora $u\in\mathbb{M}(R)$ con $u(R_{1})=0$ e $u\in H(R\setminus R_{1})$.
Dato che $f-u=BD-\lim_{n}(f-u_{n})$, e per costruzione
$(f-u_{n})\in\mathbb{M}_{0}(R)$, $f-u\in\mathbb{M}_{\Delta}(R)$.
Grazie alla formula di Green 3.30, si ha che:
$\displaystyle D_{R}(f-u_{n};u)=0$
poiché $f-u_{n}=0$ sull’insieme $R\setminus R_{n}$, e anche su $R_{1}$ (quindi
anche sui relativi bordi). Quindi anche
$D_{R}(f-u;u)=\lim_{n}D_{R}(f-u_{n};u)=0$. Quindi otteniamo che:
$\displaystyle D_{R}(f-u;u)=0\ \ \Rightarrow\ \ D_{R}(u)\equiv
D_{R}(u;u)=D_{R}(f;u)=\lim_{n}D_{R}(f_{n};u)$
Dato che $f_{n}\in\mathbb{M}_{\Delta}(R)$ per ipotesi, per ogni $n$ esiste una
successione $h_{k}$ di funzioni in $\mathbb{M}_{0}(R)$ tali che
$f_{n}=BD-\lim_{k}h_{k}$. Questo permette di osservare che
$\displaystyle D_{R}(f_{n};u)=\lim_{k}D_{R}(h_{k};u)=\lim_{k}\int_{\partial
R_{1}}h_{k}\ast du$
dove l’ultima uguaglianza è conseguenza di 3.30. Dato che
$\left\|h_{k}\right\|_{\infty,\partial R_{1}}\to 0$ (poiché la funzione
$f_{n}=0$ su $\partial R_{1}$), l’ultimo limite è $0$, quindi $D_{R}(u)=0$, il
che significa che $u\equiv 0$, e quindi $f=f-u\in\mathbb{M}_{\Delta}(R)$, come
volevasi dimostrare. ∎
Come corollario a questa proposizione, osserviamo che:
###### Osservazione 3.36.
Se $\\{f_{n}\\}\subset\mathbb{M}_{\Delta}(R)$ e $f=BD-\lim_{n}f_{n}$, allora
$f\in\mathbb{M}_{\Delta}(R)$.
### 3.3 Compattificazione di Royden
Ora siamo pronti per introdurre il concetto di compattificazione di Royden.
Data una varietà riemanniana $R$, la sua compattificazione di Royden $R^{*}$ è
uno spazio compatto che in qualche senso contiene $R$. La compattificazione di
Royden è l’iniseme dei caratteri sull’algebra $\mathbb{M}(R)$, quindi
l’integrale di Dirichlet delle funzioni gioca un ruolo fondamentale nella
costruzione di questa compattificazione. Questo fa intuire che proprietà come
parabolicità e iperbolicità vengano in qualche modo riflesse nelle proprietà
di $R^{*}$.
#### 3.3.1 Definizione
###### Definizione 3.37.
Data una varietà riemanniana $(R,g)$, si definisce compattificazione di Royden
uno spazio $R^{*}$ tale che:
1. 1.
$R^{*}$ è uno spazio compatto di Hausdorff
2. 2.
$R$ è un sottoinsieme aperto e denso di $R^{*}$
3. 3.
ogni funzione in $\mathbb{M}(R)$ può essere estesa per continuità a una
funzione definita su $R^{*}$
4. 4.
l’insieme $\overline{\mathbb{M}}(R)$ delle funzioni che sono estensione delle
funzioni in $\mathbb{M}(R)$ separa i punti di $R^{*}$
Osserviamo subito che se $R$ è compatta, necessariamente $R^{*}=R$.
###### Teorema 3.38.
Per ogni varietà riemanniana $R$ esiste $R^{*}$ ed è unica a meno di
omeomorfismi che mantengaono fissi i punti di $R$.
Spezziamo la dimostrazione di questo teorema in alcuni lemmi per facilitarne
la lettura, e riportiamo nell proposizione 3.44 lo schema riassuntivo della
dimostrazione.
Sia $R^{*}$ l’insime dei funzionali lineari moltiplicativi su $\mathbb{M}(R)$
dotato della topologia debole-* rispetto a $\mathbb{M}(R)$. Ovviamente questo
spazio è uno spazio di Hausdorff 282828sia $p\neq q$, allora per definizione
esiste $f\in\mathbb{M}(R)$ tale che $\left|p(f)-q(f)\right|=\delta>0$. Quindi
gli intorni $V(p)=\\{h\in R^{*}\ t.c.\ \left|h(f)-p(f)\right|<\delta/3\\}$ e
$W(q)=\\{h\in R^{*}\ t.c.\ \left|h(f)-q(f)\right|<\delta/3\\}$ sono intorni
disgiunti dei due caratteri $p$ e $q$., e grazie alla proposizione 1.85 è
anche uno spazio compatto. $R^{*}$ è il candidato a compattificazione di
Royden di $R$.
Per ogni punto $p\in R$, il funzionale $x_{p}(f)\equiv f(p)$ appartiene di
certo all’algebra di Royden. Definiamo la funzione $\tau:R\to R^{*}$ come
$\displaystyle\tau(p)=x_{p}\ \Longleftrightarrow\ \tau(p)(f)=f(p)$
###### Lemma 3.39.
La funzione $\tau$ è un’omeomorfismo sulla sua immagine.
###### Proof.
Infatti sia $p\neq q$, necessariamente $\tau(p)\neq\tau(q)$. Infatti poiché
$R$ è una varietà riemanniana, per ogni coppia di punti disgiunti esiste una
funzione $f$ liscia a supporto compatto (quindi $f\in\mathbb{M}(R)$) tale che
$f(p)\neq f(q)$ 292929questa funzione si può costruire ad esempio sfruttando
le partizioni dell’unità, quindi $\tau(p)(f)\neq\tau(q)(f)$.
Inoltre $\tau$ è continua. Consideriamo un qualunque aperto di $R^{*}$. Per
definizione di topologia debole-*, questo aperto contiene un’intersezione
finita di insiemi della forma
$\displaystyle V(q,f,\epsilon)=\\{p\in R^{*}\ t.c.\
\left|p(f)-q(f)\right|<\epsilon\\}$
La controimmagine $\tau^{-1}(V)$ è aperta poiché $f$ è continua.
Inoltre la mappa $\tau$ è anche una mappa aperta sulla sua immagine.
Consideriamo a questo scopo un punto $p\in R$ e un suo intorno aperto $V$
qualsiasi, e dimostriamo che $\tau(p)$ ha un intorno aperto contenuto in
$\tau(V)\cap\tau(R)$. Sia a questo proposito $\psi$ una funzione liscia tale
che $\psi(p)=0$ e $1-\psi$ abbia supporto compatto contenuto in $V$ (questo
garantisce che $\psi\in\mathbb{M}(R)$). L’insieme
$\displaystyle A=\\{q\in\tau(R)\ t.c.\ \left|\psi(q)-\psi(p)\right|<1/2\\}$
è un aperto su $\tau(R)$ ed è ovviamente contenuto in $\tau(V)$. ∎
Per quanto riguarda l’estendibilità delle funzioni $f\in\mathbb{M}(R)$,
osserviamo che:
###### Lemma 3.40.
Tutte le funzioni $f\in\mathbb{M}(R)$ possono essere estese per continuità a
$R^{*}$.
###### Proof.
Definiamo $\bar{f}$ l’estensione di $f$ a $R^{*}$, e definiamo in maniera
naturale
$\displaystyle\bar{f}(x)\equiv x(f)\ \ \forall x\in R^{*}$
Se $p\in R$, allora $\bar{f}(p)=\tau(p)(f)=f(p)$, il che dimostra che
$\bar{f}$ è un’estensione di $f$. La continuità di $\bar{f}$ segue dalla
definizione di topologia debole-*, infatti:
$\displaystyle\\{y\in R^{*}\ t.c.\
\left|\bar{f}(x)-\bar{f}(y)\right|<\epsilon\\}=\\{y\in R^{*}\ t.c.\
\left|x(f)-y(f)\right|<\epsilon\\}$
dove l’ultimo insieme è aperto in $R^{*}$ per definizione. ∎
Per dimostrare la densità di $R$ in $R^{*}$, utilizzeremo quest’altro lemma:
###### Lemma 3.41.
Lo spazio $\overline{\mathbb{M}(R)}=\\{\bar{f}\ t.c.\ f\in\mathbb{M}(R)\\}$ è
denso rispetto alla norma del sup nell’algebra delle funzioni continue da
$R^{*}$ a $\mathbb{R}$, $C(R^{*})$.
###### Proof.
Questa dimostrazione è una facile conseguenza del teorema di Stone-Weierstrass
(vedi ad esempio teorema 7.31 pag 162 di [R3]). L’algebra
$\overline{\mathbb{M}}(R)$ infatti separa i punti su $R^{*}$ e contiene la
fuzione costante uguale a 1 che non si annulla mai (quindi l’algebra non si
annulla in nessun punto di $R^{*}$). ∎
###### Lemma 3.42.
L’insieme $\tau(R)$ è denso in $R^{*}$.
###### Proof.
Dimostriamo questa affermazione per assurdo: sia $\tilde{x}\in
R^{*}\setminus\overline{\tau(R)}$. Allora per il lemma di Uryson (vedi teorema
2.12 pag 39 di [R4]) esiste una funzione continua $h:R^{*}\to\mathbb{R}$ tale
che $h(\tilde{x})=0$ e $h(\tau(R))=1$. Grazie al lemma precedente, esiste una
funzione $\bar{f}\in\overline{\mathbb{M}(R)}$ tale che $\bar{f}(\tilde{x})=0$
e $\bar{f}(\tau(R))>1/3$ 303030infatti grazie al lemma precedente esiste una
funzione $\bar{h}\in\overline{\mathbb{M}}(R)$ tale che
$\bar{h}(\tilde{x})<1/3$ e $\bar{h}(\tau/R))>2/3$. La funzione
$\bar{f}=\bar{h}-\bar{h}(\tilde{x})$ ha le caratteristiche richieste.Allora la
sua restrizione $f:R\to\mathbb{R}$ ha estremo inferiore positivo, quindi è
invertibile in $\mathbb{M}(R)$. Questo significa che esiste $g$ tale che
$(fg)(p)=1\ \forall p\in R$. Ma per definizione di carattere
$\overline{(fg)}(x)=x(fg)=1\ \forall x\in R^{*}$, quindi anche
$\overline{(fg)}(\tilde{x})=\bar{f}(\tilde{x})\bar{g}(\tilde{x})=1$ 313131dato
che $x(fg)=x(f)x(g)$, $\overline{(fg)}=\bar{f}\bar{g}$, impossibile se
$\bar{f}(\tilde{x})=0$ ∎
In seguito a questi lemmi siamo pronti a dimostrare che
###### Proposizione 3.43.
$R^{*}$ l’insieme dei caratteri su $\mathbb{M}(R)$ con la topologia debole
ereditata da questo spazio è una compattificazione di Royden per $R$.
###### Proof.
Il fatto che $R^{*}$ sia compatto e di Hausdorff è stato dimostrato a pagina
3.38. Che $R$ sia denso in $R^{*}$ è il contenuto dei lemmi 3.39 e 3.42.
Inoltre $R$ è aperto come sottoinsieme di $R^{*}$ grazie alla sua locale
compattezza. Infatti sia $p\in R$ qualsiasi, e $U$ un suo intorno aperto
relativamente compatto nella topologia di $R$, allora esiste un aperto $A\in
R^{*}$ tale che $\tau(U)=A\cap R$. Ma allora $\tau(U)=A$, infatti se per
assurdo non fosse così, $A\setminus\tau(\overline{U})$ sarebbe ancora un
insieme aperto per compattezza di $\tau(\overline{U})$, e quindi se fosse non
vuoto avrebbe intersezione non vuota con $R$ per densità, ma questo è
impossibile, quindi $A\subset\tau(\overline{U})$, e cioè $A=\tau(U)$.
Ogni funzione in $\mathbb{M}(R)$ può essere estesa per continuità a una
funzione definita su $R^{*}$ grazie al lemma 3.40, e infine l’insieme
$\overline{\mathbb{M}}(R)$ delle funzioni che sono estensione delle funzioni
in $\mathbb{M}(R)$ separa i punti di $R^{*}$ per definizione di carattere,
oppure come conseguenza del lemma 3.41. ∎
Resta da verificare l’unicità di $R^{*}$ a meno di omeomorfismi che tengano
fissi gli elementi di $R$.
###### Proposizione 3.44.
Sia $X$ uno spazio con le proprietà (1),(2),(3),(4) di 3.38. Allora la mappa
$\sigma:X\to R^{*}$ definita da
$\displaystyle\sigma(p)(f)=f(p)$
dove $f\in\overline{\mathbb{M}}(R)$, è un omeomorfismo che tiene fissi gli
elementi di $R$.
###### Proof.
È ovvio che $\sigma$ tiene fissi gli elementi di $R$, e grazie a un
ragionamento molto simile a quello del lemma 3.39, $\sigma$ è un omeomorfismo
sulla sua immagine. Essendo però $X$ compatto, $R$ denso in $R^{*}$, e anche
$R^{*}$ compatto, allora necessariamente $\sigma(X)=R^{*}$, quindi $\sigma$ è
anche suriettivo. ∎
D’ora in avanti per comodità di notazione confonderemo la scrittura $f$ e
$\bar{f}$, cioè indicheremo una funzione $f:R\to\mathbb{R}$ e la sua
estensione a tutta la compattificazione di Royden nello stesso modo.
#### 3.3.2 Esempi non banali di caratteri
Come preannunciato nel capitolo 2, utilizziamo gli ultrafiltri e in
particolare i risultati ottenuti nella sezione 2.2 per descrivere alcuni
caratteri non banali sull’algebra di Royden (quindi punti di $R^{*}$).
Dalla definizione di carattere, ci si aspetta che sull’algebra di Royden
esistano dei caratteri che in qualche senso rappresentino il limite della
funzione in una certa direzione. Ad esempio se consideriamo una successione
$x_{n}\in R$, l’operazione di limite lungo questa successione è un’operazione
lineare e moltiplicativa su $\mathbb{M}(R)$ quando definita, nel senso che
date due funzioni $f$ e $g$ tali che esista il limite $\lim_{n}f(x_{n})$ e
$\lim_{n}g(x_{n})$, allora vale che:
$\displaystyle\lim_{n}(f\cdot g)(x_{n})=\lim_{n}f(x_{n})\cdot\lim_{n}g(x_{n})$
data una funzione in $\mathbb{M}(R)$, non sempre è garantito che su ogni
successione che tende a infinito esista il limite di $f(x_{n})$. Per aggirare
questo problema, utilizziamo i limiti lungo ultrafiltri.
###### Osservazione 3.45.
Con la tecnica mostrata nella sezione 2.2 è possibile definire caratteri anche
sulle algebre di Royden. Basta considerare una qualsiasi successione
$\\{x_{n}\\}$ in $R$ e un qualsiasi ultrafiltro $\mathcal{M}$ su $\mathbb{N}$
e definire
$\displaystyle\phi(f)=\lim_{\mathcal{M}}f(x_{n})$
Se l’ultrafiltro è non costante, allora $\phi$ è un’operazione di limite,
quindi:
$\displaystyle\liminf_{n}f(x_{n})\leq\phi(f)\leq\limsup_{n}f(x_{n})$
Questo garantisce che l’operazione descritta sia un’estensione dell’operazione
di limite standard, nel senso che se esiste $\lim_{n}f(x_{n})$, allora
$\phi(f)=\lim_{n}f(x_{n})$, e anche che questo carattere non sia un carattere
banale, non sia cioè un carattere immagine attraverso $\tau$ di un punto di
$R$. Infatti è sempre possibile creare una funzione in $\mathbb{M}(R)$ che
valga $1$ su un qualsiasi punto fissato $\bar{x}$ e che valga $O$
definitivamente su una successione $x_{n}$ che tende a infinito 323232quindi
$\phi(f)=0$, mentre $\bar{x}(f)\equiv f(\bar{x})=1$.
La compattificazione di Royden quindi contiene tutti i caratteri che rendono
conto del comportamento di una funzione al “limite” lungo una successione.
In realtà non è necessario passare attraverso le successioni, con un
ragionamento molto simile a quello sviluppato nella sezione precendente, si
osserva che
###### Proposizione 3.46.
Ogni ultrafiltro $\mathcal{M}$ su $R$ definisce un carattere sull’algebra di
Royden, quindi un elemento di $R^{*}$
###### Proof.
La dimostrazione è la generalizzazione della dimostrazione di 2.24.
Consideriamo una funzione continua limitata $f:R\to\mathbb{R}$. La collezione
$f(\mathcal{M})$ è un ultrafiltro in $\mathbb{R}$, anzi un ultrafiltro in
$[\inf_{R}(f),\sup_{R}(f)]$, insieme compatto. Quindi per ogni funzione
possiamo definire
$\displaystyle\phi_{\mathcal{M}}(f)=\lim f(\mathcal{M})$
e con argomenti del tutto analoghi alla dimostrazione 2.24 ottenere che $\phi$
è un carattere su $M(R)$. ∎
Quello che manca in questo caso rispetto a sopra è la proprietà che vale per
gli ultrafiltri non costanti
$\displaystyle\liminf_{n}x(n)\leq\phi(x)\leq\limsup_{n}x(n)$
quindi una specie di controllo del carattere con il comportamento di $f$
all’infinito. Ovviamente non ha senso la definizione standard di $\liminf$ e
$\limsup$ per una funzione che come dominio ha un’insieme con la potenza del
continuo, ma comunque anche in questo caso esiste una proprietà simile, che
verrà esplorata nella sezione successiva.
#### 3.3.3 Caratterizzazione del bordo
In questa sezione ci occupiamo di due caratterizzazioni del bordo di $R^{*}$,
una di natura funzionale e una di natura topologica.
###### Definizione 3.47.
Indichiamo con $\Gamma=R^{*}\setminus R$ il bordo di $R^{*}$
Ovviamente $\Gamma$ è un’insieme compatto in $R^{*}$. Per prima cosa
dimostreremo che per un carattere $p\in\Gamma$ $p(f)=p((1-\lambda)f)$ per ogni
funzione $\lambda$ a supporto compatto in $R$, cioè il valore di un carattere
in $\Gamma$ dipende solo dal comportamento “all’infinito” della funzione a cui
è applicato.
###### Proposizione 3.48.
$p\in\Gamma$ se e solo se $p(f)=0$ $\forall f\in\mathbb{M}_{0}(R)$, o
equivalentemente se per ogni funzione $f\in\mathbb{M}(R)$ e $\forall\lambda\in
C^{\infty}(R,\mathbb{R})$ a supporto compatto
$\displaystyle p(\lambda f)=0\ \ \Longleftrightarrow\ \ p(f)=p((1-\lambda)f)$
###### Proof.
Questa dimostrazione è un’adattamento dell’esempio 11.13 (a) pag 283 di [R2].
Per prima cosa notiamo che vale una ovvia dicotomia per gli elementi di
$R^{*}$. Consideriamo un’esaustione $K_{n}$ di $R$, e consideriamo una
successione di funzioni di cut-off $\lambda_{n}$ tali che
$\lambda_{n}(K_{n})=1$ e $supp(\lambda_{n})\subset K_{n+1}$. Allora $\forall
p\in R^{*}$ vale che
$\displaystyle\forall n\ p(\lambda_{n})=0\ \vee\ \exists n\ t.c.\
p(\lambda_{n})\neq 0$
Nel primo caso, consideriamo $\lambda$ una funzione a supporto compatto.
Allora esiste $n$ tale che $supp(\lambda)\subset K_{n}$, quindi
$\lambda\lambda_{n}f=\lambda f$ da cui:
$\displaystyle p(\lambda f)=p(\lambda_{n}\lambda f)=p(\lambda_{n})p(\lambda
f)=0$
Dimostriamo che nel secondo caso necessariamente $p=\tau(x)$ per qualche $x\in
K_{n+1}$.
Per prima cosa osserviamo che nel secondo caso $\forall f\in\mathbb{M}(R)$
$\displaystyle p(f)=p(f\lambda_{n})/p(\lambda_{n})$ (3.15)
Supponiamo per assurdo che questo non sia vero, cioè che per ogni $x$ in
$K_{n+1}$, $p\neq\tau(x)$. Questo significa che esiste $f\in\mathbb{M}(R)$
tale che
$\displaystyle x(f)=f(x)\neq p(f)\ \Rightarrow\ p(f-p(f)1)=0\neq
x(f-p(f)1)=f(x)-p(f)$
cioè per ogni punto $x$ in $K_{n+1}$, esiste $f_{x}$ tale che $p(f_{x})=0$ e
$f_{x}(x)\neq 0$. Visto che tutte le funzioni $f_{x}$ sono continue, per ogni
$x$ esiste un intorno $U_{x}$ in cui $f_{x}\neq 0$. Quindi per compattezza di
$K_{n+1}$ esiste un numero finito di punti $\\{x_{k}\\}\subset K_{n+1}$ tali
che $\cup_{k}U_{k}\supset K_{n+1}$. Consideriamo la funzione
$\displaystyle F=\sum_{k}f_{k}^{2}$
questa funzione è strettamente positiva su tutto l’insieme $K_{n+1}$, quindi
anche su un suo intorno compatto $A$. Sia $\psi$ una funzione di cut-off con
$supp(\psi)\Subset A$ e $\psi(K_{n+1})=1$. La funzione
$\displaystyle\tilde{F}=F\cdot\psi+(1-\psi)$
è per costruzione una funzione strettamente positiva su tutta la varietà $R$ e
$\tilde{F}(x)=1\ \forall x\not\in A$
Quindi $\inf_{R}(\tilde{F})>0$ e grazie alla proposizione 3.9 è un’elemento
invertibile di $\mathbb{M}(R)$. Questo implica che necessariamente
$p(\tilde{F})\neq 0$, ma ciò nonostante:
$\displaystyle
p(\tilde{F})=p(\lambda_{n}\tilde{F})/p(\lambda_{n})=p(\lambda_{n}F)/p(\lambda_{n})=p(F)=\sum_{k}[p(f_{k})]^{2}=0$
dove abbiamo sfruttato la relazione 3.15 e la definizione delle varie funzioni
in gioco. ∎
Passiamo ora alla caratterizzazione topologica del bordo $\Gamma$.
###### Proposizione 3.49.
$p\in\Gamma$ se e solo se $p$ non è un insieme $G_{\delta}$, cioè se e solo se
$p$ non è intersezione numerabile di aperti.
###### Proof.
Supponiamo per assurdo che $p$ sia un $G_{\delta}$, quindi siano $V_{n}$ tali
che $\overline{V_{n+1}}\subset V_{n}$ e sia $\\{K_{n}\\}$ un’esaustione di
$R$. Dato che $p\in\Gamma$, $W_{n}\equiv V_{n}\cap K_{n}^{C}$ sono ancora
intorni di $p$, e vale ancora che $\overline{W_{n+1}}\subset W_{n}$. Per ogni
aperto $W_{n}\setminus\overline{W_{n+1}}$, scegliamo una funzione $w_{n}$ con
supporto in questo aperto, con massimo $1$ e di integrale di Dirichlet
$D_{R}(w_{n})\leq 2^{-n}$ 333333possibile grazie alla proposizione 4.10.
Allora la funzione
$\displaystyle f=\sum_{n=1}^{\infty}w_{n}$
converge localmente uniformemente a una funzione continua su $R$ con integrale
di Dirichlet finito, quindi è estendibile con contintuità a $R^{*}$. Ma visto
che per ogni intorno $W_{n}$ di $p$ la funzione oscilla tra $0$ e $1$, non è
estendibile con continuità in $p$, contraddizione. ∎
Questo prova che la topologia di $R^{*}$ non è I numerabile, quindi neanche
metrizzabile.
#### 3.3.4 Bordo armonico e decomposizione
In questa sezione definiamo il bordo armonico di $R^{*}$, concetto che sarà
utile per dimostrare una versione del principio del massimo.
###### Definizione 3.50.
Definiamo $\Delta$ in bordo armonico di $R^{*}$ come:
$\displaystyle\Delta=\\{p\in R^{*}\ t.c.\ \forall f\in\mathbb{M}_{\Delta}(R)\
f(p)=0\\}=\bigcap_{f\in\mathbb{M}_{\Delta}(R)}f^{-1}(0)$
Dalla definizione risulta evidente che $\Delta$ è un insieme chiuso, e che
$\Delta\cap R=\emptyset$, cioè $\Delta\subset\Gamma$. Infatti per ogni punto
di $R$ è sempre possibile trovare una funzione liscia a supporto compatto che
vale 1 sul punto.
Il seguente lemma dimostra l’esistena di particolari funzioni in
$\mathbb{M}_{\Delta}(R)$.
###### Lemma 3.51.
Sia $K$ un insieme compatto in $R^{*}$ tale che $K\cap\Delta=\emptyset$.
Allora esiste una funzione $f\in\mathbb{M}_{\Delta}(R)$ identicamente uguale a
$1$ su $K$.
###### Proof.
Dato che $K\cap\Delta=\emptyset$, per ogni punto $p\in K$ esiste una funzione
$f_{p}\in\mathbb{M}_{\Delta}(R)$ tale che $f_{p}(p)>1$. Per compattezza di
$K$, esiste un numero finito di punti $p_{1},\cdots,p_{n}$ tali che
$\displaystyle\bigcup_{i=1}^{n}f_{p_{n}}^{-1}(1,\infty)\supset K$
Quindi la funzione
$\displaystyle f_{1}(x)\equiv\sum_{i=1}^{n}f^{2}_{p_{n}}(x)$
è una funzione appartenente a $\mathbb{M}_{\Delta}(R)$ (che ricordiamo essere
un’ideale), e maggiore di $1$ sull’insieme $K$. La funzione
$g(x)=\max\\{1/2,f_{1}(x)\\}$ è una funzione $g\in\mathbb{M}(R)$ invertibile
grazie al lemma 3.9 e identicamente uguale a $f_{1}$ su $K$, quindi grazie al
fatto che $\mathbb{M}_{\Delta}(R)$ è un’ideale di $\mathbb{M}(R)$, si ha che
la funzione
$\displaystyle f=f_{1}\cdot g^{-1}$
soddisfa la tesi del lemma. ∎
Come corollario, osserviamo che:
###### Osservazione 3.52.
Se e solo se $\Delta=\emptyset$, allora
$\mathbb{M}_{\Delta}(R)=\mathbb{M}(R)$.
###### Proof.
L’implicazione da destra a sinistra è evidente dalla definizione dell’insime
$\Delta$. Per quanto riguarda l’altra implicazione, se $\Delta=\emptyset$,
grazie alla proposizione precedente la funzione costante uguale a $1$ su tutto
$R^{*}$ appartiene a $\mathbb{M}_{\Delta}(R)$, da cui segue la tesi. ∎
Il seguente teorema permette di scrivere ogni funzione $f\in\mathbb{M}(R)$
come la somma di due funzioni, una armonica e l’altra in
$\mathbb{M}_{\Delta}(R)$. Questa decomposizione sarà utile sia in questa
sezione per dimostrare una forma del principio del massimo che nel seguito.
###### Teorema 3.53.
Sia $f\in\mathbb{M}(R)$. Allora esistono una funzione $u\in HDB(R)$
343434$HDB(R)$ è l’insime delle funzioni armoniche su $R$, limitate e con
integrale di Dirichlet finito che indicheremo per comodità $u=\pi(f)$ e una
funzione $h\in\mathbb{M}_{\Delta}(R)$ tali che
1. 1.
$f=\pi(f)+h$
2. 2.
se $\Delta\neq\emptyset$, la decomposizione è unica, altrimenti unica a meno
di costanti
3. 3.
se $f\geq 0$, allora anche $\pi(f)\geq 0$
4. 4.
$\left\|f\right\|_{\infty}\geq\left\|\pi(f)\right\|_{\infty}$
5. 5.
per ogni funzione $\phi\in\mathbb{M}_{\Delta}(R)$, $D_{R}(\phi,\pi(f))=0$,
quindi
$\displaystyle D_{R}(f)=D_{R}(\pi(f))+D_{R}(h)$
###### Proof.
La dimostrazione di questo teorema è costruttiva. Sia $K_{n}$ un’esaustione di
$R$ con domini regolari. Definiamo $u_{n}\in\mathbb{M}(R)$ come $u_{n}=f$
sull’insieme $R\setminus K_{n}$, e $u_{n}\in H(K_{n})$. Grazie al principio
del massimo, notiamo subito che
$\left\|u_{n}\right\|_{\infty}\leq\left\|f\right\|_{\infty}$ e che se $f\geq
0$, anche $u_{n}\geq 0$. Visto che la successione $u_{n}$ è uniformemente
limitata, grazie al principio di Harnack 1.57 esiste una sua sottosuccessione
(che continueremo a indicare con $u_{n}$) che converge localmente
uniformemente a una funzione armonica $u$ definita su $R$. Inoltre, grazie
all’identità di Green 3.30, se $k>n$:
$\displaystyle D_{K_{k}}(u_{k}-u_{n},u_{k})=D_{R}(u_{n}-u_{k},u_{k})=0$
e quindi:
$\displaystyle
D_{R}(u_{n})=D_{R}(u_{k})+D_{R}(u_{n}-u_{k})+2D_{R}(u_{k},u_{n}-u_{k})=D_{R}(u_{k})+D_{R}(u_{n}-u_{k})$
in particolare, se $k>n$, $D_{R}(u_{k})\leq D_{R}(u_{n})$, quindi la
successione $D_{R}(u_{n})$ è decrescente e converge a un valore $\geq 0$.
Grazie all’ultima uguaglianza, questo significa anche che $D_{R}(u_{n}-u_{k})$
è piccolo a piacere se $n$ e $k$ sono sufficientemente grandi, cioè la
successione $\\{u_{n}\\}$ è $D-$Cauchy. Vista la completezza $BD$ dello spazio
$\mathbb{M}(R)$, la funzione $u\in\mathbb{M}(R)$, quindi
$u\in HDB(R)$
Consideriamo ora la successione $h_{n}=f-u_{n}$. Per le proprietà di $u_{n}$,
$h_{n}\in\mathbb{M}_{0}(R)$, e $BD-\lim_{n}h_{n}=f-u=h$, quindi chiaramente
$h\in\mathbb{M}_{\Delta}(R)$.
Rimanne da dimostrare l’unicità della decomposizione. Supponiamo che esistano
due decomposizioni distinte per $f$:
$\displaystyle f=u+h=\bar{u}+\bar{h}$
Definiamo $v=u-\bar{u}=\bar{h}-h$. Necessariamente $v\in
HDB(R)\cap\mathbb{M}_{\Delta}(R)$. Quindi esiste una successione
$v_{n}\in\mathbb{M}_{0}(R)$ tale che $v=BD-\lim_{n}v_{n}$, quindi:
$\displaystyle
D_{R}(v)=\lim_{n}D_{R}(v_{n},v)=\lim_{n}D_{K_{k}}(v_{n},v)=\lim_{n}\int_{\partial
K_{k}}v_{n}\ast dv=0$
Dove $K_{k}$ è scelto in modo che $supp(v_{n})\subset K_{k}$.
Questo dimostra che $v$ ha derivata nulla, quindi è costante su $R$, e se
$\Delta\neq\emptyset$, allora $v=0$ dato che $v(\Delta)=0$.
Il punto (5) è una facile applicazione della formula di Green 3.29. Infatti se
$\phi\in\mathbb{M}_{\Delta}(R)$, esiste una successione
$\phi_{n}\in\mathbb{M}_{0}(R)$ tale che $\phi=BD-\lim_{n}\phi_{n}$, quindi:
$\displaystyle D_{R}(\phi,u)=\lim_{n}D_{R}(\phi_{n},u)$
Per ogni $n$, sia $K$ un compatto dal bordo liscio in $R$ tale che
$supp(\phi_{n})\subset K$, allora per 3.29.
$\displaystyle D_{R}(\phi_{n},u)=D_{K}(\phi_{n},u)=0$
poiché $\phi_{n}=0$ su $\partial K$. ∎
Dalla dimostrazione è facile ricavare questo corollario:
###### Proposizione 3.54.
Data $f\in\mathbb{M}(R)$, se esiste una funzione subarmonica $v$ definita su
$R$ tale che $v\leq f$, allora
$\displaystyle v\leq\pi(f)$
allo stesso modo, se esiste $v$ superarmonica tale che $v\geq f$ allora
$\displaystyle v\geq\pi(f)$
Questa proposizione può essere migliorata, infatti nella composizione si può
chiedere che $\pi(f)=f$ su un predeterminato insieme compatto $K$ con bordo
regolare a tratti.
###### Teorema 3.55.
Sia $f\in\mathbb{M}(R)$ e sia $K$ un compatto non vuoto in $R$ con bordo
regolare, allora esistono uniche una funzione $u\in HBD(R\setminus
K)\cap\mathbb{M}(R)$ e $g\in\mathbb{M}_{\Delta}(R)$, $g=0$ su $K$, tali che:
1. 1.
$f=u+g$
2. 2.
$D_{R}(u,\phi)=0$ per ogni $\phi\in\mathbb{M}_{\Delta}(R)$ e $\phi=0$ su $K$
3. 3.
$D_{R}(f)=D_{R}(u)+D_{R}(g)$
4. 4.
se $v$ è superarmonica su $R\setminus K$ e su questo insieme $v\geq f$, allora
$v\geq u$ su $R\setminus K$
5. 5.
$\left\|u\right\|_{\infty,R\setminus K}\leq\left\|f\right\|_{\infty,R\setminus
K}$
###### Proof.
La dimostrazione è del tutto analoga alla dimostrazione precedente, quindi
lasciamo i dettagli al lettore. ∎
###### Osservazione 3.56.
Nelle ipotesi del teorema precedente, denotiamo la funzione $u$ con
$\pi_{K}(f)$. Osserviamo che se $K\subset C$:
$\displaystyle\pi_{K}(\pi_{C}(f))=\pi_{K}(f)$
###### Proof.
La dimostrazione è una semplice applicazione del teorema precedente. Infatti
sappiamo che:
$\displaystyle f=\pi_{K}(f)+g_{K}$ $\displaystyle f=\pi_{C}(f)+g_{C}$
$\displaystyle\pi_{C}(f)=\pi_{K}(f)+g_{K}-g_{C}\equiv\pi_{K}(\pi_{C}(f))+g$
dato che $g|_{K}\equiv(g_{K}-g_{C})|_{K}=0$ e che
$g\in\mathbb{M}_{\Delta}(R)$, per unicità della decomposizione abbiamo la
tesi. ∎
#### 3.3.5 Principio del massimo
Ora siamo pronti a presentare questa versione del principio del massimo.
###### Teorema 3.57 (Principio del massimo).
Data una funzione $u\in HBD(R)$, si ha che:
$\displaystyle\min_{p\in\Delta}u(p)=\inf_{p\in R^{*}}u(p)\ \ \
\max_{p\in\Delta}u(p)=\sup_{p\in R^{*}}u(p)$
In particolare se $\Delta=\emptyset$, allora $u\in HBD(R)\Rightarrow u=cost.$
###### Proof.
Se $\Delta=\emptyset$ allora $\mathbb{M}(R)=\mathbb{M}_{\Delta}(R)$ (vedi
osservazione 3.52), e si può dimostrare che $u=cost$ con un ragionamento del
tutto simile a quello utilizzato nel teorema 3.53 per dimostrare che $v=cost$.
Supponiamo quindi che $\Delta\neq\emptyset$. Per dimostrare la tesi
dimostriamo che $u(\Delta)\leq 0$ implica $u(R)\leq 0$. Grazie alla densità di
$R$ in $R^{*}$, l’ultima affermazione implica che $u(R^{*})\leq 0$, e se
questo è vero, la tesi si ottiene considerando la funzione
$u-\leavevmode\nobreak\ \max_{p\in\Delta}u(p)$, oppure la funzione
$-u+\min_{p\in\Delta}u(p)$.
Supponiamo quindi che $u(\Delta)\leq 0$. Fissato $\epsilon>0$, definiamo
l’insieme
$\displaystyle A=\\{p\in R^{*}\ t.c.\ u(p)\geq\epsilon\\}$
Dato che ovviamente $A\cap\Delta=\emptyset$, per ogni $p$ 353535e per
definizione di $\Delta$ esiste una funzione $f_{p}\in\mathbb{M}_{\Delta}(R)$
tale che $f_{p}(p)\geq 2$. Visto che l’algebra di Royden è chiusa rispetto
all’operazione di massimo, possiamo fare in modo che $f_{p}\geq 0$ su $R^{*}$
363636basta considerare $\max\\{f_{p},0\\}$. Allora il ricoprimento
$\displaystyle U_{p}=\\{q\in R^{*}\ t.c.\ f_{p}(q)>1\\}$
è un ricoprimento aperto di $A$, e quindi per compattezza esiste un numero
finito di punti $p_{1},\cdots,p_{n}$ tali che $U_{p_{i}}$ ricopre $A$. Questo
significa che la funzione
$\displaystyle f(q)\equiv\sum_{i=1}^{n}f_{p_{n}}(q)$
è una funzione $f\in\mathbb{M}_{\Delta}(R)$ tale che $f|_{A}\geq 1$. Visto che
per ipotesi la funzione $u$ è limitata da sopra, esiste un numero $M$ tale che
$u-Mf\leq 0$ sull’insieme $A$, e quindi per definizione di $A$,
$u-Mf-\epsilon\leq 0$ su tutta $R^{*}$.
Grazie al teorema di decomposizione appena dimostrato possiamo concludere che
esistono (uniche visto che $\Delta\neq\emptyset$) due funzioni $v\in HBD(R)$ e
$h\in\mathbb{M}_{\Delta}(R)$ tali che
$\displaystyle u-Mf-\epsilon=v+h$
e per la proposizione 3.54 $v\leq 0$ su $R^{*}$. Per unicità, $v=u-\epsilon$,
e per arbitrarietà di $\epsilon$, $u\leq 0$ su $R^{*}$. ∎
La dimostrazione di questo teorema può essere facilmente adattata per
ottenere:
###### Teorema 3.58.
Data una funzione $u\in HBD(R\setminus K)$ con $K$ compatto in $R$, si ha che:
$\displaystyle\min_{p\in(\Delta\cup\partial K)}u(p)=\inf_{p\in R^{*}\setminus
K}u(p)\ \ \ \max_{p\in(\Delta\cup\partial K)}u(p)=\sup_{p\in R^{*}\setminus
K}u(p)$
Come corollario a questo teorema possiamo facilmente dimostrare che
###### Proposizione 3.59.
Data una funzione subarmonica $u\in\mathbb{M}(R)$ limitata, allora:
$\displaystyle\max_{p\in\Delta}u(p)=\sup_{p\in R^{*}}u(p)$
Data una funzione superarmonica $v\in\mathbb{M}(R)$:
$\displaystyle\min_{p\in\Delta}v(p)=\inf_{p\in R^{*}}v(p)$
###### Proof.
Dimostriamo solo il primo caso. La dimostrazione è identica al caso armonico,
ma una volta trovata la decomposizione di $u-Mf-\epsilon=v+h$, non possiamo
concludere che $v=u-\epsilon$, ma grazie alla proposizione 3.54, possiamo
concludere che $u\leq v$. ∎
Un altro corollario di questo principio è il seguente:
###### Proposizione 3.60.
Se $u\in H(R\setminus K)$ con $K$ insieme compatto 373737possibilmente vuoto è
una funzione limitata e $u\in\mathbb{M}_{\Delta}(R)$, allora
$u=BD-\lim_{n}u_{n}$, dove $u_{n}$ sono le funzioni definite da:
$\displaystyle u_{n}(x)=\begin{cases}u(x)&se\ x\in K\\\ 0&se\ x\in
K_{n}^{C}\\\ u_{n}\in H(K_{n}\setminus K)\end{cases}$
dove $K_{n}$ è un’esaustione regolare di $R$ con $K\subset K_{1}$.
Questo implica che $\forall x\in R\setminus K$:
$\displaystyle\min\\{0,\min u|_{\partial K}\\}\leq u(x)\leq\max\\{0,\max
u|_{\partial K}\\}$
###### Proof.
Grazie alle considerazioni fatte in precedenza, sappiamo che esiste il limite
$\displaystyle v=BD-\lim_{n}u_{n}$
la funzione $u-v$ è una funzione armonica su $R\setminus K$ con $(u-v)|_{K}=0$
e $(u-v)|_{\Delta}=0$, quindi grazie al principio del massimo appena
dimostrato, $u=v$.
L’ultima considerazione segue dal principio del massimo applicato alle
funzioni $u_{n}$ sull’insieme $K_{n}\setminus K$. Per le funzioni $u_{n}$
infatti massimo e minimo sono assunti su $\partial(K_{n}\setminus K)=\partial
K\cup\partial K_{n}$. Passando al limite si ottiene la tesi. ∎
La costruzione della compattificazione di Royden trova una buona
giustificazione nel principio appena dimostrato. Su insiemi compatti con
bordo, uno strumento fondamentale per lo studio delle funzioni armoniche è il
principio del massimo che garantisce che una funzione armonica assume il suo
massimo in un punto del bordo. Questo principio non è (ovviamente) applicabile
se studiamo funzioni armoniche su varietà non compatte senza bordo. La
compattificazione di Royden è il tentativo di “rendere compatta” una varietà
in modo da poter applicare ancora i principi del massimo. Come ci si può
aspettare, un insieme compatto con bordo e la compattificazione di Royden non
si comportano esattamente nello stesso modo. Nelle ipotesi di quest’ultimo
principio infatti compare l’ipotesi che $u\in HBD(R)$, cioè che l’integrale di
Dirichlet della funzione sia finito, però questo principio dice anche dove
esattamente cercare il massimo delle funzioni armoniche. Non in un punto
qualsiasi del bordo come nel caso di insiemi compatti, ma è sufficiente
controllare il comportamento di $f$ sul bordo armonico della varietà.
Concludiamo questo capitolo con una caratterizzazione di
$\mathbb{M}_{\Delta}(R)$. Dalla definizione 3.50, il bordo armonico è
l’insieme dove tutte le funzioni in $\mathbb{M}_{\Delta}(R)$ sono nulle. Vale
anche una sorta di viceversa, nel senso che:
###### Proposizione 3.61.
$\displaystyle\mathbb{M}_{\Delta}(R)=\\{f\in\mathbb{M}(R)\ \ t.c.\
f(\Delta)=0\\}$
###### Proof.
La dimostrazione segue facilmente dalla decomposizione descritta nella
proposizione 3.53 e dal principio del massimo 3.57.
Se $\Delta=\emptyset$, $\mathbb{M}_{\Delta}(R)=\mathbb{M}(R)$ come
dall’osservazione 3.52. Supponiamo quindi che $\Delta\neq\ \emptyset$. Se
$f\in\mathbb{M}_{\Delta}(R)$, è ovvio dalla definizione che $f(\Delta)=0$.
Supponiamo quindi che $f(\Delta)=0$. Allora, grazie al teorema 3.53, esistono
due funzioni $u\in HBD(R)$ e $h\in\mathbb{M}_{\Delta}(R)$ tali che
$\displaystyle f=u+h$
Essendo $h(\Delta)=f(\Delta)=0$, si ha che $u(\Delta)=0$, quindi per il
principio del massimo 3.57, $u$ è identicamente nulla, da cui la tesi. ∎
## Chapter 4 Varietà paraboliche e iperboliche
In questo capitolo tratteremo proprietà e caratterizzazioni di varietà
paraboliche e iperboliche. Il risultato principale è l’esistenza dei
potenziali di Evans sulle varietà paraboliche. In tutto il capitolo $R$ sarà
una varietà riemanniana sensa bordo di dimesione $m$.
### 4.1 Capacità
In questa sezione introduciamo la capacità di una coppia di insiemi e
definiamo di conseguenza le varietà paraboliche. I risultati principali sono
tratti da [G2] e [PSR] capitolo 7.
###### Definizione 4.1.
Dati due insiemi $K$ compatto e $K\subset\Omega$ aperto paracompatto,
definiamo la capacità di $K$ rispetto a $\Omega$:
$\displaystyle\text{Cap}(K,\Omega)=\inf{\int_{R}(\nabla f)^{2}dV}$
dove l’inf è preso su tutte le funzioni di Tonelli 111in realtà l’insieme di
funzioni considerate può essere più generale senza cambiare il valore della
capacità, vedi [G2] tali che
$\displaystyle f(K)=1\ \ supp(f)\subset\overline{\Omega}\ \ 0\leq f\leq 1$
Data questa definizione, osserviamo subito che la capacità di una coppia di
insiemi aumenta se rimpiccioliamo $\Omega$ e se ingrandiamo $K$, cioè se
$K\subset K^{\prime}$ e $\Omega\subset\Omega^{\prime}$:
$\displaystyle\text{Cap}(K,\Omega^{\prime})\leq\text{Cap}(K,\Omega)\leq\text{Cap}(K^{\prime},\Omega)\
;\
\text{Cap}(K,\Omega^{\prime})\leq\text{Cap}(K^{\prime},\Omega^{\prime})\leq\text{Cap}(K^{\prime},\Omega^{\prime})$
Visti gli scopi della tesi, d’ora in avanti considereremo solo insiemi $K$ e
$\Omega$ con bordo regolare.
La prima domanda data questa definizione è se l’inf è raggiunto, e in tale
caso da quale funzione. La risposta dipende dalla regolarità dei bordi di $K$
e $\Omega$, in particolare se entrambi i bordi sono lisci grazie al principio
di Dirichlet possiamo dimostrare che:
###### Proposizione 4.2.
Il valore della capacità di una coppia di insiemi $K\Subset\Omega$ con bordi
lisci è equivalente all’integrale di Dirichlet della funzione soluzione del
problema
$\displaystyle\ \begin{cases}\Delta u=0\ \ in\ \Omega\setminus K\\\
u|_{K}=1\\\ u|_{\Omega^{C}}=0\end{cases}$ (4.1)
Chiamiamo la funzione $u$ potenziale di capacità della coppia $K$, $\Omega$.
###### Proof.
Osserviamo che se i bordi degli insiemi sono lisci, è possibile risolvere il
problema di Dirichlet che definisce $u$.
La dimostrazione è una diretta conseguenza del principio di Dirichlet
riportato nella proposizione 3.31. ∎
Nella seguente proposizione osserviamo che se è possibile risolvere il
problema di Dirichlet 4.1, anche se $\Omega$ e $K$ non hanno bordo liscio
$\text{Cap}(K,\Omega)=D_{\Omega}(u)$.
###### Proposizione 4.3.
Siano $K$ e $\Omega$ tali che $\Omega\setminus K$ sia regolare per il problema
di Dirichlet 222cioè tale che il problema di Dirichlet 4.1 abbia soluzione,
vedi sezione 1.9 per condizioni sulla regolarità degli insiemi. Se esiste una
funzione di Tonelli $f$ tale che $f|_{K}=1$, $f|_{\Omega^{C}}=0$ e
$D_{R}(f)<\infty$, allora $\text{Cap}(K,\Omega)=D_{R}(u)<\infty$.
###### Proof.
La dimostrazione è una diretta conseguenza del principio di Dirichlet
riportato nella proposizione 3.32. ∎
###### Proposizione 4.4.
Nella definizione di capacità, se $\Omega$ e $K$ hanno bordi regolari, allora
possiamo sostituire l’insieme delle funzioni di Tonelli con l’insieme delle
funzioni $f$ lisce in $\Omega\setminus K$ uguali a $1$ su $K$ e nulle su
$\Omega^{C}$.
###### Proof.
Questo risultato segue dalla densità delle funzioni lisce in $\Omega\setminus
K$ e uguali a $1$ su $K$ e nulle in $\Omega^{C}$ illustrato nella proposizione
3.28. ∎
Possiamo ulteriormente caratterizzare la capacità di una coppia di insiemi
grazie alle formule di Green.
###### Proposizione 4.5.
Sia $u$ il potenziale di capacità di $(K,\Omega)$. Allora si ha che:
$\displaystyle\text{Cap}(K,\Omega)=\int_{R}\left|\nabla
u\right|^{2}dV=-\int_{\partial K}\ast du=-\int_{\partial\Omega}\ast du$
###### Proof.
La dimostrazione della prima uguaglianza è una diretta conseguenza della
formula di Green 3.29. La seconda segue dalla considerazione che $u$ è
armonica in $\Omega\setminus K$, quindi grazie alla proposizione 1.11
$\displaystyle 0=\int_{\Omega\setminus K}\Delta u\
dV=\int_{\partial(\Omega\setminus K)}\ast du=\int_{\partial\Omega}\ast
du+\int_{\partial K}\ast du$
∎
Oltre alla capacità di una coppia di insiemi, è possibile definire la capacità
di un insieme compatto come:
###### Definizione 4.6.
Dato un compatto $K\Subset R$ regolare per il problema di Dirichlet, definiamo
$\displaystyle\text{Cap}(K)=\lim_{n}\text{Cap}(K,E_{n})$
dove $E_{n}$ è una qualsiasi esaustione regolare di $R$ 333per l’esistenza di
queste esaustioni, vedi 1.20. Questa definizione è ben posta, nel senso che
non dipende dalla scelta dell’esaustione.
###### Proof.
Date due esaustioni regolari $E_{n}$ e $C_{n}$, definiamo
$\displaystyle e_{n}\equiv\text{Cap}(K,E_{n})\to e\ \
c_{n}\equiv\text{Cap}(K,C_{n})\to c$
le due successioni sono monotone decrescenti grazie alle considerazioni fatte
prima. Inoltre per la compattezza di ogni $E_{n}$, esiste un intero $k$ tale
che $E_{n}\Subset C_{k}$. Questo implica che $e_{n}\geq c_{k}\geq c$. Quindi
anche $e\geq c$. Scambiando i ruoli di $E_{n}$ e $C_{n}$ si ottiene $e=c$. ∎
Possiamo definire un potenziale di capacità anche per un singolo insieme
compatto.
###### Proposizione 4.7.
Dato un insieme aperto relativamente compatto regolare per il problema di
Dirichlet $K$ e una qualsiasi esaustione regolare $K_{n}$ con $K\subset
K_{1}^{\circ}$ di $R$, sia $u_{n}$ il potenziale di capacità di $(K,K_{n})$,
allora la funzione
$\displaystyle u=\lim_{n}u_{n}$
è il potenziale di capacità per $K$, nel senso che $u$ è una funzione armonica
su $R\setminus K$, $u=1$ su $K$ tale che
$\displaystyle\text{Cap}(K)=D_{R}(u)$
###### Proof.
La successione $u_{n}$ è una successione di funzioni armoniche su $K\setminus
K_{n}$ strettamente positive su $K_{n}^{\circ}$ grazie alla proposizione 1.54.
Dato che tutte queste funzioni sono uniformemente limitate da $1$, grazie alle
proposizioni 1.57 e 1.58 $u_{n}$ converge localmente uniformemente a una
funzione $u$ armonica in $R\setminus K$. Sempre grazie alla proposizione 1.58
si ha che
$\displaystyle\lim_{n}D_{R}(u_{n})=D_{R}(u)$
come volevasi dimostrare. ∎
Osserviamo che possiamo estendere la nozione di capacità e di potenziale di
capacità anche a insiemi più generali di quelli utilizzati fino ad ora. In
particolare, dato $C\subset R$ compatto, se è possibile risolvere il problema
di Dirichlet
$\displaystyle u|_{K}=1\ \ \ u|_{\partial E_{n}}=0\ \ \ u\in H(E_{n}\setminus
K)$
dove $E_{n}$ è un’esaustione regolare di $R$ con $K\subset E_{1}$, allora ha
senso parlare di capacità di $K$ e di potenziale di capacità di $K$. In
particolare:
###### Osservazione 4.8.
Se $K$ è una sottovarietà regolare compatta di $R$ di codimensione $1$
possibilmente con bordo, allora ha senso definire la sua capacità e il suo
potenziale di capacità.
###### Proof.
Questo risultato segue dalle considerazioni della sezione 1.9.3. ∎
Prima di proseguire, riportiamo un’applicazione di quanto appena descritto. In
particolare dimostriamo che in ogni insieme aperto, esiste sempre una coppia
$(K,\Omega)$ contenuta in questo insieme con capacità grande a piacere e
piccola a piacere (quindi una funzione armonica con integrale di Dirichlet
grande a piacere e piccolo a piacere).
###### Proposizione 4.9.
Dato un qualsiasi aperto $A\subset R$, esistono $K\Subset\Omega\subset A$ tali
che $\text{Cap}(K,\Omega)$ è grande a piacere.
###### Proof.
La dimostrazione di questa proposizione è costruttiva. Consideriamo un punto
$p\in A$, allora esiste un intorno normale $B(p)\subset A$, intorno che
possiamo dotare delle coordinate geodetiche $(r,\theta)$ (fuori da $p$)
444vedi sezione 1.1.3. Consideriamo
$\displaystyle K=\\{p\in B\ t.c.\ r(p)\leq R_{1}\\}\ \ \Omega=\\{p\in B\ t.c.\
r(p)<R_{2}\\}$
dove $R_{1}<R_{2}$ e $R_{2}$ è tale che $\Omega\subset B(p)$. Consideriamo una
qualsiasi funzione di Tonelli $0\leq f\leq 1$ tale che $f(K)=1$ e $f=0$ fuori
da $\Omega$. Per comodità di notazione, utiliziamo lo stesso simbolo $f$ con
la sua rappresentazione in coordinate geodetiche. Abbiamo che
$\displaystyle\int_{R}\left|\nabla f\right|^{2}\ dV=\int_{\Omega\setminus
K}\left|\nabla f\right|^{2}\ dV=\int_{\Omega\setminus K}g^{ij}\frac{\partial
f}{\partial x^{i}}\frac{\partial f}{\partial x^{j}}\
\sqrt{\left|g\right|}drd\theta^{1}\cdots d\theta^{m-1}$
dove $r=x^{1}$ e $\theta$ rappresenta tutte le altre coordinate. Grazie alla
particolare forma che la metrica assume in coordinate polari geodetiche (vedi
1.3) si ha che:
$\displaystyle\int_{\Omega\setminus K}\left|\nabla f\right|^{2}\
dV\geq\int_{\Omega\setminus K}\left(\frac{\partial f}{\partial r}\right)^{2}\
\sqrt{\left|g\right|}drd\theta^{1}\cdots d\theta^{m-1}\geq$ $\displaystyle\geq
c\omega_{m-1}\int_{R_{1}}^{R_{2}}\left(\frac{\partial f}{\partial
r}\right)^{2}dr$
dove $c$ è un limite inferiore positivo per $\sqrt{\left|g\right|}$ su
$B(p)\setminus K$. Grazie alla disuguaglianza di Schwartz, otteniamo che:
$\displaystyle\frac{1}{R_{2}-R_{1}}=\frac{1}{R_{2}-R_{1}}\left(\int_{R_{1}}^{R_{2}}\frac{\partial
f}{\partial r}dr\right)^{2}\leq\int_{R_{1}}^{R_{2}}\left(\frac{\partial
f}{\partial r}\right)^{2}dr$
il che dimostra che a patto di scegliere $R_{2}-R_{1}$ sufficientemente
piccolo, la capacità dell’anello è abbastanza grande. Osserviamo che in questa
dimostrazione è essenziale trovare il limite inferiore per
$\sqrt{\left|g\right|}$ 555che ad esempio nel caso di $\mathbb{R}^{m}$ vale
$\omega_{m-1}r^{m-1}$, dove $\omega_{m-1}$ è l’area della sfera $m-1$
dimensionale rispetto alla metrica euclidea standard, quindi è essenziale che
$R_{1}$ non tenda a $0$ quando scegliamo $R_{2}-R_{1}$ piccolo. In pratica per
ottenere la tesi, fissato $R_{1}$ e trovata la costante $c$, scegliamo $R_{2}$
in modo che $R_{2}-R_{1}$ sia sufficientemente piccolo. ∎
###### Proposizione 4.10.
Dato un qualsiasi aperto $A\subset R$, esistono $K\Subset\Omega\subset A$ tali
che $\text{Cap}(K,\Omega)$ è piccola a piacere.
###### Proof.
Procediamo in maniera del tutto analoga a sopra, solo che in questo caso al
posto di considerare una funzione di Tonelli qualsiasi, ne consideriamo una
particolare. Infatti se vogliamo dimostrare che la capacità di un’insieme è
piccola, basta trovare una funzione che renda piccolo l’integrale di
Dirichlet.
Sia come sopra $B(p)$ un intorno normale contenuto in $A$, e siano
$\displaystyle K=\\{p\in B\ t.c.\ r(p)\leq R_{1}\\}\ \ \Omega=\\{p\in B\ t.c.\
r(p)<R_{2}\\}$
dove $R_{1}<R_{2}$ e $R_{2}$ è tale che $\Omega\subset B(p)$. Consideriamo la
funzione
$\displaystyle f(p)=\begin{cases}1&se\ p\in K\\\ 0&se\ p\in\Omega^{C}\\\
\frac{\log(r(p)/R_{2})}{\log(R_{1}/R_{2})}&se\ p\in\Omega\setminus
K\end{cases}$
Dato che $f$ dipende solo da $r(p)$, il suo integrale di Dirichlet vale:
$\displaystyle\int_{R}\left|\nabla f\right|^{2}dV=\int_{\Omega\setminus
K}\left(\frac{\partial f}{\partial
r}\right)^{2}\sqrt{\left|g\right|}drd\theta^{1}\cdots d\theta^{m-1}=$
$\displaystyle=\int_{\Omega\setminus
K}\frac{1}{\log^{2}(R_{1}/R_{2})}\frac{1}{r^{2}}\sqrt{\left|g\right|}drd\theta^{1}\cdots
d\theta^{m-1}$
Notiamo che l’integrale $\int_{r=k}\sqrt{\left|g\right|}d\theta^{1}\cdots
d\theta^{m-1}$ è la superficie della sfera di raggio $k$, quindi per le
proprietà della metrica, se $k$ è tale che $B_{k}\subset B(p)$, allora esiste
una costante $C$ tale che:
$\displaystyle\int_{r=k}\sqrt{\left|g\right|}d\theta^{1}\cdots
d\theta^{m-1}\leq Ck^{m-1}$
dove $m$ rappresenta la dimensione della varietà. Questa stima permette di
concludere che:
$\displaystyle\int_{R}\left|\nabla
f\right|^{2}dV\leq\frac{C}{\log^{2}(R_{1}/R_{2})}\int_{R_{1}}^{R_{2}}r^{m-3}dr$
Se $m=2$, si ottiene:
$\displaystyle\int_{R}\left|\nabla
f\right|^{2}dV\leq\frac{C}{\log(R_{2}/R_{1})}$
quindi fissato $R_{2}$, è possibile scegliere $R_{1}>0$ in modo che questa
quantità sia piccola a piacere.
In caso $m\geq 3$ invece si ottiene:
$\displaystyle\int_{R}\left|\nabla
f\right|^{2}dV\leq\frac{C}{\log^{2}(R_{1}/R_{2})}(R_{2}^{m-2}-R_{1}^{m-2})$
Anche in questo caso, fissato $R_{2}$ è sempre possibile scegliere $R_{1}$ in
modo che $D_{R}(f)$ sia piccolo a piacere. ∎
La capacità di una coppia di insiemi è legata al comportamento della funzione
di Green su quell’insieme, in particolare si ha che 666proposizione 4.1 pag
154 di [G2]:
###### Proposizione 4.11.
Siano $K,\ \Omega$ insiemi compatti dal bordo liscio in $R$ con
$K\Subset\Omega^{\circ}$, e sia $p\in K$. Allora:
$\displaystyle\min_{x\in\partial
K}G_{\Omega}(p,x)\leq(\text{Cap}(K,\Omega))^{-1}\leq\max_{x\in\partial
K}G_{\Omega}(p,x)$
dove $G_{\Omega}$ indica la funzione di Green relativa al dominio $\Omega$
777vedi proposizione 1.62.
###### Proof.
Siano
$\displaystyle b\equiv\min_{x\in\partial K}G_{\Omega}(p,x)\ \ \
a\equiv\max_{x\in\partial K}G_{\Omega}(p,x)$
e definiamo per ogni $c>0$ l’insieme compatto
$\displaystyle F_{c}\equiv\\{x\in\Omega\ t.c.\ G_{\Omega}(p,x)\geq c\\}$
Per prima cosa osserviamo che $F_{a}\subset K\subset F_{b}$. Questo è
conseguenza del principio del massimo, infatti la funzione
$G_{\Omega}(\cdot,p)$ è armonica su $\Omega\setminus K$, quindi assume il suo
massimo su $\partial K$, cioè $G_{\Omega}(x,p)<a$ se $x\in\Omega\setminus K$.
Inoltre dato che $G_{\Omega}$ è superarmonica su $K^{\circ}$, si ha che
$G_{\Omega}(x,p)>b$ se $x\in K$.
Grazie alla “monotonia” della capacità, si ha che:
$\displaystyle\text{Cap}(F_{a},\Omega)\leq\text{Cap}(K,\Omega)\leq\text{Cap}(F_{b},\Omega)$
la tesi segue dalla considerazione che $\text{Cap}(F_{c},\Omega)=1/c$.
Per il teorema si Sard, quasi ogni $c$ è un valore regolare per
$G_{\Omega}(\cdot,p)$, quindi $F_{c}$ ha bordo regolare. Per questi valori di
$c$, la funzione $G_{\Omega}/c$ (estesa costante uguale a $1$ dentro $F_{c}$ e
uguale a $0$ fuori da $\Omega$) è il potenziale di capacità per la coppia
$(F_{c},\Omega)$, quindi grazie alla proposizione 4.5:
$\displaystyle\text{Cap}(F_{c},\Omega)=-\frac{1}{c}\int_{\partial F_{c}}\ast
dG_{\Omega}(\cdot,p)=\frac{1}{c}$
Se $c$ non è un valore regolare di $G_{\Omega}(\cdot,p)$, allora esiste
$c_{n}\nearrow c$ e $c_{n}^{\prime}\searrow c$ successioni di valori regolari
di $G_{\Omega}(\cdot,p)$. Dalla monotonia della capacità si ottiene che per
ogni $n$:
$\displaystyle\frac{1}{c_{n}}\leq\text{Cap}(F_{c},\Omega)\leq\frac{1}{c_{n}^{\prime}}$
da cui $Cap(F_{c},\Omega)=1/c$. ∎
Grazie alla definizione di capacità appena data possiamo dividere in 2
categorie le varietà Riemanniane, quelle per cui ogni insieme aperto non vuoto
relativamente compatto 888noi considereremo solo compatti con bordo liscio per
comodità ha capacità positiva, e quelle per cui ogni insieme compatto ha
capacità nulla. In questa sezione daremo delle caratterizzazioni equivalenti
di queste proprietà.
###### Definizione 4.12.
Una varietà Riemanniana $R$ si dice parabolica se ogni insieme aperto non
vuoto relativamente compatto con bordo liscio ha capacità nulla, in caso
contrario la varietà si definisce iperbolica
Per prima cosa osserviamo che nella definizione non è necessario chiedere che
ogni insieme aperto relativamente compatto abbia capacità nulla, basta che un
solo aperto relativamente compatto possieda questa proprietà e automaticamente
tutti gli aperti non vuoti relativamente compatti hanno capacità nulla. Questo
implica anche che se un solo aperto relativamente compatto ha capacità
positiva, tuttigli aperti relativamente compatti hanno capacità postiva.
###### Proposizione 4.13.
Sia $R$ una varietà Riemanniana. Se un insieme aperto non vuoto relativamente
compatto con bordo liscio $K\Subset R$ ha capacità nulla, allora tutti gli
aperti non vuoti relativamente compatti in $R$ hanno capacità nulla.
###### Proof.
Sia $K$ un aperto relativamente compatto con bordo liscio di capacità nulla.
Allora tutti gli insiemi compatti contenuti in $K$ hanno capacità nulla grazie
alla “monotonia” della capacità.
Consideriamo un aperto relativamente compatto con bordo liscio $C\supset K$ e
chiamiamo $u$ il potenziale di capacità di $K$ e $u^{\prime}$ il potenziale di
capacità di $C$. Sia $K_{n}$ un’esaustione regolare di $R$ con $C\subset
K_{1}^{\circ}$ e sia $u_{n}$ il potenziale di capacità di $(K,K_{n})$,
$u_{n}^{\prime}$ il potenziale di capacità di $(C,K_{n})$. Grazie al principio
del massimo applicato all’insieme $K_{n}\setminus C$ (vedi 1.50), si ha che
$u_{n}\leq u_{n}^{\prime}$, quindi $u_{n}\leq u^{\prime}$, disuguaglianza
valida per ogni $n$. Passando al limite si ottiene che $u\leq u^{\prime}$. Se
$K$ ha capacità nulla, il suo potenziale di capacità è la funzione costante
uguale a $1$, quindi necessariamente anche $u^{\prime}$ è costante uguale a
$1$, cioè la capacità di $C$ è nulla.
Se $C\cap K\neq\emptyset$, allora considerando una bolla $B\subset C\cap K$,
$B$ ha capacità nulla poiché contenuta in $K$, e quindi $C$ ha capacità nulla
poiché contiene $B$. Infine se $C\cap K=\emptyset$, allora $K\subset C\cup K$,
quindi grazie a quanto appena dimostrato $C\cup K$ ha capacità e anche
$C\subset C\cup K$ ha capacità nulla. ∎
Osserviamo da questa definizione che modificando la metrica di $R$ su un
insieme compatto, la varietà rimane parabolica o non parabolica come la
varietà non modificata.
###### Proposizione 4.14.
Sia $(R,g)$ una varietà riemanniana, e sia $(R,g^{\prime})$ un’altra varietà
con $g=g^{\prime}$ fuori da un compatto $K$. Allora $(R,g)$ è parabolica se e
solo se $(R,g^{\prime})$ lo è.
###### Proof.
Consideriamo un compatto $C$ che contenga $K$ nella sua parte interna. La
capacità di questo compatto nelle due varietà è identica, perché su $C^{C}$ le
due metriche coincidono, da cui la tesi. ∎
### 4.2 Bordo Armonico
La prima caratterizzazione che diamo della parabolicità riguarda il bordo
armonico di $R^{*}$.
###### Proposizione 4.15.
La varietà $R$ è parabolica se e solo se $1\in\mathbb{M}_{\Delta}(R)$.
###### Proof.
Una parte della dimostrazione è semplice. Se la capacità di un insieme
compatto $K$ è nulla, questo implica che il suo potenziale armonico $u$ è
costante uguale a $1$. Dato che $u$ per definizione è il limite locale
uniforme del potenziale armonico $u_{n}$ della coppia $(K,K_{n})$, e dato che
$u_{n}$ è limitata da $1$, si ha che $u=B-\lim_{n}u_{n}$.
La successione $u_{n}$ inoltre è anche di Cauchy rispetto alla seminorma $D$,
infatti se $m>n$:
$\displaystyle
D_{R}(u_{n}-u_{m})=D_{R}(u_{n})-2D_{R}(u_{n};u_{m})+D_{R}(u_{m})$
Poiché la funzione di Tonelli $u_{m}-u_{n}$ è nulla sul bordo di
$K_{m}\setminus K$, grazie alla formula di Green 3.30 osserviamo che
$\displaystyle D_{R}(u_{m}-u_{n};u_{m})=D_{K_{m}\setminus
K}(u_{m}-u_{n};u_{m})=0$
Questo ci permette di concludere che:
$\displaystyle 2D_{R}(u_{m})-2D_{R}(u_{m};u_{n})=0$
da cui
$\displaystyle D_{R}(u_{n}-u_{m})=D_{R}(u_{n})-D_{R}(u_{m})$
poiché la successione $D_{R}(u_{n})$ è decrescente e converge a
$\text{Cap}(K)$, abbiamo dimostrato che $\\{u_{n}\\}$ è $D-$Cauchy. Quindi
$1=BD-\lim_{n}u_{n}$, e poiché $u_{n}\in\mathbb{M}_{0}(R)$,
$1\in\mathbb{M}_{\Delta}(R)$.
Per dimostrare l’implicazione inversa, sia $u$ il potenziale di capacità di un
compatto $K$ e sia $f_{n}$ una successione in $\mathbb{M}_{0}(R)$ tale che:
$\displaystyle 1=BD-\lim_{n}f_{n}$
Chiaramente:
$\displaystyle u=B-\lim_{n}uf_{n}$
Per dimostrare che $u=D-\lim_{n}uf_{n}$ consideriamo un compatto qualsiasi
$C\Subset R$, e osserviamo che:
$\displaystyle
D_{R}(uf_{n}-u)=\int_{R}\left|\nabla(u(f_{n}-1))\right|^{2}dV\leq$
$\displaystyle\leq
2\int_{R}\left|u\right|^{2}\left|\nabla(f_{n}-1)\right|^{2}dV+2\int_{R}\left|\nabla
u\right|^{2}\left|f_{n}-1\right|^{2}dV$
il primo termine della somma converge a $0$ per ipotesi, mentre per il secondo
termine osserviamo che:
$\displaystyle\int_{R}\left|\nabla
u\right|^{2}\left|f_{n}-1\right|^{2}dV=\int_{C}\left|\nabla
u\right|^{2}\left|f_{n}-1\right|^{2}dV+\int_{R\setminus C}\left|\nabla
u\right|^{2}\left|f_{n}-1\right|^{2}dV$
dove $C$ è un compatto qualsiasi. Dato che $u\in\mathbb{M}(R)$ (quindi ha
integrale di Dirichlet finito) e $f_{n}$ converge localmente uniformemente a
$1$, il primo addendo tende a zero indipendentemente dal compatto scelto.
Detto $b=\limsup_{n}\left|f_{n}-1\right|^{2}$ 999sicuramente finito grazie
all’uniforme limitatezza della successione $\\{f_{n}\\}$, abbiamo che per ogni
$C\Subset R$:
$\displaystyle\limsup_{n}D_{R}(uf_{n}-u)\leq 2b\int_{R\setminus C}\left|\nabla
u\right|^{2}dV$
vista l’arbitrarietà di $C$, e dato che $\int_{R}\left|\nabla
u\right|^{2}dV<\infty$, possiamo concludere che
$\displaystyle\lim_{n}D_{R}(uf_{n}-u)=0$
Sia $F_{n}$ un compatto dal bordo regolare che contiene $supp(f_{n})$, allora
grazie alla formula di Green 3.30 applicata all’insieme $A_{n}=F_{n}\setminus
K$ , osserviamo che
$\displaystyle D_{R}((1-u)f_{n},1-u)=D_{A_{n}}((1-u)f_{n},1-u)=0$
poiché la funzione di Tonelli $(1-u)f_{n}$ è nulla sul bordo di $A_{n}$.
Grazie alle ultime due considerazioni possiamo concludere che:
$\displaystyle D_{R}(u)=D_{R}(1-u)\equiv
D_{R}(1-u;1-u)=\lim_{n}D_{R}((1-u)f_{n},1-u)=0$
quindi la capacità di $K$ è nulla, il che dimostra la parabolicità di $R$. ∎
Come corollario di questa proposizione osserviamo che:
###### Proposizione 4.16.
Una varietà $R$ è parabolica se e solo se
1. 1.
$1\in\mathbb{M}_{\Delta}(R)$
2. 2.
$\mathbb{M}_{\Delta}(R)=\mathbb{M}(R)$
3. 3.
$\Delta=\emptyset$
###### Proof.
Il punto (1) è il contenuto della proposizione precedente. Dato che
$\mathbb{M}_{\Delta}(R)$ è un’ideale di $\mathbb{M}(R)$, se
$1\in\mathbb{M}_{\Delta}(R)$ necessariamente
$\mathbb{M}(R)=\mathbb{M}_{\Delta}(R)$ e viceversa. L’equivalenza tra (2) e
(3) è il contenuto dell’osservazione 3.52. ∎
Osserviamo che grazie ad una forma del principio del massimo (la forma
riportata in 3.57), una varietà parabolica non ammette funzioni armoniche
limitate con integrale di Dirichlet finito che non siano costanti. Il
viceversa non è vero, ad esempio $\mathbb{R}^{n}$ con la metrica euclidea
standard non ammette funzioni armoniche limitate non costanti pur non essendo
una varietà parabolica (per $n\geq 3$) 101010vedi teorema 2.1 pag. 31 di
[ABR].
Possiamo migliorare questa osservazione, infatti non è necessario chiedere che
la funzione armonica abbia integrale di Dirichlet finito, e neanche che sia
limitata, basta una delle due condizioni per dimostrare che la funzione è
costante.
###### Proposizione 4.17.
Una varietà Riemanniana $R$ parabolica non ammette funzioni armoniche limitate
non costanti.
###### Proof.
Sia $u$ una funzione subarmonica limitata dall’alto su $R$. A meno di una
traslazione, possiamo supporre $u\geq 0$. Sia $a=\sup_{x\in R}u(x)<\infty$.
Consideriamo un’esaustione regolare $K_{n}$ $n=0,1,\cdots$, e siano
$(1-\omega_{n})$ i potenziali armonici della coppia $(K_{0},K_{n})$, e sia
$b=\max_{x\in K_{0}}u(x)$. Allora grazie al principio del massimo applicato a
$K_{n}\setminus K_{0}$ abbiamo che:
$\displaystyle u(x)\leq b+a\omega_{n}$
per ogni $n$. Passando al limite, visto che $R$ è parabolica, otteniamo che:
$\displaystyle u(x)\leq b+a0=b$
Questo è valido per ogni $K_{0}$ dominio relativamente compatto con bordo
liscio. Se consideriamo l’insieme delle bolle coordinate centrate in un punto
qualsiasi $x_{0}$, otteniamo quindi che $u(x)\leq u(x_{0})$. Ripetendo il
ragionamento con $-u$, otteniamo che $u(x)=u(x_{0})$ per ogni $x\in R$. ∎
###### Proposizione 4.18.
Una varietà parabolica $R$ non ammette funzioni armoniche con integrale di
Dirichlet finito che non siano costanti.
###### Proof.
Sia $u$ una funzione con integrale di Dirichlet $D_{R}(u)<\infty$. Dato che
$R$ è parabolica, $1\in\mathbb{M}_{\Delta}(R)$ 111111vedi 4.15, quindi esiste
una successione di funzioni $\phi_{n}\in\mathbb{M}_{0}(R)$ tale che
$1=BD-\lim_{n}\phi_{n}$. Consideriamo la funzione
$\displaystyle u_{m}(z)\equiv\min\\{\max\\{u(z),-m\\}m\\}$
cioè $u_{m}$ è la funzione $u$ troncata in $[-m,m]$. Grazie alla formula di
Green 3.29 se consideriamo un insieme compatto dal bordo liscio $K$ tale che
$supp(\phi_{n})\subset K$, otteniamo che:
$\displaystyle D_{R}(\phi_{n}u_{m},u)=D_{K}(\phi_{n}u_{m},u)=0$
poiché la funzione $\phi_{n}u_{m}|_{\partial K}=0$. Dato che
$1=BD-\lim_{n}\phi_{n}$, $u_{m}=BD-\lim_{n}\phi_{n}u_{m}$. Infatti:
$\displaystyle
D_{R}(\phi_{n}u_{m}-u_{m})=\int_{R}\left|\nabla{(\phi_{n}-1)u_{m}}\right|^{2}dV=\int_{R}\left|\nabla(\phi_{n}-1)u_{m}+(\phi_{n}-1)\nabla
u_{m}\right|^{2}dV$
dato che $u_{m}$ è limitata e ha integrale di Dirichlet finito, si ottiene
facilmente che il limite per $n\to\infty$ di questa quantità è $0$. Questo
dimostra che $u_{m}=D-\lim_{n}\phi_{n}u_{m}$, il fatto che
$u_{m}=B-\lim_{n}\phi_{n}u_{m}$ è quasi scontato.
Dato che per ogni $n$, $D_{R}(\phi_{n}u_{m},u)=0$, si ha che per ogni $m$:
$\displaystyle D_{R}(u_{m},u)=0$
È facile verificare che $u=CD-\lim_{m}u_{m}$, quindi abbiamo che:
$\displaystyle D_{R}(u)\equiv D_{R}(u,u)=\lim_{m}D_{R}(u_{m},u)=0$
da cui $u$ è costante. ∎
### 4.3 Funzioni di Green
Un’altra caratterizzazione delle varietà paraboliche riguarda l’esistenza di
funzioni di Green definite su tutta la varietà.
###### Proposizione 4.19.
Consideriamo un’esaustione regolare $K_{n}$ di $R$, e siano $G_{n}\equiv
G_{K_{n}}$ le funzioni di Green 121212estese a $0$ fuori da $K_{n}$ rispetto a
questi compatti. La successione di funzioni
$\displaystyle G_{n}(\cdot,p)$
con $p\in K_{1}$ fissato converge a una funzione armonica positiva
$G(\cdot,p)$ su $R\setminus\\{p\\}$ se e solo se la varietà $R$ non è
parabolica.
###### Proof.
Per $m>n$, sia
$\displaystyle\delta_{n}^{m}(p)\equiv G_{m}(\cdot,p)-G_{n}(\cdot,p)$
dove $\delta_{n}^{m}$ è definita sull’insieme $K_{n}$. Grazie al fatto che per
$d(x,p)\to 0$ le due funzioni $G_{n}$ e $G_{m}$ hanno lo stesso comportamento
asintotico e grazie alla proposizione 1.65 $\delta_{n}^{m}$ è una funzione
armonica su tutto $K_{n}$ per ogni $m$.
Osserviamo che per ogni $n$, $\delta_{n}^{n+1}$ è una funzione strettamente
positiva su $K_{n}$, infatti su $\partial K_{n}$ $G_{n}(\cdot,p)=0$, mentre
per il principio del massimo $G_{n+1}(\cdot,p)>0$, quindi
$\delta_{n}^{n+1}|_{\partial K_{n}}>0$, e sempre per il principio del massimo
$\delta_{n}^{n+1}>0$.
Dato che
$\displaystyle\delta_{n}^{m}=\sum_{i=n+1}^{m}\delta_{n}^{i}$
otteniamo che al variare di $m$, $\delta_{n}^{m}$ è una successione di
funzioni armoniche positive crescenti. Per il principio di Harnack (vedi
1.57), la successione $\delta_{n}^{m}$ al variare di $m$ converge localmente
uniformemente (in $K_{n}$) o diverge localmente uniformemente. È evidente che
il comportamento delle successioni $\delta_{n}^{m}$ è indipendente dal
parametro $n$, infatti sull’intersezione dei vari insiemi di definizione vale
che
$\displaystyle\delta_{n}^{m}=\delta_{k}^{m}-\delta_{n}^{k}$
quindi se e solo se al variare di $m$ $\delta_{n}^{m}$ è limitata, anche
$\delta_{k}^{m}$ lo è.
Consideriamo $K\subset K_{1}$ un compatto contenente $p$ come punto interno,
allora dalla proposizione 4.11, sappiamo che per ogni $m$:
$\displaystyle\min_{x\in\partial
K}G_{m}(x,p)\leq\text{Cap}(K,K_{m})^{-1}\leq\max_{x\in\partial K}G_{m}(x,p)$
facendo tendere $m$ a infinito, osserviamo che se
$\text{Cap}(K)=\lim_{m}\text{Cap}(K,K_{m})>0$, allora necessariamente
$G_{m}|_{\partial K}(\cdot,p)$ è limitata, altrimenti tende a infinito. Dato
che al variare di $m$ $\delta_{n}^{m}$ è limitata se e solo se
$G_{m}(\cdot,p)|_{\partial K}$ lo è, allora indipendentemente da $n$,
$\delta_{n}^{m}$ converge su $K_{n}$ a una funzione armonica se e solo se $R$
è non parabolica.
Quindi la successione $G_{m}(\cdot,p)$, che sul compatto $K_{n}$ è uguale a
$G_{n}(\cdot,p)+\delta_{n}^{m}(\cdot)$ 131313se $m>n$, converge localmente
uniformemente in $R$ 141414anche se le funzioni $G_{m}$ non sono definite in
$p$, la loro differenza può essere estesa in $p$, e in questo senso diciamo
che la convergenza è uniforme anche su $p$ a una funzione armonica se e solo
se $\delta_{n}^{m}$ converge, quindi se e solo se $R$ non è parabolica. ∎
Da questa dimostrazione deduciamo che se $\Omega\Subset\Omega^{\prime}$,
allora $G_{\Omega}(\cdot,p)\leq G_{\Omega^{\prime}}(\cdot,p)$, quindi che la
funzione $G(\cdot,p)$ ottenuta come limite di $G_{n}$ è indipendente dalla
scelta dell’esaustione $K_{n}$.
###### Proposizione 4.20.
La funzione $G(\cdot,p)=\lim_{n}G_{n}(\cdot,p)$ è indipendente dalla scelta
dell’esaustione $K_{n}$ utilizzata per definire $G_{n}$.
###### Proof.
Siano $K_{n}$ e $K_{m}^{\prime}$ due esaustioni regolari di $R$ con $p\in
K_{1}\cap K_{1}^{\prime}$, e siano $G_{n}(\cdot,p)$ e
$G_{m}^{\prime}(\cdot,p)$ i relativi nuclei di Green, dove
$\displaystyle G(\cdot,p)\equiv\lim_{n}G_{n}(\cdot,p)\ \ \
G^{\prime}(\cdot,p)\equiv\lim_{m}G_{m}^{\prime}(\cdot,p)$
Per ogni $n$, esiste $\bar{m}$ tale che $K_{n}\Subset K^{\prime}_{\bar{m}}$,
allora grazie all’osservazione precedente:
$\displaystyle G_{n}(\cdot,p)\leq G_{\bar{m}}^{\prime}(\cdot,p)\leq
G^{\prime}(\cdot,p)$
quindi passando al limite su $n$, $G(\cdot,p)\leq G^{\prime}(\cdot,p)$, e
poiché i ruoli di $G_{n}$ e $G_{m}^{\prime}$ sono simmetrici, vale anche il
viceversa, quindi
$\displaystyle G(\cdot,p)=G^{\prime}(\cdot,p)$
∎
Osserviamo che anche per le funzioni di Green definite su tutta $R$ vale una
proprietà analoga a 1.63:
###### Proposizione 4.21.
Sia $G(\cdot,p)$ la funzione di Green ottenuta con il metodo di esaustione,
allora se $K$ e $K^{\prime}$ sono domini relativamente compatti con $p\in
K\Subset K^{\prime}$, vale che:
$\displaystyle
G|_{K^{\prime}\setminus\overline{K}}(\cdot,p)\leq\max_{x\in\partial K}G(x,p)\
\ \ \ \ \ G(\cdot,p)|_{R\setminus\overline{K}}\leq\max_{x\in\partial K}G(x,p)$
###### Proof.
Sia $K_{n}$ un’esaustione regolare di $R$ con $p\in K_{1}$ e siano
$G_{n}(\cdot,p)$ le relative funzioni di Green. Se $n$ è abbastanza grande per
cui $K^{\prime}\Subset K_{n}$, allora vale che:
$\displaystyle
G_{n}|_{K^{\prime}\setminus\overline{K}}(\cdot,p)<\max_{x\in\partial
K}G_{n}(x,p)$
poiché questa relazione vale definitivamente, passando al limite su $n$
otteniamo che:
$\displaystyle
G|_{K^{\prime}\setminus\overline{K}}(\cdot,p)\leq\max_{x\in\partial K}G(x,p)$
Data l’indipendenza da $K^{\prime}$ di questa proprietà, possiamo concludere
che:
$\displaystyle G(\cdot,p)|_{R\setminus\overline{K}}\leq\max_{x\in\partial
K}G(x,p)$
come volevasi dimostrare. ∎
Grazie alle tecniche usate, possiamo facilmente dimostrare anche alcune
proprietà della funzione $G(x,p)=\lim_{n}G_{n}(x,p)$ che corrispondono alle
proprietà delle funzioni $G_{n}$.
###### Proposizione 4.22.
Per la funzione $G(x,p)=\lim_{n}G_{n}(x,p)$ vale che:
1. 1.
per ogni dominio relativamente compatto con bordo liscio $\Omega$ contenente
$p$, vale che $G(\cdot,p)-G_{\Omega}(\cdot,p)$ è una funzione estendibile a
una funzione armonica su tutto $\Omega$.
2. 2.
$G$ è strettamente positiva su $\Omega$
3. 3.
$G$ è simmetrica, cioè $G(p,q)=G(q,p)$
4. 4.
Fissato $q\in\Omega$, la funzione $G(q,p)$ è armonica rispetto a $p$
sull’insieme $\Omega\setminus\\{q\\}$ e superarmonica su tutto $\Omega$.
5. 5.
$G$ è soluzione fondamentale dell’operatore $\Delta$, cioé per ogni funzione
liscia $f$ a supporto compatto in $R$:
$\displaystyle\Delta_{x}\int_{R}G(x,y)f(y)dy=\int_{R}G(x,y)\Delta_{y}(f)(y)dy=-f(x)$
questo significa che nel senso delle distribuzioni
$\Delta_{y}G(x,y)=-\delta_{x}$
6. 6.
Il flusso di $G_{\Omega}(\ast,p)$ attraverso il bordo di un’insieme regolare
$K\Subset\Omega$ con $p\not\in\partial K$ vale:
$\displaystyle\int_{\partial K}\ast dG(\cdot,p)=\begin{cases}-1&se\ p\in K\\\
0&se\ p\not\in K\end{cases}$
7. 7.
La funzione $G$ ha un comportamento asintotico della forma:
$\displaystyle G(x,y)\sim C(m)\begin{cases}-log(d(x,y))&m=2\\\
d(x,y)^{m-2}&m\geq 3\end{cases}$
quando $d(x,y)\to 0$. La costante $C(m)$ dipende solo dalla dimensione della
varietà e può essere determinata sfruttando la condizione (5).
###### Proof.
In tutta la dimostrazione supponiamo che $G$ esista, cioè che $R$ non sia
parabolica.
Il punto (1) è il punto chiave per dimostrare tutte le altre proprietà. Dalla
costruzione nella proposizione precedente sappiamo che per ogni $n$
sull’insieme $K_{n}$ la funzione
$G(\cdot,p)-G_{n}(\cdot,p)=\lim_{m}\delta_{n}^{m}\equiv\delta_{n}$
è una funzione armonica su $K_{n}$. Consideriamo una funzione di Green
$G_{\Omega}(\cdot,p)$ con $\Omega$ qualsiasi, e sia $\Omega\Subset K_{n}$.
Dalla proposizione 1.65 sappiamo che $G_{n}(\cdot,p)-G_{\Omega}(\cdot,p)$ è
estendibile a una funzione armonica $\phi$ definita su tutto $\Omega$. Quindi:
$\displaystyle
G(\cdot,p)-G_{\Omega}(\cdot,p)=G(\cdot,p)-G_{n}(\cdot,p)+G_{n}(\cdot,p)-G_{\Omega}(\cdot,p)=\delta_{n}(\cdot)-\phi(\cdot)$
è una funzione armonica su $\Omega$.
Questo implica in particolare che fissato $q\in\Omega$, $G(q,p)$ si può
scrivere come la somma tra la funzione $G_{\Omega}(q,p)$ e una funzione
armonica, il che dimostra anche il punto (4).
La positività e la simmetria di $G$ sono una ovvia conseguenza del fatto che
queste proprietà valgono anche per tutte le $G_{n}$.
Per dimostrare il punto (5), consideriamo una funzione $f\in
C^{\infty}_{C}(R)$. Sia $\Omega$ tale che $supp(f)\subset\Omega$, allora
grazie al punto (1) sappiamo che $G(x,y)=G_{\Omega}(x,y)+\phi(x,y)$ con
$\phi(\cdot,y)$ armonica in $\Omega$ per $y$ fissato 151515e viceversa per
simmetria, quindi:
$\displaystyle\Delta_{x}\int_{R}G(x,y)f(y)dy=\Delta_{x}\int_{\Omega}G_{\Omega}(x,y)f(y)dy+\Delta_{x}\int_{\Omega}\phi(x,y)f(y)dy=-f(x)$
Grazie al fatto che $\Delta_{x}\phi(x,y)=0$ e al teorema di derivazione sotto
al segno d’integrale.
Allo stesso modo:
$\displaystyle\int_{R}G(x,y)\Delta_{y}f(y)dy=\int_{\Omega}G_{\Omega}(x,y)\Delta_{y}f(y)dy+\int_{\Omega}\phi(x,y)\Delta_{y}f(y)dy$
grazie a un’integrazione per parti su $\Omega$ (il termine al bordo è nullo
grazie al fatto che $supp(f)\Subset\Omega$):
$\displaystyle\int_{\Omega}\phi(x,y)\Delta_{y}f(y)dy=-\int_{\Omega}\Delta_{y}\phi(x,y)\Delta_{y}f(y)dy=0$
otteniamo che:
$\displaystyle\int_{R}G(x,y)\Delta_{y}f(y)dy=\int_{\Omega}G_{\Omega}(x,y)\Delta_{y}f(y)dy=-f(x)$
In maniera del tutto analoga, si dimostra il punto (6), basta considerare il
fatto che $\ast dG=\ast dG_{\Omega}+\ast d\phi$ e il flusso di una funzione
armonica attraverso il bordo di un compatto regolare è nullo grazie alla
formula di green 1.11.
Il punto (7) è conseguenza del fatto che la funzione armonica $\phi$ è
limitata. ∎
Osserviamo che anche nel caso di varietà paraboliche è possibile definire
delle funzioni di Green, solo che in questo caso le funzioni non sono positive
161616e neanche limitate dal basso. Dall’articolo [LT] riportiamo il teorema:
###### Teorema 4.23.
Sia $R$ una varietà riemanniana completa non compatta e senza bordo. Allora
esiste un nucleo di Green simmetrico $G(x,y)$ liscio sull’insieme
$R\times\leavevmode\nobreak\ R\setminus\leavevmode\nobreak\ D$ dove
$D=\\{(x,x)\ t.c.\ x\in R\\}$ è la diagonale di $R\times R$. In particolare
$G$ soddisfa:
$\displaystyle\Delta_{x}\int_{R}G(x,y)f(y)dy=\int_{R}G(x,y)\Delta_{y}(f)(y)dy=-f(x)$
per ogni funzione $f$ liscia a supporto compatto in $R$.
#### 4.3.1 Funzioni di Green sulla compattificazione di Royden
In questa sezione dimostriamo che per una varietà iperbolica le funzioni di
Green si possono estendere alla compattificazione di Royden $R^{*}$.
Utilizzeremo i risultati ottenuti nella sezione 4.3.
###### Proposizione 4.24.
Per ogni $c>0$, la funzione
$\displaystyle G_{c}(x,p)\equiv G(x,p)\curlywedge c\equiv\min\\{G(x,p);c\\}$
appartiene a $\mathbb{M}_{\Delta}(R)$, inoltre
$\displaystyle D_{R}(G(x,p)\curlywedge c)\leq c$
###### Proof.
Data un’esaustione regolare $K_{n}$ sia $G_{n}(x,p)$ la funzione di Green
relativa a $K_{n}$ e $G(x,p)$ il suo limite.
L’obiettivo della dimostrazione è mostrare che per ogni $c>0$ 171717questo è
vero anche per $c\leq 0$, ma è privo di senso essendo $G(x,p)>0$ su $R$ la
successione $G_{n}(\cdot,p)\curlywedge c$ converge localmente uniformemente a
$G_{c}(\cdot,p)$ 181818questa considerazione è quasi ovvia e ha integrale di
Dirichlet limitato, quindi dato che ovviamente $G_{n}(x,p)\curlywedge
c\in\mathbb{M}_{0}(R)\subset\mathbb{M}_{\Delta}(R)$, grazie al teorema 3.24
$G_{c}\in\leavevmode\nobreak\ \mathbb{M}_{\Delta}(R)$.
Per dimostrare che $D_{R}(G_{n}(x,p)\curlywedge c)<M(c)$, per prima cosa
osserviamo che se $c^{\prime}\leq c$
$\displaystyle D_{R}(G_{n}(x,p)\curlywedge
c^{\prime})=\int_{G_{n}(x,p)<c^{\prime}}\left|\nabla(G_{n}(x,p))\right|^{2}dV\leq$
$\displaystyle\leq\int_{G_{n}(x,p)<c}\left|\nabla(G_{n}(x,p))\right|^{2}dV=D_{R}(G_{n}(x,p\curlywedge
c)$
quindi basta dimostrare che $D_{R}(G_{n}(x,p)\curlywedge c)<M(c)$ è valida per
un insieme illimitato di $c$.
Consideriamo un insieme aperto relativamente compatto $p\in\Omega$. Grazie a
4.21, $G(x,p)$ ristretta a $R\setminus\Omega$ assume il suo massimo $M$ in
$\partial\Omega$. Allora l’insieme $U_{c}\equiv\\{x\ t.c.\ G(x,p)>c\\}$ è
contenuto in $\Omega$ se $c>M$, e per monotonia anche gli insiemi
$\displaystyle U_{c}^{n}\equiv\\{x\ t.c.\ G_{n}(x,p)>c\\}$
sono contenuti in $\Omega$ se $c>M$, quindi sono relativamente compatti.
All’interno di $(M,\infty)$, consideriamo i valori di $c$ che sono valori
regolari per tutte le funzioni $G_{n}(\cdot,p)$. Per il teorema di Sard
191919e per il fatto che l’unione numerabile di insiemi di misura nulla ha
misura nulla questi valori sono densi in $(M,\infty)$. D’ora in avanti
consideriamo solo $c$ con queste caratteristiche.
Osserviamo ora che per ogni $n$, l’integrale:
$\displaystyle D_{R}(G_{n}(x,p)\curlywedge
c)=\int_{K_{n}\cap(U_{c}^{n})^{C}}\left|\nabla(G_{n}(x,p))\right|^{2}dV=$
$\displaystyle\int_{\partial(K_{n}\cap(U_{c}^{n})^{C})}G_{n}\ast
dG_{n}=-c\int_{\partial U_{c}^{n}}\ast dG_{n}=c$
grazie alle proprietà di $G_{n}$ descritte in 1.7.4. Questo e il teorema 3.24
danno la tesi. Inoltre osserviamo che dall’ultima considerazione possiamo
ottenere che:
$\displaystyle D_{R}(G(x,p)\curlywedge c)\leq c$
per ogni $c>0$. ∎
Grazie al fatto che $G_{c}(x,p)\in\mathbb{M}_{\Delta}(R)$ per ogni $c$,
possiamo estendere la definizione del nucleo di Green a $R\times R^{*}$. Sia a
questo scopo $U_{c}\equiv\\{x\in R\ t.c.\ G(x,z)>c\\}$, e sia $c$ tale che
$U_{c}$ è relativamente compatto in $R$. Allora grazie all’osservazione 3.48,
per ogni funzione liscia a supporto compatto $\lambda$, se $p\in\Gamma$ e
$z\in R$ fisso:
$\displaystyle p(G_{c}(\cdot,z))=p(G_{c}(\cdot,z)(1-\lambda))$
se scegliamo $\lambda$ in modo che $supp(\lambda)\supset U_{c}$, questa
relazione mostra che per ogni $c^{\prime}>c$ si ha che:
$\displaystyle p(G_{c}(\cdot,z))=p(G_{c^{\prime}}(\cdot,z))$
quindi il valore che assume $G_{c}(\cdot,z)$ sul punto $p\in\Gamma$ è
indipendente dalla scelta di $c$, e quindi ha senso definire:
$\displaystyle G(p,z)=\lim_{q\to p}G(q,z)$
Da questa definizione segue che:
###### Proposizione 4.25.
La funzione $G^{\prime}(z,p)$ con $(z,p)\in R\times R^{*}$ è una armonica in
$R\setminus\\{p\\}$, continua su $R\times R^{*}\setminus D$,
$G^{\prime}_{c}(\cdot,p)\equiv\min\\{G^{\prime}(\cdot,p),c\\}\in\mathbb{M}_{\Delta}(R)$
per ogni $c>0$, e $D_{R}(G_{c}^{\prime}(\cdot,p))\leq c$.
###### Proof.
È necessario dimostrare la continuità della funzione $G^{\prime}(\cdot,\cdot)$
solo sui punti $R\times\Gamma$, dato che $G^{\prime}$ è estensione di $G$ che
è continua su $R\times R\setminus D$. Consideriamo quindi una coppia
$(z_{0},p_{0})\in R\times\Gamma$, $\epsilon>0$, e siano $U$ e $V$ due intorni
di $z_{0}$ e $p_{0}$ aperti in $R^{*}$ e a chiusura disgiunta. Fissato
$z_{0}$, la funzione $G^{\prime}(z_{0},\cdot)$ è continua su
$R^{*}\setminus\\{z_{0}\\}$, questo significa che possiamo scegliere $V$ in
modo che per ogni $p\in W$:
$\displaystyle\left|G^{\prime}(z_{0},p)-G^{\prime}(z_{0},p_{0})\right|\leq\epsilon$
Sia $s=\sup_{p\in R\setminus U}{G^{\prime}(z_{0},p)}=\sup_{p\in R^{*}\setminus
U}{G^{\prime}(z_{0},p)}<\infty$ 202020l’ultima uguaglianza segue dal fatto che
$R$ è denso in $R^{*}$. Consideriamo la funzione di Harnack definita
sull’aperto $U\times U$. Allora vale che:
$\displaystyle k(z,z_{0})^{-1}G^{\prime}(z_{0},p)\leq G^{\prime}(z,p)\leq
k(z_{0},z)G^{\prime}(z_{0},p)$
per ogni punto $p\in W\cap R$. Data la proposizione 1.60, esiste un intorno
$U^{\prime}\subset U$ tale che $k(U^{\prime},z_{0})\subset[1,1+\epsilon/s)$.
Allora per $z\in U^{\prime}$ e per ogni $p\in W\cap R$, si ha che:
$\displaystyle\left|G^{\prime}(z,p)-G^{\prime}(z_{0},p)\right|\leq\epsilon$
Sempre per continuità su $R^{*}$ di $G^{\prime}(z_{0},\cdot)$, si ha che in
realtà questa relazione vale per ogni $p\in W$. Riassumendo otteniamo che se
$(z,p)\in U^{\prime}\times W$, si ha che:
$\displaystyle\left|G^{\prime}(z_{0},p_{0})-G^{\prime}(z,p)\right|\leq\left|G^{\prime}(z_{0},p_{0})-G^{\prime}(z_{0},p)\right|+\left|G^{\prime}(z_{0},p)-G^{\prime}(z,p)\right|\leq
2\epsilon$
data l’arbitrarietà di $\epsilon$ si ottiene la tesi.
La funzione $G^{\prime}(\cdot,p)$ è il limite di funzioni armoniche positive,
quindi grazie al principio di Harnack (vedi 1.57) è una funzione armonica.
Sappiamo che per ogni $p\in R$:
$\displaystyle D_{R}(G(x,p)\curlywedge c)\leq c$
Sia $p_{0}\in\Gamma$. Dato che $G^{\prime}(\cdot,p_{0})\curlywedge
c=C-\lim_{p\in R,\ p\to p_{0}}G(\cdot,p)\curlywedge c$, al teorema 3.24,
abbiamo che $G^{\prime}(\cdot,p_{0})\curlywedge c\in\mathbb{M}_{\Delta}(R)$ e
anche $D_{R}(G^{\prime}(\cdot,p_{0})\curlywedge c)\leq c$. ∎
Grazie alle proprietà fino a qui dimostrate, possiamo definire
$G^{*}(\cdot,\cdot)$ l’estensione di $G^{\prime}(\cdot,\cdot)$ a tutto
$R^{*}\times R^{*}\setminus D$. Infatti, essendo
$G^{\prime}(\cdot,p)\curlywedge
c\in\mathbb{M}_{\Delta}(R)\subset\mathbb{M}(R)$ per ogni $c>0$, ha senso
definire:
$\displaystyle G^{*}(z_{0},p_{0})\curlywedge c=\lim_{z\in R,\ z\to
z_{0}}G^{\prime}(z,p_{0})\curlywedge c\equiv\lim_{z\in R,\ z\to
z_{0}}\lim_{p\in R,\ p\to p_{0}}G(z,p)\curlywedge c$
per ogni $c>0$, quindi:
###### Definizione 4.26.
Definiamo $G^{*}(\cdot,\cdot):R^{*}\times R^{*}\setminus D\to[0,\infty)$ con
il doppio limite:
$\displaystyle G^{*}(z_{0},p_{0})=\lim_{z\in R,\ z\to
z_{0}}G^{\prime}(z,p_{0})\equiv\lim_{z\in R,\ z\to z_{0}}\lim_{p\in R,\ p\to
p_{0}}G(z,p)$
Definiamo inoltre il bordo essenziale irregolare di $R^{*}$ come:
###### Definizione 4.27.
L’insieme dei punti in $\Gamma$
$\displaystyle\Xi\equiv\\{p\in\Gamma\ t.c.\ G(z,p)>0\ z\in R\\}$
è detto bordo essenziale irregolare di $R$.
Riassumendo le proprietà fin qui dimostrate, per la funzione $G^{*}$ valgono
le seguenti proprietà:
###### Proposizione 4.28.
Sia $R$ una varietà non parabolica senza bordo, allora per la funzione
$G^{*}(\cdot,\cdot)$, l’estensione del nucleo di Green alla compattificazione
$R^{*}$, valgono le proprietà:
1. 1.
$G^{*}(x,y)=G(x,y)\ \forall(x,y)\in R$
2. 2.
$G^{*}(\cdot,p)$ è continua sull’insieme $R^{*}\setminus\\{p\\}$
3. 3.
$G^{*}(\cdot,p)$ è una funzione armonica su $R\setminus\\{p\\}$
4. 4.
$D_{R}(G^{*}(\cdot,p)\curlywedge c)\leq c$ per ogni $c>0$
5. 5.
$G^{*}(\cdot,\cdot)|_{(\Delta\times R^{*})}=0$
6. 6.
$G^{*}(\cdot,\cdot)|_{(R\cup\Xi)\times(R\cup\Xi)}>0$
###### Proof.
(1) e (2) e (3) seguono dal fatto che $G^{*}$ è un’estensione di $G^{\prime}$
che è un’estensione di $G$, e visto che sia $G$ che $G^{\prime}$ sono in
$\mathbb{M}(R)$, l’estensione è continua su tutta $R^{*}$ per definizione
della compattificazione di Royden.
Grazie alla 4.25, possiamo dedurre (4) e (5), infatti dato che
$G^{\prime}\in\mathbb{M}_{\Delta}(R)$, $D_{R}(G^{\prime}\curlywedge c)\leq c$
e grazie alla proposizione 3.35, possiamo dedurre che $G^{*}\curlywedge
c\in\mathbb{M}_{\Delta}(R)$, quindi, $G^{*}(\cdot,\cdot)|_{\Delta\times
R^{*}}=0$.
Rimane da dimostrare (6). Come spesso accade, questa dimostrazione è
un’applicazione del principio del massimo. Sappiamo che $G^{*}(x,y)>0$ se
$(x,y)\in R\times R$, e anche se $(x,y)\in R\times\Xi$, resta il caso
$(x,y)\in\Xi\times\Xi$. La funzione $G^{*}(\cdot,y)$ è armonica positiva su
$R$ (come dimostrato sopra). Quindi se consideriamo un aperto $U\in R$ e un
punto $p\in U$, esiste un numero positivo $a$ tale che
$aG^{*}(\cdot,y)-G(\cdot,p)|_{\partial U}>0$. Consideriamo una successione
$G_{n}(\cdot,p)$ di funzioni di Green relative a un’esaustione regolare
$R_{n}$. Allora definitivamente $aG^{*}(\cdot,y)-G_{n}(\cdot,p)|_{\partial
U}>0$ e anche $aG^{*}(\cdot,y)-G_{n}(\cdot,p)|_{\partial R_{n}}>0$. Quindi per
il principio del massimo $aG^{*}(\cdot,y)-G_{n}(\cdot,p)|_{R_{n}\setminus
U}\geq 0$. Allora questa disuguaglianza vale anche per il limite su $n$, cioè
$\displaystyle aG^{*}(\cdot,y)-G(\cdot,p)|_{\partial U}\geq 0$
dato che $G(x,p)>0$ per definizione di $\Xi$, si ha la tesi. ∎
D’ora in avanti confonderemo la notazione di $G,\ G^{\prime},\ G^{*}$ quando
non ci sia rischio di confusione.
### 4.4 Potenziali di Evans
In questa sezione diamo un’altra caratterizzazione della parabolicità di una
varietà $R$ attraverso l’esistenza di particolari funzioni armoniche, i
potenziali di Evans.
###### Definizione 4.29.
Dato un compatto $K\Subset R$, un potenziale di Evans rispetto a questo
compatto è una funzione armonica $f:(R\setminus K)\to\mathbb{R}$ tale che
$\displaystyle\lim_{x\to\infty}f(x)=\infty\ \ \ f|_{\partial K}=0$
L’ultimo limite può essere inteso equivalentemente in 2 sensi: per ogni
successione $x_{n}\to\infty$, $f(x_{n})\to\infty$, oppure per ogni $N>0$,
esiste $K\subset K_{N}\Subset R$ tale che $f(K_{N}^{C})\subset(N,+\infty)$.
Se la funzione $f$ è solo superarmonica in $R\setminus K$, viede definita
potenziale di Evans superarmonico.
I potenziali di Evans sono caratteristici delle varietà paraboliche, nel senso
che una varietà è parabolica se e solo se per ogni compatto $K$ esiste un
potenziale di Evans relativo a questo compatto. La dimostrazione di
un’implicazione è quasi immediata, mentre l’altra implicazione è il contenuto
fondamentale di questa tesi.
Osserviamo che per caratterizazione delle varietà paraboliche, è sufficiente
chiedere che il potenziale di Evans sia superarmonico.
###### Proposizione 4.30.
Se esiste un compatto $K$ che ammette un potenziale di Evans superarmonico,
allora la varietà è parabolica
###### Proof.
Sia $K$ un compatto e $f$ il relativo potenziale di Evans superarmonico.
Consideriamo un compatto $C$ con $K\Subset C^{\circ}$ e bordo liscio. La
funzione $f$ continua a essere superarmonica positiva su $C^{C}$ e tende a
infinito. Inoltre grazie al principio del massimo 1.52 $f|_{\partial C}>0$.
Dimostriamo che il potenziale armonico $u$ di $C$ è costante uguale a $1$.
Sia $b=1-u$ funzione armonica definita su $C^{C}$ 212121estendibile per
continuità a $0$ su $C$, e sia $C_{k}$ un’esaustione regolare di $R$ con
$C_{1}=C$. Per ogni $N>1$, esiste $\bar{k}$ tale che
$f(C_{\bar{k}}^{C})\subset(N,\infty)$. Dato che la funzione $b$ è
necessariamente $\leq 1$ 222222indipendendemente dal fatto che $R$ sia
parabolica, e che $Nb|_{\partial C}<f|_{\partial C}$, per il principio del
massimo abbiamo che per ogni $k>\bar{k}$, $Nb\leq f$ sull’insieme
$C^{k}\setminus C$, quindi su tutto l’insieme $R\setminus C$. Data
l’arbitrarietà di $N$, otteniamo che per ogni $N>1$
$\displaystyle Nb\leq f\ \ \Rightarrow\ \ b\leq\frac{f}{N}$
quindi necessariamente $b=0$, cioè $u=1$, come volevasi dimostrare. ∎
I paragrafi seguenti si occupano di dimostrare l’implicazione inversa.
#### 4.4.1 Diametro transfinito
In questo paragrafo definiamo due strumenti che saranno utili per costruire
particolari funzioni armoniche, tra cui i potenziali di Evans. D’ora in avanti
assumiamo che $R$ sia una varietà riemanniana non parabolica.
###### Definizione 4.31.
Detto $G^{*}(\cdot,\cdot)$ il nucleo di Green su $R^{*}$, e dato $X\subset
R^{*}$ definiamo:
$\displaystyle\binom{n}{2}\rho_{n}(X)\equiv\inf_{p_{1},\cdots,p_{n}\in
X}\sum_{i<j}^{1\cdots n}G(p_{i},p_{j})$ (4.2)
inoltre per convenzione $\rho(\emptyset)=\infty$
###### Proposizione 4.32.
Fissato l’insieme $X$, $\rho_{n}(X)$ è una successione crescente al crescere
di $n$.
###### Proof.
La dimostrazione è un semplice esercizio di algebra, non riguarda le
caratteristiche di $R$.
Siano $p_{1},\cdots,p_{n}$ punti qualsiasi in $X$, e consideriamo per ogni
$1\leq k\leq n+1$:
$\displaystyle\sum_{i<j}^{1\cdots
n+1}G(p_{i},p_{j})=\sum_{i=1}^{k-1}G(p_{i},p_{k})+\sum_{j=k+1}^{n+1}G(p_{k},p_{j})+\sum_{i<j;\
i,j\neq k}^{1\cdots n}G(p_{i},p_{j})$
e quindi dalla definizione di $\rho_{n}(X)$:
$\displaystyle\sum_{i<j}^{1\cdots
n+1}G(p_{i},p_{j})\geq\sum_{i=1}^{k}G(p_{i},p_{k})+\sum_{j=k+1}^{n+1}G(p_{k},p_{j})+\binom{n}{2}\rho_{n}(X)$
Sommando tutte queste disuguaglianze al variare di $k$ tra $1$ e $n+1$,
otteniamo che:
$\displaystyle(n+1)\sum_{i<j}^{1\cdots n+1}G(p_{i},p_{j})\geq
2\sum_{i<j}^{1\cdots n+1}G(p_{i},p_{j})+(n+1)\binom{n}{2}\rho_{n}(X)$
cioè:
$\displaystyle(n-1)\sum_{i<j}^{1\cdots
n+1}G(p_{i},p_{j})\geq(n+1)\binom{n}{2}\rho_{n}(X)$
vista l’arbitrarietà dei punti considerati, possiamo concludere che:
$\displaystyle(n-1)\binom{n+1}{2}\rho_{n+1}(X)\geq(n+1)\binom{n}{2}\rho_{n}(X)$
considerando che
$\displaystyle(n-1)\binom{n+1}{2}=(n+1)\binom{n}{2}$
otteniamo la tesi, cioè
$\displaystyle\rho_{n+1}(X)\geq\rho_{n}(X)$
∎
Questa proposizione ci permette di definire il limite al variare di $n$ di
$\rho_{n}(X)$, in particolare:
###### Definizione 4.33.
Dato $X\in R^{*}$, definiamo il diametro transfinito di $X$:
$\displaystyle\rho(X)\equiv\lim_{n}\rho_{n}(X)$ (4.3)
Assieme al diametro transfinito, definiamo la costante di Tchebycheff, e
esploriamo alcuni legami tra i due concetti.
###### Definizione 4.34.
Dato $X\subset R^{*}$, definiamo:
$\displaystyle n\tau_{n}(X)=\sup_{p_{1},\cdots,p_{n}\in X}\left(\inf_{p\in
X}\sum_{i=1}^{n}G(p,p_{i})\right)$
e per convenzione $\tau_{n}(\emptyset)=\infty$
Anche in questo caso vale una relazione che ci consente di definire il limite
di $\tau_{n}$.
Infatti osserviamo che:
$\displaystyle\sum_{i=1}^{n+m}G(p,p_{i})=\sum_{i=1}^{n}G(p,p_{i})+\sum_{i=n+1}^{n+m}G(p,p_{i})$
quindi applicando l’$\inf$ a entrambi i membri otteniamo:
$\displaystyle\inf_{p\in X}\sum_{i=1}^{n+m}G(p,p_{i})\geq\inf_{p\in
X}\sum_{i=1}^{n}G(p,p_{i})+\inf_{p\in X}\sum_{i=n+1}^{n+m}G(p,p_{i})$
cioè per definizione:
$\displaystyle(n+m)\tau_{n+m}(X)\geq n\tau_{n}(X)+m\tau_{m}(X)$ (4.4)
se consideriamo $n=m$ otteniamo il caso particolare:
$\displaystyle(qm)\tau_{qm}(X)\geq
q\tau_{m}(X)\Rightarrow\tau_{qm}(X)\geq\tau_{m}(X)$ (4.5)
Grazie a questa relazione possiamo dimostrare che:
###### Proposizione 4.35.
Sia $\alpha\equiv\sup_{n}\tau_{n}(X)$, allora $\lim_{n}\tau_{n}(X)=\alpha$.
###### Proof.
La dimostrazione segue dalla relazione 4.4. Infatti scegliamo un qualunque
$\beta<\alpha$. Per definizione di $\sup$, esiste un $m$ tale che
$\tau_{m}(X)>\beta$. È un fatto noto che ogni numero $n$ può essere scritto in
maniera univoca come:
$\displaystyle n=qm+r$
dove $q,r\in\mathbb{N}$ e $0\leq r\leq m-1$. Quindi per $\tau_{n}(X)$ vale
che:
$\displaystyle n\tau_{n}(X)=(qm+r)\tau_{qm+r}(X)\geq
qm\tau_{qm}(X)+r\tau_{r}(X)\geq qm\tau_{m}(X)$
dove abbiamo utilizzato le relazioni 4.4 ed 4.5. Quindi possiamo concludere
che
$\displaystyle\tau_{n}(X)\geq\frac{qm}{qm+r}\tau_{m}(X)>\frac{qm}{qm+r}\beta$
Se $n$ tende a infinito, $q$ tente a infinito, mentre $0\leq r\leq n-1$
continua a valere, quindi:
$\displaystyle\liminf_{n}\tau_{n}(X)\geq\liminf_{n}\frac{qm}{qm+r}\beta=\beta$
Cioè per ogni $\beta<\alpha=\sup_{n}\tau_{n}(X)$ abbiamo che:
$\displaystyle\alpha\geq\limsup_{n}\tau_{n}(X)\geq\liminf_{n}\tau_{n}(X)\geq\beta$
data l’arbitrarietà di $\beta$, otteniamo la tesi. ∎
###### Definizione 4.36.
Dato $X\subset R$ definiamo la sua costante di Tchebycheff il limite:
$\displaystyle\tau(X)\equiv\lim_{n\to\infty}\tau_{n}(X)$
Il diametro transfinito di un insieme e la costante di Tchebycheff misurano in
qualche senso la grandezza di un insieme. Più il nucleo di Green $G^{*}$ ha un
valore alto sui punti di $X$, più queste due costanti hanno valore alto, ed è
facile osservare che per insiemi di cardinalità finita
$\tau(X)=\rho(X)=\infty$, e che se $X^{\prime}\subset X$, allora
$\tau(X^{\prime})\geq\tau(X)$. Di seguito riportiamo un’altra proprietà di
queste due costanti.
###### Proposizione 4.37.
Per ogni $X\subset R^{*}$ si ha che:
$\displaystyle\tau(X)\geq\rho(X)$ (4.6)
###### Proof.
Assumiamo che $X$ abbia cardinalità infinita 232323altrimenti abbiamo visto
che $\tau(X)=\rho(X)=\infty$. Per $n>1$ sia $r\equiv\frac{1}{n-1}$. È
possibile scegliere $n$ punti $p_{1},\cdots,p_{n}$ tali che per ogni
$i=1,\cdots,n-1$ abbiamo che:
$\displaystyle\sum_{j=n+1-i}^{n}G(p_{n-i};p_{j})\leq\inf_{p\in
X}\sum_{j=n+1-i}^{n}G(p,p_{j})+r$ (4.7)
Dimostriamo questa affermazione per induzione su $i$. Per $i=1$ l’affermazione
segue direttamente dalla definizione di $\inf$. Infatti se scegliamo $p_{n}$
arbitrariamente e $p_{n-1}$ in modo che
$\displaystyle G(p_{n-1},p_{n})\leq\inf_{p\in X}G(p,p_{n})+r$
Supponiamo che la tesi sia vera per $1\leq i<n-1$ 242424quindi supponiamo di
aver determinato $p_{n},\cdots,p_{n-i+1}$ e consideriamo la funzione
$\displaystyle f(p)=\sum_{j=n-i+1}^{n}G(p,p_{i})$
osserviamo che se $p=p_{k}$ per qualche $k=n-i+1,\cdots n$, allora $f(p)$ vale
infinito. La funzione $f$ comunque è positiva, quindi possiamo trovare un
punto $p_{n-i}$ tale che:
$\displaystyle f(p_{n-i})\leq\inf_{p\in X}f(p)+r$
quindi otteniamo che esistono punti $p_{1},\cdots,p_{n}\in X$ tali che
$\displaystyle\sum_{j=n+1-i}^{n}G(p_{n-i};p_{j})\leq\inf_{p\in
X}\sum_{j=n+1-i}^{n}G(p;p_{j})\leq i\tau_{i}(X)+r$
sommando queste disuguaglianze per $i=1,\cdots,n-1$ otteniamo che
$\displaystyle\sum_{i<j}^{1\cdots
n}G(p_{i};p_{j})\leq\sum_{i=1}^{n-1}i\tau_{i}(X)+1$
e dalla definizione di $\rho_{n}$ abbiamo:
$\displaystyle\binom{n}{2}\rho_{n}(X)\leq\sum_{i=1}^{n-1}i\tau_{i}(X)+1$
quindi:
$\displaystyle\rho_{n}(X)\leq\binom{n}{2}^{-1}\left(\sum_{i=1}^{n-1}i\tau_{i}(X)+1\right)$
Passando al limite otteniamo la tesi, cioè:
$\displaystyle\rho(X)=\lim_{n}\rho_{n}(X)\leq\lim_{n}\binom{n}{2}^{-1}\left(\sum_{i=1}^{n-1}i\tau_{i}(X)+1\right)=\tau(X)$
dove l’ultimo passaggio è giustificato nel seguente lemma ∎
###### Lemma 4.38.
Sia $a_{n}$ tale che $\lim_{n}a_{n}=a$, allora:
$\displaystyle\lim_{n}\sum_{k=1}^{n-1}\frac{ka_{k}}{\binom{n}{2}}=a$
###### Proof.
Per definizione di limite, per ogni $\epsilon>0$, esiste $N$ tale che
$a_{k}>a-\epsilon$ per ogni $k>N$, quindi per ogni $\epsilon>0$ e per ogni
$n>N+1$ (quindi definitivamente):
$\displaystyle\binom{n}{2}^{-1}\sum_{k=1}^{n-1}ka_{k}=\binom{n}{2}^{-1}\left(\sum_{k=1}^{N}ka_{k}+\sum_{k=N+1}^{n-1}ka_{k}\right)>$
$\displaystyle>\binom{n}{2}^{-1}\left(\sum_{k=1}^{N}ka_{k}\right)+(a-\epsilon)\binom{n}{2}^{-1}\left(\sum_{k=N+1}^{n-1}k\right)$
applicando il $\liminf_{n}$, e osservando che $\binom{n}{2}\to\infty$,
$\sum_{k=1}^{N}ka_{k}$ è costante al variare di $n$ e
$\sum_{k=N+1}^{n-1}k=\binom{n}{2}-\binom{N+1}{2}$ otteniamo:
$\displaystyle\liminf_{n}\binom{n}{2}^{-1}\sum_{k=1}^{n-1}ka_{k}\geq
a-\epsilon$
con un ragionamento del tutto analogo si ottiene anche:
$\displaystyle\limsup_{n}\binom{n}{2}^{-1}\sum_{k=1}^{n-1}ka_{k}<a+\epsilon$
e per l’arbitrarietà di $\epsilon$, otteniamo la tesi. ∎
#### 4.4.2 Stime per il diametro transfinito
Lo scopo di questo paragrafo è ottenere la stima riportata nella proposizione
4.40, una stima tecnica che servirà nei paragrafi successivi a dimostrare che
per ogni $\Xi^{\prime}\Subset\Xi$,
$\rho(\Xi^{\prime})=\tau(\Xi^{\prime})=\infty$. L’affermazione è vuotamente
vera se $R$ è una varietà iperbolica regolare (cioè se $\Xi=\emptyset$),
quindi in tutta la sezione assumeremo che $R$ sia una varietà iperbolica
irregolare.
A questo scopo, introduciamo un insieme di funzioni ausiliarie che serviranno
a stimare il diametro di $\Xi_{n}$.
Fissato un punto $z_{0}\in R$, sia $r_{n}$ una successione di numeri reali
positivi tali che
$\displaystyle r_{n}>r_{n+1}\ \ \ \lim_{n}r_{n}=0$
inoltre chiediamo che $U_{n}\equiv\\{z\in R\ t.c.\ G(z,z_{0})>r_{n}\\}$ non
sia relativamente compatto e che $r_{n}$ sia un valore regolare di
$G(\cdot,z_{0})$ 252525in modo che $U_{n}$ sia un insieme con bordo regolare.
In questo modo possiamo descrivere l’insieme $\Xi$ come unione di $\Xi_{n}$,
dove
$\displaystyle\Xi_{n}=\overline{U_{n}}\cap\Gamma=\\{z\in\Gamma\ t.c.\
G(z,z_{0})\geq r_{n}\\}$
Osserviamo che l’insieme $U_{n}$ è necessariamente connesso, infatti se avesse
una componente connessa non contenente $z_{0}$, per il principio del massimo e
per il fatto che $G(\cdot,z_{0})|_{\Delta}=0$, allora $G(\cdot,z_{0})$ sarebbe
necessariamente minore di $r_{n}$ su questa componente (assurdo per
definizione di $U_{n}$.
Grazie alla proposizione 1.24, possiamo scegliere un’esaustione regolare
$K_{m}$ di $R$ in modo che gli insieme
$\displaystyle F_{nm}\equiv\overline{U_{n}}\cap\partial K_{m}$
siano per ogni $n$ e $m$ sottovarietà regolari di codimensione $1$ con bordo
liscio.
Fissato un indice $m$, esiste una successione di insiemi $C_{p}$ aperti
relativamente compatti in $R$ con bordo liscio tale che
$\overline{C_{p+1}}\subset C_{p}$ e $K_{m}=\cap_{p}C_{p}$. Questa affermazione
può essere dimostrata con argomentazioni simili a quelle riportate nella
proposizione 1.20. La successione $C_{p}$ è utile per dimostrare che:
###### Lemma 4.39.
Con le notazioni introdotte qui sopra, esiste una funzione $w_{n,m,p}$ tale
che:
1. 1.
$w_{n,m,p}\in H(U_{n+1}\setminus\overline{K_{m}})$
2. 2.
$w_{n,m,p}\geq 0$
3. 3.
$w_{n,m,p}|_{\partial U_{n+1}\setminus C_{p}}=0$
4. 4.
$w_{n,m,p}|_{\partial U_{n+1}\setminus K_{m}}\leq 1$
5. 5.
$w_{n,m,p}|_{F_{n+1.m}}=1$
6. 6.
$w_{n,m,p}|_{\Xi_{n}}\geq\sigma_{n}$
7. 7.
$D_{R}(w)<\infty$
dove $\sigma_{n}$ è un numero strettamente positivo indipendente da $m$ e $p$.
###### Proof.
La dimostrazione di questo lemma è molto tecnica. La sua utilità sarà
illustrata nella proposizione seguente.
Costruiamo le funzioni $w_{n,m,p}$ su $U_{n+1}\setminus K_{m}$ per esaustione,
dimostriamo che è possibile estendere queste funzioni a una funzioni in
$\mathbb{M}(R)$ (quindi ha senso parlare di $w_{n,m,p}|_{\Xi_{n}}$) e
dimostriamo l’ultima disuguaglianza confrontando queste funzioni con funzioni
di Green opportunamente modificate.
In tutta la dimostrazione considereremo fissati i valori di $n$, $m$, $p$,
quindi $w_{n,m,p}\equiv w$, $U_{n+1}\equiv U$, $K_{m}\equiv K$ e $C_{p}\equiv
C$.
Fissiamo una funzione liscia a supporto compatto $\lambda:R\to[0,1]$ tale che:
$\displaystyle\lambda|_{(\partial K)\cap U}=1,\ \ \
\text{supp}(\lambda)\subset C$
Per $k>m$, sia $u_{k}$ la soluzione del problema di Dirichlet
$\displaystyle u_{k}\in H(U\cap(K_{k}\setminus\overline{K})),\ \ \
u_{k}|_{\partial K\cup[(\partial U)\cap K_{k}\setminus K]}=\lambda|_{\partial
K\cup[(\partial U)\cap K_{k}\setminus K]},\ \ \ u_{k}|_{\partial K_{k}}=0$
Per il principio del massimo, la successione $u_{k}$ è una successione
crescente e limitata da $1$, quindi grazie al principio di Harnack, $u_{k}$
ammette come limite una funzione armonica su $U\setminus\overline{K}$, in
particolare:
$\displaystyle\lim_{k}u_{k}=w$
Osserviamo che grazie al principio di Dirichlet 3.32, $D_{R}(u_{k})\leq
D_{R}(\lambda)<\infty$ per ogni $k$, e inoltre se $i>k$
$\displaystyle D_{R}(u_{k})=D_{R}(u_{i})+D_{R}(u_{k}-u_{i})$
da cui con un ragionamento simile a quello riportato nella dimostrazione del
teorema 3.53, otteniamo che la successione $\\{u_{k}\\}$ è $D$-Cauchy in
$\mathbb{M}(R)$, quindi per completezza $u_{k}$ ammette limite $CD$, e per
unicità del limite
$\displaystyle CD-\lim_{k}u_{k}=w$
da cui $D_{R}(w)<\infty$, quindi (7) è dimostrata.
Per quanto riguarda le altre proprietà, (2), (3), (4) e (5) seguono
direttamente dal fatto che tutte le funzioni $u_{k}$ soddisfano queste
proprietà. Per dare senso alla richiesta $w|_{\Xi_{n}}\geq\sigma_{n}$,
estendiamo la funzione $w$ a una funzione $\tilde{w}$ definita su tutta $R$ in
questo modo:
$\displaystyle\tilde{w}(x)=\begin{cases}w(x)&\text{se }x\in U\setminus K\\\
\lambda(x)&\text{se }x\in(U\setminus K)^{C}\end{cases}$
Osserviamo che $\Xi_{n}\subset\overline{U}$, quindi il valore di
$\tilde{w}|_{\Xi_{n}}$ è indipendente dall’estensione di $w$ che si sceglie.
È facile osservare che $\tilde{w}$ è continua e di Tonelli grazie a un
ragionamento simile a quello riportato in 3.28, e il suo integrale di
Dirichlet è finito poiché gli integrali di Dirichlet di $\lambda$ e $w$ lo
sono.
Questo dimostra che ha senso parlare di $w|_{\Xi_{n}}$ grazie al fatto che $w$
può essere estesa a tutto $R^{*}$ e che $w$ è definita su $U_{n+1}\setminus
K_{m}$, quindi in un intorno di $\Xi_{n}$.
Per dimostrare (5), consideriamo la funzione
$\displaystyle h(x)=\frac{G(x,z_{0})-r_{n+1}}{b-r_{n+1}}$
dove $b$ è un numero positivo sufficientemente grande da rendere l’insieme
$\displaystyle B\equiv\\{z\in R\ \ r.c.\ \ G(z,z_{0})>b\\}$
contenuto nell’insieme $K_{1}\cap U_{n+1}$. In questo modo $h(x)\leq 1$
sull’insieme $U\setminus K$. Confrontiamo ora le funzioni $w$ e $h$
sull’insieme $U\setminus\overline{K}$. Osserviamo prima di tutto che queste
sono entrambe funzioni armoniche su $U\setminus\overline{K}$ e continue fino
al bordo di questo insieme. Per definizione di $U\equiv U_{n+1}$, la funzione
$h$ è negativa su $\partial U$, e come osservato in precedenza $h\leq 1=w$
sull’insieme $\partial K\cap U$. Inoltre entrambe le funzioni sono nulle
sull’insieme $\Delta\cap U$, quindi grazie al principio del massimo 3.58,
$w\geq h$ su tutto l’insieme di definizione, quindi in particolare anche su
$\Xi_{n}$.
Dato che $h|_{\Xi_{n}}\geq\frac{r_{n}-r_{n+1}}{b-r_{n+1}}\equiv\sigma_{n}$, si
ha la tesi. ∎
Grazie alla funzione $w_{nm}$ possiamo ottenere una stima su $\rho(\Xi_{n})$,
infatti vale che:
###### Proposizione 4.40.
Secondo le notazioni fino a qui introdotte:
$\displaystyle\rho(\Xi_{n})\geq\sigma_{n}^{2}\rho(F_{n+1,m})$ (4.8)
per ogni $m$.
###### Proof.
In tutta la dimostrazione, sottointendiamo che le sommatorie che scorrono su
nessun indice sono nulle, nel senso che ad esempio:
$\displaystyle\sum_{i=1}^{0}a_{i}\equiv 0$
Assumiamo inoltre che $\Xi_{n}$ abbia cardinalità infinita, in caso contrario
la tesi è ovvia essendo $\rho(A)=\infty$ per ogni insieme $A$ di cardinalità
finita.
Sia $k\geq 4$ un intero fissato e $p_{1},\cdots,p_{k}$ punti arbitrari in
$\Xi_{n}$. Ci prefiggiamo di trovare $k$ punti $z_{1},\cdots,z_{k}\in
F_{n+1,m}$ tali che
$\displaystyle\sigma^{2}_{n}\sum_{i<j}^{1\cdots
t}G(z_{i},z_{j})+\sigma_{n}\sum_{i=1}^{t}\sum_{j=t+1}^{k}G(z_{i},p_{j})+\sum_{i<j}^{t+1,\cdots,k}G(p_{i},p_{j})\leq\sum_{i<j}^{1,\cdots,k}G(p_{i},p_{j})$
(4.9)
per ogni $t=1,\cdots,k$. Una volta dimostrato questo otteniamo in particolare
per $t=k$ che:
$\displaystyle\sigma^{2}_{n}\sum_{i<j}^{1\cdots
k}G(z_{i},z_{j})\leq\sum_{i<j}^{1,\cdots,k}G(p_{i},p_{j})$
e per definizione di $\rho(F_{n+1,m})$ questo implica che:
$\displaystyle\sigma^{2}_{n}\binom{k}{2}\rho_{k}(F_{n+1,m})\leq\sum_{i<j}^{1,\cdots,k}G(p_{i},p_{j})$
inoltre data l’arbitrarietà della scelta dei punti $p_{1},\cdots,p_{k}$, si ha
che:
$\displaystyle\sigma^{2}_{n}\rho_{k}(F_{n+1,m})\leq\binom{k}{2}\rho_{k}(\Xi_{n})$
passando al limite per $k$ che tende a infinito, si ottiene la tesi.
Resta da dimostrare la parte tecnica della prova, cioè la relazione 4.9.
A questo scopo utilizzeremo l’induzione sull’indice $1\leq t\leq k$.
Supponiamo di aver trovato dei punti $z_{1},\cdots,z_{h-1}$ con $1\leq h\leq
k-1$ per cui vale 4.9 262626ovviamente l’ipotesi di induzione garantisce che
questa relazione valga solo per $1\leq t\leq h-1$, perché per indici più
grandi le equazioni non hanno senso non avendo determinato tutti punti
$z_{i}$. Definiamo
$\displaystyle
u_{h}(z)\equiv\sum_{j=h+1}^{k}G(z,p_{j})+\sigma_{n}\sum_{i=1}^{h-1}G(z_{i},z)$
Dato che $u_{h}$ è continua e positiva su
$R\setminus\\{z_{1},\cdots,z_{h-1}\\}$, ammette un minimo positivo su
$F_{n+1,m}$ assunto in $z_{h}$. Osserviamo che per ogni $\delta>0$:
$\displaystyle u_{h}(z)-(u_{h}(z_{h})-\delta)>\delta$
sull’insieme $\partial K_{m}\cap U_{n+1}$. Questo implica che esiste un
intorno di questo insieme su cui questa funzione è strettamente positiva.
Denotiamo questo intorno con il simbolo $A$. Per definizione della successione
$C_{p}$, esiste un indice $p$ per il quale l’insieme $(C_{p}\setminus
K_{m})\cap U_{n+1}\subset A$. Consideriamo quindi la funzione $w_{n,m,p}$
definita nel lemma precedente. Sappiamo che
$\displaystyle\phi_{h}^{\delta}(z)\equiv
u_{h}(z)-(u_{h}(z_{h})-\delta)w_{n,m,p}\geq 0$
sull’insieme $\partial(U_{n+1}\setminus K_{m})$, quindi per il principio del
massimo questa funzione è positiva su tutto l’insieme $U_{n+1}\setminus
K_{m}$, e in particolare sull’insieme $\Xi_{n}$, cioé:
$\displaystyle u_{h}(z)\geq(u_{h}(z_{h})-\delta)\sigma_{n}$
per ogni $z\in\Xi_{n}$. Data l’arbitrarietà di $\delta$, otteniamo che
$\displaystyle u_{h}(z)\geq u_{h}(z_{h})\sigma_{n}$
per ogni $z\in\Xi_{n}$.
Applicando le definizioni otteniamo che:
$\displaystyle\sum_{j=h+1}^{k}G(p_{h},p_{j})+\sigma_{n}\sum_{i=1}^{h-1}G(z_{i},p_{h})\geq\sigma_{n}\sum_{j=h+1}^{k}G(z_{h},p_{j})+\sigma_{n}^{2}\sum_{i=1}^{h-1}G(z_{i},z_{h})$
(4.10)
questa relazione per $h=1$ è la dimostrazione di 4.9 per $t=1$. Per gli altri
valori di $t$, dall’ipotesi induttiva sappiamo che per $t=h-1$:
$\displaystyle\sum_{i<j}^{1\cdots,k}G(p_{i},p_{j})\geq\sigma^{2}_{n}\sum_{i<j}^{1\cdots
h-1}G(z_{i},z_{j})+\sigma_{n}\sum_{i=1}^{h-1}\sum_{j=h}^{k}G(z_{i},p_{j})+\sum_{i<j}^{h,\cdots,k}G(p_{i},p_{j})=$
$\displaystyle=\sigma^{2}_{n}\sum_{i<j}^{1\cdots
h-1}G(z_{i},z_{j})+\sigma_{n}\sum_{i=1}^{h-1}\sum_{j=h+1}^{k}G(z_{i},p_{j})+$
$\displaystyle+\sum_{i<j}^{h+1,\cdots,k}G(p_{i},p_{j})+\left\\{\sigma_{n}\sum_{i=1}^{h-1}G(z_{i},p_{h})+\sum_{j=h+1}^{k}G(p_{h},p_{j})\right\\}$
applicando la relazione 4.10 nelle parentesi graffe, otteniamo la 4.9 per
$t=h$, il che completa il ragionamento di induzione. ∎
#### 4.4.3 Pricipio dell’energia
Anche questo paragrafo, come il precedente, riporta alcuni risultati tecnici
utili per stimare il diametro transfinito degli insiemi compatti contenuti in
$\Xi$. In particolare introduciamo l’energia e il potenziale di Green relativo
a una misura, e troveremo come queste nozioni sono legate alla capacità di un
insieme.
Alcuni risultati della teoria del potenziale sono stati cortesemente segnalati
dal professor. Wolfhard Hansen (University of Bielefeld), che ringraziamo.
In tutto il paragrafo, $K$ indicherà una sottovarietà regolare $m-1$
dimensionale di $R$ possibilmente con bordo liscio.
###### Definizione 4.41.
Sull’insieme delle misure di Borel regolari con supporto in $K$ definiamo il
prodotto 272727il cui risultato potrebbe anche essere $\infty$:
$\displaystyle\left\langle\mu\middle|\nu\right\rangle_{B}\equiv\int_{K}d\mu(x)\
\int_{K}\ d\nu(y)G(x,y)$
la simmetria di questo prodotto segue dalla simmetria di $G(x,y)$ rispetto a
$x$ e $y$.
Chiamiamo $\left\langle\mu\middle|\nu\right\rangle_{B}$ l’energia relativa tra
le misure $\mu$ e $\nu$, e definiamo energia di $\mu$ la quantità
$\left\langle\mu\middle|\mu\right\rangle_{B}$.
###### Proposizione 4.42.
$\left\langle\mu\middle|\mu\right\rangle_{B}=0$ se e solo se $\mu=0$·
###### Proof.
Supponiamo per assurdo che $\mu\neq 0$. Allora la funzione
$\displaystyle f(x)\equiv\int G(x,y)d\mu(y)$
è strettamente positiva per ogni $x$, infatti su $K$, $G(x,\cdot)$ assume un
minimo positivo $\lambda>0$, e quindi:
$\displaystyle f(x)\geq\lambda\mu(K)$
con un ragionamento analogo, si ottiene che anche
$\int f(x)d\mu(x)=\int d\mu(x)\int d\mu(y)G(x.y)>0$
∎
###### Definizione 4.43.
Dato un insieme $K$ con le caratteristiche descritte all’inizio del paragrafo,
definiamo l’energia minima di $K$ la quantità
$\displaystyle\epsilon(K)\equiv\inf_{\mu\in
m_{K}}\left\langle\mu\middle|\mu\right\rangle_{B}$
dove $m_{K}$ è l’insieme delle misure di Borel regolari positive unitarie con
supporto su $K$ (cioè $\mu(K)=1$).
Per convenzione $\epsilon(\emptyset)=\infty$, e osserviamo immediatamente che
per definizione di $\inf$ se $K\subset K^{\prime}$,
$\epsilon(K)\geq\epsilon(K^{\prime})$.
###### Osservazione 4.44.
Per gli insiemi $K$ con le caratteristiche descritte all’inizio del paragrafo,
$\epsilon(K)<\infty$.
###### Proof.
Per dimostrare questa affermazione, basta trovare una misura $\mu$ di Borel
regolare finita 282828il fatto che $\mu(R)\neq 1$ non è importante, infatti è
sufficiente riscalare la misura per la quale
$\left\langle\mu\middle|\mu\right\rangle_{B}<\infty$.
Sia $(U,\phi)$ un insieme coordinato che interseca $K$ per il quale
$\phi(K)=(x_{1},\cdots,x_{m-1},0)$
e sia $B$ una bolla di raggio $\bar{r}$ in $\mathbb{R}^{m-1}$ contenuta in
$\phi(K)$. Consideriamo la misura di superficie data dalla metrica riemanniana
su questo insieme, misura che ha la forma
$\displaystyle dV=\sqrt{\left|g\right|}d\lambda$
dove $\lambda$ è la misura di Legesue standard. Grazie alle proprietà del
nucleo di Green, sappiamo che sull’insieme $B$ $G$ in coordinate locali si può
scrivere come
$\displaystyle G(x,y)=f(x,y)+C(m)\gamma(x,y)$
dove $f(x,y)$ è una funzione continua nelle due variabili $x$ e $y$, $C(m)$
una costante che dipende solo dalla dimensione $m$ e $\gamma(x,y)$ è una
funzione che dipende dalla dimensione $m$ della varietà $R$, in particolare:
$\displaystyle\gamma(x,y)=\begin{cases}-\log(d(x,y))&m=2\\\ d(x,y)^{m-2}&m\geq
3\end{cases}$
Per limitatezza di $f$ sull’insieme $B\times B$, è ovvio che:
$\displaystyle\int_{B}f(x,y)dV(x)dV(y)=\int_{B}f(x,y)\sqrt{\left|g\right|}d\lambda(x)\sqrt{\left|g\right|}d\lambda(y)<\infty$
Consideriamo quindi l’integrale
$\displaystyle\int_{B}\gamma(x,y)dV(x)dV(y)=\int\gamma(x,y)\sqrt{\left|g\right|}d\lambda(x)\sqrt{\left|g\right|}d\lambda(y)\leq
M^{2}\int_{B}\gamma(x,y)d\lambda(x)d\lambda(y)$
dove $M$ è un limite superiore per la funzione $\sqrt{\left|g\right|}$
sull’insieme relativamente compatto $B$. La funzione
$\displaystyle\bar{\gamma}(x)\equiv\int_{B}\gamma(x,y)d\lambda(y)$
è una funzione limitata in $x$, infatti scelto $x\in B$, detto $2B(x)$ la
bolla di raggio $2\bar{r}$ centrata in $x$, si ha che:
$\displaystyle\bar{\gamma}(x)=\int_{B}\gamma(x,y)d\lambda(y)\leq\int_{2B(x)}\gamma(x,y)d\lambda(y)$
per $m=2$, si ha che:
$\displaystyle\int_{2B(x)}-\log(\left|x-y\right|)d\lambda(y)=\int_{-\bar{r}}^{\bar{r}}-\log(r)dr=2\bar{r}(\log(\bar{r})-1)<\infty$
mentre per $m\geq 3$, passando alle coordinate polari centrate in $x$, si ha
che
$\displaystyle\int_{2B(x)}\frac{1}{\left|x-y\right|^{m-2}}d\lambda(y)=\int_{S_{m-2}}d\theta\int_{0}^{\bar{r}}dr\
r^{m-2}\frac{1}{r^{m-2}}=\omega_{m-2}\bar{r}$
dove $\omega_{m}$ è la superficie della sfera $m$ dimensionale. Essendo la
funzione $\bar{\gamma}$ limitata, abbiamo che:
$\displaystyle\int_{B\times
B}\gamma(x,y)d\lambda(x)d\lambda(y)=\int_{B}\bar{\gamma}(x)d\lambda(x)\leq\lambda(B)\sup_{B}(\bar{\gamma}(x))<\infty$
∎
###### Definizione 4.45.
Per una misura $\mu\in m_{K}$, definiamo il relativo potenziale di Green la
funzione:
$\displaystyle G_{\mu}(x)\equiv\int_{K}G(x,y)d\mu(y)$
Per prima cosa esploriamo le proprietà di $G_{\mu}$ per una qualsiasi misura
di Borel positiva regolare a supporto compatto $S\in R$.
###### Proposizione 4.46.
$G_{\mu}\in H(R\setminus S)$, $G_{\mu}$ è positiva, semicontinua
inferiormente, e può essere estesa a tutto $R^{*}$ con $G_{\mu}|_{\Delta}=0$.
Inoltre $G_{\mu}$ è superarmonica su $R$.
###### Proof.
La positività di $G_{\mu}$ è un’ovvia conseguenza della positività della
misura e del nucleo di Green. Per dimostrare che $G_{\mu}\in H(R\setminus S)$,
sfruttiamo il teorema di derivazione sotto al segno di integrale 1.37. Sia
$x\in R\setminus S$. Allora esiste un intorno compatto $V$ di $x$ disgiunto da
$S$, e per le proprietà di $G(\cdot,\cdot)$, sappiamo che
$\displaystyle G(x,y)\leq\sup_{z\in\partial V}G(x,z)<\infty$
per ogni $y\in S$. Grazie al fatto che $G\in C^{\infty}(R\times R\setminus
D)$, sappiamo che tutte le derivate $\partial_{i}G(x,y)$ e
$\partial_{i}\partial_{j}G(x,y)$ sono uniformemente limitate se $(x,y)\in
V\times S$, quindi possiamo applicare il teorema 1.37 e ottenere che:
$\displaystyle\Delta
G_{\nu}(x)=g^{ij}(x)\partial_{i}\partial_{j}\int_{S}G(x,y)d\mu(y)=$
$\displaystyle=g^{ij}(x)\partial_{i}\int_{S}\partial_{j}G(x,y)d\mu(y)=\int_{S}g^{ij}(x)\partial_{i}\partial_{j}G(x,y)d\mu(y)=0$
Con un raginamento del tutto analogo possiamo dimostrare che la funzione
$G_{\mu}$ è continua sull’insieme $R\setminus S$.
Il fatto che $G_{\mu}$ è semicontinua inferiormente su $R$ segue dalla
considerazione che grazie al teorema di convergenza monotona:
$\displaystyle
G_{\mu}(x)=\int_{S}G(x,y)d\mu(y)=\lim_{n}\int_{S}(G(x,y)\curlywedge
n)d\mu(y)\equiv\lim_{n}G_{\mu}^{n}(x)$
La successione $G_{\mu}^{n}$ è una successione crescente di funzioni continue
292929la continuità segue dalla continuità della funzione $G(x,y)\curlywedge
n$ e dal teorema di convergenza dominata, quindi $G_{\mu}(x)$ è
necessariamente una funzione semicontinua inferiormente.
Sempre grazie al teorema di convergenza monotona, sappiamo che:
$\displaystyle G_{\mu}(x)\curlywedge c=\int_{S}(G(x,y)\curlywedge c)d\mu(y)$
Queste funzioni appartengono tutte a $\mathbb{M}(R)$, infatti sono continue,
di Tonelli, e:
$\displaystyle D_{R}(G_{\mu}(x)\curlywedge
c)=\int_{R}dx\left|\nabla\int_{S}(G(x,y)\curlywedge c)d\mu(y)\right|^{2}\leq$
$\displaystyle\leq\int_{R}dx\int_{S}\left|\nabla(G(x,y)\curlywedge
c)\right|^{2}d\mu(y)=\int_{S}d\mu(y)\int_{R}dx\left|\nabla(G(x,y)\curlywedge
c)\right|^{2}\leq$ $\displaystyle\leq\int_{S}d\mu(y)c=c$
dove abbiamo sfruttato il fatto che $\mu(R)=1$ 303030quindi
$\left|\int_{S}f(y)d\mu(y)\right|^{2}\leq\int_{S}\left|f(x)\right|^{2}d\mu(y)$
e il teorema di Fubuni per lo scambio di integrali quando l’integrando è
positivo.
Sempre per il teorema di convergenza monotona, detto $G_{n}(x,y)$ i nuclei di
Green rispetto a un’esaustione regolare di $R$, sappiamo che:
$\displaystyle G_{\mu}(x)\curlywedge c=\lim_{n}\int_{S}(G_{n}(x,y)\curlywedge
c)d\mu(y)$
Poiché tutte le funzioni $G_{n}(x,y)\curlywedge c$ appartengono a
$\mathbb{M}_{\Delta}(R)$ e
$\displaystyle D_{R}(G_{n}(x,y)\curlywedge c)\leq c$ (4.11)
grazie al teorema 3.35, $G_{\mu}(x)\curlywedge c\in\mathbb{M}_{\Delta}(R)$,
quindi $G_{\mu}|_{\Delta}=0$.
La superarmonicità di $G_{\mu}$ segue dalle proposizioni 1.76 e 1.77. Infatti
sappiamo che:
$\displaystyle
G_{\mu}(x)=\int_{K}G(x,y)d\mu(y)=\lim_{n}\int_{K}(G(x,y)\curlywedge n)d\mu(y)$
Le funzioni $\int_{K}(G(x,y)\curlywedge n)d\mu(y)$ sono superarmoniche grazie
alla proposizione 3.54, e anche il loro limite è superarmonico grazie a 1.77.
∎
###### Proposizione 4.47.
Se $\epsilon(K)<\infty$, esiste una misura $\nu\in m_{K}$ tale che:
$\displaystyle\epsilon(K)=\left\langle\nu\middle|\nu\right\rangle_{B}$
Cioè esiste una misura che realizza il minimo dell’energia.
###### Proof.
Dalla definizione di $\inf$, esiste una successione $\nu_{n}\in m_{K}$ tale
che:
$\displaystyle\lim_{n}\left\langle\nu_{n}\middle|\nu_{n}\right\rangle_{B}=\epsilon(K)$
Dalla teoria della misura e degli spazi di Banach, sappiamo che esiste una
sottosuccessione di $\\{\nu_{n}\\}$ (che per comodità continueremo a indicare
con lo stesso indice) tale che $\nu_{n}$ converge debolmente a $\nu\in m_{K}$,
cioè per ogni funzione $f\in C(K)$:
$\displaystyle\lim_{n}\int fd\nu_{n}=\int fd\nu$
Ora consideriamo le funzioni
$\displaystyle\phi_{n}^{c}(x)\equiv\int_{K}(G(x,y)\curlywedge c)d\nu_{n}(y)\ \
\ \ \ \ \phi^{c}(x)\equiv\int_{K}(G(x,y)\curlywedge c)d\nu(y)$
Ricordiamo che per convergenza monotona:
$\displaystyle\int_{K}G(x,y)d\nu(y)=\lim_{c\to\infty}\phi^{c}(x)$
Osserviamo che la successione $\\{\phi_{n}^{c}\\}$ converge uniformemente su
$K$ a $\phi^{c}$, infatti grazie alla definizione di $\nu$, c’è convergenza
puntuale. Inoltre:
$\displaystyle\left|\phi_{n}^{c}(x)\right|=\int_{K}(G(x,y)\curlywedge
c)d\nu_{n}(y)\leq c\nu_{n}(K)\leq cM$
$\displaystyle\left|\phi_{n}^{c}(x_{1})-\phi_{n}^{c}(x_{2})\right|\leq\int_{K}\left|(G(x_{1},y)\curlywedge
c)-G(x_{2},y)\curlywedge c\right|d\nu_{n}(y)$
Data la continuità uniforme della funzione $G(\cdot,\cdot)\curlywedge c$
sull’insieme $K\times K$, si ha che per ogni $\epsilon>0$, esiste $\delta$ per
cui
$d(x_{1},x_{2})<\delta\ \ \Rightarrow\ \ \left|G(x_{1},y)\curlywedge
c-G(x_{2},y)\curlywedge c\right|<\epsilon$
quindi se $d(x_{1},x_{2})<\delta$:
$\displaystyle\left|\phi_{n}^{c}(x_{1})-\phi_{n}^{c}(x_{2})\right|\leq\epsilon\nu_{n}(K)\leq\epsilon
M$
Queste osservazioni dimostrano l’uniforme limitatezza e l’equicontinuità della
successione $\phi_{n}^{c}$ sull’insieme $K$, quindi grazie al teorema di
Ascoli-Arzelà (vedi ad esempio appendice A5 pag 394 di [R2]) per ogni
sottosuccessione di $\\{\phi_{n}^{c}\\}$, esiste una sua sottosottosuccessione
convergente uniformemente su $K$. Dato che $\\{\phi_{n}^{c}\\}$ converge
puntualmente a $\phi^{c}$, allora la convergenza è uniforme su $K$.
Questo implica che:
$\displaystyle\left|\int_{K}\phi_{n}^{c}d\nu_{n}-\int_{K}\phi^{c}d\nu\right|\leq\left|\int_{K}(\phi_{n}^{c}-\phi^{c})d\nu_{n}\right|+\left|\int_{K}\phi^{c}d\nu_{n}-\int_{K}\phi^{c}d\nu\right|\leq$
(4.12)
$\displaystyle\leq\left\|\phi_{n}^{c}-\phi^{c}\right\|_{\infty,K}\nu_{n}(K)+\left|\int_{K}\phi^{c}d\nu_{n}-\int_{K}\phi^{c}d\nu\right|\to
0$
Da queste considerazioni otteniamo che:
$\displaystyle\left\langle\nu\middle|\nu\right\rangle_{B}=\lim_{c\to\infty}\int_{K}d\nu(x)\int_{K}d\nu(y)(G(x,y)\curlywedge
c)=\lim_{c\to\infty}\lim_{n\to\infty}\int_{K}\phi_{n}^{c}d\nu_{n}=$
$\displaystyle\lim_{c\to\infty}\liminf_{n\to\infty}\int_{K}\phi_{n}^{c}d\nu_{n}\leq\liminf_{n\to\infty}\lim_{c\to\infty}\int_{K}\phi_{n}^{c}d\nu_{n}=\liminf_{n\to\infty}\left\langle\nu_{n}\middle|\nu_{n}\right\rangle$
dove il penultimo passaggio, lo scambio tra limite e liminf, è giustificato
nel lemma seguente (lemma 4.48).
Questa disuguaglianza ci permette di concludere che:
$\displaystyle\epsilon(K)\leq\left\langle\nu\middle|\nu\right\rangle_{B}\leq\liminf_{n}\left\langle\nu_{n}\middle|\nu_{n}\right\rangle_{B}=\epsilon(K)$
da cui segue la tesi. ∎
###### Lemma 4.48.
Sia $a_{n,m}$ una successione con due indici crescente in $m$ per ogni $n$
fissato. Allora:
$\displaystyle\lim_{m\to\infty}\liminf_{n\to\infty}a_{n,m}\leq\liminf_{n\to\infty}\lim_{m\to\infty}a_{n,m}$
###### Proof.
Sia per definizione
$\displaystyle L\equiv\lim_{m\to\infty}\liminf_{n\to\infty}a_{n,m}\ \ \ \ \ \
\ b_{m}\equiv\liminf_{n\to\infty}a_{n,m}$
Dalla definizione di limite, sappiamo che per ogni $\epsilon>0$, esiste $M$
tale che per ogni $m\geq M$, $b_{m}>L-\epsilon$, quindi dalla definizione di
$\liminf$ sappiamo che fissato $\bar{m}$, esiste $N(\bar{m})$ tale che per
ogni $n\geq N(\bar{m})$:
$\displaystyle a_{n,\bar{m}}>L-\epsilon$
Data la monotonia di $a_{n,m}$ rispetto all’indice $m$, sappiamo che per ogni
$m\geq\bar{m}$ e per ogni $n\geq N(\bar{m})$, $a_{n,m}>L-\epsilon$, quindi
passando al limite in $m$:
$\displaystyle\lim_{m}a_{n,m}\geq L-\epsilon$
quindi passando al $\liminf$:
$\displaystyle\liminf_{n}\lim_{m}a_{n,m}\geq L-\epsilon$
la tesi si ottiene per arbitrarietà di $\epsilon$. ∎
Ora introduciamo una particolare misura positiva di Borel regolare
sull’insieme $K$, ragionando con le funzioni armoniche.
Sia $K$ una sottovarietà regolare compatta di codimensione $1$ (possibilmente
con bordo liscio). $R\setminus K$ avrà $n$ componenti connesse
$R_{1},\cdots,R_{n}$ 313131ad esempio, se $K$ è una sfera, $R\setminus K$ avrà
due componenti connesse, se invece $K$ è una parte di piano, $R\setminus K$ è
connesso, inoltre ovviamente $K=\cup\partial R_{i}$. Su ogni insieme $R_{i}$
possiamo risolvere un problema di Dirichlet, nel senso che data una funzione
$f$ continua su $\partial R_{i}$, esiste una funzione armonica in $R_{i}$ che
indicheremo $H^{i}(f)$ che ristretta al bordo di $R_{i}$ uguagli $f$.
Questa caratterizzazione però è sufficiente solo se $R_{i}$ è relativamente
compatto. In caso contrario indichiamo con $H^{i}(f)$ la funzione armonica
ottenuta per esaustione di $R_{i}$. In dettaglio, sia $C_{n}$ un’esaustione
regolare di $R$, e sia $H^{i}_{n}(f)$ la funzione
$\displaystyle H^{i}_{n}(f)\in H(R_{i}\cap C_{n}),\ \ H^{i}_{n}(f)|_{\partial
R_{i}\cap C_{n}}=f,\ \ H^{i}_{n}(f)|_{\partial C_{n}\cap R_{i}}=0$
grazie al principio del massimo si ottiene che $H^{i}_{n}(f)$ è una
successione crescente in $n$ di funzioni armoniche, e poiché sempre per il
principio del massimo
$\displaystyle\left\|H^{i}_{n}(f)\right\|_{\infty}\leq\left\|f\right\|_{\infty}$
grazie al principio di Harnack $H^{i}_{n}(f)$ converge a una funzione
$H^{i}(f)$ armonica su $R_{i}$ e continua su $\overline{R_{i}}$.
Ora se scegliamo un punto qualsiasi $z_{0}\in R^{i}$, per quanto appena visto
possiamo definire un funzionale lineare continuo positivo 323232per
definizione, $\phi$ è positivo se $\phi(f)\geq 0$ per ogni $f\geq 0$ sullo
spazio delle funzioni continue in $K$ in questo modo:
$\displaystyle\phi_{z_{0}}(f)\equiv H^{i}(f)(z_{0})$
dalla teoria 333333vedi teorema di rappresentazione di Riestz, ad esempio su
[R4], teorema 2.14 pag 40 sappiamo che esiste unica una misura di Borel
positiva regolare $\xi_{z_{0}}$ con supporto su $\partial R^{i}$ 343434quindi
con supporto in $K$ a patto di estenderla a nulla su $K\setminus\partial
R^{i}$ tale che:
$\displaystyle\phi_{z_{0}}(f)\equiv H^{i}(f)(z_{0})=\int_{K}fd\xi_{z_{0}}$
per ogni funzione $f$ continua su $K$.
###### Definizione 4.49.
La misura $\xi_{z_{0}}$ appena caratterizzata è detta la misura armonica
dell’insieme $K$ rispetto a $z_{0}$.
Nel seguente lemma ci occupiamo di un aspetto tecnico legato a questa misura,
in particolare a cosa succede su un insieme $F\subset K$ di misura nulla
rispetto a una di queste $\xi_{z_{0}}$.
###### Lemma 4.50.
Se $\xi_{z_{0}}(F)=0$, esiste una funzione armonica positiva
$h:R^{i}\to[0,\infty)$ tale che per ogni $x_{0}\in F$:
$\displaystyle\lim_{x\to x_{0}}h(x)=\infty$
Come corollario di questo lemma, possiamo dimostrare che la proprietà
$\xi_{z_{0}}(F)=0$ NON dipende dalla scelta di $z_{0}$ (se $z_{0}$ viene
scelto tra gli elementi dello stesso insieme $R^{i}$).
###### Proof.
Dato che $\xi_{z_{0}}\equiv\xi_{0}$ è una misura regolare, esiste una
successione di insiemi $\\{F_{n}\\}$ aperti nella topologia di $K$ tale che
$F\subset F_{n}\subset\overline{F_{n}}\subset F_{n+1}\ \ \ \
\bigcap_{n}F_{n}=F\ \ \ \ \xi_{0}(F_{n})\leq 2^{-n}$
Costruiamo le funzioni $q_{n}:K\to\mathbb{R}$ con queste proprietà:
1. 1.
$q_{n}\in C(K,[0,1])$ per ogni $n$
2. 2.
$supp(q_{n})\subset F_{n}$, e $q_{n}|_{F_{n+1}}=1$
Sia $s_{n}\equiv\sum_{i=1}^{n}q_{i}$. Visto quanto abbiamo appena osservato,
possiamo costruire la successione di funzioni armoniche positive
$\displaystyle h_{n}\equiv H^{i}(s_{n})$
Per definizione di $s_{n}$, sappiamo che $h_{n}|_{F}=s_{n}|_{F}=n$, inoltre la
successione di funzioni armoniche positive crescenti $h_{n}$ è tale che:
$\displaystyle
h_{n}(z_{0})=H^{i}(s_{n})(z_{0})=\int_{K}s_{n}d\xi_{z_{0}}=\sum_{j=1}^{n}\int_{K}q_{j}d\xi_{z_{0}}\leq\sum_{i=j}^{n}\int_{K}\chi_{F_{j}}d\xi_{z_{0}}\leq\sum_{j=1}^{n}2^{-j}\leq
1$
quindi per il principio di Harnack 353535vedi 1.57 la successione $h_{n}$
converge a una funzione armonica $h$ su $R^{i}$ semicontinua inferiormente su
$\overline{R^{i}}$ 363636dato che la successione $h_{n}$ è una successione
crescente di funzioni continue su $\overline{R^{i}}$ tale che $h(z)=\infty$
per ogni $z\in F$.
Consideriamo ora un punto $z_{1}\in R^{i}$, e sia $\xi_{z_{1}}$ la relativa
misura armonica. Per la successione $h_{n}$ vale che;
$\displaystyle h_{n}(z_{1})=\int_{K}s_{n}d\xi_{z_{1}}$
quindi passando al limite su $n$ otteniamo che:
$\displaystyle\lim_{n}\int_{K}s_{n}d\xi_{z_{1}}=h(z_{1})<\infty$
se per assurdo $\xi_{z_{1}}(F)>0$, allora necessariamente
$\displaystyle\int_{K}s_{n}d\xi_{z_{1}}\geq n\xi_{z_{1}}(F)\to\infty$
che contraddice quanto appena scoperto, quindi se $\xi_{z_{0}}(F)=0$, allora
per ogni $z_{1}\in R^{i}$ si ha che $\xi_{z_{1}}(F)=0$. Scambiando i ruoli di
$z_{0}$ e $z_{1}$ si ottiene che per un insieme $F\subset K$, il fatto di
avere misura armonica $\xi$ nulla è indipendente dalla scelta del punto
$z_{0}\in R^{i}$. ∎
La misura armonica è legata all’energia dal fatto che
###### Proposizione 4.51.
Sia $z_{0}\not\in K$ e $\xi_{z_{0}}=\xi$ la misura armonica di $K$ relativa a
$z_{0}$. Allora se un insieme di Borel $F\subset K$ ha misura armonica non
nulla ($\xi(F)\neq 0$), allora ha energia finita.
###### Proof.
Consideriamo la misura di Borel $\xi_{1}\equiv\xi|_{F}$. Il potenziale di
Green di questa misura è caratterizzato dal fatto che per ogni $x\not\in K$:
$\displaystyle
G_{\xi_{1}}(x)=\int_{K}G(x,y)d\xi_{1}(y)\leq\int_{K}G(x,y)d\xi(y)=H(G(x,\cdot)|_{z_{0}}\leq
G(x,z_{0})$
infatti per superarmonicità, la funzione $G(x,\cdot)$ è maggiore della
funzione armonica che assume valore $G(x,\cdot)$ sull’insieme $K$. Data la
continutà di $G(\cdot,z_{0})$ su $R\setminus\\{z_{0}\\}$ e data la
semicontinuità inferiore di $G_{\xi_{1}}$, la disuguaglianza continua a valere
anche se $x\in K$, dimostrando che:
$\displaystyle G_{\xi_{1}}(x)\leq G(z_{0},x)\leq M$
dove $M$ è un maggiorante per la funzione continua $G(z_{0},\cdot)|_{K}$,
quindi:
$\displaystyle\left\langle\xi_{1}\middle|\xi_{1}\right\rangle_{B}\leq
M\xi_{1}(K)<\infty$
∎
Con la seguente proposizione ci occupiamo di alcune proprietà fini del
potenziale $G_{\nu}$, dove $\nu$ è la misura che minimizza l’energia su $K$.
La proposizione e la relativa dimostrazione sono tratte dal paragrafo 9G di
[SN] e dal teorema III.12 di [T].
###### Proposizione 4.52.
Sia $K$ un’insieme con le caratteristiche descritte all’inizio del paragrafo,
sia $\xi$ la misura armonica di $K$ rispetto a un punto qualsiasi
$z_{0}\not\in K$ e sia $\nu\in m_{K}$ una misura tale che
$\displaystyle E\equiv\epsilon(K)=\left\langle\nu\middle|\nu\right\rangle_{B}$
Allora:
1. 1.
$G_{\nu}(x)\leq E$ sul supporto $S_{\nu}$ della misura $\nu$
2. 2.
$G_{\nu}(x)\geq E$ a meno di un insieme di misura armonica $\xi$ nulla
###### Proof.
Sia $A_{\delta}=\\{x\in K\ t.c.\ G_{\nu}(x)>E+2\delta\\}$, dove $\delta>0$.
Data la semicontinuità inferiore di $G_{\nu}$, $A_{\delta}$ sono insiemi
aperti per ogni $\delta$, quindi Borel misurabili. Supponiamo per assurdo che
$\nu(A_{\delta})>0$ per qualche $\delta>0$. Per la regolarità della misura
$\nu$, esiste un insieme $C\subset A_{\delta}$ compatto tale che $\nu(C)\neq
0$. Dato che
$\displaystyle
E=\epsilon(K)=\left\langle\nu\middle|\nu\right\rangle_{B}=\int_{S_{\nu}}G_{\nu}(x)d\nu(x)$
esiste un punto $x_{0}\in S_{\nu}$ tale $G_{\nu}(x_{0})\leq E+\delta$, inoltre
l’insieme
$B=\\{x\in K\ t.c.\ G_{\nu}(x)\leq E+\delta\\}$
è chiuso e $\nu(B_{\delta})\neq 0$. Consideriamo la misura di Borel
$\displaystyle\sigma(\cdot)=-\frac{\nu(\cdot\cap
C)}{\nu(C)}+\frac{\nu(\cdot\cap B)}{\nu(B)}$
Dato che
$\displaystyle 0\leq\left\langle\nu(\cdot\cap C)\middle|\nu(\cdot\cap
C)\right\rangle_{B}=\int_{K}G_{\nu(\cdot\cap C)}d\nu{\cdot\cap
C}\leq\int_{K}G_{\nu}d\nu(\cdot\cap C)=$
$\displaystyle=\int_{K}G_{\nu(\cdot\cap
C)}d\nu\leq\int_{K}G_{\nu}d\nu=\left\langle\nu\middle|\nu\right\rangle_{B}<\infty$
e con un ragionamento analogo anche $\left\langle\nu(\cdot\cap
B)\middle|\nu(\cdot\cap B)\right\rangle_{B}<\infty$, e dato che gli insiemi
$B$ e $C$ sono compatti disgiunti, allora
$\displaystyle 0\leq\left\langle\nu(\cdot\cap B)\middle|\nu(\cdot\cap
C)\right\rangle_{B}=\int_{K}G_{\nu(\cdot\cap B)}d\nu(\cdot\cap C)\leq M\nu(C)$
dove $M$ è un maggiorante per la funzione $G_{\nu(\cdot\cap B)}$ continua sul
compatto $C$,. Inoltre notiamo che
$\left\langle\sigma\middle|\sigma\right\rangle_{B}\neq\infty$, infatti:
$\displaystyle\left\langle\sigma\middle|\sigma\right\rangle_{B}=\left\langle-\frac{\nu(\cdot\cap
C)}{\nu(C)}+\frac{\nu(\cdot\cap B)}{\nu(B)}\middle|-\frac{\nu(\cdot\cap
C)}{\nu(C)}+\frac{\nu(\cdot\cap B)}{\nu(B)}\right\rangle_{B}=$
$\displaystyle=\frac{1}{\nu(C)^{2}}\left\langle\nu(\cdot\cap
C)\middle|\nu(\cdot\cap
C)\right\rangle_{B}-\frac{2}{\nu(C)\nu(B)}\left\langle\nu(\cdot\cap
C)\middle|\nu(\cdot\cap B)\right\rangle_{B}+$
$\displaystyle+\frac{1}{\nu(B)^{2}}\left\langle\nu(\cdot\cap
B)\middle|\nu(\cdot\cap B)\right\rangle_{B}$
Consideriamo ora per $0\leq\eta<\nu(C)$ la misura $\nu_{\eta}=\nu+\eta\sigma$.
È facile verificare che $\nu_{\eta}$ è una misura di Borel regolare positiva e
$\nu_{\eta}(K)=1$, quindi $\nu_{\eta}\in m_{K}$. D’altronde osserviamo che:
$\displaystyle\left\langle\nu_{\eta}\middle|\nu_{\eta}\right\rangle_{B}-\left\langle\nu\middle|\nu\right\rangle_{B}=\left\langle\nu+\eta\sigma\middle|\nu+\eta\sigma\right\rangle_{B}-\left\langle\nu\middle|\nu\right\rangle_{B}=2\eta\left\langle\nu\middle|\sigma\right\rangle_{B}+\eta^{2}\left\langle\sigma\middle|\sigma\right\rangle_{B}$
Inoltre:
$\displaystyle\left\langle\nu\middle|\sigma\right\rangle_{B}=-\frac{1}{\nu(C)}\int_{C}G_{\nu}d\nu+\frac{1}{\nu(B)}\int_{B}G_{\nu}d\nu\leq-E-2\delta+E+\delta<-\delta<0$
Quindi per $\eta$ sufficientemente piccolo,
$\left\langle\nu_{\eta}\middle|\nu_{\eta}\right\rangle_{B}<\left\langle\nu\middle|\nu\right\rangle_{B}$,
contraddicendo la definizione di $\nu$.
La seconda parte della proposizione si dimostra in maniera analoga alla prima.
Supponiamo per assurdo che esista $\delta>0$ tale che l’insieme
$A_{\delta}=\\{x\in K\ t.c.\ G_{\nu}(x)\leq E-2\delta\\}$
abbia misura armonica positiva. Grazie alla proposizione 4.51, l’energia di
$A$ è finita, quindi esiste una misura positiva unitaria $\sigma_{1}$ con
supporto in $A$ tale che
$\left\langle\sigma_{1}\middle|\sigma_{1}\right\rangle_{B}<\infty$. Sia
$B=\\{x\in Kt.c.\ G_{\nu}(x)>E-\delta\\}$. Dato che
$\displaystyle\int_{K}G_{\nu}(x)d\nu(x)=E$
$\nu(B)\neq 0$. Se definiamo
$\sigma(\cdot)=\sigma_{1}(\cdot)-\frac{\nu(\cdot\cap B)}{\nu(B)}$,
analogamente a quanto dimostrato sopra, si dimostra che
$\sigma(K)=\sigma(R)=0$ e che per ogni $0\leq\eta<\nu(B)$, la misura
$\nu+\eta\sigma\in m_{K}$.
La contraddizione segue dal fatto che:
$\displaystyle\left\langle\nu_{\eta}\middle|\nu_{\eta}\right\rangle_{B}-\left\langle\nu\middle|\nu\right\rangle_{B}=2\eta\left\langle\sigma\middle|\nu\right\rangle_{B}+\eta^{2}\left\langle\sigma\middle|\sigma\right\rangle_{B}$
$\displaystyle\left\langle\sigma\middle|\nu\right\rangle_{B}=\int_{K}G_{\nu}(x)d\sigma=$
$\displaystyle=\int_{A_{\delta}}G_{\nu}(x)d\sigma_{1}-\frac{1}{\nu(B)}\int_{B}G_{\nu}(x)d\nu\leq
E-2\delta-E+\delta\leq-\delta<0$
∎
Nella seguente proposizione dimostriamo una proprietà di continuità del
potenziale di Green $G_{\mu}$ rispetto a una qualsiasi $\mu\in m_{K}$.
###### Proposizione 4.53.
Sia $\mu$ una misura di Borel positiva finita e $G_{\mu}$ il relativo
potenziale di Green. Se $G_{\mu}$ ristretto al supporto $S_{\mu}$ della misura
$\mu$ è continuo, allora $G_{\mu}$ è una funzione continua su tutta la varietà
$R$.
###### Proof.
Dato che $G_{\mu}\in H(R\setminus S_{\mu})$, è sufficiente dimostrare che
$G_{\mu}$ è continua sull’insieme $S_{\mu}$, quindi che per ogni $x_{0}\in
S_{\mu}$:
$\displaystyle\lim_{x\to x_{0}}G_{\mu}(x)=G_{\mu}(x_{0})$
Sappiamo dalle proposizioni precedenti che $G_{\mu}$ è semicontinua
inferiormente, quindi basta dimostrare che:
$\displaystyle\limsup_{x\to x_{0}}G_{\mu}(x)\leq G_{\mu}(x_{0})$ (4.13)
inoltre possiamo assumere che $G_{\mu}(x_{0})<\infty$, quindi
$\mu(\\{x_{0}\\})=0$.
Consideriamo un intorno coordinato di $U$ di $x_{0}$. Su questo intorno
possiamo scrivere:
$\displaystyle G(x,y)=f(x,y)+C(m)h(x,y)$
dove $f(x,y)$ è una funzione continua in entrambe le sue variabili, $C(m)$ una
costante che dipende solo da $m=dim(R)$, mentre
$\displaystyle h(x,y)=\begin{cases}-\log(d(x,y))\ &\ se\ m=2\\\ d(x,y)^{-m+2}\
&\ se\ m\geq 3\end{cases}$
Decomponendo la misura $\mu$ in $\mu_{0}(\cdot)\equiv\mu(\cdot\cap U)$ e
$\mu^{\prime}\equiv\mu-\mu_{0}$ possiamo scrivere il potenziale di Green come:
$\displaystyle G_{\mu}(x)=G_{\mu_{0}}(x)+G_{\mu^{\prime}}(x)$
e dato che $G_{\mu^{\prime}}(x)$ è una funzione continua in $x_{0}$, basta
dimostrare 4.13 per il solo potenziale $G_{\mu_{0}}$. Inoltre
$\displaystyle
G_{\mu_{0}}(x)=\int_{K}f(x,y)d\mu_{0}(y)+C(m)\int_{K}h(x,y)d\mu_{0}(y)$
e dato che la misura $\mu_{0}$ è finita, il primo integrale è una funzione
continua della variabile $x$, quindi ancora basta dimostrare 4.13 per la
funzione
$H(x)\equiv\int_{K}h(x,y)d\mu_{0}(y)$
Poiché $\mu(\\{x_{0}\\})=0$, e dato che $\mu$ è una misura regolare, per ogni
$\epsilon>0$, esiste un aperto $U_{\epsilon}\subset U$ contenente $x_{0}$ tale
che $\mu(U_{\epsilon})=\mu_{0}(U_{\epsilon})<\epsilon$, quindi vale che:
$\displaystyle H(x)=\int_{U_{\epsilon}}h(x,y)d\mu_{0}(y)+\int_{U\setminus
U_{\epsilon}}h(x,y)d\mu_{0}(y)$
Per ogni $x\in U$, definiamo $\pi(x)\in S_{\mu}$ un punto tale che
$d(x,\pi(x))=\min_{y\in S_{\mu}}d(x,y)$. Osserviamo che in generale il punto
$\pi(x)$ non è unico, ma comunque
$\lim_{x\to x_{0}}\pi(x)=x_{0}$
se $x_{0}\in S_{\mu}$, inoltre vale che per ogni $y\in S_{\mu}$:
$\displaystyle d(y,\pi(x))\leq d(x,y)+d(x,\pi(x))\leq 2d(x,y)$
quindi:
$\displaystyle h(x,y)\leq\begin{cases}\log(2)+h(\pi(x),y)&se\ m=2\\\
2^{m-2}h(\pi(x),y)&se\ m\geq 3\end{cases}$
Dividiamo la dimostrazione in due casi: se $n=2$:
$\displaystyle
H(x)\leq\int_{U_{\epsilon}}h(\pi(x),y)d\mu_{\epsilon}+\epsilon\log(2)+\int_{U\setminus
U_{\epsilon}}h(x,y)d\mu_{0}(y)=$
$\displaystyle=H(\pi(x))+\epsilon\log(2)+\int_{U\setminus
U_{\epsilon}}(h(x,y)-h(\pi(x),y)d\mu_{0}(y)$
Applicando il $\limsup_{x\to x_{0}}$ a entrambi i membri e tenedo conto
dell’ipotesi di continuità di $G_{\mu}$ (quindi di $H$) ristretta all’insieme
$S_{\mu}$ otteniamo:
$\displaystyle\limsup_{x\to x_{0}}H(x)\leq
H(x_{0})+\epsilon\log(2)+\limsup_{x\to x_{0}}\int_{U\setminus
U_{\epsilon}}(h(x,y)-h(\pi(x),y)d\mu_{0}(y)$
Per convergenza dominata, l’ultimo addendo è nullo, e data l’arbitrarietà di
$\epsilon$, otteniamo la tesi.
In maniera analoga, se $m\geq 3$:
$\displaystyle
H(x)\leq\int_{U_{\epsilon}}h(\pi(x),y)d\mu_{\epsilon}+\alpha_{m}\int_{U_{\epsilon}}h(\pi(x),y)d\mu(y)+\int_{U\setminus
U_{\epsilon}}h(x,y)d\mu_{0}(y)$
dove $\alpha_{m}=2^{m-2}-1$. Applicando il $\limsup_{x\to x_{0}}$ a entrambi i
membri e tenendo conto della continuità della funzione $H$ ristretta
all’insieme $S_{\mu}$, otteniamo:
$\displaystyle\limsup_{x\to x_{0}}H(x)\leq
H(x_{0})+\alpha_{m}\int_{U_{\epsilon}}h(x_{0},y)d\mu(y)+$
$\displaystyle+\limsup_{x\to x_{0}}\int_{U\setminus
U_{\epsilon}}(h(x,y)-h(\pi(x),y)d\mu_{0}(y)$
Come nel caso bidimensionale, l’ultimo limite è nullo per convergenza dominata
373737infatti dato che $x\to x_{0}$, $x$ e $\pi(x)$ appartengono
definitivamente a un intorno compatto di $x_{0}$ contenuto in $U_{\epsilon}$,
quindi se $y\in U\setminus U_{\epsilon}$, sia $h(x,y)$ che $h(\pi(x),y)$ sono
uniformemente limitate da una costante, mentre dato che
$\int_{U}h(x_{0},y)d\mu(y)<\infty$, grazie a una nota proprietà degli
integrali se $\epsilon$ è sufficientemente piccolo
$\int_{U_{\epsilon}}h(x_{0},y)d\mu(y)$ può essere reso piccolo a piacere.
Quindi ancora una volta otteniamo che:
$\displaystyle\limsup_{x\to x_{0}}H(x)\leq H(x_{0})$
da cui la tesi. ∎
###### Osservazione 4.54.
Osserviamo che nel caso $m=2$, la relazione 4.13 segue dal “principio di
Frostman” (vedi ad esempio la sezione 9E pag 320 di [SN], o il teorema III.1
di [T]) e non è necessario assumere che $G_{\mu}$ sia continuo quando
ristretto all’insieme $S_{\mu}$. In dimensione maggiore, però, la
dimostrazione di questo principio non è facilmente estendibile, per questa
ragione riportiamo solo la versione generale della proposizione (che comunque
è sufficiente per i nostri scopi).
Questa proposizione e l’armonicità di $G_{\mu}$ su $R\setminus S_{\mu}$ ci
permettono di dimostrare che:
###### Proposizione 4.55.
Se $G_{\mu}$ è continua quando ristretta all’insieme $S_{\mu}$, allora il suo
massimo è raggiunto su $S_{\mu}$, quindi se $G_{\mu}$ è continua quando
ristretta a $S_{\mu}$ e $G_{\mu}(x)\leq c$ per ogni $x\in S_{\mu}$, allora
$G_{\mu}(x)\leq c$ per ogni $x\in R$.
###### Proof.
La dimostrazione è una semplice applicazione del teorema 3.58 (ricordiamo che
$G_{\mu}|_{\Delta}=0$). ∎
Questa proposizione può essere migliorata, nel senso che non è necessario
chiedere la continuità di $G_{\mu}$ sull’insieme $S_{\mu}$. A questo scopo
riportiamo i seguenti lemmi, tratto dai teoremi 3.6.2 e 3.6.3 di [H1], e
cortesemente segnalati dal professor. Wolfhard Hansen (University of
Bielefeld):
###### Lemma 4.56.
Data una misura positiva regolare finita $\mu$ con supporto compatto $S_{\mu}$
tale che $G_{\mu}<\infty$ su $S_{\mu}$, per ogni $\epsilon>0$, esiste un
insieme compatto $C\subset S_{\mu}$ tale che, detta
$\mu|(\cdot)C\equiv\mu(\cdot\cap C)$, $G_{\mu|C}$ è continua su $C$ e
$\mu(S_{\mu}\setminus C)<\epsilon$.
###### Proof.
Grazie al teorema di Lusin (vedi ad esempio teorema 2.23 pag 53 di [R4]), per
ogni $\epsilon>0$ esiste un compatto $C\subset S_{\mu}$ tale che
$G_{\mu}|_{C}$ è una funzione continua. Dato che:
$\displaystyle G_{\mu|C}=G_{\mu}-G_{\mu-\mu|C}$
e dato che su $C$ $G_{\mu}$ è continua, $G_{\mu|C}$ è semicontinua
superiormente su $C$, e quindi continua su $C$ e automaticamente continua su
tutto $R$ grazie alla proposizione 4.53. ∎
###### Lemma 4.57.
Se $G_{\mu}<\infty$ su $S_{\mu}$, allora esiste una successione di misure di
Borel regolari $\mu_{n}$ tali che $\\{G_{\mu_{n}}\\}$ è una successione
crescente di funzioni continue su $R$ e:
$\displaystyle G_{\mu}(x)=\lim_{n}G_{\mu_{n}}(x)$
###### Proof.
Scegliamo per induzione una successione di insiemi compatti $C_{n}$ tali che
$C_{n}\subset C_{n+1}$, $\mu(S_{\mu}\setminus C_{n})\leq 2^{-n}$ e $G_{\mu}$
continua sull’insieme $C_{n}$, e consideriamo $\mu_{n}=\mu|C_{n}$.
Dal lemma precedente, sappiamo che $G_{\mu_{n}}$ è una funzione continua, e
per costruzione delle misure $\mu_{n}$, ovviamente $\\{G_{\mu_{n}}(x)\\}$ è
una successione crescente $\forall x\in R$. Inoltre:
$\displaystyle G_{\mu_{n}}(x)=\int_{K}G(x,y)\chi_{C_{n}}(y)d\mu(y)$
e quindi per convergenza monotona, $\lim_{n}G_{\mu_{n}}(x)=G_{\mu}(x)$. ∎
Ora siamo pronti per dimostrare che:
###### Proposizione 4.58.
Se $G_{\mu}(x)\leq c$ per ogni $x\in S_{\mu}$, allora $G_{\mu}(x)\leq c$ per
ogni $x\in R$.
###### Proof.
Sia $\\{\mu_{n}\\}$ una successione di misure con le caratteristiche descritte
nell’ultimo lemma, allora $G_{\mu_{n}}(x)\leq G_{\mu}(x)\leq c$ per ogni $x\in
S_{\mu}$, e dato che $G_{\mu_{n}}$ sono funzioni continue, grazie alla
proposizione 4.55, sappiamo che per ogni $x$ per ogni $n$:
$G_{\mu_{n}}(x)\leq c$
Passando al limite su $n$ otteniamo la tesi. ∎
Grazie a questa proposizione, possiamo dimostrare questo teorema che
garantisce lega l’energia di un insieme alla sua capacità:
###### Teorema 4.59.
Sia $K$ una sottovarietà regolare di codimensione $1$ di $R$ possibilmente con
bordo liscio. Esiste una misura di Borel regolare unitaria 383838cioè
$\nu(R)=1$ $\nu$ con supporto in $K$ tale che:
$\displaystyle\epsilon(K)=\left\langle\nu\middle|\nu\right\rangle_{B}<\infty$
Inoltre detto $u$ il potenziale armonico dell’insieme $K$, per il potenziale
di Green relativo a $\nu$ vale che:
1. 1.
$G_{\nu}\in H(R\setminus K)$
2. 2.
$G_{\nu}(x)=\epsilon(K)u(x)$ per ogni $x\in R$
3. 3.
$G_{\nu}\in\mathbb{M}_{\Delta}(R)$
4. 4.
$D_{R}(G_{\nu})=\epsilon(K)$
5. 5.
$S_{\nu}=K$
Inoltre la misura $\nu$ è unica.
###### Proof.
Sia $u$ il potenziale di capacità dell’insieme $K$ (vedi osservazione 4.8).
Sia $\nu$ la misura che risolve il problema 4.47. Sappiamo che il potenziale
$G_{\nu}$ soddisfa
$G_{\nu}|_{S_{\nu}}\leq\epsilon(K)u|_{S_{\nu}}=\epsilon(K)$, e grazie alla
proposizione 4.58, sappiamo anche che $G|_{\nu}\leq\epsilon(K)$ su tutta la
varietà $R$. Dato che entrambe le funzioni $G_{\nu}$ e $u$ sono armoniche in
$R\setminus K$, su annullano su $\Delta$ e
$G_{\nu}|_{K}\leq\epsilon(K)=\epsilon(K)u|_{K}$, grazie al principio 3.58,
possiamo concludere che
$\displaystyle G_{\nu}(x)\leq\epsilon(K)u(x)\ \ \ \forall x\in R$
Per dimostrare questa disuguaglianza con il verso opposto, sappiamo che
$G_{\nu}(x)\geq\epsilon(K)u(x)$ per $x\in K\setminus F$ dove $F$ è un’insieme
di misura armonica $\xi$ nulla, quindi grazie al lemma 4.50 esiste una
funzione $w$ armonica su $R\setminus K$ che tende a infinito nei punti di $F$.
Allora per ogni $\epsilon>0$, la funzione
$\displaystyle G_{\nu}(x)+\epsilon w(x)\geq\epsilon(K)u(x)$
per ogni $x\in R\setminus K$ grazie al teorema 3.58. Data l’arbitrarietà di
$\epsilon$, possiamo concludere che
$\displaystyle G_{\nu}(x)=\epsilon(K)u(x)\ \ \ \forall x\in R\setminus K$
Dato che $u$ è continua su $R$ e $G_{\nu}$ è superarmonica, quindi
semicontinua inferiormente, l’uguaglianza vale su tutto $R$. Sia infatti
$x_{0}\in K$. Allora esiste una successione $\\{x_{n}\\}\subset R\setminus K$
che converge a $x_{0}$ e per la quale:
$\displaystyle
G_{\nu}(x_{0})\geq\liminf_{n}G_{\nu}(x_{n})\geq\lim_{n}\epsilon(K)u(x_{n})=\epsilon(K)$
dato che $G_{\nu}\leq\epsilon(K)$, necessariamente
$G_{\nu}(x_{0})=\epsilon(K)=\epsilon(K)u(x_{0})$.
Per quanto riguarda la proprietà (4), osserviamo che per ogni aperto
relativamente compatto con bordo liscio $C$ tale che $K\subset C$:
$\displaystyle D_{R}(u)=-\int_{\partial C}\ast du$
Consideriamo infatti una successione di aperti con bordo liscio $\\{A_{n}\\}$
tali che $A_{n}\subset A_{n-1}$ e $K=\cap_{n}A_{n}$ 393939un esempio di
insiemi con queste caratteristiche è descritto nell’osservazione 1.27. Allora
sappiamo che:
$\displaystyle D_{R}(u)=\lim_{n}D_{R\setminus
A_{n}}(u)=\lim_{n}\int_{\partial(R\setminus A_{n})}u\ast
du=-\lim_{n}\int_{\partial A_{n}}u\ast du$
Grazie al fatto che $u$ è continua, è anche uniformemente continua su ogni
insieme compatto, quindi ad esempio sull’insieme $\overline{A_{1}}$. Quindi
per ogni $\epsilon>0$, esiste $\delta$ tale che
$\displaystyle d(x,y)<\delta\ \ \Rightarrow\ \
\left|u(x)-u(y)\right|<\epsilon$
In particolare, se $d(x,K)<\delta$, otteniamo che $0\leq 1-u(x)\leq\epsilon$.
Dato che $\cap_{n}A_{n}=K$, $A_{n}$ è contenuto definitivamente nell’aperto
$\displaystyle K+\delta\equiv\\{x\in R\ t.c.\ d(x,K)<\delta\\}$
quindi definitivamente in $n$ vale che:
$\displaystyle\int_{\partial A_{n}}(1-u)\ast du=\int_{\partial
A_{n}}(1-u)(\ast du)^{+}-\int_{\partial A_{n}}(1-u)(\ast du)^{-}\leq$
$\displaystyle\leq\epsilon\int_{\partial A_{n}}(\ast
du)^{+}\leq\epsilon\int_{\partial A_{n}}\ast du$ $\displaystyle\int_{\partial
A_{n}}(1-u)\ast du=\int_{\partial A_{n}}(1-u)(\ast du)^{+}-\int_{\partial
A_{n}}(1-u)(\ast du)^{-}\geq$ $\displaystyle\geq-\epsilon\int_{\partial
A_{n}}(\ast du)^{-}\geq-\epsilon\int_{\partial A_{n}}\ast du$
cioé:
$\displaystyle-\epsilon\int_{\partial A_{n}}\ast du\leq\int_{\partial
A_{n}}(1-u)\ast du\leq\epsilon\int_{\partial A_{n}}\ast du$
dato che $u\in H(\overline{A_{n}}\setminus A_{m})$:
$\int_{\partial A_{n}}\ast du=\int_{\partial A_{m}}\ast du$
per ogni $n$ e $m$, quindi possiamo concludere che
$\displaystyle\lim_{n}\int_{\partial A_{n}}(1-u)\ast du=0\ \ \Rightarrow$
$\displaystyle\Rightarrow\ \ -D_{R}(u)=\lim_{n}\int_{\partial A_{n}}u\ast
du=\lim_{n}\int_{\partial A_{n}}\ast du=\int_{\partial C}\ast du$
Passiamo ora a considerare $D_{R}(G_{\mu})$
$\displaystyle-
D_{R}(G_{\nu})=-D_{R}(\epsilon(K)u)=\epsilon(K)^{2}\int_{\partial C}\ast
du=\epsilon(K)\int_{\partial C}\ast d(G_{\nu})=$
$\displaystyle=\epsilon(K)\int_{\partial C}\ast
d\left(\int_{K}G(x,y)d\mu(y)\right)=\epsilon(K)\int_{\partial C}\int_{K}\ast
dG(\cdot,y)d\mu(y)=$ $\displaystyle=\epsilon(K)\int_{K}d\mu(y)\int_{C}\ast
dG(\cdot,y)=-\epsilon(K)$
dove abbiamo sfruttato il fatto che $G(x,y)\in C^{\infty}(K\times C)$ per
scambiare tra loro i segni di integrali e derivate, e la relazione (6) di
1.61.
Resta da dimostrare la proprietà (5). A questo scopo notiamo che per il
principio del massimo forte, una funzione armonica su $R$ non può assumere il
suo massimo in un punto interno del dominio, quindi se $x\in K\setminus
S_{\nu}$, $G_{\nu}(x)<\max_{x\in R}G_{\nu}(x)=\epsilon(K)$. Ma dato che
$G_{\nu}(x)=\epsilon(K)u(x)$, e dato che il potenziale di capacità $u$ è
identicamente uguale a $1$ su $S_{\nu}$, necessariamente l’insieme $K\setminus
S_{\nu}$ è vuoto.
Una dimostrazione alternativa di questa affermazione si può ottenere
sfruttando una tecnica del tutto analoga a quella utilizzata nella
dimostrazione della proposizione 4.52. Supponiamo per assurdo che
$S_{\nu}\not=K$. Allora, dato che $S_{\nu}$ è chiuso, sulla sottovarietà $K$
esiste una bolla chiusa $m-1$ dimensionale contenuta in $K$ ma disgiunta da
$S_{\nu}$. Grazie all’osservazione 4.44, esiste una misura $\mu\in m_{K}$ con
supporto contenuto nella bolla tale che
$\left\langle\mu\middle|\mu\right\rangle_{B}<\infty$. Per ogni
$0\leq\lambda\leq 1$, consideriamo la misura:
$\displaystyle\nu_{\lambda}\equiv(1-\lambda)\nu+\lambda\mu$
È facile verificare che $\nu_{\lambda}\in m_{K}$ per ogni $0\leq\lambda\leq 1$
inoltre:
$\displaystyle\left\langle\nu_{\lambda}\middle|\nu-\lambda\right\rangle_{B}-\left\langle\nu\middle|\nu\right\rangle_{B}=$
$\displaystyle=[(1-\lambda)^{2}-1]\left\langle\nu\middle|\nu\right\rangle_{B}+\lambda^{2}\left\langle\mu\middle|\mu\right\rangle_{B}+2\lambda(1-\lambda)\left\langle\nu\middle|\mu\right\rangle_{B}=$
$\displaystyle=\lambda^{2}(\left\langle\nu\middle|\nu\right\rangle_{B}+\left\langle\mu\middle|\mu\right\rangle_{B}-2\left\langle\nu\middle|\mu\right\rangle_{B})+2\lambda(\left\langle\nu\middle|\mu\right\rangle_{B}-\left\langle\nu\middle|\nu\right\rangle_{B})$
Dato che
$\displaystyle\left\langle\nu\middle|\mu\right\rangle_{B}=\int_{K}G_{\nu}(x)d\mu\leq
M<\epsilon(K)<\infty$
dove $M$ è il massimo della funzione $G_{\nu}$ sull’insieme $S_{\mu}$, che è
strettamente minore di $\epsilon(K)$ grazie al principio del massimo,
$\left\langle\nu\middle|\mu\right\rangle_{B}-\left\langle\mu\middle|\mu\right\rangle_{B}<0$,
quindi esiste un valore di $\lambda$ sufficientemente piccolo per il quale
$\displaystyle\left\langle\nu_{\lambda}\middle|\nu_{\lambda}\right\rangle_{B}<\left\langle\nu\middle|\nu\right\rangle_{B}$
contraddicendo l’ipotesi di minimalità di $\nu$.
Concludiamo con la dimostrazione dell’unicità della misura $\nu$. Supponiamo
per assurdo che esista un’altra misura di minimo $\bar{\nu}$. Vale comunque
che $G_{\nu}(x)=\epsilon(K)u(x)=G_{\bar{\nu}}(x)$. Grazie al punto (5) della
proposizione 4.28, per ogni funzione liscia a supporto compatto in $R$,
abbiamo che:
$\displaystyle\int_{R}f(y)d\nu(y)=\int_{R}\left(\int_{R}G(x,y)\Delta
f(x)d\lambda(x)\right)d\nu(y)$
dato che
$\displaystyle\int_{R}\left|\Delta
f(x)\right|\int_{R}G(x,y)d\nu(y)d\lambda(x)\leq M\epsilon(K)\lambda(S)<\infty$
dove $M$ è un maggiorante di $\left|\Delta f(x)\right|$ e $S$ il supporto
della funzione $f$, grazie al teorema di Fubini possiamo scambiare l’ordine di
integrazione e ottenere:
$\displaystyle\int_{R}f(y)d\nu(y)=\int_{R}\Delta
f(x)G_{\nu}(x)d\lambda(x)=\int_{R}f(y)d\bar{\nu}(y)$
Per densità delle funzioni lisce a supporto compatto nelle funzioni continue a
supporto compatto, necessariamente $\nu=\bar{\nu}$. ∎
###### Osservazione 4.60.
Osserviamo subito che nelle ipotesi del teorema precedente, otteniamo che la
capacità di $K$ è precisamente l’inverso di $\epsilon(K)$, infatti:
$\displaystyle
Cap(K)=D_{R}(u)=\frac{D_{R}(G_{\mu})}{\epsilon(K)^{2}}=\epsilon(K)^{-1}$
Con la seguente stima leghiamo l’energia al diametro transfinito di un
insieme.
###### Proposizione 4.61.
Sia $K\subset R$. Allora $\rho(K)\geq\epsilon(K)$.
###### Proof.
Grazie alla definizione di $\rho_{n}(K)$, per ogni $n$ possiamo scegliere $n$
punti $p_{1}\cdots,p_{n}$ in $K$ tali che
$\displaystyle\binom{n}{2}\rho_{n}(K)\geq\sum_{i<j}^{1,\cdots,n}G(p_{i},p_{j})-\frac{1}{n}$
(4.14)
Sia $\mu_{n}$ la misura data da
$\displaystyle\mu_{n}=\frac{1}{n}\sum_{i=1}^{n}\delta_{p_{i}}$
dove $\delta_{x}$ è la misura che associa $1$ agli insiemi contenenti $x$ e
$0$ agli altri. Quindi $\mu_{n}\in m_{K}$. Allora esiste una sottosuccessione
di $\\{\mu_{n}\\}$ (che per comodità continueremo a indicare con lo stesso
simbolo) che converge debolmente nel senso della misura a una misura $\mu$,
cioè esiste una misura $\mu$ tale che per ogni funzione $\phi$ continua su
$K$:
$\displaystyle\lim_{n}\int_{K}\phi d\mu_{n}=\int_{K}\phi d\mu$
quindi la misura $\mu$ è di Borel, positiva e unitaria. Definiamo le funzioni:
$\displaystyle\phi_{n}^{c}(x)\equiv\int_{K}(G(x,y)\curlywedge c)d\mu_{n}(y)\ \
\ \ \ \ \phi^{c}(x)\equiv\int_{K}(G(x,y)\curlywedge c)d\mu(y)$
e seguendo lo stesso ragionamento della dimostrazione della proposizione 4.47,
otteniamo la validità dell’equazione 4.12, cioè otteniamo:
$\displaystyle\lim_{n}\int_{K}d\mu_{n}(x)\int_{K}d\mu_{n}(y)(G(x,y)\curlywedge
c)=\int_{K}d\mu(x)\int_{K}d\mu(y)(G(x,y)\curlywedge c)$
Moltiplicando ambo i membri di 4.14 per $\binom{n}{2}^{-1}$ otteniamo:
$\displaystyle\rho_{n}(K)+\frac{1}{n}\binom{n}{2}^{-1}\geq\binom{n}{2}^{-1}\sum_{i<j}^{1,\cdots,n}G(p_{i},p_{j})\geq\frac{2}{n^{2}}\sum_{i<j}^{1,\cdots,n}(G(p_{i},p_{j})\curlywedge
c)$
dove abbiamo sfruttato il fatto che $\binom{n}{2}\leq\frac{n^{2}}{2}$. Ora
dalla definizione di $\mu_{n}$ otteniamo che:
$\displaystyle\sum_{i,j}^{1,\cdots,n}(G(p_{i},p_{j})\curlywedge
c)=n^{2}\int_{K}d\mu_{n}\int_{K}d\mu_{n}(G(x,y)\curlywedge c)$
considerando che la funzione $G$ è simmetrica nei suoi due argomenti e che:
$\displaystyle\sum_{i=j=1}^{n}(G(p_{i},p_{j})\curlywedge
c)=\sum_{i=1}^{n}(G(p_{i},p_{i})\curlywedge c)=nc$
otteniamo:
$\displaystyle\sum_{i<j}^{1,\cdots,n}(G(p_{i},p_{j})\curlywedge
c)=\frac{n^{2}}{2}\int_{K}d\mu_{n}\int_{K}d\mu_{n}(G(x,y)\curlywedge
c)-\frac{nc}{2}$
quindi:
$\displaystyle\rho_{n}(K)+\frac{1}{n}\binom{n}{2}^{-1}\geq\int_{K}d\mu_{n}\int_{K}d\mu_{n}(G(x,y)\curlywedge
c)-\frac{c}{2n}$
facendo tendere $n$ a infinito, otteniamo:
$\displaystyle\rho(K)\geq\lim_{n}\int_{K}d\mu_{n}\int_{K}d\mu_{n}(G(x,y)\curlywedge
c)=\int_{K}d\mu\int_{K}d\mu(G(x,y)\curlywedge c)$
grazie all’arbitrarietà del parametro $c$, concludiamo:
$\displaystyle\rho(K)\geq\int_{K}d\mu\int_{K}d\mu G(x,y)\geq\epsilon(K)$
come volevasi dimostrare. ∎
L’ultima proposizione “tecnica” della dimostrazione è:
###### Proposizione 4.62.
La capacità di $F_{n+1,m}$ tende a $0$ se $m$ tende a infinito.
###### Proof.
Ricordiamo che la capacità di $F_{n+1,m}$ (insieme che in questa dimostrazione
indicheremo per comodità con $F_{m}$) equivale per definizione all’integrale
di Dirichlet della funzione $u_{n+1,m}$ (che indicheremo per semplicità
$u_{m}$), funzione armonica su $R\setminus F_{m}$,
$u_{m}\in\mathbb{M}_{\Delta}(R)$ e $u_{m}|_{\partial F_{m}}=1$.
Grazie al lemma 3.51, sappiamo che esiste una funzione
$f_{m}\in\mathbb{M}_{\Delta}(R)$, $f_{m}=1$ su
$\displaystyle E_{n+1,m}\equiv\overline{U_{n+1}\cap K_{m}^{C}}$
Infatti per definizione:
$\displaystyle U_{n}\equiv\\{z\in R\ t.c.\ G(z,z_{0})>r_{n}\\}$
e per 4.28, $G(\cdot,z_{0})|_{\Delta}=0$, cioè $U_{n}\cap\Delta=\emptyset\
\forall n$. Possiamo scegliere questa funzione armonica su $R\setminus
E_{n+1,m}$, infatti definiamo $W_{n+1,m,p}$ con $p>m$ l’insieme
$\displaystyle W_{n+1,m,p}\equiv U_{n+1}\cap(\overline{K_{p}}\setminus K_{m})$
e chiamiamo $\pi_{m,p}$ l’operatore definito in 3.55 rispetto all’insieme
$W_{n+1,m,p}$. La successione $\\{w_{m,p}\equiv\pi_{m,p}(f_{m})\\}$ è una
successione di funzioni armoniche su $R\setminus E_{n+1,m}$, appartenenti
all’insieme $\mathbb{M}_{\Delta}(R)$.
Grazie al punto (2) del teorema 3.55, si ha che:
$\displaystyle D(w_{m,p+q}-w_{m,p},w_{m,p})=0\ \ \Rightarrow\ \
D(w_{m,p+q}-w_{m,p})=D(w_{m,p})-D(w_{m,p+q})$
Con un ragionamento analogo a quello esposto in 3.35, otteniamo che la
successione $\\{w_{m.p}\\}$ è D-cauchy rispetto all’indice $p$.
Osserviamo che la successione $\\{w_{m,p}\\}$ è una successione crescente in
$p$, infatti dato che $w_{m,p}=1$ su $W_{n+1,m,p}$ e $w_{m,p}|_{\Delta}=0$,
grazie all’osservazione 3.60 $w_{m,p}\leq 1$ su $R$, quindi in particolare su
$W_{n+1,m,p+1}$ si ha che $w_{m,p}\leq w_{m,p+1}$. Sempre grazie alla
proposizione 3.60, otteniamo che questa disuguaglianza è valida su tutto $R$.
Quindi la successione $w_{m,p}$ converge monotonamente rispetto a $p$ a una
funzione $w_{m}$ armonica su $R\setminus E_{n+1,m}$ per il principio di
Harnack e identicamente uguale a $1$ su $E_{n+1,m}$ per costruzione. È facile
verificare che la convergenza è locale uniforme su $R$.
Dimostriamo che:
$\displaystyle 0=BD-\lim_{m}w_{m}$
Il punto (2) di 3.55 garantisce che per ogni $p$:
$\displaystyle D_{R}(w_{m+q,\ p}-w_{m},w_{m+q,\ p})=0$
infatti $w_{m+q,\ p}-w_{m}=0$ su $W_{n+1,m,p}$ e questa funzione è in
$\mathbb{M}_{\Delta}(R)$. Passando al limite su $p$, otteniamo che:
$\displaystyle
0=D_{R}(w_{m+q}-w_{m},w_{m+q})=D_{R}(w_{m+q})-D_{R}(w_{m+q},w_{m})$
da cui:
$\displaystyle D_{R}(w_{m+q}-w_{m})=D_{R}(w_{m})-D_{R}(w_{m+q})$
quindi grazie a un ragionamento simile a quello riportato in 3.35, otteniamo
che la successione $\\{w_{m}\\}$ è D-cauchy.
La successione $w_{n}$ inoltre è una successione decrescente di funzioni.
Infatti per ogni $p$ abbiamo che $w_{m+1,p}\leq 1$ sull’insieme $E_{n+1,m}$
grazie alla proposizione 3.60, e quindi $w_{m+1,p}\leq w_{m}$ sull’insieme
$E_{n+1,m}$. La disuguaglianza vale su tutta la varietà $R$ (quindi per
continuità anche su tutta $R^{*}$) grazie al teorema 3.58. Questo garantisce
che esiste il limite
$\displaystyle w=BD-\lim_{m}w_{m}$
Grazie al principio di Harnack, questa funzione è armonica su tutta $R$, e per
l’osservazione 3.36 $w\in\mathbb{M}_{\Delta}(R)$. Grazie al principio del
massimo 3.57, otteniamo che $w=0$, quindi:
$\displaystyle\lim_{m}D_{R}(w_{m})=0$
Questo risultato è utile in quanto
$\displaystyle D_{R}(u_{m})\leq D_{R}(w_{m})$
dove questa osservazione segue dal teorema 3.55. Consideriamo infatti
l’insieme $F_{n+1,m}$ come l’insieme $K$ del teorema. La proiezione della
funzione $w_{m}$ su questo insieme è la funzione $u_{m}$ dato che
$(w_{m}-u_{m})|_{F_{n+1,m}}=0$ e questa funzione è in
$\mathbb{M}_{\Delta}(R)$. Quindi grazie al punto (3) del teorema abbiamo che:
$\displaystyle D_{R}(w_{m})=D_{R}(u_{m})+D_{R}(w_{m}-u_{m})$
da cui la tesi. ∎
#### 4.4.4 Il diametro transfinito $\rho(\Xi_{n})=\infty$
Mettendo assieme le varie proposizioni e lemmi visti fino ad ora, siamo pronti
per dimostrare che:
###### Proposizione 4.63.
Per ogni $n$, $\tau(\Xi_{n})=\rho(\Xi_{n})=\infty$. Questo implica che ogni
compatto contenuto nel bordo irregolare $\Xi$ ha diametro transfinito
infinito.
###### Proof.
Grazie alla proposizione 4.37 sappiamo che
$\displaystyle\tau(X)\geq\rho(X)$
per ogni insieme $X\subset R^{*}$. Quindi basta dimostrare che
$\rho(\Xi_{n})=\infty$. La relazione 4.8 combinata con la proposizione 4.61 ci
permette di scrivere:
$\displaystyle\rho(\Xi_{n})\geq\sigma_{n}^{2}\rho(F_{n+1,m})\geq\sigma_{n}^{2}\epsilon(F_{n+1,m})$
(4.15)
Questo assicura che se $\lim_{m}\epsilon(F_{n+1,m})=\infty$ otteniamo la tesi.
Per dimostrare questa uguaglianza, ricordiamo che $F_{n+1,m}$ è una
sottovarietà di codimensione $1$ di $R$ con bordo liscio, quindi grazie a
4.60:
$\displaystyle\epsilon(F_{n+1,m})=Cap(F_{n+1,m})^{-1}$
e grazie alla proposizione 4.62, che assicura che:
$\displaystyle\lim_{m}Cap(F_{n+1,m})=0$
otteniamo la tesi. ∎
#### 4.4.5 Funzioni armoniche che tendono a infinito sul bordo di R
Utilizzando in fatto che $\tau(\Xi_{n})=\infty$ per ogni $n$, in questa
sezione costruiremo una funzione armonica su $R$ che converge a infinito su
$\Xi$ (in un senso che analizzeremo meglio in seguito).
###### Proposizione 4.64.
Esiste una funzione armonica $E:R\to\mathbb{R}$ tale che $E(p)=\infty$ per
ogni $p\in\Xi$.
###### Proof.
La dimostrazione è costruttiva. Dato che $\tau(\Xi_{n})=\infty$ per ogni $n$,
per definizione di $\tau$ 404040vedi 4.34 esiste una successione di interi
$n_{k}$ tale che
$\displaystyle\tau_{n_{k}}(\Xi_{n})\geq 2^{k}$
cioè sempre per definizione di $\tau_{n}$, esistono $n_{k}$ punti
$p_{1},\cdots,p_{n_{k}}\in\Xi_{n}$ tali che
$\displaystyle\inf_{p\in\Xi_{n}}\sum_{i=1}^{n_{k}}G(p,p_{i})>2^{k}n_{k}$
definiamo la funzione
$\displaystyle
E_{n,k}(x)\equiv\sum_{i=1}^{n_{k}}\frac{1}{2^{k}n_{k}}G(x,p_{n_{k}})$
Osserviamo che $E_{n,k}(p)>1$ per ogni $p\in\Xi_{n}$. Se definiamo
$\displaystyle E_{n}(x)\equiv\sum_{k=1}^{\infty}E_{n,k}(x)$
osserviamo che in ogni punto di $R$ la somma converge. Infatti per come è
definita $E_{n,k}$, $E_{n}$ è una combinazione convessa delle funzioni
$G(x,p_{n,k})$, quindi una serie di funzioni armoniche positive. Per il
principio di Harnack, questa successione o converge localmente uniformemente
in $R$ o diverge in tutti i punti di $R$.
Consideriamo $z_{0}\in R$ un punto qualsiasi. Sia $V$ un suo intorno
relativamente compatto. Sappiamo che
$\displaystyle\sup_{x\in R^{*}\setminus V}G(x,z_{0})\leq\sup_{x\in\partial
V}G(x,z_{0})<\infty$
questo vuol dire che la successione $G(z_{0},p_{k})$ è uniformemente limitata,
quindi qualunque sua combinazione convessa converge 414141e dal principio di
Harnack converge localmente uniformemente a una funzione armonica. Un modo
meno banale ma più veloce di dimostrare la stessa cosa, è osservare che
$G(z_{0},\cdot)$ è una funzione continua su $\Gamma$ insieme compatto, quindi
assume massimo finito su questo insieme.
Inoltre osserviamo che per ogni $p\in\Xi_{n}$, $E_{n}(p)=\infty$. Per definire
la funzione $E$ ripetiamo un ragionamento simile a quello fin qui esposto, in
particolare definiamo:
$\displaystyle E(x)\equiv\sum_{n=1}^{\infty}\frac{1}{2^{n}}E_{n}(x)$
anche in questo caso $E$ è una combinazione convessa di funzioni della forma
$G(\cdot,p)$ con $p\in\Xi$ 424242$p$ può stare in un $\Xi_{n}$ qualsiasi,
quindi in generale $p\in\Xi$, cioè esiste una successione di numeri positivi
$t_{k}$ con somma $\sum_{k=1}^{\infty}t_{k}=1$ e una successione di punti
$p_{k}$ in $\Xi$ tale che:
$\displaystyle E(x)=\sum_{k=1}^{\infty}t_{k}G(x,p_{k})$ (4.16)
Grazie al principio di Harnack, $E(x)$ è una funzione armonica su $R$ e per
ogni $p\in\Xi$, $E(p)=\infty$. ∎
Il fatto che $E(p)=\infty$ per ogni $p\in\Xi$ è assolutamente inutile se non
consideriamo che:
###### Osservazione 4.65.
La funzione $E$ è semicontinua inferiormente su $R^{*}$, nel senso che
$\displaystyle\liminf_{p\to p_{0}}E(p)\geq E(p_{0})$
per ogni $p$ in $R^{*}$.
Ricordiamo la definizione di $\liminf$ in una topologia non I numerabile:
$\displaystyle\liminf_{p\to p_{0}}f(p)=L\ \Longleftrightarrow\
\forall\epsilon>0,\ \exists V(p_{0})\ t.c.\ E(p)>L-\epsilon\ \forall p\in
V(p_{0})\ \wedge$ $\displaystyle\wedge\ \forall\epsilon>0,\ \forall V(p_{0}),\
\exists p\in V(p_{0})\ t.c.\ E(p)<L+\epsilon$
###### Proof.
Se $p\in R\cup\Delta$, la dimostrazione è ovvia essendo $E$ continua su $R$,
positiva ovunque e uguale a $0$ su $\Delta$ 434343infatti tutte le funzioni
$G(\cdot,p)$ si annullano sul bordo armonico $\Delta$, vedi 4.28.
Data la positività delle funzioni di Green, abbiamo anche che:
$\displaystyle\inf_{q\in
V\setminus\\{q_{0}\\}}\sum_{i=1}^{\infty}\alpha_{i}G(q,p_{i})\geq\inf_{q\in
V\setminus\\{q_{0}\\}}\sum_{i=1}^{N}\alpha_{i}G(q,p_{i})$
per ogni scelta di $\alpha_{i}\geq 0$ e per ogni interno $N$.
Dato che le funzioni di Green sono semicontinue inferiormente 444444infatti se
$q_{0}\neq p_{i}$, questo segue dalla continuità della funzione $G$,
altrimenti dal fatto che $\lim_{V\to x}G(y,x)=\infty$, vedi sezione 1.7.4,
cioè:
$\displaystyle\lim_{V\to
q_{0}}\inf_{V\setminus\\{q_{0}\\}}G(q,p_{i})=G(q_{0},p_{i})$
possiamo osservare che:
$\displaystyle\lim_{V\to q_{0}}\inf_{q\in
V\setminus\\{q_{0}\\}}\sum_{i=1}^{\infty}\alpha_{i}G(q,p_{i})\geq\lim_{V\to
q_{0}}\inf_{q\in
V\setminus\\{q_{0}\\}}\sum_{i=1}^{N}\alpha_{i}G(q,p_{i})=\sum_{i=1}^{N}\alpha_{i}G(q,p_{i})$
visto che la relazione vale per ogni $n$, si ha la tesi. ∎
Questa osservazione garantisce che per $p\to p_{0},\ p\in R$, $E(p)\to\infty$.
Riassumendo, abbiamo dimostrato l’esistenza di una funzione
$E:R^{*}\to\mathbb{R}\cup\\{\infty\\}$ continua su $R$ e semicontinua
inferiormente su $R^{*}$, tale che $E(p)=\infty$ per ogni $p\in\Xi$. Ora ci
concentriamo sul dimostrare altre proprietà di questa funzione, come ad
esempio caratterizzare l’integrale di Dirichlet $D_{R}(E\curlywedge c)$ per
$c>0$ 454545come avevamo fatto per le funzion di Green $G^{*}(\cdot,p)$.
###### Proposizione 4.66.
Per ogni $c>0$ vale che
$\displaystyle D_{R}(E(\cdot)\curlywedge c)\leq c$
###### Proof.
Siano $t_{k}$ e $p_{k}$ tali che valga la relazione 4.16. Definiamo la
successione di funzioni
$\displaystyle\psi_{n}(p)\equiv\sum_{k=1}^{n}t_{k}G(p,p_{k})$
cioè la successione delle somme parziali che definiscono $E(\cdot)$. Grazie
alle considerazioni precedenti, sappiamo che $\psi_{n}$ converge localmente
uniformemente a $E$, e se dimostriamo che:
$\displaystyle D_{R}(\psi_{n}\curlywedge c)\leq c$
allora grazie al teorema 3.24 464646e all’osservazione 3.26 abbiamo la tesi.
Resta da dimostrare la parte tecnica del teorema, cioè l’ultima relazione.
Fissiamo $n$ e definiamo per comodità $\psi_{n}(p)\equiv\psi(p)$ 474747fino a
quando non ci può essere confusione con l’indice $n$. Sia $\alpha$ un valore
regolare per la funzione $\psi$. Definiamo la quantità:
$\displaystyle L(\alpha)\equiv\sum_{j=1}^{n}\int_{\psi=\alpha}\left|\ast
dG(\cdot,p_{j})\right|=\sum_{j=1}^{m}\int_{\phi(\partial\Omega)}\left|\frac{\partial
G(\cdot,p_{j})}{\partial x}(y)\right|\sqrt{\left|g\right|}dy^{1}\cdots
dy^{m-1}$
dove $(x,y^{1}\cdots,y^{m-1})$ sono coordinate per la sottovarietà
$\psi=\alpha$ dove
$\\{\psi=\alpha\\}=\\{y^{m}=0\\}$
e dove la metrica assume la forma particolare 484848cioè per coordinatizzare
un intorno di $\\{\psi=\alpha\\}$ scegliamo come coordinate $x$ e altre tutte
ortogonali a $x$, possibile grazie al fatto che $\alpha$ è un valore regolare
per la funzione $x$
$\displaystyle g=\begin{bmatrix}1&0\\\ 0&A(y)\end{bmatrix}$
Ricordando che per ogni $n$-upla di numeri positivi
$\displaystyle\left(\sum_{i=1}^{n}a_{i}\right)^{2}\leq
n\sum_{i=1}^{n}a_{i}^{2}$
e la disuguaglianza di Schwartz, cioè:
$\displaystyle\left(\int\left|f\right|gdx\right)^{2}\leq\int f^{2}dx\int
g^{2}dx$
abbiamo che:
$\displaystyle L(\alpha)^{2}\leq$ $\displaystyle\leq
n\sum_{j=1}^{m}\int_{\phi(\partial\Omega)}\left(\frac{\partial
G(\cdot,p_{j})}{\partial x}(y)\right)^{2}\sqrt{\left|g\right|}dy^{1}\cdots
dy^{m-1}\int_{x=\alpha}\sqrt{\left|g\right|}dy^{1}\cdots dy^{m-1}\leq$
$\displaystyle\leq n\sum_{j=1}^{n}\int_{\psi=\alpha}\left|\nabla
G(\cdot,p_{j})\right|^{2}\sqrt{\left|g\right|}dy^{1}\cdots
dy^{m-1}\int_{\psi=\alpha}\ast dx$
Dimostreremo nel lemma seguente (lemma 4.67 e osservazione seguente) che vale
la proprietà:
$\displaystyle D_{R}(\psi\curlywedge\alpha)=\alpha\int_{\psi=\alpha}\ast dx$
(4.17)
Consideriamo due numeri reali $c,\ c^{\prime}$ con $0<c<\alpha<c^{\prime}$
dove $c$ e $c^{\prime}$ sono valori regolari della funzione $\psi$. Allora
vale che:
$\displaystyle\int_{\psi=\alpha}\ast dx=\alpha
D_{R}(\psi\curlywedge\alpha)\leq c^{\prime}D_{R}(\psi\curlywedge c^{\prime})$
quindi
$\displaystyle L(\alpha)\leq nc^{\prime}D_{R}(\psi\curlywedge
c^{\prime})\sum_{j=1}^{n}\int_{\psi=\alpha}\left|\nabla
G(\cdot,p_{j})\right|^{2}\sqrt{\left|g\right|}dy^{1}\cdots dy^{m-1}$
Inoltre dato che l’insieme dei valori non regolari per $\psi$ ha misura nulla
(vedi teorema di Sard), vale che:
$\displaystyle\int_{c}^{c^{\prime}}d\alpha\sum_{j=1}^{n}\int_{\psi=\alpha}\left|\nabla
G(\cdot,p_{j})\right|^{2}\sqrt{\left|g\right|}dy^{1}\cdots
dy^{m-1}=\sum_{j=1}^{n}\int_{c<\psi<c^{\prime}}\left|\nabla
G(\cdot,p_{j})\right|^{2}dV\leq$
$\displaystyle\leq\sum_{j=1}^{n}\int_{\psi<c^{\prime}}\left|\nabla
G(\cdot,p_{j})\right|^{2}dV$
Per definizione della funzione $\psi$, se $\psi(z)<c^{\prime}$, allora
necessariamente per ogni $j=1\cdots n$
$\displaystyle G(z,p_{j})<\frac{c^{\prime}}{t_{j}}$
quindi grazie alla proprietà (5) di 4.28, si la stima sull’integrale di
Dirichlet:
$\displaystyle\int_{\psi<c^{\prime}}\left|\nabla
G(\cdot,p_{j})\right|^{2}dV\leq D_{R}(G(\cdot,p_{j})\curlywedge
c^{\prime}t_{j}^{-1})\leq\frac{c^{\prime}}{t_{j}}$
Da questa stima, ricordando che $\psi\curlywedge c\in\mathbb{M}_{\Delta}(R)$,
possiamo ottenere che:
$\displaystyle\int_{c<\psi<c^{\prime}}L(\alpha)^{2}d\alpha\leq
nc^{\prime}D_{R}(\psi\curlywedge
c^{\prime})\sum_{j=1}^{n}\frac{c^{\prime}}{t_{j}}<\infty$
Il ragionamento fino a qui discusso ha l’utilità di dimostrare (grazie al
teorema di Fubini) che a meno di un insieme di $\alpha$ di misura nulla
rispetto alla misura di Lebesgue su $\mathbb{R}$, vale che:
$\displaystyle\int_{\psi=\alpha}\left|\ast dG(\cdot,p_{j})\right|<\infty$
D’ora in avanti considereremo valori di $\alpha$ regolari per la funzione
$\psi$ e per i quali vale la relazione appena scritta per ogni $j=1\cdots,n$
494949grazie alle considerazioni fatte fino ad ora, risulta evidente che
questo insieme abbia complementare di misura nulla in $\mathbb{R}$, quindi sia
denso in $\mathbb{R}$.
Grazie alla relazione 4.17 e alla definizione della funzione $\psi$, sappiamo
che:
$\displaystyle D_{R}(\psi\curlywedge\alpha)=\alpha\int_{\psi=\alpha}\ast
dx=\alpha\sum_{k=1}^{n}t_{k}\int_{\psi_{\alpha}}\ast dG(\cdot,p_{k})$ (4.18)
Sia $c_{k}>\alpha/t_{k}$ un valore regolare per la funzione $G(\cdot,p_{k})$.
Grazie alla positività di tutte le funzioni $G(\cdot,p_{k})$, abbiamo che:
$\displaystyle K\equiv\\{G(\cdot,p_{k})\geq c_{k}\\}\subset
K^{\prime}\equiv\\{\psi(\cdot)\geq\alpha\\}$
applicando ancora il lemma seguente (lemma 4.67) alla funzione
$(G(\cdot,p_{k})\curlywedge c_{k})/c_{k})$, otteniamo con l’aiuto della
proposizione 4.28
$\displaystyle\int_{\psi=\alpha}\ast
d(c_{k}^{-1}G(\cdot,p_{k}))=\int_{\partial K^{\prime}}\ast
d(c_{k}^{-1}G(\cdot,p_{k}))=\int_{\partial K}\ast
d(c_{k}^{-1}G(\cdot,p_{k}))=$ $\displaystyle=\int_{\partial K}\ast
d(c_{k}^{-1}(G(\cdot,p_{k})\curlywedge
c_{k}))=D_{R}(c_{k}^{-1}(G(\cdot,p_{k})\curlywedge c_{k}))\leq
c_{k}^{-2}c_{k}$
e quindi:
$\displaystyle\int_{\psi=\alpha}\ast dG(\cdot,p_{k})\leq 1$
da cui utilizzando 4.18:
$\displaystyle
D_{R}(\psi\curlywedge\alpha)\leq\alpha\sum_{i=k1}^{n}t_{k}\int_{\psi_{\alpha}}\leq\alpha$
Se $\alpha$ non è un valore regolare per $\psi$, comunque esiste una
successione $\alpha_{n}$ che converge dall’alto a $\alpha$ di valori regolari
di $\psi$, quindi:
$\displaystyle D_{R}(\psi\curlywedge\alpha)\leq
D_{R}(\psi\curlywedge\alpha_{n})\leq\alpha_{n}$
dato che la relazione vale $\forall n$, abbiamo la tesi. ∎
###### Lemma 4.67.
Sia $K$ un’insieme compatto in $R^{*}$ il cui bordo in $R$ sia liscio liscio e
tale che $K\cap\Delta=\emptyset$, e sia $u$ una funzione limitata tale che
$u\in\mathbb{M}_{\Delta}(R)$, $u|_{K}=1$, $u\in H(R\setminus K)$. Allora si ha
che:
1. 1.
$D_{R}(u)=-\int_{\partial K}\ast du$
2. 2.
se $K^{\prime}$ è un insieme compatto in $R^{*}$ con bordo liscio, disgiunto
da $\Delta$ e la cui parte interna contiene $K$, allora se
$\int_{\partial(K^{\prime}\setminus K)}\left|\ast du\right|<\infty$:
$\displaystyle D_{R}(u)=-\int_{\partial K}\ast du=-\int_{\partial
K^{\prime}}\ast du$
###### Proof.
Sia $R_{n}$ un’esaustione regolare di $R$, definiamo $K_{n}\equiv
K\cap\overline{R_{n}}$. Con le notazioni dell’osservazione 3.56, abbiamo che:
$\displaystyle\pi_{K_{n}}(\pi_{K_{n+p}}(u))=\pi_{K_{n}}(u)$
per ogni $n$ e $p$ naturali. Grazie al punto (2) del teorema 3.55, si ha che:
$\displaystyle D(\pi_{K_{n+p}}(u)-\pi_{K_{n}}(u),\pi_{K_{n}}(u))=0\ \
\Rightarrow$ $\displaystyle\Rightarrow\ \
D(\pi_{K_{n+p}}(u)-\pi_{K_{n}}(u))=D(\pi_{K_{n+p}}(u))-D(\pi_{K_{n}}(u))$
Con un ragionamento analogo a quello esposto in 3.35, otteniamo che la
successione $\\{\pi_{K_{n}}(u)\\}$ è D-cauchy.
Osserviamo che la successione $\pi_{K_{n}}(u)$ è una successione crescente in
$n$, infatti dato che $\pi_{K_{n}}(u)=1$ su $K_{n}$ e
$\pi_{K_{n}}(u)|_{\Delta}=0$, grazie all’osservazione 3.60 $\pi_{K_{n}}(u)\leq
1$ su $R$, quindi in particolare su $K_{n+1}$ si ha che
$\pi_{K_{n}}(u)\leq\pi_{K_{n+1}}(u)$. Sempre grazie alla proposizione 3.60,
otteniamo che questa disuguaglianza è valida su tutto $R$. Quindi la
successione $\pi_{K_{n}}(u)$ converge monotonamente a una funzione $f$
armonica su $R\setminus K$ per il principio di Harnack e identicamente uguale
a $1$ su $K$ per costruzione. È facile verificare che la convergenza è locale
uniforme su $R$.
Dato che $(f-u)|_{K}=0$ e $(f-u)|_{\Delta}=0$, per la proposizione 3.60, $f=u$
su $R$, quindi
$\displaystyle u=BD-\lim_{n}\pi_{K_{n}}(u)$
Fissato $n$, per ogni $m>n$, definiamo la funzione $v^{(n)}_{m}$ tale che:
1. 1.
$v^{(n)}_{m}|_{K_{n}}=1$
2. 2.
$v^{(n)}_{m}|_{R\setminus R_{m}}=0$
3. 3.
$v^{(n)}_{m}\in H(R_{m}\setminus K_{n})$
dalla dimostrazione del teorema 3.53 505050o meglio dal suo adattamento alla
dimostrazione di 3.55, si ha che
$\displaystyle\pi_{K_{n}}(u)=BD-\lim_{m}v^{(n)}_{m}$
Applicando la formula di Green 3.30 all’insieme $R_{m}\setminus K$ otteniamo
che:
$\displaystyle D_{R}(v^{(n)}_{m},u)=D_{R_{m}\setminus
K}(v^{(n)}_{m},u)=-\int_{\partial K}v_{m}\ast du$
dove il segno $-$ viene dall’orientazione di $\partial K$, contraria a quella
di $\partial(R_{m}\setminus K)$.
Grazie al fatto che $u|_{K}=1$ e $u|_{R\setminus K}\leq 1$, possiamo dedurre
che $\ast du\leq 0$ su $\partial K$, e dato che $v^{(n)}_{m}$ è una
successione crescente in $m$ 515151semplice applicazione del principio del
massimo, grazie al teorema di convergenza monotona possiamo concludere che:
$\displaystyle
D_{R}(\pi_{K_{n}}(u),u)=\lim_{n}D_{R}(v^{(n)}_{m},u)=\lim_{n}-\int_{\partial
K}v^{(n)}_{m}\ast du=-\int_{\partial K}\pi_{K_{n}}(u)\ast du$
La successione $\pi_{K_{n}}(u)$ è crescente sull’insieme $\partial K_{n}$ e in
particolare tende a $1$ su questo insieme, quindi sempre per convergenza
monotona:
$\displaystyle D_{R}(u)\equiv
D_{R}(u,u)=\lim_{n}D_{R}(\pi_{K_{n}}(u),u)=\lim_{n}-\int_{\partial
K}\pi_{K_{n}}(u)\ast du=-\int_{\partial K}1\ast du$
Per dimostrare il punto (3), sfruttiamo il lemma 3.51. Sia
$f\in\mathbb{M}_{\Delta}(R)$ tale che $f_{K^{\prime}}=1$. Allora per
definizione di $\mathbb{M}_{\Delta}(R)$, esiste una successione di funzioni
$f_{n}\in\mathbb{M}_{0}(R)$ tale che $f=BD-\lim_{n}f_{n}$. Quindi vale che:
$\displaystyle 0=D_{K^{\prime}\setminus K}(1,u)=D_{K^{\prime}\setminus
K}(f,u)=\lim_{n}D_{K^{\prime}\setminus K}(f_{n},u)$
Sia $S_{n}\equiv supp(f_{n})$ 525252compatto in $R$ per definizione di
$\mathbb{M}_{0}(R)$, allora:
$\displaystyle D_{K^{\prime}\setminus K}(f_{n},u)=D_{(K^{\prime}\setminus
K)\cap S_{n}}(f_{n},u)$
applicando la formula di Green 3.30 otteniamo:
$\displaystyle D_{(K^{\prime}\setminus K)\cap
S}(f_{n},u)=\int_{\partial(K^{\prime}\setminus K)}f_{n}\ast du$
Per ipotesi, sappiamo che $\int_{\partial(K^{\prime}\setminus K)}\left|\ast
du\right|<\infty$, quindi per uniforme limitatezza della successione $f_{n}$
vale anche che $\int_{\partial(K^{\prime}\setminus
K)}\left|f_{n}\right|\left|\ast du\right|<M<\infty$ dove $M$ è indipendente da
$n$. Quindi possiamo applicare il teorema di convergenza dominata e ottenere
che:
$\displaystyle\lim_{n}\int_{\partial(K^{\prime}\setminus K)}f_{n}\ast
du=\int_{\partial(K^{\prime}\setminus K)}\ast du$
Riassumendo, abbiamo ottenuto:
$\displaystyle 0=\int_{\partial(K^{\prime}\setminus K)}\ast du=\int_{\partial
K^{\prime}}\ast du-\int_{\partial K}\ast du$
∎
Riassumendo abbiamo dimostrato l’esistenza di funzioni con queste
caratteristiche:
###### Proposizione 4.68.
Data una varietà $R$ iperbolica irregolare ($\Xi(R)\neq\emptyset$), esiste una
funzione armonica $E:R\to[0,\infty)$ tale che:
1. 1.
$E\in H(R)$
2. 2.
$E(\cdot)$ è la combinazione convessa di funzioni $G(p_{i},\cdot)$ dove
$p_{i}\in\Xi$
3. 3.
$E(z)=\infty$ se $z\in\Xi$, $E(z)=0$ se $z\in\Delta$
4. 4.
$E(z)$ è semicontinua inferiormente su $R^{*}$
5. 5.
$D_{R}(E(z)\curlywedge c)\leq c$ per ogni $c>0$
#### 4.4.6 Varietà iperboliche irregolari
Data una varietà iperbolica $R$, sappiamo che esiste una funzione di Green
$G(\cdot,\cdot):R^{*}\times R^{*}\to\mathbb{R}\cup\\{\infty\\}$
con le proprietà descritte nella proposizione 4.28. L’insieme $\Xi$ è definito
come il sottoinsieme di $\Gamma$ tale che
$\displaystyle\Xi\equiv\\{p\ t.c.\ G(z,p)>0\ z\in R\\}$
dove la definizione non dipende dalla scelta del punto $z$. In funzione del
fatto che $\Xi$ sia vuoto o meno definiamo una varietà iperbolica regolare o
irregolare
###### Definizione 4.69.
Data una varietà iperbolica $R$, diciamo che è regolare se $\Xi=\emptyset$,
altrimento diciamo che è irregolare.
Una facile caratterizzazione delle varietà regolari è la seguente:
###### Proposizione 4.70.
Sia $G(z,\cdot)$ un nucleo di Green su $R$ iperbolica. $R$ è regolare se e
solo se per ogni $c>0$ l’insieme
$\displaystyle A_{c}\equiv\\{p\in R\ t.c.\ G(z.p)\geq c\\}$
è compatto in $R$.
###### Proof.
La dimostrazione di questa affermazione è abbastanza immediata. Supponiamo che
gli insiemi $A_{c}$ siano compatti. Allora per ogni $p\in\Gamma$, sappiamo
che:
$\displaystyle G(z,p)=p(G(z,\cdot))=p(G(z,\cdot)(1-\lambda_{c}))$
dove $\lambda_{c}$ è una funzione liscia a supporto compatto tale che
$\lambda_{c}=1$ sull’insieme $A_{c}$. Grazie all’osservazione 3.48 vale la
seconda uguaglianza scritta in formula, e inoltre grazie all’osservazione 4.21
possiamo concludere che la funzione $G(z,\cdot)(1-\lambda_{c})$ è compresa tra
$0$ e $c$. Quindi per continuità, anche:
$\displaystyle 0\leq p(G(z,\cdot)(1-\lambda_{c}))=G(z,p)\leq c$
data l’arbitrarietà di $c>0$, abbiamo che se $A_{c}$ è compatto per ogni $c$,
allora fissato $z$, $G(z,p)=0$ per ogni $p\in\Gamma$, quindi $\Xi=\emptyset$,
cioè la varietà iperbolica è regolare.
Supponiamo ora che esiste un insieme $A_{c}$ non compatto, allora esiste una
successione $p_{n}\in R$, $p_{n}\to\infty$ tale che
$\displaystyle G(z,p_{n})\geq c\ \ \ \forall n$
Consideriamo un qualunque carattere $p$ su $R$ descritto da un ultrafiltro su
$z_{n}$. Dall’ultima equazione sappiamo che
$\displaystyle G(z,p)\geq c$
da cui $p\in\Xi\neq\emptyset$. ∎
###### Osservazione 4.71.
Per le proprietà della funzione $G$, il fatto che per ogni $c>0$ $A_{c}$ sia
compatto, è equivalente a chiedere che per ogni $c$ l’insieme $\partial A_{c}$
sia compatto.
Per le varietà iperboliche irregolari, definiamo l’insieme delle successioni
irregolari, cioé
###### Definizione 4.72.
Data una varietà iperbolica $R$, definiamo:
$\displaystyle\Sigma(R)=\\{\\{z_{n}\\}\subset R\ t.c\ z_{n}\to\infty\ \ e\
\liminf_{n}G(z_{0},z_{n})>0\\}$
Grazie a un ragionamento simile a quello riportato nella dimostrazione di
4.28, si ottiene che questa condizione è indipendente dalla scelta di $z_{0}$,
quindi $\Sigma(R)$ è ben definito. Inoltre dalla caratterizzazione appena
dimostrata, è immediato verificare che $R$ è irregolare se e solo se
$\Sigma(R)\neq\emptyset$.
Possiamo riformulare la proposizione 4.68 con questa nuova terminologia. In
particolare:
###### Proposizione 4.73.
Sia $R$ una varietà iperbolica irregolare, e sia $E$ la funzione descritta in
4.68. Allora per ogni successione $\\{z_{n}\\}\in\Sigma(R)$:
$\displaystyle\lim_{n\to\infty}E(z_{n})=\infty$
###### Proof.
Questa osservazione segue dalla semicontinuità di $E$ su $\Xi$.
Per prima cosa osserviamo che la proprietà appena enunciata è equivalente a:
$\displaystyle\lim_{n\to\infty}\inf_{z\in V(z_{0},a,n)}E(z)=\infty$ (4.19)
per ogni $a>0$, $a$ tale che $G(z_{0},\cdot)\geq a$ non sia un insieme
compatto in $R$, dove $V(z_{0},a,n)\equiv\\{z\in R\ t.c.\
G(z_{0},z)>a\\}\setminus K_{n}$ e $K_{n}$ è un’esaustione di $R$ 535353dalle
considerazioni precedenti, sappiamo che questa proprietà è indipendente dalla
scelta di $z_{0}\in R$. Infatti, se questo è vero e consideriamo una
successione $\\{z_{m}\\}\in\Sigma(R)$, dato che
$\liminf_{m}G(z_{0},z_{m})=\lambda>0$
$z_{m}$ appartiene definitivamente a tutti gli insiemi $V(z_{0},\lambda/2,n)$,
e per $m$ che tende a infinito, esiste una successsione $n_{m}$ che tende a
infinito tale che $z_{n}\in V(z_{0},\lambda/2,n_{m})$.
Per dimostrare l’altra implicazione, supponiamo che esista $a>0$ tale che:
$\displaystyle\lim_{n\to\infty}\inf_{z\in V(z_{0},a,n)}E(z)=M<\infty$
questo significa che per ogni $n$ esiste un $z_{n}\in V(z_{0},a,n)$ tale che
$E(z_{n})\leq M+1$. Per costruzione, la successione $z_{n}$ tende a infinito,
e $G(z_{0},z_{n})\geq a$, quindi $\\{z_{n}\\}\in\Sigma(R)$. Questo completa la
dimostrazione dell’equivalenza tra le due proprietà.
Dato che:
$\displaystyle\bigcap_{n}V(z_{0},a,n)=\Xi_{a}\equiv\\{p\in\Gamma\ t.c.\
G(z_{0},p)>a\\}\subset\Xi$
allora l’equazione 4.19 è vera perché $E$ è semicontinua inferiormente
sull’insieme $R\cup\Xi$ e $E(\Xi)=\infty$. ∎
#### 4.4.7 Potenziali di Evans su varietà paraboliche
Osserviamo che tutti gli spazi $R^{n}$ con $n\geq 3$ dotati della metrica
euclidea standard sono iperbolici regolari, e gli stessi spazi privati di un
punto qualsiasi sono iperbolici irregolari. Consideriamo ad esempio l’insieme
$\mathbb{R}^{3}\setminus\\{(0,0,1)\\}$. Sappiamo che la funzione di Green su
$\mathbb{R}^{3}$ è proporzionale all’inverso della distanza:
$\displaystyle G(x,y)=C(3)\frac{1}{d(x,y)}$
Si vede facilmente che la stessa funzione ristretta all’insieme
$\mathbb{R}^{3}\setminus\\{(0,0,1)\\}$ è ancora un nucleo di Green. Fissiamo
il punto $x=(0,0,0)$ e consideriamo la funzione $G(0,y)$. L’insieme
$\\{y\in\mathbb{R}^{3}\setminus\\{(0,0,1)\ t.c.\ G(0,y)\geq C(3)/2\\}\\}$ NON
è un insieme compatto in $\mathbb{R}^{3}\setminus\\{(0,0,1)\\}$, quindi questa
varietà è iperbolica irregolare.
Altri esempi di varietà iperboliche irregolari possono essere costruiti
togliendo un insieme compatto da una varietà riemanniana parabolica.
Illustriamo questo risultato in due proposizioni.
###### Proposizione 4.74.
Sia $R$ una varietà Riemanniana e $K$ un compatto che sia chiusura della sua
parte interna in $R$. Allora la varietà $R\setminus K$ dotata della metrica di
sottospazio è una varietà iperbolica.
###### Proof.
Consideriamo un compatto $C$ in $R^{\prime}\equiv R\setminus K$ con bordo
regolare, e dimostriamo che la capacità di questo compatto è necessariamente
diversa da $0$. A questo scopo consideriamo un altro insieme compatto con
bordo regolare $K^{\prime}$ tale che:
$\displaystyle K\Subset K^{\prime\circ}\subset K^{\prime}\ \ \ K^{\prime}\cap
C=\emptyset$
Per ora trattiamo il caso di $K$ compatto con bordo regolare. Sia per
definizione $w$ la funzione armonica in $K^{\prime}\setminus K$ tale che:
$\displaystyle w|_{\partial K}=0\ \ \ w|_{\partial K^{\prime}}=1$
Consideriamo un’esaustione regolare $V_{n}$ di $R^{\prime}$ tale che
$C^{\prime}\Subset V_{1}$, e sia $v_{n}$ il potenziale armonico della coppia
$(C,V_{n})$. Sappiamo che la capacità di $C$ è per definizione l’integrale di
Dirichlet del limite di $v_{n}$, ed è nulla se e solo se $v=1$ costantemente.
Applicando il principio del massimo sull’insieme $V_{n}\cap K^{\prime}$,
otteniamo che per ogni $n$, $v_{n}\leq w$. Infatti sul bordo di $K^{\prime}$,
$w=1$ mentre $v_{n}\leq 1$, e sul bordo di $V_{n}$, $v_{n}=0$ mentre $w\geq
0$. Questo garantisce che
$\displaystyle v\leq w\ \ \ su\ \ K^{\prime}\setminus K$
quindi $v$ non può essere costante uguale a $1$.
Se $K$ non ha bordo liscio, è sufficiente ripetere la costruzione di $w$ con
un insieme $K^{\prime\prime}\Subset K$ dal bordo liscio. Tutte le
considerazioni fatte si applicano a questo caso senza complicazioni. ∎
###### Proposizione 4.75.
Sia $R$ una varietà parabolica e sia $K$ un insieme compatto che sia chiusura
della sua parte interna e abbia bordo regolare. Allora la varietà
$R^{\prime}\equiv R\setminus K$ è iperbolica irregolare. Inoltre tutte le
successioni $\\{z_{n}\\}\subset R^{\prime}\subset R$ che tendono a infinito in
$R$ sono irregolari, appartengono all’insieme $\Sigma(R^{\prime})$, e dato
$z_{0}\in R^{\prime}$, la funzione $G(z_{0},\cdot)$ può essere estesa per
continuità anche a $\partial K$ e su questo insieme è nulla.
###### Proof.
Dalla proposizione precedente sappiamo che $R^{\prime}$ è una varietà
iperbolica, quindi ammette nucleo di Green positivo. Sia $z_{0}\in R^{\prime}$
qualsiasi, e consideriamo un compatto con bordo regolare $K^{\prime}\Subset R$
545454questo insieme NON è compatto in $R^{\prime}$ tale che $K\subset
K^{\prime\circ}$ e $z_{0}\in K^{\prime\circ}$. Sia inoltre $K_{n}$
un’esaustione regolare di $R$ con $K^{\prime}\subset K_{1}$.
Osserviamo che il nucleo di Green $G(z_{0},\cdot)$ su $R^{\prime}$ è una
funzione armonica sull’insieme $R\setminus K^{\prime}$, e su $\partial
K^{\prime}$ assume minimo strettamente positivo $\lambda$ 555555$\lambda>0$
perchè il principio del massimo assicura che $G(z_{0},\cdot)$ non può assumere
il suo minimo in un punto interno all’insieme di definizione. Allora grazie al
principio del massimo possiamo confrontare $G(z_{0},\cdot)$ con $v_{n}$, i
potenziali armonici della coppia $(K^{\prime},K_{n})$. Per ogni $n$ si ha
infatti che sull’insieme $R\setminus K^{\prime}$:
$\displaystyle G(z_{0},\cdot)\geq\lambda v_{n}(\cdot)$
passando al limite e ricordando che $R$ è parabolica (quindi
$\lim_{n}v_{n}=1$) otteniamo che su $R\setminus K^{\prime}$:
$\displaystyle G(z_{0},\cdot)\geq\lambda$
Questo dimostra anche che qualunque successione tendente a infinito in $R$ è
in $\Sigma(R^{\prime})$.
Per dimostrare l’ultimo punto, consideriamo il massimo $\Lambda$ della
funzione $G(\cdot,z_{0})$ sul bordo di un dominio compatto 565656rispetto alla
topologia di $R$ con bordo liscio $C$ tale che $K\subset C$ e $z_{0}\not\in
C$. Sia $v$ il potenziale di capacità della coppia $(K,C)$. Grazie al
principio del massimo, $G(z_{0},\cdot)\leq\Lambda(1-v(\cdot))$, infatti
$G(z_{0},\cdot)$ è costruita come limite crescente di nuclei di Green $G_{n}$
su domini compatti, ed è facile verificare che
$\displaystyle G_{n}(z_{0},\cdot)\leq\Lambda(1-v(\cdot))$
passando al limite su $n$ si ottiene la disuguaglianza desiderata. Questo
dimostra che $G(z_{0},\cdot)$ tende a zero se l’argomento tende a un punto
qualsiasi di $\partial K$.
Osserviamo anche che grazie a questa proprietà, tutte le successioni che
tendono a $\partial K$ 575757con questo si intendono tutte le successioni di
elementi di $R^{\prime}$ tali che per ogni intorno $U$ di $\partial K$,
intorno rispetto alla topologia di $R$, esiste $N$ tale che $x_{n}\in U$ per
ogni $n\geq N$. Ad esempio tutte le successioni che convergono rispetto alla
topologia di $R$ a un punto di $\partial K$ non appartengono a
$\Sigma(R^{\prime})$. ∎
Per dimostrare l’esistenza di potenziali di Evans relativi a un qualsiasi
dominio compatto con bordo liscio $K$ in $R$ varietà parabolica, sfruttiamo
quest’ultima proposizione, l’esistenza delle funzioni $E$ descritte in 4.68 e
la proposizione 4.73.
###### Teorema 4.76.
Data una varietà parabolica $R$ e un dominio compatto con bordo liscio $K$,
esiste un potenziale di Evans su $R$ rispetto a $K$, cioè una funzione
$E:R\setminus K^{\circ}\to[0,\infty)$ tale che:
1. 1.
$E\in H(R\setminus K)$
2. 2.
$E(z)=0$ se $z\in\partial K$
3. 3.
$E(z_{n})\to\infty$ per ogni successione $z_{n}$ che tende a infinito in $R$.
4. 4.
$D_{R}(E\curlywedge c)\leq c$ per ogni $c>0$
###### Proof.
La dimostrazione è semplicemente una raccolta di risultati precedentemente
ottenuti.
Grazie a quanto appena dimostrato, $R\setminus K$ è una varietà iperbolica
irregolare, quindi considerando la funzione $E$ costruita nella proposizione
4.68 e grazie alla proposizione 4.73, otteniamo che $E$ soddisfa (1), (3) e
(4).
Il punto (2) si ricava considerando che $E$ è una combinazione convessa di
nuclei di Green sulla varietà $R\setminus K$, che grazie alla proposizione
4.75 si annullano sull’insieme $\partial K$. ∎
## Appendix A Glossario
$A^{C}$ | complementare dell’insieme $A$
---|---
$\overline{A}$ | chiusura topologica dell’insieme $A$
d$\lambda$, d$\lambda^{m}$ | misura di Lebesgue su $\mathbb{R}$ o su $\mathbb{R}^{m}$
$(R,g)$ | varietà Riemanniana $R$ con tensore metrico $g$
$\Omega$ | dominio (insieme aperto connesso) in $R$
$\sqrt{\left|g\right|}$ | radice quadrata del determinante di $g$
$dV$ | $dV=\sqrt{\left|g\right|}dx^{1}\cdots dx^{n}$ elemento di volume su $(R,g)$
$L^{2}(\Omega)$ | spazio delle funzioni a quadrato integrabile su $\Omega$
$\mathcal{L}^{2}(\Omega)$ | spazio delle $1$-forme a quadrato integrabile su $\Omega$
| (vedi sezione 1.3)
$\left\|f\right\|_{\infty}$ | norma del sup della funzione $f$
$D_{\Omega}(f)$ | integrale di Dirichlet della funzione $f$ sull’insieme $\Omega$.
$\left\|f\right\|_{R}$ | norma di $f$ nell’algebra di Royden $\mathbb{M}(R)$
| (vedi teorema 3.23)
$\mathbb{M}(R)$ | Algebra di Royden su $R$ (vedi definizione 3.7)
$\mathbb{M}_{0}(R)$ | Insieme delle funzioni a supporto compatto in $\mathbb{M}(R)$
$\mathbb{M}_{\Delta}(R)$ | Completamento di $\mathbb{M}_{0}(R)$ nella topologia $BD$
| (vedi paragrafo 3.2.5)
$R^{*}$ | Compattificazione di Royden di $R$ (vedi definizione 3.37)
$\Gamma$ | $\Gamma=R^{*}\setminus R$ è il bordo di $R^{*}$ (vedi definizione 3.47)
$\Delta$ | bordo armonico di $R$ (vedi definizione 3.50)
$\Xi$ | bordo irregolare di $R$ (vedi definizione 4.27)
$f\ast h$ | convoluzione della funzione $f$ con $h$ (vedi sezione 1.5)
$\int f\ast du$ | vedi definizione 1.10
$H(\Omega)$ | spazio delle funzioni armoniche su $\Omega$
$HP(\Omega)$ | spazio delle funzioni armoniche positive su $\Omega$
$HD(\Omega)$ | spazio delle funzioni armoniche con $D_{\Omega}(u)<\infty$
$C(\Omega)=C(\Omega,\mathbb{R})$ | spazio delle funzioni continue a valori reali su $\Omega$
$C(\overline{\Omega})=C(\overline{\Omega},\mathbb{R})$ | spazio delle funzioni continue definite in un intorno di $\overline{\Omega}$
$\text{Cap}(K,\Omega)$ | capacità della coppia $(K,\Omega)$ (vedi definizione 4.1)
$\text{Cap}(K)$ | capacità di $K$ (vedi definizione 4.6)
## Bibliography
* [1]
* [A] Arkhangel’skii, A.V.; Pontryagin, L.S. General Topology I Springer-Verlag, New York (1990) ISBN 3-540-18178-4
* [ABR] Axler, Bourdon, Ramey Harmonic Function Theory, II edition Springer Verlag http://www.axler.net/HFT.pdfhttp://www.axler.net/HFT.pdf
* [C1] do Carmo Riemannian geometry Birkhäuser
* [CSL] Chang, John; Sario, Leo Royden’s Algebra on Riemannian Spaces Math. Scand. 28 (1971), 139-158 http://www.mscand.dk/article.php?id=2001http://www.mscand.dk/article.php?id=2001
* [C2] Conlon, Lawrence Differentiable Manifolds, Second edition Birkhäuser
* [C3] Conway, John A First Course in Functional Analysis Springer-Verlag
* [D] Dugundji, James Topology Allyn and Bacon, Inc.
* [F1] Folland, G. B. Real Analysis, Modern techniques and their applications (II ed.) Wiley-Interscience Publication
* [F2] Friedman, Avner Partial Differential Equations Holt, Rinehart and Winston Inc.
* [G1] Gray, Alfred Tubes Addison Wesley
* [G2] Grigor’yan, Alexander Analytic and Geometric Background of Recurrence and non-explosion of the Brownian Motion on Riemannian Manifolds Bullettin of the american mathematical society
* [GP] Guillemin, Victor; Pollack, Alan Differential Topology Prentice Hall
* [GT] Gilbard, Trudinger Elliptic partial differential equations Springer
* [H1] Helms Potential theory Springer
* [H2] Hervé, Rose-Marie Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel
http://archive.numdam.org/article/AIF_1962__12__415_0.pdfhttp://archive.numdam.org/article/AIF_1962__12__415_0.pdf
* [H3] Hirsch, Morris W. Differential topology Springer-Verlag
* [L] Lang, Serge Real Analysis, II ed Addison Wesley
* [LT] Li, Peter; Tam, Luen-Fai Symmetric Green’s Functions on complete manifolds American Journal of Mathematics, Vol 109, n6, Dic 1987, pp. 1129-1154 http://www.jstor.org/stable/2374588?origin=JSTOR-pdfstable link Jstor
* [M1] Magginson, Robert E. An Introduction to Banach Space Theory Graduate texts in mathematics
disponibile in anteprima limitata su
http://books.google.it/http://books.google.it/books?id=fD-
GeCsqoqkC&printsec=frontcoverdq=megginson+banach
* [MZ] Malỳ, Jean; Ziemer, William P. Fine Regularity of Solutions of Elliptic Partial Differential Equations American Mathematical Society
disponibile in anteprima limitata su
http://books.google.it/http://books.google.it/
* [M2] Munkres, James R. Analysis on Manifolds Addison-Wesley Pubblishing Company
* [M3] Munkres, James R. Elementary differential topology Princeton University Press
* [N] Nagy, Gabriel “Strange” Limits Notes from the Functional Analysis Course (Fall 07 - Spring 08)
www.math.ksu.edu/ nagy/func-an-2007-2008/strange-limits-
HB.pdfhttp://www.math.ksu.edu/ nagy/func-an-2007-2008/strange-limits-HB.pdf
* [P1] Petersen, Peter Riemannian Geometry, II ed Springer
* [PSR] Pigola; Setti; Rigoli Vanishing and Finiteness Results in Geometric Analysis Birkhäuser
* [R1] Royden, H. L. Real Analysis The Macmillan Company
* [R2] Rudin, Walter Functional Analysis, II ed. McGraw-Hill
* [R3] Rudin, Walter Principles of Mathematical Analysis, III ed McGraw-Hill
* [R4] Rudin, Walter Real and Complex Analysis McGraw-Hill
* [SN] Sario, L.; Nakai, M. Classification theory of Riemann Surfaces Springer Verlag
* [SY] Schoen, R.; Yau, S.T. Lectures on Differential Geometry International Press
* [S1] Springer, George Introduction to Riemann Surfaces Addison-Wesley Pubblishing Company
* [S2] Strauss, Walter A. Partial Differential Equations An Introduction John Wiley and Sons
* [T] Tsuji, M. Potential theory in modern function theory (2ed) Chelsea
* [Z] Ziemer, William P.Weakly differentiable functions. Sobolev spaces and functions of bounded variation Springer Verlag
### Ringraziamenti… e non solo
I miei ringraziamenti vanno innanzitutto al prof. Alberto Giulio Setti e al
prof. Stefano Pigola. In particolare ringrazio Alberto soprattutto per i
consigli che mi ha dato e per l’estrema gentilezza, pazienza e diponibilità
dimostrate, ben oltre quanto richiesto e non solo dal punto di vista
accademico. Ringrazio anche il prof. Wolfhard Hansen (University of Bielefeld)
per la disponibilità e per i suggerimenti dati riguardo alla teoria del
potenziale.
Vuoi studiare a Princeton? - …
Ovviamente ringrazio tutti i miei amici dell’università, la compagnia del
$4^{\circ}$ piano e della Norvegia, nanetti compresi. Non vedo l’ora del
prossimo ciclo!
C’è uno spiffero…
Non posso dimenticare di ringraziare Madesimo, che mi ha fornito un posto
idilliaco dove pensare e scrivere questa tesi, e non posso dimenticare tutti
gli amici che mi hanno accompagnato in questa impresa.
Ai larici a mezzogiorno?
Un ringraziamento importante va a Internet e a tutti quelli che non credono
nel diritto d’autore (senza i quali non avrei potuto studiare tutto quello che
ho studiato), da Wikipediahttp://www.wikipedia.org a Ubuntu -
Linuxhttp://www.ubuntulinux.org passando per l’indispensabile
Gigapediahttp://www.gigapedia.org.
A journey of a thousand sites begins with a single click.
Ringrazio l’Unione Europea, per quello che è ma soprattutto per quello che
potrebbe essere. Ringrazio in particolare tutti quelli che ci hanno lavorato
seriamente, perché hanno aperto una strada che ora noi abbiamo la
responsabilità di portare avanti e soprattutto perché ci hanno lasciato
qualcosa che sprecare sarebbe una follia.
European Union - United in diversity
Un pensiero speciale va a Lucy, e a suo padre scomparso di recente.
Cosa, cosa?
Ringrazio Homer, Marge, Bart, Lisa e Maggie, che mi hanno divertito,
intrattenuto, e dato da pensare.
Farfalla vendetta!
Ringrazio Kathryn, Seven, Benjamin, Jean-Luc e rispettive compagnie perché mi
hanno fatto sognare.
I look forward to it. Or should I say backward? - Don’t get started!
Un ringraziamento speciale a Oriana Fallaci, e alla maga.
[…] Ad esempio perché si trovasse qui, perché avesse scelto un mestiere che
non si addiceva al suo carattere e alla sua struttura mentale cioè il mestiere
di soldato, perché con quel mestiere avesse tradito la matematica. Quanto gli
mancava la matematica, quanto la rimpiangeva! Massaggia le meningi come un
allenatore massaggia i muscoli di un atleta, la matematica. Le irrora di
pensiero puro, le lava dai sentimenti che corrompono l’intelligenza, le porta
in serre dove crescono fiori stupendi. I fiori di un’astrazione composta di
concretezza, di una fantasia composta di realtà […] No, non è vero che è una
scienza rigida, la matematica, una dottrina severa. È un’arte seducente,
estrosa, una maga che può compiere mille incantesimi e mille prodigi. Può
mettere ordine nel disordine, dare un senso alle cose prive di senso,
rispondere ad ogni interrogativo. Può addirittuta fornire ciò che in sostanza
cerchi: la formula della Vita. Doveva tornarci, ricominciare da capo con
l’umiltà d’uno scolaro che nelle vacanze ha dimenticato la tavola pitagorica.
Due per due fa quattro, quattro per quattro fa sedici, sedici per sedici fa
duecentocinquantasei, e la derivata di una costante è uguale a zero. La
derivata di una variabile è uguale a uno, la derivata di una potenza di una
variabile… Non se ne ricordava? Sì che se ne ricordava! La derivata di una
potenza di una variabile è uguale all’esponente della potenza moltiplicata per
la variabile con lo stesso esponente diminuito di uno. E la derivata di una
divisione? È uguale alla derivata del dividendo moltiplicato per il divisore
meno la derivata del divisore moltiplicata per il dividendo, il tutto diviso
il dividendo moltiplicato per sé stesso. Semplice! Bè, naturalmente trovare la
formula della Vita non sarebbe stato così semplice. Trovare una formula
significa risolvere un problema, e per risolvere un problema bisogna
enunciarlo, e per enunciarlo bisogna partire da un presupposto… Ah perché
aveva tradito la maga? Che cosa lo aveva indotto a tradirla? […]
Oriana Fallaci: Insciallah. Atto primo, Capitolo primo
|
arxiv-papers
| 2011-01-13T17:38:05 |
2024-09-04T02:49:16.432306
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Daniele Valtorta",
"submitter": "Daniele Valtorta Mr.",
"url": "https://arxiv.org/abs/1101.2618"
}
|
1101.2913
|
# Hypercontractivity and its Applications
Punyashloka Biswal
###### Abstract
Hypercontractive inequalities are a useful tool in dealing with extremal
questions in the geometry of high-dimensional discrete and continuous spaces.
In this survey we trace a few connections between different manifestations of
hypercontractivity, and also present some relatively recent applications of
these techniques in computer science.
## 1 Preliminaries and notation
#### Fourier analysis on the hypercube.
We define the inner product $\langle
f,g\rangle=\operatorname*{\mathbb{E}}_{x}f(x)g(x)$ on functions
$f,g\colon\\{-1,1\\}^{n}\to\mathbb{R}$, where the expectation is taken over
the uniform (counting) measure on $\\{-1,1\\}^{n}$. The multilinear
polynomials $\chi_{S}(x)=\prod_{i\in S}x_{i}$ (where $S$ ranges over subsets
of $[n]$) form an orthogonal basis under this inner product; they are called
the Fourier basis. Thus, for any function
$f\colon\\{-1,1\\}^{n}\to\mathbb{R}$, we have
$f=\sum_{S\subseteq[n]}\hat{f}(S)\chi_{S}(x)$, where the Fourier coefficients
$\hat{f}(S)=\langle f,\chi_{S}\rangle$ obey Plancherel’s relation
$\sum\hat{f}(S)^{2}=1$. It is easy to verify that
$\operatorname*{\mathbb{E}}_{x}f(x)=\hat{f}(0)$ and
$\operatorname*{Var}_{x}f(x)=\sum_{S\neq\emptyset}\hat{f}(S)^{2}$.
#### Norms.
For $1\leq p<\infty$, define the $\ell_{p}$ norm
$\|f\|_{p}=(\operatorname*{\mathbb{E}}_{x}|f(x)|^{p})^{1/p}$. These norms are
monotone in $p$: for every function $f$, $p\geq q$ implies
$\|f\|_{p}\geq\|f\|_{q}$. For a linear operator $M$ carrying functions
$f\colon\\{-1,1\\}^{n}\to\mathbb{R}$ to functions
$Mf=g\colon\\{-1,1\\}^{n}\to\mathbb{R}$, we define the $p$-to-$q$ operator
norm $\|M\|_{p\to q}=\sup_{f}\|Mf\|_{q}/\|f\|_{p}$. $M$ is said to be a
contraction from $\ell_{p}$ to $\ell_{q}$ when $\|M\|_{p\to q}\leq 1$. Because
of the monotonicity of norms, a contraction from $\ell_{p}$ to $\ell_{p}$ is
automatically a contraction from $\ell_{p}$ to $\ell_{q}$ for any $q<p$. When
$q>p$ and $\|M\|_{p\to q}\leq 1$, then $M$ is said to be hypercontractive.
#### Convolution operators.
Letting $xy$ represent the coordinatewise product of $x,y\in\\{-1,1\\}^{n}$,
we define the convolution $(f*g)(x)=\operatorname*{\mathbb{E}}_{y}f(x)g(xy)$
of two functions $f,g\colon\\{-1,1\\}^{n}\to\mathbb{R}$, and note that it is a
linear operator $f\mapsto f*g$ for every fixed $g$. Convolution is commutative
and associative, and the Fourier coefficients of a convolution satisfy the
useful property $\widehat{f*g}=\hat{f}\hat{g}$. We shall be particularly
interested in the convolution properties of the following functions
* •
The Dirac delta $\delta\colon\\{-1,1\\}^{n}\to\mathbb{R}$, given by
$\delta(1,\dotsc,1)=1$ and $\delta(x)=0$ otherwise. It is the identity for
convolution and has $\hat{\delta}(S)=1$ for all $S\subseteq[n]$.
* •
The edge functions $h_{i}\colon\\{-1,1\\}^{n}\to\mathbb{R}$ given by
$h_{i}(x)=\begin{cases}\phantom{-}1/2&x=(1,\dotsc,1)\\\
-1/2&x_{i}=-1,x_{[n]\setminus\\{i\\}}=(1,\dotsc,1)\\\
\phantom{-}0&\text{otherwise.}\end{cases}$
$\hat{h}_{i}(S)$ is $1$ or $0$ according as $S$ contains or does not contain
$i$, respectively. For any function $f\colon\\{-1,1\\}^{n}\to\mathbb{R}$,
$(f*h_{i})(x)=(f(x)-f(y))/2$, where $y$ is obtained from $x$ by flipping just
the $i$th bit. Convolution with $h_{i}$ acts as an orthogonal projection (as
we can easily see in the Fourier domain), so for any functions
$f,g\colon\\{-1,1\\}^{n}\to\mathbb{R}$, we have $\langle
f*h_{i},g\rangle=\langle f,h_{i}*g\rangle=\langle f*h_{i},g*h_{i}\rangle$
* •
The Bonami-Gross-Beckner noise functions
$\operatorname{BG}_{\rho}\colon\\{-1,1\\}^{n}\to\mathbb{R}$ for $0\leq\rho\leq
1$, where $\widehat{\operatorname{BG}}_{\rho}(S)=\rho^{|S|}$ and we define
$0^{0}=1$. These operators form a semigroup, because
$\operatorname{BG}_{\sigma}*\operatorname{BG}_{\rho}=\operatorname{BG}_{\sigma\rho}$
and $\operatorname{BG}_{1}=\delta$. Note that
$\operatorname{BG}_{\rho}(x)=\sum_{S}\rho^{|S|}\chi_{S}(x)=\prod_{i}(1+\rho
x_{i})$. We define the noise operator $T_{\rho}$ acting on functions on the
discrete cube by $T_{\rho}f=\operatorname{BG}_{\rho}*f$. In combinatorial
terms, $(T_{\rho}f)(x)$ is the expected value of $f(y)$, where $y$ is obtained
from $x$ by independently flipping each bit of $x$ with probability $1-\rho$.
###### Lemma 1.
$\frac{d}{d\rho}\operatorname{BG}_{\rho}=\frac{1}{\rho}\operatorname{BG}_{\rho}*\sum
h_{i}$
###### Proof.
This is easy in the Fourier basis:
$\widehat{\operatorname{BG}}_{\rho}^{\prime}=(\rho^{|S|})^{\prime}=|S|\rho^{|S|-1}=\sum_{i\in[n]}\hat{h}_{i}\frac{\widehat{\operatorname{BG}}_{\rho}}{\rho}.\qed$
## 2 The Bonami-Gross-Beckner Inequality
### 2.1 Poincaré and Log-Sobolev inequalities
The Poincaré and logarithmic Sobolev inequalities both relate a function’s
global non-constantness to how fast it changes “locally”. The amount of local
change is quantified by the _energy_ $\operatorname{\mathbb{D}}(f,f)$, where
the Dirichlet form $\operatorname{\mathbb{D}}$ is defined as
$\operatorname{\mathbb{D}}(f,g)=\tfrac{1}{2}\operatorname*{\mathbb{E}}_{xy\in
E}(f(x)-f(y))(g(x)-g(y))$
($E$ is the set of pairs $x,y$ that differ in a single coordinate). In terms
of the edge functions $h_{i}$, observe that
$\operatorname{\mathbb{D}}(f,g)=\frac{2}{n}\sum_{i}\langle
f*h_{i},g*h_{i}\rangle$.
In the case of the Poincaré inequality, we measure the distance of $f$ to a
constant by its variance
$\operatorname*{Var}(f)=\operatorname*{\mathbb{E}}(f-\operatorname*{\mathbb{E}}f)^{2}=\operatorname*{\mathbb{E}}f^{2}-(\operatorname*{\mathbb{E}}f)^{2}$.
Then the Poincaré constant (of the discrete cube) is the supremal $\lambda$
such that the inequality
$\operatorname{\mathbb{D}}(f,f)\geq\lambda\operatorname*{Var}(f)$
holds for all $f\colon\\{-1,1\\}^{n}\to\mathbb{R}$. This quantity is also the
smallest nonzero eigenvalue of the Laplacian of the discrete cube, viewed as a
graph (i.e., its spectral expansion).
Another way of measuring the non-constantness of a function is to consider its
entropy
$\operatorname*{Ent}(f)=\operatorname*{\mathbb{E}}[f\log\frac{f}{\operatorname*{\mathbb{E}}f}]$
(where we assume $f\geq 0$ and use the convention that $0\log 0=0$). Note that
$\operatorname*{Ent}(cf)=c\operatorname*{Ent}(f)$ for any $c\geq 0$, so the
entropy is homogenous of degree $1$ in its argument. Because we are comparing
the entropy with the energy (which is homogenous of degree $2$) we use the
entropy of the _square_ of the function to define the Log-Sobolev constant:
the largest $\alpha$ such that the inequality
$\operatorname{\mathbb{D}}(f,f)\geq\alpha\operatorname*{Ent}(f^{2})$
holds for all $f\colon\\{-1,1\\}^{n}\to\mathbb{R}$. For the discrete cube
$\\{-1,1\\}^{n}$, we have $\lambda=2/n$ and $\alpha=1/n$, as we shall see
below. It is interesting to ask how these quantities are related when we
consider other probability spaces equipped with a suitable Dirichlet form (for
example, $d$-regular graphs with
$\operatorname{\mathbb{D}}(f,g)=\operatorname*{\mathbb{E}}_{xy\in
E}(f(x)-f(y))(g(x)-g(y))$, where the expectation is taken over all edges). Set
$f=1+\epsilon g$ for a sufficiently small $\epsilon$ and observe that
$\operatorname*{Var}(f)=\epsilon^{2}\operatorname*{Var}(g)$ and
$\operatorname{\mathbb{D}}(f,f)=\epsilon^{2}\operatorname{\mathbb{D}}(g,g)$,
whereas
$\displaystyle\operatorname*{Ent}(f^{2})$
$\displaystyle=\operatorname*{\mathbb{E}}\left[(1+\epsilon
g)^{2}(2\log(1+\epsilon g)-\log\operatorname*{\mathbb{E}}[(1+\epsilon
g)^{2}])\right]$
$\displaystyle=2\epsilon^{2}\operatorname*{Var}(g)+O(\epsilon^{3})$
This shows that $\alpha\leq\lambda/2$, which is tight in the case of the cube.
However, for constant-degree expander families (in particular, for random
$d$-regular graphs with high probability) we have [DSC96, Example 4.2]
$\lambda=\Omega(1)$ but $\alpha=O(\log\log n/\log n)\ll\lambda$.
### 2.2 Hypercontractivity and the log-Sobolev inequality
When $\rho\in[0,1]$, the noise operator $T_{\rho}$ is easily seen to contract
$\ell_{2}$: for any $f\colon\\{-1,1\\}^{n}\to\mathbb{R}$, we have
$\|T_{\rho}f\|_{2}^{2}=\sum_{S}\rho^{|S|}\hat{f}(S)^{2}\leq\sum_{S}\hat{f}(S)^{2}=\|f\|_{2}^{2}$.
Now consider its behavior from $\ell_{2}$ to $\ell_{q}$ for some $q>2$. When
$\rho=1$, we have $T_{1}f=f$; in particular, for $g(x)=(1+x_{1})/2$,
$\|g\|_{q}=1/2^{1/q}>1/2^{1/2}=\|g\|_{2}$. On the other hand,
$T_{0}f=\operatorname*{\mathbb{E}}f$, so
$\|T_{0}f\|_{q}=|\operatorname*{\mathbb{E}}f|\leq\|\operatorname*{\mathbb{E}}f^{2}\|^{1/2}$.
By the intermediate value theorem, there must be some $\rho\in(0,1)$ such that
$\|T_{0}\|_{2\to q}=1$. A theorem of Gross [Gro75] connects this critical
$\rho$ with the Log-Sobolev constant $\alpha$ of the underlying space:
###### Theorem 2.
$\|T_{\rho}f\|_{p\to q}\leq 1$ if and only if $\rho^{-2\alpha
n}\geq\frac{q-1}{p-1}$.
Stated differently, $\|T_{1-\epsilon}f\|_{q}\leq\|f\|_{2}$ when
$q\leq(1-\epsilon)^{-2}+1\approx 2+2\epsilon$. Thus to prove hypercontractive
inequalities on the discrete cube, it suffices to bound the log-Sobolev
constant. We shall prove this claim for $p=2$, which turns out to imply the
general version.
###### Proof of Theorem 2.
We shall prove that $\|T_{\rho}f\|_{q}\leq\|f\|_{2}$ for $q=1+\rho^{-2\alpha
n}$; the remainder of the theorem can be shown using similar techniques. As we
observed before, this inequality is tight when $\rho=1$, so it suffices to
show that $\frac{d}{d\rho}\|T_{\rho}f\|_{q}\geq 0$ for $0\leq\rho\leq 1$. For
notational convenience, let $G=\|T_{\rho}f\|_{q}^{q}$. Then
$\|T_{\rho}f\|_{q}^{\prime}=(G^{1/q})^{\prime}=q^{-2}G^{(1/q)-1}\left(qG^{\prime}-q^{\prime}G\log
G\right).$
Now we use the fact that $G=\operatorname*{\mathbb{E}}(T_{\rho}f)^{q}$ to get
$G^{\prime}=q\operatorname*{\mathbb{E}}\left[(T_{\rho}f)^{q-1}(T_{\rho}f)^{\prime}\right]+q^{\prime}\operatorname*{\mathbb{E}}\left[(T_{\rho}f)^{q}\log(T_{\rho}f)\right].$
Applying Lemma 3 and simplifying, we get
$qG^{\prime}-q^{\prime}G\log
G=q^{\prime}\operatorname*{Ent}\left((T_{\rho}f)^{q}\right)+\frac{nq^{2}}{2\rho}\operatorname{\mathbb{D}}\left((T_{\rho}f)^{q-1},T_{\rho}f\right).$
We use Lemma 4 to handle the second term, and plug in $q=1+\rho^{-2\alpha n}$
to get
$qG^{\prime}-q^{\prime}G\log G=n\rho^{-2\alpha
n-1}\bigl{[}\operatorname{\mathbb{D}}\bigl{(}(T_{\rho}f)^{q/2},(T_{\rho}f)^{q/2}\bigr{)}-\operatorname*{Ent}\left((T_{\rho}f)^{q}\right)\bigr{]},$
whose positivity we are guaranteed by the log-Sobolev inequality applied to
$(T_{\rho}f)^{(q-1)/2}$. ∎
###### Lemma 3.
For any $f,g\colon\\{-1,1\\}^{n}\to\mathbb{R}$, $\langle
g,\frac{d}{d\rho}(T_{\rho}f)\rangle=\frac{n}{2\rho}\operatorname{\mathbb{D}}(g,T_{\rho}f)$.
###### Proof.
Recalling Lemma 1 and the projection property of the $h_{i}$s, we have
$\langle g,(T_{\rho}f)^{\prime}\rangle=\langle
g,\operatorname{BG}_{\rho}^{\prime}*f\rangle=\biggl{<}g,\frac{1}{\rho}\operatorname{BG}_{\rho}*f*\sum_{i}h_{i}\biggr{>}=\frac{1}{\rho}\sum_{i}\langle
g*h_{i},\operatorname{BG}_{\rho}*f\rangle=\frac{n}{2\rho}\operatorname{\mathbb{D}}(g,T_{\rho}f).\qed$
###### Lemma 4.
For any $f\colon\\{-1,1\\}^{n}\to\mathbb{R}$ and $q\geq 2$,
$\operatorname{\mathbb{D}}(f,f^{q-1})\geq\frac{4(q-1)}{q^{2}}\operatorname{\mathbb{D}}\left(f^{q/2},f^{q/2}\right)$.
###### Proof.
It suffices to show that
$(a^{q-1}-b^{q-1})(a-b)>\frac{4(q-1)}{q^{2}}(a^{q/2}-b^{q/2})^{2}$ for all
$a>b\geq 0$ and $q\geq 2$. But observe that
$\displaystyle\left(\int_{a}^{b}t^{q/2-1}dt\right)^{2}$
$\displaystyle=\frac{4}{q^{2}}(a^{q/2}-b^{q/2})^{2}$
$\displaystyle\int_{a}^{b}t^{q-2}dt\ \int_{a}^{b}dt$
$\displaystyle=\frac{1}{q-1}(a^{q-1}-b^{q-1})(a-b)$
and the inequality between the integrals follows from convexity. ∎
### 2.3 Two-point inequality
We begin by showing that the log-Sobolev inequality holds for the uniform
distribution on the two-point space $\\{-1,1\\}$ with $\alpha=2$. Without loss
of generality, consider $f(x)=1+sx$. Then
$\operatorname*{Ent}(f^{2})=\tfrac{1}{2}(1+s)^{2}\log(1+s)^{2}+\tfrac{1}{2}(1-s)^{2}\log(1-s)^{2}-(1+s^{2})\log(1+s^{2})$
and $\operatorname{\mathbb{D}}(f,f)=2s^{2}$. We shall show that
$\phi(s)=\operatorname{\mathbb{D}}(f,f)-\alpha\operatorname*{Ent}(f^{2})$ is
non-negative for $-1\leq s\leq 1$. By symmetry it suffices to consider $s\geq
0$. But $\phi(0)=0$ and
$\phi^{\prime}(s)=4s+2s\log(1+s^{2})+2(1-s)\log(1-s)-2(1+s)\log(1+s),$
which is non-negative because $\phi^{\prime}(0)=0$ and
$\phi^{\prime\prime}(s)=\frac{4s^{2}}{s^{2}+1}+2\log\frac{1+s^{2}}{1-s^{2}}\geq
0.$
### 2.4 Tensoring property
###### Theorem 5.
Let $\alpha$ be the log-Sobolev constant of $\\{-1,1\\}^{n}$. Then the log-
Sobolev constant of $\\{-1,1\\}^{2n}$ is $\alpha/2$.
When $n$ is a power of $2$, we can conclude inductively that $\alpha=1/n$; a
proof along similar lines works for arbitrary $n$ as well.
###### Proof of Theorem 5.
For any $f\colon\\{-1,1\\}^{n}\times\\{-1,1\\}^{n}\to\mathbb{R}$, set
$g(x)=\|f(x,\cdot)\|_{2}$. Then by the conditional entropy formula,
$\operatorname*{Ent}(f^{2})\leq\operatorname*{Ent}(g^{2})+\operatorname*{\mathbb{E}}_{x}\operatorname*{Ent}_{y}(f(x,y)^{2})\leq\frac{\operatorname{\mathbb{D}}(g,g)+\operatorname*{\mathbb{E}}_{x}\operatorname{\mathbb{D}}_{y}(f(x,y),f(x,y))}{\alpha}$
and by convexity,
$\displaystyle\operatorname{\mathbb{D}}(g,g)$
$\displaystyle=\tfrac{1}{2}\operatorname*{\mathbb{E}}_{x\sim
x^{\prime}}(g(x)-g(x^{\prime}))^{2}\leq\tfrac{1}{2}\operatorname*{\mathbb{E}}_{x\sim
x^{\prime}}\operatorname*{\mathbb{E}}_{y}\left[(f(x,y)-f(x^{\prime},y))^{2}\right]=\operatorname*{\mathbb{E}}_{y}\operatorname{\mathbb{D}}_{x}(f(x,y),f(x,y))$
where the notation $x\sim x^{\prime}$ ranges over edges of $\\{-1,1\\}^{n}$.
Taken together, these give
$\operatorname*{Ent}(f^{2})\leq\frac{\operatorname*{\mathbb{E}}_{x}\operatorname{\mathbb{D}}_{y}(f(x,y),f(x,y))+\operatorname*{\mathbb{E}}_{y}\operatorname{\mathbb{D}}_{x}(f(x,y),f(x,y))}{\alpha}\leq\frac{2\operatorname{\mathbb{D}}(f)}{\alpha}=\frac{\operatorname{\mathbb{D}}(f)}{\alpha/2}.$
as claimed. ∎
### 2.5 Non-product groups
Recall that we defined the Dirichlet form
$\operatorname{\mathbb{D}}(f,g)=\tfrac{1}{2}\operatorname*{\mathbb{E}}_{u\sim
v}(f(u)-f(v))(g(u)-g(v))$
for functions $f,g\colon\\{-1,1\\}^{n}\to\mathbb{R}$, but it makes sense for
any regular graph if we sample $u,v$ uniformly from the edges. Thus, given any
family of regular graphs, we can ask if they satisfy a log-Sobolev inequality
of the form $\operatorname{\mathbb{D}}(f,f)\geq\alpha\operatorname*{Ent}(f)$
for all suitable $f$.
It turns out that the relationship between logarithmic Sobolev inequalities
and hypercontractive noise operator subgroups, as stated by Gross [Gro75],
holds for a wide class of spaces, not just the hypercube $\\{-1,1\\}^{n}$.
Diaconis and Saloff-Coste [DSC96] explored an intermediate between these two
extremes of specialization to give improved mixing time results for Markov
chains on various graphs.
One of the first discrete applications of hypercontractivity was a celebrated
theorem of Kahn, Kalai and Linial [KKL88] relating the maximum influence of a
function on the hypercube to its variance. In Theorem 7, we discuss some
recent work [OW09b] of O’Donnell and Wimmer generalizing the KKL theorem to
apply to the wider class of Schreier graphs associated with group actions
(defined below).
An action of a group $G$ on a set $X$ is a homomorphism from $G$ to the group
of bijections on $X$, and we write $x^{g}$ for the image of $x$ under the
bijection for $g$. If $S$ is a set of generators for $G$, then the Schreier
graph $\operatorname{Sch}(G,S,X)$ has vertex set $X$ and edges $(x,x^{g})$ for
all $x\in X$ and $g\in S$. It is known that every connected regular graph of
even degree can be obtained in this way [Gro77]. The definition of the
Dirichlet form $\operatorname{\mathbb{D}}$ generalizes without change, but to
be able to derive a log-Sobolev inequality for this space, we must define the
noise operator $T_{\rho}$ in an appropriate fashion to satisfy the claim of
Lemma 1: $\langle
g,\frac{d}{d\rho}(T_{\rho}f)\rangle\propto\frac{1}{\rho}\operatorname{\mathbb{D}}(g,T_{\rho}f)$.
## 3 Boolean-Valued Functions
### 3.1 Influences
Write $x_{-i}$ for the collection of random variables
$\\{x_{1},\dotsc,x_{n}\\}\setminus\\{x_{i}\\}$. The influence of the $i$th
coordinate on a function $f\colon\\{-1,1\\}^{n}\to\mathbb{R}$ is given by
$\operatorname{Inf}_{i}(f)=\operatorname*{\mathbb{E}}_{x_{-i}}\operatorname*{Var}_{x_{i}}f(x)=\operatorname*{\mathbb{E}}_{x_{-i}}\left[\operatorname*{\mathbb{E}}_{x_{i}}f(x)^{2}-(\operatorname*{\mathbb{E}}_{x_{i}}f(x))^{2}\right].$
When $f$ is Boolean-valued, this quantity is just the probability that
changing $x_{i}$ changes $f(x)$. Writing $f$ in the Fourier basis, we have
$\operatorname*{\mathbb{E}}_{x_{-i}}\operatorname*{\mathbb{E}}_{x_{i}}f(x)^{2}=\operatorname*{\mathbb{E}}_{x}f(x)^{2}=\sum_{S}\hat{f}(S)^{2}$
and
$\operatorname*{\mathbb{E}}_{x_{-i}}(\operatorname*{\mathbb{E}}_{x_{i}}f(x))^{2}=\sum_{S\not\ni
i}\hat{f}(S)^{2}$, so that $\operatorname{Inf}_{i}(f)=\sum_{S\ni
i}\hat{f}(S)^{2}=\operatorname*{\mathbb{E}}(f*h_{i})^{2}$. In addition, we
define the total influence
$\operatorname{Inf}(f)=\sum_{i}\operatorname{Inf}_{i}(f)=\sum_{S}|S|\hat{f}(S)^{2}$.
### 3.2 Structural results
Boolean functions are natural combinatorial objects, but they were first
studied from an analytical viewpoint in work on voting and social choice. In
this setting, a function $f\colon\\{-1,1\\}^{n}\to\\{-1,1\\}$ is viewed as a
way to combine the preferences of $n$ voters to yield the result of the
election. This explains the notions of dictator or junta functions, which
depend on only one or a few of their coordinates, respectively. In this
context it is also natural to consider functions where no coordinate (“voter”)
has a very large influence. Kahn, Kalai, and Linial [KKL88] first introduced
the Fourier analysis of Boolean functions as a technique in computer science.
Their theorem establishes that if a function is far from a constant (i.e., has
variance at least a constant), then it must have a variable of influence
$\Omega(\frac{\log n}{n})$. We state a strengthening of their original
inequality due to Talagrand [Tal95]:
###### Theorem 6 ([KKL88, Tal95]).
For any $f\colon\\{-1,1\\}^{n}\to\\{-1,1\\}$,
$\sum_{i}\frac{\operatorname{Inf}_{i}(f)}{\log(1/\operatorname{Inf}_{i}(f))}\geq\Omega(1)\cdot\operatorname*{Var}(f).$
We can compare this to the Poincaré inequality on the cube, which can be
stated as
$\sum_{i}\operatorname{Inf}_{i}(f)\geq\Omega(1)\cdot\operatorname*{Var}(f).$
(In particular, there exists a variable of influence
$\Omega\bigl{(}\tfrac{1}{n}\bigr{)}\operatorname*{Var}(f)$.) The KKL theorem
is a stronger result of the same form: it is a comparison between a local and
a global measure of variation. The proofs of KKL and Talagrand used the
hypercontractivity of the cube, but we present here a more recent proof due to
Rossignol that uses the log-Sobolev inequality instead. For simplicity we’ll
just show the weaker statement that the maximum influence is
$\Omega\bigl{(}\tfrac{\log n}{n}\bigr{)}\operatorname*{Var}(f)$.
###### Proof.
Write $f-\operatorname*{\mathbb{E}}f=f_{1}+\dotsb+f_{n}$, where
$f_{j}=\sum_{S:\max S=j}\hat{f}(S)\chi_{S}$. For each $f_{j}$, the log-Sobolev
inequality states that
$\operatorname{\mathbb{D}}(f_{j},f_{j})\geq\alpha\operatorname*{Ent}(f_{j}^{2})=\frac{1}{n}\operatorname*{Ent}(f_{j}^{2})$.
By writing $\operatorname{\mathbb{D}}(f_{j},f_{j})$ in terms of the Fourier
coefficients $\hat{f}(S)$, we can check that
$\operatorname{\mathbb{D}}(f,f)=\sum_{j=0}^{n}\operatorname{\mathbb{D}}(f_{j},f_{j})$,
so that we can sum all these inequalities to obtain
$n\operatorname{\mathbb{D}}(f,f)\geq\sum_{j}\operatorname*{Ent}(f_{j}^{2})=\underbrace{\sum_{j}\operatorname*{\mathbb{E}}\left[f_{j}^{2}\log(f_{j}^{2})\right]}_{A}+\underbrace{\sum_{j}\operatorname*{\mathbb{E}}f_{j}^{2}\log\frac{1}{\operatorname*{\mathbb{E}}f_{j}^{2}}}_{B}.$
In order to bound $B$, we begin by noting that
$\operatorname*{\mathbb{E}}f_{j}^{2}=\sum_{S:\max
S=j}\hat{f}(S)^{2}\leq\sum_{S\ni
j}\hat{f}(S)^{2}=\operatorname*{\mathbb{E}}(f*h_{j})^{2}$
where the $h_{j}$s are the edge functions we defined earlier. Letting
$M(f)=\max_{j}\operatorname*{\mathbb{E}}(f*h_{j})^{2}=\max_{j}\operatorname{Inf}_{j}(f)$,
we have
$B=\sum_{j}\operatorname*{\mathbb{E}}f_{j}^{2}\log\frac{1}{\operatorname*{\mathbb{E}}f_{j}^{2}}\geq\sum_{j}\operatorname*{\mathbb{E}}f_{j}^{2}\log\frac{1}{M(f)}=\operatorname*{Var}(f)\log\frac{1}{M(f)}$
where we have used the orthogonality of the $f_{j}$s and the fact that
$\operatorname*{Var}(f)=\sum_{S\neq\emptyset}\hat{f}(S)^{2}$.
To bound $A$, we split it up further:
$A=\underbrace{\sum_{j}\operatorname*{\mathbb{E}}\left[f_{j}^{2}\log(f_{j}^{2})\cdot
1_{f_{j}^{2}\leq
t}\right]}_{A_{1}}+\underbrace{\sum_{j}\operatorname*{\mathbb{E}}\left[f_{j}^{2}\log(f_{j}^{2})\cdot
1_{f_{j}^{2}>t}\right]}_{A_{2}}.$
For $0\leq t\leq 1/e^{2}$, we have that $\sqrt{t}\log\sqrt{t}$ is a
nonpositive decreasing function and therefore,
$A_{1}=2\sum_{j}\operatorname*{\mathbb{E}}\left[|f_{j}|\log|f_{j}|\cdot|f_{j}|1_{f_{j}^{2}\leq
t}\right]\geq
2\sqrt{t}\log\sqrt{t}\sum_{j}\operatorname*{\mathbb{E}}|f_{j}\cdot
1_{f_{j}^{2}\leq t}|\geq\sqrt{t}\log
t\sum_{j}\operatorname*{\mathbb{E}}|f_{j}|.$
By comparing Fourier coefficients, it is easy to verify that
$f_{j}=\operatorname*{\mathbb{E}}_{x_{j+1},\dotsc,x_{n}}(f*h_{j})$. Therefore,
by convexity,
$\operatorname*{\mathbb{E}}|f_{j}|\leq\operatorname*{\mathbb{E}}|f*h_{j}|.$
Until now, the proof has made no use of the fact that $f$ takes on only
Boolean values. Now we argue that because $f(x)\in\\{-1,1\\}$, we must have
$(f*h_{j})(x)\in\\{-1,0,1\\}$, so that
$\operatorname*{\mathbb{E}}|f*h_{j}|=\operatorname*{\mathbb{E}}(f*h_{j})^{2}$.
Plugging this into our bound for $A_{1}$ yields
$A_{1}\geq\sqrt{t}\log
t\sum_{j}\operatorname*{\mathbb{E}}(f*h_{j})^{2}=\frac{n}{2}\sqrt{t}\log
t\cdot\operatorname{\mathbb{D}}(f,f).$
For $A_{2}$, note that $\log(\cdot)$ is increasing, so
$A_{2}\geq\log t\sum_{j}\operatorname*{\mathbb{E}}f_{{}_{j}}^{2}=\log
t\operatorname*{Var}f.$
Summing all these bounds gives us
$n\operatorname{\mathbb{D}}(f,f)\geq\log\frac{1}{M(f)}\operatorname*{Var}(f)+\frac{n}{2}\sqrt{t}\log
t\cdot\operatorname{\mathbb{D}}(f,f)+\log t\cdot\operatorname*{Var}(f).$
By the Poincaré inequality,
$\operatorname{\mathbb{D}}(f,f)\geq\frac{2}{n}\operatorname*{Var}(f)$, so we
can set
$t=\bigl{(}\frac{2\operatorname*{Var}(f)}{ne\operatorname{\mathbb{D}}(f,f)}\bigr{)}^{2}\leq
1/e^{2}$. With this substitution, the above inequality becomes
$\frac{2}{e\sqrt{t}}\geq\log\frac{t^{1+1/e}}{M(f)}.$
Suppose $t\leq(\frac{4}{e\log n})^{2}$. Then
$\operatorname{\mathbb{D}}(f,f)\geq\frac{2\operatorname*{Var}(f)}{en}\cdot\frac{e\log
n}{4}=\Omega\Bigl{(}\frac{\log n}{n}\Bigr{)},$
and we know that $M(f)\geq 2\operatorname{\mathbb{D}}(f,f)$. On the other
hand, if $t>(\frac{4}{e\log n})^{2}$, then
$M(f)>t^{1+1/e}\exp\Bigl{(}\frac{-2}{e\sqrt{t}}\Bigr{)}=\Bigl{(}\frac{4}{e\log
n}\Bigr{)}^{2+2/e}\exp\Bigl{(}\frac{-\log n}{2}\Bigr{)}\gg\frac{\log
n}{n}.\qed$
We are now in a position to state the recent result of O’Donnell and Wimmer
[OW09b] generalizing the KKL theorem to Schreier graphs satisfying a certain
technical property.
###### Theorem 7 ([OW09b]).
Let $G$ be a group acting on a set $X$, $U\subseteq X$ be a union of conjugacy
classes that generates $G$, and $\alpha$ be the log-Sobolev constant of
$\operatorname{Sch}(G,X,U)$. Then for any $f\colon X\to\\{-1,1\\}$,
$\frac{\sum_{U}\operatorname{Inf}_{u}(f)}{\log(1/\max_{U}\operatorname{Inf}_{u}(f))}\geq\Omega(\alpha\operatorname*{Var}(f)).$
In particular, there is some $u\in U$ such that
$\operatorname{Inf}_{u}(f)\geq\Omega(\alpha\log\frac{1}{\alpha})\operatorname*{Var}(f)$.
For an Abelian group such as $\mathbb{Z}_{2}^{n}$ (the cube), every group
element is in a conjugacy class by itself, so the extra condition on $U$ is
vacuous. Using $\alpha=\Omega(\frac{1}{n})$ for the cube, we recover the
original KKL theorem. O’Donnell et al. apply the generalized result to the
non-Abelian group $S_{n}$ of permutations on $[n]$, generated by
transpositions and acting on the family $\binom{[n]}{k}$ of $k$-subsets of
$[n]$. By viewing these families as sets of $n$-bit strings, they recover a
“rigidity” version of the Kruskal-Katona theorem that states (roughly) that if
a subset of a layer of a cube has a small expansion to the layer above it,
then it must be correlated to some dictator function.
#### Coding theoretic interpretation.
In the _long code_ , an integer $i\in[n]$ is encoded as the dictator function
$(x_{1},\dotsc,x_{n})\mapsto x_{i}$. By using many more bits ($2^{n}$ rather
than $\log n$) of redundant storage, we hope to be able to recover from
corruptions in the data. The theorem tells us that as long as the corrupted
version of an encoding is far from a constant function, it can be decoded to a
coordinate whose influence is $\Omega(\log n)$ times the average influence.
Since every coordinate’s influence is nonnegative, only $O(\log n)$
coordinates can have influence this large. Thus, we have a “small” set of
candidate long codes to which we might decode the word. To complete this
picture, we’d like to understand how far the word can be from functions that
depend only on these coordinates; the following theorem of Friedgut, which we
state without proof, furnishes this information.
###### Theorem 8 ([Fri98]).
For every $f\colon\\{-1,1\\}^{n}\to\\{-1,1\\}$ and $0<\epsilon<1$, there is a
function $g\colon\\{-1,1\\}^{n}\to\\{-1,1\\}$ depending on at most
$\exp\bigl{(}\frac{2+o(1)}{\epsilon n}\operatorname{Inf}(f)\bigr{)}$ variables
such that $\operatorname*{\mathbb{E}}|f-g|\leq\epsilon$.
## 4 Gaussian isoperimetry and an algorithmic application
Hypercontractive inequalities were first investigated in the context of
Gaussian probability spaces, for their applications to quantum field theory.
The following simple proof reduces the continuous Gaussian hypercontractive
inequality to its discrete counterpart on the cube.
### 4.1 From the central limit theorem to Gaussian hypercontractivity
###### Theorem 9 ([Gro75]).
Let $x\in\mathbb{R}$ be normally distributed, i.e.,
$\Pr[x\in
A]=\frac{1}{\sqrt{2\pi}}\int_{A}\exp\left(-\frac{x^{2}}{2}\right)\,dx.$
Then for a smooth function $f\colon\mathbb{R}\to\mathbb{R}$, the random
variable $F=f(x)$ satisfies
$\operatorname{\mathbb{D}}(F,F)\geq\alpha\operatorname*{Ent}(F^{2})$
with $\alpha=1$ and
$\operatorname{\mathbb{D}}(F,G)=\frac{1}{2}\left<{\frac{dF}{dx},\frac{dG}{dx}}\right>.$
###### Proof.
We shall approximate the Gaussian distribution by a weighted sum of Bernoulli
variables. Let $y\in\\{-1,1\\}^{k}$ be uniformly distributed, and set
$g(y)=\frac{y_{1}+\dotsb+y_{k}}{\sqrt{k}}$. By the log-Sobolev inequality
applied to $f\circ g(y)$, we have $\operatorname{\mathbb{D}}\left(f\circ
g(y),f\circ g(y)\right)\geq\operatorname*{Ent}(f\circ g(y)^{2})$. By the
central limit theorem, the right side converges to
$\operatorname*{Ent}(f(x)^{2})=\operatorname*{Ent}(F^{2})$ as $k\to\infty$, so
it remains to show that the left side converges to
$\operatorname{\mathbb{D}}(F,F)$ as well. Let $y|_{y_{i}=\theta}$ be the value
obtained by replacing the $i$th coordinate of $y$ with the value $\theta$, and
observe that $g(y|_{y_{i}=1})-g(y|_{y_{i}=-1})=2/\sqrt{k}$. Then, using the
smoothness of $f$, we have
$\left|\left(h_{i}*(f\circ g)\right)(y)\right|=\frac{1}{2}\left|f\circ
g(y|_{y_{i}=1})-f\circ
g(y|_{y_{i}=-1})\right|=\frac{1}{\sqrt{k}}\left|f^{\prime}\circ
g(y)\right|+o\left(\frac{1}{\sqrt{k}}\right),$
so that
$\operatorname{\mathbb{D}}\left(f\circ g(y),f\circ
g(y)\right)=\frac{1}{2}\operatorname*{\mathbb{E}}_{y}\left[\sum_{i}\left(h_{i}*(f\circ
g)\right)(y)^{2}\right]=\frac{1}{2}\operatorname*{\mathbb{E}}_{y}\left[f^{\prime}\circ
g(y)^{2}+o(1)\right].$
The second term vanishes as $k\to\infty$, and the first term converges to
$\operatorname{\mathbb{D}}(F,F)$ by the Central Limit Theorem. ∎
The tensoring property of log-Sobolev inequalities lets us extend this result
to Gaussian distributions over $\mathbb{R}^{d}$. We are also interested in the
corresponding noise operator $S_{\rho}$, known as the Ornstein-Uhlenbeck
operator, which is given by
$S_{\rho}f(x)=\operatorname*{\mathbb{E}}_{z\sim\mathcal{N}(0,1)^{d}}f(\rho
x+(1-\rho^{2})^{1/2}z).$
Theorem 2 has an analog in this setting, which lets us conclude that every
function $f\colon\mathbb{R}^{d}\to\mathbb{R}$ satisfies
$\|S_{\rho}f\|_{q}\leq\|f\|_{p}$ where $q>p\geq 1$ and
$\rho^{-2}\geq(p-1)/(q-1)$.
### 4.2 Reverse hypercontractivity and isoperimetry
In 1982, Borell showed a _reversed_ inequality of a similar form when $q<p<1$:
###### Theorem 10 (Reverse hypercontractivity, [Bor82]).
Fix $q<p\leq 1$ and $\rho\geq 0$ such that $\rho^{-4}\geq(p-1)/(q-1)$. Then
for any positive-valued function $f\colon\mathbb{R}^{d}\to\mathbb{R}^{+}$, we
have $\|S_{\rho}f\|_{q}\geq\|f\|_{p}$.
Note that the expressions $\|\cdot\|_{p}$ are not norms when $p<1$; in
particular, they are not convex. However, this theorem can be proved by means
similar to our proof for the Gaussian log-Sobolev inequality: we start with a
base result for the 2-point space, proceed by tensoring to the hypercube, and
use the central limit theorem to cover Gaussian space.
As an application of Borell’s result, consider the following strong
isoperimetry theorem for Gaussian space (due to Sherman).
###### Theorem 11 (Gaussian isoperimetry, [She09]).
Let $u,u^{\prime}\in\mathbb{R}^{d}$ be independent Gaussian random variables.
Then for any set $A\subseteq\mathbb{R}^{d}$ and any $\tau>0$, we have
$\Pr_{u}\left[\Pr_{u^{\prime}}[\rho u+(\sqrt{1-\rho^{2}})u^{\prime}\in
A]\leq\tau\right]\leq\frac{\tau^{1-\rho}}{\mu(A)}$
###### Proof.
When $\mu(A)\leq\tau^{1-\delta}$, there is nothing to prove. Otherwise, let
$f$ be the indicator function of $A$ and observe that
$\Pr_{u^{\prime}}\bigl{[}\rho u+(1-\rho^{2})^{1/2}u^{\prime}\in
A\bigr{]}=S_{\rho}f(u)$. Therefore, for $q=1-1/\rho<0$, we have
$\displaystyle\Pr_{u}\left[\Pr_{u^{\prime}}[u^{\prime}\in A]\leq\tau\right]$
$\displaystyle=\Pr_{u}[S_{\rho}f(u)\leq\tau]$
$\displaystyle=\Pr_{u}[S_{\rho}f(u)^{q}\geq\tau^{q}]$
$\displaystyle\leq\frac{\operatorname*{\mathbb{E}}_{u}(S_{\rho}f(u))^{q}}{\tau^{q}}$
by an application of Markov’s inequality. But
$\operatorname*{\mathbb{E}}_{u}(S_{\rho}f(u))^{q}$ is just
$\|S_{\rho}f\|_{q}^{q}$, and we know by Borell’s theorem that
$\|S_{\rho}f\|_{q}\geq\|f\|_{p}$ for $p=1-\rho$. Thus
$\Pr_{u}\left[\Pr_{u^{\prime}}[u^{\prime}\in
A]\leq\tau\right]\leq\frac{\|f\|_{p}^{q}}{\tau^{q}}=\frac{\mu(A)^{q/p}}{\tau^{q}}=\left(\frac{\tau^{1-\rho}}{\mu(A)}\right)^{1/\rho}\leq\frac{\tau^{1-\rho}}{\mu(A)}$
where we have used the facts that $q<0$ and $\rho\leq 1$. ∎
### 4.3 Fast graph partitioning and the constructive Big Core Theorem
#### Problem and SDP rounding algorithm.
In the $c$-balanced separator problem, we are given a graph $G=(V,E)$ on $n$
vertices and asked to find the smallest set of edges such that their removal
disconnects the graph into pieces of size at most $cn$. The problem is NP-
hard, and the best known approximation ratio111For technical reasons, it is
actually a pseudo-approximation: the algorithm’s output for $c$ is compared to
the optimal value for $c^{\prime}\neq c$. is $\Theta(\sqrt{\log n})$.
The first algorithm to achieve this bound was based on a semidefinite program
that assigns a unit vector to each vertex and minimizes the total embedded
squared length of the edges subject to the constraint that the vertices are
spread out and that the squared distances between the points form a metric:
minimize $\displaystyle\textstyle\sum_{i\sim j}\|x_{i}-x_{j}\|_{2}^{2}$
subject to $\displaystyle\|x_{i}\|_{2}^{2}=1$ $\displaystyle\forall i\in V$
$\displaystyle\textstyle\sum_{i,j}\|x_{i}-x_{j}\|^{2}\geq c(1-c)n$
$\displaystyle\|x_{i}-x_{j}\|_{2}^{2}+\|x_{j}-x_{k}\|_{2}^{2}\geq\|x_{i}-x_{k}\|^{2}$
$\displaystyle\forall i,j,k\in V$
To round this SDP, Arora, Rao and Vazirani [ARV09] pick a random direction $u$
and project all the points along $u$. They then define sets $A$ and $B$
consisting of points $x$ whose projections are sufficiently large, i.e.,
$A=\\{x\mid\langle x,u\rangle<-K\\}$ and similarly $B=\\{x\mid\langle
x,u\rangle>K\\}$, where $K$ is chosen to make $A$ and $B$ have size
$\Theta(n)$ with high probability. Next, they discard points $a\in A,b\in B$
such that $\|a-b\|$ is much smaller than expected for a pair whose projections
are $\geq 2K$ apart. Finally, if the resulting pruned sets $A^{\prime}\subset
A$ and $B^{\prime}\subset B$ are large enough, they show that greedily growing
$A$ yields a good cut.
#### Matchings and cores.
The key step in making this argument work is to ensure that not too many pairs
$(a,b)$ are removed in the pruning step. To bound the probability of this bad
event, we consider the possibility that for a large fraction
$\delta=\Omega(1)$ of directions $u$, there exists a matching of points
$M_{u}$ such that each pair $(a,b)\in M_{u}$ is short (i.e.,
$\|a-b\|\leq\ell=O(1/\sqrt{\log n})$) but stretched along $u$ (i.e., $|\langle
a-b,u\rangle|\geq\sigma=\Omega(1)$). Such a set of points is called a
_$(\sigma,\delta,\ell)$ -core_. The big core theorem (first proved with
optimal parameters by Lee [Lee05]) asserts that this situation can’t arise:
for a fixed $\sigma,\delta$, and $\ell$, we must have
$n\gg\exp(\sigma^{6}/\ell^{4}\log^{2}(1/\delta))$, which is a contradiction
for our chosen values of $\sigma,\delta,\ell$.
In order to prove the big core theorem, Lee concatenates pairs that share a
point and belong in matchings for nearby directions. The existence of a long
chain of such concatenations is what leads to a contradiction: if we consider
the endpoints $a,b$ of a chain of length $p$, the projection $|\langle
a-b,u\rangle|$ grows linearly in $p$ whereas the distance $\|a-b\|$ grows only
as $\sqrt{p}$ (recall that the SDP constrained the _squared_ distances to form
a metric).
#### Boosting.
The matching chaining argument we have just presented in its simple form
doesn’t work, for the following reason. At each chaining step, the fraction of
nearby directions available for our use reduces by roughly $1-\delta$ (by a
union bound) so that we are rapidly left with no direction to move in. To
remedy this situation, we need to boost the fraction of usable directions at
each step, say from $\delta/2$ to $1-\delta/2$, so that we can carry on
chaining in spite of a $1-\delta$ loss. Lee’s proof uses the standard
isoperimetric inequality for the sphere to show that this boosting can be
performed with no change in $\ell$ and a very small penalty in $\sigma$. In
other words, we take advantage of the fact that a very small dilation of a set
of constant measure (i.e., the set of available directions) has measure close
to $1$.
#### Faster algorithms.
Lee’s big core theorem is non-constructive in the sense that it only shows the
_existence_ of such a long chain of matched pairs in order to give a
contradiction. While this form suffices to bound the approximation ratio of
the ARV rounding scheme, other variants of their technique require a way to
_efficiently sample_ long chains, not just show their existence. Sherman
constructs a distribution over directions that does not depend on the point
set at all, yet is guaranteed to always have a non-trivial probability of
producing long chains of stretched pairs. More precisely,
###### Theorem 12 (Constructive big core [She09]).
For any $1\leq R\leq\Theta(\sqrt{\log n})$, there is $P\geq\Theta(R^{2}/\log
n)$ and an efficiently sampleable distribution $\mu$ over the set of sequences
of $\leq P$ direction vectors (each in $\mathbb{R}^{d}$), such that: for any
$(\sigma,\delta,\ell)$-core $M$, if the string of directions is sampled from
$\mu$, the expected number of chains whose endpoints are $\geq P\ell$ apart is
at least $\exp(-O(P^{2})n)$.
We sketch some of the ideas of the proof here. Sherman constructs two
sequences of Gaussian directions $u_{1},\dotsc,u_{P}$ and
$w_{1},\dotsc,w_{P}$. Each $w_{i}$ is an independent Gaussian vector, whereas
each $u_{i}$ for $i>1$ is a Gaussian vector $\rho$-correlated with $u_{i-1}$.
Finally, the distribution $\mu$ is given by randomly shuffling together the
$u_{i}$ and $w_{i}$, picking a uniformly random $R$ between $1$ and $P$, and
returning the first $R$ elements of the shuffled sequence. The correlated
directions $u_{i}$ correspond to the steps in which Lee’s proof chained pairs
from similar directions, whereas the independent $w_{i}$ correspond to the
region-growing steps necessary for boosting. By randomly interleaving these
two types of moves, Sherman’s sampling algorithm can be oblivious to the
actual point set it is acting on.
## 5 Complexity theoretic applications
### 5.1 Dictatorship testing with perfect completeness
#### Definitions.
A function $f\colon\\{-1,1\\}^{n}\to\mathbb{R}$ is said to be
$(\epsilon,\delta)$-quasirandom if $\hat{f}(S)\leq\epsilon$ whenever $|S|\leq
1/\delta$. In order to show that a given problem is hard to approximate, we
often need to design a test that
* •
performs $q$ _queries_ on a black-box function $f$,
* •
accepts every dictator function with probability $\geq c$ (the _completeness_
probability), and
* •
accepts every $(\epsilon,\delta)$-quasirandom function with probability $\leq
s$ (the _soundness_ probability).
A test is said to be _adaptive_ if each query is allowed to depend on the
result of the queries so far.
While dictatorship tests for the $c<1$ setting have been known for over a
decade (first from the work of Håstad and more recently via the Unique Games
Conjecture of Khot), there were no nontrivial bounds for $c=1$ until some
recent results of O’Donnell and Wu. Their analysis, which we show below,
relies heavily on the hypercontractive inequality.
###### Theorem 13 ([OW09a]).
For every $n>0$, there is a $3$-query non-adaptive test that accepts every
dictator function $(x_{1},\dotsc,x_{n})\mapsto x_{i}$ with probability $c=1$
but accepts any $(\delta,\delta/\log(1/\delta))$-quasirandom odd function
$f\colon\\{-1,1\\}^{n}\to[-1,1]$ with probability $\leq
s=5/8+O(\sqrt{\delta})$.
The proof uses the following strengthening of the hypercontractive inequality
for restricted parameter values.
###### Lemma 14.
If $0\leq\rho\leq 1$, $q\geq 1$, and $0\leq\lambda\leq 1$ satisfy
$\rho^{\lambda}\leq 1/\sqrt{q-1}$, then for all
$f\colon\\{-1,1\\}^{n}\to\mathbb{R}$,
$\|T_{\rho}f\|_{q}\leq\|T_{\rho}f\|_{2}^{1-\lambda}\|f\|_{2}^{\lambda}$.
###### Proof.
$\displaystyle\|T_{\rho}f\|_{q}^{2}$
$\displaystyle=\|T_{\rho^{\lambda}}T_{\rho^{1-\lambda}}f\|_{q}^{2}$
$\displaystyle\leq\|T_{\rho^{1-\lambda}}f\|_{2}^{2}$
$\displaystyle=\sum_{S}|\rho\hat{f}(S)|^{2(1-\lambda)}|\hat{f}(S)|^{2\lambda}$
$\displaystyle=\|T_{\rho}f\|_{2}^{2(1-\lambda)}\|f\|_{2}^{2\lambda}\qquad\qed$
###### Proof of Theorem 13.
Define the “not-two” predicate
$\operatorname{\texttt{NTW}}\colon\\{-1,1\\}^{3}\to\\{-1,1\\}$ as follows:
$\operatorname{\texttt{NTW}}(a,b,c)=1$ if exactly two of $a,b,c$ equal $-1$,
and $\operatorname{\texttt{NTW}}(a,b,c)=-1$ otherwise. Explicitly,
$\begin{array}[]{rrrrrrrrr}a&-1&-1&-1&-1&1&1&1&1\\\ b&-1&-1&1&1&-1&-1&1&1\\\
c&-1&1&-1&1&-1&1&-1&1\\\
\hline\cr\operatorname{\texttt{NTW}}(a,b,c)&-1&1&1&-1&1&-1&-1&-1\end{array}$
Let $\delta\in[0,1]$ be a parameter to be fixed later. For $i=1,\dotsc,n$, we
pick bits $x_{i},y_{i},z_{i}\in\\{-1,1\\}$ as follows:
* •
with probability $1-\delta$: we choose $x_{i},y_{i}$ uniformly and
independently, then set $z_{i}=-x_{i}y_{i}$;
* •
with probability $\delta$: we choose $x_{i}$ uniformly, then set
$y_{i}=z_{i}=x_{i}$.
Note that for $i\neq j$, $(x_{i},y_{i},z_{i})$ is independent of
$(x_{j},y_{j},z_{j})$. We accept if
$\operatorname{\texttt{NTW}}(f(x),f(y),f(z))=-1$. It is immediate from the
construction of $x_{i},y_{i},z_{i}$ that
$\operatorname{\texttt{NTW}}(x_{i},y_{i},z_{i})=-1$ for $i=1,\dotsc,n$.
Therefore, if $f$ is a dictator function, it follows that
$\operatorname{\texttt{NTW}}(f(x),f(y),f(z))$ must also equal $-1$.
#### Soundness.
It remains to analyze the test when $f$ is pseudorandom. We begin by writing
$\operatorname{\texttt{NTW}}$ in the Fourier basis:
$\operatorname{\texttt{NTW}}=-\frac{1}{4}\chi_{\emptyset}-\frac{1}{4}(\chi_{\\{1\\}}+\chi_{\\{2\\}}+\chi_{\\{3\\}})-\frac{1}{4}(\chi_{\\{1,2\\}}+\chi_{\\{2,3\\}}+\chi_{\\{1,3\\}})+\frac{3}{4}\chi_{\\{1,2,3\\}}$.
Therefore, by symmetry,
$\operatorname*{\mathbb{E}}_{x,y,z}\operatorname{\texttt{NTW}}(f(x),f(y),f(z))=-\tfrac{1}{4}-\tfrac{3}{4}\operatorname*{\mathbb{E}}_{x}f(x)-\tfrac{3}{4}\operatorname*{\mathbb{E}}_{x,y}f(x)f(y)+\tfrac{3}{4}\operatorname*{\mathbb{E}}_{x,y,z}f(x)f(y)f(z).$
We shall systematically rewrite the right-hand side in terms of the Fourier
coefficients of $f$. By our assumption that $f$ is odd, we have $\hat{f}(S)=0$
whenever $S$ has even cardinality. Therefore
$\operatorname*{\mathbb{E}}f(x)=\hat{f}(\emptyset)=0$. Also,
$\operatorname*{\mathbb{E}}_{x,y}f(x)f(y)=\sum_{S,T}\hat{f}(S)\hat{f}(T)\operatorname*{\mathbb{E}}_{x,y}\chi_{S}(x)\chi_{T}(y).$
Consider a summand where $S\neq T$, and without loss of generality fix $i\in
S\setminus T$. It is easy to see that the contributions due to $x_{i}=\pm 1$
cancel each other. Thus, the only terms that remain are of the form $S=T$,
i.e.,
$\operatorname*{\mathbb{E}}_{x,y}f(x)f(y)=\sum_{S}\hat{f}(S)^{2}\operatorname*{\mathbb{E}}_{x,y}\chi_{S}(x)\chi_{S}(y)=\sum_{S}\hat{f}(S)^{2}\left(\operatorname*{\mathbb{E}}_{x_{i},y_{i}}x_{i}y_{i}\right)^{|S|}=\sum_{S}\hat{f}(S)^{2}\delta^{|S|},$
where we have used the fact that
$\operatorname*{\mathbb{E}}(x_{i}y_{i})=(1-\delta)\cdot 0+\delta\cdot
1=\delta$. But $\hat{f}(S)$ is nonzero only for $|S|$ odd, and
$\sum_{S}\hat{f}(S)^{2}=1$, so we can upper-bound the above sum by $\delta$.
#### Bounding the cubic term.
We proceed similarly:
$\operatorname*{\mathbb{E}}_{x,y,z}f(x)f(y)f(z)=\sum_{S,T,U}\hat{f}(S)\hat{f}(T)\hat{f}(U)\operatorname*{\mathbb{E}}_{x,y,z}\chi_{S}(x)\chi_{T}(y)\chi_{U}(z).$
(1)
Each of the expectations can be written as a product over coordinates
$i\in[n]$ using the fact that individual coordinates of $x,y,z$ are chosen
independently. When $i$ belongs to exactly one of $S,T,U$ (say $S$), then it
contributes a factor $\operatorname*{\mathbb{E}}x_{i}=0$, making the product
zero. Similarly, when $i$ belongs to two of the sets (say $S,T$), then the
contribution is $\operatorname*{\mathbb{E}}x_{i}y_{i}=\delta$ by our earlier
calculation. Finally, when $i$ belongs to all three of the sets, we have
$\operatorname*{\mathbb{E}}x_{i}y_{i}z_{i}=(1-\delta)\cdot(-1)+\delta\cdot(0)=-(1-\delta)$.
In light of this calculation, any triple $S,T,U$ that makes a nonzero
contribution to the sum (1) must be of the form
$\displaystyle S$ $\displaystyle=A\cup B\cup C$ $\displaystyle T$
$\displaystyle=A\cup C\cup D$ $\displaystyle U$ $\displaystyle=A\cup D\cup B$
for suitable sets $A,B,C,D\subseteq[n]$ where $A$ is disjoint from $B,C,D$.
Also $|S|,|T|,|U|$ must be odd, from which we can show that $|A|$ must be odd.
In terms of these new sets we can rewrite
$\operatorname*{\mathbb{E}}_{x,y,z}f(x)f(y)f(z)=-\sum_{\begin{subarray}{c}B,C,D\text{
disj. from }A\\\ |A|\text{ odd}\end{subarray}}\hat{f}(A\cup B\cup
C)\hat{f}(A\cup C\cup D)\hat{f}(A\cup D\cup
B)(1-\delta)^{|A|}\delta^{|B|+|C|+|D|}.$
For a fixed $A$, define the function $g_{A}\colon\\{-1,1\\}^{[n]\setminus
A}\to\mathbb{R}$ by $\hat{g}_{A}(X)=\hat{f}(A\cap X)$. Then we have
$\displaystyle\operatorname*{\mathbb{E}}_{x,y,z}f(x)f(y)f(z)$
$\displaystyle=-\sum_{|A|\text{
odd}}(1-\delta)^{|A|}\sum_{\begin{subarray}{c}B,C,D\\\ \text{disj. from
}A\end{subarray}}\hat{g}_{A}(B\cup C)\sqrt{\delta}^{|B\cup
C|}\cdot\hat{g}_{A}(C\cup D)\sqrt{\delta}^{|C\cup D|}\cdot\hat{g}_{A}(D\cup
B)\sqrt{\delta}^{|D\cup B|}$ $\displaystyle=-\sum_{|A|\text{
odd}}(1-\delta)^{|A|}\sum_{\begin{subarray}{c}B,C,D\\\ \text{disj. from
}A\end{subarray}}\widehat{T_{\sqrt{\delta}}g_{A}}(B\cup
C)\cdot\widehat{T_{\sqrt{\delta}}g_{A}}(C\cup
D)\cdot\widehat{T_{\sqrt{\delta}}g_{A}}(D\cup B)$
$\displaystyle=-\sum_{|A|\text{
odd}}(1-\delta)^{|A|}\|T_{\sqrt{\delta}}g_{A}\|_{3}^{3}.$
Write
$g_{A}(u)=\operatorname*{\mathbb{E}}_{x}g_{A}(u)+\tilde{g}_{A}(u)=\hat{f}(A)+\tilde{g}_{A}(u)$.
Then, using the inequality $|a+b|^{3}\leq 4(|a|^{3}+|b|^{3})$, we have
$\|T_{\sqrt{\delta}}g_{A}\|_{3}^{3}=\|\hat{f}(A)+T_{\sqrt{\delta}}\tilde{g}_{A}\|_{3}^{3}\leq
4|\hat{f}(A)|^{3}+4\|T_{\sqrt{\delta}}\tilde{g}_{A}\|_{3}^{3}$
and therefore,
$\displaystyle\sum(1-\delta)^{|A|}\|T_{\sqrt{\delta}}g_{A}\|^{3}\leq
4\sum(1-\delta)^{|A|}|\hat{f}(A)|^{3}+4\sum(1-\delta)^{|A|}\|T_{\sqrt{\delta}}\tilde{g}_{A}\|_{3}^{3}.$
To bound the first term, note that
$\sum(1-\delta)^{|A|}|\hat{f}(A)|^{3}\leq\sum\hat{f}(A)^{2}\cdot\max\\{(1-\delta)^{|A|}|\hat{f}(A)|)\\}$.
The sum of the squared Fourier coefficients is just $1$ (by Parseval’s
identity) and we can use the
$(\delta,\frac{\delta}{\log(1/\delta)})$-pseudorandomness property to bound
the quantity in the maximum: when $|A|<\frac{1}{\delta}\log\frac{1}{\delta}$,
then $|\hat{f}(A)|\leq\sqrt{\delta}$ and when
$|A|\geq\frac{1}{\delta}\log\frac{1}{\delta}$ then
$(1-\delta)^{|A|}\leq\delta$. Thus the entire first summand is
$O(\sqrt{\delta})$.
#### Hypercontractivity.
It remains to bound
$\sum(1-\delta)^{|A|}\|T_{\sqrt{\delta}}\tilde{g}_{A}\|_{3}^{3}$. Fix
$\lambda=\frac{\log 2}{\log(1/\delta)}$ and apply the modified
hypercontractive inequality:
$\displaystyle\sum(1-\delta)^{|A|}\|T_{\sqrt{\delta}}\tilde{g}_{A}\|_{3}^{3}$
$\displaystyle\leq\sum(1-\delta)^{|A|}\|T_{\sqrt{\delta}}\tilde{g}_{A}\|_{2}^{3-3\lambda}\|\tilde{g}_{A}\|_{2}^{3\lambda}$
Now, $\|\tilde{g}_{A}\|_{2}^{3\lambda}\leq 1$ and
$\|T_{\sqrt{\delta}}\tilde{g}_{A}\|_{2}^{3-3\lambda}=O(\sqrt{\delta})\sum_{\emptyset\neq
B\subseteq\overline{A}}\delta^{|B|}\hat{f}(A\cup B)^{2}$. The contribution of
the corresponding term to the sum we were trying to bound is
$O(\sqrt{\delta})\cdot\hat{f}(A\cup B)^{2}\cdot(1-\delta)^{|A|}\delta^{|B|}$.
For each choice of $A\cup B$, the $(1-\delta)^{|A|}\delta^{|B|}$ terms sum to
at most one, and all the $\hat{f}(A\cup B)^{2}$ terms themselves sum to at
most one. Therefore, we have bounded the entire sum by $O(\sqrt{\delta})$ as
desired. ∎
### 5.2 Integrality gap for Unique Label Cover SDP
#### Problem and SDP relaxation.
In the Unique Label Cover problem, we are given a label set $L$ and a weighted
multigraph $G=(V,E)$ whose edges are labeled by permutations $\\{\pi_{e}\colon
L\to L\\}_{e\in E}$, and are asked to find an assignment $f\colon V\to L$ of
labels to edges that maximizes the fraction of edges $e\\{u,v\\}$ that are
“consistent” with our labeling, i.e., $\pi_{e}(f(u))=f(v)$. If there exists a
labeling that satisfies all the edges, then it is easy to find such a
labeling. However, when all we can guarantee is that $99\%$ fraction of the
edges can be satisfied, it is not known how to find a labeling satisfying even
$1\%$ of them. At the same time, present techniques cannot show that finding a
$1\%$-consistent labeling is NP-hard.
One approach to solving this problem is to use an extension of the Goemans-
Williamson SDP for Max-Cut, where we set up a vector $v_{i}$ for every vertex
$v$ and label $i$:
maximize
$\displaystyle\textstyle\operatorname*{\mathbb{E}}_{e\\{u,v\\}}\sum_{i\in
L}\langle u_{i},v_{\pi_{e}(i)}\rangle$ subject to $\displaystyle\langle
u_{i},v_{j}\rangle\geq 0$ $\displaystyle\forall u,v\in V,\forall i,j\in L$
$\displaystyle\textstyle\sum_{i\in L}\langle v_{i},v_{i}\rangle=1$
$\displaystyle\forall v\in V$ $\displaystyle\langle\textstyle\sum_{i\in
L}u_{i},\textstyle\sum_{j\in L}v_{j}\rangle=1$ $\displaystyle\forall u,v\in L$
$\displaystyle\langle v_{i},v_{j}\rangle=0$ $\displaystyle\forall v\in
V,\forall i\neq j\in L$
(The expectation in the objective is over a distribution where $e\\{u,v\\}$ is
picked with probability proportional to its weight.) The intent is that
$\|v_{i}\|^{2}$ should be the probability that $v$ receives label $i$, and
$\langle u_{i},v_{j}\rangle$ should be the corresponding joint probability. It
is easy to see that this SDP is a relaxation of the original problem.
#### Gap instance.
In an influential paper, Khot and Vishnoi [KV05] constructed an integrality
gap for this SDP: for a label set of size $2^{k}$ and an arbitrary parameter
$\eta\in[0,\frac{1}{2}]$, a graph whose optimal labeling satisfies $\leq
1/2^{\eta k}$ fraction of the edges, but for which the SDP optimum is at least
$1-\eta$. The hypercontractive inequality plays a central role in the
soundness analysis, which we present below.
Let $\tilde{V}$ be the set of all functions
$f\colon\\{-1,1\\}^{k}\to\\{-1,1\\}$ and $L$ be the Fourier basis
$\\{\chi_{S}\mid S\subseteq[k]\\}$; clearly, $|L|=2^{k}$. Observe that
$\tilde{V}$ is an Abelian group under pointwise multiplication, and $L$ is a
subgroup. We take the quotient $V=\tilde{V}/L$ to be the vertex set. Fix an
arbitrary representative for each coset and write
$V=\\{f_{1}L,f_{2}L,\dotsc,f_{|V|}L\\}$. We shall define a weighted edge
between every pair of these representative functions, then show how to extend
this definition to all pairs of functions, and finally map these edges to
edges between cosets.
* •
The edge $\tilde{e}\\{f,g\\}$ has weight equal to
$\Pr_{h,h^{\prime}}[(f,g)=(h,h^{\prime})]$, where $h,h^{\prime}\in V$ are
drawn to be $\rho$-correlated on every bit with uniform marginals, where
$\rho=1-2\eta$.
* •
With every edge $\tilde{e}\\{f_{i},f_{j}\\}$ between representative functions,
we associate the identity permutation.
* •
A non-representative function acts as if its label is assigned according to
its coset’s representative. Thus, the permutation associated with
$\tilde{e}\\{f_{i}\chi_{S},f_{j}\chi_{T}\\}$ is
$\chi_{U}\chi_{S}\mapsto\chi_{U}\chi_{T}$.
* •
In the actual graph under consideration, every edge
$\tilde{e}\\{f_{i}\chi_{S},f_{j}\chi_{T}\\}$ appears as an edge
$e\\{f_{i}L,f_{j}L\\}$ (with the same permutation and weight).
#### Soundness analysis.
Given a labeling $R\colon V\to L$ on the cosets, we consider the induced
labeling $\tilde{R}\colon\tilde{V}\to L$ given by
$\tilde{R}(f_{i}\chi_{S})=R(f_{i}L)\chi_{S}$. From our definitions, it is
clear that the objective value attained by $\tilde{R}$ is precisely
$\Pr_{h,h^{\prime}}[\tilde{R}(h)=\tilde{R}(h^{\prime})]$, where $h,h^{\prime}$
are chosen as before. Fix any label $\chi_{S}$ and consider the indicator
function $\phi\colon\tilde{V}\to\\{0,1\\}$ of functions that $\tilde{R}$
labels with $\chi_{S}$. Since exactly one function in each coset gets labeled
$\chi_{S}$, we know that $\operatorname*{\mathbb{E}}\phi=1/2^{k}$. Therefore,
$\Pr_{h,h^{\prime}}[\tilde{R}(h)=\tilde{R}(h^{\prime})=\chi_{S}]=\operatorname*{\mathbb{E}}_{h,h^{\prime}}[\phi(h)\phi(h^{\prime})]=\langle
h,T_{\rho}h\rangle=\|T_{\sqrt{\rho}}h\|^{2}_{2},$
which we can upper-bound (using hypercontractivity) by
$\|h\|^{2}_{1+\rho}=1/2^{\frac{2k}{1+\rho}}\leq 1/2^{\eta k}$.
## References
* [ARV09] Sanjeev Arora, Satish Rao, and Umesh V. Vazirani. Expander flows, geometric embeddings and graph partitioning. J. ACM, 56(2), 2009.
* [Bor82] C. Borell. Positivity improving operators and hypercontractivity. Mathematische Zeitschrift, 180(3):225–234, 1982.
* [DSC96] P. Diaconis and L. Saloff-Coste. Logarithmic Sobolev inequalities for finite Markov chains. The Annals of Applied Probability, 6(3):695–750, 1996.
* [Fri98] E. Friedgut. Boolean functions with low average sensitivity depend on few coordinates. Combinatorica, 18(1):27–35, 1998.
* [Gro75] L. Gross. Logarithmic sobolev inequalities. American Journal of Mathematics, 97(4):1061–1083, 1975.
* [Gro77] J.L. Gross. Every connected regular graph of even degree is a Schreier coset graph. Journal of Combinatorial Theory, Series B, 22(3):227–232, 1977\.
* [KKL88] J. Kahn, G. Kalai, and N. Linial. The influence of variables on Boolean functions. In Proceedings of 29th IEEE Symp. Foundations of Computer Science (FOCS), 1988.
* [KV05] Subhash Khot and Nisheeth K. Vishnoi. The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into $\ell_{1}$. In FOCS, pages 53–62. IEEE Computer Society, 2005.
* [Lee05] James R. Lee. On distance scales, embeddings, and efficient relaxations of the cut cone. In SODA, pages 92–101. SIAM, 2005.
* [OW09a] Ryan O’Donnell and Yi Wu. 3-bit dictator testing: 1 vs. 5/8. In Claire Mathieu, editor, SODA, pages 365–373. SIAM, 2009.
* [OW09b] R. O’Donnell and K. Wimmer. KKL, Kruskal-Katona, and monotone nets. In FOCS, 2009.
* [She09] J. Sherman. Breaking the Multicommodity Flow Barrier for $O(\sqrt{\log n})$-Approximations to Sparsest Cut. University of California, Berkeley, 2009.
* [Tal95] M. Talagrand. Concentration of measure and isoperimetric inequalities in product spaces. Publications Mathematiques de l’IHES, 81(1):73–205, 1995.
|
arxiv-papers
| 2011-01-14T21:30:47 |
2024-09-04T02:49:16.481227
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Punyashloka Biswal",
"submitter": "Punyashloka Biswal",
"url": "https://arxiv.org/abs/1101.2913"
}
|
1101.3257
|
# The Chromospheric Activity, Age, Metallicty and Space Motions of 36 Wide
Binaries
J. K. Zhao11affiliation: Florida Institute of Technology, Melbourne, USA,
32901 22affiliation: Key Laboratory of Optical Astronomy, National
Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100012,
China , T. D. Oswalt11affiliation: Florida Institute of Technology, Melbourne,
USA, 32901 , M. Rudkin11affiliation: Florida Institute of Technology,
Melbourne, USA, 32901 , G. Zhao22affiliation: Key Laboratory of Optical
Astronomy, National Astronomical Observatories, Chinese Academy of Sciences,
Beijing, 100012, China , Y. Q. Chen22affiliation: Key Laboratory of Optical
Astronomy, National Astronomical Observatories, Chinese Academy of Sciences,
Beijing, 100012, China jzhao@fit.edu toswalt@fit.edu mrudkin@fit.edu
gzhao@bao.ac.cn cyq@bao.ac.cn
###### Abstract
We present the chromospheric activity (CA) levels, metallicities and full
space motions for 41 F, G, K and M dwarf stars in 36 wide binary systems.
Thirty-one of the binaries, contain a white dwarf component. In such binaries
the total age can be estimated by adding the cooling age of the white dwarf to
an estimate of the progenitor’s main sequence lifetime. To better understand
how CA correlates to stellar age, 14 cluster member stars were also observed.
Our observations demonstrate for the first time that in general CA decays with
age from 50 Myr to at least 8 Gyr for stars with 1.0 $\leq$ V-I $\leq$ 2.4.
However, little change occurs in CA level for stars with V-I $<$ 1.0 between 1
Gyr and 5 Gyr, consistent with the results of Pace et al. (2009). Our sample
also exhibits a negative correlation between stellar age and metallicity, a
positive correlation between stellar age and W space velocity component and
the W velocity dispersion increases with age. Finally, the population
membership of these wide binaries is examined based upon their U, V, W
kinematics, metallicity and CA. We conclude that wide binaries are similar to
field and cluster stars in these respects. More importantly, they span a much
more continuous range in age and metallicity than is afforded by nearby
clusters.
activity: Stars —late-type: Stars—white dwarfs: Stars
## 1 Introduction
Age is one of the most difficult to determine properties of a star. The Vogt-
Russell theorem asserts that the structure of a star is uniquely determined by
its mass and composition. Nucleosynthesis in the core results in changes in
composition and this implies at least some measurable property(ies) of a star
must vary with age. Unfortunately, these changes are subtle and difficult to
measure. It is ironic that the age of the Universe (13.7 $\pm$ 0.2 Gyr;
Bennett et al. 2003) is known to better precision than the age of any star
other than the Sun. The present methods by which stellar ages can be estimated
are seldom consistent within 50% (Soderblom 2010). Even the Sun does not
reveal its age directly; this key calibration point is determined from the
decay of radioisotope samples to be 4,566${}_{-1}^{+2}$ Myr (Chaussidon 2007).
One of the ‘semifundamental’ methods of stellar age determination is isochrone
fitting the position of a star in the Hertzsprung-Russell diagram (HRD).
However, because of the degeneracy of theoretical isochrones, this technique
does not work well for the vast majority of stars-those on the lower main
sequence (MS). Here small errors in luminosity or metallicity translate into
large errors in age.
CaII H$\&$K features in the violet spectra of MS stars are one of the more
well-studied indicators of CA. Early work by Wilson (1963; 1968) and Vaughan
$\&$ Preston (1980) established CaII H$\&$K emission as a useful marker of CA
in lower MS stars. In F to early M stars Skumanich (1972) found that CaII
H$\&$K emission, magnetic field strength and rotation all decay as the inverse
square root of stellar age.
Mamajek $\&$ Hillenbrand (2008) and others have shown that the CA vs. age
relation is much more complex than Skumanich envisioned; such factors as
metallicity, photospheric contamination of CA indices and variation in CA must
be considered. Clusters provide only a limited range of ages and metallicities
to investigate these effects.
A self-sustaining magnetic dynamo driven by rotation and convection is
believed to be the source of CA in MS stars of spectral type F, G, K and early
M. According to this paradigm, due to magnetic breaking, rotational velocity
decreases with age, which leads to a decrease in CA as well, unless angular
momentum is sustained by tidal interaction, as in the case of short-period
binaries, or maintained by convection as in late M type dwarfs.
Using a lower resolution analog of the Mount Wilson S index in eight clusters
and the Sun, Barry, Cromwell $\&$ Hege (1987) found that the decay of CA is
well-represented by an exponential over the age range from $10^{7}$ to 6
$\times$ $10^{9}$ yr. Barry (1988) adjusted the age of three clusters from
Barry, Cromwell $\&$ Hege (1987). Using a color correction C${}_{cf}^{\prime}$
from Noyes et al. (1984), Barry found empirically that CA $\sim$ t1/e. In
addition, he concluded that the star formation rate in the solar neighborhood
has not been constant, suggesting a recent burst of star formation because of
the large number of stars in his youngest age bins. Working toward a more
detailed understanding of the CA vs. age relation, Soderblom, Duncan $\&$
Johnson (1991) found that, while a power law is generally the best fit to the
CaII H$\&$K vs. age relation, it implies a constant star formation rate (SFR).
Any different SFR causes the Skumanich relation to indicate an excess of young
stars in the solar neighborhood. They also found that the calibrated cluster
data presented in Barry, Cromwell $\&$ Hege (1987) were consistent with a
constant SFR.
Pace $\&$ Pasquini (2004) found that for several clusters older than 1 Gyr
there appeared to be a constant activity level. Pace et al. (2009) believed
stars change from active to inactive, crossing the activity range
corresponding to the so-called ‘Vaughan-Preston gap’, on a time-scale that
might be as short as 200 Myr. If true, this would bring into question whether
the Skumanich relation is valid for MS stars in all age regimes.
Among wide white dwarf (WD)+dM binaries Silvestri, Hawley $\&$ Oswalt (2005)
used the cooling ages of WD components, plus an average estimate for MS
lifetime, to explore the activity vs. age relation among lower MS stars. This
study confirmed that stars later than spectral type dM3 do not exhibit a
Skumanich-style CA vs. age relation. In a study of activity among unresolved
WD+dM cataclysmic variable candidates Silvestri et al (2006) again used the
white dwarf (WD) cooling times alone as lower age limit. Both of these studies
found the general trend seen in clusters, i.e, that later dM spectral types
remain active at a roughly constant level for a longer period of time than
earlier spectral types, whereupon each star becomes inactive. The transition
from active to inactive appears to take place over a relatively short period
of time. However, in both studies some dM stars in binaries were found to
exhibit activity more characteristic of brighter, bluer and more massive M
dwarfs than seen in clusters. West et al. (2008) examined the age-activity
relation among a sample of over 38,000 low-mass (M0-M7) stars drawn from the
Sloan Digital Sky Survey (SDSS) Data Release 5 (DR5) and also found later
spectral types remain active longer. It is important to note that late type
stars do not seem to decline in activity monotonically, they are either ‘on’
if young, or ‘off’ if old. While the activity turnoff point on the lower MS
constrains a cluster’s age it is not a useful means for estimating the age of
a field star.
This paper describes our analysis of a sample of common proper motion binary
(CPMB) systems. Most of the systems selected have a late MS star paired with a
WD component. All have relatively wide orbital separations ($<a>$ $\sim$ 103
AU; Oswalt et al. 1993). Thus, each component is assumed to have evolved
independently, unaffected by mass exchange or tidal coupling that complicates
the evolution of closer pairs (Greenstein 1986). It is also assumed that
members of such binaries are coeval. Essentially, each may be regarded as an
open cluster with only two components. Such binaries are far more numerous
than clusters and span a much broader and more continuous range in age. Thus,
they are potentially very valuable to investigations of phenomena that depend
upon age. The total age of a CPMB can be estimated from the cooling age of the
WD component added to an estimate of its progenitor’s MS lifetime. Since some
CPMBs in our sample have ages well beyond the present $\sim$4 Gyr nearby
cluster limit, this provides an opportunity to test the CA vs. age relation in
much older MS stars.
Wide binaries present another opportunity. From the spectra of the MS
components, the metallicity can be measured. Presumably this is also the
original metallicity of the WD progenitor. In addition, the radial velocities
of the MS stars are easy to determine. Since proper motions are available,
with trigonometric or photometric parallaxes, full space motions for all
systems in the observed sample can be estimated. Thus, CPMBs present an
opportunity to investigate the relations among age, metallicity and space
motion, as well as population membership even for WDs, which often have weak
or no spectral feature.
In section 2 we present an overview of the observations and reductions for our
sample. A discussion of the CA vs. age relation is given in section 3. Our age
and metallicity relation is presented in section 4. In section 5, we describe
our analysis of population membership. We conclude with a discussion of the
implications of our findings in section 6.
## 2 Observations and Data Reduction
Most stars chosen for this study are components of wide MS+WD pairs from the
Luyten (1979) and Giclas, Burnham $\&$ Thomas (1971) proper motion catalogs
chosen by Oswalt, Hintzen $\&$ Luyten (1988). A key impetus for using such
pairs in this study is that the total lifetime of each pair should be
approximately the age derived from measurements of the MS component. In
addition the total age of a pair should be approximately the sum of the WD
component’s cooling time and the MS lifetime of its progenitor.
Table 1 gives the observed data for 36 wide binaries. Column 1 is a unique ID
number. Columns 2 - 4 list each component’s name, right ascension, and
declination (coordinates are for epoch 1950). The V magnitudes and original
low-resolution spectroscopic identifications are given in columns 5 - 8.
Column 9 is the observation date for the high resolution ($\sim$ 2$\rm\AA$)
spectra in the present study. Columns 10 - 13 list the proper motion,
direction of proper motion (measured east of north), position angle (centered
on the primary measured east of north), and separation of the components (in
arcseconds), respectively. Of the 36 wide binaries, 5 systems consist of two
MS stars, 1 system is a triple WD+dK+dM, and the remaining 30 are MS+WD pairs.
A sample of cluster MS stars of known age, such as IC 2391, IC 2602 and M67,
previously studied by Patten $\&$ Simon (1993, 1996), Barrado y Navascués,
Stauffer $\&$ Jayawardhana (2004) and Giampapa et al. (2006), were adopted as
‘CA standards’. These stars were routinely observed in the course of our
observing program for the CPMBs. Table 2 provides the observational
information for these cluster member stars. Column 1 is a unique ID. Columns 2
- 4 list the name, V magnitude and observation date. The colors V-I and B-V
are given in column 5 - 6. The CA flux ratio SHK (see Section 3.1), age and
cluster membership are given in columns 7 - 9. The last column provides the
corresponding literature source for each star.
### 2.1 BVRI Photometry
We used BVRI photometric data for our wide binaries from Smith (1997) whenever
available. Photometric data for some stars that were not in this source were
taken from the literature identified by the Simbad Astronomical Database
(Genova 2006). Photometric colors of other stars were estimated from our
spectra by empirical calibrations. For example, Fig.1 is the relation between
V-I and CaI4227 flux ratio. The feature of this index is within 4211 - 4242
$\rm\AA$ and the continuum ranges within 4152 - 4182 $\rm\AA$. The filled
circles are stars with known V-I color. The dotted line is a least-squares
fit. The scatter $\rm\sigma_{V-I}\approx$ 0.22. We used this relation to
estimate the V-I for stars with no published colors. For the stars in cluster
IC 2602, IC 2391 and M67, photometric colors were taken from Barnes, Sofia
$\&$ Prosser (1999 and references therein), Patten $\&$ Simon (1996) and
Giampapa et al. (2006).
### 2.2 Spectroscopic Observations
In the southern hemisphere, observations were conducted at Cerro Tololo
Interamerican Observatory (CTIO) using the Blanco 4-meter telescope. The
Ritchey-Chr$\acute{e}$tien (RC) Cassegrain spectrograph was used on two
separate observing runs (February 2004 and February 2005) to obtain optical
spectroscopy of CPMBs, as well as the CA standard stars. During the two
observation runs, the KPGL1 grating was used to obtain spectra with a scale of
0.95 $\rm{\AA}$/pixel. A Loral 3K CCD (L3K) was used with the RC spectrograph.
It is a thinned 3K$\times$1K CCD with 15 $\mu$m pixels. A spectral range of
approximately 3800 $\rm{\AA}$ to 6700 $\rm{\AA}$ was achieved.
Northern hemisphere observations were conducted at Kitt Peak National
Observatory (KPNO) using the Mayall 4-meter telescope. The RC spectrograph,
with the BL450 grating set for the 2nd order to yield a resolution of 0.70
$\rm{\AA}$/pixel, was used to obtain optical spectra during the July 2005 and
November 2006 observing runs. The 2K$\times$2K T2KB CCD camera with 24 $\mu$m
pixels was used to image the spectra. An 8-mm CuSO4 order-blocking filter was
added to decrease 1st-order overlap at the blue end of the spectrum. A
spectral range of approximately 3800 $\rm{\AA}$ to 5100 $\rm{\AA}$ was
achieved.
### 2.3 Data Reduction
The data were reduced with standard IRAF111IRAF is distributed by the National
Optical Astronomy observatories, which are operated by the Association of
Universitites for Research in Astronomy, Inc., under cooperative agreement
with the National Science Foundation (http://iraf.noao.edu). reduction
procedures. In all cases, program objects were reduced with calibration data
(bias, flat, arc, flux standard) taken on the same night. Data were bias-
subtracted and flat-fielded, and one-dimensional spectra were extracted using
the standard aperture extraction method. A wavelength scale was determined for
each spectrum using HeNeAr arc lamp calibrations. Flux standard stars were
used to place the spectra on a calibrated flux scale. We emphasize that these
are only relative fluxes, as most nights were not spectrophotometric.
The radial velocity of each MS star was determined by cross-correlation
between the observed spectra and a set of MS template spectra. The F, G and K
template spectra were generated from a theoretical atmosphere grid (Castelli
$\&$ Kurucz 2003). The dM template spectra were compiled using observed M
dwarf spectra from the Sloan Digital Sky Survey
(SDSS)222http://www.astro.washington.edu/slh/templates. The wavelength ranged
from roughly 3900 - 9200 $\rm\AA$ (see Bochanski et al. 2007). Our typical
internal measurement uncertainties in radial velocity were $\sigma\rm_{v_{r}}$
= $\pm$ 4.6 km s-1. The final radial velocities listed in column 5 of Table 3
were corrected to the heliocentric frame.
## 3 CA-Age Relation
### 3.1 Measurement of SHK
The flux ratio
$\displaystyle\rm{S_{HK}}$ $\displaystyle=$
$\displaystyle\alpha\rm{\frac{H+K}{R+V}}$ (1)
was determined for each MS star, where H and K are the fluxes measured in 2
$\rm{\AA}$ rectangular windows centered on the line cores of CaII H$\&$K. Here
R and V are the fluxes measured in 20 $\rm{\AA}$ rectangular ‘pseudocontinuum’
windows on either side. Although these are not strictly equivalent to the
triangular windows Wilson (1968) used with his photomultiplier-based
spectrometer, Hall, Lockwood $\&$ Skiff (2007) have shown that using 1.0
$\rm{\AA}$ rectangular H$\&$K windows produces results that are easily
calibrated to the Baliunas et al. (1995) analysis of Wilson’s (1968) original
survey of bright MS stars. Our choice of 2 $\rm{\AA}$ windows is set by the
resolution of the CTIO and KPNO instrumentation, but it detects CaII H$\&$K
emission nearly as well and allows fainter stars to be observed. Gray et al.
(2003) have shown that even a resolution of $\sim$4$\rm\AA$ can produce useful
measure of SHK. In our measurement, the scale $\alpha$ is 10.0, reflecting the
fact that the continuum windows are 10 times wider than the H$\&$K windows.
In our sample some stars were observed two or three times. In such cases, the
mean of these measurements was adopted as the star’s $\rm{S_{HK}}$ and the
scatter as the uncertainty. For those stars observed only once, we adopted the
average uncertainty derived from those stars having more than one observation
($\pm$4.6%). The $\rm{S_{HK}}$ indices of all the MS stars are shown in column
4 of Table 3. For the purpose of this study we need only a calibration of CA
vs. age on our instrumental system. Some stars were observed both at CTIO and
KPNO. We found an empirical calibration: Sctio=Skpno+0.095. To remove this
small instrumental effect, all the SHK measured at KPNO were transformed into
the CTIO instrumental system with this relation.
### 3.2 The age determination
Our sample includes 14 cluster member stars: 6 in IC2602, 6 in IC2391 and 2 in
M67. The ages of IC2602 and IC 2391 are approximately the same ($\sim$
50$\pm$5 Myr; Barrado y Navascues, Stauffer $\&$ Jayawardhana 2004). They
obtained intermediate-resolution optical spectroscopy of 44 potentially very
low mass members of IC2391 and derived the cluster age from a comparison of
several theoretical models. The most recent age determination for M67 is
4.05$\pm$0.05 Gyr (Jorgensen $\&$ Lindegren 2005).
Among our 31 systems containing WD components, the ages of 23 were determined
by using computed cooling times of WD companions added to estimates of their
progenitor’s MS lifetimes. The $T$eff and log $g$ of each WD companion was
obtained from the literature (see Table 4). The ages of the remaining 8 wide
binaries were not obtained because the $T$eff and log $g$ of WD companions
could not be obtained from our spectra (3: DC type; 1: DQ; 4: low S/N) and
could not be found in the literature. In cases where uncertainties were not
given, we adopted 200 K and 0.05 as the average uncertainty for $T$eff and log
$g$, respectively. This decision was based on the recommendation of Bergeron,
Saffer $\&$ Liebert (1992) who believe the internal errors are typically
100-300 K in $T$eff and 0.02-0.06 in log $g$. From the $T$eff and log $g$ of
each WD, its mass (MWD) and cooling time (tcool) were estimated from
Bergeron’s cooling sequences333The cooling sequences can be downloaded from
the website: http://www.astro.umontreal.ca/ bergeron/CoolingModels/.. For the
pure hydrogen model atmospheres above $T$eff = 30,000 K we used the carbon-
core cooling models of Wood (1995), with thick hydrogen layers of qH = MH/M∗ =
10-4. For $T$eff below 30,000 K we used cooling models similar to those
described in Fontaine, Brassard $\&$ Bergeron (2001) but with carbon-oxygen
cores and qH = 10-4 (see Bergeron, Leggett $\&$ Ruiz 2001). For the pure
helium model atmospheres we used similar models but with qH = 10-10. MWD and
tcool were then calculated by spline interpolation based on the $T$eff and log
$g$. In Table 4, columns 6 and 7 list the final MWD and tcool for these 23
WDs. Although $T$eff and log $g$ are from different literature sources, the
parameters are consistent for common stars in these references. Thus, there
appears to be no systematic uncertainties expected among the $T$eff and log
$g$ we used.
Using the Initial-Final Mass Relation (IFMR; equations 2 - 3 presented in
Catalán et al. 2008b), we then estimated the progenitor masses Mi of the WDs.
There are two WDs whose masses are lower than 0.5M⊙ and the current IFMR does
not extend to such low mass objects. From Fig. 2 in Catalán et al. (2008b), we
adopted Mi $\sim$ 1.25M⊙ for MWD $<$ 0.5M⊙. Next, using the third-order
polynomial of Iben $\&$ Laughlin (1989),
$\displaystyle\rm{log~{}t_{evol}}$
$\displaystyle=\rm{9.921-3.6648(logM_{i})+1.9697(logM_{i})^{2}-0.9369(logM_{i})^{3}}$
(2)
we determined the MS lifetime tevol (in years) corresponding to each Mi,
progenitor mass of the WD (in M⊙). Columns 8 - 9 in Table 4 are the Mi and
tevol we derived for 23 WDs. Finally, the total ages of these wide binaries
were computed by adding tevol to the cooling ages of the WDs (tcool; column 11
in Table 3).
Independent age determination for 4 pairs were found in the literature. These
pairs are listed in Table 5. Column 1 gives their identifications. Columns 2-3
list the ages included in Holmberg, Nordström $\&$ Andersen (2009) and Valenti
$\&$ Fischer (2005), respectively. It can be seen that our ages are younger
than isochrone fitting ages in all four cases. For 40 Eri A the isochone
fitting age is unreasonably large, while our age of this star is consistent
with the rotation age 4.75$\pm$0.75 Gyr from Barnes (2007). For CD-37 10500,
the error bar is too large for isochrone fitting age to be useful. It is very
difficult to estimate the age for K and M dwarfs with the isochrone fitting
method because of the narrow vertical dispersion of isochrones within the MS
in an HR diagram. Small uncertainties in luminosity and metallicity cause
large uncertainties in age. Using the white dwarf cooling times plus the
estimated progenitor lifetimes should give a more consistent age for such
stars. Note that the internal uncertainties of our age determinations are
smaller than those derived from isochrone fitting.
Although we have taken a similar approach, there are a few differences between
our age determinations and those by Silvestri, Hawley $\&$ Oswalt (2005) and
Silvestri et al. (2006). Silvestri et al. (2006) only used the WD cooling time
as a lower limit to the age of each binary without considering the WD
progenitor’s lifetime in evaluating the age-activity relation among close
binaries. Therefore, the ages were somewhat less than actual ages. In
Silvestri, Hawley $\&$ Oswalt (2005), each age was estimated from the WD’s
cooling time plus an estimate of its progenitor’s lifetime. The Mi of each WD
was computed from the IFMR of Weidemann(2000) using an adopted mean WD mass of
0.61 M⊙. We explicitly computed the Mi of each WD from the IFMR of
Catal$\rm\acute{a}$n (2008b) using our own estimate of its current mass (Mf).
In view of this, our method should provide more precise age estimates.
### 3.3 Discussion
All MS stars are plotted in Fig. 2, which displays our empirical relation
among log(SHK), age and V-I. Asterisks represent the young cluster stars in
IC2602 and IC2391 that have the same age 50$\pm$5 Myr (Barrado y Navascues,
Stauffer $\&$ Jayawardhana 2004). It is clear that log(S$\rm{}_{HK})$ tends to
be larger in red stars and smaller in blue stars at the same age. Two M67
cluster member stars are plotted with squares. M67 is much older than IC2602
and IC2391 and these points clearly demonstrate the expectation that CA
declines with age. The plus ( + ) signs represent field stars whose ages are
between 1.0 Gyr and 5.0 Gyr. Diamonds represent stars whose ages are between
5.0 Gyr and 8.0 Gyr. Our sample includes five known flare stars (40 Eri C,
LP891-13, LP888-63, G216-B14A, BD+26 730) which are displayed as filled
circles. These clearly show enhanced CA for their supposed age.
Fig. 2 can be divided into four arbitrary age domains, represented by dotted
lines. The top domain mainly consists of young active stars and flare stars.
The other three domains consist of less active stars whose ages are,
respectively, 1.0 - 5.0 Gyr, 5.0 - 8.0 Gyr and $>$ 8.0 Gyr. The typical
uncertainty in V-I and log(SHK) for inactive stars are displayed at the right-
top of Fig. 2 as an error bar. A least-squares fit to stars in IC2602 and
IC2391 (dash-dot line) illustrates that this group conforms to a consistent
age in the log(S$\rm{}_{HK})$ vs. V-I plane.
Stars to the right of the vertical dashed line at V-I = 2.4 are later in
spectral type than $\sim$dM3. Silvestri, Hawley $\&$ Oswalt (2005) found no
Skumanich-style CA vs. age relation among stars later than dM3, in accord with
the expectation that a so-called turbulent dynamo drives CA in such stars (See
Reid $\&$ Hawley 2000 and reference therein). Thus, stars in this region are
not expected to follow the CA vs. age relation seen in earlier spectral type
MS stars.
Overall, it can be seen that CA generally decays with age from 50 Myr to at
least 8 Gyr for stars with 1.0 $\leq$ V-I $\leq$ 2.4. However, for stars bluer
than V-I $\approx$ 1.0, CA shows little variation between 1.0-5.0 Gyr, which
is consistent with the results of Pace et al. (2009) who found that CA from
1.4 Gyr (NGC3680) to 4.0 Gyr (M67) remains almost constant. Also note that
lines of constant age do not have the same slope. For young stars the lines of
constant age have a relatively steep slope, while they appear to be much
flatter for old stars.
Some comments on a few individual stars in Fig. 2 are appropriate. The star
labeled as 1 in Fig. 2 is G163-B9A. Its companion, G163-B9B, was identified as
an sdB star by Wegner $\&$ Reid (1991) and Catalán et al. (2008a). The CA of
G163-B9A is very strong. Thus, it is either a very young star or perhaps was
observed during a flare event. The latter possibility was eliminated based on
a detailed examination of its spectrum. Thus we conclude G163-B9A/B is
probably not a physical pair. Stars labeled as 2-3 are possible halo stars;
star 4 has the weakest CA. These will be discussed in Section 5.
There are 5 wide binaries (CD-31 1454/LP888-25; G114-B8A/B; CD-31
7352/LP902-30; LP684-1/2 and LP387-1/2) that consist of pairs of MS stars.
Only LP387-1/2 was observed at KPNO while all others were observed at CTIO.
Since the two components of coeval binaries are expected to have consistent
levels of CA, this provides a good reality check on our CA, V-I and age
relation. Unfortunately, the spectrum of LP888-25, the companion to CD-31
1454, was unusable because of low signal-to-noise (S/N). In addition, we
concluded that G114-B8A/B is not a physical binary because the components’
radial velocities are inconsistent. The solid lines in Fig. 2 connect the MS
components in the other three pairs CD-31 7352/LP902-30, LP684-1/2 and
LP387-1/2, that most likely are physical pairs. The components in CD-31
7352/LP902-30 and LP387-1/2 have CA consistent with the same age. Some age
difference is implied in LP684-1/2 though the difference is within the
uncertainty implied by the error bar in the upper right corner of Fig. 2.
Clearly, CA, along with radial velocity, provides a very useful way for
filtering out nonphysical pairs.
## 4 The age-[Fe/H] Relation
The metallicity of each MS star was estimated by comparing the observed
spectra to a set of template spectra. Initially, a library of low resolution
theoretical spectra was generated using the SYNTH program, based on Kurucz’s
New ODF atmospheric models (Castelli $\&$ Kurucz 2003). The atmosphere models
assume local thermodynamic equilibrium (LTE). A mixing-length of l$/$Hp = 1.25
and a microturbulence $\xi$ = 2 km s-1 were adopted. The line list included
the atoms and molecules from Kurucz (1993). The maximum correlation method was
applied to find the closest matching theoretical spectra to each observed
spectrum. Since our objects are MS stars, we adopted log $g$ = 4.5 and [Fe/H]
= 0.0 as initial values. The effective $T$eff was then estimated based on the
maximum correlation method. Once an estimated $T$eff was obtained, a new
estimated [Fe/H] could be obtained with the same procedure. After several
iterations, the best-fitting parameters stabilized and computations were
terminated. The left panel of Fig. 3 displays the correlation coefficient vs.
[Fe/H] in a typical fit. Each open circle represents one template. The filled
circle is the maximum correlation coefficient obtained from polynomial fitting
(solid line). The value of [Fe/H] at this point is regarded as our best
estimate for this star’s metallicity. The diamond points mark the templates
yielding the maximum correlation (‘+’ : highest template point above; ‘-’: the
highest template point below). The difference in [Fe/H] between the two
diamond points and the filled circle is taken as the estimated uncertainty in
metallicity. The right panel of Fig. 3 compares our [Fe/H] estimates to those
which could be found in the literature. The mean difference is smaller than
0.15 dex, suggesting our metallicity is basically consistent with that of
other work. [Fe/H] measurements were not made for stars later than M3 because
we do not have templates for the latest spectra nor could we expect the CA vs.
age determination to be valid in such late type MS stars. The resulting
metallicities estimated for 37 MS stars in our sample are given in column 10
of Table 3.
Although one of the key consequences of the stellar evolution theory is the
gradual increase in the metal content of the interstellar medium (ISM) and the
progressive enrichment of subsequent stellar generations, some studies have
found little, if any, indication that an age-metallicity relation (AMR) exists
amongst solar neighborhood late-type stars. For example, Rocha-Pinto et al.
(2000) studied the AMR using a sample of 552 late-type dwarfs. For those
stars, metallicities were estimated from uvby data, and ages were calculated
from their chromospheric emission levels using a metallicity-dependent CA vs.
age relation developed by Rocha-Pinto $\&$ Maciel (1998). The resulting AMR
was found to be a smooth function in their analysis. Feltzing, Holmberg $\&$
Hurley (2001) found that the solar neighborhood age-metallicity diagram is
well populated at all ages in a sample of 5828 dwarf and sub-dwarf stars from
the Hipparcos Catalogue. Bensby, Feltzing $\&$ Lundström (2004) investigated
the AMR using a sample of 229 nearby thick disk stars. Their results indicate
that there is indeed an AMR in the thick disk. They found that the median age
decreases by about 5-7 Gyr when going from [Fe/H]$\approx$ -0.8 to
[Fe/H]$\approx$ -0.1.
Fig. 4 is our [Fe/H] vs. age relation for 21 stars. An asterisk represents one
likely halo star (see next section). Disk stars are displayed as filled
circles. The typical uncertainties in age and [Fe/H] are shown at the left-
bottom of this figure. The dotted line is a least-squares fit for only disk
stars, while the dashed line is the fit for all stars. The expected trend in
age-metallicity is apparent: old stars tend to be of lower metallicity.
Our [Fe/H] vs. age relation differs somewhat from the early work on single
stars by Barry (1988; see Fig. 5 in that reference). Our Fig. 4 shows a more
clear trend, quite similar in fact to the newer study of field stars by Rocha-
Pinto $\&$ Maciel (2000); the slope of their relation and ours are
approximately the same. We conclude that wide binaries exhibit a [Fe/H] $\sim$
age relation similar to field stars. However, at present our sample contains
too few stars to support a detailed examination of the nearby star formation
history.
## 5 Population Membership
Column 6 of Table 3 lists the parallaxes of our wide binaries. Some
trigonometric parallaxes were obtained from the Simbad Astronomical Database
(Genova 2006). For 11 wide binaries lacking trigonometric parallaxes, we
computed photometric parallaxes using equation 3-5 which were derived from the
data in Bergeron, Leggett $\&$ Ruiz (2001).
$\displaystyle\rm{\pi}$ $\displaystyle=\rm{10^{-(\frac{V-M_{v}+5}{5}})}$ (3)
$\displaystyle\rm{M_{v}}$
$\displaystyle=\rm{12.2199+1.8152(V-I)+2.9704(V-I)^{2}-1.7082(V-I)^{3}}$ (4)
$\displaystyle\rm{M_{v}}$
$\displaystyle=\rm{11.7099+6.6038(B-V)-3.7273(B-V)^{2}+0.8066(B-V)^{3}}$ (5)
The rectangular velocity components relative to the Sun for 41 MS stars were
then computed and transformed into Galactic velocity components U, V, and W,
and corrected for the peculiar solar motion (U, V, W) = (-9, +12, +7) km s-1
(Wielen 1982). The UVW-velocity components are defined as a right-handed
system with U positive in the direction radially outward from the Galactic
center, V positive in the direction of Galactic rotation, and W positive
perpendicular to the plane of the Galaxy in the direction of the north
Galactic pole. The typical uncertainties in U, V and W are no more than
$\sim$10 km s-1. Columns 7-9 of Table 3 list our computed U, V and W velocity
components.
The top panel of Fig. 5 shows contours, centered at (U, V) = (0, -220) km s-1,
that represent 1$\sigma$ and 2$\sigma$ velocity ellipsoids for stars in the
Galactic stellar halo as defined by Chiba $\&$ Beers (2000). Six stars lie
outside the 2$\sigma$ velocity contour centered on (U, V) = (0, -35) km s-1
defined for disk stars (Chiba $\&$ Beers 2000). The bottom of Fig. 5 shows the
Toomre diagram of our stars. Venn et al. (2004) suggest stars with Vtotal $>$
180km s-1 are possible halo members. There are two stars that meet this
criterion. Taking metallicity, space motion and CA into account, they have a
high probability of belonging to the halo population. One is LHS300A ([Fe/H]=
-0.95 $\pm$ 0.25) which is identified as a thick disk star in Monteiro et al.
(2006). Considering the metallicity and space motion, we think it is a halo
star. The other is CD-31 1454 ([Fe/H] = -0.48 $\pm$ 0.02) which is regarded as
a halo star by Chanamé $\&$ Gould (2004). The two likely halo stars are also
labeled in Fig. 2 as number 2 and 3 respectively. Their CA is weak, suggesting
ages in excess of 5 Gyr. These two plausible halo stars are displayed as
asterisks plus filled circles in the bottom pane of Fig. 5. The star numbered
4 in Fig. 2 is G114-B8B ([Fe/H] = -0.40 $\pm$ 0.08) which may be the oldest
star in our sample. Its age appears to be older than 8 Gyr as suggested by its
location in Fig. 2. The other stars are most likely members of the disk. The
above analysis demonstrates the difficulty of making a population assignment
on the basis of only space velocity or metallicity or CA. Ideally all three
should be used.
The left panel of Fig. 6 displays the computed absolute value of the W
components of our stars’ space motions vs. estimated ages for 21 stars for
which complete information is available. As expected, old stars tend to have
larger W velocity. A weak positive correlation between the vertical velocity
(W) of stars in CPMBs and age is expected based on the standard paradigm for
stellar encounters in the disk. The right panel of Fig. 6 presents the
dispersion in W as a function of age. It is clear that in general the W
dispersion becomes larger with age from 1 Gyr to 8 Gyr.
## 6 Conclusion
In this study we presented the CA levels, ages, metallicities and space
motions for components of 36 wide binaries. WD components were identified in
31 wide binaries. We also observed a sample of cluster member stars with well-
determined ages in order to test the expected CA vs. age relation. The ages of
23 wide binaries were derived by the cooling time of each WD companion added
to the lifetime estimate of its progenitor.
We first examined the relation among log (SHK), V-I and age. Our results
support the expected hypothesis established among single and cluster MS stars,
i.e., in general CA declines with age for stars with 1.0 $\leq$ V-I $\leq$ 2.4
from 50 Myr to at least 8 Gyr. However, for stars with V-I $<$ 1.0, the CA
varies little between 1 Gyr to 5 Gyr. This is consistent with results of Pace
et al. (2009) who found nearly constant CA level from 1.4 Gyr (NGC3680) to 4.0
Gyr (M67). Apparently the slope in the log(SHK) vs. V-I plane for young stars
is relatively steep, while for old stars it appears to be flatter. Additional
observations will be required to determine whether this slope changes
monotonically or discontinuously with age. These limitations will need to be
taken carefully into account by anyone attempting to use CA to determine ages
for single stars.
The metallicities of stars earlier in spectral type than M3 were measured by
template matching. Our sample generally supports the expected paradigm, i.e.
older stars tend to have lower metallicity. However, it also underscores the
fact that there is much variation in metallicity at all ages, precluding its
use for determining ages for single stars. Also, the AMR among wide binaries
appears to be quite comparable to that found in single field stars (Rocha-
Pinto $\&$ Maciel 2000).
With trigonometric parallaxes from the literature and photometric parallaxes
derived from B, V, R, I data, proper motions and our measured vr values, we
calculated full space motions U, V and W for as many of our MS stars as
possible. Our results clearly show that the W dispersion increases with age.
In general, the W velocity component is also relatively larger for old stars.
Using our measurements of CA and metallicity, we concluded that 2 wide
binaries ( CD-31 1454/LP888-25 and LHS300A/B) are probably halo members, while
the others are disk stars. This low fraction of halo members among wide
binaries is consistent with the earlier results of Silvestri, Oswalt $\&$
Hawley (2002) who found only one halo binary in their sample of WD+dM pairs.
We estimated the oldest disk star in our sample is G114-B8B ($>$ 8 Gyr) based
on its weak CA.
Five of our wide binaries consist of two MS stars. Two (G114-B8A/B;
G163-B9A/B) apparently are not physical pairs, since the two companions have
inconsistent CA levels, radial velocities and/or metallicities. CD-31
7352/LP902-30 and LP387-1/2 are physical pairs because the two MS components
have very similar velocities, metallicty and CA level. LP684-1/2 is probably a
physical pair since the radial velocity difference between two components is
within the range of uncertainty and its components display comparable SHK
value.
In conclusion, our study affirms the assumption that wide binaries (CPMBs)
share the same kinematic $\&$ spectroscopic properties as nearby single field
and cluster stars. Thus they are very promising resources for studying stellar
populations and age groups that are not well sampled by nearby clusters.
TDO acknowledges supported from NSF grant AST-0807919 to Florida Institute of
Technology. JKZ, GZ and YQC acknowledge support from NSFC grant No. 10821061,
11073026 and 11078019. We are also grateful for constructive comments by the
reviewer that substantially improved our paper.
## References
* Baliunas et al. (1995) Baliunas, S. L., Donahue, R. A., Soon, W. H., Horne, J. H., Frazer, J., Woodard-Eklund, L., Bradford, M., Rao, L. M., Wilson, O. C., Zhang, Q., Bennett, W., Briggs, J., Carrol, S. M., Duncan, D. K., Figueroa, D., Lanning, H. H., Misch, A., Mueller, J., Noyes, R. W., Poppe, D., Porter, A. C., Robinson, C. R., Russell, J., Shelton, J. C., Soyumber, T., Vaughan, A. H., $\&$ Whitney, J. H., 1995, ApJ, 438, 269
* Barnes et al. (1999) Barnes, S. A., Sofia, S., $\&$ Prosser, C. F., 1999, ApJ, 516, 265
* Barnes (2007) Barnes S. A., 2007, ApJ, 669, 1167
* Barrado et al. (2004) Barrado y Navascu$\acute{e}$s, D., Stauffer, J.R., $\&$ Jayawardhana, R., 2004, ApJ, 614, 386
* Barry (1988) Barry, D. C., 1988, ApJ, 334, 436
* Barry et al. (1987) Barry, D. C., Cromwell, R. H., $\&$ Hege, E. K., 1987, ApJ, 315, 264
* Bennett et al. (2003) Bennett, C. L., Hill, R. S., Hinshaw, G., Nolta, M. R., Odegard, N., Page, L., Spergel, D. N., Weiland, J. L., Wright, E. L., Halpern, M., Jarosik, N., Kogut, A., Limon, M., Meyer, S. S., Tucker, G. S., $\&$ Wollack, E., 2003, ApJS, 148, 97
* Bensby et al. (2004) Bensby, T., Feltzing, S., $\&$ Lundström, I., 2004, A&A, 421, 969
* Bergeron et al. (1992) Bergeron, P., Saffer, R. A., $\&$ Liebert, J., 1992, ApJ, 394, 228
* Bergeron et al. (1995) Bergeron, P., Wesemael, F., Lamontagne, R., Fontaine, G., Saffer, R. A., $\&$ Allard, N. F., 1995, ApJ, 449, 258
* Bergeron et al. (2001) Bergeron, P., Leggett, S. K., $\&$ Ruiz, M. T., 2001, ApJS, 133, 413
* Bochanski et al. (2007) Bochanski, J. J., West, A. A., Hawley, S. L., $\&$ Covey, K. R., 2007, AJ, 133, 531
* Castelli et al. (2003) Castelli, F., $\&$ Kurucz, R. L. 2003, Modelling of Stellar Atmospheres (IAU Symp. 210)
* Catalán et al. (2008a) Catalán, S., Isern, J., García-Berro, E., Ribas, I., Allende Prieto, C., $\&$ Bonanos, A. Z., 2008a, A&A, 477, 213
* Catalán et al. (2008b) Catalán, S., Isern, J., García-Berro, E., Ribas, I., 2008b, MNRAS, 387, 1693
* Chanamé et al. (2004) Chanamé, J., $\&$ Gould, A., 2004, ApJ, 601, 289
* Chiba $\&$ Beers (2000) Chiba, M., $\&$ Beers, T. C., 2000, AJ, 119, 2843
* Chaussidon (2007) Chaussidon, M., 2007. In Lectures in Astrobiology II, ed.MGargaud,HMartin, P Claeys, p. 45. Berlin: Springer-Verlag
* D’Orazi et al. (2009) D’Orazi, V., $\&$ Randich, S., 2009, A&A, 501, 553
* Feltzing et al. (2001) Feltzing, S., Holmberg, J., $\&$ Hurley, J. R., 2001, A&A, 377, 911
* Fontaine et al. (2001) Fontaine, G., Brassard, P., $\&$ Bergeron, P., 2001, PASP, 113, 409
* Genova (2006) Genova F., 2006, Centre de Donn es astronomiques de Strasbourg $<$http://simbad.u-strasbg.fr/$>$
* Giampapa et al. (2006) Giampapa, M. S., Hall, G. C., Radick, R. R., $\&$ Baliunas, S. L., 2006, ApJ, 651, 44
* Giclas et al. (1971) Giclas, H. L., Burnham, R., $\&$ Thomas, N. G., 1971, Lowell Proper Motion Survey, Northern Hemisphere: The G Numbered Stars (Flagstaff: LowellObs.)
* Gray et al. (1986) Gray, R. O., Corbally, C. J., Garrison, R. F., Mcfadden, M. T., $\&$ Robinson, P. E., 2003, AJ, 126, 2048
* Greenstein (1986) Greenstein, J. L. 1986, AJ, 92, 859
* Hall et al. (2007) Hall, J. C., Lockwood, G. W., $\&$ Skiff, B. A., 2007, AJ, 133, 862
* Holmberg et al. (2009) Holmberg, J., Nordström, B., $\&$ Andersen, J., 2009, A&A, 501, 941
* Iben et al. (1989) Iben, I., $\&$ Laughlin, G., 1989, ApJ, 341, 312
* Jørgensen $\&$ Lindegren (2005) Jorgensen, B. R., $\&$ Lindegren, L., 2005, A&A, 436, 127
* Koester et al. (2001) Koester, D., Napiwotzki, R., Christlieb, N., Drechsel, H., Hagen, H.-J., Heber, U., Homeier, D., Karl, C., Leibundgut, B., Moehler, S., Nelemans, G., Pauli, E.-M., Reimers, D., Renzini, A., $\&$ Yungelson, L, 2001, A&A, 378, 556
* Koester et al. (2009) Koester, D., Voss, B., Napiwotzki, R., Christlieb, N., Homeier, D., Lisker T., Reimers, D., $\&$ Heber, U., 2009, A&A, 505, 441
* Kurucz (1993) Kurucz, R. L. 1993, Kurucz CD-ROM 13, ATLAS9 Stellar Atmosphere Programs and 2 km/s grid (Cambridge: SAO)
* Luyten (1979) Luyten, W. J., 1979, Proper Motion Survey with the 48 Inch Telescope (Minneapolis: Univ. Minnesota Press)
* Mallik (1998) Mallik, S. V., 1998, A&A, 338, 623
* Mamajek et al. (2008) Mamajek, E. E., $\&$ Hillenbrand, L. A., 2008, ApJ, 687, 1264
* Monteiro et al. (2006) Monteiro, H., Jao, W. C., Henry, T., Subasavage, J., $\&$ Beaulieu, T., 2006, ApJ, 638, 446
* Oswalt et al. (1988) Oswalt, T. D., Hintzen, P. M., $\&$ Luyten, W. J., 1988, ApJ, 66, 3910
* Oswalt et al. (1991) Oswalt, T. D., Sion, E. M., Hintzen, P. M., $\&$ Liebert, J. W. 1991, in White Dwarfs, ed. G. Vauclair $\&$ E. M. Sion (NATO ASI Ser. C, 336; Dordrecht: Kluwer), 379
* Oswalt et al. (1993) Oswalt, T. D., Smith, J. A., Shufelt, S., Hintzen, P. M., Leggett, S. K., Liebert, J., $\&$ Sion, E. M. 1993, in White Dwarfs: Advances in Observation and Theory, ed. M. A. Barstow (NATO ASI Ser. C, 403; Dordrecht: Kluwer), 419
* Pace et al. (2004) Pace, G., $\&$ Pasquini, L., 2004, A&A, 426, 1021
* Pace et al. (2009) Pace, G., Melendez, J., Pasquini, L., Carraro, G., Danziger, J., François, P., Matteucci, F., $\&$ Santos, N. C., 2009, A&A, 499, L9
* Pancino et al. (2010) Pancino, E., Carrera, R., Rossetti, E., $\&$ Gallart, C., 2010, A&A, 511, 56
* Patten et al. (1993) Patten, B. M., $\&$ Simon, T., 1993, ApJ, 415, L123
* Patten et al. (1996) Patten, B. M., $\&$ Simon, T., 1996, ApJS, 106, 489
* Reid et al. (2000) Reid, N. I., $\&$ Hawley, S. L., 2000, New Light on Dark Stars
* Rocha-Pinto et al. (1998) Rocha-Pinto, H. J., $\&$ Maciel, W. J., 1998, MNRAS, 298, 332
* Rocha-pinto et al. (2000) Rocha-Pinto, H. J., Maciel, W. J., Scalo J., $\&$ Flynn C., 2000, A&A, 358, 850
* Silvestri et al. (2005) Silvestri, N. M., Hawley, S. L., $\&$ Oswalt, T. D., 2005, AJ, 129, 2428
* Silvestri et al. (2006) Silvestri, N. M., Hawley, S. L., West, A. A., Szkody P., Bochanski, J. J., Eisenstein, D. J., McGehee, P., Schmidt, G. D., Smith, J. A., Wolfe, M. A., Harris, H. C., Kleinman, S. J., Liebert, J., Nitta, A., Barentine, J. C., Brewington, H. J., Brinkmann, J., Harvanek, M., Krzesi$\acute{n}$ski, J., Long, D., Neilsen, E. H., Jr., Schneider, D. P., $\&$ Snedden, S. A., 2006, AJ, 131, 1674
* Sion et al. (2009) Sion, E. M., Holberg, J. B., Oswalt T. D., McCook G. P., $\&$ Wasatonic, R., 2009, AJ, 138, 1681
* Skumanich (1972) Skumanich, A., 1972, ApJ, 171, 565
* Smith (1997) Smith, J. A., 1997, Ph.D. Dissertation, Florida Institute of Technology
* Soderblom et al. (1991) Soderblom, D. R., Duncan, D. K., $\&$ Johnson, D. H. R., 1991, ApJ, 375, 722
* Soderblom (2010) Soderblom, D. R., 2010, Annu. Rev. Astron. Astrophys., 48, 581
* Sousa et al. (2008) Sousa, S. G., Santos, N. C., Mayor, M., Udry, S., Casagrande, L., Israelian, G., Pepe, F., Queloz, D., $\&$ Monteiro, M. J. P. F. G., 2008, A&A, 487, 373
* Valenti et al. (2005) Valenti, J. A., $\&$ Fischer, D. A., 2005, ApJS, 159, 141
* Vaughan et al. (1980) Vaughan, A. H., $\&$ Preston, G. W., 1980, PASP, 92, 385
* Venn et al. (1980) Venn, K. A., Irwin, M., Shetrone, M. D., Tout, C. A., Hill, V., $\&$ Tolstoy, E., 2004, AJ, 128, 1177
* Voss et al. (2007) Voss, B., Koester, D., Napiwotzki, R., Christlieb, N., $\&$ Reimers, D., 2007, A&A, 470, 1079
* Weidemann & Koester (1984) Weidemann, V., $\&$ Koester, D. 1984, A&A, 132, 195
* Weidemann (2000) Weidemann, V., 2000, A&A, 363, 647
* Wegner $\&$ Reid (1991) Wegner, G., $\&$ Reid, I. N., 1991, ApJ, 375, 674
* West et al. (2008) West, A. A., Hawley, S. L., Bochanski, J. J., Covey, K. R., Reid, I. N., Dhital, S., Hilton, E. J., $\&$ Masuda M., 2008, AJ, 135, 785
* Wielen (1982) Wielen, R. 1982, Landolt-Börnstein Tables, Astrophysics, Vol. 2C, Sec. 8.4 (Berlin: Springer), 202
* Wilson (1963) Wilson, O. C., 1963, ApJ, 138, 832
* Wilson (1968) Wilson, O. C., 1968, ApJ, 153, 221
* Wood (1995) Wood, M., 1995, White Dwarfs, ed. D. Koester $\&$ K.Werner, vol. 443, ( Berlin: Springer), 41
Figure 1: The relation between V-I and flux ratio of CaI4227. The dotted line
is a least-squares fit. We used this relation to estimate V-I color for a few
stars whose photometric data was unavailable. See text for details. Figure 2:
The relation among log($\rm{S_{HK}}$), V-I and age. Open circles represent
CPMB stars with unknown age. Filled circles are flare stars. Stars in cluster
IC2602 + IC2391 and M67 are shown as asterisks and squares respectively. Plus
signs and diamonds represent stars with different ages: 1Gyr $<$ age $<$ 5Gyr
and 5Gyr $<$ age $<$ 8Gyr, respectively. Four domains are divided by dotted
lines. The dash-dot line is the least-squares fit of young clusters
IC2602+IC2391. The vertical dashed line is V-I = 2.4. To the right of this, CA
is not expected to depend on age. Solid lines connect the MS components in
CD-31 7352/LP902-30, LP684-1/2 and LP387-1/2. The star indicated as number 1
is G163-B9A. Numbers 2-3 denote halo star candidates (LHS300A and CD-31 1454).
Number 4 is the star with the weakest activity in our sample (G114-B8B). The
typical error bar for our measurement is in the upper right corner. Separate
V-I uncertainties are given for 4 stars whose V-I were estimated by the
relation in Fig.1. A square with a plus sign is an outlier (G95-B5A) that
seems overactive for its age domain. This star is perhaps a close binary and
it warrants follow-up observations to determine the cause of its high CA.
Figure 3: Left: An example of how [Fe/H] estimates were made with the maximum
correlation method. The filled circle is the maximum of fitted polynomial. The
[Fe/H] correlated to this maximum is regarded as our final result. + and -
signs indicate our estimate of uncertainty as discussed in the text. Right:
The comparison between our [Fe/H] results and those in the literature. Filled
circles represent stars in Catalán et al. 2008a. Single open circles represent
stars in Holmberg, Nordström $\&$ Andersen (2009). Double open circles, X
signs and squares represent the clusters IC2391, IC2602 and M67, respectively.
We adopted the average [Fe/H] of member stars as the final cluster [Fe/H]. The
[Fe/H] of IC2391 and IC2602 are from D’Orazi $\&$ Randich (2009), while that
of M67 is from Pancino et al. 2010. Diamonds and triangles represent stars in
Mallik (1998) and Sousa et al. (2008), respectively. Note that many of these
sources did not give uncertainties, hence no vertical error bar could be
plotted. Figure 4: Our [Fe/H] vs. age relation for 21 MS stars. Typical
uncertainties in age and [Fe/H] are displayed at the lower left of this
figure. Filled circles represent disk stars. Only one star indicated by an
asterisk (LHS300A) is a likely halo star. The dotted line is a least-squares
fit for only disk stars, while the dashed line includes all stars. Figure 5:
Top: UV-velocity distribution of our sample with measured vr (km s-1). The
ellipsoids indicate the 1 $\sigma$ (inner) and 2 $\sigma$ (outer) contours for
Galactic thick-disk and stellar halo populations, respectively. Typical error
bars are given in both panels. Bottom: Toomre diagram of our stars. Dashed
line is Vtotal = 180 km s-1. Figure 6: Left: age vs. absolute value of the W
component of space velocity for 21 MS stars in our CPMB sample. Solid line is
a least-squares fit. Typical uncertainties in age and W are displayed at the
upper right of this figure. Right: W dispersion vs. age derived by binning the
data in the left panel in five age group ranging from 1-8 Gyr.
Table 1: Observed wide binaries
ID | Name | R.A. | Decl. | | | | | UT | $\mu$ | $\theta_{\mu}$ | pos | Sep | Location
---|---|---|---|---|---|---|---|---|---|---|---|---|---
| (Star 1/Star 2) | (B1950.0) | (B1950.0) | V1 | Sp1aaSpectral types for the WD and MS stars were determined from low - resolution ($\sim$ 7 - 15 $\rm{\AA}$) spectra (Oswalt et al. 1988, 1991, 1993). | V2 | Sp2aaSpectral types for the WD and MS stars were determined from low - resolution ($\sim$ 7 - 15 $\rm{\AA}$) spectra (Oswalt et al. 1988, 1991, 1993). | (mm/yy) | (arcsec yr${}^{-1})$ | (deg) | (deg) | (arcsec) |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13) |
1 | CD-31 1454/LP888-25 | 03 32 36 | -31 14 00 | 11.8 | dG | 15.9 | dK | 02/05 | 0.51 | 186 | 264 | 223 | CTIO
2 | G114-B8A/B | 08 58 17 | -04 10 24 | 11.0 | dK0 | 15.9 | dG2 | 02/05 | 0.10 | 150 | 151 | 54 | CTIO
3 | BD+12 937/G102-39 | 05 51 05 | 12 23 48 | 7.6 | dF8 | 15.7 | DC | 02/04 | 0.28 | 184 | 47 | 91 | CTIO
4 | G272-B5A/B | 02 00 31 | -17 07 30 | 12.5 | dG | 15.8 | DA | 02/05 | 0.05 | 181 | 79 | 72 | CTIO
5 | BD-03 2935/LP670-9 | 10 27 24 | -03 57 00 | 11.2 | dG | 18.7 | | 02/05 | 0.18 | 103 | 139 | 35 | CTIO
6 | G163-B9A/B | 10 43 39 | -03 24 06 | 12.6 | dF9 | 15.6 | | 02/05 | 0.08 | 115 | 122 | 76 | CTIO
7 | BD+23 2539/LP378-537 | 13 04 48 | 22 43 00 | 9.8 | dK0 | 16.2 | DA | 02/04 | 0.11 | 300 | 106 | 20 | CTIO
8 | CD-25 8487/LP849-59 | 11 07 00 | -25 43 00 | 9.3 | sdM0 | 16.8 | DC | 02/04 | 0.25 | 106 | 181 | 100 | CTIO
9 | CD-28 3361/LP895-41 | 06 42 34 | -28 30 48 | 11.2 | dK | 16.8 | DA | 02/04 | 0.16 | 227 | 75 | 16 | CTIO
10 | BD-5 3450/LP674-29 | 12 09 48 | -06 05 00 | 12.0 | dK5 | 17.2 | DC | 02/05 | 0.44 | 220 | 102 | 202 | CTIO
11 | BD-18 2482/LP786-6 | 08 45 18 | -18 48 00 | 12.8 | dK3 | 15.1 | DB | 02/04 | 0.16 | 268 | 236 | 31 | CTIO
12 | 40 Eri A/B/C | 04 13 03 | -07 44 06 | 5.3 | dG | 9.5 | DA | 02/04,02/05 | 4.08 | 213 | 105 | 82 | CTIO
13 | CD-31 7352/LP902-30 | 09 28 20 | -31 53 12 | 9.3 | dK | 14.5 | dM | 02/05 | 0.34 | 348 | 206 | 12 | CTIO
14 | LP684-1/2 | 15 54 00 | -04 41 00 | 12.7 | dM | 15.5 | dM | 02/05 | 0.32 | 244 | 202 | 5 | CTIO
15 | BD-18 3019/LP791-55 | 10 43 30 | -18 50 00 | 12.9 | dM0 | 16.6 | DQ | 02/04 | 1.98 | 250 | 356 | 7.5 | CTIO
16 | LP856-54/53 | 13 48 30 | -27 19 00 | 13.9 | dM | 15.1 | DA | 02/04 | 0.24 | 166 | 233 | 9 | CTIO
17 | LP498-25/26 | 13 36 45 | 12 23 48 | 13.9 | dM | 14.5 | DB | 02/04 | 0.19 | 134 | 307 | 87 | CTIO
18 | LP672-2/1 | 11 05 30 | -04 53 00 | 12.6 | dM6 | 13.8 | DA | 02/04 | 0.44 | 184 | 160 | 279 | CTIO
19 | LP916-26/27 | 15 42 18 | -27 30 00 | 15.5 | dM | 16.3 | DB | 02/04 | 0.24 | 235 | 330 | 52 | CTIO
20 | LP891-13/12 | 04 43 18 | -27 32 00 | 15.6 | dM | 15.9 | DQ | 02/05 | 0.24 | 246 | 62 | 49 | CTIO
21 | LP783-2/3 | 07 38 02 | -17 17 24 | 12.9 | dM | 17.6 | DB | 02/04,02/05 | 1.26 | 117 | 276 | 21 | CTIO
22 | CD-37 10500/L481-60 | 15 44 12 | -37 46 00 | 6.8 | dG | 13.2 | DA | 02/05 | 0.48 | 243 | 131 | 15 | CTIO
23 | CD-59 1275/L182-61 | 06 15 36 | -59 11 24 | 7.0 | dG0 | 13.7 | DB | 02/04,02/05 | 0.33 | 190 | 302 | 41 | CTIO
24 | LP888-63/64 | 03 26 45 | -27 18 36 | 13.9 | | 15.6 | | 02/05 | 0.83 | 63 | 227 | 7 | CTIO
25 | CD-38 10983/10980 | 16 20 38 | -39 04 42 | 6.1 | dG | 10.7 | DA | 02/04,02/05 | 0.08 | 95 | 248 | 345 | CTIO
26 | LHS193A/B | 04 30 50 | -39 08 55 | 11.7 | dM | 17.7 | DB | 02/05 | 1.023 | 44.5 | | | CTIO
27 | LHS300A/B | 11 08 58 | -40 49 05 | 13.2 | dK | 17.8 | DB | 02/05 | 1.277 | 264.5 | | | CTIO
28 | LP387-2/1 | 16 44 18 | 24 06 00 | 16.8 | dG | 17.6 | DG | 07/05 | 0.11 | 163 | 296 | 37 | KPNO
29 | BD-8 0980/G156-64 | 22 53 12 | -08 05 24 | 9.0 | dG | 16.4 | DA | 07/05 | 0.59 | 91 | 168 | 43 | KPNO
30 | G171-62/G172-4 | 00 30 17 | 44 27 18 | 10.3 | dK | 16.6 | DA | 11/06 | 0.16 | 285.9 | | | KPNO
31 | BD-1 469/LP592-80 | 03 15 48 | -01 06 18 | 6.6 | dG | 17.2 | DA | 11/06 | 0.18 | 192 | 50 | 49 | KPNO
32 | G216-B14A/B | 22 58 49 | 40 40 12 | 12.0 | | 15.5 | | 11/06 | 0.07 | 215 | 261 | 23 | KPNO
33 | BD+44 1847/G116-16 | 09 11 51 | 44 15 36 | 10.2 | | 15.5 | | 11/06 | 0.28 | 174 | 95 | 1020 | KPNO
34 | G273-B1A/B | 23 50 54 | -08 21 06 | 12.0 | dG | 16.4 | DA | 11/06 | 0.12 | 75 | 210 | 36 | KPNO
35 | G95-B5A/B | 02 20 44 | 22 14 00 | 9.2 | | 15.6 | | 11/06 | 0.15 | 113 | 94 | 26 | KPNO
36 | BD+26 730/LP358-525 | 04 33 42 | 27 02 00 | 9.4 | dK | 16.3 | DA | 11/06 | 0.28 | 122 | 338 | 128 | KPNO
Note. — Units of right ascension are hours, minutes, and seconds, and units of
declination are degrees, arcminutes, and arcseconds. $V$ magnitude values
(except 26 and 27) are $m_{pg}$ magnitudes from Oswalt, Hintzen $\&$ Luyten
1988.
Table 2: The list of cluster member stars ID | Name | V | UT | V-I | B-V | SHK | age (Gyr) | Cluster | Refaa1: Barnes, Sofia $\&$ Prosser (1999); 2: Patten $\&$ Simon (1996); 3: Giampapa et al. (2006)
---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10)
37 | [RSP95] 15 | 11.75 | 02/05 | 1.0600 | 0.9300 | 0.6819 | 0.050${}_{0.045}^{0.055}$ | IC2602 | 1
38 | [RSP95] 32 | 15.06 | 02/05 | 2.1600 | 1.6300 | 1.5799 | 0.050${}_{0.045}^{0.055}$ | IC2602 | 1
39 | [RSP95] 66 | 11.07 | 02/04 | 0.8300 | 0.6800 | 0.4241 | 0.050${}_{0.045}^{0.055}$ | IC2602 | 1
40 | [RSP95] 70 | 10.92 | 02/04 | 0.7100 | 0.6900 | 0.3553 | 0.050${}_{0.045}^{0.055}$ | IC2602 | 1
41 | [RSP95] 72 | 10.89 | 02/05 | 0.7600 | 0.6400 | 0.5110 | 0.050${}_{0.045}^{0.055}$ | IC2602 | 1
42 | [RSP95] 80 | 11.75 | 02/04 | 1.0900 | 0.9300 | 0.5738 | 0.050${}_{0.045}^{0.055}$ | IC2602 | 1
43 | VXR PSPC 12 | 11.86 | 02/04 | 0.9100 | 0.8300 | 0.5429 | 0.050${}_{0.045}^{0.055}$ | IC2391 | 2
44 | VXR PSPC 14 | 10.45 | 02/04 | 0.6900 | 0.5700 | 0.4302 | 0.050${}_{0.045}^{0.055}$ | IC2391 | 2
45 | VXR PSPC 70 | 10.85 | 02/04 | 0.7500 | 0.6400 | 0.3716 | 0.050${}_{0.045}^{0.055}$ | IC2391 | 2
46 | VXR PSPC 72 | 11.46 | 02/04 | 0.8400 | 0.7300 | 0.4984 | 0.050${}_{0.045}^{0.055}$ | IC2391 | 2
47 | VXR PSPC 76a | 12.76 | 02/04 | 1.2414 | 1.0400 | 0.8793 | 0.050${}_{0.045}^{0.055}$ | IC2391 | 2
48 | VXR PSPC 77a | 9.91 | 02/04 | 0.6000 | 0.5000 | 0.4184 | 0.050${}_{0.045}^{0.055}$ | IC2391 | 2
49 | Cl* NGC 2682 SAND 785 | 14.8 | 02/05 | 0.8315 | 0.6500 | 0.3233 | 4.000${}_{3.800}^{4.300}$ | M67 | 3
50 | Cl* NGC 2682 SAND 1477 | 14.6 | 02/05 | 0.8497 | 0.6700 | 0.2981 | 4.000${}_{3.800}^{4.300}$ | M67 | 3
Table 3: The V-I, B-V, SHK, vr, full space motions and age for 27 wide binaries ID | V-I | B-V | SHK | vr | $\pi$ | U | V | W | [Fe/H] | age
---|---|---|---|---|---|---|---|---|---|---
| | | | (km s${}^{-1})$ | (mas) | (km s-1) | (km s-1) | (km s-1) | | (Gyr)
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11)
1a | 0.5210 | 0.8230 | 0.2303 | 76.1$\pm$7.3 | 5.89$\pm$2.77 | -329.0 | -242.5 | -101.4 | -0.48$\pm$0.02 |
2a | 0.6595 | 0.5660 | 0.2843 | 16.1$\pm$4.3 | 9.06$\pm$2.68 | -42.5 | -38.6 | 11.7 | -0.17$\pm$0.07 |
2b | 1.0701 | 0.9111 | 0.2413 | 114.3$\pm$2.1 | 9.06$\pm$2.68 | 10.3 | -109.3 | 54.9 | -0.40$\pm$0.08 |
3 | 0.6624 | 0.5699 | 0.2891 | 58.8$\pm$9.7 | 18.70$\pm$0.80 | 26.4 | -65.8 | -39.0 | -0.25$\pm$0.02 |
4 | 0.6800 | 0.5230 | 0.2689 | 28.3$\pm$5.4 | 62.95 | -10.3 | 16.5 | -19.1 | -0.34$\pm$0.06 |
5 | 0.6901 | 0.6762 | 0.2666 | -54.3$\pm$6.1 | 28.3 | -51.2 | 44.0 | -18.6 | 0.10$\pm$0.01 |
6 | 0.7420 | 0.5970 | 0.4582 | -20.8$\pm$6.9 | 83.3 | -18.5 | 18.6 | -7.1 | -0.25$\pm$0.15 |
7 | 0.7754 | 0.6422 | 0.3150 | -7.7$\pm$4.5 | 56.7 | 0.4 | 4.4 | 0.4 | 0.06$\pm$0.04 | 1.0${}_{0.9}^{1.2}$
8 | 0.7892 | 0.6963 | 0.2856 | 28.5$\pm$4.9 | 26.76$\pm 1.09$ | -59.4 | -8.6 | 30.9 | -0.16$\pm$0.06 |
9 | 1.0776 | 0.9720 | 0.3454 | 27.0$\pm$1.6 | 28.17$\pm 0.06$ | -6.4 | -15.4 | -26.4 | -0.13$\pm$0.13 |
10 | 1.1181 | 0.9884 | 0.3180 | 55.9$\pm$1.6 | 22.94$\pm 1.63$ | -0.3 | -101.2 | 5.9 | -0.48$\pm$0.02 |
11 | 1.1316 | 1.0362 | 0.4074 | 69.2$\pm$5.6 | 29.49 | 35.6 | -53.3 | 3.5 | 0.19$\pm$0.09 | 2.0${}_{1.4}^{2.8}$
12a | 1.1469 | 0.7750 | 0.3051 | -37.9$\pm$1.7 | 198.25$\pm$0.84 | -104.0 | -8.5 | -37.4 | -0.17$\pm$0.17 | 5.0${}_{4.0}^{6.1}$
12c | 2.8297 | 1.3938 | 3.0759 | -54.0$\pm$1.7 | 198.25$\pm$0.84 | -104.0 | -8.5 | -37.4 | -0.17$\pm$0.17 | 5.0${}_{4.0}^{6.1}$
13a | 1.1891 | 0.9822 | 0.3140 | 21.8$\pm$2.0 | 51.71$\pm$0.91 | 19.6 | -9.4 | 28.7 | -0.30$\pm$0.10 |
13b | 2.3761 | 1.4432 | 0.5585 | 33.8$\pm$3.1 | 51.71$\pm$0.91 | 21.6 | -21.6 | 31.7 | -0.22$\pm$0.08 |
14a | 1.4140 | 1.2120 | 0.4563 | 67.2$\pm$1.8 | 42.69 | -58.5 | -23.0 | 58.5 | 0.19$\pm$0.09 |
14b | 2.2270 | 0.6870 | 0.5730 | 60.3$\pm$6.9 | 42.69 | -52.8 | -23.5 | 54.5 | 0.05$\pm$0.09 |
15 | 1.8888 | 1.4484 | 0.6202 | 57.7$\pm$2.0 | 56.92 | 236.9 | -187.8 | -229.5 | -0.25$\pm$0.15 | 3.6${}_{2.7}^{5.0}$
16 | 1.9870 | 1.4550 | 0.74958 | 14.4$\pm$4.2 | 49.63 | -28.6 | -11.1 | -4.2 | 0.15$\pm$0.05 | 1.2${}_{0.9}^{1.4}$
17 | 2.3380 | 1.5710 | 1.0492 | 11.7$\pm$5.2 | 46.12 | -33.3 | 3.1 | 11.8 | -0.24$\pm$0.10 | 1.5${}_{0.8}^{2.9}$
18 | 2.4790 | 1.5020 | 0.7738 | 28.6$\pm$4.2 | 57.70$\pm$14.40 | -22.9 | -39.0 | 8.8 | -0.02$\pm$0.03 | 3.5${}_{2.5}^{4.5}$
19 | 2.8165 | 1.6318 | 0.9172 | -53.4$\pm$4.0 | 40.05 | 48.1 | -9.6 | -7.5 | | 2.2${}_{1.6}^{4.8}$
20 | 2.8474 | 1.5111 | 2.5424 | 24.0$\pm$3.2 | 38.80 | -14.0 | 3.4 | -33.0 | |
21 | 4.2570 | 1.8760 | 1.0893 | -24.6$\pm$3.9 | 102.00$\pm$14.00 | -65.8 | -3.1 | 40.6 | | 2.4${}_{1.8}^{4.2}$
22 | 0.8933 | 0.7180 | 0.3031 | -18.3$\pm$10.1 | 65.1$\pm$0.4 | 24.8 | -24.1 | 15.5 | 0.02$\pm$0.08 | 1.1${}_{0.9}^{1.3}$
23 | 0.7769 | 0.5900 | 0.27162 | -41.9$\pm$4.2 | 27.5$\pm$0.5 | -63.3 | 12.9 | -17.8 | 0.03$\pm$0.07 | 1.4${}_{0.4}^{2.4}$
24 | 1.8366 | 1.5000 | 1.2597 | 3.2$\pm$3.8 | 57.6 | 29.6 | -4.0 | 83.1 | | 3.6${}_{2.2}^{5.6}$
25 | 0.8133 | 0.630 | 0.32504 | -2.5$\pm$4.1 | 77.69$\pm$0.86 | -9.5 | 9.7 | 2.1 | -0.10$\pm$0.12 | 1.4${}_{0.9}^{2.3}$
26 | 1.5700 | 1.1600 | 0.3815 | 56.6$\pm$4.4 | 32.06$\pm$1.65 | 131.7 | -50.6 | 48.2 | -0.45$\pm$0.26 | 7.5${}_{6.1}^{8.7}$
27 | 1.6900 | | 0.3835 | 145.0$\pm$1.8 | 32.3 | 100.3 | -195.3 | -46.3 | -0.95$\pm$0.25 | 7.9${}_{5.5}^{9.0}$
28a | 1.0215 | 0.8394 | 0.2264 | -29.9$\pm$2.2 | 45.0 | -14.0 | -19.3 | 27.4 | 0.04$\pm$0.06 |
28b | 0.901 | 0.8470 | 0.2054 | -32.1$\pm$5.8 | 45.0 | -14.2 | -20.4 | 29.3 | 0.03$\pm$0.07 |
29 | 0.7100 | 0.5300 | 0.2189 | -28.1$\pm$4.1 | 28.7$\pm$1.3 | 80.4 | -38.6 | -12.3 | -0.40$\pm$0.02 | 3.5${}_{3.0}^{3.9}$
30 | 0.8800 | 0.9800 | 0.2389 | 44.4$\pm$1.5 | 9.52$\pm$1.63 | 101.3 | -20.3 | -36.3 | -0.23$\pm$0.03 | 1.6${}_{1.0}^{2.4}$
31 | 0.3758aaV-I is from the relation in Fig.1 | 1.0400 | 0.1585 | 33.2$\pm$1.9 | 14.68$\pm$0.96 | -19.7 | -29.4 | -49.3 | -0.16$\pm$0.04 | 2.6${}_{1.3}^{3.9}$
32 | 2.0889aaV-I is from the relation in Fig.1 | | 1.0034 | 1.6$\pm$1.2 | | | | | -0.15$\pm$0.05 | 1.2${}_{1.0}^{1.3}$
33 | 0.7160 | 0.6600 | 0.1820 | -61.4$\pm$12.0 | 19.36$\pm$1.30 | -57.6 | -65.3 | -27.8 | -0.40$\pm$0.10 | 1.6${}_{1.4}^{1.8}$
34 | 0.6861aaV-I is from the relation in Fig.1 | | 0.2081 | 28.9$\pm$1.4 | | | | | -0.16$\pm$0.06 | 3.4${}_{2.7}^{4.2}$
35 | 0.9195aaV-I is from the relation in Fig.1 | | 0.3607 | 27.3$\pm$3.6 | 21.65 | 26.4 | -11.3 | -7.7 | -0.50$\pm$0.03 | 3.4${}_{2.6}^{4.2}$
36 | 1.52 | 1.1200 | 1.7876 | 42.7$\pm$1.3 | 56.02$\pm$1.21 | 35.8 | -11.6 | 4.4 | -0.21$\pm$0.03 | 4.5${}_{3.6}^{5.4}$
Table 4: The $T$eff, log $g$, mass, cooling time and reference of fifteen wide dwarfs ID | Name | Sp | $T$eff | log $g$ | MWD | cooling time | Mi | tevol | RefaaThis column lists the source of $T$eff, log $g$. 1: Catalán et al. 2008; 2: Voss et al. 2007; 3: Sion et al. 2009; 4: Bergeron et al. 1995; 5: Koester et al. 2009; 6: Koester et al. 2001; 7: Monteiro et al. 2006; 8: Weidemann $\&$ Koester 1984
---|---|---|---|---|---|---|---|---|---
| | | (K) | | (M⊙) | (Gyr) | (M⊙) | (Gyr) |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10)
7b | LP378-537 | DA | 10800$\pm$120 | 8.21$\pm$0.05 | 0.732$\pm$0.032 | 0.6760$\pm$0.0626 | 3.02$\pm$0.23 | 0.3${}_{0.2}^{0.5}$ | 1
11b | LP786-6 | DB | 17566$\pm$200 | 7.97$\pm$0.05 | 0.579$\pm$0.028 | 0.1182$\pm$0.0135 | 1.56$\pm$0.29 | 1.9${}_{1.3}^{2.7}$ | 2
12b | 40Eri B | DA | 16570$\pm$350 | 7.86$\pm$0.05 | 0.540$\pm$0.019 | 0.1122$\pm$0.0116 | 1.15$\pm$0.20 | 4.9${}_{3.9}^{6.0}$ | 4
15b | LP791-55 | DQ | 6190$\pm$200 | 8.09$\pm$0.05 | 0.630$\pm$0.029 | 2.8244$\pm$0.6080 | 2.09$\pm$0.31 | 0.8${}_{0.3}^{2.1}$ | 3
16b | LP856-53 | DA | 10080$\pm$200 | 8.17$\pm$0.05 | 0.705$\pm$0.032 | 0.7572$\pm$0.0911 | 2.82$\pm$0.23 | 0.4${}_{0.2}^{0.6}$ | 4
17b | LP498-26 | DB | 16779$\pm$200 | 8.00$\pm$0.05 | 0.595$\pm$0.027 | 0.1475$\pm$0.0155 | 1.73$\pm$0.28 | 1.4${}_{0.6}^{2.7}$ | 2
18b | LP672-1 | DA | 15996$\pm$11 | 7.753$\pm$0.002 | 0.486$\pm$0.001 | 0.1066$\pm$0.0004 | 1.25$\pm$0.01 | 3.4${}_{2.4}^{4.4}$ | 5
19b | LP916-27 | DB | 10826$\pm$200 | 8.00$\pm$0.05 | 0.585$\pm$0.029 | 0.5365$\pm$0.0531 | 1.62$\pm$0.30 | 1.7${}_{1.2}^{4.3}$ | 2
21b | LP783-3 | DZ | 7590$\pm$200 | 8.07$\pm$0.05 | 0.619$\pm$0.031 | 1.4854$\pm$0.1856 | 1.93$\pm$0.31 | 0.9${}_{0.3}^{2.7}$ | 3
22b | L481-60 | DA | 10613$\pm$18 | 8.12$\pm$0.03 | 0.675$\pm$0.018 | 0.6167$\pm$0.0254 | 2.56$\pm$0.19 | 0.5${}_{0.3}^{0.7}$ | 6
23b | L182-61 | DB | 16714$\pm$200 | 8.07$\pm$0.05 | 0.605$\pm$0.029 | 0.1539$\pm$0.0160 | 1.83$\pm$0.30 | 1.2${}_{0.4}^{2.2}$ | 2
24b | LP888-63 | DA | 9408$\pm$8 | 7.93$\pm$0.02 | 0.559$\pm$0.011 | 0.6515$\pm$0.0159 | 1.35$\pm$0.11 | 3.0${}_{1.6}^{5.0}$ | 3
25b | CD-38 10980 | DA | 24276$\pm$200 | 8.01$\pm$0.05 | 0.641$\pm$0.028 | 0.0279$\pm$0.0043 | 2.21$\pm$0.29 | 0.7${}_{0.3}^{1.6}$ | 6
26b | LHS193B | DA | 4394$\pm$200 | 8.10$\pm$0.05 | 0.632$\pm$0.032 | 6.6900$\pm$0.8100 | 2.11$\pm$0.33 | 0.8${}_{0.3}^{1.3}$ | 7
27b | LHS300B | DA | 4705$\pm$200 | 7.80$\pm$0.05 | 0.456$\pm$0.026 | 4.5385$\pm$0.7298 | 1.25$\pm$0.01 | 3.4${}_{2.4}^{4.4}$ | 7
29b | G156-64 | DA | 7165$\pm$165 | 8.43$\pm$0.07 | 0.869$\pm$0.046 | 3.2983$\pm$0.4292 | 4.02$\pm$0.34 | 0.1${}_{0.1}^{0.3}$ | 4
30b | G172-4 | DA | 10440$\pm$240 | 8.02$\pm$0.07 | 0.613$\pm$0.043 | 0.5566$\pm$0.0803 | 1.92$\pm$0.45 | 1.1${}_{0.3}^{1.9}$ | 8
31b | LP592-80 | DA | 7520$\pm$260 | 8.01$\pm$0.45 | 0.600$\pm$0.256 | 1.2822$\pm$0.7289 | 1.78$\pm$0.20 | 1.3${}_{0.3}^{2.0}$ | 1
32b | G216-B14B | DA | 9860$\pm$226 | 8.20$\pm$0.07 | 0.724$\pm$0.045 | 0.8443$\pm$0.1246 | 2.96$\pm$0.33 | 0.3${}_{0.2}^{0.4}$ | 4
33b | G116-16 | DA | 8750$\pm$201 | 8.29$\pm$0.07 | 0.780$\pm$0.045 | 1.3397$\pm$0.1785 | 3.37$\pm$0.33 | 0.2${}_{0.1}^{0.3}$ | 4
34b | G273-B1B | DA | 18529$\pm$37 | 7.79$\pm$0.01 | 0.509$\pm$0.004 | 0.0617$\pm$0.0136 | 0.83$\pm$0.04 | 3.4${}_{2.7}^{4.0}$ | 5
35b | G94-B5B | DA | 15630$\pm$65 | 7.89$\pm$0.01 | 0.555$\pm$0.005 | 0.1456$\pm$0.0384 | 1.31$\pm$0.05 | 3.3${}_{2.4}^{4.0}$ | 5
36b | LP358-525 | DA | 5620$\pm$110 | 8.14$\pm$0.07 | 0.673$\pm$0.044 | 4.0693$\pm$0.8147 | 2.54$\pm$0.46 | 0.5${}_{0.2}^{0.8}$ | 4
Table 5: The comparison between our ages and those from literature Name | agenaaages are from Holmberg, Nordström $\&$ Andersen (2009) | agevbbages are from Valenti $\&$ Fischer (2005) | age
---|---|---|---
| (Gyr) | (Gyr) | (Gyr)
(1) | (2) | (3) | (4)
40 Eri A | $\sim$ | 12.2${}_{8.5}^{14.5}$ | 5.0${}_{4.0}^{6.1}$
CD-38 10983 | 2.5${}^{7.0}_{\sim}$ | 2.0${}_{0.4}^{3.9}$ | 1.2${}_{0.4}^{2.2}$
CD-59 1275 | 5.9${}_{5.4}^{6.6}$ | 3.7${}_{3.4}^{4.7}$ | 1.4${}_{0.4}^{2.4}$
CD-37 10500 | 7.4${}_{1.9}^{13.0}$ | 4.4${}_{1.4}^{7.0}$ | 1.1${}_{0.9}^{1.3}$
|
arxiv-papers
| 2011-01-17T16:28:58 |
2024-09-04T02:49:16.498241
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. K. Zhao, T. D. Oswalt, M. Rudkin, G. Zhao, Y. Q. Chen",
"submitter": "Jingkun Zhao",
"url": "https://arxiv.org/abs/1101.3257"
}
|
1101.3260
|
2009 Vol. 9 No. XX, 000–000
11institutetext: Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100012, China;
gzhao@bao.ac.cn
Received [year] [month] [day]; accepted [year] [month] [day]
# Metallicity calibration for solar type stars based on red spectra
J. K. Zhao G. Zhao Y. Q. Chen A. L. Luo
###### Abstract
Based on high resolution and high signal-to-noise ratio (S/N) spectra analysis
of 90 solar type stars, we have established several new metallicity
calibrations in $T\rm{{}_{eff}}$ range [5600, 6500] K based on red spectra
with the wavelength range of 560-880 nm. The new metallicity calibrations are
applied to determine the metallicity of solar analogs selected from SDSS
spectra. There is a good consistent result with the adopted value presented in
SDSS-DR7 and a small scatter of 0.26 dex for stars with S/N $>$ 50 is
obtained. This study provides a new reliable way to derive the metallicity for
solar-like stars with low resolution spectra. In particular, our calibrations
are useful for finding metal-rich stars, which are missing in SSPP.
###### keywords:
techniques: radial velocities - stars: temperatures - stars: abundances
## 1 Introduction
The stellar spectroscopic survey with the Large Area sky Multi-Object fiber
Spectroscopic Telescope (LAMOST) will provide a huge amount of data, which can
be used for the study of chemical and kinematical evolution of our Galaxy. In
this respect, stellar metallicity and radial velocity, being two main
parameters, can be derived from spectra. The determination of radial velocity
is generally easier mainly by using either cross-correlation of the template
spectra or Doppler shift through line calibration. The consistency is usually
quite good depending on the quality of the spectra. The metallicity estimation
from stellar spectra is based on various methods as shown in Lee et al.
(2008a) (hereafter Lee08). However, for solar type stars, these values can be
underestimated by up to 0.5 dex in the previous version of SSPP (SEGUE Stellar
Parameter Pipeline; Lee et al. 2008a). The current version of SSPP has made
great improvement, reaching about 0.1 dex in the underestimation (Lee et al.
2008b). From Fig A1 in Bond et al. (2009), the largest difference between the
SDSS (Sloan Digital Sky Survey; York et al. 2001) spectroscopic metallicity
values with DR6 (Data Release 6; Adelman-McCarthy et al. 2008) and DR7 (Data
Release 7; Abazajian et al. 2009) is shown for solar type stars, so it is
worth the effort to do more research about deriving the metallicty of those
stars.
In this work, we attempt to establish a new metallicity calibration for low
resolution solar type stars based on the result from high resolution and high
signal-to-noise ratio spectral analysis performed by Chen et al. (2000,
hereafter Chen00). In comparison with the methods presented in Lee08, this
work has some advantages. Firstly, the calibration is based on the real
stellar (empirical) spectra and their metallicity is derived from fine
analysis of high resolution and high S/N spectra. Secondly, we have used the
red spectral coverage of 560-880 nm but most of the methods in Lee08 are based
on blue spectra with $\lambda<600$ nm. As is well known, the advantage of the
red spectra is easier to define continuum, which is not possible for blue
spectra due to the heavy line blanketing at low resolution observations. In
view of this advantage, we adopted the equivalent widths (EW) of individual
lines in the calibration instead of the line index. As for line index, there
might be different definition for different authors while EW is a fixed value.
For example, CaII K line, there are K6, K12 and K18 among its definition. If
we can define the continuum well, the EW is better than line index. Thirdly,
we have adopted only Fe lines for metallicity calibration and avoid
contributions from other elements, which do not exactly trace iron evolution
at different times and different nucleosynthesis sites. In Lee08, the
wavelength ranges of templates match include all lines from different
elements. The KP line index is also an indicator of [Ca/H]. Although these
weak Fe lines are undetectable in metal-poor stars because of the noise, we
can recognize them in solar type stars where the S/N is higher than 20.
Moreover, the EW of weak lines is more sensitive to abundance than that of
strong lines. For high resolution spectra analysis, strong lines, e.g. Na
5895$\rm\AA$ and Na 5890$\rm\AA$ are saturated and in the growth curve the
increasing EWs do not give higher abundance. In general, the EW of strong
lines does not change a lot with the degradation of resolution. In view of
this, it is not optimal to establish relation between abundance and EW in
combination with colors using very strong line. Finally, the calibration is
internally consistent, while Lee08 adopted the average of different values
from various methods.
In Section 2, a description of the data for the calibration is presented. The
template matching analysis is described in Section 3. The detailed analysis
procedure to get the calibration formula is given in Section 4. The
application of the calibration to SDSS spectra is illustrated in Section 5.
Finally, the conclusion is given in Section 6.
## 2 Data
The real spectra are taken from Chen00 which has a resolving power of 37 000
and S/N of 150-300 obtained with the Coud$\acute{e}$ Echelle Spectrograph
mounted on the 2.16 m telescope of the National Astronomical Observatories
(Zhao $\&$ Li 2001). The sample has $T\rm{{}_{eff}}$, log $g$ and [Fe/H]
distributions as shown in Fig. 1. It is clear that $T\rm{{}_{eff}}$ ranges
between [5600, 6500], log $g$ is in [3.98, 4.43] and [Fe/H] is in [-1.04,
0.06]. Moreover, the range of b-y is within [0.28, 0.43], B-V is within [0.40,
0.67], V-I is within [0.46, 0.73] and V-K is within [0.9, 1.65]. We convolved
the normalized spectra to low resolution of 2000 with a Gaussian Function. In
addition, the spectra were rebinned to 1.5$\rm{\AA}/pix$ after smoothed to
R$\sim$2000.
Figure 1: Stellar parameter distribution of the sample in Chen00 Table 1: [Fe/H] results of template match with different wavelength ranges wavelength range(nm) | mean deviation | scatter
---|---|---
570-653 | 0.448 | 0.175
570-684 | 0.226 | 0.275
570-700 | 0.016 | 0.230
640-670 | 0.470 | 0.280
651-662 | 0.599 | 0.390
690-713 | -0.07 | 0.590
735-756 | 0.350 | 0.290
772-810 | -0.100 | 0.260
## 3 The template match analysis
Following one method of Lee08, we have performed the template spectra match
(also see Allende et al 2006, Re Fiorentin et al 2007, Zwitter et al. 2005)
for the normalized low resolution spectra of 90 stars and derived the stellar
temperature, gravity and metallicity. In this method, we generate a library of
low resolution theoretical spectra by using the SYNTH program based on Kurucz
New ODF (Castelli $\&$ Kurucz 2003) atmospheric models. The atmosphere models
are under the assumption of local thermodynamic equilibrium (LTE). The mixing-
length is adopted to be l$/$Hp = 1.25 and microturbulence is 1.5 km s-1. The
line list, including the atoms and molecules, are all from Kurucz (1993). The
molecular species include CH, CN, OH, and TiO. Solar abundances are from
Asplund (2005). As for those grids, $T\rm{{}_{eff}}$ covers the range
[3500-9750]K with 250K interval; log $g$ is within [1.0, 5.0] dex with 0.5 dex
interval; [Fe/H] is within [-4.0, -3.0] dex with 0.25 dex interval and 0.1 dex
interval in [-3.0,+0.5] dex. The minimum distance method is applied to obtain
the parameters by interpolation among several of the closest theoretical
spectra with the observed one.
Figure 2: [Fe/H] and $T\rm{{}_{eff}}$ comparison between those of Chen00 and
the results from the template match with spectral range of 570nm-750nm
We have adopted different wavelength ranges in the matching procedure and
obtained different results as shown in Table 1. As compared with the
‘standard’ values presented in Chen00 paper, we have found that the spectral
range of 570-700 nm is the best choice with a mean deviation of 0.016 dex and
scatter of 0.23 dex in [Fe/H]. The comparison of temperature and metallicity
of the 570-700 nm spectral range is shown in Fig. 2. It is clear that the
temperature estimation has systematical deviation with a lower value in the
present work. Since the high resolution spectra in Chen00 are obtained with
echelle spectrograph. The order is not wide enough to include the whole H
alpha line region and the normalization is implemented order by order. Thus,
the continuum is not very reliable in H alpha region, which might be the
reason of systematical deviation in temperature estimation. The metallicity is
quite consistent with an rms of 0.23 dex.
## 4 [Fe/H] vs. EW of the FeI line - An empirical calibration
Although the template match method can be used to obtain accurate stellar
parameters, it may give different results with different wavelength ranges.
Hence, we will determine stellar metallicity based on the strength of some FeI
lines.
Figure 3: A portion of the spectra of HD94280, HD100446 and HD184601. Solid line is the high resolution spectra while thick dotted line is the low resolution spectra. The two FeI lines are those that meet the stated requirements. Table 2: The definition of FeI lines. Left | Right | Center | Element
---|---|---|---
6060.917 | 6069.954 | 6065.492 | FeI
6217.769 | 6221.788 | 6219.292 | FeI
6391.147 | 6395.953 | 6393.605 | FeI
6398.256 | 6402.209 | 6400.232 | FeI
6675.000 | 6682.326 | 6678.256 | FeI
### 4.1 The spectral lines selection
What we want is to choose some Fe lines that are not heavily blended and have
better profiles in the low resolution (R$\sim$2000) spectra. At the beginning,
we draw the original spectrum, then overplot the low resolution spectrum on
it. After checking the lines one by one, we select five FeI lines as our
indices of metallicity. The spectral lines for three stars with different
metallicity values are given in Fig. 3: HD94280 has [Fe/H] of 0.06; HD100446
has -0.48; HD184601 has -0.81. In Fig. 3, the solid line is the high
resolution spectrum while the thick dotted line is that of the low resolution
one. In this segment of the spectrum, only two FeI lines meet our requirements
since they are detectable; they have good shapes and are not seriously blended
in the spectra with a resolution of R$\sim$2000\. From the top to the bottom
of Fig. 3, it is obvious that the strength of FeI lines decreases. Also, for
stars with [Fe/H] $>$ -0.8, the two lines can be identified. Table 2 presents
the definition of five FeI line indices used for our metallicity calibration.
There are three parts for each line including red, center and blue spectra.
The EW of each line can be measured by using a direct integration method.
Figure 4: The relation between [Fe/H] and EW for five FeI lines for the stars with [Fe/H] $>$ -0.8 Table 3: The coefficient and $\sigma$ of the fitting between [Fe/H] and EW for each FeI line Lines | a | b | $\sigma$
---|---|---|---
FeI1 | -0.922 | 5.063 | 0.170
FeI2 | -0.866 | 6.559 | 0.143
FeI3 | -1.010 | 5.960 | 0.167
FeI4 | -0.970 | 4.958 | 0.163
FeI5 | -1.026 | 6.056 | 0.204
Table 4: The coefficient and $\sigma$ of [Fe/H] calibration based on the EW of five FeI lines and temperature Lines | a | b | c | $\sigma$
---|---|---|---|---
FeI1 | 1.731 | 5.929 | -3.281 | 0.150
FeI2 | 2.138 | 7.726 | -3.692 | 0.112
FeI3 | 2.282 | 7.386 | -4.116 | 0.136
FeI4 | 2.793 | 6.381 | -4.702 | 0.122
FeI5 | 0.974 | 6.949 | -2.505 | 0.194
### 4.2 The metallicity calibration based on EWs
Since our spectra are normalized and there are no spectra with flux
calibration, it is difficult to derive reliable temperature measurements. In
order to improve the metallicity determination, we resort to using the
strengths of iron lines and establish a calibration between metallicity and
EWs of iron lines. The EWs are derived with Equation 1.
$\displaystyle\rm{EW}$ $\displaystyle=$ $\displaystyle\int_{\lambda
1}^{\lambda 2}{\frac{f_{c}-f_{\lambda}}{f_{c}}d_{\lambda}}$ (1)
In Equation1, the fλ represents the flux of wavelength $\lambda$, while fc
means the continuum of wavelength $\lambda$. It is shown in Fig. 4 that there
is a good correlation between [Fe/H] and EWs for the five FeI lines for stars
whose [Fe/H] $>$ -0.8. The calibration (Equation 2) for each line is shown in
Table 3.
$\displaystyle\rm[Fe/H]$ $\displaystyle=$ $\displaystyle a+b*\rm{EW(FeI)}$ (2)
As seen in Fig. 4, FeI2 and FeI4 show the best result with the lowest scatter
in the relation. To show the temperature effect of EW, we divide the
temperature range into three parts. The first part is the range of [5265,
5900]; the second part is [5900, 6200] and the last part is [6200, 6496]. In
Fig. 5, the stars in first part are given with the sign of the dot and
asterisks represent the stars in second part, while the stars in last part are
plotted with diamonds. From Fig. 5, it is clear that the relation between EW
and [Fe/H] changes with temperature. FeI2 and FeI4 have lower scatter than
other FeI lines and this may be due to the lower sensitivity of line strength
with temperature. In order to understand this issue, we added the temperature
term in the calibration in Fig. 6. The coefficients and scatter of each line
are given in Table 4. It is obvious that the $\sigma$ becomes small for the
calibration of each line after considering the effect of temperature. Since
the temperature is usually unknown in the spectra analysis, it may be good to
replace the temperature term with the color index. Thus, we collected (b-y),
(B-V), (V-I) and (V-K) and performed a similar calibration (Equation 3). log
$g$ also will bring more or less uncertainty on metallicity determination.
However, the gravity range considered in the calibration sample is narrow,
hence its effect is very little and could be ignored.
Table 5 is the coefficients and scatter of the calibration of [Fe/H] through
EW and B-V.
Table 5: The coefficient, $\sigma$ and the EW range of the fitting between [Fe/H] and the EW plus B-V for each FeI line Lines | a | b | c | $\sigma$ | EW range
---|---|---|---|---|---
FeI1 | -0.790 | 5.351 | -0.315 | 0.169 | 0.025$\sim$0.225
FeI2 | -0.290 | 8.144 | -1.342 | 0.132 | 0.006$\sim$0.158
FeI3 | -0.555 | 7.352 | -1.167 | 0.160 | 0.044$\sim$0.185
FeI4 | -0.321 | 6.622 | -1.643 | 0.151 | 0.029$\sim$0.224
FeI5 | -1.200 | 5.497 | 0.453 | 0.202 | 0.046$\sim$0.193
$\displaystyle\rm[Fe/H]$ $\displaystyle=$ $\displaystyle
a+b*\rm{EW(FeI)}+c*(B-V)$ (3)
Figure 5: The relation between the metallicity and EW of FeI lines in
different $T\rm{{}_{eff}}$ ranges. Dots represent stars in [5265, 5900]K;
asterisks are those in [5900, 6200]K; diamonds are those in [6200, 6500]K.
Figure 6: The comparison between [Fe/H] in Chen00 and that of calibration from
each FeI line and the temperature Figure 7: The application of our [Fe/H]
calibration in Miles spectra library. The x axis is [Fe/H] from the Miles
catalog and the y axis is [Fe/H] obtained by our calibration. The solid line
is x=y Figure 8: Plots of (a) number vs. S/N (b) log $g$ vs. $T\rm{{}_{eff}}$
Figure 9: The spectra of 53738-2051-030. ‘a’ is the original spectrum while
‘b’ is normalized spectrum. ‘c’,‘d’,‘e’,‘f’ and ‘g’ present the spectral lines
in our calibration
### 4.3 Calibration in Miles spectra library
To make an external calibration, we selected 107 spectra from the Miles
spectral library (Sánchez-Blázquez et al. 2006; Cenarro et al. 2007) which
meet the following conditions: -0.8 $\leq$ [Fe/H] $\leq$ 0.5; 4.0 $\leq$ log
$g$ $\leq$ 4.5; 5600 $\leq$ $T\rm{{}_{eff}}$ $\leq$ 6500\. Thus, it is
available to estimate the metallicity of these 107 spectra using above
calibrations. The resolution of Miles spectra is about 2.3$\rm\AA$ and the
wavelength has already been calibrated with radial velocity. First, we do
normalization for these 107 spectra. The continuum is determined by
iteratively smoothing with a Gaussian profile, and then clipping off points
that lie beyond 1 $\sigma$ or 4 $\sigma$ above the smooth curve. Second, the
EWs of five FeI lines are measured. B-V of those 107 stars are taken from the
literature identified by the Simbad Astronomical Database (Genova 2006).
Finally, the metallicities are derived by EW of FeI4 line using equation 3
(the coefficients is shown in Table 5). Fig. 7 is [Fe/H] comparison between
our results and those from Miles library. The mean error is about 0.08 dex,
and the scatter is about 0.21 dex. [Fe/H] of Miles library is obtained by the
compilation from the literature. This suggests that our metallicity is
basically consistent with that of other work.
## 5 Application of these calibrations
In order to check the accuracy of [Fe/H] calibration from the EW of five FeI
lines, we implement this calibration to determine [Fe/H] for solar-like stars
with SDSS spectra. The selection limitation is as follows:
$0.4<(g-r)_{0}<0.5$, $0.10<(r-i)_{0}<0.14$, $0.02<(i-z)_{0}<0.06$, and
$g_{0}<20$. The above color ranges come from the color of the Sun and its
error bars. Based on this limitation, 4356 stars are extracted from the SDSS
DR7 archive. Fig. 8 presents the information of this sample, from which we can
see the peak of S/N is 20; the effective temperature of most stars is located
in the range of [5500, 6000]K; log $g$ is in [4.0, 4.5]. By transforming
equation of Bilir et al. (2005), the g-r range of Chen00 is [0.197, 0.501], so
it is available to estimate the metallicity of these solar-like stars using
Equation 2 and Equation 3.
### 5.1 Preprocess
Our first preprocessing procedures mainly include radial velocity correction
and normalization. The value of radial velocity comes from the FIT head of
each spectra. The pseudocontinuum is determined with the same method
illustrated in Sec. 4.3. Although the method of pseudocontinuum determination
is different with that in Chen00, it is good enough for solar type stars in
the red spectral region. Since the SDSS spectra are relative flux calibrated,
the pseudocontinuum in red region is easier represented by a lower order
polynomial. Thus, iteratively smoothing with a Gaussian profile to original
spectrum will get a good continuum shape. After normalization, then the EW of
the lines can be measured by the direct integration method. Fig. 9 is an
example of the SDSS spectrum. ‘a’ is the original spectrum while ‘b’ is the
normalized spectrum. ‘c’, ‘d’, ‘e’ and ‘f’ present the FeI lines in our
calibration. It is clear that these FeI lines are detectable and show a good
profile in the SDSS spectra.
Figure 10: [Fe/H] comparison of the sdss solar-like stars between those from
SSPP and those derived from the calibration using five FeI lines
### 5.2 Calibration
After obtaining the EW of five FeI lines, [Fe/H] can be determined by our
calibration formula. We do some comparison between [Fe/H] from our calibration
and those from SSPP. Equation 4 is the calibration formula based on the FeI4
line. B-V can be obtained from g-r transformation (Bilir et al. 2005). Fig. 10
is [Fe/H] comparison between those from SSPP and the results from Equation 4.
The comparison of stars with S/N $>$ 50 is shown in the top panel and the
difference distribution is given in bottom panel. It is clear that our
calibration from Equation 4 is very consistent with [Fe/H] of SSPP for those
with S/N $>$ 50\. The mean difference is about 0.018 dex and the scatter is
around 0.26 dex.
$\displaystyle\rm[Fe/H]=-0.321+6.622*\rm{EW(FeI4)}-1.643*(B-V)$ (4)
Moreover, we extract 51 stars which meet these conditions: [Fe/H]$\geq$0 from
our result; [Fe/H]$<0$ from SSPP; S/N$>50$. So these 51 stars are metal rich
stars if our result is right. As for these 51 stars, the temperature range is
about [5650, 5865]K and the gravity range is in [4.3, 4.5] dex. To check the
reliability of our result, we make a comparison between the strength of some
spectral lines in these stars and those in the Sun since $T\rm{{}_{eff}}$ and
log $g$ of these stars are very close to those of the Sun. If the strength is
stronger than that of the Sun then this star can be regarded as a metal rich
star and its [Fe/H] is larger than 0. Since the NaI and CaII line are strong
in the red band, these lines, as well as two FeI lines, are selected for
comparison. There are three cases: one is that the strengths of these lines
are all stronger than those of the Sun (see Fig. 11); one is that the
strengths of these lines are all close to those of the Sun (See Fig. 12); the
others are taken as the third case (See Fig. 13). After comparison, there are
33 stars in first case, 10 stars in second case and 8 stars in third case. In
view of this, the metal rich stars account for 84% of these 51 stars. So our
calibration also provides a reliable way to identify metal rich stars.
Figure 11: The comparison of the strength of spectral lines. The solid line is
the object spectrum and the dotted line is the solar spectrum Figure 12: The
comparison of the strength of spectral lines. The solid line is the object
spectrum and the dotted line is the solar spectrum
Figure 13: The comparison of the strength of spectral lines. The solid line is
the object spectrum and the dotted line is the solar spectrum Figure 14:
Left:[Fe/H] change vs. S/N for seven stars. The original spectra of these
seven stars were extracted from DR7 with S/N $\sim$ 100\. By introducing
Gaussian noise in original spectra, we degraded them to S/N of 50, 30, and 20,
respectively. Filled squares are [Fe/H] changes of these seven stars with
different S/N, while asterisks represent average [Fe/H] change for the given
S/N. Right: [Fe/H] differences between our calibration results and those from
SSPP vs. S/N. Asterisks represent average [Fe/H] differences and vertical
lines represent the scatter of [Fe/H] differences at the given S/N.
### 5.3 The effect of S/N
To investigate the S/N effect on our calibration, we selected seven spectra
from DR7 with S/N (r band) $\sim$ 100\. By introducing Gaussian noise in these
spectra, we degraded them to S/N of 50, 30, and 20, respectively. Then,
normalization and EW measurement were carried out to all spectra. Finally, the
metallicity were derived with our above calibration. The left panel of Fig. 14
presents the S/N effect on metallicity change. Filled squares represent [Fe/H]
changes for these seven stars with different S/N. Asterisks represent the
average [Fe/H] changes for the given S/N. It can be seen that [Fe/H] changes
about 0.22 dex from S/N = 100 to 20. [Fe/H] differences between our
calibration results and those of SSPP vs. S/N are shown in right panel of Fig.
14. Asterisks are average differences and vertical lines represent scatters
for the given S/N for these seven stars. Average [Fe/H] differences nearly
keep the same while the scatter will be smaller than 0.4 dex when S/N is
higher than 30. To sum up, our metallicity determination is quite robust to
reductions in S/N.
## 6 Conclusions
For solar type stars, although template matching can derive reliable results
with a suitable wavelength range, it is very difficult to determine the most
appropriate wavelength range for matching.
We selected five FeI lines from the red part of the R$\sim$2000 resolution
spectra. These lines, which have a good profile, are not seriously blended and
could be detectable with [Fe/H] $>$ -0.8. At the beginning, the metallicity
calibrations are set up only through the EW and the scatters are from 0.14 to
0.20 dex. The dispersion becomes small after adding the temperature into the
calibrations. Since the temperature is usually unknown in the spectra
analysis, it may be good to replace the temperature term with the color index.
In view of this, several metallicity calibrations are constructed by the EW of
FeI lines and colors based on the 90 solar type stars. The dispersion of all
the calibrations is smaller than 0.21 dex. Among the five FeI lines, FeI2 and
FeI4 have contributed the better calibrations (Equation 5-6) which have
smaller scatters (0.13 dex, 0.15 dex).
$\displaystyle\rm[Fe/H]=-0.290+8.144*\rm{EW(FeI2)}-1.342*(B-V),~{}~{}~{}0.006<EW(FeI2)<0.158$
(5)
$\displaystyle\rm[Fe/H]=-0.321+6.622*\rm{EW(FeI4)}-1.643*(B-V),~{}~{}~{}0.029<EW(FeI4)<0.224$
(6)
Moreover, we use the calibration from the EW of FeI4 and the B-V to estimate
[Fe/H] of the solar type stars in DR7. After comparing with the value from
SSPP, our method gives a good consistency for S/N larger than 50. In addition,
we analyze the stars for which [Fe/H] $\geq 0$ by the spectral lines
comparison and found that 84% of them are reliable. Usually, [Na/Fe]=0 $\&$
[Ca/Fe]=0 for most stars with [Fe/H] $>$ -0.4 in the solar neighborhood. In
view of this, Na and Ca lines are stronger in Fe-rich stars. So this provides
a new formula to estimate [Fe/H] with the red band and presents a reliable way
to identify metal rich stars.
###### Acknowledgements.
This work is supported by the National Natural Science Foundation of China
under grant Nos. 10673015, 10821061, 10973021, 11078019 and 11073026, the
National Basic Research Program of China (973 program) No.
2007CB815103/815403, the Academy program No. 2006AA01A120 and the Youth
Foundation of the National Astronomical Observatories of China. Many thanks to
James Wicker for his help revising English grama of this paper.
## References
* Abazajian (2009) Abazajian K., et al. 2009, ApJS, 182, 543
* Ademan-McCarthy (2008) Adelman-McCarthy J. K., et al., 2008, ApJS, 175, 297
* Allende (2006) Allende P. C., Beers T. C., Wilhelm R., Newberg H. J., Rockosi C. M., Yanny B., Lee Y. S., 2006, ApJ, 636,804
* Allende (2008) Allende P. C., Sivarani T., Beers T. C., Lee Y. S., Koesterke L., Shetrone M., Sneden C., Lambert D. L., Wilhelm R., Rockosi C. M., Lai D. K., Yanny B., Ivans I. I., Johnson J. A., Aoki W., Bailer-Jones C. A. L., Re Fiorentin P., 2008, AJ ,136, 2070
* Asplund (2005) Asplund M., 2005, AR$\rm{A\&A}$, 43, 481
* Bilir (2005) Bilir S., Karaali S., Tuncel S., 2005, AN, 326, 321
* Bond (2009) Bond N. A., Ivezic Z., Sesar B., Juric M., Munn J., 2009, arXiv:0909.0013
* Castelli (2003) Castelli F., Kurucz, R. L. 2003, Modelling of Stellar Atmospheres (IAU Symp. 210), ed. N. Piskunov, W. W. Weiss, D. F. Gray (San Francisco, CA: ASP), 20P
* Cenarro (2007) Cenarro A. J., Peletier R. F., Sánchez-Blázquez P., Selam S. O., Toloba E., Cardiel N., Falcón-Barroso J., Gorgas J., Jiménez-Vicente J., Vazdekis, A., 2007, MNRAS, 374, 664
* Chen (2000) Chen Y. Q., Nissen P. E., Zhao G., Zhang H. W., Benoni T., 2000, A&AS, 141, 491
* Genova (2006) Genova F., 2006, Centre de Donnes astronomiques de Strasbourg $<$http://simbad.ustrasbg.fr$>$
* Lee (2008a) Lee Y. S., Beers T. C., Sivarani T., Allende P.C., Koesterke L., Wilhelm R., Re Fiorentin P., Bailer-Jones C. A. L., Norris J. E., Rockosi C. M., Yanny B., Newberg H. J., Covey K. R.,Zhang H. T., Luo A. L., 2008, AJ, 136, 2022
* Lee (2008b) Lee Y. S., Beers T. C., Sivarani T., Johnson J. A., An D., Wilhelm R., Allende P. C., Koesterke L., Re Fiorentin P., Bailer-Jones C. A. L., Norris J. E., Yanny B., Rockosi C., Newberg H. J., Cudworth K. M., Pan K., 2008, AJ, 136, 2050
* Re Fiorentin (2007) Re Fiorentin P., Bailer-Jones C. A. L., Lee Y. S., Beers, T. C., Sivarani T., Wilhelm R., Allende P. C., Norris J. E., 2007, A&A, 467, 1373
* York (2000) York D. G., et al., AJ, 2000, 120, 1579
* Sánchez-Blázquez (2003) Sánchez-Blázquez P., Peletier R. F., Jiménez-Vicente J., Cardiel N., Cenarro A. J., Falcón-Barroso J., Gorgas J., Selam S., Vazdekis A, 2006, MNRAS, 371, 703
* Zwitter (2005) Zwitter T., Munari U., Siebert A., 2005, ESASP, 576, 623
* Zhao (2001) Zhao G., & Li H. B., 2001, Chinese J. Astron. Astrophys., 1, 555
|
arxiv-papers
| 2011-01-17T16:36:38 |
2024-09-04T02:49:16.506636
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. K. Zhao, G. Zhao, Y. Q. Chen, A. L. Luo",
"submitter": "Jingkun Zhao",
"url": "https://arxiv.org/abs/1101.3260"
}
|
1101.3373
|
# Effects of interaction and polarization on spin-charge separation: A time-
dependent spin-density-functional theory study
Gao Xianlong gaoxl@zjnu.edu.cn Department of Physics, Zhejiang Normal
University, Jinhua 321004, Zhejiang Province, China
###### Abstract
We calculate the nonequilibrium dynamic evolution of a one-dimensional system
of two-component fermionic atoms after a strong local quench by using a time-
dependent spin-density-functional theory. The interaction quench is also
considered to see its influence on the spin-charge separation. It is shown
that the charge velocity is larger than the spin velocity for the system of
on-site repulsive interaction (Luttinger liquid), and vise versa for the
system of on-site attractive interaction (Luther-Emery liquid). We find that
both the interaction quench and polarization suppress the spin-charge
separation.
###### pacs:
71.15.Mb, 03.75.Ss, 71.10.Pm
## I Introduction
While the nonequilibrium dynamic evolution of quantum systems has long been
extensively studied, Mattis progress is hindered by the tremendous
difficulties in solving the nonequilibrium quantum many-body Schrödinger
equation. This situation is going to be changed due to the progress in
experiments and the development in numerical methods.
On the experimental side, the development in manipulating ultracold atomic
gases makes it feasible to study strongly correlated systems with time-varying
interactions and external potentials and in out-of-equilibrium situations. The
high controllability in ultracold atomic-gases’ systems provides an ideal
testbed to observe the long-time evolution of strongly correlated quantum
many-body systems, and to test theoretical predictions, such as the Bloch
oscillation, Bloch the absence of thermalization in nearly integrable one-
dimensional (1D) Bose gases, Kinoshita and the expansion of BEC in a random
disorder after switching off the trapping potential. Anderson These efforts
allow us to study the nonequilibrium dynamics of strongly correlated systems
from a new perspective.
Numerically, many techniques have been developed, such as, the time-adaptive
density-matrix renormalization group (t-DMRG), Schollwoek the time-dependent
numerical renormalization group, Costi continuous-time Monte Carlo algorithm,
Eckstein and time-evolving block decimation method. Vidal Time-dependent
spin-density-functional theory (TDSDFT) has been proved to be a powerful
numerical tool beyond the linear-response regime in studying the interplay
between interaction and the time-dependent external potential. gaoprb78Liwei ;
verdozzi_2007 More tests of the performance of TDSDFT will be done in this
paper on the polarized system with attractive or repulsive interactions.
Compared to the algorithms, such as the t-DMRG, this technique gives
numerically inexpensive results for large lattice systems and long-time
evolution, but with difficulties in calculating some properties, such as, the
correlation functions.
The 1D bosonic or fermionic systems accessible by the present ultracold
experiments, moritz ; bosonic are exactly solvable in some cases Gaudin and
can be used to obtain a thorough understanding of the many-body ground-state
and the dynamical properties. The nonequilibrium problems in 1D system are
especially remarkable in which the 1D systems are strongly interacting, weakly
dissipative, and lack of thermalization. Sutherland The 1D systems, belonging
to the universality class described by the Luttinger-liquid theory, have its
particularity in its low-energy excitations, characterized by charged,
spinless excitations and neutral, spin-carrying collective excitations.
Generically, the different dynamics is determined by the velocities of the
charge and spin collective excitations, which has been verified experimentally
in semiconductor quantum wires by Auslaender et al.. Auslaender The
possibility of studying these phenomena experimentally in 1D two-component
cold Fermi gases, moritz where ”spin” and ”charge” refer, respectively, to
the density difference and the total atomic mass density of the two internal
atomic states, was first highlighted by Recati et al. recati_prl_2003 . The
different velocities for spin and charge in the propagation of wave packets
have been demonstrated by Kollath et al. kollath ; Kollath2 in a numerical
t-DMRG study of the 1D Fermi-Hubbard model, by Kleine et al. kleine in a
similar study of the two-component Bose-Hubbard model, and, analytically, by
Kecke et al. kecke_prl_2005 for interacting fermions in a 1D harmonic trap.
Exact diagonalization and quantum Monte Carlo simulations are also used in
studying the spin and charge susceptibilities of the Hubbard model. Jagla ;
Zacher Dynamic structure factors of the charge density and spin are analyzed
for the partially spin-polarized 1D Hubbard model with strong attractive
interactions using a time-dependent density-matrix renormalization method.
Huse The spin-charge separation is well addressed for this system. Huse We
would like to mention here that a genuine observation of spin-charge
separation requires one to explore the single-particle excitation, which is
studied recently in simulating the excitations created by adding or removing a
single particle. Kollath2 ; Ulbricht
The nonequilibrium dynamics in 1D systems has attracted a growing attention in
the possible equilibrium properties after an external perturbation and the
changes in physical quantities after the quench. quench ; kollath ; Karlsson
The dynamic phase transition and different relaxation behavior are studied
with a sudden interaction quench Eckstein ; Barmettler . The relation between
the thermalization and the integrability in 1D system is well addressed. Rigol
The real-time evolution for the magnetization in the 1D spin chain is also
studied in great details using the t-DMRG. Langer
In this paper, we study the 1D system under an instantaneous switching off a
strong local potential or on-site interactions, namely, a sudden quantum
quench is considered. The strong local potential creates Gaussian-shaped
charge and/or spin accumulations at some position in space. After the quantum
quench, the time-evolution of spin and charge densities is then calculated at
later times. We tackle this problem using TDSDFT based on an adiabatic local
spin density approximation (ALSDA).
The contents of the paper are as follows. In Sec. II, we introduce the model:
a time-dependent lattice Hamiltonian that we use to study spin-charge
separation and quench dynamics. Then we briefly summarize the self-consistent
lattice TDSDFT scheme that we use to deal with the time-dependent
inhomogeneous system. In Sec. III, we report and discuss our main numerical
results. At last, a concluding section summarizes our results.
## II Model and the method
We consider a two-component repulsive/attractive Fermi gas with $N_{f}$ atoms
loaded in a 1D optical lattice with $N_{s}$ lattice sites. At time $t\leq 0$,
a localized spin- and charge-density perturbation is created by switching on
slowly the local potential, such that the system is in the ground state of the
system with the additional potential. At $t=0^{+}$, the localized potential is
removed abruptly and/or the on-site interaction is switched off
instantaneously. This system is modeled by a time-dependent Fermi-Hubbard
Hamiltonian as follows:
$\displaystyle{\hat{\cal H}}(t)$ $\displaystyle=$
$\displaystyle-\gamma\sum_{i,\sigma}({\hat{c}}^{\dagger}_{i\sigma}{\hat{c}}_{i+1\sigma}+{\rm
H}.{\rm c}.)+U(t)\sum_{i}{\hat{n}}_{i\uparrow}{\hat{n}}_{i\downarrow}$ (1)
$\displaystyle+$
$\displaystyle\sum_{i,\sigma}V_{i\sigma}(t){\hat{n}}_{i\sigma}~{}.$
Here $\gamma$ is the hopping parameter, ${\hat{c}}^{\dagger}_{i\sigma}$
(${\hat{c}}_{i\sigma}$) creates (annihilates) a fermion in the $i$th site
($i\in[1,N_{s}]$), $\sigma=\uparrow,\downarrow$ is a pseudospin-$1/2$ degree-
of-freedom (hyperfine-state label), $U(t)$ is the time-dependent on-site
Hubbard interaction of negative or attractive nature, and
${\hat{n}}_{i\sigma}={\hat{c}}^{\dagger}_{i\sigma}{\hat{c}}_{i\sigma}$. We
also introduce for future purposes the local number operator
${\hat{n}}_{i}=\sum_{\sigma}{\hat{n}}_{i\sigma}$ and the local spin operator
${\hat{s}}_{i}=\sum_{\sigma}\sigma{\hat{n}}_{i\sigma}/2$.
The external time-dependent potential $V_{i\sigma}(t)=V^{\rm
ext}_{i\sigma}\Theta(-t)$, which simulates the spin-selective focused laser-
induced potential. $\Theta(t)$ is the Heaviside step function which relates
the quench dynamics to the modification of the local potential. $\Theta(-t)=0$
for $t>0$. $V^{\rm ext}_{i\sigma}$ is taken to be of the following Gaussian
form:
$\displaystyle V^{\rm ext}_{i\sigma}$ $\displaystyle=$ $\displaystyle
W_{\sigma}\exp{\left\\{-\frac{[i-(N_{s}+1)/2]^{2}}{2\alpha^{2}}\right\\}}~{}.$
(2)
Here $W_{\sigma}$ is the amplitude of the local potential. We discuss the
system of conserved particle number in the canonical ensemble. The number of
atoms for spin up and spin down is, $N_{\uparrow}$ and $N_{\downarrow}$,
respectively. The polarization is defined as
$p=(N_{\uparrow}-N_{\downarrow})/N_{f}$. The on-site interaction and
$W_{\sigma}$ are scaled in units of $\gamma$ as, $u=U/\gamma$ and
$w_{\sigma}=W_{\sigma}/\gamma$, respectively.
A powerful theoretical tool to investigate the dynamics of many-body systems
in the presence of time-dependent inhomogeneous external potentials, such as
that in Eq. (1), is TDSDFT, Giuliani_and_Vignale ; marques_2006 based on the
Runge-Gross theorem rgt and on the time-dependent single-particle Kohn-Sham
equations. The complication of the problem is hidden in the unknown time-
dependent exchange and correlation (xc) potential. Most applications of TDSDFT
use the simple adiabatic local spin-density approximation for the dynamical xc
potential, Giuliani_and_Vignale ; zangwill which has often been proved to be
successful in studying the real-time evolution. marques_2006 In this
approximation, one assumes that the time-dependent xc potential is just the
static xc potential evaluated at the instantaneous density, where the xc
potential is local in time and space. The static xc potential is then treated
within the static local spin-density approximation. Very attractive features
of the ALSDA are its extreme simplicity, the ease of implementation, and the
fact that it is not restricted to mean-field approximation and small
deviations from the ground-state density, i.e., to the linear response regime.
The dynamics induced by the strong local perturbation discussed here cannot be
dealt with the theory based on the linear response while TDSDFT is a good
candidate.
We here employ a lattice version of spin-density-functional theory (SDFT) and
TDSDFT. gaoprb78Liwei In short, for times $t\leq 0$, the spin-resolved site-
occupation profiles can be calculated by means of a static SDFT. For times
$t>0$, we calculate the time evolution of spin-resolved site-occupation
profiles $n_{i\sigma}(t\leq 0)$ by means of a TDSDFT scheme in which the time-
dependent xc potential is determined exactly at the ALSDA level (details see,
Ref. [gaoprb78Liwei, ]). The performance of this method has been tested
systematically against accurate t-DMRG simulation data for the repulsive
Hubbard model. gaoprb78Liwei It is found that, the simple ALSDA for the time-
dependent xc potential is surprisingly accurate in describing collective
density and spin dynamics in strongly correlated 1D ultracold Fermi gases in a
wide range of coupling strengths and spin polarizations. The performance of
TDSDFT in describing the nonequilibrium behavior of strongly correlated
lattice models has also been recently addressed in Ref. [verdozzi_2007, ].
In this work, we use this method to mainly discuss the nature of the
interactions on the velocities of the density and spin evolution. The spin-
charge dynamics after a local quench is discussed in Luttinger liquids (for
$U>0$, gapless spin and charge excitations) and in Luther-Emery liquids (for
$U<0$, gapless charge and gapful spin excitations). We consider at the same
time the influence of polarization on the spin-charge dynamics. For attractive
interactions, we limit our discussion on the weak-interaction case because for
strong attractive interactions we found our SDFT code overestimates the
amplitude of the bulk atomic density waves, which will greatly influence the
TDSDFT results based on that.
Experimentally the strong local potential can be obtained by a blue- or red-
detuned laser beam tightly focused perpendicular to the 1D atomic wires, which
generates locally repulsive or attractive potentials for the atoms in the
wires, corresponding to $W_{\sigma}>0$ or $W_{\sigma}<0$. In this paper, we
are interested in the repulsive potential for the atoms. The charge and spin
densities can be observed by using in situ sequential absorption imaging,
electron beams, or noise interference, Shin which, in principle, gives an
unambiguous information on the spin-charge separation.
## III Numerical results and discussion
In this section, we report on the results calculated by solving the time-
dependent Kohn-Sham equations. Mathematically the solution of the time-
dependent Kohn-Sham equations is an initial value problem. A given set of
initial orbitals calculated from the static Kohn-Sham equations is propagated
forward in time. No self-consistent iterations are required as in the static
case.
For times $t\leq 0$, the system is in the presence of a strong local
potential, which creates a strong local disturbance in ultracold gases and
makes the total density and spin-density distributions in the center of the
system locally different (up to a few lattice sites). We are interested in two
kinds of quench dynamics. The first one is that, at time $t=0^{+}$, the local
potential is quenched with the time-independent on-site interaction $U(t)=U$.
The second is that, at time $t=0^{+}$, the local potential is switched off and
at the same time the on-site interaction is quenched instantaneously with
$U(t)=U\Theta(-t)$. After the quench, excitations are produced. We concern in
this paper the subsequent real-time evolution of the spin-resolved densities
after the quench,
$n_{i\sigma}(t)=\langle\Psi(t)|\hat{n}_{i\sigma}|\Psi(t)\rangle$ with
$|\Psi(t)\rangle$ the state of the system at time $t$. Charge density and spin
density are defined accordingly as
$n_{i}(t)=n_{i\uparrow}(t)+n_{i\downarrow}(t)$ and
$s_{i}(t)=[n_{i\uparrow}(t)-n_{i\downarrow}(t)]/2$.
If not mentioned otherwise, the numerical results presented below correspond
to a system with $N_{f}=30$ atoms on $N_{s}=100$ sites, and with open (hard
wall) boundary conditions imposed at the sites $i=0$ and $i=101$. The external
potential is chosen to be spin dependent: $w_{\uparrow}=-1$ and
$w_{\downarrow}=0$, used to form a local density and spin density occupations
in the center of the system.
### III.1 $u>0$ and $p=0$
In Fig. 1, we show results for a spin-unpolarized system
($N_{\uparrow}=N_{\downarrow}=15$) with repulsive interaction of $u=2$. At
$t\leq 0$, a dominant local charge- and spin-density profiles in the center of
the system are generated by the strong local potential. After the quench of
the local potential, the charge and spin densities evolve and split into two
counterpropagating density wave packets. The propagation in time is in fact
due to the nonequilibrium initial condition. The charge density evolves with a
quicker velocity than the spin, which is in agreement with the general picture
of spin-charge separation. giamarchi_book A qualitative analysis based on the
continuity equation for the momentum density can also well explain the
phenomena of spin-charge separation. gaoprl102
Figure 1: (Color online) Charge $n_{i}(t)$ and spin $s_{i}(t)$ occupations as
functions of lattice site $i$ and time $t$ for $N_{s}=100$,
$N_{\uparrow}=N_{\downarrow}=15$, $w_{\uparrow}=-1$, $w_{\downarrow}=0$,
$\alpha=2$, and repulsive interaction of $u=+2$. Top panel: ground-state
charge and spin occupations for times $t\leq 0$ (solid line) and at time
$t=5~{}\hbar/\gamma$ (dashed-dotted line). Bottom panel: same as in the top
panels but at time $t=10~{}\hbar/\gamma$ (solid line) and
$t=20~{}\hbar/\gamma$ (dashed-dotted line). The charge and spin densities are
plotted in the top and bottom of the panel, respectively. The arrows in the
plot indicate the positions where the wave packets propagate. In the inset, we
show the velocities of the charge $v_{c}$ (open circles) and spin $v_{s}$
(solid circles) density wave packets as a function of the amplitude of the
local potential $|w_{\uparrow}|$. Both velocities are increasing functions of
$|w_{\uparrow}|$.
We notice a common feature in almost all the figures in this paper, that is,
the spin and charge densities have an asymmetric forward-leaning shape. This
is caused by a nonlinear effect, i.e., the different local velocities in the
center and at the edges. Since the local velocity is proportional to the
density, the higher density in the center gains larger velocity than that at
the edges, which qualitatively explains why the asymmetric forward-leaning
shape happens during the density propagation. For perturbations with small
amplitude, the charge velocity is studied in details by t-DMRG and compared to
the Bethe-ansatz results with good agreement. kollath For the strong local
potential studied here, the spin and charge velocities, determined from the
propagation of the maximum of the charge and spin wave packets away from the
center, vary with time. We thus calculate and compare the velocities
determined at the fixed time $t=10\hbar/\gamma$. In the inset of Fig. 1, we
show the spin and charge velocities as a function of the amplitude of the
local potential $|w_{\uparrow}|$. We find both velocities are increasing
functions of $|w_{\uparrow}|$. For the charge background density ($\sim 0.3$)
in Fig. 1, the charge and spin velocities by the Bethe-ansatz method are
$v_{c}=1.15$ and $v_{s}=0.75$. In the limit of $w_{\uparrow}\rightarrow 0$,
but $w_{\downarrow}\equiv 0$, our results give $v_{c}=1.3$ and $v_{s}=0.65$.
The differences are possibly caused by the simultaneous local perturbations in
the charge and spin densities used here, which break the spin-charge scenario
and couple the spin and charge modes, similar to the effects caused by the
finite spin polarization (see Secs. III-C and III-D).
Figure 2: (Color online) Charge $n_{i}(t)$ and spin $s_{i}(t)$ occupations as
functions of lattice site $i$ and time $t$ with quenches for the local
potential and on-site interaction, i.e., $V_{i\sigma}(t)=V^{\rm
ext}_{i\sigma}\Theta(-t)$ and $U(t)=U\Theta(-t)$. The other parameters are the
same as that in Fig. 1. The static density (solid line) is shown together with
two time shots for $t=5~{}\hbar/\gamma$ (dash line) and $t=10~{}\hbar/\gamma$
(dashed-dotted line).
Figure 3: (Color online) 3D plots for the Charge density $n_{i}(t)$ (Top
panel) and spin density $s_{i}(t)$ (Bottom panel) as functions of lattice site
$i$ and time $t$ (in units of $\hbar/\gamma$) for a harmonically trapped
system with $N_{s}=200$, $N_{\uparrow}=N_{\downarrow}=15$,
$V_{2}/\gamma=5\times 10^{-4}$, $w_{\uparrow}=-1$, $w_{\downarrow}=0$,
$\alpha=2$, and repulsive interaction of $u=2$.
In Fig. 2, we study the local potential quench together with an on-site
interaction quench, i.e., $V_{i\sigma}(t)=V^{\rm ext}_{i\sigma}\Theta(-t)$ and
$U(t)=U\Theta(-t)$. We find that, the spin- and charge-density wave packets
split and counterpropagate as usual but the phenomena of the spin-charge
separation completely disappears. That is, the spin and charge densities
evolve with the same velocity. From the Luttinger-liquid theory based on the
bosonization method Coll or from the Bethe-ansatz solution, Schulz one can
derive that the spin velocity $v_{s}$ and the charge velocity $v_{c}$ satisfy
$v_{c}=v_{s}=v_{F}$ in the noninteracting limit, with $v_{F}=2\gamma\sin(\pi
n/2)$ the Fermi velocity. The interaction between the different species is one
of the important ingredients for the spin-charge separation, which explains
the suppression of the spin-charge separation after the interaction quench.
Making use of the techniques from the cold atomic gases, two different ways of
quenching, used in Figs. 1 and 2, respectively, can give a clear signal that
different collective spin and charge dynamics happens when starting from the
same initial strong local perturbation. We would like to mention that Kollath
proposed to repeat the dynamics in Fig. 1 in higher dimensions where no
separation of spin and charge should be seen. Kollath2 We notice that in Fig.
2 already at short time, some density waves coming from the sharp edges begin
to influence the charge- and spin-density wave packets from the center. At
larger time, they will mix with the original packets.
Figure 4: (Color online) Charge $n_{i}(t)$ and spin $s_{i}(t)$ occupations as
functions of lattice site $i$ and time $t$ for $N_{s}=100$,
$N_{\uparrow}=N_{\downarrow}=15$, $w_{\uparrow}=-1$, $w_{\downarrow}=0$,
$\alpha=2$, and attractive interaction of $u=-1$. Top panel: ground-state
charge and spin occupations for times $t\leq 0$ (solid line) and at time
$t=5~{}\hbar/\gamma$ (dashed-dotted line). Bottom panel: same as in the top
panels but at time $t=10~{}\hbar/\gamma$ (solid line) and
$t=20~{}\hbar/\gamma$ (dashed-dotted line). The inset shows the velocities of
the charge $v_{c}$ (open circles) and spin $v_{s}$ (solid circles) density
wave packets as a function of the amplitude of the local potential
$|w_{\uparrow}|$.
Figure 5: (Color online) Contour plots for the Charge density $n_{i}(t)$ (Top
panel) and spin density $s_{i}(t)$ (Bottom panel) as functions of lattice site
$i$ and time $t$ (in units of $\hbar/\gamma$) for a harmonically trapped
system with $N_{s}=200$, $N_{\uparrow}=N_{\downarrow}=15$,
$V_{2}/\gamma=5\times 10^{-4}$, $w_{\uparrow}=-1$, $w_{\downarrow}=0$,
$\alpha=2$, and attractive interaction of $u=-1$.
In practice, an additional trapping potential is unavoidable in the present
experimental set-ups. We thus present our simulations for the system in the
presence of an additional weak superimposed harmonic trapping potential,
namely, $V^{\rm ext}_{i\sigma}$ in Eq. (2) is changed into,
$\displaystyle V^{\rm
ext}_{i\sigma}=W_{\sigma}e^{-\frac{[i-(N_{s}+1)/2]^{2}}{2\alpha^{2}}}+V_{2}\left(i-\frac{N_{s}+1}{2}\right)^{2}.$
(3)
Here we take $V_{2}/\gamma=5\times 10^{-4}$. The three-dimensional (3D) plots
of the time evolution of the spin- and charge-density wave packets are shown
in Fig. 3. From the figure, we observe that, in the presence of the harmonic
potential the charge and spin wave packets are highly inhomogeneous, but the
counter-propagation and the separation of the charge- and spin-density wave
packets are still visible in the background of the inverted parabola.
### III.2 $u<0$ and $p=0$
In one-dimensional Hubbard model, away from half filling, the spin and charge
velocities of the low-energy collective excitations satisfy, Coll ; Schulz
$v_{s,c}=v_{F}\sqrt{1\mp\frac{U}{\pi v_{F}}}\,.$
This gives a qualitative explanation that for the positive-$U$ Hubbard model,
the charge velocity is larger than the spin velocity, while for the
negative-$U$ Hubbard model, the charge velocity is smaller than the spin
velocity. In Fig. 4, the quench dynamics for the attractive Hubbard model,
which belongs to the Luther-Emery universality class, illustrates that spin-
wave packets evolve with a faster speed than the charge branches. In the inset
of Fig. 4, we show the spin and charge velocities evaluated at
$t=10\hbar/\gamma$ as a function of the amplitude of the local potential
$|w_{\uparrow}|$. We notice that an abrupt change appears in the charge
velocity at $|w_{\uparrow}|\approx 0.55$. For attractive interactions, Luther-
Emery paring induces a prominent density wave characterized by the dip-hump
structure. While the charge velocity is determined from the propagation of the
maximum of the charge wave packets located at one of the humps of the density
wave. The increase in the amplitude of the local potential makes the maximum
of the charge wave packets move from the lattice site $i=44$ to $i=39$, which
explains the discontinuity of the charge velocity for attractive interactions
at $|w_{\uparrow}|\approx 0.55$. However, this discontinuous change has
artifacts because the way of extracting $v_{c,s}$ used here is not an optimum
one. In Fig. 5, we present the contour plots of the time evolution of the
density and spin packets for the system in the presence of a harmonic trapping
potential with $V_{2}/\gamma=5\times 10^{-4}$. The different evolution
velocities for the charge- and spin-density wave packets are clearly visible.
Figure 6: (Color online) The ground-state charge and spin occupations as
functions of lattice site $i$ and time $t$ for the system of repulsive
interaction of $u=2$ in the polarized case of $P=0.47$
($N_{\uparrow}=22,N_{\downarrow}=8$). Besides the ground-state density and
spin density (solid line), three time shots are shown with
$t=5~{}\hbar/\gamma$ (dash line,) $t=10~{}\hbar/\gamma$ (dashed-dotted line),
and $t=15~{}\hbar/\gamma$ (dotted line). Figure 7: (Color online) Same as
Fig. 6 but for the polarized system of $P=0.87$
($N_{\uparrow}=28,N_{\downarrow}=2$). Figure 8: (Color online) The ground-
state charge and spin occupations as functions of lattice site $i$ and time
$t$ for the system of attractive interaction of $u=-1$ in the polarized case
of $p=0.47$ ($N_{\uparrow}=22,N_{\downarrow}=8$). Besides the ground-state
density and spin density (solid line), three time shots are shown with
$t=5~{}\hbar/\gamma$ (dash line,) $t=10~{}\hbar/\gamma$ (dashed-dotted line),
and $t=15~{}\hbar/\gamma$ (dotted line). Figure 9: (Color online) Same as
Fig. 8, but for the polarized system of $p=0.87$
($N_{\uparrow}=28,N_{\downarrow}=2$).
### III.3 $u>0$ and $p\neq 0$
The spin-charge separation in a spin-polarized one-dimensional system is quite
different from the fully polarized one. The spin-charge-coupled dynamics in a
polarized system formulated with the first-quantized path-integral formalism
and bosonization techniques provides us a new non-Tomanaga-Luttinger-liquid
universality class. Akhanjee For the Luther-Emery liquid of unpolarized
attractive Fermi gases, the spin and charge degrees of freedom are decoupled.
In contrast, in the system with finite spin imbalance, spin-charge mixing is
found based on an effective-field theory for the long-wavelength and low-
energy properties. Erhai In Figs. 6 and 7, the quench dynamics for spin- and
charge-density waves is shown for the system of repulsive interaction ($u=2$)
with polarization of $p=0.47$ and $0.87$, respectively. For $p\geq 0.47$,
there is only small difference between spin and charge velocities. In the case
of a large polarization, the same propagating velocities for spin and charge
are obtained.
### III.4 $u<0$ and $p\neq 0$
The quench dynamics for spin and charge density waves of the attractive case
for $u=-1$ is shown in Figs. 8 and 9. We find with the increasing of the
polarization, the spin-charge separation is strongly suppressed due to the
interplay between charge and spin degrees of freedom. Theoretically, for the
partially polarized system, the spin and charge modes are coupled. In this
case, there is no strict spin-charge separation scenario, namely, the spin-
charge separation breaks down. Numerically, we observe that, at small
polarization the spin and charge wave packets still evolve at different
velocities although they are coupled and influence each other. At large
polarization, the spin-charge separation disappears and evolves at the same
velocities for both the repulsive and the attractive systems we studied.
## IV Conclusions
In summary, we have calculated the non-equilibrium dynamic evolution of a one-
dimensional system of two-component fermionic atoms after a strong local
quench with or without interaction quench by using a time-dependent density-
functional theory with a suitable Bethe-ansatz based adiabatic local spin-
density approximation. A test of the performance of TDSDFT is provided for the
unpolarized systems with attractive or repulsive interactions in the presence
of a harmonic trapping potential. Under the same local perturbation, the
charge velocity is larger than the spin velocity for the system of repulsive
interaction and vice versa for the attractive case, which is compatible with
the low-energy collective dynamics from the Bethe-ansatz solution or the
bosonization techniques. We found the spin-charge separation is strongly
suppressed when the interaction quench is forced together with the local
potential quench. Spin-charge mixing is found for the system of polarization
signaling by the disappearance of the spin-charge separation. Numerically we
observe that the spin-charge separation disappears for large polarizations in
both the repulsive and the attractive Hubbard model we studied.
###### Acknowledgements.
This work was supported by NSF of China under Grants No. 10974181 and No.
10704066, Qianjiang River Fellow Fund 2008R10029, Program for Innovative
Research Team in Zhejiang Normal University, and partly by the Project of
Knowledge Innovation Program (PKIP) of Chinese Academy of Sciences under Grant
No. KJCX2.YW.W10.
## References
* (1) The Many-Body Problem: An Encyclopedia of Exactly Solved Models in One Dimension, edited by D. C. Mattis (World Scientific, Singapore, 1995), p. 845.
* (2) M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, Phys. Rev. Lett. 76, 4508 (1996).
* (3) T. Kinoshita, T. Wenger, and D. S. Weiss, Science 305, 1125 (2004).
* (4) J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clément, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature (London) 453, 891 (2008); G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
* (5) A. J. Daley, C. Kollath, U. Schollwöck, and G. Vidal, J. Stat. Mech. 2004, P04005; S. R. White, Phys. Rev. Lett. 69, 2863 (1992); U. Schollwöck and S. R. White, in Effective Models for Low-Dimensional Strongly Correlated Systems, edited by G. Batrouni and D. Poilblanc (AIP, Melville, NY, 2006), p. 155.
* (6) T. A. Costi, Phys. Rev. B 55, 3003 (1997); F. B. Anders and A. Schiller, Phys. Rev. Lett. 95, 196801 (2005).
* (7) M. Eckstein, M. Kollar, and P. Werner, Phys. Rev. Lett. 103, 056403 (2009).
* (8) G. Vidal, Phys. Rev. Lett. 91, 147902 (2003); Y.-Y. Shi, L.-M. Duan, and G. Vidal, Phys. Rev. A. 74, 022320 (2006),
* (9) C. Verdozzi, Phys. Rev. Lett. 101, 166401 (2008); M. Dzierzawa, U. Eckern, S. Schenk, and P. Schwab, Phys. Status Solidi B 246, 941 (2009).
* (10) W. Li, G. Xianlong, C. Kollath, and M. Polini, Phys. Rev. B 78, 195109 (2008).
* (11) H. Moritz, T. Stöferle, K. Güenter, M. Köhl, and T. Esslinger, Phys. Rev. Lett. 94, 210401 (2005); T. Stöferle, H. Moritz, C. Schori, M. Köhl, and T. Esslinger, Phys. Rev. Lett. 92, 130403 (2004).
* (12) B. Paredes, A. Widera, V. Murg, O. Mandel, S. Fölling, I. Cirac, G. V. Shlyapnikov, T. W. Hänsch, and I. Bloch, Nature 429, 277 (2004).
* (13) M. Gaudin, Phys. Lett. 24A, 55 (1967); C. N. Yang, Phys. Rev. Lett. 19, 1312 (1967); E. H. Lieb and F. Y. Wu, Phys. Rev. Lett 20, 1445 (1968).
* (14) B. Sutherland, Beautiful Models: 70 Years of Exactly Solved Quantum Many-body Problems (World Scientific Press,Singapore, 2004), p. 7.
* (15) O. M. Auslaender, A. Yacoby, R. de Picciotto, K. W. Baldwin, L. N. Pfeiffer, and K. W. West, Science 295, 825 (2002); O. M. Auslaender, H. Steinberg, A. Yacoby, Y. Tserkovnyak, B. I. Halperin, K. W. Baldwin, L. N. Pfeiffer, and K. W. West, Science 308, 88 (2005).
* (16) A. Recati, P. O. Fedichev, W. Zwerger, and P. Zoller, Phys. Rev. Lett. 90, 020401 (2003); J. Opt. B: Quantum Semiclassical Opt. 5, S55 (2003).
* (17) C. Kollath, U. Schollwöck, and W. Zwerger, Phys. Rev. Lett. 95, 176401 (2005); C. Kollath and U. Schollwöck, New J. Phys. 8, 220 (2006); A. Kleine, C. Kollath, I. P. McCulloch, T. Giamarchi, and U. Schollwöck, New J Phys 10, 045025 (2008).
* (18) C. Kollath, J. Phys. B 39, S65 (2006); C. Lavalle, M. Arikawa, S. Capponi, and A. Muramatsu, Proceedings of NIC Symposium 2004 NIC Series, 2003 (unpublished), Vol 20, p. 281.
* (19) A. Kleine, C. Kollath, I. P. McCulloch, T. Giamarchi, and U. Schollwöck, Phys. Rev. A77, 013607 (2008).
* (20) L. Kecke, H. Grabert, and W. Häusler, Phys. Rev. Lett. 94, 176802 (2005).
* (21) E. A. Jagla, K. Hallberg, and C. A. Balseiro, Phys. Rev. B 47, 5849 (1993).
* (22) M. G. Zacher, E. Arrigoni, W. Hanke, and J. R. Schrieffer, Phys. Rev. B 57, 6370 (1998).
* (23) A. E. Feiguin and D. A. Huse, Phys. Rev. B 79, 100507(R) (2009).
* (24) T. Ulbricht and P. Schmitteckert, EPL 86, 57006 (2009).
* (25) P. Calabrese and J. Cardy, Phys. Rev. Lett. 96 136801 (2006); M. A. Cazalilla, Phys. Rev. Lett. 97, 156403 (2006); S. R. Manmana, S. Wessel, R. M. Noack, and A. Muramatsu, Phys. Rev. Lett. 98, 210405 (2007).
* (26) D. Karlsson, C. Verdozzi, M. Odashima, and K. Capelle, arXiv:0905.1398; F. Heidrich-Meisner, M. Rigol, A. Muramatsu, A. E. Feiguin, and E. Dagotto, Phys. Rev. A 78, 013620 (2008); F. Heidrich-Meisner, S. R. Manmana, M. Rigol, A. Muramatsu, A. E. Feiguin, and E. Dagotto, Phys. Rev. A 80, 041603(R) (2009).
* (27) P. Barmettler, M. Punk, V. Gritsev, E. Demler, and E. Altman, Phys. Rev. Lett. 102, 130603 (2009).
* (28) M. Rigol, Phys. Rev. Lett. 103, 100403 (2009); M. Rigol, V. Dunjko and M. Olshanii, Nature (London) 452, 854 (2008).
* (29) S. Langer, F. Heidrich-Meisner, J. Gemmer, I. P. McCulloch, and U. Schollwöck, Phys. Rev. B 79, 214409 (2009).
* (30) G.F. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid (Cambridge University Press, Cambridge, 2005); G. Vignale and W. Kohn, in Electronic Density Functional Theory, edited by J. Dobson, M. K. Das, and G. Vignale (Plenum Press, New York, 1996).
* (31) Time-Dependent Density Functional Theory, Lecture Notes in Physics, edited by M.A.L. Marques, F. Noguiera, A. Rubio, K. Burke, C.A. Ullrich, and E.K.U. Gross (Springer, Berlin, 2006), Vol. 706.
* (32) E. Runge and E.K.U. Gross, Phys. Rev. Lett. 52, 997 (1984); R. van Leeuwen, Phys. Rev. Lett. 82, 3863 (1999).
* (33) A. Zangwill and P. Soven, Phys. Rev. Lett. 45, 204 (1980); Phys. Rev. B24, 4121 (1981).
* (34) Y. Shin, C. Schunck, A. Schirotzek, and W. Ketterle, Nature 451, 689 (2008).
* (35) T. Giamarchi, Quantum Physics in One Dimension (Clarendon Press, Oxford, 2004).
* (36) G. Xianlong, M. Polini, D. Rainis, M. P. Tosi, and G. Vignale, Phys. Rev. Lett. 101, 206402 (2008); M. Polini and G. Vignale, Phys. Rev. Lett. 98, 266403 (2007).
* (37) C. F. Coll, Phys. Rev. B 9, 2150 (1974).
* (38) H. J. Schulz, Int. J. Mod. Phys. B 5, 57 (1991).
* (39) S. Akhanjee and Y. Tserkovnyak, Phys. Rev. B 76, 140408(R) (2007); A. Imambekov and L. I. Glazman, Science 323, 228 (2009).
* (40) E. Zhao and W. V. Liu, Phys. Rev. A 78, 063605 (2008); E. Zhao, X.-W. Guan, W. V. Liu, M. T. Batchelor, and M. Oshikawa, Phys. Rev. Lett. 103, 140404(2009); S. Rabello and Q. Si, Europhys. Lett. 60, 882 (2002); T. Vekua, S. I. Matveenko, and G. V. Shlyapnikov, JETP Lett. 90, 289 (2009); M. Rizzi, M. Polini, M. A. Cazalilla, M. R. Bakhtiari, M. P. Tosi, and R. Fazio, Phys. Rev. B 77, 245105 (2008).
|
arxiv-papers
| 2011-01-18T03:01:48 |
2024-09-04T02:49:16.513969
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gao Xianlong",
"submitter": "Gao Xianlong",
"url": "https://arxiv.org/abs/1101.3373"
}
|
1101.3398
|
11institutetext: National Key Lab. of ISN, Xidian University , Xi’an 710071,
P.R.China
11email: wp_ma@mail.xidian.edu.cn
# New Quadriphase Sequences families with Larger Linear Span and Size
Wenping Ma
###### Abstract
In this paper, new families of quadriphase sequences with larger linear span
and size have been proposed and studied. In particular, a new family of
quadriphase sequences of period $2^{n}-1$ for a positive integer $n=em$ with
an even positive factor $m$ is presented, the cross-correlation function among
these sequences has been explicitly calculated. Another new family of
quadriphase sequences of period $2(2^{n}-1)$ for a positive integer $n=em$
with an even positive factor $m$ is also presented, a detailed analysis of the
cross-correlation function of proposed sequences has also been presented.
## 1 Introduction
Family of pseudorandom sequences with low cross correlaton and large linear
span has important application in code-division multiple access communications
and cryptology. Quadriphase sequences are the one most often used in practice
because of their easy implementation in modulators. However, up to now, only
few families of optimal quadriphase sequences are found [1],[5, 6, 7, 8, 9,
10, 11, 12].
Among the known optimal quadriphase sequence families, the most famous ones
are the families $\mathcal{A}$ and $\mathcal{B}$ investigated by Boztas,
Hammons, and Kummar in[5]. The family $\mathcal{A}$ has period $2^{n}-1$ and
family size $2^{n}+1$, while the two corresponding parameters of the family
$\mathcal{B}$ are $2(2^{n}-1)$ and $2^{n-1}$, respectively. Another optimal
family $\mathcal{C}$ was discussed in [10], and this family has the same
correlation properties as the family $\mathcal{B}$. Families $S(t)$ were
defined by Kumar et.[6], and when $t=0$ or $m$ is odd, the correlation
distributions of families $S(t)$ are established by Kai-Uwe Schmidt[7]. Tang,
Udaya, and Fan generalized the family $\mathcal{A}$ and proposed a new family
of quadriphase sequences with low correlation in [12]. By utilizing a
variation of family $\mathcal{B}$ and $\mathcal{C}$, Tang and Udaya obtained
the family $\mathcal{D}$, which has period $2(2^{n}-1)$ and a larger family
size $2^{n}$[9]. Recently, Wenfeng Jiang, Lei Hu, Xiaohu Tang, and Xiangyong
Zeng proposed two new families $\mathcal{S}$ and $\mathcal{U}$ of quadriphase
sequences with larger linear spans for a positive integer $n=em$ with an odd
positive factor $m$. Both families are asymptotically optimal with respect to
the Wech and Sidelnikov bounds. The family $\mathcal{S}$ has period $2^{n}-1$,
family size $2^{n}+1$, and maximum correlation magnitude $2^{\frac{n}{2}}+1$.
The family $\mathcal{U}$ has period $2(2^{n}-1)$, family size $2^{n}$, and
maximum correlation magnitude $2^{\frac{n+1}{2}}+2$ [1].
In this paper, motivated by the constructions proposed in [1, 2, 3, 4, 6], the
new families of quadriphase sequences with larger linear span and size have
been presented. As a special case of the sequence families, a new family of
quadriphase sequences of period $2^{n}-1$ for a positive integer $n=em$ with
an even positive factor $m$ is presented, the cross-correlation function among
these sequences has been explicitly calculated. Another new family of
quadriphase sequences of period $2(2^{n}-1)$ for a positive integer $n=em$
with an even positive factor $m$ is also presented, a detailed analysis of the
cross-correlation function of proposed sequences has been presented. The
sequences have low correlations and are useful in code division multiple
access communication systems and cryptography.
This paper is organized as follows. Section 2 introduces the preliminaries and
notations. In section 3, we give the constructions and properties of the new
sequences families $\mathcal{L}$ and $\mathcal{V}$ with period $2^{n}-1$. The
constructions and correlation properties of the new sequences family
$\mathcal{W}$ with period $2(2^{n}-1)$ are presented in section 4. The
conclusions and acknowledgement are presented in section 5 and 6 respectively.
## 2 Preliminaries
### 2.1 Basic Concepts
Let $a=\\{a(t)\\}$ and $b=\\{b(t)\\}$ be two quadriphase sequences of period
$L$, the correlation function $R_{a,b}(\tau)$ between them at a shift
$0\leq\tau\leq L-1$ is defined by
$R_{a,b}(\tau)=\sum_{t=0}^{L-1}\omega^{a(t)-b(t+\tau)}$
where $\omega^{2}=-1$.
Let $\mathcal{F}$ be a family of $M$ quadriphase sequences
$\mathcal{F}=\\{a_{i}=\\{a(t)\\}:1\leq i\leq M\\}.$
The maximum correlation magnitude $R_{max}$ of $\mathcal{F}$ is
$R_{max}=\max\\{|R_{a_{i},a_{j}}(\tau)|:1\leq i,j\leq M,\ \ i\neq j\ or\
\tau\neq 0\\}.$
### 2.2 Galois Ring
Let $Z_{4}[x]$ be the ring of all polynomials over $Z_{4}$ . A monic
polynomial $f(x)\in Z_{4}[x]$ is said to basic primitive if its projection
$\overline{f(x)}$
$\overline{f(x)}=f(x)\ mod\ 2$
is primitive over $Z_{2}[x].$
Let $f(x)$ be a basic primitive polynomial of degree $n$ over $Z_{4}$, and
$Z_{4}[x]/(f(x))$ denotes the ring of residue classes of polynomials over
$Z_{4}$ modulo $f(x)$. It can be shown that this quotient ring is a
commutative ring with identity called Galois ring, denoted as $GR(4,n)$[11].
As a multiplicative group, the units $GR^{*}(4,n)$ have the following
structure:
$GR^{*}(4,n)=G_{A}\otimes G_{C}$
where $G_{C}$ is a cyclic group of order $2^{n}-1$ and $G_{A}$ is an Abelian
group of order $2^{n}$. Naturally, the projection map $\overline{a}$ from
$Z_{4}$ to $Z_{2}$ induces a homomorphism from $GR(4,n)$ to finite field
$GF(2^{n})$.
Let $\beta\in GR^{*}(4,n)$ be a generator of the cyclic group $G_{C}$ , then
$\alpha=\overline{\beta}$ is a primitive root of $GF(2^{n})$ with primitive
polynomial $\overline{f(x)}$ over $Z_{2}$.
For each element $x\in GR(4,n)$ has a unique $2-adic$ representation of the
form
$x=x_{0}+2x_{1},x_{0},x_{1}\in G_{C}.$ (1)
Let $n=em$, The $Frobenius$ automorphism of $GR(4,n)$ over $GR(4,e)$ is given
by
$\sigma(x)=x_{0}^{2^{e}}+2x_{1}^{2^{e}}$
for any element $x$ expressed as (1), and the trace function $Tr_{e}^{n}$ from
$GR(4,n)$ to $GR(4,e)$ is defined by
$Tr^{n}_{e}(x)=x+\sigma(x)+\sigma^{2}(x)+\cdots+\sigma^{m-1}(x)$
where $\sigma^{i}(x)=\sigma^{i-1}(\sigma(x))$ for $1<i\leq m-1$.
Let $GF(q)$ is the finite field with $q$ elements, $tr^{n}_{e}(x)$ is the
trace function from $GF(2^{n})$ to $GF(2^{e})$, i.e.,
$tr_{e}^{n}(x)=x+x^{2^{e}}+\cdots=x^{2^{e(\frac{n}{e}-1)}},x\in GF(2^{n}).$
We have $\overline{Tr_{e}^{n}(x)}=tr_{e}^{n}(\overline{x})$, where $x\in
GR(4,n)$.
Throughout this paper, we suppose (1) $n=em$ with $e\geq 2\ and\ m\geq 2$,(2)
$\lambda\in GR(4,e)$ such that $\overline{\lambda}\in
GF(2^{e})\setminus\\{1,0\\}$.
### 2.3 Linear Span
Let
$f(x)=Tr_{1}^{n}[(1+2\alpha)x]+2\sum_{i=1}^{r}Tr_{1}^{n_{i}}(A_{i}x^{v_{i}}),\alpha\in
G_{C},A_{i}\in GF(2^{n_{i}}),x\in G_{C},$
where $v_{i}$ is a coset leader of a cyclotomic coset modulo $2^{n_{i}}-1$,
and $n_{i}|n$ is the size of the cyclotomic coset containing $v_{i}$. For
sequence $a=\\{a_{i}\\}$ such that
$a_{i}=f(\beta),i=0,1,2,\cdots$
where $\beta$ is a primitive element of $G_{C}$.
Linear span of a sequence $a$ is equal to $n+\sum_{i,A_{i}\neq 0}n_{i}$ , or
equivalently, the degree of the shortest linear feedback shift register that
can generates $a$ [1, 4].
## 3 New Quadriphase Sequences with Larger Size and Linear Span
Define a function $P(x)$ over $GR(4,n)$ as
$P(x)=\left\\{\begin{array}[]{ll}$$\sum_{j=1}^{l-1}Tr_{1}^{n}(x^{2^{ej}+1})+Tr_{1}^{le}(x^{2^{le}+1}),if\
m=2l$$,\\\ \\\ $$\sum_{j=1}^{l}Tr_{1}^{n}(x^{2^{ej}+1}),if\
m=2l+1.$$\end{array}\right.$
For any $x,y\in GR(4,n)$, it is easy to check that[1, 2, 3]
$\begin{array}[]{lll}2P(x)+2P(y)+2P(x+y)=2Tr_{1}^{n}[y(x+Tr^{n}_{e}(x))].\end{array}$
(2)
###### Definition 1
Let $\rho$ be an integer such that
$1\leq\rho<\displaystyle{\lfloor\frac{n}{2}\rfloor}$ , a family of quaternary
sequences of period $2^{n}-1$, $\mathcal{L}=\\{s_{i}(t):0\leq t<2^{n}-1,1\leq
i\leq 2^{\rho n}+1\\}$ is defined by
$s_{i}(t)=\left\\{\begin{array}[]{ll}$$Tr^{n}_{1}[(1+2\lambda_{0}^{i})\beta^{t}]+2\sum_{k=1}^{\rho-1}Tr^{n}_{1}(\lambda_{k}^{i}\beta^{t(1+2^{k})})+2P(\lambda\beta^{t}),1\leq
i\leq 2^{\rho n},$$\\\ \\\ $$2Tr^{n}_{1}(\beta^{t}),i=2^{\rho
n}+1$$\end{array}\right.$ where
$\\{(\lambda_{0}^{i},\lambda_{1}^{i},\cdots,\lambda_{\rho-1}^{i}),i=1,2,\cdots,2^{\rho
n}\\}$ is an enumeration of the elements of $G_{C}\times
G_{C}\times\cdots\times G_{C}$, $\beta$ is a generator element of group
$G_{C}$.
###### Lemma 1
All sequence in $\mathcal{L}$ are cyclically distinct. Thus, the family size
of $\mathcal{L}$ is $2^{n\rho}+1$ .
###### Proof
The proof of lemma 1 is similar to the proofs of Lemma 1 and Lemma 6 in [4],
we cancel the details.
### 3.1 The Correlation Function of the Sequence Family
(1) Suppose $s_{i},s_{j}$,$1\leq i,j\leq 2^{\rho n}$, are two sequences, the
correlation function between $s_{i}$ and $s_{j}$ is
$R_{s_{i},s_{j}}(\tau)=\sum_{x\in
G_{C}}\omega^{Tr^{n}_{1}[(1+2\gamma^{i}_{0}-(1+2\gamma^{j}_{0})\delta)x]+2\sum_{k=1}^{\rho-1}Tr_{1}^{n}(\eta_{k}x^{1+2^{k}})+2(P(\lambda
x)+P(\lambda\delta x))}-1$ (3)
where $\delta=\beta^{\tau}$, $\tau\neq 0$,
$\lambda_{k}^{i}-\delta^{1+2^{k}}\lambda^{j}_{k}=\eta_{k}$,$k=1,2,\cdots,\rho-1$.
$\displaystyle(R_{s_{i},s_{j}}(\tau)+1)(R_{s_{i},s_{j}}(\tau)+1)^{*}$
$\displaystyle=\sum_{s\in G_{C}}\sum_{y\in
G_{C}}\omega^{Tr_{1}^{n}[(1+2\gamma_{0}^{i}-(1+2\gamma_{0}^{j})\delta)x+2\sum_{k=1}^{\rho-1}Tr_{1}^{n}(\eta_{k}x^{1+2^{k}})+2(P(\lambda
x)+P(\lambda\delta x))]}$ $\displaystyle\verb+
+\cdot\omega^{-Tr_{1}^{n}[(1+2\gamma_{0}^{i}-(1+2\gamma_{0}^{j})\delta)y+2\sum_{k=1}^{\rho-1}Tr_{1}^{n}(\eta_{k}y^{1+2^{k}})+2(P(\lambda
y)+P(\lambda\delta y))]}$ $\displaystyle=\sum_{x\in G_{C}}\sum_{y\in
G_{C}}\omega^{Tr_{1}^{n}((1+2\gamma_{0}^{i}-(1+2\gamma_{0}^{j})\delta)(x+3y))}$
$\displaystyle\verb+
+\cdot\omega^{2[\sum_{k=1}^{\rho-1}Tr_{1}^{n}[\eta_{k}(x^{1+2^{k}}+y^{1+2^{k}})]+P(\lambda
x)+P(\lambda\delta x)+P(\lambda y)+P(\lambda\delta y)]}$
$\displaystyle=\sum_{x\in G_{C}}\sum_{y\in
G_{C}}\omega^{Tr_{1}^{n}((\Delta(x+3y))+2(\sum_{k=1}^{\rho-1}Tr_{1}^{n}[\eta_{k}(x^{1+2^{k}}+y^{1+2^{k}})]+P(\lambda
x)+P(\lambda\delta x)+P(\lambda y)+P(\lambda\delta y))}$
$\displaystyle=\sum_{z\in G_{C}}\sum_{y\in G_{C}}\omega^{Tr_{1}^{n}(\Delta
z)+2[\sum_{k=1}^{\rho-1}Tr_{1}^{n}[\eta_{k}((y+z+2\sqrt{yz})^{1+2^{k}}+y^{1+2^{k}})]+2Tr_{1}^{n}(\Delta\sqrt{yz})}$
$\displaystyle\verb+
+\cdot\omega^{2P(\lambda(y+z+2\sqrt{yz}))+2P(\lambda\delta(y+z+2\sqrt{yz}))+2P(\lambda
y)+2P(\lambda\delta y)]}$ $\displaystyle=\sum_{z\in
G_{C}}\omega^{\phi(z)}\sum_{y\in
G_{C}}\omega^{2[Tr_{1}^{n}(y(\Delta^{2}z))+v(y,z)]}$
where $\Delta=1+2\gamma_{0}^{i}-(1+2\gamma_{0}^{j})\delta$,
$x=y+z+2\sqrt{yz}$,
$\phi(z)=Tr_{1}^{n}(\Delta z)+2[P(\lambda z)+2P(\lambda\delta
z)]+2\sum_{k=1}^{\rho-1}Tr^{n}_{1}(\eta_{k}z^{1+2^{k}})$,
$v(y,z)=\sum_{k=1}^{\rho-1}Tr_{1}^{n}[\eta_{k}((y+z+2\sqrt{yz})^{1+2^{k}}+y^{1+2^{k}}+z^{1+2^{k}})]+P(\lambda(y+z+$
$2\sqrt{yz}))+P(\lambda\delta(y+z+2\sqrt{yz}))+P(\lambda y)+P(\lambda\delta
z)+P(\lambda z)+P(\lambda\delta z).$
Then, by (2), we have
$2v(y,z)=2Tr_{1}^{n}[\lambda y(\lambda z+Tr^{n}_{e}(\lambda z))+\lambda\delta
y(\lambda\delta z+Tr^{n}_{e}(\lambda\delta
z))]+2Tr_{1}^{n}\sum_{k=1}^{\rho-1}[\eta_{k}(zy^{2^{k}}+z^{2^{k}}y)]$
$=2Tr^{n}_{1}[y(\lambda^{2}z+\lambda Tr_{e}^{n}(\lambda
z)+\lambda^{2}\delta^{2}z+\lambda\delta Tr^{n}_{e}(\lambda\delta
z)+\sum_{k=1}^{\rho-1}(\eta_{k}^{-2^{k}}z^{-2^{k}}+\eta_{k}z^{2^{k}}))].$
Define
$L(z)=\overline{\delta}tr^{n}_{e}(\overline{\delta}z)+tr^{n}_{e}(z)+\frac{1}{\overline{\lambda}^{2}}(\overline{\lambda}^{2}+1)(\overline{\delta}^{2}+1)z+\frac{1}{\overline{\lambda}^{2}}\sum_{k=1}^{\rho-1}(\overline{\eta}^{-2^{k}}_{k}z^{-2^{k}}+\overline{\eta}_{k}z^{2^{k}}),$
(4)
where $z\in GF(2^{n})$, then $L(z)$ is a linear equation over $GF(2^{n})$.
For $L(z)=0$, we have to count the number of solutions in the equation
$\overline{\delta}tr^{n}_{e}(\overline{\delta}z)+tr^{n}_{e}(z)+\frac{1}{\overline{\lambda}^{2}}(\overline{\lambda}^{2}+1)(\overline{\delta}^{2}+1)z+\frac{1}{\overline{\lambda}^{2}}\sum_{k=1}^{\rho-1}(\overline{\eta}^{-2^{k}}_{k}z^{-2^{k}}+\overline{\eta}_{k}z^{2^{k}})=0$
(5)
for given $\eta_{i}$’s in $G_{C}$,$\lambda\in G_{C}$ such that
$\overline{\lambda}\in GF(2^{e})\backslash\\{0,1\\}$, and
$\overline{\delta}\in GF(2^{n})\backslash\\{0\\}$.
It is easy to verify that
$\frac{1}{\overline{\lambda}^{2}}(\overline{\lambda}^{2}+1)(\overline{\delta}^{2}+1)z+\frac{1}{\overline{\lambda}^{2}}\sum_{k=1}^{\rho-1}(\overline{\eta}^{-2^{k}}_{k}z^{-2^{k}}+\overline{\eta}_{k}z^{2^{k}})$
is not a constant polynomial of z, and the maximum number of solutions of
equation $L(z)=0$ is at most $2^{2(\rho-1)+2e}$. Thus
$|R_{s_{i},s_{j}}(\tau)+1|\leq 2^{\frac{n+2(\rho-1)+2e}{2}}.$ (6)
(2) If $1\leq i\leq 2^{\rho n}$ and $j=2^{n\rho}+1$ are two sequences, then
the correlation function between $s_{i}$ and $s_{j}$ is
$R_{s_{i},s_{j}(\tau)}=\sum_{x\in
G_{C}}\omega^{Tr_{1}^{n}[(1+2\gamma_{1}-2\delta)x]+2\sum_{k=1}^{\rho-1}Tr_{1}^{n}(\lambda_{k}^{i}x^{1+2^{k}})+2P(\lambda
x)}-1,$
similar to analysis above, the equation (4) become the following equation.
$L(z)=\frac{1+\bar{\lambda}^{2}}{\bar{\lambda}^{2}}z+tr^{n}_{e}(z)+\frac{1}{\overline{\lambda}^{2}}\sum_{k=1}^{\rho-1}((\overline{\lambda}_{k}^{i})^{-2^{k}}z^{-2^{k}}+\overline{\lambda}^{i}_{k}z^{2^{k}}).$
Thus
$|R_{s_{i},s_{j}}(\tau)+1|\leq 2^{\frac{n+2(\rho-1)+e}{2}}.$ (7)
(3) If $i=j=2^{\rho n}+1$, then $s_{i}$ is essentially a binary $m-$sequence,
Then $R_{s_{i},s_{j}}(\tau)=-1$ for $\tau\neq 0$.
(4) Suppose that $s_{i}$,$s_{j}$, $1\leq i,j\leq 2^{\rho n}$, are two
sequences, then
$R_{s_{i},s_{j}}(0)=\sum_{x\in
T}\omega^{2Tr_{1}^{n}[(\gamma^{i}_{0}+\gamma^{j}_{0})x]+2\sum^{\rho-1}_{k=1}Tr_{1}^{n}(\eta_{k}x^{1+2^{k}})}-1,$
similar to the analysis above, we have
$|R_{s_{i},s_{j}(0)}+1|\leq 2^{\frac{n+2(\rho-1)}{2}}.$ (8)
It seems difficult to get tighter bound for inequality(6)-(8), thus we propose
the following open problem.
Open problem:For $n=em$ , how many solutions exist exactly for the equation
(5) over finite field $GF(2^{n})$.
Following the discussion above, we have the following theorem.
###### Theorem 3.1
For $n=em$ and an integer $\rho$ such that
$1\leq\rho<\displaystyle{\lfloor\frac{n}{2}\rfloor}$, the proposed quadriphase
family has $2^{n\rho}+1$ cyclically distinct binary sequences of period
$2^{n}-1$. The maximum correlation magnitude of sequences is smaller than
$1+2^{\frac{n+2(\rho-1)+2e}{2}}$. Therefore, the sequences family constitutes
a $(2^{n}-1,2^{n\rho}+1,1+2^{\frac{n+2(\rho-1)+2e}{2}})$ quadriphase signal
set.
### 3.2 Linear Spans of the Sequence
In order to express clearly, let
$s(\lambda_{0},\Lambda,t)=Tr^{n}_{1}[(1+2\lambda_{0})\beta^{t}]+2\sum_{k=1}^{\rho-1}Tr^{n}_{1}(\lambda_{k}\beta^{t(1+2^{k})})+2P(\lambda\beta^{t})$,
where $\Lambda=(\lambda_{1},\cdots,\lambda_{\rho-1})$.
We divide the set $\Delta=\\{1,2,\cdots,\rho-1\\}$ into two sets $A$ and $B$
such that $\Delta=A\bigcup B$, where $A=\\{ke+r:1\leq
k\leq\displaystyle{\lfloor\frac{\rho-1}{e}\rfloor,0<r<e}\\}$, $B=\\{ke:1\leq
k\leq\displaystyle{\lfloor\frac{\rho-1}{e}\rfloor}\\}$.
###### Theorem 3.2
(1)Consider a sequence represented by $s(\lambda_{0},\Lambda,t)$ where $j\
\lambda_{i}$’s with $i\in A$ in
$\Lambda=(\lambda_{1},\cdots,\lambda_{\rho-1})$ are equal to 0 and $l\
\lambda_{i}$’s with $i\in B$ in
$\Lambda=(\lambda_{1},\cdots,\lambda_{\rho-1})$ are equal to
$\overline{\lambda}$.Let $LS_{j,l}(\rho)$ be the linear span of the
sequence.Then
$LS_{j,l}(\rho)=n(\frac{m-1}{2}+\rho-1-\lfloor\frac{\rho-1}{e}\rfloor+1-j-l),0\leq
j\leq|A|,0\leq l\leq|B|.$
and there are $(^{\rho-1-\lfloor\frac{\rho-1}{e}\rfloor}_{\verb+ +j})(_{\verb+
+l}^{\lfloor\frac{\rho-1}{e}\rfloor})2^{n}(2^{n}-1)^{\rho-1-j-l}$ sequences
having linear span $LS_{j,l}(\rho)$.
(2) The linear span of the sequences $s_{2^{n\rho}+1}(t)$ is $n$.
###### Proof
First, consider the linear span of sequences with $m$ is odd. A sequence
constructed above has a total of
$\displaystyle{\frac{m-1}{2}+\rho-1-\lfloor\frac{\rho-1}{e}\rfloor+1}$ trace
terms and each trace term has the linear span of $n$. If $j\ \lambda$’s with
$i\in A$ in $\Lambda=(\lambda_{1},\cdots,\lambda_{\rho-1})$ are equal to 0,
and $l\ \lambda_{i}$’s with $i\in B$ in
$\Lambda=(\lambda_{1},\cdots,\lambda_{\rho-1})$ are equal to
$\overline{\lambda}^{2^{i}+1}$, it has
$\displaystyle{\frac{m-1}{2}+\rho-1-\lfloor\frac{\rho-1}{e}\rfloor+1-j-l}$
nonzero trace terms and the corresponding linear span of the sequences is
given by
$LS_{j,l}(\rho)=n(\frac{m-1}{2}+\rho-1-\lfloor\frac{\rho-1}{e}\rfloor+1-j-l),0\leq
j\leq|A|,0\leq l\leq|B|.$
Since (1) $j\ \lambda_{i}$’s with $i\in A$ are 0 and $(|A|-j)\ \lambda$ ’s are
nonzero, (2) $l\ \lambda_{i}$’s with $i\in B$ are
$\overline{\lambda}^{2^{i}+1}$ and $(|B|-l)\ \lambda_{i}$’s are not equal to
$\overline{\lambda}^{2^{i}+1}$, (3) the number of $\lambda_{0}$ is $2^{n}$.
Therefore, the number of corresponding sequences given above is
$\displaystyle(_{\verb+
+j}^{\rho-1-\lfloor\frac{\rho-1}{e}\rfloor})(2^{n}-1)^{\rho-1-\lfloor\frac{\rho-1}{e}\rfloor-j}\cdot(_{\verb+
+l}^{\lfloor\frac{\rho-1}{e}\rfloor})(2^{n}-1)^{\lfloor\frac{\rho-1}{e}\rfloor-l}\cdot
2^{n}$ $\displaystyle=(_{\verb+
+j}^{\rho-1-\lfloor\frac{\rho-1}{e}\rfloor})\cdot(_{\verb+
+l}^{\lfloor\frac{\rho-1}{e}\rfloor})2^{n}(2^{n}-1)^{\rho-1-l-j}.$
Applying this result to each $j$ and each $l$, we obtain the linear span of
the proposed sequence. Using a similar approach to the odd case, we see that
the linear span of both sequences is same.
For the sequences families above, some special conditions had already been
discussed, for example, the case with $n=em$, where $m$ is an odd, and
$\rho=1$ had been discussed in [1]. In the following, we will discuss another
special case with $n=em$, where $m$ is an even, and $\rho=1$, we call this
special sequence family as family $\mathcal{V}$.
### 3.3 Correlation Function of the Sequence family for even $m$ and $\rho=1$
If $\rho=1$, then the equation (5) becames
$\overline{\delta}Tr^{n}_{e}(\overline{\delta}z)+Tr^{n}_{e}(z)+\frac{(\overline{\lambda}^{2}+1)(\overline{\delta}^{2}+1)}{\overline{\lambda}^{2}}z=0$
(9)
In the following, we study the solution of the equation (9).
Let $Tr_{e}^{n}(\overline{\delta}z)=a$, $Tr_{e}^{n}(z)=b$, then
$z=\frac{\overline{\lambda}^{2}}{\overline{\lambda}^{2}+1}\frac{\overline{\delta}a+b}{\overline{\delta}^{2}+1}.$
By computing $Tr^{n}_{e}(z)$ and $Tr^{n}_{e}(\overline{\delta}z)$, we have
$\left\\{\begin{array}[]{lll}$$aTr^{n}_{e}(\displaystyle{\frac{\overline{\delta}}{\overline{\delta}^{2}+1}})+b[Tr^{n}_{e}(\frac{1}{\overline{\delta}^{2}+1})-\frac{\overline{\lambda}^{2}+1}{\overline{\lambda}^{2}}]=0$$\\\
\\\
$$\displaystyle{a[tr^{n}_{e}(\frac{1}{\overline{\delta}^{2}+1})-\frac{\overline{\lambda}^{2}+1}{\overline{\lambda}^{2}}]+bTr^{n}_{e}(\frac{\overline{\delta}}{\overline{\delta}^{2}+1})=0}$$\end{array}\right.$
(10)
The determinant of corresponding coefficient matrix of (10) is equal to
$\displaystyle[Tr^{n}_{e}(\frac{1}{\overline{\delta}^{2}+1})-\frac{\overline{\lambda}^{2}+1}{\overline{\lambda}^{2}}]^{2}+[Tr^{n}_{e}(\frac{\overline{\delta}}{1+\overline{\delta}^{2}})]^{2}$
$\displaystyle=(\frac{\overline{\lambda}^{2}+1}{\overline{\lambda}^{2}})^{2}+[Tr^{n}_{e}(\frac{1}{1+\overline{\delta}})]^{2}$
(1) If
$\displaystyle{Tr^{n}_{e}(\frac{1}{1+\overline{\delta}})\neq\frac{\overline{\lambda}^{2}+1}{\overline{\lambda}^{2}}}$,
then the determinant of coefficient matrix (10) is not equal to zero, the
equation (10) has unique solution $a=0$, $b=0$, then $z=0$.
(2) If
$\displaystyle{Tr^{n}_{e}(\frac{1}{1+\overline{\delta}})=\frac{\overline{\lambda}^{2}+1}{\overline{\lambda}^{2}}}$,
then the determinant of coefficient matrix (10) is equal to zero,
$\displaystyle
aTr^{n}_{e}(\frac{1}{\overline{\delta}+1})+aTr^{n}_{e}(\frac{1}{\overline{\delta}^{2}+1})+b[Tr^{n}_{e}(\frac{1}{\overline{\delta}^{2}+1})-\frac{\overline{\lambda}^{2}+1}{\overline{\lambda}^{2}}]=0,$
thus $a=b$, the equation (10) has $2^{e}$ solutions, then
$z=\frac{\overline{\lambda}^{2}}{\overline{\lambda}^{2}+1}\frac{1}{\overline{\delta}+1}a$.
$\displaystyle
2P(\overline{\lambda}z)+2P(\overline{\lambda}\overline{\delta}z)=2\sum^{l-1}_{j=1}Tr^{n}_{1}[(\overline{\lambda}z)^{2^{ej}+1}+(\overline{\lambda}\overline{\delta}z)^{2^{ej}+1}]+2Tr^{le}_{1}[(\overline{\lambda}x)^{2^{le}+1}+(\overline{\lambda}\overline{\delta}z)^{2^{el}+1}]$
$\displaystyle=2\sum^{l-1}_{j=1}Tr^{n}_{1}[(\frac{\overline{\lambda}^{3}a}{1+\overline{\lambda}^{2}})^{2^{ej}+1}(\frac{1}{1+\overline{\delta}})^{2^{ej}+1}+(\frac{\overline{\lambda}^{3}a}{1+\overline{\lambda}^{2}})^{2^{ej}+1}(\frac{\overline{\delta}}{1+\overline{\delta}})^{2^{ej}+1}]$
$\displaystyle\verb+
++2Tr^{el}_{1}[(\frac{\overline{\lambda}^{3}a}{1+\overline{\lambda}^{2}})^{2^{el}+1}(\frac{1}{1+\overline{\delta}})^{2^{el}+1}+(\frac{\overline{\lambda}^{3}a}{1+\overline{\lambda}^{2}})^{2^{el}+1}(\frac{\overline{\delta}}{1+\overline{\delta}})^{2^{el}+1}]$
$\displaystyle=2Tr^{e}_{1}\\{(\frac{\overline{\lambda}^{3}a}{1+\overline{\lambda}^{2}})^{2}[\sum^{l-1}_{j=1}Tr^{n}_{e}[(\frac{1}{1+\overline{\delta}})^{2^{ej}+1}+(\frac{\overline{\delta}}{1+\overline{\delta}})^{2^{ej}+1}]$
$\displaystyle\verb+
++Tr^{le}_{e}[(\frac{1}{1+\overline{\delta}})^{2^{el}+1}+(\frac{\overline{\delta}}{1+\overline{\delta}})^{2^{el}+1}]]\\}$
$\displaystyle=2Tr^{e}_{1}{[(\frac{\overline{\lambda}^{3}a}{1+\overline{\lambda}^{2}})^{2}}(Tr^{le}_{e}1+Tr^{n}_{e}(\frac{1}{1+\overline{\delta}}))]$
$\displaystyle=\left\\{\begin{array}[]{ll}$$2Tr^{n}_{1}(\frac{\overline{\lambda}^{2}z}{1+\overline{\lambda}^{2}}),\
for\ odd\ l$$,\\\ \\\
$$2Tr^{n}_{1}(\frac{\overline{\lambda}^{2}z}{1+\overline{\lambda}}),\ for\
even\ l.$$\end{array}\right.$
Thus, $\phi(z)=Tr^{n}_{1}(\Delta z)+2[P(\lambda z)+P(\lambda\delta z)]$
$=\left\\{\begin{array}[]{ll}$$2Tr^{n}_{1}[(\overline{\gamma}_{0}^{i}+(\overline{\gamma}_{0}^{j}+1)\overline{\delta}+\frac{\overline{\lambda}^{2}}{1+\overline{\lambda}^{2}})z],\
\ for\ odd\ l$$,\\\ \\\
$$2Tr^{n}_{1}[(\overline{\gamma}_{0}^{i}+(\overline{\gamma}_{0}^{j}+1)\overline{\delta}+\frac{\overline{\lambda}^{2}}{1+\overline{\lambda}})z],\
for\ even\ l.$$\end{array}\right.$
Because the solutions space of equation (9) is a linear subspace, following
the discussions above, we have
###### Theorem 3.3
for $m$ is even, $\rho=1$, the nontrivial correlation function of the proposed
sequences family $\mathcal{V}$ takes values in $\\{-1,-1\pm
2^{\frac{n}{2}},-1\pm 2^{\frac{n}{2}}\omega,-1\pm 2^{\frac{n+e}{2}},-1\pm
2^{\frac{n+e}{2}}\omega\\}$.
## 4 Quadriphase Sequences with period $2(2^{n}-1)$
Similar to the [1, 2], for an even $m$, we propose the following sequence
family, the correlation function of the sequences family is calculated.
In this section, let $G=\\{\eta_{1},\eta_{2},\cdots,\eta_{2^{n-1}}\\}$ be a
maximum subset of $G_{C}$ such that $2\eta_{i}\neq 2(\eta_{j}+1)$ for
arbitrary $1\leq i,j\leq 2^{n-1}$. By convention, denote
$\beta^{\frac{1}{2}}=\beta^{2^{n-1}}$. We present another family of
quadriphase sequences with period $2(2^{n}-1)$ as follows.
###### Definition 2
A family $\mathcal{W}$ of quadriphase sequences with period $2(2^{n}-1)$ is
defined as $\mathcal{W}=\\{u_{i}(t),v_{i}(t):0\leq i<2^{n-1}\\}$ is given by
1)
$u_{i}(t)=$$\left\\{\begin{array}[]{lll}Tr^{n}_{1}[(1+2\eta_{i})\beta^{t_{0}}]+2P(\lambda\beta^{t_{0}}),t=2t_{0}\\\
\\\
Tr^{n}_{1}[(1+2(\eta_{i}+1))\beta^{t_{0}+\frac{1}{2}}]+2P(\lambda\beta^{t_{0}+\frac{1}{2}}),t=2t_{0}+1\end{array}\right.$
for $0\leq i<2^{n-1}$, where $\eta_{i}\in G$.
2)
$v_{i}(t)=$$\left\\{\begin{array}[]{lll}Tr^{n}_{1}[(1+2\eta_{i})\beta^{t_{0}}]+2P(\lambda\beta^{t_{0}})+2,t=2t_{0}\\\
\\\
Tr^{n}_{1}[(1+2(\eta_{i}+1))\beta^{t_{0}+\frac{1}{2}}]+2P(\lambda\beta^{t_{0}+\frac{1}{2}}),t=2t_{0}+1\end{array}\right.$
for $0\leq i<2^{n-1}$, where $\eta_{i}\in G$.
###### Theorem 4.1
the correlation functions of the family $\mathcal{W}$ satisfy the following
properties. 1) if $\tau=2^{n}-1$, then $R_{u_{i},u_{j}}(\tau)=-2$,
$R_{v_{i},v_{j}}(\tau)=2$, $R_{u_{i},v_{j}}(\tau)=0$.
2) if $\tau=0$, then $R_{u_{i},v_{j}}(\tau)=0$ and
$R_{u_{i},u_{j}}(\tau)=R_{v_{i},v_{j}}(\tau)=\left\\{\begin{array}[]{ll}2(2^{n}-1),i=j\\\
-2,i\neq j,\end{array}\right.$
3)If $\tau=2\tau_{0}+1\neq 2^{n}-1$, then
$\displaystyle a)\ R_{u_{i},u_{j}}(\tau)\ takes\ values\ in\ \\{-2,-2\pm
2^{\frac{n}{2}+1},-2\pm 2^{\frac{n+e}{2}+1}\\},$ $\displaystyle b)\
R_{v_{i},v_{j}}(\tau)\ takes\ values\ in\ \\{2,2\pm 2^{\frac{n}{2}+1},2\pm
2^{\frac{n+e}{2}+1}\\},$ $\displaystyle c)\ R_{u_{i},v_{j}}(\tau)\ takes\
values\ in\ \\{\pm 2^{\frac{n}{2}+1}\omega,\pm 2^{\frac{n+e}{2}+1}\omega\\}.$
4)If $\tau=2\tau_{0}$ and $\tau_{0}\neq 0$, then
$\displaystyle a)\ R_{u_{i},u_{j}}(\tau)\ takes\ values\ in\ \\{-2,-2\pm
2^{\frac{n}{2}+1},-2\pm 2^{\frac{n+e}{2}+1}\\},$ $\displaystyle b)\
R_{v_{i},v_{j}}(\tau)\ takes\ values\ in\ \\{-2,-2\pm 2^{\frac{n}{2}+1},-2\pm
2^{\frac{n+e}{2}+1}\\},$ $\displaystyle c)\ R_{u_{i},v_{j}}(\tau)\ takes\
values\ in\ \\{\pm 2^{\frac{n}{2}+1}\omega,\pm 2^{\frac{n+e}{2}+1}\omega\\}.$
###### Proof
In order to analysis easily, let
$\varsigma(\gamma_{1},\gamma_{2},\delta)=\sum_{x\in
G_{C}}\omega^{Tr^{n}_{1}[(1+2\gamma_{1}-(1+2\gamma_{2})\delta)x]+2(P(\lambda
x)+P(\lambda\delta x))}.$ (11)
It is easy to check that
$\varsigma(\gamma_{1}+1,\gamma_{2},\delta)=\varsigma(\gamma_{1},\gamma_{2}+1,\delta)^{*}$,
where $*$ denotes complex conjugate.
Similar to [1], the following facts can be easily checked.
1) if $\tau=2\tau_{0}+1$, then
$\displaystyle
R_{u_{i},u_{j}}(\tau)=\varsigma(\eta_{i},\eta_{j}+1,\delta)+\varsigma(\eta_{i}+1,\eta_{j},\delta)-2,$
$\displaystyle
R_{v_{i},v_{j}}(\tau)=-\varsigma(\eta_{i},\eta_{j}+1,\delta)-\varsigma(\eta_{i}+1,\eta_{j},\delta)+2,$
$\displaystyle
R_{u_{i},v_{j}}(\tau)=\varsigma(\eta_{i},\eta_{j}+1,\delta)-\varsigma(\eta_{i}+1,\eta_{j},\delta).$
2) if $\tau=2\tau_{0}$, then
$\displaystyle
R_{u_{i},u_{j}}(\tau)=\varsigma(\eta_{i},\eta_{j}+1,\delta)+\varsigma(\eta_{i}+1,\eta_{j},\delta)-2,$
$\displaystyle
R_{v_{i},v_{j}}(\tau)=\varsigma(\eta_{i},\eta_{j}+1,\delta)+\varsigma(\eta_{i}+1,\eta_{j},\delta)-2,$
$\displaystyle
R_{u_{i},v_{j}}(\tau)=-\varsigma(\eta_{i},\eta_{j}+1,\delta)+\varsigma(\eta_{i}+1,\eta_{j},\delta).$
Due to (3),(11) and the theorem 3, the theorem 4 is proved.
Similar to the proof of the theorem 2 above, or the proof of theorem 3 and
theorem 7 [1] the following theorem is obtained.
###### Theorem 4.2
the linear spans of the sequences in $\mathcal{W}$ are given as follows
(1) For $u_{i}\in\mathcal{W}$, the linear span $LS(u_{i})$ of $u_{i}$ is given
by $\displaystyle{LS(u_{i})=\frac{n(n+e)}{2e}}$.
(2) For $v_{i}\in\mathcal{W}$, the linear span $LS(v_{i})$ of $v_{i}$ is given
by $\displaystyle{LS(u_{i})=\frac{n(n+e)}{2e}+2}$.
## 5 Conclusions
In this paper, we have proposed the new families of quadriphase sequences with
larger linear span and size. The maximum correlation magnitude of proposed
sequences family is bigger then that of the related sequence in [1], and is
smaller than that of the related binary sequences family in [2, 3] with same
parameters. The proposed two families of quadriphase sequences with period
$2^{n}-1$ and $2(2^{n}-1)$ respectively for a positive integer $n=em$ where
$m$ is an even positive can be take as an extensions of the results in [1]
where $m$ is an odd positive.
## 6 Acknowledgment
This work was supported by National Science Foundation of China under grant
No.60773002 and 61072140, the Project sponsored by $SRF$ for $ROCS$, $SEM$,
863 Program (2007AA01Z472), and the 111 Project (B08038).
## References
* [1] Jiang W.F.,Hu L.,Tang X.H.,Zeng X.Y.: New optimal quadriphase sequences with larger linear span. IEEE Trans. Inform. Theory, Vol.55, No.1, pp.458-470, Jan. 2009.
* [2] Tang X.H.,Udaya P.,Fan P.Z.:Generalized binary Udaya-Siddiqi sequences. IEEE Trans. Inform. Theory, Vol.53, No.3, pp.1225-1230, Mar. 2007.
* [3] KimS H.,No J.S.: New families of binary sequences with low cross correlation property.IEEE Trans. Inform. Theory, Vol.49, No.1, pp.3059-3065, Jan. 2009.
* [4] Yu N.Y.,Gong G.: A new binary sequence family with low correlation and large size. IEEE Trans. Inform. Theory, Vol.52, No.4, pp.1624-1636, Mar.2006.
* [5] Boztas S.,Hammons R.,Kumar P.V.: 4-phase sequences with near optimum correlation properties. IEEE Trans. Inform. Theory, Vol.14, No.3, pp.1101-1113, May 1992.
* [6] Kumar P.V.,Helleseth T.,Calderbank A.R.,Hammons A.R.Jr.: Large families of quaternary sequences with low correlation. IEEE Trans. Inform. Theory, Vol.42, No.2, pp.579-592, Mar.1996.
* [7] Schmidt K.-U.:$Z_{4}$-valued quadratic forms and quaternary sequence families. IEEE Trans. Inform. Theory, Vol.55, No.12, pp.5803-5810, Dec.2009.
* [8] Sole P.: A quaternary cyclic code, and a family of quadriphase sequences with low correlation properties.Lecture Notes in Computer Science, Vol.388, pp.193-201,1989.
* [9] Tang X.H.,Udaya P.:A note on the optimal quadriphase sequences families. IEEE Trans. Inform. Theory, Vol.53, No.1, pp433-436, Jan. 2007\.
* [10] Udaya P.,Siddiqi M.U.: Optimal and suboptimal quadriphase sequences derived from maximal length sequences over $Z_{4}$.Appl. Algebra Eng. Commun.Comput., Vol.9, no.2, pp.161-191,1998.
* [11] Hammons A.R.Jr.,Kumar P.V.,Calderbank A.R.,Sloane N.J.A.,Sole P.: The $Z_{4}$ linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inform. Theory, Vol.40, No.2, pp.301-319, Mar. 1994\.
* [12] Tang X.T.,Udaya P.,Fan P.Z.:Quadriphase sequences obtained from binary quadratic form sequences. Lecture Note in Computer Science, Vol.3486, pp.243-254, 2005.
|
arxiv-papers
| 2011-01-18T08:35:50 |
2024-09-04T02:49:16.519842
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wenping Ma",
"submitter": "Wenping Ma",
"url": "https://arxiv.org/abs/1101.3398"
}
|
1101.3652
|
# Three-dimensional topological insulators in the octahedron-decorated cubic
lattice
Jing-Min Hou jmhou@seu.edu.cn Wen-Xin Zhang Guo-Xiang Wang Department of
Physics, Southeast University, Nanjing, 211189, China
###### Abstract
We investigate a tight-binding model of the octahedron-decorated cubic lattice
with spin-orbit coupling. We calculate the band structure of the lattice and
evaluate the $Z_{2}$ topological indices. According to the $Z_{2}$ topological
indices and the band structure, we present the phase diagrams of the lattice
with different filling fractions. We find that the $(1;111)$ and $(1;000)$
strong topological insulators occur in some range of parameters at $1/6,1/2$
and $2/3$ filling fractions. Additionally, the $(0;111)$ weak topological
insulator is found at $1/6$ and $2/3$ filing fractions. We analyze and discuss
the characteristics of these topological insulators and their surfaces states.
###### pacs:
73.43.-f, 71.10.Fd, 73.20.-r, 72.25.-b
## I Introduction
Usually, different phases of matter can be classified using Landau’s approach
according to their underling symmetriesLandau . In 1980s, the discovery of the
quantum Hall effect changed physicists’ viewpoint on the classification of
matterKlitzing . The quantum Hall states can be classified by a topological
invariant, now named the TKNN numberThouless (equivalent to the first Chern
number), which is directly connected to the quantized Hall conductivity, but
they have the same symmetry. Since the Hall conductivity is odd under time
reversal, the topological non-trivial quantum Hall states can only occur when
time reversal symmetry is broken, which is performed by a magnetic field. In
1988, Haldane also proposed a time reversal symmetry broken toy model without
a magnetic field to realize quantum Hall statesHaldane . All the quantum Hall
states have a gapped band structure in bulk and chiral gapless edge states
that are topologically protected.
Recently, the promising prospect of spintronics in technology stimulates
physicists to generate spin current. Quantum spin Hall effect was proposed to
create spin currentBernevig ; Kane . The quantum spin Hall states are non-
trivial topological phases with time reversal symmetry, which have a bulk gap
and topologically protected gapless helical edge states. For the above reason,
the quantum spin Hall states also called topological insulators. Two-
dimensional topological insulators are characterized by a $Z_{2}$ topological
index $\nu=0,1$Kane2 . For a non-trivial topological insulator the topological
index has a value $\nu=1$ while $\nu=0$ for a trivial band insulator.
Therefore, a topological insulator always has a metallic boundary when placed
next to a vacuum or an ordinary band insulator because topological invariants
cannot change as long as a material remains insulating. The remarkable
metallic boundaries of topological insulators may result in new spintronic or
magnetoelectric devices and a new architecture for topological quantum bits.
In quantum spin Hall phases, the spin-orbit coupling plays the role of the
spin-dependent effective magnetic field. The first real material, a HgTe
quantum well, supporting two-dimensional topological insulators was predicted
by Bernevig, et al.Bernevig2 and experimentally conformed by König et
al.Konig .
Figure 1: (Color online). (a) The octahedron-decorated cubic lattice which can
be obtained by replacing every lattice site of a cubic lattice with an
octahedral cluster as shown in (b). (c) The three-dimensional Brillouin zone
and high symmetry points. (d) The two-dimensional Brillouin zone of a slab
with two $001$ surfaces.
Soon after the quantum spin Hall insulator was discovered, time-reversal
invariant topological insulators were generalized to three dimensionsFu ;
Moore ; Roy . Three-dimensional time-reversal invariant band insulators are
classified according to four $Z_{2}$ topological indices
$(\nu_{0};\nu_{1}\nu_{2}\nu_{3})$ with $\nu_{i}=0,1$Fu . In three dimensions,
the time-reversal invariant band insulators can be classified into 16 phases
according to the four $Z_{2}$ topological indices. A band insulator with
$\nu_{0}=1$ is called a strong topological insulator(STI), a band insulator
with $\nu_{0}=0$ and at least one non-zero $\nu_{i}(i=1,2,3)$ is called a weak
topological insulator(WTI), while an ordinary trivial band insulator has an
index $(0;000)$. For an STI phase, the surface states have an odd number of
Dirac points, which are topologically protected and for a WTI or trivial band
insulator phase, the surface states have an even number of Dirac points. Fu
and Kane firstly predicted that Bi1-x Sbx supports a three-dimensional
topological insulatorFu2 , which was conformed experimentally by Hsieh, et al.
in 2008Hsieh . Later, Bi2Se3 was discovered to be a three-dimensional
insulator experimentally as a second generation materialXia , which also was
supported by theoretical calculationsXia ; H.Zhang . Additionally, reference
H.Zhang also predicted that Bi2Te3 and Sb2Te3 are second generation materials
supporting three-dimensional topological insulators. The later experimental
studies on Bi2Te3Chen ; Hsieh2 ; Hsieh3 and Sb2Te3Hsieh3 identified their
topological band structures.
To help experimental physicists find more topological insulator materials,
theoretical physicists have investigated several models that support non-
trivial topological insulators. Theoretical studies have demonstrated that,
within the tight-binding approximation and with the spin-orbit coupling, the
honeycombKane , kagomeGuo , checkerboardSun , decorated honeycombRuegg ,
LiebWeeks , and square-octagonKargarian lattices support two-dimensional
topological insulators and the diamondFu , pyrochloreGuo2 , and
perovskiteWeeks lattices support three-dimensional topological insulators.
In this paper, we shall show that a new lattice, the octahedron-decorated
cubic lattice as shown in Fig.1 (a), supports three-dimensional topological
insulators with the spin-orbit coupling existing. This lattice can be regarded
as a three-dimensional generalization of the square-octagon latticeRuegg . We
find that this model supports STI and WTI phases for $1/6$ and $2/3$ filling
and STI phases for $1/2$ filling as well as ordinary band insulator and metal
phases.
## II Model
We consider the octahedron-decorated cubic lattice as shown in Fig.1 (a),
which can be obtained by replacing every lattice site of a cubic lattice with
an octahedral cluster as shown in Fig.1(b). This lattice has a unit cell with
six different lattice sites as denoted in Fig.1(b) so that it contains six
sublattices. Here, we assume that the distance between the centers of two
nearest-neighbor octahedral clusters is $a$, which is the same with the
lattice constant of all sublattices, the distance of every lattice site of an
octahedral cluster from its center is $a/4$, and the distance of two nearest-
neighbor lattice sites in different octahedral clusters is $a/2$. With the
tight-binding approximation, we can write the second quantized Hamiltonian of
the lattice as follows,
$\displaystyle H_{0}=-t\sum_{\langle
i,j\rangle,\sigma}c_{i\sigma}^{\dagger}c_{j\sigma}-t_{1}\sum_{[i,j],\sigma}c_{i\sigma}^{\dagger}c_{j\sigma}$
(1)
where $c_{i\sigma}$ is the annihilation operator destructing an electron with
spin $\sigma$ on the site ${\bf r}_{i}$ of the octahedron-decorated cubic
lattice, $\langle i,j\rangle$ represents nearest-neighbor hopping in the same
octahedral cluster with amplitude $t$ and $[i,j]$ denotes nearest-neighbor
hopping between two different octahedral clusters with amplitude $t_{1}$.
Figure 2: (Color online). Phase diagrams of the octahedron-decorated cubic
lattice for (a) $1/6$ filling, (b) $1/2$ filling, and (c) $2/3$ filling. Here,
BI denotes a trivial band insulator; STI1 and STI2 denote $(1;111)$ and
$(1;000)$ strong topological insulators, respectively; WTI denotes a $(0;111)$
weak topological insulator; and M denotes a metal phase.
In momentum space, the Hamiltonian (1) can be represented by $H_{0}=\sum_{{\bf
k}\sigma}\Psi^{\dagger}_{{\bf k}\sigma}{\cal H}^{(0)}_{\bf k}\Psi_{{\bf
k}\sigma}$ with $\Psi_{{\bf k}\sigma}=(c_{1{\bf k}\sigma},c_{2{\bf
k}\sigma},c_{3{\bf k}\sigma},c_{4{\bf k}\sigma},c_{5{\bf k}\sigma},c_{6{\bf
k}\sigma})^{T}$, which are ordered according to the sequence denoted in
Fig.1(b). Here, ${\cal H}^{(0)}_{\bf k}$ takes the following form,
$\displaystyle{\cal H}_{\bf k}^{0}=$
$\displaystyle-\left(\matrix{0&t&t&t&t&t_{1}e^{ik_{z}}\cr
t&0&t&t_{1}e^{ik_{x}}&t&t\cr t&t&0&t&t_{1}e^{ik_{y}}&t\cr
t&t_{1}e^{-ik_{x}}&t&0&t&t\cr t&t&t_{1}e^{-ik_{y}}&t&0&t\cr
t_{1}e^{-ik_{z}}&t&t&t&t&0}\right)$ (2)
Since $H_{0}$ is spin-decoupling, ${\cal H}_{\bf k}^{0}$ is spin-independent,
i.e. it is the same for both spin-up and spin-down electrons. Fig.1(c) shows
the first Brillouin zone of the octahedron-decorated cubic lattice. The
spectrum of Eq.(2) with $t_{1}=t$ is calculated and shown in Fig.3(d). The
spectrum contains six bands which come from the six sites in every unit cell.
A gap exists between the first and second bands. The second, third and fourth
bands touch together at points $\Gamma,R$ and $M$. The third, fourth and fifth
band touch at point $X$, near which a Dirac cone occurs. Five bands including
the second, third, fourth, fifth and sixth bands meet at point $\Gamma$. Along
the $\Gamma\rightarrow R$ line in momentum space, the second and third bands
are degenerate and the fifth and sixth bands are degenerate.
Now, in order to find non-trivial topological insulators in the octahedron-
decorated cubic lattice, we proceed to introduce the spin-orbit interactions
between next-nearest-neighbor sites as follows,
$\displaystyle H_{\rm SO}=i\frac{8\lambda_{\rm SO}}{a^{2}}\sum_{\langle\langle
i,j\rangle\rangle\alpha\beta}({\bf d}_{ij}^{1}\times{\bf
d}_{ij}^{2})\cdot\boldmath{\mbox{$\sigma$}}_{\alpha\beta}c_{i\alpha}^{\dagger}c_{j\beta},$
where $\langle\langle i,j\rangle\rangle$ represents two next-nearest-neighbor
sites $i,j$, and $\lambda_{\rm SO}$ is the amplitude of spin-orbit coupling of
the two next-nearest-neighbor sites.
${\boldmath{\mbox{$\sigma$}}}=(\sigma_{x},\sigma_{y},\sigma_{z})$ is the
vector of Pauli spin matrices. ${\bf d}_{ij}^{1,2}$ are the two nearest
neighbor bond vectors traversed between sites $i$ and $j$ with $8|{\bf
d}_{ij}^{1}\times{\bf d}_{ij}^{2}|/a^{2}=1$. In momentum space, the
Hamiltonian for spin-orbit coupling (II) can be expressed as $H_{\rm
SO}=\sum_{\bf k}\Psi_{\bf k}^{\dagger}{\cal H}_{\bf k}^{\rm SO}\Psi_{\bf k}$
with $\Psi_{\bf k}=(c_{1{\bf k}\uparrow},c_{2{\bf k}\uparrow},$ $c_{3{\bf
k}\uparrow},c_{4{\bf k}\uparrow},c_{5{\bf k}\uparrow},c_{6{\bf
k}\uparrow},c_{1{\bf k}\downarrow},c_{2{\bf k}\downarrow},c_{3{\bf
k}\downarrow},c_{4{\bf k}\downarrow},c_{5{\bf k}\downarrow},c_{6{\bf
k}\downarrow})^{T}$. Since ${\cal H}_{\bf k}^{\rm SO}$ does not decouple for
the two spin projections, it is a $12\times 12$ matrix. In momentum space, the
total single particle Hamiltonian is ${\cal H}_{\bf k}={\cal H}_{\bf
k}^{0}+{\cal H}_{\bf k}^{\rm SO}$. The bands and eigenstates can be obtained
by exactly diagonalizing ${\cal H}_{\bf k}$.
## III Three-dimensional topological insulators
Figure 3: (Color online). Band structures of the octahedron-decorated cubic
lattice for various parameters $t_{1}$ and $\lambda_{\rm SO}$. Here, the
horizontal axis represents the wave vectors along the path in the first
Brillouin zone indicated by the red lines in Fig.1(c). (a)
$t_{1}=t,\lambda_{\rm SO}=0.5t$, (b) $t_{1}=-t,\lambda_{\rm SO}=-0.5t$, (c)
$t_{1}=t,\lambda_{\rm SO}=t$, (d) $t_{1}=t,\lambda_{\rm SO}=0$, (e)
$t_{1}=3t,\lambda_{\rm SO}=-0.4t$, (f) $t_{1}=-3t,\lambda_{\rm SO}=0.4t$, (g)
$t_{1}=3.2t,\lambda_{\rm SO}=-0.2t$, (h) $t_{1}=t,\lambda_{\rm SO}=0.2t$, (i)
$t_{1}=t,\lambda_{\rm SO}=-0.2t$, (j) $t_{1}=-t,\lambda_{\rm SO}=0.2t$, (k)
$t_{1}=2.5t,\lambda_{\rm SO}=-0.2t$, and (l) $t_{1}=0.5t,\lambda_{\rm SO}=0$.
The classification of three-dimensional topological insulators is presented in
Ref.Fu . For three-dimensional lattices there eight distinct time reversal
invariant momenta (TRIM), which can be expressed in terms of primitive
reciprocal lattice vectors as $\Gamma_{i=(n_{1},n_{2},n_{3})}=(n_{1}{\bf
b}_{1}+n_{2}{\bf b}_{2}+n_{3}{\bf b}_{3})/2$ with $n_{j}=0,1$. Three-
dimensional topological insulators can be distinguished by four $Z_{2}$
topological invariants $(\nu_{0};\nu_{1}\nu_{2}\nu_{3})$, which are defined as
$(-1)^{\nu_{0}}=\prod_{n_{j}=0,1}\delta_{n_{1}n_{2}n_{3}}$ and
$(-1)^{\nu_{i=1,2,3}}=\prod_{n_{j\neq
i}=0,1;n_{i}=1}\delta_{n_{1}n_{2}n_{3}}$, where
$\delta_{n_{1}n_{2}n_{3}}=\sqrt{\det[w(\Gamma_{n_{1}n_{2}n_{3}})]}/{\rm
Pf}[w(\Gamma_{n_{1}n_{2}n_{3}})]=\pm 1$. Here the unitary matrix $w$ is
defined as $w_{ij}({\bf k})=\langle u_{i}(-{\bf k})|\Theta|u_{j}({\bf
k}\rangle$ with $\Theta$ being the time reversal operator and $|u_{j}({\bf
k})\rangle$ being the Bloch wave functions for occupied bands. Fu and Kane
have found a simple method to identify the $Z_{2}$ invariants for the system
with the presence of inversion symmetryFu2 . In this case,
$\delta_{n_{1}n_{2}n_{3}}$ can be calculated by
$\delta_{n_{1}n_{2}n_{3}}=\prod_{m=1}^{N}\xi_{2m}(\Gamma_{n_{1}n_{2}n_{3}})$,
where $N$ is the number of occupied bands and
$\xi_{2m}(\Gamma_{n_{1}n_{2}n_{3}})=\pm 1$ is the parity eigenvalue of the
$2m$th occupied band at $\Gamma_{n_{1}n_{2}n_{3}}$. Our model is inversion
symmetric so we will adopt this method to evaluate the $Z_{2}$ invariants
$\nu_{i}(i=0,1,2,3)$. We select the center of an octahedron in the lattice as
the center of inversion, then the parity operator acts as ${\cal
P}[\psi_{1}({\bf r}),\psi_{2}({\bf r}),\psi_{3}({\bf r}),\psi_{4}({\bf
r}),\psi_{5}({\bf r}),\psi_{6}({\bf r})]^{T}=[\psi_{6}(-{\bf
r}),\psi_{4}(-{\bf r}),\psi_{5}(-{\bf r}),\psi_{2}(-{\bf r}),\psi_{3}(-{\bf
r}),\psi_{1}(-{\bf r})]^{T}$, where $[\psi_{1}({\bf r}),\psi_{2}({\bf
r}),\psi_{3}({\bf r}),\psi_{4}({\bf r}),\psi_{5}({\bf r}),\psi_{6}({\bf
r})]^{T}$ is the six-component wave function. Taking Fourier transformation,
we can write the six-component wave function as $[\psi_{1}({\bf
r}),\psi_{2}({\bf r}),\psi_{3}({\bf r}),\psi_{4}({\bf r}),\psi_{5}({\bf
r}),\psi_{6}({\bf r})]=\sum_{\bf k}[\phi_{1}({\bf k}),\phi_{2}({\bf
k}),\phi_{3}({\bf k}),\phi_{4}({\bf k}),\phi_{5}({\bf k}),\phi_{6}({\bf
k})]e^{i{\bf k}\cdot{\bf r}}$ and the parity operator as ${\cal P}=\sum_{\bf
k}e^{i{\bf k}\cdot{\bf r}}{\cal P}_{\bf k}e^{-i{\bf k}\cdot{\bf r}}$. Then, in
momentum space, we obtain the equation ${\cal P}_{\bf k}[\phi_{1}({\bf
k}),\phi_{2}({\bf k}),\phi_{3}({\bf k}),\phi_{4}({\bf k}),\phi_{5}({\bf
k}),\phi_{6}({\bf k})]^{T}=[\phi_{6}(-{\bf k}),\phi_{4}(-{\bf
k}),\phi_{5}(-{\bf k}),\phi_{2}(-{\bf k}),\phi_{3}(-{\bf k}),\phi_{1}(-{\bf
k})]^{T}$. Considering the degree of spin, we can express the parity operator
at the time reversal invariant momenta $\Gamma_{n_{1}n_{2}n_{3}}$ as follows,
$\displaystyle{\cal P}_{\Gamma_{n_{1}n_{2}n_{3}}}=\left(\matrix{1&0\cr
0&1}\right)\otimes\left(\matrix{0&0&0&0&0&1\cr 0&0&0&1&0&0\cr 0&0&0&0&1&0\cr
0&1&0&0&0&0\cr 0&0&1&0&0&0\cr 1&0&0&0&0&0}\right)$ (3)
where the $4\times 4$ matrix is the unit matrix in spin space.
We diagonalize the total single-particle Hamiltonian ${\cal H}_{\bf k}$ and
calculate the $Z_{2}$ topological invariants for different filling fractions.
We find that non-trivial topological insulators exist for $1/6,1/2$ and $2/3$
filling while only metal phase occurs for $1/3$ and $5/6$ filling. Thus, we
will focus on and discuss the cases with $1/6,1/2$ and $2/3$ filling fractions
in the following part of the paper. We identify phases for different
parameters $t_{1}$ and $\lambda_{\rm SO}$ with $1/6,1/2$ and $2/3$ filling
fractions and draw phase diagrams as shown in Fig.2. Figs.2(a), 2(b) and 2(c)
show the phase diagrams for $1/6,1/2$ and $2/3$ filling, respectively. For
$1/6$ and $2/3$ filling, there are $(1;111)$ and $(1;000)$ STI phases,
$(0;111)$ WTI phase as well as trivial band insulator and metal phases. For
$1/2$ filling, there are $(1;111)$ and $(1;000)$ STI phases, trivial band
insulator and metal phases except $(0;111)$ WTI phase.
To clearly manifest the bulk band structure of different phases for various
filling factions, we calculate the bulk energy bands for several cases with
different parameters $t_{1}$ and $\lambda_{\rm SO}$, which are shown in Fig.3.
In order to investigate the characteristics of surface states for various
phases, we evaluate the energy bands in a slab geometry with two $001$
surfaces. The Brillouin zone of the slab is shown in Fig.1(d). The energy
bands are present along lines that connect the four surface TRIM as shown in
Fig.4. With the assistance of the bulk energy bands shown in Fig.3 and the
two-dimensional energy bands for a slab shown in Fig.4, we will sequentially
analyze various phases, identify three-dimensional topological insulators, and
discuss their characteristics for $1/6,1/2$ and $2/3$ filling.
Figure 4: (Color online). Band structures of a slab with two $001$ surfaces
for various parameters $t_{1}$ and $\lambda_{\rm SO}$. Here, the horizontal
axis represents the wave vectors along the path in the surface Brillouin zone
indicated by the red lines in Fig.1(d). (a) $t_{1}=t,\lambda_{\rm SO}=0.5t$,
(b) $t_{1}=-t,\lambda_{\rm SO}=-0.5t$, (c) $t_{1}=t,\lambda_{\rm SO}=t$, (d)
$t_{1}=t,\lambda_{\rm SO}=0$, (e) $t_{1}=3t,\lambda_{\rm SO}=-0.4t$, (f)
$t_{1}=-3t,\lambda_{\rm SO}=0.4t$, (g) $t_{1}=3.2t,\lambda_{\rm SO}=-0.2t$,
(h) $t_{1}=t,\lambda_{\rm SO}=0.2t$, (i) $t_{1}=t,\lambda_{\rm SO}=-0.2t$, (j)
$t_{1}=-t,\lambda_{\rm SO}=0.2t$, (k) $t_{1}=2.5t,\lambda_{\rm SO}=-0.2t$, and
(l) $t_{1}=0.5t,\lambda_{\rm SO}=0$.
### III.1 $1/6$ filling
Fig.2(a) shows the phase diagram of the octahedron-decorated cubic lattice for
$1/6$ filling. In this case, the $(1;111)$ and $(1;000)$ STI phases are
discovered. The non-trivial STI phases have a gap between the first and second
bands as shown in Fig.3(a) and 3(b) corresponding to $(1;111)$ and $(1;000)$
STI phases, respectively. We note that for $1/6$ filling there is only one
Dirac point on TRIM as shown in Fig.4(a) and (b), that is, only a pair of
robust spin-filtered states exists. We also find a $(0;111)$ WTI phase for
$1/6$ filling e.g., as shown in Fig.3(c). Fig.4(c) shows the surface states
for a $(0;111)$ WTI phase that has two Dirac points between the first and
second bands on TRIM. We note that trivial band insulators occur for smaller
$t_{1}$ and smaller $\lambda_{\rm SO}$ parameters, which is easily understood
for when $t_{1}$ and $\lambda_{\rm SO}$ approaches to zero the lattice becomes
separated octahedral clusters. For a trivial band insulator there is a gap
between the first and second bands as shown in Fig.3(d), but there are not
surface states as shown in Fig.4(d). For a metal phase, the gap vanishes.
### III.2 $1/2$ filling
Fig.2(b) shows the phase diagram of the octahedron-decorated cubic lattice for
$1/2$ filling. For $1/2$ filling, $(1;111)$ and $(1;000)$ STI phases, trivial
band insulators, and metal phases occur, but WTI phases are not found.
Figs.3(e) and 3(f) show the band structure for $(1;111)$ and $(0;111)$ STI
phases, respectively. We can find from these diagrams that a gap opens between
the third and fourth bands. For STI phases, there only one Dirac point on TRIM
as shown in Fig.4(e) and (f). For trivial band insulators, there is also a gap
between the third and fourth bands as shown in Fig.3(g), but even number of
Dirac points exist on TRIM as shown in Fig.4(g). For smaller $t_{1}$ and
smaller $\lambda_{\rm SO}$, a metal phase occurs except a special point
$t_{1}=0$ and $\lambda_{\rm SO}=0$. For $t_{1}=0$ and $\lambda_{\rm SO}=0$,
the second, third and fourth bands are degenerate and become a flat band,
which means that electrons are localized. In other works, the system with
$t_{1}$ and $\lambda_{\rm SO}$ for $1/2$ filling is a trivial band insulator.
However, a tiny change from $t_{1}=0$ and $\lambda_{\rm SO}=0$ for parameters
$t_{1}$ and $\lambda_{\rm SO}$ makes the flat band become three dispersive
bands that are crossover each other, then the lattice with three bands
occupied becomes a metal.
### III.3 $2/3$ filling
Fig.2(c) shows the phase diagram of the octahedron-decorated cubic lattice for
$2/3$ filling. We note that, similar to $1/6$ filling, $(1;111)$ and $(1;000)$
STI phases, $(0;111)$ WTI phase, trivial band insulator, and metal phase occur
in different ranges of parameters $t_{1}$ and $\lambda_{\rm SO}$. Fig.3(i)
shows the band structure for $t_{1}=t,\lambda_{\rm SO}=-0.2t$ at which a
$(1;111)$ STI phase occurs. We can find that a gap opens between the fourth
and fifth bands as shown in Fig.3(i). There is an odd number of surface states
which traverse the gap as shown in Fig.4(i). For the $(1;000)$ STI phase, the
similar characteristics are exemplified in Figs.3(j) and 4(j). The $(0;111)$
WTI phase is found as well. Fig.3(k) and Fig.4(k) show the $(0;111)$ WTI phase
has a gap between the fourth and fifth bands and an even number of surface
states traversing the gap. For smaller $t_{1}$ and smaller $\lambda_{\rm SO}$,
the system for $2/3$ filling is a trivial band insulator, which is feathered
by a gap between the fourth and fifth bands combined with an even number of
surface states traversing the gap as shown in Fig.3(l) and Fig.4(l),
respectively.
## IV Conclusion
In summary, we have shown that the octahedron-decorated cubic lattice with
spin-orbit coupling supports three-dimensional topological insulators at
$1/6,1/2$ and $2/3$ filling fractions. For $1/6$ and $2/3$ filling, $(1;111)$
and $(1;000)$ STI phases, $(0;111)$ WTI phase, trivial band insulator, and
metal phase are found, while for $1/2$ filling, $(1;111)$ and $(1;000)$ STI
phases, trivial band insulator, and metal phase occur except $(0;111)$ WTI
phase. We have calculated the band structure and surface band structure for
the tight-binding model of the octahedron-decorated cubic lattice with spin-
orbit coupling and evaluated the $Z_{2}$ topological invariants. We have
analyzed and discussed the characters of the band structures and the surface
states of different phases. Although the octahedron-decorated cubic lattice we
considered is a toy model, our study points out an alternative path to search
for real topological materials. On the other hand, it might as well be built
from optical lattices due to their diversity and controllability.
###### Acknowledgements.
This work was supported by the National Natural Science Foundation of China
under Grant No. 11004028 and the Science and Technology Foundation of
Southeast University under Grant No. KJ2010417
## References
* (1) L. D. Landau, Phys. Z. Sowjetunion 11, 26 (1937).
* (2) K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980).
* (3) D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).
* (4) F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).
* (5) B. A. Bernevig and S. C. Zhang, Phys. Rev. Lett. 96, 106802 (2006).
* (6) C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005).
* (7) C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802 (2005).
* (8) B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Science, 314, 1757 (2006).
* (9) M. König, S. Wiedmann, C. Brüne, et al., Science 318, 766 (2007).
* (10) L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98, 106803 (2007).
* (11) J. E. Moore and L. Balents, Phys. Rev. B 75, 121306 (2007).
* (12) R. Roy, Phys. Rev. B 79, 195322 (2009).
* (13) L. Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007).
* (14) D. Hsieh, D. Qian, L. Wray, et al. Nature 542, 970 (2008).
* (15) Y. Xia, D. Qian, D. Hsieh, et al., Nature Phys. 5, 398 (2009).
* (16) H. Zhang, C. X. Liu, X. L. Qi, et al., Nature Phys. 5, 438 (2009).
* (17) Y. L. Chen, J. G. Analytis, J. H. Chu, et al., Science 325, 178 (2009).
* (18) D. Hsieh, Y. Xia, D. Qian, et al., Nature 460, 1101 (2009).
* (19) D. Hsieh, Y. Xia, D. Qian, et al., Phys. Rev. Lett. 103 146401 (2009).
* (20) H. M. Guo and M. Franz, Phys. Rev. B 80, 113102 (2009).
* (21) K. Sun, H. Yao, E. Fradkin, and S. A. Kivelson, Phys. Rev. Lett. 103, 046811 (2009).
* (22) A. Rüegg, J. Wen, and G. A. Fiete, Phys. Rev. B 81, 205115 (2010).
* (23) C. Weeks and M. Franz, Phys. Rev. B 82, 085310 (2010).
* (24) M. Kargarian and G. A. Fiete, Phys. Rev. B 82, 085106 (2010).
* (25) H. M. Guo and M. Franz, Phys. Rev. Lett. 103, 206805 (2009).
|
arxiv-papers
| 2011-01-19T10:07:03 |
2024-09-04T02:49:16.530060
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jing-Min Hou, Wen-Xin Zhang, and Guo-Xiang Wang",
"submitter": "Jing-Min Hou",
"url": "https://arxiv.org/abs/1101.3652"
}
|
1101.3766
|
# Quantum coherence between two atoms beyond $Q=10^{15}$
C. W. Chou chinwen@nist.gov D. B. Hume M. J. Thorpe D. J. Wineland T.
Rosenband Time and Frequency Division, National Institute of Standards and
Technology, Boulder, Colorado 80305
###### Abstract
We place two atoms in quantum superposition states and observe coherent phase
evolution for $3.4\times 10^{15}$ cycles. Correlation signals from the two
atoms yield information about their relative phase even after the probe
radiation has decohered. This technique was applied to a frequency comparison
of two 27Al+ ions, where a fractional uncertainty of $3.7^{+1.0}_{-0.8}\times
10^{-16}/\sqrt{\tau/s}$ was observed. Two measures of the Q-factor are
reported: The Q-factor derived from quantum coherence is
$3.4^{+2.4}_{-1.1}\times 10^{16}$, and the spectroscopic Q-factor for a Ramsey
time of 3 s is $6.7\times 10^{15}$. As part of this experiment, we demonstrate
a method to detect the individual quantum states of two Al+ ions in a
Mg+-Al+-Al+ linear ion chain without spatially resolving the ions.
Coherent evolution of quantum superpositions follows directly from
Schrödinger’s equation, and is a hallmark of quantum mechanics. Quantum
systems with a high degree of coherence are desirable for sensitive
measurements and for studies in quantum control. Typically, quantum
superposition states quickly decohere due to uncontrolled interactions between
the system and its environment. However, through careful isolation and system
preparation, quantum coherence has been observed in naturally occurring
systems including photons and atoms, as well as in engineered macroscopic
systems Haroche and Raimond (2006); Nat ; Leggett (2002); O’Connell et al.
(2010). In order to observe the coherence time of a system, it must be
compared to a reference system that is at least as coherent, a requirement
that can be difficult to satisfy, particularly in systems with the highest
degree of coherence. In atomic physics, quality (Q-) factors as high as
$1\times 10^{14}$ to $4\times 10^{14}$ Rafac et al. (2000); Boyd et al.
(2006); Chou et al. (2010a) have been observed with laser spectroscopy, where
the linewidths are often limited by laser noise rather than atomic
decoherence. In this report we apply a recent spectroscopic technique Chwalla
et al. (2007) to directly observe atomic coherence beyond the laser limit and
probe an atomic resonance with a Q-factor above $10^{15}$.
Historically, Mössbauer spectroscopy with $\gamma$-rays has exhibited the
highest relative coherence, as quantified by the spectroscopic Q-factor (the
ratio of oscillation frequency to observed resonance linewidth). Values as
high as $8.3\times 10^{14}$ are observed Potzel et al. (1976) in the 93.3 keV
radioactive decay of 67Zn, limited by the nuclear lifetime of 13.4 $\mu$s. In
those measurements, two separate crystals that contained 67Zn nuclei were
compared. One sample provided the probe radiation, while the other served as
the resonant absorber. The Mössbauer method might be extended to characterize
optical transitions in atoms Dehmelt and Nagourney (1988), but here we use a
method based on Ramsey spectroscopy in which the phase fluctuations of the
probe source are rejected as common-mode noise Chwalla et al. (2007), enabling
Ramsey times much longer than the probe coherence time. Other experiments that
compare pairs of microwave Bize et al. (2000) or optical clocks Katori et al.
(2010) use a related technique to reduce the Dick-effect noise Dick et al.
(1987); Lodewyck et al. (2010) that can limit the stability of frequency
comparisons.
Figure 1: (Color online) Illustration of the protocol. (a) The detected states
from the previous Ramsey experiment serve as the initial states for the the
current measurement. (b) The first $\pi/2$-pulse is applied. This is
accomplished by a laser beam whose axis coincides with the axis of the ion
array. (c) The clock state superpositions freely evolve. (d) The spacing is
adjusted at the end of the free-evolution period to vary the differential
phase $\Delta\phi$. This is followed immediately by the second $\pi/2$-pulse.
At the end of the sequence, the final states are detected to obtain the
correlation.
In the experiment reported here, atomic superposition states evolve coherently
for up to 5 s at a frequency of $1.12\times 10^{15}$ Hz. Following Chwalla et
al. Chwalla et al. (2007), a Ramsey pulse sequence Ramsey (1956) is
simultaneously applied to two trapped 27Al+ ions, labeled $i\in\\{1,2\\}$ (see
Fig. 1). The probe radiation for both ions is derived from the same source.
Each ion is initialized in one of the two quantum states that make up the
clock transition (clock states), which need not be the same for both ions.
Immediately prior to the second $\pi/2$-pulse, a variable displacement
$\mathbf{r_{i}}$ is applied to the ions. This Ramsey sequence induces a state
change with probability $p_{i}=(1+\cos{\delta\phi_{i}})/2$, where
$\delta\phi_{i}=\phi_{L}+\mathbf{k}\cdot\mathbf{r_{i}}-\phi_{i}$ is the
difference between the phase accumulated by the laser
($\phi_{L}$+$\mathbf{k}\cdot\mathbf{r_{i}}$) and ion ($\phi_{i}$) during the
free-evolution period $T$, and $\mathbf{k}$ is the laser beam wavevector,
$\mathbf{k}=\hat{z}2\pi/(267\text{ nm})$. The correlation probability (the
probability that both ions make a transition, or both do not make a
transition) is then
$P=[2+\cos{(\delta\phi_{1}-\delta\phi_{2})}+\cos{(\delta\phi_{1}+\delta\phi_{2})}]/4$.
Here the relative phase, $\delta\phi_{1}-\delta\phi_{2}$, is independent of
$\phi_{L}$, which is uniformly randomized over the interval $[0,2\pi)$ with a
pseudo-random number generator. Without knowledge of $\phi_{L}$, the
probability of correlated transitions is
$P_{c}=\frac{1}{2\pi}\int^{2\pi}_{0}Pd\phi_{L}=\frac{1}{2}+\frac{C}{2}\cos{\Delta\phi},$
(1)
where $\Delta\phi=\phi_{2}-\phi_{1}+\mathbf{k}\cdot(\mathbf{r_{1}-r_{2}})$ and
$C\equiv P_{c,\text{ max}}-P_{c,\text{ min}}\leq\frac{1}{2}$ is the contrast.
The correlation signal $P_{c}$ provides a measurement of the differential
phase evolution of the two Al+ “clock” ions similar to the measurement of
differential phase between source and absorber in Mössbauer spectroscopy. Its
statistical properties are equivalent to that of a single-ion Ramsey
experiment with reduced contrast, and the ultimate measurement uncertainty is
determined by quantum projection noise Itano et al. (1993). When
$|\Delta\phi|$ is kept near $\pi/2$, the statistical uncertainty of the ion-
ion fractional frequency difference, or measurement instability, is
$\sigma(\tau)\equiv\sigma_{\nu}/\nu=(2\pi\nu C\sqrt{T\tau})^{-1}$, where
$\tau$ is the total measurement duration, $\sigma_{\nu}$ is the uncertainty in
the measured frequency difference $(\phi_{2}-\phi_{1})/(2\pi T)$, and
$\nu\approx 1.12$ PHz is the transition frequency. Importantly, the free-
evolution period $T$ is not limited by laser phase noise.
In the experiment, a linear Paul trap confines one 25Mg+ ion and two Al+ ions
in an array Chou et al. (2010b); Hume et al. (2007) along the trap z-axis
(Fig. 1). The motional frequencies of a single Mg+ in the trap are
$\\{f_{x},f_{y},f_{z}\\}=\\{5.13,6.86,3.00\\}$ MHz. The ions are maintained in
the order of Mg+-Al+-Al+ (inter-ion spacing 2.69 $\mu$m) by periodically
adjusting the trap conditions and verifying via Mg+ spectroscopy the frequency
of the “stretch” mode of motion, whose value is 5.1 MHz for the correct order
DBH .
The two states involved in the Al+ clock transition,
$|\\!\\!\downarrow\rangle\equiv|^{1}S_{0}$, $m_{F}=5/2\rangle$ and
$|\\!\\!\uparrow\rangle\equiv|^{3}P_{0}$, $m_{F}=5/2\rangle$, are detected
with an adaptive quantum non-demolition process Hume et al. (2007). The
present implementation distinguishes all four states
$|\\!\\!\downarrow_{1}\downarrow_{2}\rangle$,
$|\\!\\!\downarrow_{1}\uparrow_{2}\rangle$,
$|\\!\\!\uparrow_{1}\downarrow_{2}\rangle$, and
$|\\!\\!\uparrow_{1}\uparrow_{2}\rangle$ by observing Mg+ fluorescence after
controlled interactions between the Al+ and Mg+ ions. Individual state
detection relies on the two Al+ ions having different amplitudes in several
motional eigenmodes, which affects the state-mapping probability onto the Mg+
ion. Information from several measurements is combined in a Bayesian process
Hume et al. (2007), to determine the most likely state of the two Al+ ions
with typically 99 % probability in an average of 30 detection cycles
(approximately 50 ms total duration). We note that this technique allows
individual state detection of two ions in the same trap, without the need for
high spatial-resolution optics.
The Ramsey experiments use $\pi/2$-pulse durations of 1.2 ms and are carried
out for various free-evolution periods $T$. For each $T$,
$\Delta\phi_{z}\equiv\mathbf{k}\cdot(\mathbf{r_{1}-r_{2}})$ is varied from 0
to beyond $2\pi$ to characterize the correlation. The duration required to
shift the positions by $\mathbf{r_{i}}$ is approximately 10 ms. Figure 2 shows
the correlation signals for $T$ between 0.1 and 5 s. Currently, collisions
between the ions and background gas make it impractical to generate sufficient
statistics for $T$ greater than 5 s. The collisions result in changes of ion
order and loss of ions due to chemical reactions.
Figure 2: (Color online) Correlation probabilities $P_{c}$ versus
$\Delta\phi_{z}$ for various Ramsey times: (a) 0.1 s, 1500 probes; (b) 0.5 s,
600 probes;(c) 1 s, 600 probes; (d) 2 s, 360 probes; (e) 3 s, 300 probes; (f)
5 s, 100 probes. Dots: measurement outcomes; lines: maximum-likelihood fits to
the fringes.
The phase difference $\phi_{2}-\phi_{1}$, and thus the frequency difference,
between the two Al+ ions can be determined from the phases of the $P_{c}$
fringes in Fig. 2. In the experiment, we apply a magnetic field gradient of
$dB/dz=1.32\pm 0.33$ mT/m, as measured by monitoring the frequency of the
$|F=3\text{, }m_{F}=-3\rangle\rightarrow|F=2\text{, }m_{F}=-2\rangle$
magnetic-field dependent transition in the 25Mg+ $3s\text{ }S_{1/2}$ ground
state hyperfine manifold, when the Mg+ position along the trap axis is
adjusted. This gradient induces a fractional frequency shift
$(\nu_{2}-\nu_{1})/\nu=1.32\pm 0.33\times 10^{-16}$ between the
$|\\!\\!\downarrow\rangle\leftrightarrow|\\!\\!\uparrow\rangle$ transitions of
the two Al+ ions. The phases of the $P_{c}$ fringes, determined by maximum-
likelihood fits Sivia and Skilling (2006), increase linearly with $T$, as
shown in Fig. 3a. A linear fit has a slope of $0.84\pm 0.06$ rad/s,
corresponding to a measured shift of $1.19\pm 0.08\times 10^{-16}$, in
agreement with the shift caused by the magnetic-field gradient. All reported
uncertainties represent a 68 % confidence interval.
We derive the contrast $C$ from the maximum-likelihood fits to the data in
Fig. 2. An exponential fit of $C$ versus $T$ yields a relative coherence time
$T_{C}$ of $9.7^{+6.9}_{-3.1}$ s, corresponding to a Q-factor ($Q=\pi\nu
T_{C}$ Vion et al. (2002)) of $3.4^{+2.4}_{-1.1}\times 10^{16}$. A uniform
prior distribution of $T_{C}$ on the interval 0 s to 25 s is assumed. The
measured coherence time is compatible with the expected result, which is given
by the lifetime $T^{\prime}=20.6\pm 1.4$ s Rosenband et al. (2007) of the Al+
$|^{3}P_{0}\rangle$ state. When viewed in terms of Ramsey spectroscopy, for
$T=3$ s, the full-width-at-half-maximum of the Ramsey signal corresponds to a
Q-factor of $2\nu T=6.7\times 10^{15}$.
Figure 3: (Color online) (a) Differential phase $\phi_{2}-\phi_{1}$ versus
Ramsey time $T$. The solid line is a linear fit, with slope $0.84\pm 0.06$
rad/s. (b) Measurement uncertainty extrapolated to 1 s averaging time as a
function of Ramsey time. Dots: measurement results, where the uncertainties
are derived from the uncertainties in the contrast $C$; solid line:
theoretical lifetime-limited instability, where only phases corresponding to
$\Delta\phi\approx\pm\pi/2$ are probed; dashed line: expected experimental
instability, with $\Delta\phi$ uniformly distributed over $[0,2\pi)$. The
dashed line is derived from the measured coherence time of 9.7 s, and an
approximate overhead of 100 ms per Ramsey measurement, which reduces the duty
cycle.
The current protocol could significantly reduce the total duration of future
high-precision measurements with atomic clocks. Figure 3b shows the
measurement uncertainties extrapolated to 1 s ($\sigma_{1s}$) versus Ramsey
time $T$. The long-term statistical uncertainty is then
$\sigma(\tau)=\sigma_{1s}/\sqrt{\tau/s}$ for a measurement duration $\tau$.
Note that, for $T=3$ s, the frequency difference between the two Al+ ions can
be determined with a fractional uncertainty $\sigma=1.1\times 10^{-17}$ in a
1126 s measurement (900 s integrated free-evolution time), which can be
extrapolated to infer a relative measurement uncertainty
$\sigma_{1s}=3.7\times 10^{-16}$. This result may be compared to a recent
frequency difference measurement of two Al+ clocks, where 65,000 s were
required to reach the same uncertainty of $1.1\times 10^{-17}$ Chou et al.
(2010c). In general, the lifetime-limited contrast is
$C=\frac{1}{2}\exp(-T/T^{\prime})$, yielding an instability of
$\sigma(\tau)=\exp(T/T^{\prime})/(\pi\nu\sqrt{T\tau})$, which is shown for Al+
in Fig. 3b (solid line). The optimal probe time of $T=T^{\prime}/2$ yields
$\sigma_{1s}=1.4\times 10^{-16}$.
Although we have used the technique to measure two ions in the same trap, it
may also be applied to clocks at different locations. A proposed frequency
comparison of remote optical clocks is depicted in Fig. 4. Note, however, that
due to the requirement that $\phi_{L}$ be the same for both clocks, this
technique is limited to comparisons between clocks operating at similar
frequencies. Although the clocks need not be identical, the differential
phase, $\Delta\phi$, must be known well enough to make phase errors of $\pi$
unlikely. In order to retain control over the differential phase, the
individual paths (paths 1 and 2 in Fig. 4) need to be phase-stabilized and the
Ramsey pulses at the two locations need to be synchronized so that the two
clocks experience the same laser phase noise. For ions with very long
radiative lifetimes, the same technique could be used to compare two ion
samples, each composed of maximally entangled states Leibfried et al. (2005);
Monz et al. .
Figure 4: (Color online) Proposed frequency cmparison of remote optical
clocks, here based on Al+ ions. The paths 1 and 2 that direct the clock laser
light to the ions need to be controlled so that they can faithfully transmit
the light without introducing additional phase noise. Local frequency
fluctuations, such as those caused by fluctuating magnetic fields, should be
minimized. The free evolution periods need to be synchronized so that the
atoms are subjected to the same phase noise in the Ramsey pulses, the effect
of which cancels in the protocol.
A similar approach can be taken in comparisons of two clocks composed of many
unentangled atoms. The measurement protocol is again based on synchronized
Ramsey pulses where the free-evolution time $T$ exceeds the laser coherence
time. The two clocks (labeled $X\in\\{A,B\\}$) measure transition
probabilities $p_{X}=\frac{1}{2}[1+\cos{(\phi_{X}-\phi_{L}-\theta_{X})}]$, and
the quantity of interest $\delta\phi_{AB}=\phi_{A}-\phi_{B}$ is determined
from
$\delta\phi_{AB}=\cos^{-1}{(2p_{A}-1)}-\cos^{-1}{(2p_{B}-1)}+\theta_{A}-\theta_{B}$,
where $\theta_{X}$ are the controlled laser phase offsets at the two clocks.
If we consider only atomic projection noise in $p_{A}$ and $p_{B}$, this
measurement has a variance of
$var(\delta\phi_{AB})=\frac{1}{N_{A}}+\frac{1}{N_{B}}$, where $N_{A}$ and
$N_{B}$ are the numbers of atoms in clocks $A$ and $B$. The fractional
frequency stability of the clock comparison is then
$\sigma_{y}(\tau)=\sqrt{var(\delta\phi_{AB})}/(2\pi\nu\sqrt{T\tau})$.
A complication is introduced by the fact that $\phi_{L}$ will be initially
unknown, which leads to ambiguities in the trigonometric inversions from which
$\delta\phi_{AB}$ is calculated. Such ambiguities will be absent in the
majority of measurements, if an approximate value of $\delta\phi_{AB}$ can be
determined through prior calibrations (with $var(\delta\phi_{AB})\ll 1)$, and
phase offsets $\theta_{X}$ are adjusted such that
$\phi_{A}-\phi_{B}-(\theta_{A}-\theta_{B})\approx\pi/2$. After this
calibration procedure, $p_{A}$ and $p_{B}$ represent approximate quadratures
of the laser-atom phase difference, and for most values of $\phi_{L}$ the
trigonometric inversions are unambiguous. In such a measurement the Ramsey
free-evolution time is no longer constrained by laser decoherence, and the
Dick effect due to the probe source is absent. Therefore, more rapid frequency
comparisons of similar-frequency many-atom optical clocks should also be
possible.
Small values of $\sigma(\tau)$ in frequency comparisons are useful for
evaluating and improving the performance of optical clocks and for
metrological applications. For example, comparison of clocks in geographically
distinct locations can be used to evaluate spatial and temporal variations in
the geoid Chou et al. (2010a). More generally, any physical process that leads
to small, constant frequency shifts in an optical clock can be studied in this
way. This includes relativistic effects as well as shifts caused by electric
fields, magnetic fields and atom collisions. Our observation of a Q-factor
beyond $10^{15}$ and a frequency ratio measurement instability of $3.7\times
10^{-16}/\sqrt{\tau/s}$ highlights the intrinsic sensitivity of optical clocks
as a metrological tool.
This work is supported by ONR, AFOSR, DARPA, and IARPA. We thank D. Leibrandt
and J. Sherman for comments on the manuscript. Publication of NIST, not
subject to U.S. copyright.
## References
* Haroche and Raimond (2006) S. Haroche and J. M. Raimond, _Exploring the quantum_ (Oxford University Press, Oxford, U.K., 2006).
* (2) Quantum coherence experiments in several technologies are reviewed in R. Blatt and D. Wineland, Nature 453, 1008 (2008); I. Bloch, ibid. 453, 1016 (2008); H. J. Kimble, ibid. 453, 1023 (2008); J. Clarke and F. K. Wilhelm, ibid. 453, 1031 (2008); R. Hanson and D. D. Awschalom, ibid. 453, 1043 (2008).
* Leggett (2002) A. J. Leggett, J. Phys.: Condens. Matter 14, R415 (2002).
* O’Connell et al. (2010) A. D. O’Connell et al. Nature 464, 697 (2010).
* Rafac et al. (2000) R. J. Rafac, B. C. Young, J. A. Beall, W. M. Itano, D. J. Wineland, and J. C. Bergquist, Phys. Rev. Lett. 85, 2462 (2000).
* Boyd et al. (2006) M. M. Boyd, T. Zelevinsky, A. D. Ludlow, S. M. Foreman, S. Blatt, T. Ido, and J. Ye, Science 314, 1430 (2006).
* Chou et al. (2010a) C. W. Chou, D. B. Hume, D. J. Wineland, and T. Rosenband, Science 329, 5999 (2010a).
* Chwalla et al. (2007) M. Chwalla, K. Kim, T. Monz, P. Schindler, M. Riebe, C. F. Roos, and R. Blatt, Appl. Phys. B 89, 483 (2007).
* Potzel et al. (1976) W. Potzel, A. Forster, and G. M. Kalvius, J. De Physique C6, 691 (1976).
* Dehmelt and Nagourney (1988) H. Dehmelt and W. Nagourney, Proc. Nad. Acad. Sci. 85, 7426 (1988).
* Bize et al. (2000) S. Bize, Y. Sortais, P. Lemonde, S. Zhang, P. Laurent, G. Santarelli, C. Salomon, and A. Clairon, Ultrasonics, Ferroelectrics and Frequency Control, IEEE Transactions on 47, 1253 (2000).
* Katori et al. (2010) H. Katori, T. Takano, and M. Takamoto, in _Proceedings of the 22nd International Conference on Aomic Physics_ (2010), to be published.
* Dick et al. (1987) G. J. Dick, J. D. Prestage, C. A. Greenhall, and L. Maleki, in _Proc. 19th Precise Time and Time interval Mtg._ (1987), p. 133.
* Lodewyck et al. (2010) J. Lodewyck, P. G. Westergaard, A. Lecallier, L. Lorini, and P. Lemonde, New J. Phys. 12, 065026 (2010).
* Ramsey (1956) N. F. Ramsey, _Molecular Beams_ (Oxford University Press, New York, 1956).
* Itano et al. (1993) W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, D. J. Heinzen, F. L. Moore, M. G. Raizen, and D. J. Wineland, Phys. Rev. A 47, 3554 (1993).
* Chou et al. (2010b) C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T. Rosenband, Phys. Rev. Lett. 104, 070802 (2010b).
* Hume et al. (2007) D. B. Hume, T. Rosenband, and D. J. Wineland, Phys. Rev. Lett. 99, 120502 (2007).
* (19) D. B. Hume, Ph.D. Thesis, University of Colorado (2010).
* Sivia and Skilling (2006) D. S. Sivia and J. Skilling, _Data Analysis: A Bayesian Tutorial, Second Edition_ (Oxford University Press, Oxford, U.K., 2006).
* Vion et al. (2002) D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, D. Esteve, and M. H. Devoret, Science 296, 886 (2002).
* Rosenband et al. (2007) T. Rosenband et al. Phys. Rev. Lett. 98, 220801 (2007).
* Chou et al. (2010c) C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T. Rosenband, Phys. Rev. Lett. 104, 070802 (2010c).
* Leibfried et al. (2005) D. Leibfried et al. Nature 438, 639 (2005).
* (25) T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Han̈sel, M. Hennrich, and R. Blatt, arXiv:1009.6126v1.
|
arxiv-papers
| 2011-01-19T20:13:11 |
2024-09-04T02:49:16.536033
|
{
"license": "Public Domain",
"authors": "C. W. Chou, D. B. Hume, M. J. Thorpe, D. J. Wineland, and T. Rosenband",
"submitter": "Chin-wen Chou",
"url": "https://arxiv.org/abs/1101.3766"
}
|
1101.3828
|
# Mean-field Study of Charge, Spin, and Orbital Orderings
in Triangular-lattice Compounds $A$NiO2 ($A$=Na, Li, Ag)
Hiroshi Uchigaito E-mail address: uchigaito@aion.t.u-tokyo.ac.jp Masafumi
Udagawa and Yukitoshi Motome Department of Applied PhysicsDepartment of
Applied Physics University of Tokyo University of Tokyo Hongo Hongo Bunkyo-ku
Bunkyo-ku Tokyo 113-8656 Tokyo 113-8656 Japan Japan
###### Abstract
We present our theoretical results on the ground states in layered triangular-
lattice compounds $A$NiO2 ($A$=Na, Li, Ag). To describe the interplay between
charge, spin, orbital, and lattice degrees of freedom in these materials, we
study a doubly-degenerate Hubbard model with electron-phonon couplings by the
Hartree-Fock approximation combined with the adiabatic approximation. In a
weakly-correlated region, we find a metallic state accompanied by
$\sqrt{3}\times\sqrt{3}$ charge ordering. On the other hand, we obtain an
insulating phase with spin-ferro and orbital-ferro ordering in a wide range
from intermediate to strong correlation. These phases share many
characteristics with the low-temperature states of AgNiO2 and NaNiO2,
respectively. The charge-ordered metallic phase is stabilized by a compromise
between Coulomb repulsions and effective attractive interactions originating
from the breathing-type electron-phonon coupling as well as the Hund’s-rule
coupling. The spin-orbital-ordered insulating phase is stabilized by the
cooperative effect of electron correlations and the Jahn-Teller coupling,
while the Hund’-rule coupling also plays a role in the competition with other
orbital-ordered phases. The results suggest a unified way of understanding a
variety of low-temperature phases in $A$NiO2. We also discuss a keen
competition among different spin-orbital-ordered phases in relation to a
puzzling behavior observed in LiNiO2.
multi-orbital Hubbard model, electron-phonon coupling, charge order, orbital
order, metal-insulator transition, triangular lattice, NaNiO2, LiNiO2, AgNiO2
## 1 Introduction
One of the most distinctive features of strongly-correlated electron systems
is diverse cooperative phenomena. [1, 2] Correlated electron systems generally
show intricate phase diagrams full of competing or coexisting states, and the
phase competition often leads to exotic many-body phenomena. Among many
aspects, there are two factors which promote such complexity, i.e., multiple
degrees of freedom [2, 3, 4] and geometrical frustration [5]. The multiple
degrees of freedom are composed of charge, spin, and orbital of electrons.
Strong electron correlations induce interplay among them, resulting in a
variety of phases. In addition, the interplay often causes exotic response to
external perturbations, such as the colossal magneto-resistance in perovskite
manganites [6]. On the other hand, the geometrical frustration promotes the
phase competition. In general, the geometrical frustration results in a huge
number of low-energy degenerate states, by suppressing conventional long-range
orders. The degeneracy yields nontrivial phenomena such as complicated
ordering, glassy behavior, and spin-liquid states. It is a long-standing
problem in condensed-matter physics to understand a variety of phenomena which
emerge from synergetic effects between the multiple degrees of freedom and the
geometrical frustration.
The family of compounds $A$NiO2 ($A$=Li, Na, Ag) is a typical example of such
geometrically-frustrated systems with multiple degrees of freedom. $A$NiO2
takes two different lattice structures depending on the cation $A$, that is,
the delafossite structure [7] (AgNiO2) and the ordered rock-salt structure [8]
(NaNiO2 and LiNiO2). Both structures are quasi-two-dimensional, composed of
stacking of Ni, O, and $A$ layers. The magnetic and transport properties are
dominated by Ni cations, which are surrounded by the octahedron of oxygens.
The NiO6 octahedra share their edges so that the Ni sites constitute the
frustrated triangular layers as shown in Fig. 1. Ni3+ cation has seven $3d$
electrons in the low-spin configuration: Six out of seven fully occupy the
lower $t_{2g}$ levels and the remaining one electron enters in the higher
$e_{g}$ levels. Hence the doubly-degenerate $e_{g}$ orbital degree of freedom
is active in these systems.
Figure 1: (Color online). Schematic picture of NiO6 layer in $A$NiO2. NiO6
octahedra share their edges with neighbors to form the triangular lattice of
Ni cations. $\gamma$ denotes the Ni-Ni bond directions. Two $e_{g}$ orbitals,
$3z^{2}-r^{2}$ and $x^{2}-y^{2}$, are shown.
$A$NiO2 shows a variety of behaviors depending on the cation $A$ in spite of
the similarity in lattice and electronic structure. NaNiO2 is a Mott insulator
with a gap of 0.24eV [9]. This compound shows a first-order structural
transition accompanied by cooperative Jahn-Teller distortions at 480K [8, 10]
and a second-order magnetic transition at 20K [10, 12, 11]. At the lowest
temperature($T$), the system exhibits orbital-ferro and A-type antiferro-spin
order (antiferromagnetic stacking of spin-ferro ordered layers)[13, 11].
LiNiO2 is also a Mott insulator with a gap of 0.2eV [14], however, it shows no
clear phase transition down to 1.4K in contrast to NaNiO2 [15]. There is
strong sample dependence in $T$ dependence of the magnetic susceptibility; the
field-cool and zero-field-cool bifurcation appears in different ways depending
on samples. This sample dependence strongly suggests the relevance of
extrinsic disorder [16]. The ground state as well as the finite-$T$ properties
remain controversial [16, 17, 18, 19, 20, 21, 22, 23, 24]. In addition to
these insulating materials, recently, a new metallic compound AgNiO2 was
synthesized [7, 25]. It shows a structural transition associated with a
$\sqrt{3}\times\sqrt{3}$ charge ordering at 365K and antiferromagnetic
transition at 20K [26, 27]. The system remains metallic down to the lowest
$T$. It was claimed that in the low-$T$ phase the system separates into rather
localized spins at 1/3 Ni sites and itinerant electrons at the remaining sites
[26, 27]. Magnetic properties at low $T$ were analyzed by considering the
competing nearest- and second-neighbor exchange couplings between localized
spins [28, 29].
So far, $A$NiO2 has been theoretically studied by the first-principle
calculations and the strong-coupling analyses. The orbital and spin ordering
observed in NaNiO2 was reproduced by the LSDA+$U$ first-principle calculations
[30, 31]. The metallicity as well as the charge ordering in AgNiO2 was also
reproduced by the first-principle calculations [25, 26]. On the other hand,
effective models in the limit of strong electron correlation, the so-called
Kugel-Khomskii models, were studied to understand the orbital and spin
ordering in NaNiO2 and the peculiar disordered state in LiNiO2 [20, 32, 33].
For AgNiO2, recently, the magnetic phase diagram was investigated by the
classical spin model with ignoring the itinerant electrons [34].
Despite the extensive studies so far, comprehensive understanding of the
ground states of $A$NiO2 has not been reached yet. Although the low-$T$ states
of NaNiO2 and AgNiO2 are reproduced by the first-principle calculations, the
mechanism of stabilizing these states is not fully clarified. In addition, the
effective-model approach does not fully succeed in reproducing the ground
states of NaNiO2 and LiNiO2 in an unified way. A possible way to explore the
comprehensive understanding is to investigate a model in a wide region of
interaction parameters systematically beyond the strong-coupling approach. In
fact, in both NaNiO2 and LiNiO2, the Mott gap is not large; the gap is
comparable to the transfer integrals. Furthermore, the newly-synthesized
AgNiO2 shows metallic behavior. These facts suggest an importance of charge
fluctuations in weakly- or intermediately-correlated regions. Electron-phonon
couplings may be another essential factors, which have not been considered
seriously in spite of the experimental facts that the structural transitions
are observed in NaNiO2 and AgNiO2.
In this study, aiming at a unified picture of these compounds $A$NiO2, we
investigate a multi-orbital Hubbard model with electron-phonon couplings on a
two-dimensional triangular lattice. Our purpose is to elucidate the
microscopic mechanism for the variety of phases in these compounds $A$NiO2. In
particular, we focus on electron-phonon couplings and charge degrees of
freedom, both of which have not been carefully examined in the previous
studies. We clarify that these two elements play an important role in the
phase competition in these complicated systems. We obtain a
$\sqrt{3}\times\sqrt{3}$ charge-ordered metallic (COM) phase in the weakly-
correlated region and an insulating phase with ferro-type spin and orbital
ordering in the intermediately- to strongly-correlated region. These two
phases reproduce many aspects of the low-$T$ states in AgNiO2 and NaNiO2,
respectively. We discuss the peculiar disordered state in LiNiO2 in relation
with a keen phase competition in the obtained phase diagram.
The organization of this paper is as follows. In Sec. 2, we describe our model
and method. We introduce the Hamiltonian term by term, and present the
approximations adopted in the calculations. We present our results in Sec. 3.
We discuss the parameter region and the mechanism to stabilize the
$\sqrt{3}\times\sqrt{3}$ COM phase and the spin-orbital-ordered insulating
phase. Finally, Sec. 4 is devoted to summary.
## 2 Model and Method
### 2.1 Model
In the present study, to elucidate the phase competition in $A$NiO2, we
investigate the ground state of the multi-orbital Hubbard model with electron-
phonon couplings. Among the five $3d$ orbitals, we consider only the twofold
degenerate $e_{g}$ orbitals, by taking account of the low-spin state of Ni3+
cations. Our Hamiltonian is written as
$\mathcal{H}=\mathcal{H}_{\rm{kin}}+\mathcal{H}_{\rm{int}}+\mathcal{H}_{\text{el-
ph}}+\mathcal{H}_{\rm{ph}},$ (1)
where $\mathcal{H}_{\rm{kin}}$, $\mathcal{H}_{\rm{int}}$,
$\mathcal{H}_{\text{el-ph}}$, and $\mathcal{H}_{\rm{ph}}$ represent the
kinetic term of electrons, the electron-electron interactions, the electron-
phonon couplings, and the elastic term of phonons, respectively. We describe
the detailed forms of each term in the following.
#### 2.1.1 Kinetic term
Due to the spatial anisotropy of the $e_{g}$-orbital wave functions, transfer
integrals between Ni sites depend on the bond direction as well as orbital
types (see Fig. 1). The kinetic term in eq. (1) is written as
$\mathcal{H}_{\rm{kin}}=-\sum_{\langle
ij\rangle}\sum_{\alpha,\beta}\sum_{\sigma}t^{\gamma_{ij}}_{\alpha\beta}\bigl{(}c_{i\alpha\sigma}^{\dagger}c_{j\beta\sigma}+\rm{H.c}.\bigr{)}.$
(2)
Here, $i$ and $j$ denote the site indices, $\alpha$ and $\beta$ represent the
orbital indices with $\alpha=a(b)$ corresponding to the $3z^{2}-r^{2}$
($x^{2}-y^{2}$) orbital, $\sigma$ is the spin, and $\gamma_{ij}$ denotes the
direction of bond between the site $i$ and $j$, as shown in Fig. 1. The sum
over $\langle ij\rangle$ is taken for the nearest-neighbor sites on the
triangular lattice. The transfer integrals are given by the following
matrices;
$t^{\gamma=1}=\begin{pmatrix}t&0\\\ 0&t^{\prime}\end{pmatrix},\
t^{\gamma=2}=\begin{pmatrix}\displaystyle t_{2}&t_{3}\\\
t_{3}&t_{4}\end{pmatrix},\ t^{\gamma=3}=\begin{pmatrix}t_{2}&-t_{3}\\\
-t_{3}&t_{4}\end{pmatrix},$ (3)
for the two bases of $3z^{2}-r^{2}$ and $x^{2}-y^{2}$ orbitals. From the
symmetry of orbitals, we obtain the following relations:
$t_{2}=t/4+3t^{\prime}/4$, $t_{3}=\sqrt{3}(t-t^{\prime})/4$, and
$t_{4}=3t/4+t^{\prime}/4$, with two independent parameters, $t$ and
$t^{\prime}$. We set $t=1$ as an energy scale. The value of $t^{\prime}$
depends on both $d$-$d$ direct transfer integrals and $d$-$p$-$d$ indirect
ones in a complicated manner [32, 33]. In the following, we show the results
for $t^{\prime}=-1$ by noting that the orbital overlaps between atomic
orbitals at the neighboring sites lead to $t^{\prime}/t\sim-1$ when one
consider both contributions. An extended study in wider range of
$t^{\prime}/t$ for an effective model without phonon is found in Ref. Vernay.
The choice of $t$ and $t^{\prime}$ gives the non-interacting bandwidth $8t$.
#### 2.1.2 Electron-electron interactions
Next we introduce the electron-electron interaction term
$\mathcal{H}_{\rm{int}}$ in eq. (1). We consider only the on-site Coulomb
interactions. For the doubly-degenerate $e_{g}$ orbital system,
$\mathcal{H}_{\rm{int}}$ is written as
$\displaystyle\mathcal{H}_{\rm{int}}=\mathcal{H}_{U}+\mathcal{H}_{U^{\prime}}+\mathcal{H}_{J_{\rm{H}}}+\mathcal{H}_{J_{\rm{H}}^{\prime}},$
(4)
where
$\displaystyle\mathcal{H}_{U}=$ $\displaystyle\
U\sum_{i}\sum_{\alpha}n_{i\alpha\uparrow}n_{i\alpha\downarrow},$ (5)
$\displaystyle\mathcal{H}_{U^{\prime}}=$ $\displaystyle\
U^{\prime}\sum_{i}\sum_{\sigma\sigma^{\prime}}n_{ia\sigma}n_{ib\sigma^{{}^{\prime}}},$
(6) $\displaystyle\mathcal{H}_{J_{\rm{H}}}=$ $\displaystyle\
J_{\rm{H}}\sum_{i}\sum_{\sigma\sigma^{\prime}}c^{\dagger}_{ia\sigma}c^{\dagger}_{ib\sigma^{\prime}}c_{ia\sigma^{\prime}}c_{ib\sigma},$
(7) $\displaystyle\mathcal{H}_{J_{\rm{H}}^{\prime}}=$ $\displaystyle\
J_{\rm{H}}^{\prime}\sum_{i}\sum_{\alpha\neq\alpha^{\prime}}c^{\dagger}_{i\alpha\uparrow}c^{\dagger}_{i\alpha\downarrow}c_{i\alpha^{\prime}\downarrow}c_{i\alpha^{\prime}\uparrow}.$
(8)
Here $n_{i\alpha\sigma}=c^{\dagger}_{i\alpha\sigma}c_{i\alpha\sigma}$, $U$ and
$U^{\prime}$ denote the intra- and inter-orbital Coulomb repulsions, and
$J_{\rm{H}}$ and $J_{\rm{H}}^{\prime}$ denote the exchange interaction and the
pair hopping, respectively. $\mathcal{H}_{J_{\rm{H}}}$ and
$\mathcal{H}_{J_{\rm{H}}^{\prime}}$ are called the Hund’s-rule couplings.
Hereafter we assume the relations $\ U^{\prime}=U-2J_{\rm{H}}$ and
$J_{\rm{H}}=J_{\rm{H}}^{\prime}$ to retain the rotational symmetry of the
Coulomb interaction.
#### 2.1.3 Electron-phonon couplings
As to the electron-phonon couplings, we consider two relevant distortions of
NiO6 octahedra in $A$NiO2, namely, the $A_{1g}$ breathing mode and the $E_{g}$
Jahn-Teller modes. $\mathcal{H}_{\text{el-ph}}$ in eq. (1) is given by the sum
of these two contributions as
$\mathcal{H}_{\text{el-ph}}=\mathcal{H}_{\text{el-
ph}}^{\rm{br}}+\mathcal{H}_{\text{el-ph}}^{\rm{JT}}.$ (9)
The $A_{1g}$ mode corresponds to the isotropic expansion (contraction) of NiO6
octahedron [Fig. 2(a)], which lowers (raises) two $e_{g}$ energy levels
without lifting their degeneracy. Namely, the $A_{1g}$ mode couples to the
local charge on each Ni site, written in the form
$\mathcal{H}_{\text{el-ph}}^{\rm{br}}=-\gamma_{\rm{br}}\sum_{i}x_{{\rm
br},i}\left(n_{ia}+n_{ib}-1\right),$ (10)
where $n_{i\alpha}=\sum_{\sigma}n_{i\alpha\sigma}$, $x_{{\rm br},i}$ is the
amplitude of the $A_{1g}$ lattice distortion, and $\gamma_{\rm{br}}>0$ is the
corresponding coupling constant. A positive (negative) $x_{{\rm br},i}$
corresponds to an expansion (contraction).
The $E_{g}$ mode has two components, $E_{g,u}$ and $E_{g,v}$, as shown in
Figs. 2(b) and 2(c), respectively. The $E_{g,u}$ mode corresponds to the
$z$-axis elongation of NiO6 octahedron, which splits the energy levels of
$x^{2}-y^{2}$ and $3z^{2}-r^{2}$ orbitals, while the $E_{g,v}$ mode causes a
mixing of the two orbitals: The coupling to the $E_{g}$ modes is written as
$\displaystyle\mathcal{H}_{\text{el-ph}}^{\rm{JT}}=-\gamma_{\rm{JT}}$
$\displaystyle\sum_{i,\sigma}\Bigl{\\{}x_{{\rm
JT},i}\left(n_{ia\sigma}-n_{ib\sigma}\right)$
$\displaystyle\quad+\bar{x}_{{\rm
JT},i}(c_{ia\sigma}^{\dagger}c_{ib\sigma}+c_{ib\sigma}^{\dagger}c_{ia\sigma})\Bigr{\\}},$
(11)
where $x_{{\rm JT},i}$ and $\bar{x}_{{\rm JT},i}$ are the amplitudes of
$E_{g,u}$ and $E_{g,v}$ modes, respectively, and $\gamma_{\rm{JT}}$ represents
the common coupling constant.
Figure 2: Schematic pictures of the displacement of oxygens (open circles)
around a Ni cation at the origin in (a) $A_{1g}$ mode, (b) $E_{g,u}$ mode, and
(c) $E_{g,v}$ mode.
#### 2.1.4 Phonon term
The phonon term $\mathcal{H}_{\rm{ph}}$ in eq. (1) consists of the on-site
term ($\mathcal{H}_{\rm{ph}}^{\rm{elastic}}$) and the inter-site term
($\mathcal{H}_{\rm{ph}}^{\rm{coop}}$) as,
$\displaystyle\mathcal{H}_{\rm{ph}}=$
$\displaystyle\mathcal{H}_{\rm{ph}}^{\rm{elastic}}+\mathcal{H}_{\rm{ph}}^{\rm{coop}}.$
(12)
Each term is given by the sum of contributions from the $A_{1g}$ and $E_{g}$
mode phonons. The first term $\mathcal{H}_{\rm{ph}}^{\rm{elastic}}$ is the
elastic energy of lattice distortions, which is given by the sum of the
following two terms;
$\displaystyle\mathcal{H}_{\rm{ph}}^{\rm{elastic,br}}=$
$\displaystyle\frac{1}{2}\sum_{i}x_{{\rm br},i}^{2},$ (13)
$\displaystyle\mathcal{H}_{\rm{ph}}^{\rm{elastic,JT}}=$
$\displaystyle\frac{1}{2}\sum_{i}\left(x_{{\rm JT},i}^{2}+\bar{x}_{{\rm
JT},i}^{2}\right).$ (14)
The elastic constants of $A_{1g}$ mode and $E_{g}$ modes are taken as unity
without losing generality, by normalizing the amplitudes of lattice
distortions ($x_{{\rm br},i}$, $x_{{\rm JT},i}$, and $\bar{x}_{{\rm JT},i}$)
and the coupling constants ($\gamma_{\rm{br}}$ and $\gamma_{\rm{JT}}$).
The second term $\mathcal{H}_{\rm{ph}}^{\rm{coop}}$ describes the cooperative
couplings of lattice distortions. The precise form of the couplings is, in
principle, determined by the phonon dispersion in the materials, but here, for
simplicity, we consider the nearest-neighbor couplings only. Then
$\mathcal{H}_{\rm{ph}}^{\rm{coop}}$ is defined by the sum of two terms;
$\displaystyle\mathcal{H}_{\rm{ph}}^{\rm{coop,br}}=$
$\displaystyle\lambda_{\rm{br}}\sum_{\langle ij\rangle}x_{{\rm br},i}x_{{\rm
br},j},$ (15) $\displaystyle\mathcal{H}_{\rm{ph}}^{\rm{coop,JT}}=$
$\displaystyle-\lambda_{\rm{JT}}\sum_{\langle ij\rangle}x_{{\rm JT},i}x_{{\rm
JT},j}-\bar{\lambda}_{\rm{JT}}\sum_{\langle ij\rangle}\bar{x}_{{\rm
JT},i}\bar{x}_{{\rm JT},j},$ (16)
with the coupling constants $\lambda_{\rm{br}}$, $\lambda_{\rm{JT}}$, and
$\bar{\lambda}_{\rm{JT}}$. Although the values of $\lambda_{\rm{JT}}$ and
$\bar{\lambda}_{\rm{JT}}$ are generally different, we take
$\lambda_{\rm{JT}}=\bar{\lambda}_{\rm{JT}}$ for simplicity. It is reasonable
to assume $\lambda_{\rm{br}}$ to be positive, i.e., the “antiferro”-type
coupling, since the $A_{1g}$-type expansion (contraction) of NiO6 octahedron
tends to shrink (expand) the neighboring octahedra due to the edge-sharing
network of octahedra. On the other hand, the tendency is opposite for the
$E_{g}$ modes; the Jahn-Teller distortion of an octahedra induces the same
distortion in the neighboring octahedra. Hence we consider the “ferro”-type
coupling $\lambda_{\rm{JT}}>0$ in the following study. In the following
calculations, the “antiferro”-type $A_{1g}$ coupling tends to stabilize a
charge ordering by differentiating charge density between the neighboring
sites, while the “ferro”-type Jahn-Teller coupling favors “ferro”-type orbital
ordering.
### 2.2 Method
In order to study the ground state of the model (1) in a wide range of
parameters, we adopt the Hartree-Fock approximation to decouple the electron-
electron interactions, and the adiabatic approximation to treat the electron-
phonon couplings. Within the Hartree-Fock approximation, the two-body
interaction terms in $\mathcal{H}_{\mathrm{int}}$ are decoupled by introducing
mean fields, $\langle
c_{i\alpha\sigma}^{\dagger}c_{i\alpha^{\prime}\sigma^{\prime}}\rangle$. The
amplitudes of lattice distortions are determined by the adiabatic
approximation. Within this approximation, the equilibrium values of $x_{{\rm
br},i}$, $x_{{\rm JT},i}$, and $\bar{x}_{{\rm JT},i}$ are determined by using
the Hellmann-Feynman theorem as
$\Big{\langle}\frac{\partial\mathcal{H}}{\partial x_{{\rm
br},i}}\Big{\rangle}=\Big{\langle}\frac{\partial\mathcal{H}}{\partial x_{{\rm
JT},i}}\Big{\rangle}=\Big{\langle}\frac{\partial\mathcal{H}}{\partial\bar{x}_{{\rm
JT},i}}\Big{\rangle}=0.$ (17)
These relations lead to the set of equations in the form
$\displaystyle x_{{\rm br},i}+\lambda_{\rm{br}}{\sum_{j}}^{\prime}x_{{\rm
br},j}=$ $\displaystyle\gamma_{\rm{br}}\left(\langle n_{ia}\rangle+\langle
n_{ib}\rangle-1\right),$ (18) $\displaystyle x_{{\rm
JT},i}-\lambda_{\rm{JT}}{\sum_{j}}^{\prime}x_{{\rm JT},j}=$
$\displaystyle\gamma_{\rm{JT}}\sum_{\sigma}\left(\langle
n_{ia\sigma}\rangle-\langle n_{ib\sigma}\rangle\right),$ (19)
$\displaystyle\bar{x}_{{\rm
JT},i}-\lambda_{\rm{JT}}{\sum_{j}}^{\prime}\bar{x}_{{\rm JT},j}=$
$\displaystyle\gamma_{\rm{JT}}\sum_{\sigma}(\langle
c_{ia\sigma}^{\dagger}c_{ib\sigma}\rangle+\langle
c_{ib\sigma}^{\dagger}c_{ia\sigma}\rangle),$ (20)
where the sum $\sum^{\prime}_{j}$ is taken over the nearest neighbors of the
site $i$.
We determine the mean fields and the lattice distortions in a self-consistent
way. For a given set of $\\{\langle
c_{i\alpha\sigma}^{\dagger}c_{i\alpha^{\prime}\sigma^{{}^{\prime}}}\rangle,x_{{\rm
br},i},x_{{\rm JT},i},\bar{x}_{{\rm JT},i}\\}$, we diagonalize the Hamiltonian
under the Hartree-Fock approximation and obtain one-particle eigenenergies and
eigenstates, which are used to calculate a new set of $\\{\langle
c_{i\alpha\sigma}^{\dagger}c_{i\alpha^{\prime}\sigma^{{}^{\prime}}}\rangle\\}$.
These new mean fields are substituted in Eqs. (18)-(20) to determine the new
set of $\\{x_{{\rm br},i}$, $x_{{\rm JT},i}$, $\bar{x}_{{\rm JT},i}\\}$. These
procedures are repeated until the convergence is reached within the precision
less than $10^{-4}$ for all the variables.
In the following calculations, we take the unit cell which includes six Ni3+
sites in the triangular lattice as shown in Fig. 3. To incorporate different
orbital orderings depending on the way of taking the $\gamma$ direction in
Fig. 1, we consider two different ways of embedding the unit cell as shown in
Fig. 3. These unit cells accommodate a $\sqrt{3}\times\sqrt{3}$ charge
ordering and a two-sublattice ordering such as a stripe-type antiferromagnetic
state. Note that the ordering patterns observed in NaNiO2 and AgNiO2 are both
included by taking the unit cells. For the initial state in the iteration, we
consider more than 30 different states with different symmetry in spin,
orbital, and charge sectors, which are relevant in the parameter space we
study. The different initial configurations are adopted for each type of the
unit cells in Fig. 3.
The integration over the wave number in the calculations of the mean fields is
replaced by the sum over $24\times 24$ grids in the Brillouin zone for the
supercell. Hereafter, we focus on the quarter-filling case, i.e., one electron
per site on average, corresponding to one $e_{g}$ electron in the low-spin
state of Ni3+.
Figure 3: Two different ways of taking the unit cell with six Ni sites on the
triangular lattice used in the calculations. The axis $\gamma$ for the
orbitals is shown (see also Fig. 1).
## 3 Results and Discussion
In this section, we show the results obtained for the ground state of the
model (1). In particular, we focus on the roles of $U$, $J_{\rm H}/U$, and
$\gamma_{\rm{JT}}$, and discuss how these parameters affect the ground state.
We take $\gamma_{\rm{br}}=1.6$ and $\lambda_{\rm{br}}=\lambda_{\rm{JT}}=0.05$,
which result in reasonable energy gain in forming CO, as we will see below.
As a result, we find that the tendency to charge ordering becomes pronounced
by a compromise among $U$, $U^{\prime}$, $J_{\rm H}$, and $\gamma_{\rm br}$.
On-site repulsions $U$ and $U^{\prime}$ suppress charge disproportionation,
while the Hund’s-rule coupling $J_{\rm H}$ as well as the breathing-type
coupling $\gamma_{\rm br}$ works as an inter-orbital effective attractive
interaction and promotes charge disproportionation. On the other hand, for
larger $U$ and $U^{\prime}$, the system becomes insulating, and the Jahn-
Teller coupling $\gamma_{\rm{JT}}$ becomes important and enhances the tendency
to orbital ordering, concomitant with magnetic ordering. In order to
characterize the orbital-ordered phases, we introduce the pseudospin operators
in the orbital sector, defined as
$\boldsymbol{\tau}_{i}\equiv\sum_{\sigma}c^{{\dagger}}_{i\alpha\sigma}\boldsymbol{\sigma}_{\alpha\beta}c_{i\beta\sigma},$
(21)
where $\boldsymbol{\sigma}_{\alpha\beta}$ denotes the pauli matrix. For
example, $\tau_{z}$-OF means a ferro-type order of $\tau_{iz}$, which is the
$z$ component of $\boldsymbol{\tau}_{i}$.
The main result is summarized as the phase diagrams shown in Fig. 4. In the
following, we will focus on the $\sqrt{3}\times\sqrt{3}$-type charger-ordered
metallic (COM) phase found in the weak-coupling region [Fig. 4(a)], and the
orbital-ferro spin-ferro insulating ($\tau_{z}$-OF SF I) phase obtained in the
intermediately- to strongly-correlated region [Figs. 4(b) and 4(c)]. These two
phases are candidates for the low-$T$ states of AgNiO2 and NaNiO2,
respectively. We will identify the parameter range for these phases, and
discuss the origin and the nature of them in the following.
Figure 4: (Color online). Ground-state phase diagrams for the model (1) in the
plane of $J_{\rm H}/U$ and $\gamma_{\rm JT}$ at (a) $U=4$, (b) $U=8$, and (c)
$U=20$. We take $t^{\prime}=-1$, $\gamma_{\rm br}=1.6$, and $\lambda_{\rm
br}=\lambda_{\rm JT}=0.05$. The ordering patterns of each phase are shown in
Fig. 5. Figure 5: Schematic pictures of charge, spin, and orbital ordering
patterns in the six-site unit cell for the phases in Fig. 4. The size of the
circles schematically indicates the charge density at each Ni site. The length
of the arrows denotes the magnitude of the spin moment at each site. The open
circles denote the orbital para state, while the filled or shaded circles show
orbitally polarized states. The different patterns denote different orbitally
polarized states. See the text for details of (a)-(d). Both $\tau_{z}$-OF SF I
and $\tau_{y}$-OF SF are represented by Fig. 5(b) because these phases have
the same symmetry (OF SF). Both OF SAF I and $\tau_{x}$-OF SAF M (I) are
represented by Fig. 5(e). Typical values of of the charge disproportionation
and orbital polarization in (e)-(i) are as follows. (e) OF SAF I:
$\langle\tau_{x}\rangle\simeq\langle\tau_{z}\rangle\simeq 0.6$ at all sites.
$\tau_{x}$-OF SAF M: $\langle\tau_{x}\rangle\simeq 0.2$ and
$\langle\tau_{z}\rangle\simeq-0.06$ at all sites. (f): $\langle n\rangle\simeq
3.0$ and $\langle n\rangle\simeq 1.7$ at the two charge-rich sites and
$\langle n\rangle\simeq 0.3$ at the other charge-poor sites. (g):
$\langle\tau_{x}\rangle\simeq 0.2$, $\langle\tau_{z}\rangle\simeq 0.7$, and
$\langle n\rangle\simeq 1.2$ at the charge rich sites;
$\langle\tau_{x}\rangle\simeq 0.1$, $\langle\tau_{z}\rangle\simeq-0.2$, and
$\langle n\rangle\simeq 0.8$ at the charge poor sites. (h):
$\langle\tau_{x}\rangle\simeq 1.4$, $\langle\tau_{z}\rangle\simeq 0.6$, and
$\langle n\rangle\simeq 1.7$ at the charge rich sites and
$\langle\tau_{x}\rangle\simeq 0.2$, $\langle\tau_{z}\rangle\simeq 0.0$, and
$\langle n\rangle\simeq 0.3$ at the charge poor sites. (i): $\langle
n\rangle\simeq 1.02$ at the charge rich sites (1 and 4 in Fig. 3), $\langle
n\rangle\simeq 0.98$ at the charge poor sites (3 and 6), and $\langle
n\rangle\simeq 1.00$ at the other charge-neutral sites (2 and 5).
### 3.1 Weakly-correlated region
#### 3.1.1 Phase diagram
A tendency toward charge ordering is widely observed in a weakly-correlated
region. Figure 4(a) shows the phase diagram at $U=4$. Among the competing
phases, a COM phase with $\sqrt{3}\times\sqrt{3}$-type charge ordering is
stabilized in a wide range of $\gamma_{\rm JT}$ at $J_{\rm H}/U\sim 0.25$. The
$\sqrt{3}\times\sqrt{3}$ charge ordering is a three-sublattice order, in which
one is charge rich (the density is almost two) and the other two charge poor
(the density is almost 0.5 per site). This $\sqrt{3}\times\sqrt{3}$ COM state,
shown in Fig. 5(a), is remarkable, since it has the same charge ordering
pattern as the low-$T$ state of AgNiO2 [26, 27]. The spin state is also
interesting; large moments appear at charge-rich sites ($S\simeq 0.6$), while
the moments are suppressed at charge-poor sites ($S\simeq 0.05$). A similar
differentiation was proposed for the low-$T$ state of AgNiO2 [26, 27],
although the calculated spin pattern does not fully agree with the
experimental result. We note that importance of interlayer coupling is
experimentally suggested for the magnetic ordering [28, 26, 27], which is not
taken into account in our model. We will discuss the nature of this COM phase
in Sec. 3.1.3 in detail.
Around $J_{\rm{H}}/U=0.24$ and $\gamma_{\rm{JT}}=1.1$, we find another COM
phase, i.e., the sixfold COM phase. Although this phase is stabilized in a
narrow region in the phase diagram, it remains to be metastable in a wide
parameter range, with a slightly higher energy than the ground state, as we
will discuss in the next section 3.1.2. This phase has the consistent ordering
structure with the low-$T$ state of AgNiO2 in terms of both charge and spin,
as shown in Fig. 5(c). Strictly speaking, the charge pattern of this phase has
lower symmetry compared with AgNiO2, due to the superposition of stripe-type
charge modulation onto the $\sqrt{3}\times\sqrt{3}$ charge ordering. However,
the magnitude of this modulation is very small: For charge rich sites (sites 1
and 4 in Fig. 3), we have $\langle n_{1}\rangle=1.5040$ and $\langle
n_{4}\rangle=1.5027$, while for charge poor sites (sites 2, 3, 5, and 6 in
Fig. 3), we have $\langle n_{3}\rangle=\langle n_{5}\rangle=0.7479$ and
$\langle n_{2}\rangle=\langle n_{6}\rangle=0.7487$, at $J_{\rm H}/U=0.24$ and
$\gamma_{\rm{JT}}=1.1$. The modulation gives very small charge
disproportionation of the order of $\sim 0.001$ within charge rich and poor
sites. [The small disproportionations are exaggerated in the schematic picture
in Fig. 5(c).]
In addition to the two COM states, we obtain a variety of ordered phases in
the weakly-correlated region. Among them, we focus on two phases which compete
with the COM states; the sixfold charge-ordered insulating (sixfold COI) phase
in the large $J_{\rm H}/U$ region, and the spin-ferro metallic phase with a
weak charge ordering (SF-COM) stabilized for smaller $J_{\rm H}/U$ [Fig.
4(a)]. We argue the stability of the $\sqrt{3}\times\sqrt{3}$ COM as well as
the sixfold COM in comparison with the two competing phases in the next
section.
#### 3.1.2 Stability of the $\sqrt{3}\times\sqrt{3}$ COM phase
In order to clarify the competition among the $\sqrt{3}\times\sqrt{3}$ COM,
sixfold COM, sixfold COI, and SF-COM phases, we investigate the internal
energy in detail by comparing the contributions from different terms in the
Hamiltonian; $E_{\rm{kin}}\equiv\langle\mathcal{H}_{\rm{kin}}\rangle$,
$E_{U}\equiv\langle\mathcal{H}_{U}\rangle$,
$E_{U^{\prime}}\equiv\langle\mathcal{H}_{U^{\prime}}\rangle$,
$E_{J_{\rm{H}}}\equiv\langle\mathcal{H}_{J_{\rm{H}}}+\mathcal{H}_{J_{\rm{H}}^{\prime}}\rangle$,
$E_{\rm{br}}\equiv\langle\mathcal{H}_{\text{el-
ph}}^{\rm{br}}+\mathcal{H}_{\rm{ph}}^{\rm{elestic,br}}+\mathcal{H}_{\rm{ph}}^{\rm{coop,br}}\rangle$,
and the total energy $E_{\rm{tot}}\equiv\langle\mathcal{H}\rangle$. We show
the comparison as a function of $U$ at $J_{\rm H}/U=0.27$ and $\gamma_{\rm
JT}=0.5$ in Fig. 6.
For $U\lesssim 3.9$, the sixfold COI state has the lowest energy. As shown in
Fig. 5(f), this phase has a polaronic nature, namely, one site is almost fully
occupied (the local density is almost 4), and another one site accommodates
almost two electrons. At the latter site, spins of two electrons are aligned
parallel by the Hund’s-rule coupling. This phase is stabilized in a region
where the repulsive Coulomb interactions are compensated by effective
attractive interactions originating in the breathing-type electron-phonon
coupling as well as the inter-orbital Hund’s-rule coupling. In fact, it is
clearly observed in Figs. 6(d) and 6(e) that the energy gain in
$E_{J_{\rm{H}}}$ and $E_{\rm{br}}$ contributes to the stabilization of the
sixfold COI phase.
On the other hand, for $U\gtrsim 4.9$, the SF-COM state is most stabilized. As
schematically shown in Fig. 5(d), the charge ordering in this phase is a
stripe type, but the charge disproportionation is very small ($\langle
n\rangle\sim 1.03-1.09$ at charge rich sites, while $\langle n\rangle\sim
0.91-0.97$ at charge poor sites): the main feature is the spin ferromagnetic
ordering. The origin of this phase can be attributed to the Stoner mechanism
[35]. As shown in the inset of Fig. 7, the non-interacting Fermi level is
located in the vicinity of the steep peak of the density of states (DOS).
Consequently, a ferromagnetic instability is caused at a relatively small
$U\simeq 3.4$, according to the Stoner’s criterion. The characteristics of
Stoner ferromagnet are observed in the energy comparison in Fig. 6(c); $E_{U}$
becomes smallest among the competing phases.
The $\sqrt{3}\times\sqrt{3}$ COM state intervenes these two, and has the
lowest energy for $4.0\lesssim U\lesssim 4.8$. In the same parameter range,
the sixfold COM state appears as a metastable state and stays very close to
the ground state, as shown in the inset of Fig. 6(a) (the energy difference is
less than 0.02). These two COM states are stabilized by a compromise between
the different stabilization mechanisms for the sixfold COI and the SF-COM
phases. According to Fig. 6, the COM phases have higher (lower)
$E_{J_{\rm{H}}}$ and $E_{\rm{br}}$, while they have lower (higher) $E_{U}$ and
$E_{U^{\prime}}$, compared with the sixfold COI phase (the SF-COM phase).
Namely, the two COM phases are stabilized in a subtle balance between
repulsive Coulomb interactions and effective attractive interactions due to
the Hund’s-rule coupling and the breathing-type electron-phonon coupling.
Since the COM phases are stabilized in a delicate compromise, it is important
to consider their stability against the elements which are ignored in our
current analysis, such as fluctuations beyond the mean-field level and the
long-range part of electron interactions. First, we note that the Stoner
ferromagnetism is fragile when considering the electron correlation effect
beyond the mean-field approximation[36]. Therefore, the COM phases are
expected to extend to larger $U$ or smaller $J_{\rm H}/U$. Second, the
amplitudes of breathing-type distortions are fairly large in the sixfold COI
phase compared to those in the other phases. Hence this phase will be
suppressed by considering more realistic contributions from phonons, e.g.,
anharmonic terms of phonons. This may give a chance for the COM phases to
become wider also in smaller $U$ or larger $J_{\rm H}/U$ regions. Finally, the
long-range part of electron interactions generally works in favor of the
charge ordering, in particular, the $\sqrt{3}\times\sqrt{3}$ type and the
sixfold COM, as is discussed in several transition metal compounds and organic
materials [37]. Therefore, we expect that the COM phases relevant to AgNiO2
become more stable in a wider parameter range when extending the analyses
beyond the present model and method. Although the COM phases are robust in
this parameter region, the energy difference between the sixfold COM phase and
the $\sqrt{3}\times\sqrt{3}$ COM phase is very small, implying that the
magnetic ordering pattern might be affected by small perturbations, such as
inter-layer coupling. More accurate studies beyond the mean-field
approximation are necessary for fully determining the spin pattern of the
ground state.
Figure 6: (Color online). Energy comparisons among the
$\sqrt{3}\times\sqrt{3}$ COM (circle), sixfold COM (triangle), sixfold COI
(square), and SF-COM states (diamond): $U$ dependences of (a) the total
energy, and the contribution from (b) kinetic term, (c) Coulomb repulsions
[closed (open) symbols denote $E_{U}$ ($E_{U^{\prime}}$)], (d) Hund’-rule
coupling, and (e) breathing-type electron-phonon coupling. The inset of (a)
shows the energy differences between the $\sqrt{3}\times\sqrt{3}$ COM phase
and the other competing phases. The parameters are $t^{\prime}=-1$,
$J_{\rm{H}}/U=0.27$, $\gamma_{\rm{br}}=1.6$, $\gamma_{\rm{JT}}=0.5$, and
$\lambda_{\rm{br}}=\lambda_{\rm{JT}}=0.05$.
#### 3.1.3 Nature of the $\sqrt{3}\times\sqrt{3}$ COM phase
Reflecting the subtle balance between the attractive and repulsive
interactions, the $\sqrt{3}\times\sqrt{3}$ COM phase shows peculiar electronic
properties. The density of states (DOS) in the $\sqrt{3}\times\sqrt{3}$ COM
phase is shown in Fig. 7. The site- and spin-resolved DOS indicates that the
system exhibits a half-metallic nature: Up-spin electrons are localized at
charge-rich sites, showing a gap at the Fermi level, on the other hand, down-
spin electrons remain conductive, with a finite DOS at the Fermi level.
Electron correlations dominantly affect up-spin electrons; down-spin
conductive electrons preserve the non-interacting band structure. For
comparison, we show DOS for the non-interacting case in the inset of Fig. 7,
which is quite similar to DOS of down-spin conductive electrons. DOS in the
$\sqrt{3}\times\sqrt{3}$ COM phase reproduces several aspects of the result
obtained by the first-principle band calculation [26]: The electrons at charge
rich site tend to localize and the electronic structure at the other two
charge poor sites resembles each other.
This peculiar electronic state can be attributed to the delicate balance
between the attractive and repulsive interactions. We plot the effective one-
body potential in Fig. 8, which is defined as the sum of the terms in
$\mathcal{H}_{\rm{int}}$ and $\mathcal{H}_{\text{el-ph}}$, which couple to the
density operator $n_{i}$ at each site under the Hartree-Fock approximation.
Figure 8 shows that the charge-rich (-poor) sites bear attractive (repulsive)
potentials for up-spin electrons. In contrast, the cancellation between the
breathing-type electron-phonon coupling and the repulsive Coulomb interactions
leads to an almost flat potential for down-spin electrons. Consequently, these
interactions only work as a shift of chemical potential, and the down-spin
electrons retain the non-interacting band structure.
To conclude the discussions for the weakly-correlated region, the
$\sqrt{3}\times\sqrt{3}$ COM phase is stabilized by a compromise between
repulsive Coulomb interactions and attractive interactions originating from
the breathing-type electron-phonon coupling as well as the Hund’s-rule
coupling. This phase shows a half-metallic behavior with large magnetic
moments almost localized at charge-rich sites and conduction electrons moving
almost freely in the entire lattice. We note that this phase is distinguished
from the so-called pinball liquid state, in which the electrons at the charge-
rich sites exclude the conduction electrons as hard core potentials and
confine them to the honeycomb network of charge-poor sites, as discussed in a
spinless tight-binding model with intersite Coulomb repulsion on the
triangular lattice [38].
Figure 7: (Color online). DOS per site for the $\sqrt{3}\times\sqrt{3}$ COM
phase. The Fermi level is set to be the origin of energy. As to the
$\sqrt{3}\times\sqrt{3}$ COM phase, the up-spin (down-spin) DOS is drawn on
the upper (lower) side. The parameters are chosen as $U=4$,
$J_{\rm{H}}/U=0.27$, $\gamma_{\rm{br}}=1.6$, $\gamma_{\rm{JT}}=0.5$, and
$\lambda_{\rm{br}}=\lambda_{\rm{JT}}=0.05$. The inset shows DOS for the non-
interacting case. Figure 8: (Color online). (a) Effective one-body
potentials at each site for up-spin and down-spin electrons (see the text for
details). The horizontal axis denotes the site indices in the unit cell, shown
in Fig. 3. The charge-rich (-poor) sites are the site 1 and 4 (2, 3, 5, and
6). The parameters are chosen as $U=4$, $J_{\rm{H}}/U=0.27$,
$\gamma_{\rm{br}}=1.6$, $\gamma_{\rm{JT}}=0.5$, and
$\lambda_{\rm{br}}=\lambda_{\rm{br}}=0.05$, consistent with Fig. 7.
### 3.2 Intermediately- to strongly-correlated region
Next, let us consider the intermediately- to strongly-correlated region.
Representative phase diagrams are shown in Figs. 4(b) and 4(c).
#### 3.2.1 Phase diagram
We show the ground-state phase diagram at $U=8$ in Fig. 4(b). Note that the
value of $U$ is comparable with the non-interacting bandwidth., i.e., the
system is in the intermediately-correlated region. In this region, we find
three dominant orbital-ordered phases; $\tau_{y}$-ordered spin-ferro insulator
($\tau_{y}$-OF SF I), $\tau_{x}$-ordered spin-antiferro insulator
($\tau_{x}$-OF SAF I), and $\tau_{z}$-ordered spin-ferro insulator
($\tau_{z}$-OF SF I). The ordering pattern of $\tau_{y}$\- or $\tau_{z}$-OF SF
I ($\tau_{x}$-OF SAF I) is schematically shown in Fig. 5(b) [Fig. 5(e)]. For
the small $\gamma_{\rm JT}$ and $J_{\rm H}/U$ region, $\tau_{y}$-OF SF I is
stabilized, while it is replaced by $\tau_{x}$-OF SAF I for larger
$\gamma_{\rm JT}$ or by a charge-ordered state for larger $J_{\rm H}/U$.
Meanwhile, when both $\gamma_{\rm JT}$ and $J_{\rm H}/U$ become large,
$\tau_{z}$-OF SF I is stabilized. Among these phases, the $\tau_{z}$-OF SF I
phase deserves attention, since the spin and orbital pattern of this phase is
consistent with the low-$T$ phase of NaNiO2.
Remarkably, the three OF phases remain stable in a wide range of $U$ toward
the strongly-correlated regime. Figure 4(c) shows an example of the phase
diagram at large $U$. The result indicates that the three phases remain in
similar parameter regions of $\gamma_{\rm JT}$ and $J_{\rm H}/U$ compared to
the intermediate-$U$ case. This implies a possibility to understand the origin
of these phases from the strong-coupling analysis, i.e., by starting from the
Mott insulating state at $U=\infty$.
In fact, the $\tau_{y}$-OF SF I state was obtained for an effective spin-
orbital model in the strong-coupling limit (F3 phase of Fig. 8 in Ref.
Vernay). Our result is consistent with the previous study. Meanwhile, the
competition between the $\tau_{x}$-OF SAF I phase and the $\tau_{z}$-OF SF I
phase is obtained for the first time by explicitly taking account of electron-
phonon couplings. In the following, we will consider the mechanism of
stabilization of these phases through the detailed study of energetics.
#### 3.2.2 Stability of the $\tau_{z}$-OF SF insulating phase
In order to understand the stability condition, it is instructive to rewrite
$\mathcal{H}^{\rm{JT}}_{\text{el-ph}}$, $\mathcal{H}_{J_{\rm{H}}}$, and
$\mathcal{H}_{J^{\prime}_{\rm{H}}}$ by using the pseudospin operators in eq.
(21) as
$\mathcal{H}_{\text{el-ph}}^{\rm{JT}}=-\gamma_{\rm{JT}}\sum_{i}(x_{{\rm
JT},i}\tau_{iz}+\bar{x}_{{\rm JT},i}\tau_{ix}),$ (22)
$\mathcal{H}_{J_{\rm{H}}}+\mathcal{H}_{J_{\rm{H}}^{\prime}}=\frac{J_{\rm
H}}{2}\sum\limits_{i}\\{\tau_{ix}^{2}-(n_{ia}+n_{ib})\\}.$ (23)
These equations clearly show that the Jahn-Teller coupling stabilizes the
$\tau_{x}$ and $\tau_{z}$ orbital orderings, while the Hund’s-rule coupling
destabilizes the $\tau_{x}$ orbital ordering. In Fig. 9, we compare the energy
contributions including these terms,
$E_{\rm{JT}}\equiv\langle\mathcal{H}_{\text{el-
ph}}^{\rm{JT}}+\mathcal{H}_{\rm{ph}}^{\rm{elastic,JT}}+\mathcal{H}_{\rm{ph}}^{\rm{coop,JT}}\rangle$
and $E_{J_{\rm H}}$, together with other relevant energy contributions, for
the three orbital-ordered phases.
Figure 9(b) shows that the stability of the $\tau_{y}$-OF SF I phase in the
small $\gamma_{\rm{JT}}$ region is attributed to the energy gain in the
kinetic energy $E_{\rm{kin}}$. This is consistent with the result of strong-
coupling analysis, where the kinetic energy gain through the spin-orbital
superexchange interactions is claimed to be the origin of this phase. [33] It
is also observed that all energy contributions in this phase are fairly
insensitive to $\gamma_{\rm JT}$, as shown in Fig. 9. This is expected from
the absence of coupling between $\tau_{y}$ and Jahn-Teller distortions, as is
clear from eq. (22).
On the other hand, the $\tau_{x}$-OF SAF I and $\tau_{z}$-OF SF I states lower
their energy through the coupling to the Jahn-Teller distortions [Fig. 9(e)],
as expected from eq. (22); thus they replace the $\tau_{y}$-OF SF I phase and
become the ground state for larger $\gamma_{\rm JT}$, as shown in Fig. 9(a).
The Hund’s-rule coupling plays an important role in the relative stability
between the $\tau_{x}$-OF SAF I and the $\tau_{z}$-OF SF I phases. As is
evident from eq. (23), the Hund’s-rule coupling affects only the $\tau_{x}$
ordering, and destabilizes it [Fig. 9(d)]. Furthermore, the $\tau_{z}$-OF SF I
phase is stabilized by the kinetic energy gain from the interorbital hoppings,
compared with the $\tau_{x}$-OF SAF I phase as shown in Fig. 9(b). In fact,
according to the second-order perturbation from the strong coupling limit
$U\to\infty$, the $\tau_{z}$-OF SF I phase has lower energy than $\tau_{x}$-OF
SAF I phase for $J_{\rm H}/U>\frac{8-\sqrt{10}}{18}\simeq 0.27$. The phase
boundary between the two phases in Fig. 4(c) is roughly located around this
critical value, which indicates that the phase competition is essentially
understood from the strong coupling picture.
Figure 9: (Color online). $\gamma_{\rm JT}$ dependences of (a) the total
energy, and the contribution from (b) kinetic term, (c) Coulomb repulsions
[closed (open) symbols denote $E_{U}$ ($E_{U^{\prime}}$)], (d) Hund’s-rule
coupling, and (e) Jahn-Teller coupling. Comparison is made for the
$\tau_{z}$-OF SF I (circle), $\tau_{x}$-OF SAF I (square), and $\tau_{y}$-OF
SF I states (triangle). The parameters are $t^{\prime}=-1$, $U=20$,
$J_{\rm{H}}/U=0.25$, $\gamma_{\rm{br}}=1.6$, and
$\lambda_{\rm{br}}=\lambda_{\rm{JT}}=0.05$.
#### 3.2.3 Nature of the $\tau_{z}$-OF SF insulating phase
Due to the strong electron repulsion, an excitation gap opens at the Fermi
level in the $\tau_{z}$-OF SF I phase. Hence, this state is insulating,
consistent with the low-$T$ insulating phase of NaNiO2. Figure 10 shows DOS in
the $\tau_{z}$-OF SF I phase. DOS is composed of four sectors (two in each
spin component), and the total weight of each sector is equal to one electron
per site, as indicated in the integrated DOS in the figure. This structure can
be qualitatively understood by considering the atomic limit with ignoring the
hoppings, $t$ and $t^{\prime}$. Let us assume the perfect $\tau_{z}$-OF SF
order, i.e., $\langle n_{i,3z^{2}-r^{2},\uparrow}\rangle=1$ and let other mean
fields to be zero. Then the excitation energies in the atomic limit are
estimated as
$\displaystyle E_{x^{2}-y^{2},\uparrow}$
$\displaystyle=U^{\prime}-J_{\rm{H}}-2E^{*}_{\rm{JT}},$ (24) $\displaystyle
E_{x^{2}-y^{2},\downarrow}$ $\displaystyle=U^{\prime}-2E^{*}_{\rm{JT}},$ (25)
$\displaystyle E_{3z^{2}-r^{2},\downarrow}$
$\displaystyle=U+2E^{*}_{\rm{JT}},$ (26)
where $E_{\alpha\sigma}$ signifies the energy necessary to add one electron
with orbital $\alpha$ and spin $\sigma$ to the $(3z^{2}-r^{2},\uparrow)$
ground state. $E^{*}_{\rm{JT}}$ is the energy gain from the Jahn-Teller
distortion per site, estimated as
$E^{*}_{\rm{JT}}=-\frac{\gamma_{\rm{JT}}^{2}}{2(1-6\lambda_{\rm{JT}})}.$ (27)
Substituting the parameters used in Fig. 10 ($U=20$, $U^{\prime}=10$, $J_{{\rm
H}}=5$, $\gamma_{\rm JT}=1.0$, and $\lambda_{\rm JT}=0.05$) into these
equations, we obtain $E_{x^{2}-y^{2},\uparrow}\simeq 6.4$,
$E_{x^{2}-y^{2},\downarrow}\simeq 11$, and $E_{3z^{2}-r^{2},\downarrow}\simeq
19$. These values well correspond to the mean energy of each sector of DOS
measured from that for the lowest one in Fig. 10.
Eq. (24) gives an estimate of the energy gap $\Delta$ in the atomic limit.
This atomic value is reduced for finite $t$ and $t^{\prime}$, since each
atomic level is broadened by a renormalized bandwidth $\tilde{W}$: The
estimate of energy gap is modified as $\Delta\sim
E_{x^{2}-y^{2},\uparrow}-\tilde{W}$. We use this simple estimate with
replacing $\tilde{W}$ by the bare bandwidth $W=8t$ for a comparison with the
Hartree-Fock solutions. As shown in Fig. 11, our simple estimate from the
atomic limit is qualitatively consistent with Hartree-Fock results. This fact
also supports that the electronic spectrum of the $\tau_{z}$-OF SF I phase is
adiabatically continued from the strong-coupling limit.
Figure 10: (Color online). DOS for the $\tau_{z}$-OF SF I state. The
parameters are $t^{\prime}=-1.0$, $U=20$, $J_{\rm{H}}/U=0.25$,
$\gamma_{\rm{JT}}=1.0$, $\gamma_{\rm{br}}=1.6$, and
$\lambda_{\rm{br}}=\lambda_{\rm{JT}}=0.05$. The vertical line denotes the
Fermi level. The up-spin (down-spin) DOS is represented on the upper (lower)
side. The integrated DOS is also shown. Figure 11: (Color online). Energy
gap in the $\tau_{z}$-OF SF I phase as a function of $U$ at $t^{\prime}=-1.0$,
$J_{\rm{H}}/U=0.25$, $\gamma_{\rm{JT}}=1.0$, $\gamma_{\rm{br}}=1.6$, and
$\lambda_{\rm{br}}=\lambda_{\rm{br}}=0.05$. The line is the simple estimate
from the strong-coupling analysis. See the text for details.
To conclude this part, three orbital-ordered insulating phases appear
dominantly in the intermediately- to strongly-correlated region. Among them,
the $\tau_{z}$-OF SF I phase, which is relevant to NaNiO2, becomes stable in
the region where both the Jahn-Teller coupling and the Hund’-rule coupling are
substantial. This phase is understood by the strong-coupling picture under the
Jahn-Teller type electron-phonon couplings.
### 3.3 Comparison to experiments
In our calculations, we successfully reproduce the $\sqrt{3}\times\sqrt{3}$
COM phase and the $\tau_{z}$-OF SF I phase, whose ordering patterns are
consistent with the low-$T$ phases in AgNiO2 and NaNiO2, respectively. These
two phases appear in close parameter regions of $\gamma_{\rm JT}$ and $J_{\rm
H}/U$, but for different range of the on-site repulsion $U$. The
$\sqrt{3}\times\sqrt{3}$ COM phase is stabilized in the weakly-correlated
region, where $U$ is smaller than the bare bandwidth, while the $\tau_{z}$-OF
SF I phase is stabilized in the intermediately- to strongly-correlated region.
The difference in $U$ may be attributed to the structure of cation bands in
these compounds. The magnitude of the effective on-site repulsion for Ni $3d$
electrons is not solely determined by its atomic value, but it is considerably
affected by the screening effect brought about by electrons in the $A$ cation
and oxygen $p$ bands. According to the first-principle calculations, Ag bands
in AgNiO2 reside in the vicinity of the Fermi level [25, 26], while the Na
bands in NaNiO2 are located about 4eV above the Fermi level [30, 31].
Consequently, a larger screening effect is expected for AgNiO2, which reduces
the magnitude of $U$ considerably, compared to that for NaNiO2. Our results
are consistent with this trend. In the first-principle calculations [30, 31,
25, 26], the bandwidth of Ni 3$d$ bands was roughly estimated to be $2\sim
3$eV, leading to a rough estimate of $t=0.25\sim 0.4$eV. In the LSDA+$U$
calculation for NaNiO2 [30, 31], the value of $U$ was taken to be 5eV to
reproduce the correct size of excitation gap. On the other hand, a cluster-
model analysis of the photoemission spectra gave an estimate of $U$=7.0eV
[39]. These studies suggest that $U\simeq 10-30t$ is reasonable, consistent
with our results. It is noteworthy that the gap in our calculation at $U=20t$
corresponds to $0.5\sim 0.8$eV, which is in the same order of magnitude as the
experimental value $\sim 0.24$eV in NaNiO2 [9]. We also note that the CO
stabilization energy, which is estimated from the energy difference between
the $\sqrt{3}\times\sqrt{3}$ COM phase and a para phase, is $\sim$ $0.05t$ for
the present parameters $\gamma_{\rm br}$ and $\lambda_{\rm br}$: This result
leads a rough estimate that the CO stabilization energy is $\sim$ 0.01 - 0.02
eV, which is in the same order of magnitude as the CO transition temperature
observed in AgNiO2 (365K [26, 27]).
Finally, we make a brief comment on the absence of any explicit ordering in
LiNiO2. Since LiNiO2 is also a Mott insulator with a gap of 0.2eV [14], we
suppose that the compound is in the strongly-correlated region similar to
NaNiO2. In our results, there exists phase competition among three insulating
phases with different spin and orbital patterns, the $\tau_{z}$-OF SF,
$\tau_{x}$-OF SAF, and $\tau_{y}$-OF SF orderings. The competition brings
about a frustration in the spin and orbital sectors in the vicinity of the
phase boundaries. To argue the consequence of such frustration, we need to go
beyond the present mean-field analysis; however, we can expect severe
suppression of the orderings and some liquid-like or glassy behavior in the
spin-orbital coupled system. Hence, one possibility is that LiNiO2 is located
in such competing regime. It is noteworthy that the competition is brought
about by explicitly taking account of the electron-phonon couplings, which
have not been considered in the previous effective model approaches [20, 32,
33]. In addition to this intrinsic phase competition, extrinsic defects on Li
sites may play an important role in the glassy behavior. Furthermore, a long-
range strain effect might also play a role through the frustrating orbital and
lattice sectors [24]. It is interesting to take account of these factors
explicitly, by extending our model and analysis. We leave this problem for a
future study.
## 4 Summary
We have investigated the ground state of the multi-orbital Hubbard model with
electron-phonon couplings by the Hartree-Fock approximation and the adiabatic
approximation, in order to elucidate the origin of various phases observed in
$A$NiO2 in a unified way. We found the $\sqrt{3}\times\sqrt{3}$ charge-ordered
metallic phase in the weakly-correlated region and the orbital-ferro spin-
ferro ordered insulating phase in the strongly-correlated region. The
$\sqrt{3}\times\sqrt{3}$ charge-ordered metallic phase is stabilized by a
compromise between Coulomb repulsions and effective attractive interactions
from the breathing-type electron-phonon coupling as well as the Hund’s-rule
coupling. The electronic state is half metallic; up-spin electrons are
localized at the charge-rich sites, but down-spin electrons are extended and
almost free. On the other hand, the orbital-ferro spin-ferro ordered
insulating phase is stabilized by the Jahn-Teller coupling under strong
electron correlation, with a help by the Hund’s-rule coupling in the
competition with other orbital-ordered phases. These two phases are promising
candidates for the low-$T$ phases in AgNiO2 and NaNiO2, respectively. A
possible origin of the quite different electron repulsion between AgNiO2 and
NaNiO2 might be a screening effect from the cation and oxygen $p$ bands. The
puzzling glassy behavior in LiNiO2 might be ascribed to the competition among
different spin and orbital ordered states in the strongly-correlated region,
which occurs under a substantial Jahn-Teller type electron-phonon coupling.
## Acknowledgements
The authors thank M. Imada, S. Watanabe, and Y. Yamaji for fruitful
discussions. This work was supported by Grants-in-Aid for Scientific Research
(No. 17071003, 17740244, 19014020, and 19052008), Global COE Program “the
Physical Sciences Frontier”, the Next Generation Super Computing Project, and
Nanoscience Program, from MEXT, Japan.
## References
* [1] M. Imada, A. Fujimori, and Y. Tokura: Rev. Mod. Phys. 70 (1998) 1039.
* [2] Y. Tokura and N. Nagaosa: Science 288 (2000) 462.
* [3] K. I. Kugel and D. I. Khomskii: Sov. Phys. JETP 37 (1973) 725.
* [4] K. I. Kugel and D. I. Khomskii: Sov. Phys. Solid State 17 (1975) 285.
* [5] “Frustrated Spin Systems”, edited by H. T. Diep (World Scientific, Singapore, 2005).
* [6] “Colossal Magnetoresitive Oxides”, edited. by Y. Tokura (G & B Science Pub, 2000).
* [7] T. Sorgel and M. Jansen: Z. Anorg. Allg. Chem. 631 (2005) 2970.
* [8] E. Chappel, M. D. Nunez-Regueiro, G. Chouteau, O. Isnard, and C. Darie: Eur. Phys. J. B 17 (2000) 615.
* [9] J. Molenda and A. Stoklosa: Solid State Ionics 38 (1990) 1.
* [10] P. F. Bongers and U. Enz: Solid State Comm. 4 (1966) 153.
* [11] C. Darie, P. Bordet, S. de Brion, M. Holzapfel, O. Isnard, A. Lechi, J. E. Lorenzo, and E. Suard: Eur. Phys. J. B 43 (2005) 159.
* [12] P. J. Baker, T. Lancaster, S.J. Blundell, M. L. Brooks, W. Hayes, D. Prabhakaran, and F. L. Pratt: Physica B 374-375 (2006) 47-50.
* [13] M. J. Lewis, B. D. Gaulin, L. Filion, C. Kallin, A. J. Berlinsky, H. A. Dabkowska, Y. Qiu, and J. R. D. Copley: Phys. Rev. B 72 (2005) 014408.
* [14] R. Kanno, H. Kubo, Y. Kawamoto, T. Kamiyama, F. Izumi, Y. Takeda, M. Takano: J. Solid State Chem. 110 (1994) 216.
* [15] K. Hirakawa, H. Kadowaki, and K. Ubukoshi: J. Phys. Soc. Jpn. 54 (1985) 3526.
* [16] J. N. Reimers, J. R. Dahn, I. Davidson, and U. Von Sacken: J. Solid State Chem. 102 (1993) 542.
* [17] K. Yamaura, M. Takano, A. Hirano, and R. Kanno: J. Solid State Chem. 127 (1996) 109.
* [18] Y. Q. Li, M. Ma, D. N. Shi, and F. C. Zhang: Phys. Rev. Lett. 81 (1998) 3527.
* [19] Y. Kitaoka, T. Kobayashi, A. Koda, H. Wakabayashi, Y. Niino, H. Yamakage, S. Taguchi, K. Amaya, K. Yamaura, M. Takano, A. Hirano, and R. Kanno: J. Phys. Soc. Jpn. 67 (1998) 3703.
* [20] M. V. Mostovoy and D. I. Khomskii: Phys. Rev. Lett. 89 (2002) 227203.
* [21] H. Yoshizawa, H. Mori, K. Hirota, and M. Ishikawa: J. Phys. Soc. Jpn. 59 (1990) 2631.
* [22] J. Sugiyama, K. Mukai, Y. Ikedo, P. L. Russo, H. Nozaki, D. Andreica, A. Amato, K. Ariyoshi, and T. Ohzuku: Phys. Rev. B 78 (2008) 144412.
* [23] A. Rougier, C. Delmas, A. V. Chadwick: Solid. State. Comm. 94 (1995) 123.
* [24] J. -H. Chung, Th. Proffen, S. Shamoto, A. M. Ghorayeb, L. Croguennec, W. Tian, B. C. Sales, R. Jin, D. Mandrus, and T. Egami: Phys. Rev. B 71 (2005) 064410.
* [25] T. Sorgel and M. Jansen: J. Solid State Chem. 180 (2007) 8.
* [26] E. Wawrzynska, R. Coldea, E. M. Wheeler, I. I. Mazin, M. D. Johannes, T. Sorgel, M. Jansen, R. M. Ibberson, and P. G. Radaelli: Phys. Rev. Lett. 99 (2007) 157204.
* [27] E. Wawrzynska , R. Coldea, E. M. Wheeler, T. Sorgel, M. Jansen, R. M. Ibberson, P. G. Radaelli, and M. M. Koza: Phys. Rev. B 77 (2008) 094439.
* [28] E. M. Wheeler, R. Coldea, E. Wawrzynska, T. Sorgel, M. Jansen, M. M. Koza, J. Taylor, P. Adroguer, and N. Shannon: Phys. Rev. B 79 (2009) 104421.
* [29] A. I. Coldea, A. Carrington, R. Coldea, L. Malone, A.F. Bangura, M. D. Johannes, I. I. Mazin, E.A. Yelland, J. G. Analytis, J.A.A.J. Perenboom, C. Jaudet, D. Vignolles, T. Sorgel, M. Jansen: preprint (arXiv:0908.4169).
* [30] H. Meskine and S. Satpathy: Phys. Rev. B 72 (2005) 224423.
* [31] H. Meskine and S. Satpathy: J. Appl. Phys. 97 (2005) 10A314.
* [32] A.-M. Dare, R. Hayn, and J.-L. Richard: Europhys. Lett. 61 (2003) 803.
* [33] F. Vernay, K. Penc, P. Fazekas, and F. Mila: Phys. Rev. B 70 (2004) 014428.
* [34] L. Seabra and N. Shannon: Phys. Rev. Lett. 104 (2010) 237205.
* [35] E. C. Stoner: Rep. Prog. Phys. 11 (1946) 43.
* [36] J. Kanamori: Prog. Theor. Phys. 30 (1963) 275.
* [37] H. Seo, J. Phys. Soc. Jpn. 75 (2006) 051009.
* [38] C. Hotta and N. Furukawa: Phys. Rev. B 74 (2006) 193107.
* [39] T. Mizokawa and A. Fujimori: Phys. Rev. B 54 (1996) 5368.
|
arxiv-papers
| 2011-01-20T06:33:14 |
2024-09-04T02:49:16.542274
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hiroshi Uchigaito, Masafumi Udagawa, and Yukitoshi Motome",
"submitter": "Hiroshi Uchigaito",
"url": "https://arxiv.org/abs/1101.3828"
}
|
1101.4028
|
# Who is the best player ever? A complex network analysis of the history of
professional tennis
Filippo Radicchi Chemical and Biological Engineering, Northwestern
University, 2145 Sheridan Road, Evanston, IL 60208, US
###### Abstract
We consider all matches played by professional tennis players between $1968$
and $2010$, and, on the basis of this data set, construct a directed and
weighted network of contacts. The resulting graph shows complex features,
typical of many real networked systems studied in literature. We develop a
diffusion algorithm and apply it to the tennis contact network in order to
rank professional players. Jimmy Connors is identified as the best player of
the history of tennis according to our ranking procedure. We perform a
complete analysis by determining the best players on specific playing surfaces
as well as the best ones in each of the years covered by the data set. The
results of our technique are compared to those of two other well established
methods. In general, we observe that our ranking method performs better: it
has a higher predictive power and does not require the arbitrary introduction
of external criteria for the correct assessment of the quality of players. The
present work provides a novel evidence of the utility of tools and methods of
network theory in real applications.
## I Introduction
Social systems generally display complex features castellano09 . Complexity is
present at the individual level: the behavior of humans often obeys complex
dynamical patterns as for example demonstrated by the rules governing
electronic correspondence barabasi05 ; malmgren08 ; radicchi09a ; wu10 . At
the same time, complexity is present also at the global level. This can be
seen for example when social systems are mathematically represented in terms
of graphs or networks, where vertices identify individuals and edges stand for
interactions between pairs of social agents. Social networks are in most of
the cases scale-free Barabasi1999 , indicating therefore a strong degree of
complexity from the topological and global points of view.
During last years, the analysis of social systems has become an important
topic of interdisciplinary research and as such has started to be not longer
of interest to social scientists only. The presence of a huge amount of
digital data, describing the activity of humans and the way in which they
interact, has made possible the analysis of large-scale systems. This new
trend of research does not focus on the behavior of single agents, but mainly
on the analysis of the macroscopic and statistical properties of the whole
population, with the aim to discover regularities and universal rules. In this
sense, professional sports also represent optimal sources of data. Soccer
onody04 ; duch10 ; heuer10 , football girvan02 ; naim05 , baseball petersen08
; sire09 ; saavedra09 ; Petersen2011 and basketball naim07 ; skinner10 are
some remarkable cases in which network analysis revealed features not visible
with traditional approaches. These are practical examples of the general
outcome produced by the intense research activity of last years: network tools
and theories do not serve only for descriptive purposes, but have also wide
practical applicability. Representing a real system as a network allows in
fact to have a global view of the system and simultaneously use the entire
information encoded by its complete list of interactions. Particularly
relevant results are those regarding: the robustness of networks under
intentional attacks albert00 ; the spreading of viruses in graphs satorras01 ;
synchronization processes arenas08 , social models castellano09 , and
evolutionary and coevolutionary games Szabo2007 ; perc10 taking place on
networks. In this context fall also ranking techniques like the PageRank
algorithm Brin1 , where vertices are ranked on the basis of their “centrality”
in a diffusion process occurring on the graph. Diffusion algorithms,
originally proposed for ranking web pages, have been recently applied to
citation networks price65 . The evaluation of the popularity of papers chen07
, journals west08 ; west10 and scientists radicchi09 is performed not by
looking at local properties of the network (i.e., number of citations) but by
measuring their degree of centrality in the flow of information diffusing over
the entire graph. The use of the whole network leads to better evaluation
criteria without the addition of external ingredients because the complexity
of the citation process is encoded by the topology of the graph.
Figure 1: Properties of the data set. In panel a, we report the total number
of tournaments (top panel) and players (bottom panel) as a function of time.
In panel b, we plot the fraction of players having played (black circles), won
(red squares) and lost (blue diamonds) a certain number of matches. The black
dashed line corresponds to the best power-law fit with exponent consistent
with the value $1.2(1)$.
In this paper we continue in this direction of research and present a novel
example of a real system, taken from the world of professional sports,
suitable for network representation. We consider the list of all tennis
matches played by professional players during the last $43$ years
($1968$-$2010$). Matches are considered as basic contacts between the actors
in the network and weighted connections are drawn on the basis of the number
of matches between the same two opponents. We first provide evidence of the
complexity of the network of contacts between tennis players. We then develop
a ranking algorithm similar to PageRank and quantify the importance of tennis
players with the so-called “prestige score”. The results presented here
indicate once more that ranking techniques based on networks outperform
traditional methods. The prestige score is in fact more accurate and has
higher predictive power than well established ranking schemes adopted in
professional tennis. More importantly, our ranking method does not require the
introduction of external criteria for the assessment of the quality of players
and tournaments. Their importance is self-determined by the various
competitive processes described by the intricate network of contacts. Our
algorithm does nothing more than taking into account this information.
## II Methods
### II.1 Data set
Data were collected from the web site of the Association of Tennis
Professionals (ATP, www.atpworldtour.com). We automatically downloaded all
matches played by professional tennis players from January $1968$ to October
$2010$. We restrict our analysis only to matches played in Grand Slams and ATP
World Tour tournaments for a total of $3\,640$ tournaments and $133\,261$
matches. For illustrative purposes, in the top plot of the panel a of Figure
1, we report the number of tournaments played in each of the years covered by
our data set. With the exception of the period between $1968$ and $1970$, when
ATP was still in its infancy, about $75$ tournaments were played each year.
Two periods of larger popularity were registered around years $1980$ and
$1992$ when more than $90$ tournaments per year were played. The total number
of different players present in our data set is $3\,700$, and in the bottom
plot of panel a of Figure 1 we show how many players played at least one match
in each of the years covered by our analysis. In this case, the function is
less regular. On average, $400$ different players played in each of the years
between $1968$ and $1996$. Large fluctuations are anyway visible and a very
high peak in $1980$, when more than $500$ players participated in ATP
tournaments, is also present. Between $1996$ and $2000$, the number of players
decreased from $400$ to $300$ in an almost linear fashion. After that, the
number of participants in ATP tournaments started to be more constant with
small fluctuations around an average of about $300$ players.
### II.2 Network representation
Figure 2: Top player network and scheme for a single tournament. In panel a,
we draw the subgraph of the contact network restricted only to those players
who have been number one in the ATP ranking. Intensities and widths are
proportional to the logarithm of the weight carried by each directed edge. In
panel b, we report a schematic view of the matches played during a single
tournament, while in panel c we draw the network derived from it.
We represent the data set as a network of contacts between tennis players.
This is a very natural representation of the system since a single match can
be viewed as an elementary contact between two opponents. Each time the player
$i$ plays and wins against player $j$, we draw a directed connection from $j$
to $i$ [$j\to i$, see Figure 2]. We adopt a weighted representation of the
contacts Barrat2004 , by assigning to the generic directed edge $j\to i$ a
weight $w_{ji}$ equal to the number of times that player $j$ looses against
player $i$. Our data are flexible and allow various levels of representation
by including for example only matches played in a certain period of time, on a
certain type of surface, etc. An example is reported in panel a of Figure 2
where the network of contacts is restricted only to the $24$ players having
been number one in the official ATP ranking. In general, networks obtained
from the aggregation of a sufficiently high number of matches have topological
complex features consistent with the majority of networked social systems so
far studied in literature Reka:RevModPhys2002 ; Newman2003 . Typical measures
revealing complex structure are represented by the probability density
functions of the in- and out-strengths of vertices Barrat2004 , both following
a clear power-law behavior [see Figure 1, panel b]. In our social system, this
means that most of the players perform a small number of matches (won or lost)
and then quit playing in major tournaments. On the other hand, a small set of
top players performs many matches against worse opponents (generally beating
them) and also many matches (won or lost) against other top players. This
picture is consistent with the so-called “Matthew effect” in career longevity
recently observed also in other professional sports petersen08 ; Petersen2011
.
### II.3 Prestige score
The network representation can be used for ranking players. In our
interpretation, each player in the network carries a unit of “tennis prestige”
and we imagine that prestige flows in the graph along its weighted
connections. The process can be mathematically solved by determining the
solution of the system of equations
$P_{i}\;=\;\left(1-q\right)\,\sum_{j}\,P_{j}\,\frac{w_{ji}}{s_{j}^{out}}\;+\;\frac{q}{N}\;+\;\frac{1-q}{N}\,\sum_{j}\,P_{j}\,\delta\left(s_{j}^{out}\right)\;\;,$
(1)
valid for all nodes $i=1,\ldots,N$, with the additional constraint that
$\sum_{i}P_{i}=1$. $N$ indicates the total number of players (vertices) in the
network, while $s_{j}^{out}=\sum_{i}\,w_{ji}$ is the out-strength of the node
$j$ (i.e., the sum of the weight of all edges departing from vertex $j$).
$P_{i}$ is the “prestige score” assigned to player $i$ and represents the
fraction of the overall tennis prestige sitting, in the steady state of the
diffusion process, on vertex $i$. In Eqs. (1), $q\in\left[0,1\right]$ is a
control parameter which accounts for the importance of the various terms
contributing to the score of the nodes. The term
$\left(1-q\right)\,\sum_{j}\,P_{j}\,\frac{w_{ji}}{s_{j}^{out}}$ represents the
portion of score received by node $i$ in the diffusion process: vertices
redistribute their entire credit to neighboring nodes proportionally to the
weight of the connections linking to them. $\frac{q}{N}$ stands for a uniform
redistribution of tennis prestige among all nodes according to which each
player in the graph receives a constant and equal amount of credit. Finally
the term $\frac{1-q}{N}\,\sum_{j}\,P_{j}\,\delta\left(s_{j}^{out}\right)$
[with $\delta\left(\cdot\right)$ equal to one only if its argument is equal to
zero, and zero otherwise] serves as a correction in the case of existence of
dandling nodes (i.e., nodes with null out-strength), which otherwise would
behave as sinks in the diffusion process. Our prestige score is analogous to
the PageRank score Brin1 , originally formulated for ranking web pages and
more recently applied in different contexts.
In general topologies, analytical solutions of Eqs. (1) are hard to find. The
stationary values of the scores $P_{i}$s can be anyway computed recursively,
by setting at the beginning $P_{i}=1/N$ (but the results do not depend on the
choice of the initial value) and iterating Eqs. (1) until they converge to
values stable within a priori fixed precision.
#### II.3.1 Single tournament
Figure 3: Prestige score in a single tournament. Prestige score $P_{r}$ as a
function of the number of victories $r$ in a tournament with $\ell=7$ rounds
(Grand Slam). Black circles are obtained from Eqs. (7) and valid for $q=0$.
All other values of $q>0$ have been calculated from Eqs. (6): red squares
stand for $q=0.15$, blue diamonds for $q=0.5$, violet up-triangles for
$q=0.85$ and green down-triangles for $q=1$.
In the simplest case in which the graph is obtained by aggregating matches of
a single tournament only, we can analytically determine the solutions of Eqs.
(1). In a single tournament, matches are hierarchically organized in a binary
rooted tree and the topology of the resulting contact network is very simple
[see Figure 2, panels b and c]. Indicate with $\ell$ the number of matches
that the winner of the tournament should play (and win). The total number of
players present at the beginning of the tournament is $N=2^{\ell}$. The
prestige score is simply a function of $r$, the number of matches won by a
player, and can be denoted by $P_{r}$. We can rewrite Eqs. (1) as
$P_{r}=P_{0}\;+\;\left(1-q\right)\,\sum_{v=1}^{r}P_{v-1}\;\;,$ (2)
where $P_{0}=\frac{1-q}{N}\,P_{\ell}\;+\;\frac{q}{N}$ and $0\leq r\leq\ell$.
The score $P_{r}$ is given by the sum of two terms: $P_{0}$ stands for the
equal contribution shared by all players independently of the number of
victories; $\left(1-q\right)\,\sum_{v=1}^{r}P_{v-1}$ represents the score
accrued for the number of matches won. The former system of equations has a
recursive solution given by
$P_{r}\;=\;\left(2-q\right)P_{r-1}\;=\ldots\;=\;\left(2-q\right)^{r}\,P_{0}\;\;,$
(3)
which is still dependent on a constant that can be determined by implementing
the normalization condition
$\sum_{r=0}^{\ell}\,n_{r}\,P_{r}=1\;\;.$ (4)
In Eq. (4), $n_{r}$ indicates the number of players who have won $r$ matches.
We have $n_{r}=2^{\ell-r-1}$ for $0\leq r<\ell$ and $n_{\ell}=1$ and Eqs. (3)
and (4) allow to compute
$\begin{array}[]{ccc}\left(P_{0}\right)^{-1}=&\sum_{r=0}^{\ell-1}\,\left(2-q\right)^{r}\,2^{\ell-1-r}&+\;\left(2-q\right)^{\ell}\\\
&2^{\ell-1}\,\sum_{r=0}^{\ell-1}\,\left(\frac{2-q}{2}\right)^{r}&+\;\left(2-q\right)^{\ell}\\\
&2^{\ell-1}\,\frac{1-\left[\left(2-q\right)/2\right]^{\ell}}{\left[\left(2-q\right)/2\right]}&+\;\left(2-q\right)^{\ell}\\\
&\frac{2^{\ell}-\left(2-q\right)^{\ell}}{q}&+\;\left(2-q\right)^{\ell}\end{array}\;\;.$
In the former calculations, we have used the well known identity
$\sum_{r=0}^{v}\,x^{r}\,=\,\frac{1-x^{v+1}}{1-x}$, valid for any
$\left|x\right|<1$ and $v\geq 0$, which respectively means $0<q\leq 1$ and
$\ell>0$ in our case. Finally, we obtain
$P_{0}\;=\;\frac{q}{2^{\ell}\,+\,\left(2-q\right)^{\ell}\,\left(q-1\right)}\;\;,$
(5)
which together with Eqs. (3) provides the solution
$P_{r}\;=\;\frac{q\;\left(2-q\right)^{r}}{2^{\ell}\,+\,\left(2-q\right)^{\ell}\,\left(q-1\right)}\;\;.$
(6)
It is worth to notice that for $q=1$, Eqs. (6) correctly give
$P_{r}=2^{-\ell}$ for any $r$, meaning that, in absence of diffusion, prestige
is homogeneously distributed among all nodes. Conversely, for $q=0$ the
solution is
$P_{r}\;=\;\frac{2^{r}}{2^{\ell-1}\,\left(\ell+2\right)}\;\;.$ (7)
In Figure 3, we plot Eqs. (6) and (7) for various values of $q$. In general,
sufficiently low values of $q$ allow to assign to the winner of the tournament
a score which is about two order of magnitude larger than the one given to
players loosing at the first round. The score of the winner is an exponential
function of $\ell$, the length of the tournament. Grand Slams have for
instance length $\ell=7$ and their relative importance is therefore two or
four times larger than the one of other ATP tournaments, typically having
lengths $\ell=6$ or $\ell=5$.
## III Results
Rank | Player | Country | Hand | Start | End
---|---|---|---|---|---
$1$ | Jimmy Connors | United States | L | $1970$ | $1996$
$2$ | Ivan Lendl | United States | R | $1978$ | $1994$
$3$ | John McEnroe | United States | L | $1976$ | $1994$
$4$ | Guillermo Vilas | Argentina | L | $1969$ | $1992$
$5$ | Andre Agassi | United States | R | $1986$ | $2006$
$6$ | Stefan Edberg | Sweden | R | $1982$ | $1996$
$7$ | Roger Federer | Switzerland | R | $1998$ | $2010$
$8$ | Pete Sampras | United States | R | $1988$ | $2002$
$9$ | Ilie Năstase | Romania | R | $1968$ | $1985$
$10$ | Björn Borg | Sweden | R | $1971$ | $1993$
$11$ | Boris Becker | Germany | R | $1983$ | $1999$
$12$ | Arthur Ashe | United States | R | $1968$ | $1979$
$13$ | Brian Gottfried | United States | R | $1970$ | $1984$
$14$ | Stan Smith | United States | R | $1968$ | $1985$
$15$ | Manuel Orantes | Spain | L | $1968$ | $1984$
$16$ | Michael Chang | United States | R | $1987$ | $2003$
$17$ | Roscoe Tanner | United States | L | $1969$ | $1985$
$18$ | Eddie Dibbs | United States | R | $1971$ | $1984$
$19$ | Harold Solomon | United States | R | $1971$ | $1991$
$20$ | Tom Okker | Netherlands | R | $1968$ | $1981$
$21$ | Mats Wilander | Sweden | R | $1980$ | $1996$
$22$ | Goran Ivanišević | Croatia | L | $1988$ | $2004$
$23$ | Vitas Gerulaitis | United States | R | $1971$ | $1986$
$24$ | Rafael Nadal | Spain | L | $2002$ | $2010$
$25$ | Raúl Ramirez | Mexico | R | $1970$ | $1983$
$26$ | John Newcombe | Australia | R | $1968$ | $1981$
$27$ | Ken Rosewall | Australia | R | $1968$ | $1980$
$28$ | Yevgeny Kafelnikov | Russian Federation | R | $1992$ | $2003$
$29$ | Andy Roddick | United States | R | $2000$ | $2010$
$30$ | Thomas Muster | Austria | L | $1984$ | $1999$
Table 1: Top $30$ players in the history of tennis. From left to right we indicate for each player: rank position according to prestige score, full name, country of origin, the hand used to play, and the years of the first and last ATP tournament played. Players having been at the top of ATP ranking are reported in boldface. Figure 4: Relation between prestige rank and other ranking techniques. In panel a, we present a scatter plot of the prestige rank versus the rank based on the number of victories (i.e., in-strength). Only players ranked in the top $30$ positions in one of the two lists are reported. Rank positions are calculated on the network corresponding to all matches played between $1968$ and $2010$. In panel b, a similar scatter plot is presented, but now only matches of year $2009$ are considered for the construction of the network. Prestige rank positions are compared with those assigned by ATP. Year | Prestige | ATP year-end | ITF
---|---|---|---
$1968$ | Rod Laver | - | -
$1969$ | Rod Laver | - | -
$1970$ | Rod Laver | - | -
$1971$ | Ken Rosewall | - | -
$1972$ | Ilie Năstase | - | -
$1973$ | Tom Okker | Ilie Năstase | -
$1974$ | Björn Borg | Jimmy Connors | -
$1975$ | Arthur Ashe | Jimmy Connors | -
$1976$ | Jimmy Connors | Jimmy Connors | -
$1977$ | Guillermo Vilas | Jimmy Connors | -
$1978$ | Björn Borg | Jimmy Connors | Björn Borg
$1979$ | Björn Borg | Björn Borg | Björn Borg
$1980$ | John McEnroe | Björn Borg | Björn Borg
$1981$ | Ivan Lendl | John McEnroe | John McEnroe
$1982$ | Ivan Lendl | John McEnroe | Jimmy Connors
$1983$ | Ivan Lendl | John McEnroe | John McEnroe
$1984$ | Ivan Lendl | John McEnroe | John McEnroe
$1985$ | Ivan Lendl | Ivan Lendl | Ivan Lendl
$1986$ | Ivan Lendl | Ivan Lendl | Ivan Lendl
$1987$ | Stefan Edberg | Ivan Lendl | Ivan Lendl
$1988$ | Mats Wilander | Mats Wilander | Mats Wilander
$1989$ | Ivan Lendl | Ivan Lendl | Boris Becker
$1990$ | Stefan Edberg | Stefan Edberg | Ivan Lendl
$1991$ | Stefan Edberg | Stefan Edberg | Stefan Edberg
$1992$ | Pete Sampras | Jim Courier | Jim Courier
$1993$ | Pete Sampras | Pete Sampras | Pete Sampras
$1994$ | Pete Sampras | Pete Sampras | Pete Sampras
$1995$ | Pete Sampras | Pete Sampras | Pete Sampras
$1996$ | Goran Ivanišević | Pete Sampras | Pete Sampras
$1997$ | Patrick Rafter | Pete Sampras | Pete Sampras
$1998$ | Marcelo Ríos | Pete Sampras | Pete Sampras
$1999$ | Andre Agassi | Andre Agassi | Andre Agassi
$2000$ | Marat Safin | Gustavo Kuerten | Gustavo Kuerten
$2001$ | Lleyton Hewitt | Lleyton Hewitt | Lleyton Hewitt
$2002$ | Lleyton Hewitt | Lleyton Hewitt | Lleyton Hewitt
$2003$ | Roger Federer | Andy Roddick | Andy Roddick
$2004$ | Roger Federer | Roger Federer | Roger Federer
$2005$ | Roger Federer | Roger Federer | Roger Federer
$2006$ | Roger Federer | Roger Federer | Roger Federer
$2007$ | Rafael Nadal | Roger Federer | Roger Federer
$2008$ | Rafael Nadal | Rafael Nadal | Rafael Nadal
$2009$ | Novak Djoković | Roger Federer | Roger Federer
$2010$ | Rafael Nadal | Rafael Nadal | Rafael Nadal
Table 2: Best players of the year. For each year we report the best player
according to our ranking scheme and those of ATP and ITF. Best year-end ATP
players are listed for all years from $1973$ on. ITF world champions have
started to be nominated since $1978$ only.
We set $q=0.15$ and run the ranking procedure on several networks derived from
our data set. The choice $q=0.15$ is mainly due to tradition. This is the
value originally used in the PageRank algorithm Brin1 and then adopted in the
majority of papers about this type of ranking procedures chen07 ; radicchi09 ;
west08 ; west10 . It should be stressed that $q=0.15$ is also a reasonable
value because it ensures a high relative score for the winner of the
tournament as stated in Eqs. (6).
In Table 1, we report the results obtained from the analysis of the contact
network constructed over the whole data set. The method is very effective in
finding the best players of the history of tennis. In our top $10$ list, there
are $9$ players having been number one in the ATP ranking. Our ranking
technique identifies Jimmy Connors as the best player of the history of
tennis. This could be a posteriori justified by the extremely long and
successful career of this player. Among all top players in the history of
tennis, Jimmy Connors has been undoubtedly the one with the longest and most
regular trend, being in the top $10$ of the ATP year-end ranking for $16$
consecutive years ($1973$-$1988$). Prestige score is strongly correlated with
the number of victories, but important differences are evident when the two
techniques are compared. Panel a of Figure 4 shows a scatter plot, where the
rank calculated according to our score is compared to the one based on the
number of victories. An important outlier is this plot is represented by the
Rafael Nadal, the actual number one of the ATP ranking. Rafael Nadal occupies
the rank position number $40$ according to the number of victories obtained in
his still young career, but he is placed at position number $24$ according to
prestige score, consistently with his high relevance in the recent history of
tennis. A similar effect is also visible for Björn Borg, whose career length
was shorter than average. He is ranked at position $17$ according to the
number of victories. Prestige score differently is able to determine the
undoubted importance of this player and, in our ranking, he is placed among
the best $10$ players of the whole history of professional tennis.
In general, players still in activity are penalized with respect to those who
have ended their careers. Prestige score is in fact strongly correlated with
the number of victories [see panel a of Figure 4] and still active players did
not yet played all matches of their career. This bias, introduced by the
incompleteness of the data set, can be suppressed by considering, for example,
only matches played in the same year. Table 2 shows the list of the best
players of the year according to prestige score. It is interesting to see how
our score is effective also here. We identify Rod Laver as the best tennis
player between $1968$ and $1971$, period in which no ATP ranking was still
established. Similar long periods of dominance are also those of Ivan Lendl
($1981-1986$), Pete Sampras ($1992-1995$) and Roger Federer ($2003-2006$). For
comparison, we report the best players of the year according to ATP (year-end
rank) and ITF (International Tennis Federation, www.itftennis.com) rankings.
In many cases, the best players of the year are the same in all lists.
Prestige rank seems however to have a higher predictive power by anticipating
the best player of the subsequent year according to the two other rankings.
John McEnroe is the top player in our ranking in $1980$ and occupies the same
position in the ATP and ITF lists one year later. The same happens also for
Ivan Lendl, Pete Sampras, Roger Federer and Rafael Nadal, respectively best
players of the years $1984$, $1992$, $2003$ and $2007$ according to prestige
score, but only one year later placed at the top position of ATP and ITF
rankings. The official ATP rank and the one determined on the basis of the
prestige score are strongly correlated, but small differences between them are
very interesting. An example is reported in panel b of Figure 4, where the
prestige rank calculated over the contact network of $2009$ is compared with
the ATP rank of the end of the same year (official ATP year-end rank as of
December $28$, $2009$). The top $4$ positions according to prestige score do
not corresponds to those of the ATP ranking. The best player of the year, for
example, is Novak Djoković instead of Roger Federer.
We perform also a different kind of analysis by constructing networks of
contacts for decades and for specific types of playing surfaces. According to
our score, the best players per decade are [Table 3 lists the top $30$ players
in each decade] : Jimmy Connors ($1971-1980$), Ivan Lendl ($1981-1990$), Pete
Sampras ($1991-2000$) and Roger Federer ($2001-2010$). Prestige score
identifies Guillermo Vilas as the best player ever in clay tournaments, while
on grass and hard surfaces the best players ever are Jimmy Connors and Andre
Agassi, respectively [see Table 4 for the list of the top $30$ players of a
particular playing surface].
## IV Discussion
Tools and techniques of complex networks have wide applicability since many
real systems can be naturally described as graphs. For instance, rankings
based on diffusion are very effective since the whole information encoded by
the network topology can be used in place of simple local properties or pre-
determined and arbitrary criteria. Diffusion algorithms, like the one for
calculating the PageRank score Brin1 , were first developed for ranking web
pages and more recently have been applied to citation networks chen07 ;
radicchi09 ; west08 ; west10 . In citation networks, diffusion algorithms
generally outperform simple ranking techniques based on local network
properties (i.e., number of citations). When the popularity of papers is in
fact measured in terms of mere citation counts, there is no distinction
between the quality of the citations received. In contrast, when a diffusion
algorithm is used for the assessment of the quality of scientific
publications, then it is not only important that popular papers receive many
citations, but also that they are cited by other popular articles. In the case
of citation networks however, possible biases are introduced in the absence of
a proper classification of papers in scientific disciplinesradicchi08 . The
average number of publications and citations strongly depend on the popularity
of a particular topic of research and this fact influences the outcome of a
diffusion ranking algorithm. Another important issue in paper citation
networks is related to their intrinsic temporal nature: connections go only
backward in time, because papers can cite only older articles and not vice
versa. The anisotropy of the underlying network automatically biases any
method based on diffusion. Possible corrections can be implemented: for
example, the weight of citations may be represented by an exponential decaying
function of the age difference between citing and cited papers chen07 . Though
these corrections can be reasonable, they are ad hoc recipes and as such may
be considered arbitrary.
Here we have reported another emblematic example of a real social system
suitable for network representation: the graph of contacts (i.e., matches)
between professional tennis players. This network shows complex topological
features and as such the understanding of the whole system cannot be achieved
by decomposing the graph and studying each component in isolation. In
particular, the correct assessment of players’ performances needs the
simultaneously consideration of the whole network of interactions. We have
therefore introduced a new score, called “prestige score”, based on a
diffusion process occurring on the entire network of contacts between tennis
players. According to our ranking technique, the relevance of players is not
related to the number of victories only but mostly to the quality of these
victories. In this sense, it could be more important to beat a great player
than to win many matches against less relevant opponents. The results of the
analysis have revealed that our technique is effective in finding the best
players of the history of tennis. The biases mentioned in the case of citation
networks are not present in the tennis contact graph. Players do not need to
be classified since everybody has the opportunity to participate to every
tournament. Additionally, there is not temporal dependence because matches are
played between opponents still in activity and the flow does not necessarily
go from young players towards older ones. In general, players still in
activity are penalized with respect to those who already ended their career
only for incompleteness of information (i.e., they did not play all matches of
their career) and not because of an intrinsic bias of the system. Our ranking
technique is furthermore effective because it does not require any external
criteria of judgment. As term of comparison, the actual ATP ranking is based
on the amount of points collected by players during the season. Each
tournament has an a priori fixed value and points are distributed accordingly
to the round reached in the tournament. In our approach differently, the
importance of a tournament is self-determined: its quality is established by
the level of the players who are taking part of it.
In conclusion, we would like to stress that the aim of our method is not to
replace other ranking techniques, optimized and almost perfected in the course
of many years. Prestige rank represents only a novel method with a different
spirit and may be used to corroborate the accuracy of other well established
ranking techniques.
###### Acknowledgements.
We thank the Association of Tennis Professionals for making publicly available
the data set of all tennis matches played during last $43$ years. Helpful
discussions with Patrick McMullen are gratefully acknowledged as well.
| $1971-1980$ | $1981-1990$ | $1991-2000$ | $2001-2010$
---|---|---|---|---
Rank | Player | Country | Player | Country | Player | Country | Player | Country
$1$ | Jimmy Connors | United States | Ivan Lendl | United States | Pete Sampras | United States | Roger Federer | Switzerland
$2$ | Björn Borg | Sweden | John McEnroe | United States | Andre Agassi | United States | Rafael Nadal | Spain
$3$ | Ilie Năstase | Romania | Mats Wilander | Sweden | Michael Chang | United States | Andy Roddick | United States
$4$ | Guillermo Vilas | Argentina | Stefan Edberg | Sweden | Goran Ivanišević | Croatia | Lleyton Hewitt | Australia
$5$ | Arthur Ashe | United States | Jimmy Connors | United States | Yevgeny Kafelnikov | Russian Federation | Nikolay Davydenko | Russian Federation
$6$ | Brian Gottfried | United States | Boris Becker | Germany | Jim Courier | United States | Ivan Ljubičić | Croatia
$7$ | Manuel Orantes | Spain | Andrés Gómez | Ecuador | Richard Krajicek | Netherlands | Juan Carlos Ferrero | Spain
$8$ | Eddie Dibbs | United States | Yannick Noah | France | Thomas Muster | Austria | Novak Djoković | Serbia
$9$ | Harold Solomon | United States | Brad Gilbert | United States | Wayne Ferreira | South Africa | David Nalbandian | Argentina
$10$ | Stan Smith | United States | Tomáš Šmíd | Czech Republic | Thomas Enqvist | Sweden | Tommy Robredo | Spain
$11$ | Roscoe Tanner | United States | Henri Leconte | France | Boris Becker | Germany | David Ferrer | Spain
$12$ | Raúl Ramírez | Mexico | Tim Mayotte | United States | Stefan Edberg | Sweden | Fernando González | Chile
$13$ | Tom Okker | Netherlands | Anders Jarryd | Sweden | Sergi Bruguera | Spain | Andy Murray | Great Britain
$14$ | John Alexander | Australia | Miloslav Mečíř Sr. | Slovakia | Marc Rosset | Switzerland | Carlos Moyá | Spain
$15$ | Vitas Gerulaitis | United States | Kevin Curren | United States | Petr Korda | Czech Republic | Mikhail Youzhny | Russian Federation
$16$ | Ken Rosewall | Australia | Aaron Krickstein | United States | Todd Martin | United States | James Blake | United States
$17$ | John Newcombe | Australia | Guillermo Vilas | Argentina | Cédric Pioline | France | Tommy Haas | United States
$18$ | Wojtek Fibak | Poland | Joakim Nystrom | Sweden | Michael Stich | Germany | Fernando Verdasco | Spain
$19$ | Dick Stockton | United States | Emilio Sánchez | Spain | Àlex Corretja | Spain | Marat Safin | Russian Federation
$20$ | John McEnroe | United States | Johan Kriek | United States | Patrick Rafter | Australia | Tomás̆ Berdych | Czech Republic
$21$ | Adriano Panatta | Italy | Martin Jaite | Argentina | Magnus Gustafsson | Sweden | Juan Ignacio Chela | Argentina
$22$ | Jan Kodeš | Czech Republic | Jakob Hlasek | Switzerland | Andrei Medvedev | Ukraine | Radek Štěpánek | Czech Republic
$23$ | Jaime Fillol Sr. | Chile | Jimmy Arias | United States | Francisco Clavet | Spain | Andre Agassi | United States
$24$ | Robert Lutz | United States | Pat Cash | Australia | Marcelo Ríos | Chile | Robin Söderling | Sweden
$25$ | Marty Riessen | United States | Ramesh Krishnan | India | Greg Rusedski | Great Britain | Rainer Schüttler | Germany
$26$ | Rod Laver | Australia | José-Luis Clerc | Argentina | Fabrice Santoro | France | Feliciano López | Spain
$27$ | Tom Gorman | United States | Eliot Teltscher | United States | Magnus Larsson | Sweden | Tim Henman | Great Britain
$28$ | Vijay Amritraj | India | Thierry Tulasne | France | Tim Henman | Great Britain | Jarkko Nieminen | Finland
$29$ | Mark Cox | Great Britain | Scott Davis | United States | Alberto Berasategui | Spain | Mardy Fish | United States
$30$ | Onny Parun | New Zealand | Vitas Gerulaitis | United States | Albert Costa | Spain | Gastón Gaudio | Argentina
Table 3: Top $30$ players per decade. | Clay | Grass | Hard
---|---|---|---
Rank | Player | Country | Player | Country | Player | Country
$1$ | Guillermo Vilas | Argentina | Jimmy Connors | United States | Andre Agassi | United States
$2$ | Manuel Orantes | Spain | Boris Becker | Germany | Jimmy Connors | United States
$3$ | Thomas Muster | Austria | Roger Federer | Switzerland | Ivan Lendl | United States
$4$ | Ivan Lendl | United States | John Newcombe | Australia | Pete Sampras | United States
$5$ | Carlos Moyá | Spain | John McEnroe | United States | Roger Federer | Switzerland
$6$ | Eddie Dibbs | United States | Pete Sampras | United States | Stefan Edberg | Sweden
$7$ | José Higueras | Spain | Tony Roche | Australia | Michael Chang | United States
$8$ | Björn Borg | Sweden | Stefan Edberg | Sweden | John McEnroe | United States
$9$ | Ilie Năstase | Romania | Roscoe Tanner | United States | Andy Roddick | United States
$10$ | Andrés Gómez | Ecuador | Lleyton Hewitt | Australia | Lleyton Hewitt | Australia
$11$ | Àlex Corretja | Spain | Ken Rosewall | Australia | Brad Gilbert | United States
$12$ | Rafael Nadal | Spain | Arthur Ashe | United States | Jim Courier | United States
$13$ | José-Luis Clerc | Argentina | Stan Smith | United States | Brian Gottfried | United States
$14$ | Sergi Bruguera | Spain | Phil Dent | Australia | Thomas Enqvist | Sweden
$15$ | Mats Wilander | Sweden | Björn Borg | Sweden | Stan Smith | United States
$16$ | Albert Costa | Spain | Goran Ivanišević | Croatia | Boris Becker | Germany
$17$ | Gastón Gaudio | Argentina | Pat Cash | Australia | Wayne Ferreira | South Africa
$18$ | Juan Carlos Ferrero | Spain | Andy Roddick | United States | Ilie Năstase | Romania
$19$ | Harold Solomon | United States | Ivan Lendl | United States | Roscoe Tanner | United States
$20$ | Emilio Sánchez | Spain | Tim Henman | Great Britain | Tommy Haas | United States
$21$ | Adriano Panatta | Italy | Rod Laver | Australia | Rafael Nadal | Spain
$22$ | Félix Mantilla | Spain | Mark Edmondson | Australia | Tim Henman | Great Britain
$23$ | Francisco Clavet | Spain | John Alexander | Australia | Mats Wilander | Sweden
$24$ | Balázs Taróczy | Hungary | Hank Pfister | United States | Yevgeny Kafelnikov | Russian Federation
$25$ | Željko Franulović | Croatia | Wally Masur | Australia | Andy Murray | Great Britain
$26$ | Tomáš Šmíd | Czech Republic | Tim Mayotte | United States | Fabrice Santoro | France
$27$ | Jimmy Connors | United States | Vijay Amritraj | India | Harold Solomon | United States
$28$ | Raúl Ramirez | Mexico | Kevin Curren | United States | Ivan Ljubičić | Croatia
$29$ | Alberto Berasategui | Spain | Tom Okker | Netherlands | Marat Safin | Russian Federation
$30$ | Victor Pecci Sr. | Paraguay | Greg Rusedski | Great Britain | Aaron Krickstein | United States
Table 4: Top $30$ players of the history of tennis in tournaments played on
clay, grass and hard surfaces.
## References
* (1) Castellano C, Fortunato S, Loreto V (2009) Statistical physics of social dynamics. Rev Mod Phys 81: 591–646.
* (2) Barabási AL (2005) The origin of bursts and heavy tails in human dynamics. Nature 435: 207–211.
* (3) Malmgren RD, Stouffer DB, Motter AE, Amaral LAN (2008) A poissonian explanation for heavy tails in e-mail communication. Proc Natl Acad Sci USA 105: 18153–18158.
* (4) Radicchi F (2009) Human activity in the web. Phys Rev E 80: 026118.
* (5) Wu Y, Zhou C, Xiao J, Kurths J, Schellnhuber HJ (2010) Evidence for a bimodal distribution in human communication. Proc Natl Acad Sci USA 107: 18803–18808.
* (6) Barabási AL, Albert R (1999) Emergence of scaling in random networks. Science 286: 509-512.
* (7) Onody RN, de Castro PA (2004) Complex network study of brazilian soccer players. Phys Rev E 70: 037103.
* (8) Duch J, Waitzman JS, Amaral LAN (2010) Quantifying the performance of individual players in a team activity. PLoS ONE 5: e10937.
* (9) Heuer A, Müller C, Rubner O (2010) Soccer: Is scoring goals a predictable poissonian process? EPL 89: 38007.
* (10) Girvan M, Newman MEJ (2002) Community structure in social and biological networks. Proc Natl Acad Sci USA 99: 7821–7826.
* (11) Ben-Naim E, Vazquez F, Redner S (2007) What is the most competitive sport? J Korean Phys Soc 50: 124.
* (12) Petersen AM, Jung WS, Stanley HE (2008) On the distribution of career longevity and the evolution of home-run prowess in professional baseball. EPL 83: 50010.
* (13) Sire C, Redner S (2009) Understanding baseball team standings and streaks. Eur Phys J B 67: 473-481.
* (14) Saavedra S, Powers S, McCotter T, Porter MA, Mucha PJ (2009) Mutually-antagonistic interactions in baseball networks. Physica A 389: 1131–1141.
* (15) Petersen AM, Jung WS, Yang JS, Stanley HE (2011) Quantitative and empirical demonstration of the Matthew effect in a study of career longevity. Proc Natl Acad Sci USA 108: 18–23.
* (16) Ben-Naim E, Redner S, Vazquez F (2007) Scaling in tournaments. EPL 77: 30005.
* (17) Skinner B (2010) The price of anarchy in basketball. Journal of Quantitative Analysis in Sports 6: 3.
* (18) Albert R, Jeong H, Barabási AL (2000) Error and attack tolerance in complex networks. Nature 406: 378–382.
* (19) Pastor-Satorras R, Vespignani A (2001) Epidemic spreading in scale-free networks. Phys Rev Lett 86: 3200–3203.
* (20) Arenas A, Díaz-Guilera A, Kurths J, Moreno Y, Zhou C (2008) Synchronization in complex networks. Phys Rep 469: 93-153.
* (21) Szabò G, Fáth G (2007) Evolutionary games on graphs. Phys Rep 446: 97–216.
* (22) Perc M, Szolnoki A (2010) Coevolutionary games - a mini review. BioSystems 99: 109–125.
* (23) Brin S, Page L (1998) The anatomy of a large-scale hypertextual web search engine. Comput Netw ISDN Syst 30: 107–117.
* (24) de Solla Price DJ (1965) Networks of Scientific Papers. Science 149: 510–515.
* (25) Chen P, Xie H, Maslov S, Redner S (2007) Finding scientific gems with Google’s PageRank algorithm. Journal of Informetrics 1: 8–15.
* (26) Bergstrom CT, West J (2008) Assessing citations with the EigenfactorTM Metrics. Neurology 71: 1850–1851.
* (27) West J, Bergstrom T, Bergstrom CT (2010) Big Macs and Eigenfactor scores: Don’t let correlation coefficients fool you. J Am Soc Inf Sci 61: 1800–1807.
* (28) Radicchi F, Fortunato S, Markines B, Vespignani A (2009) Diffusion of scientific credits and the ranking of scientists. Phys Rev E 80: 056103.
* (29) Barrat A, Barthélemy M, Pastor-Satorras R, Vespignani A (2004) The architecture of complex weighted networks. Proc Natl Acad Sci USA 101: 3747–3752.
* (30) Albert R, Barabási AL (2002) Statistical mechanics of complex networks. Rev Mod Phys 74: 47–97.
* (31) Newman MEJ (2003) The structure and function of complex networks. SIAM Review 45: 167–256.
* (32) Radicchi F, Fortunato S, Castellano C (2008) Universality of citation distributions: Toward an objective measure of scientific impact. Proc Natl Acad Sci USA 105: 17268–17272.
|
arxiv-papers
| 2011-01-20T21:01:27 |
2024-09-04T02:49:16.551913
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Filippo Radicchi",
"submitter": "Filippo Radicchi",
"url": "https://arxiv.org/abs/1101.4028"
}
|
1101.4072
|
# Early phase of massive star formation: A case study of Infrared dark cloud
G084.81$-$01.09
S. B. Zhang11affiliation: Purple Mountain Observatory, Chinese Academy of
Sciences, Nanjing 210008, China; shbzhang@pmo.ac.cn , J. Yang11affiliation:
Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008,
China; shbzhang@pmo.ac.cn , Y. Xu11affiliation: Purple Mountain Observatory,
Chinese Academy of Sciences, Nanjing 210008, China; shbzhang@pmo.ac.cn , J. D.
Pandian22affiliation: Institute for Astronomy, 2680 Woodlawn Drive, Honolulu,
HI 96822, USA , K. M. Menten33affiliation: Max-Planck-Institut f$\ddot{u}$r
Radioastronomie, Auf dem H$\ddot{u}$gel 69, 53121 Bonn, Germany , C.
Henkel33affiliation: Max-Planck-Institut f$\ddot{u}$r Radioastronomie, Auf dem
H$\ddot{u}$gel 69, 53121 Bonn, Germany
###### Abstract
We mapped the MSX dark cloud G084.81$-$01.09 in the NH3 (1,1) - (4,4) lines
and in the $J$ = 1-0 transitions of 12CO, 13CO, C18O and HCO+ in order to
study the physical properties of infrared dark clouds, and to better
understand the initial conditions for massive star formation. Six ammonia
cores are identified with masses ranging from 60 to 250 $M_{\sun}$, a kinetic
temperature of 12 K, and a molecular hydrogen number density $n({\rm
H_{2}})\sim 10^{5}$ cm-3. In our high mass cores, the ammonia line width of 1
km s-1 is larger than those found in lower mass cores but narrower than the
more evolved massive ones. We detected self-reversed profiles in HCO+ across
the northern part of our cloud and velocity gradients in different molecules.
These indicate an expanding motion in the outer layer and more complex motions
of the clumps more inside our cloud. We also discuss the millimeter wave
continuum from the dust. These properties indicate that our cloud is a
potential site of massive star formation but is still in a very early
evolutionary stage.
stars: formation – ISM: molecules – ISM: kinematics and dynamics – ISM:
individual (G084.81$-$01.09)
## 1 INTRODUCTION
Infrared dark clouds (IRDCs) are identified as small areas with high
extinction against the bright diffuse mid-infrared background along the
Galactic Plane. These “flecks” in the sky were first discovered by the 15
$\mu$m ISOGAL survey (the inner Galactic disk survey of the Infrared Space
Observatory, Pérault et al. 1996) and later cataloged and studied using the
8.3 $\mu$m survey data of the Mid-course Space Experiment (MSX, Egan et al.
1998). Egan et al. found about 2000 IRDCs and note that these objects are cold
and dense molecular cores. These results were refined by later molecular line
observations (Carey et al., 1998; Teyssier et al., 2002) and radio continuum
observations (Carey et al., 2000). These authors concluded that these clouds
have low kinetic temperatures of 10-20 K and a gas density over $10^{5}$ cm-3.
Their small line widths of 1 - 3 km s-1 can be related to a very early phase
of star formation. Therefore, the IRDCs are good candidates for pre-
protostellar cores and may be the primary sites for future massive star
formation, which plays an important role in the evolution of our galaxy.
However, IRDCs remain mysterious in at least some aspects. A controversial
issue being discussed is whether the cloud evolution is quasi-static or
whether it is undergoing a dynamical process caused by lack of equilibrium
(Bergin & Tafalla, 2007). Understanding the role of supersonic turbulence in
dark clouds is important in settling this argument. Evidence from current
observations of IRDCs seems to support both views. Most IRDC cores are
supported by turbulent pressure but appear to be virialised (Pillai et al.,
2006). Furthermore, the origin of the turbulence in the molecular clouds and
the transition from the turbulent cloud to the quiescent core regime is still
unclear. Also, more systematic studies need to be done to compare them with
other relatively nearby massive star-forming clouds. The answers will provide
a critical element in our general understanding of star formation.
In this study, we have obtained spectroscopic data in several molecules
towards the dark cloud G048.81–01.09, a source in the catalog of Simon et al.
(2006). The cloud has an elongated morphology (N-S) with an extent of $\sim
15$ arcmin, and is situated in a region of high extinction ($A_{v}>20$,
Camberésy et al. 2002) of the dark cloud LDN935 (Lynds, 1962) which separates
the W80 HII region into the North America (NGC 7000) and Pelican (IC 5070)
nebulae (Comeron & Pasquali, 2005). Comeron & Pasquali (2005) proposed that
the ionizing source of the complex is 2MASS J205551.25+435224.6, an O5V star
located behind LDN935. The IRDC is almost starless with no associated IRAS or
MSX point sources. Previous molecular line observations indicated the presence
of large amounts of gas with different velocity components and signs of
ongoing star formation in LDN935 around our dark cloud (Feldt & Wendker,
1993). A single pointed CO observation towards our cloud conducted by Milman
(1975) suggests however that the gas is not appreciably heated.
There is no direct distance measurement for G084.81$-$01.09, but it is
associated with dark cloud LDN935 for which there are a number of distance
estimates. A radio continuum study carried out by Wendker et al. (1983) gave a
distance estimate of 500 pc. Straižys et al. (1993) obtained a consistent
result of 580 pc on the basis of photometric results of 564 stars toward three
areas near the North America and Pelican Nebulae complexes. New measurements
done by Cersosimo et al. (2007) derived a distance of about 0.7 kpc, which was
obtained from radio recombination line observations at a frequency around 1.4
GHz and by applying a flat rotation-curve model. Comeron & Pasquali (2005)
obtained a distance of 610 pc to the ionizing star, 2MASS J205551.25+435224.6.
Here we adopt a distance of 580 pc in our calculation of physical parameters.
## 2 OBSERVATIONS AND DATA REDUCTION
### 2.1 Effelsberg 100 m Observations
We mapped the infrared dark cloud G084.81$-$01.09 in the NH3 (1,1), (2,2),
(3,3) and (4,4) transitions using the Effelsberg 100 m telescope of the Max-
Planck-Institut für Radioastronomie (MPIfR) during December 12 - 14, 2007 (see
Table 1 for adopted line frequencies). The spectrometer backend employed was
the 8192 channel AK90 auto-correlator which consists of 8 individual 1024
channel correlators of 20 MHz bandwidth each. Additional data of NH3 (1,1),
(2,2) and (3,3) were obtained on February 21 - 24, 2008 using the Fast Fourier
Transform Spectrometer (FFTS), which has 16382 channels covering a bandwidth
of 500 MHz. The data were taken in using frequency switching with a frequency
throw of 7.5 MHz. The map covers the area with notable extinction in the MSX
8.3 $\mu$m image with a grid spacing of 30″. The pointing was checked roughly
every hour by observations of nearby continuum sources and was found to be
accurate to better than 10″.
Continuum scans were used for calibrating the absolute flux density. The raw
data were scaled from arbitrary units to main beam brightness temperature by
applying gain-elevation corrections and the flux scale was set using NGC 7027
by assuming it to have a flux density of 5.4 Jy at our observing frequency Ott
et al. (1994). Our absolute calibration uncertainty is $\pm$10%.
### 2.2 PMODLH 13.7 m Observations
The 12CO (1$-$0), 13CO (1$-$0), C18O (1$-$0) and HCO+ (1$-$0) observations of
the MSXDC G084.81$-$01.09 (line frequencies shown in Table 1) were carried out
at the Purple Mountain Observatory Delingha (PMODLH) 13.7 m telescope from
March to June in 2008. The 3 mm SIS receiver was used in double sideband mode.
Using three 1024 channel AOS spectrometers with bandwidths of 145.43 MHz,
42.87 MHz and 43.3 MHz, the three CO lines were observed simultaneously. The
source was mapped using position switched observations, with the standard
chopper wheel method for calibration (Penzias & Burrus, 1973). Standard
sources were checked roughly every 2 hours.
To compare with the NH3 data, our CO map covered the area mapped with NH3 and
extended a few arcmins to the northeast. The grid spacing was 30″ in the
center and 60″ in other regions. The HCO+ map covered a region similar to that
of the NH3 map with a typical grid spacing of 30″.
A summary of all observation parameters is provided in Table 1.
### 2.3 Data Reduction
We used the CLASS software package111CLASS is part of the GILDAS software
suite available from http://www.iram.fr/IRAMFR/GILDAS for the spectral data
reduction. After discarding the bad scans, the spectra of each position were
averaged and a polynomial baseline of orders 1-3 was subtracted. Bad channels
were excluded from the baseline fitting.
The method “NH3(1,1)” in CLASS (Buisson et al., 2005) was used to fit the
hyperfine structure of the NH3 (1,1) spectra and to derive optical depths and
line widths. The (2,2) line is so weak that only the main line could be
dectected towards a few positions. Therefore, we simply fitted the (2,2) line
with a single Gaussian to derive its main beam brightness temperature and an
optical depth was not calculated in this case. Table 2 summarizes the fit
parameters toward the cores detected in the data. No NH3 (3,3) or (4,4) lines
were detected in our observations.
The excitation temperature, rotational temperature, kinetic temperature and
ammonia column density toward the cores are listed in Table 3. These physical
parameters were derived using the standard formulae for NH3 spectra (Ho &
Townes, 1983).
The CO and HCO+ main beam brightness temperatures were derived from the
antenna temperature, $T_{A}^{*}$, using a main beam efficiency, $\eta_{\rm
mb}$, of 61%. A first order baseline was applied for the CO spectra and
1st-3rd order baselines were used for the HCO+ spectra. A single Gaussian fit
to the spectra provided the line velocity with respect to the local standard
of rest (LSR) and the line width for later discussion. The fit results are
given in Table 4.
## 3 RESULTS
### 3.1 Identification and Morphology of the Cloud
Fig. 1 shows the NH3 integrated intensity maps created from the velocity range
-2.5 to 4.5 km s-1 (which excludes the satellite lines). Our images of ammonia
emission toward G084.81$-$01.09 reveal extended, filamentary molecular
emission that closely matches the morphology of the Spitzer mid-infrared
extinction. Also, the ammonia maps (both NH3 (1,1) and (2,2)) allow us to
identify two main molecular condensations: one in the northeast, containing
ammonia cores 1, 2 and 6, and one in the south, consisting of cores 3, 4 and
5. All cores defined in the NH3 (1,1) map follow the standards given by
Tachihara et al. (2000) and the name of the cores were designated in
descending order of their integrated intensities, as tabulated in Table 2. The
NH3 (1,1) core sizes range from 0.25 to 0.42 pc.
The NH3 molecular emission also matches the 1.1 mm dust emission of Bolocam
Galactic Plane Survey (BGPS, Aguirre et al. 2010) remarkably well as shown in
Fig. 2a. Furthermore, with the blue circles marking the clump peaks of
continuum emission identified by Rosolowsky et al. (2009), we note that the
ammonia cores are coincident with the peaks in 1.1 mm emission.
The integrated intensity maps of 13CO (1$-$0) and C18O (1$-$0) are shown in
Fig. 2b and 2c. The 13CO and C18O emission lines show an extended feature
along north-south direction, with a sharp cutoff toward the northwest. The
extent of C18O emission is larger than that of both the ammonia and continuum
emission. The 13CO line, which tends to trace a lower gas density than C18O,
appears to be more extended. The strongest part of 13CO emission coincides
with the northern condensation (peak at the position of ammonia core 1) and
shows a tail to the east, while the 13CO emission in the southern part is flat
with no peak associated with that in the C18O map.
Fig. 2d shows the integrated intensity of HCO+ (1$-$0) overlaid on dust
emission. The HCO+ emission shows a different distribution compared to CO.
Since the line is optically thick (see §3.3), it is likely to trace the
envelope surrounding a dense core, while the C18O line is expected to trace
the dense core itself.
A candidate massive young stellar object (MYSO), G084.7847$-$01.1709,
identified by Urquhart et al. (2009) in their 6 cm VLA survey is marked with
cross symbol on Fig. 1. It is spatially coincident with the north condensation
of our cloud. However, no compact source can be found near this object in the
Spitzer 24 or 70 $\mu$m data. The lack of infrared emission could be due to
the source being an extragalactic object rather than a Galactic MYSO. Further
observations at centimeter wavelengths are needed to determine the spectral
index of emission and the nature of this object.
### 3.2 Line intensities and kinetic temperature
The excitation temperature, $T_{\rm ex}$, of the NH3 (1,1) transition (Column
$T_{\rm ex}$ in Table 3) was obtained from the optical depth via the relation
$T_{\rm mb}={h\nu\over k}[J(T_{\rm ex})-J(T_{\rm
bg})](1-e^{-\tau}),J(T)=(e^{h\nu\over kT}-1)^{-1},$
where $T_{\rm mb}$ and $\tau$ represent the temperature and the optical depth
from CLASS fitting procedures and $T_{\rm bg}$ equals 2.7 K. The rotational
temperature between the NH3 (1,1) and (2,2) inversion doublets were then
derived from the excitation temperature using the method given by Ho & Townes
(1983). We then estimate the kinetic temperature using the expression of
Tafalla et al. (2004).
The typical kinetic temperature in the cores is about 12 K as seen in Table 3.
No obvious difference in temperature appears among the six cores. The kinetic
temperature in the envelope is about 2 K higher than in the cores. Désert et
al. (2008) carried out a low resolution ($\sim$12 arcmin) survey at four
(sub)millimeter wavelengths and derived a dust temperature of 8.5 K with a
fixed emissivity law exponent of 2 toward our cloud, which is slightly lower
than our gas temperature, probably because of the different beam size.
The physical parameters of CO are calculated under the assumption of Local
Thermodynamic Equilibrium (LTE), in which the 13CO and C18O lines are assumed
to reach the same excitation temperature as 12CO line. The 12CO (1-0) line is
optically thick, so we can derive the kinetic temperature directly from its
brightness temperature. The kinetic temperature is found to be 12 – 14 K as
listed in Table 5 and agrees with that derived from ammonia.
The 13CO and C18O line intensity ratio indicates the different abundances and
optical depths in the cloud. As shown in Fig. 3, the 13CO to C18O ratios in
regions with strong emission are low (about 1.4-2.5), while the ratios of
faint emission areas are high and close to the local interstellar
[13CO]/[C18O] ratio of about 7.3 (Wilson & Rood, 1994; Teyssier et al., 2002).
The observation of variable line ratios as a function of intensities is
consistent with that of Teyssier et al. (2002).
### 3.3 Line widths and profiles
The NH3 (1,1) line widths in the cloud vary from 0.4 to 2.8 km s-1, which is
consistent with the value reported from massive dense cores (Bensen & Myers,
1989; Pillai et al., 2006). The distribution of ammonia line widths is shown
in Fig. 4. The line widths in the peripheral regions ($>$1.3 km s-1) are
generally larger than those in the clump centers (0.7 - 1.4 km s-1). The line
width of Core 6 is influenced and broadened by the extended part of core 1 and
double peaked spectra appear in the central part of core 6 (shown in Fig. 5,
P6). For cores 2, 4, 5 and 6, the (2,2) line widths are larger than the (1,1)
line widths, which suggests that the (2,2) line may not trace the same gas as
the (1,1) line toward these cores (Pillai et al., 2006).
As seen in Fig. 5, the CO lines show multiple velocity components. Most of the
13CO spectra in the cores are blended with a component around 5.5 km s-1,
which also appears in some C18O spectra. This component is consistent with the
extended part of another cloud to the west of core 1, reported by Feldt &
Wendker (1993). A few of the 13CO spectra are flat topped, indicating that the
line is optically thick at these locations. As with the 13CO line profiles,
the C18O spectra are also slightly asymmetric, especially in core 2. The line
widths over the core regions range between 2.3 and 4.6 km s-1 for 13CO and
between 1.5 and 2.7 km s-1 for C18O.
The thermal line width, $[(8\ln 2)kT_{kin}/m]^{1/2}$ ($k$ is the Boltzmann
constant and $m$ is the mean molecular mass), at a kinetic temperature,
$T_{kin}$, of 12 K is 0.18 km s-1 for NH3 and 0.14 km s-1 for 13CO and C18O.
Therefore, the non-thermal line widths are significantly larger than the
thermal line widths and near the observed line widths. This suggests that non-
thermal broadening mechanisms (rotation, turbulence, etc) play a dominant role
in producing the observed line profiles.
The HCO+ (1-0) spectra as shown in Fig. 6 display three types of profiles: an
asymmetric, double-peaked shape in the northern C18O condensation (e.g., B, C,
P1, P2, etc.), spectra with a blue shoulder in the far north (e.g., A, P6,
etc.) and a single blue-shifted peak in the south (e.g., D, etc.). The single
peaked C18O line appears to bisect the HCO+ profiles in velocity, indicating
that the HCO+ lines are likely optically thick and self-absorbed.
The blue-shifted dip due to self-absorption may indicates that this occurs in
an expanding envelope. By fitting Gaussian line profiles to these spectra and
comparing the peak velocity of HCO+ and C18O spectra, one can roughly estimate
an expansion velocity (Aguti et al., 2007). We measure a velocity of $\delta
V=|v_{\rm peak}({\rm C^{18}O})-v_{\rm peak}({\rm HCO^{+}})|\approx 1.05$ km
s-1. This value is less than the C18O line width at corresponding positions
but is supersonic. On the contrary, the blue peaked profile in the south can
be associated with infall asymmetry and may suggest infall motions of the
outer cloud material. However, due to the low signal to noise ratio of the
HCO+ spectra nearby and the relatively large beam sizes involved, this motion
can not be confirmed at all positions with blue-shifted peak. In order to
better understand the spatial distribution of the HCO+ line profile, we have
compared the velocity of peak emission in the HCO+ and C18O lines for each
location in the northern condensation, and mapped the velocity difference,
$\delta V=v_{\rm peak}({\rm C^{18}O})-v_{\rm peak}({\rm HCO^{+}})$ (Fig. 6).
It can be seen that the outflow asymmetry (coded blue) dominates the central
and northern regions of the cloud, while the infall asymmetry (coded red)
appears on the southern edge.
### 3.4 Velocity structure
From Table 3, we note that the line width of core 2 is rather small (0.95 km
s-1) but remarkably increases to 1.2 km s-1 when the spectra are averaged over
the core area, which reflects the existence of velocity gradients in the core.
This is also seen in the NH3 (1,1) channel maps shown in Fig. 7 for velocities
from $-$1.0 to 3.0 km s-1 where different clumps appear at separate
velocities. The clump structure is complex and covers a wide velocity range.
In the lower velocity channels ($-$1.0 to 0.0 km s-1), the gas is concentrated
in a clump in the east of northern condensation, while the higher velocity
channels (0.5 to 2.0 km s-1) indicate the clump in the two peak position and a
southern clump.
The CO presents multiple peaks along the line of sight (cf. CO spectra in Fig.
5). The channel maps of 13CO (1-0) and C18O (1-0) (Fig. 8) show that the
different velocity components present various shapes and orientations. C18O
shows a velocity structure that is similar to that of ammonia, shown in Fig.
7. The velocity increases from the position of core 1 at about 1 km s-1 to 2
km s-1 towards the north, which can be seen in position-velocity maps (Fig. 9)
and the 13CO, C18O and NH3 channel maps.
To estimate the magnitude and direction of the velocity gradient in each core,
we adopt the method of Goodman et al. (1993) by fitting a linear velocity
gradient to the projected velocity field along the line of sight. Thus the
observed velocity $v_{\rm LSR}$ can be expressed as
$v_{\rm
LSR}=v_{0}+\mathscr{G}\Delta\alpha\sin\Theta_{\mathscr{G}}+\mathscr{G}\Delta\delta\cos\Theta_{\mathscr{G}},$
where $\mathscr{G}$ is the magnitude of the velocity gradient, $\Delta\alpha$
and $\Delta\delta$ represent offsets in right ascension and declination in
arcseconds, $v_{0}$ is the systemic velocity of the cloud, and
$\Theta_{\mathscr{G}}$ is the direction of increasing velocity, measured east
of north. We carry out a least-squares fit to the two dimensional velocity
field of 13CO (1-0), C18O (1-0), NH3 (1,1) and NH3 (2,2) weighted by
$1/\sigma_{\rm LSR}^{2}$, where $\sigma_{\rm LSR}$ is the uncertainty in
fitting $v_{\rm LSR}$. In Table 6, we list our fitting results ($\mathscr{G}$
and $\Theta_{\mathscr{G}}$) with their formal errors ($\sigma_{\mathscr{G}}$
and $\sigma_{\Theta_{\mathscr{G}}}$). We exclude fits that fail the $3\sigma$
criterion ($\mathscr{G}\geq 3\sigma_{\mathscr{G}}$ mentioned by (Goodman et
al., 1993)) or those with a random velocity field as seen from the velocity
map.
Both cores 1 and 6 present significant velocity gradients in CO and NH3 lines.
The magnitude of the velocity gradient is greater in NH3, and the direction of
the gradient is somewhat different from that seen in CO. In cores 3, 4 and 5,
the velocity gradient is detected only in NH3 (1,1). The NH3 gas associated
with core 2 also exhibits a clear velocity gradient in Fig. 4. The peak lies
on a velocity ridge with a gradient of 3.15 km s-1 pc-1 to the east and 1.93
km s-1 pc-1 to the west.
The distinct velocity gradients given by different molecules are probably due
to the different densities they trace (Goodman et al., 1993). Velocity
gradients seen in the lower density tracers, 13CO or C18O, probably arise in
the envelope of the northern condensations. This gradient is dominated by
unresolved clump-clump motions within the envelope, and may also be
contaminated by the higher velocity component. The higher density tracers
(e.g., NH3), on the other hand, reveal the gradients within the small clumps
themselves.
### 3.5 Density and abundance
The column densities of CO are estimated using the equation given by Scoville
et al. (1986):
$N={3k^{2}\over 4h\pi^{3}\mu^{2}\nu^{2}}\exp\left({h\nu J\over kT_{\rm
ex}}\right)\times{T_{\rm ex}+h\nu/6k(J+1)\over{\rm exp}(-h\nu/kT_{\rm
ex})}{\tau\over 1-{\rm exp}(-\tau)}\int{T_{R}^{*}dV},$
where $J$ is the rotational quantum number of the lower state in the observed
transition, $\mu$ is the permanent dipole moment and $\tau$ is the optical
depth derived from the observed ratios of 12CO and 13CO (or C18O) emission.
The derived physical properties are listed in Table 5. We also derive the
column densities of NH3 for the cores, which range from $15-34\times 10^{14}$
cm-2 as shown in Table 3. Since 13CO is not optically thin in some cores, we
estimate the H2 column densities based on the C18O column densities where a
factor $N({\rm H_{2}})/N({\rm C^{18}O})=7\times 10^{6}$ was adopted (Frerking
et al., 1982; Kramer et al., 1999). This gives H2 column densities around
$\sim 10^{22}$ cm-2 as listed in column (2) of Table 7.
With the H2 column densities derived from CO emission, we determine the
ammonia abundances and list them in Table 7. The ammonia abundance in the six
peak positions is $3-5.5\times 10^{-8}$, which agrees with the parameters
reported by Pillai et al. (2006), though their abundances are derived from the
dust emission of SCUBA. When averaged over the cores, the abundances are
reduced to $2.3\times 10^{-8}$. The ammonia abundance and the correlation of
its spatial distribution with the morphology of the dust emission is
consistent with a chemical model given by Bergin & Langer (1997), in which NH3
is not depleted. We then compare its abundance with that of 13CO toward cores
in Fig. 10. They differ slightly from each other with core 1 showing the
highest abundance among the cores. This reveals the complex chemical
environment in the cloud.
We obtain the gas mass of cores by integrating the total H2 column density
over the cores and show the results in Table 8. The mass range from 60
$M_{\sun}$ to 250 $M_{\sun}$. The gas density can be derived by assuming the
cores to be spherical and dividing the mass by the volume of the cores. An
average density of $\sim 1.1\times 10^{5}$ cm-3 was derived as shown in Table
8.
The total H2 column density can also be related to the 1.1 mm flux by assuming
the continuum emission to be optically thin. After convolving the data to the
resolution of the 100 m Effelsberg (40″) and 13.7 m Delingha (60″) beams, the
dust based column density is given by
$N({\rm H_{2}})_{\rm 1.1~{}mm}={S_{\rm 1.1~{}mm}\over\Omega_{\rm
beam}B_{\nu}(T_{d})\kappa_{\rm 1.1~{}mm}m},$
where $\Omega_{\rm beam}$ is the telescope beam solid angle, $B_{\nu}(T_{d})$
is the Planck function at the dust temperature $T_{d}$ (assumed to be equal to
the gas temperature deduced from NH3), $\kappa_{\rm 1.1~{}mm}$ is the dust
opacity at 1.1 mm, and $m$ is the mean molecular mass. Here we adopt
$\kappa_{\rm 1.1~{}mm}$ to be $0.0114$ cm2 g-1 (Ossenkopf & Henning, 1994;
Enoch et al., 2008) and assume a gas to dust mass ratio of 100. The results
have the same magnitude as those derived from C18O and are listed in Table 7.
We also compare our molecular emission with the BGPS survey, as shown in Fig.
11. The integrated intensity of NH3 (1,1) shows a better correlation with the
1.1 mm dust emission compared to C18O as mentioned previously.
The diversity in column densities derived from molecular gas and dust may be
due to different structures and components they trace. The CO column density,
because of the low density it traces, is highly affected by the structure of
the cloud along the line of sight, especially at the positions where clumps
with similar velocities blend together. Also the molecular observations are
affected by high opacity or chemical depletion at high densities. On the other
hand, calculations related to the dust emission are based on the value of dust
opacity and assumptions about the dust to gas ratio. The dust opacity changes
between different dust models and in environments of different density
(Ossenkopf & Henning, 1994).
## 4 DISCUSSION
### 4.1 Evidence of massive star formation
In this section, we will compare our results with those from other low-mass
star forming regions, and discuss the evidence for massive star formation and
the evolutionary state of the cloud. The mean intrinsic line width for NH3
(1,1) in the samples given by Jijina et al. (1999) (most of the objects are
low-mass cores) is 0.5 km s-1. A recent work by Foster et al. (2009) also
reported a small observed line width of 0.3 km s-1 for low mass NH3 cores in
the Perseus Molecular cloud. Crapsi et al. (2005) undertook a survey toward
low-mass starless cores and also reported a narrow line width of 0.5 km s-1
for C18O (1-0). A broader line width of 1.2 km s-1 for C18O (1-0) is derived
from a JCMT observation by Jørgensen et al. (2002), which is still less than
those given in our observed region. Meanwhile, the dominant non-thermal line
widths in our cloud indicate more non-thermal support than those in low-mass
star forming regions. Although line widths can be broadened by active outflows
in low-mass protostellar regions to 2 km s-1 for C18O (1-0) (Swift & Welch,
2008) and 1.2 km s-1 for NH3 (1,1 ) (Rudolph et al., 2001), at this stage, an
active outflow has not developed yet. The NH3 line widths here are comparable
to those IRDCs of Pillai et al. (2006). Moreover, our derived column densities
of NH3, a few times 1015 cm-2, are higher than those of the low-mass cores
($\sim 10^{13}-10^{15}$ cm-2) given by Suzuki et al. (1992), and are also
comparable to those of the massive IRDCs given by Pillai et al. (2006). In
addition, the supersonic motions of the envelope and the large velocity
gradients in our cloud also indicate a more dynamic motion compared to low-
mass cores. Therefore, we suggest that MSXDC G084.81$-$01.09 is potentially at
an early evolutionary state of massive star formation.
### 4.2 Gravitational stability of the cores
The expanding envelope detected from the line profiles described in §3.3 lead
us to consider the gravitational stability of MSXDC G084.81$-$01.09. By using
the molecular mass, $M_{Molecular}$, derived from C18O (1-0) and the core
radius $R$ determined from the NH3 (1,1) maps, we obtain an escape velocity
($\sqrt{2GM/R}$) which ranges from $\sim 1.9$ km s-1 in core 6 to a maximum of
$\sim 3.2$ km s-1 in core 2 (Table 8). The three-dimensional velocity
dispersion of the cores provided by the NH3 line width lies in the range of
1.2 – 2.4 km s-1 which is less than the escape velocity in each core. However,
the C18O velocity dispersions (2.5 - 4 km s-1) exceed the escape velocity in
some regions of the cloud, especially for core 6, which has a small escape
velocity. This may lead to instability for core 6. Additionally, an expansion
speed of 1.0 km s-1 in the envelope of the core (cf. §3.3), though
significantly less than the escape velocity, may aggravate the situation.
To examine the gravitational stability of the cores, it is useful to calculate
the virial mass of the cores as well as the virial parameter, the ratio
between the virial mass and core mass. The virial parameter given by Bertoldi
& McKee (1992) can be expressed as $\alpha=5\sigma^{2}R/GM$, where R is the
radius of the core, $\sigma=\sqrt{3/(8\ln 2)}\times FWHM$, and $M$ is the core
mass. The virial mass and virial parameter can be found in Table 8. The
average virial parameter is about 1.1 which suggests that most of the cores
are virialised. However, the virial parameters of cores 2 and 6 deviate
significantly from unity. This can be caused by a number of factors. One
explanation is the potential for the presence of spatially unresolved stellar
clusters in the cores, which can be identified from the color temperature of
the dust continuum map. For $M_{\rm Virial}>M_{\rm Molecular}$, beam filling
factors smaller than unity or streaming motions may adversely affect the
estimate and may lead to an underestimated column density or an overestimated
line width. Alternatively, it is possible that the cores may not actually be
in virial equilibrium, and may be transient entities (Ballesteros-Paredes,
2006), which is probably the case in core 6.
We derive a Jeans mass greater than 170 $M_{\sun}$ for each core using the
equation $M_{Jeans}=17.3~{}{T_{kin}}^{1.5}n^{-0.5}M_{\sun}$, where $T_{kin}$
is the kinetic temperature derived from NH3 data, and $n$ is the average
density as listed in Table. 8. We note that the masses of cores 1 and 2 are
comparable to their Jeans masses, but the other cores are significantly less
massive. However, considering the clumping in cores 1 and 2, it is still
possible for them to form several individual protostars. Therefore, we suggest
that cores 1 - 5 are gravitationally bound, among which cores 1 and 2 are
marginally stable against collapse. However, core 6 is probably transient.
### 4.3 Comparison with more evolved clouds
Our dark cloud has a lower kinetic temperature than other evolved clouds. Wu
et al. (2006) reported a mean temperature of 19 K in massive water maser
sources excluding known HII regions, and in ultra-compact HII (UC HII)
regions. Churchwell et al. (1990) give a significantly higher temperature
spread from 15 K to $>$60 K for UC HII regions. The temperature of the cloud
is a good tracer to determine its evolutionary stage in the context of massive
star formation. The low temperatures in our cloud suggests that it is not an
active high-mass star forming region yet.
The C18O line widths are typically a factor of two to five times smaller than
the line widths in more evolved massive star forming regions traced by other
molecules with similar densities. The NH3 line widths we find are comparable
to the massive cores reported by Pillai et al. (2006) and Wu et al. (2006),
but are smaller than those associated with water masers or are in proximity to
UC HII regions. More specifically, the line widths of our cloud are similar to
those of the most quiescent NH3 cores with no methanol maser or 24 GHz
continuum emission described by Longmore et al. (2007) and smaller than those
in more evolved cores with maser or continuum emission. Our cloud also
presents similar NH3 line width to those bright-rimmed clouds which are
undergoing recently initiated star formation and being subjected to intense
levels of ionizing radiation (Morgan et al., 2010). After removing the thermal
broadening contributions to the line widths, the velocity dispersions
associated with turbulence are supersonic and lie between those of the sources
triggered by the radiatively driven implosion and the non-triggered ones. For
the more evolved clouds in the UC HII phase, Churchwell et al. (1990) reported
significantly larger line widths of $\sim 3$ km s-1. The large line widths in
more evolved stages are likely due to a combination of warmer temperatures and
broadening from dynamics such as outflows.
The HCO+ (1$-$0) spectra associated with hyper-compact H II regions have a
considerably larger line width (15-60 km s-1) reported by Churchwell et al.
(2010). For our cloud, our HCO+ line profiles are similar to the samples of
Brand et al. (2001) and Cesaroni et al. (1999), which are all in the pre-UC
HII stage of star formation, though some sources in Cesaroni et al. (1999) are
proven to be more evolved than those in Brand et al. (2001). We find similar
line widths and profiles as theirs, with a self-absorption dip between the
blue and red peak. However, our cloud is colder than the pre-UC HII sources
and do not have any associated IRAS point source.
The optical depths of NH3 (1,1) are comparable to the cores given by Pillai et
al. (2006) and Longmore et al. (2007). Our core 4, 5 and 6 have a lower
optical depth than the NH3 sources ($\sim 2.7-2.9$) associated with methanol
masers or 24-GHz continuum emission in Longmore et al. (2007), while core 1, 2
and 3 near the cloud center gives higher optical depth but still lower than
those samples ($\sim 3.1$) with only NH3 association. However, due to the
large uncertainty, without further observation, it would be premature to
classify these sources into more detailed evolutionary stages based on optical
depth. Our cloud density is about an order of magnitude lower than the typical
value in more evolved massive star-forming regions: an average density of
$10^{6}$ cm-3 is given by Beuther et al. (2002) using LVG calculations in
clouds prior to building up an UC HII region, and a similar value is given by
Pillai et al. (2007) in clouds harboring a UC HII region.
Our NH3 (1,1) core sizes lie between the mean value of 0.28 pc given by
Longmore et al. (2007) and 0.57 pc given by Pillai et al. (2006), but are
significantly smaller than the mean size of 1.6 pc reported by Wu et al.
(2006). The small values of Longmore et al. (2007) probably result from their
small beam size (11″), but the large values of Wu et al. (2006) are likely a
result of evolution. All these aspects suggest that our cloud is in an early
evolutionary stage of massive star formation. These quiescent cores are likely
the candidates for pre-protostellar cores.
### 4.4 Rotation in the cores
One of the possible generators of the velocity gradients mentioned in §3.4 is
rotation in the cores. Goodman et al. (1993) has discussed the relation
between rotation and the geometric properties of the cores. They found no
causal relation between velocity gradient direction and core elongation, nor
any relationship between the magnitude of the gradient and core shape, on the
size scale of $10^{17}$ cm. An examination of the gradients in our clouds also
reveals no obvious relation of this kind. In view of the early evolutionary
state and the dominant role of non-thermal line broadening in the cloud, we
believe that the velocity gradients in the cloud are more affected by
stochastic processes, such as fragmentation, collisions, and nonuniform
magnetic fields rather than ordered motions such as rotation.
We use the parameter $\beta$ as defined by Goodman et al. (1993) to compare
the rotational kinetic energy to the gravitational energy. Thus $\beta$ can be
written as
$\beta={(1/2)I\omega^{2}\over qGM^{2}/R}={1\over 2}{p\over
q}{\omega^{2}R^{3}\over GM},$
where $I$ is the moment of inertia given by $I=pMR^{2}$, $qGM^{2}/R$
represents the gravitational potential energy of the mass $M$ within a radius
$R$, and $\omega=\mathscr{G}/\sin i$, where $i$ is the inclination of the core
along the line of sight. We assume $p/q=0.22$ as for a sphere with an $r^{-2}$
density profile and $\sin i=1$. Our $\beta$ values as listed in Table 6 are
consistent with the result of Goodman et al. (1993) that most clouds have
$\beta\leq 0.05$. The small values of $\beta$ show that the effect of rotation
is not significant in maintaining the overall dynamical stability for the
cloud. The results also indicate that these cores are unlikely to experience
instabilities driven by rotation (e.g., bars, fission, or rings), if the
magnetic fields are not taken into account.
## 5 Summary
The multiple molecular line observations of the infrared dark cloud MSXDC
G084.81$-$01.09 were analyzed. The results are summarized as below.
The cloud has a low gas temperature of 12 K, a column density of $10^{22}$
cm-2 and masses of the cores range from 60 to 250 $M_{\sun}$. The spatial
distribution of the ammonia emission correlates well with the mid-infrared
absorption and millimeter dust emission. All 6 ammonia cores are associated
with their corresponding dust peaks. The line width of 1 km s-1 for ammonia
indicates the dominant role of non-thermal broadening in the cloud. The
abundances of ammonia range from $2-3.5\times 10^{-8}$. These facts show that
the cloud is a site of potential massive star formation at a very early
evolutionary stage.
13CO and C18O trace a more extended cloud structure. Together with the HCO+
emission, we detect an expanding envelope in part of the cloud with a
expansion velocity of 1 km s-1. Five of the cores are gravitationally bound
(four are virialised) with Jeans masses comparable or exceeding their
molecular mass except a possible unbound transient core. We found velocity
gradients in the cores using different molecular tracers. The different
directions and magnitudes of the gradients using the different tracers
indicate clumping motions on different scales.
The observations presented here are part of a larger program designed to
search and investigate potential massive star forming regions. Higher
resolution observations with interferometers would be needed to study
turbulence and fragmentation inside our cores. Additional submillimeter
mapping observations could derive the dust temperature and structure
associated with the gas. We could also use radio polarization observation to
study the relationship between magnetic field and cloud structure at the early
phrases of massive star forming.
This work is made based on observations with the 100-m telescope of the MPIfR
(Max-Planck-Institut für Radioastronomie) at Effelsberg and Delingha 13.7-m
telescope of Purple Mountain Observatory. This work is also based on
observations made with the Spitzer Space Telescope, which is operated by the
Jet Propulsion Laboratory, California Institute of Technology under a contract
with NASA. The authors appreciate all the staff members of the observatories
for their help during the observations. This work was supported by the Chinese
NSF through grants NSF 11073054, NSF 10733030, NSF 10703010 and NSF 10621303,
and NBRPC (973 Program) under grant 2007CB815403.
## References
* Aguirre et al. (2010) Aguirre, J. et al. 2010, in prep
* Aguti et al. (2007) Aguti, E. D., Lada, C. J., Bergin, E. A., Alves, J. F., & Birkinshaw, M. 2007, ApJ, 665, 457
* Arons & Max (1975) Arons, J. & Max, C. E. 1975, ApJ, 196, 77
* Bachiller et al. (1987) Bachiller, R., Guilloteau, S., & Kahane, C. 1987, A&A, 173, 324
* Ballesteros-Paredes (2006) Ballesteros-Paredes, J. 2006, MNRAS, 372, 443
* Bergin & Langer (1997) Bergin, E. A., & Langer, W. D. 1997, ApJ, 486, 316
* Bertoldi & McKee (1992) Bertoldi, F., & McKee, C. F. 1992, ApJ, 395, 140
* Bensen & Myers (1989) Benson, P. J., & Myers, P. C. 1989, ApJS, 71, 89
* Bergin & Tafalla (2007) Bergin, E. A. & Tafalla, M. 2007, ARA&A, 45, 339
* Buisson et al. (2005) Buisson, G., et al. 2005, CLASS Continuum and Line Analysis Single-dish Software, 4th edn., Observatoire de Grenoble/IRAM, http://www.iram.fr/IRAMFR/GILDAS/doc/html/class-html/class.html
* Beuther et al. (2002) Beuther, H., Schilke, P., Menten, K. M., Motte, F., Sridharan, T. K., & Wyrowski, F. 2002, ApJ, 566, 945
* Beuther & Sridharan (2007) Beuther, H., & Sridharan, T. K. 2007, 668, 348
* Brand et al. (2001) Brand J., Cesaroni R., Palla F., & Molinari S. 2001, A&A, 370, 230
* Camberésy et al. (2002) Cambrésy, L., Beichman, C. A., Jarrett, T. H., & Cutri, R. M. 2002, AJ, 123, 2559
* Carey et al. (1998) Carey, S. J., Clark, F. O., Egan, M. P., Price, S. D., Shipman, R. F., & Kuchar, T. A. 1998, ApJ, 508, 721
* Carey et al. (2000) Carey, S. J., Feldman, P. A., Redman, R. O., Egan, M. P., MacLeod, J. M., & Prick, S. D. 2000, ApJ, 543, L157
* Caselli et al. (2002) Caselli, P., Benson, P. J., Myers, P. C., & Tafalla, M. 2002, ApJ, 572, 238
* Cersosimo et al. (2007) Cersosimo, J. C., Muller, R. J., Figueroa Vélez, S., Santiago Figueroa, N., Baez, P., & Testori, J. C. 2007, ApJ, 656, 248
* Cesaroni et al. (1999) Cesaroni R., Felli M., & Walmsley C. M. 1999, A&AS, 136, 333
* Churchwell et al. (2010) Churchwell, E., Sievers, A., & Thum, C. 2010, A&A, 513, 9
* Churchwell et al. (1990) Churchwell E., Walmsley C. M., & Cesaroni R. 1990, A&AS, 83, 119
* Comeron & Pasquali (2005) Comeron, F., & Pasquali, A. 2005, A&A, 430, 541
* Crapsi et al. (2005) Crapsi, A., Caselli, P., Walmsley, C. M., Myers, P. C., Tafalla, M., Lee, C. W., & Bourke, T. L. 2005, ApJ, 619, 379
* Crapsi et al. (2004) Crapsi, A., Caselli, P., Walmsley, C. M., Tafalla, M., Lee, C. W., Bourke, T. L., & Myers, P. C. 2004, A&A, 420, 957
* Désert et al. (2008) Désert, F.-X., Macías-Pérez, J. F., Mayet, F., Giardino, G., Renault, C., Aumont, J., Benoît, A., Bernard, J.-Ph., Ponthieu, N., & Tristram, M. 2008, A&A, 481, 411
* Egan et al. (1998) Egan, M. P., Shipman, R. F., Price, S. D., Carey, S. J., & Clark, F. O. 1998, ApJ, 494, L199
* Enoch et al. (2008) Enoch, M. L., Evans, N. J., II, Sargent, A. I., Glenn, J., Rosolowsky, E., & Myers, P. 2008, ApJ, 684, 1240
* Feldt & Wendker (1993) Feldt, C., Wendker, H. J. 1993, A&AS, 100, 287
* Foster et al. (2009) Foster, J. B., Rosolowsky, E. W., Kauffmann, J., Pineda, J. E., Borkin, M. A., Caselli, P., Myers, P. C., & Goodman, A. A. 2009, ApJ, 696, 298
* Frerking et al. (1982) Frerking, M. A., Langer, W. D., & Wilson, R. W. 1982, ApJ, 262, 590
* Gammie & Ostriker (1996) Gammie, C. F., & Ostriker, E. C. 1996, ApJ, 466, 814
* Goodman et al. (1993) Goodman, A. A., Benson, P. J., Fuller, G. A., & Myers, P. C. 1993, ApJ, 406, 528
* Hily-Blant (2000) Hily-Blant, P. 2000, Manual for the HFS fit method of CLASS, Observatoire de Grenoble/IRAM, http://www.iram.es/IRAMES/otherDocuments/postscripts/classHFS.ps
* Ho & Townes (1983) Ho, P. T. P., & Townes, C. H. 1983, ARA&A, 21, 239
* Jijina et al. (1999) Jijina, J., Myers, P. C., & Adams, F. C. 1999, ApJS, 125, 161
* Jørgensen et al. (2002) Jørgensen, J. K., Schöier, F. L., & van Dishoeck, E. F. 2002, A&A, 389, 908
* Kramer et al. (1999) Kramer, C., Alves, J., Lada, C. J., Lada, E. A., Sievers, A., Ungerechts, H., & Walmsley, C. M. 1999, A&A, 342, 257
* Longmore et al. (2007) Longmore, S. N., Burton, M. G., Barnes, P. J., Wong, T., Purcell, C. R., & Ott, J. 2007, MNRAS, 379, 535
* Lynds (1962) Lynds, B. T. 1962, ApJS, 7, 1
* Milman (1975) Milman, A. S. 1975, ApJ, 202, 673
* Morgan et al. (2010) Morgan, L. K., Figura, C. C., Urquhart, J. S., & Thompson, M. A. 2010, MNRAS, 408, 157
* Myers et al. (1996) Myers, P. C., Mardones, D., Tafalla, M., Williams, J. P., & Wilner, D. J. 1996, ApJ, 465, 133
* Ott et al. (1994) Ott, M., Witzel, A., Quirrenbach, A., Krichbaum, T. P., Standke, K. J., Schalinski, C. J., & Hummel, C. A. 1994, A&A, 284, 331
* Ossenkopf & Henning (1994) Ossenkopf, V., & Henning, Th. 1994, A&A, 291, 943
* Penzias & Burrus (1973) Penzias A. A., & Burrus C.A. 1973, ARA&A11, 51
* Pérault et al. (1996) Pérault, M., Omont, A., Simon, G., et al. 1996, A&A, 315, L165
* Pillai et al. (2006) Pillai, T., Wyrowski, F., Carey, S. J., & Menten, K. M. 2006, A&A, 450, 569
* Pillai et al. (2007) Pillai, T., Wyrowski, F., Hatchell, J., Gibb, A. G., & Thompson, M. A. 2007, A&A, 467, 207
* Ragan et al. (2006) Ragan, S. E., Bergin, E. A., Plume, R., Gibson, D. L., Wilner, D. J., O Brien, S., & Hails, E. 2006, ApJS, 166, 567
* Rosolowsky et al. (2009) Rosolowsky, E. et al. 2009, ApJ, in prep
* Rudolph et al. (2001) Rudolph, A. L., Bachiller, R., Rieu, N. Q., Van Trung, D., Palmer, P., & Welch, W. J. 2001, ApJ, 558, 204
* Schnee et al. (2009) Schnee, S., Rosolowsky, E., Foster, J., Enoch, M., & Sargent, A. 2009, ApJ, 619, 1754
* Scoville et al. (1986) Scoville, N.Z., Sargent, A. I., Sanders, D. B., Masson, C. R., Lo, K. Y., & Phillips, T. G. 1986, ApJ, 303, 416
* Simon et al. (2006) Simon, R., Jackson, J. M., Rathborne, J. M., & Chambers, E. T. 2006, ApJ, 629, 227
* Straižys et al. (1993) Straižys, V., Kazlauskas, A., Vansevičius, V., & Černis, K. 1993, Baltic Astron., 2, 171
* Suzuki et al. (1992) Suzuki, H., Yamamoto, S., Ohishi, M., Kaifu, N., Ishikawa, S., Hirahara, Y., & Takano, S. 1992, ApJ, 392, 551
* Swift & Welch (2008) Swift, J. J., & Welch, W. J. 2008, ApJS, 174, 202
* Tachihara et al. (2000) Tachihara, K., Mizuno, A., & Fukui, Y. 2000, ApJ, 528, 817
* Tafalla et al. (2004) Tafalla, M., Myers, P. C., Caselli, P., & Walmsley, C. M. 2004, A&A, 416, 191
* Teyssier et al. (2002) Teyssier, D., Hennebelle, P., & Pérault, M. 2002, A&A, 382, 624
* Urquhart et al. (2009) Urquhart, J. S., Hoare, M. G., Purcell, C. R., Lumsden, S. L., Oudmaijer, R. D., Moore, T. J. T., Busfield, A. L., Mottram, J. C., & Davies, B. 2009, A&A, 501, 539
* Wendker et al. (1983) Wendker, H. J., Benz, D., & Baars, J. W. M. 1983, A&A, 124, 116
* Wilson & Rood (1994) Wilson, T. L. & Rood, R.T. 1994, ARA&A, 32, 191
* Wu et al. (2006) Wu, Y., Zhang, Q., Yu, W., Miller, M., Mao, R., Sun, K., & Wang, Y. 2006, A&A, 450, 607
* Zhou et al. (1989) Zhou, S., Wu, Y., Evans, N. J. II, Fuller, G. A., & Myers, P. C. 1989, ApJ, 346, 168
Figure 1: Spitzer MIPS 24 $\mu$m image overlaid with an NH3 (1,1) contour map
of integrated intensity from the MSX dark cloud G084.81$-$01.09. The spectra
were integrated over $-$2.5 to 4.5 km s-1. The peak contour flux is 8 K km s-1
with contour interval 1 K km s-1. The red star indicates the ionizing star of
W80 complex, 2MASS J205551.25+435224.6, reported by Comeron & Pasquali (2005).
The cross indicates the MYSO candidate, G084.7847$-$01.1709, identified by
Urquhart et al. (2009). The green triangles indicate the star clusters
detected by Camberésy et al. (2002). The white circle shows the beamsize of
the NH3 (1,1) observation.
Figure 2: CSO BOLOCAM 1.1 mm continuum image overlaid with an integrated
intensity contour map of observed molecular lines. (a) NH3 (1,1) integrated
over $-$2.5 to 4.5 km s-1, with a bold line at 50% of maximum intensity of 8.7
K km s-1 in steps of 9%. (b) 13CO(1$-$0) integrated over $-$0.5 to 2.5 km s-1,
with a bold line at 70% of maximum intensity of 38.2 K km s-1 in steps of 5%.
(c) C18O(1$-$0) integrated over $-$0.5 to 2.5 km s-1, with bold line at 70% of
the maximum intensity of 8.3 K km s-1 in steps of 6%. (d) HCO+(1$-$0)
integrated over $-$2.5 to 4.5 km s-1, with bold line at 70% of the maximum
intensity of 7.7 K km s-1 in steps of 8%. Four peaks in the HCO+ map are
designated as A-D. The cross indicates the MYSO candidate,
G084.7847$-$01.1709, identified by Urquhart et al. (2009). The small circles
indicate the peak positions of clumps identified from the continuum map
(Rosolowsky et al., 2009). The white circle at the bottom left shows the
beamsize of the NH3 (1,1) observation. Figure 3: $\rm T_{MB}$ of 13CO(1$-$0)
vs. that of C18O(1$-$0) for G084.81$-$01.09. The data are extracted within a
velocity range of $-$1.5 to 3.7 km s-1 and smoothed to 0.2 km s-1 spectral
resolution. Each dot represents the $\rm T_{MB}$ of the same channel at same
position for 13CO and C18O. Data points from the northern and southern clumps
are shown in black and grey colors respectively. The dashed-line represents
the local ISM ratio. Note that the northern and southern clumps have a
different trend for the intensity ratio at high intensities.
Figure 4: Left panel: Color-coded map of the NH3 (1,1) full width at half
maximum (FWHM) line width in the MSX dark cloud G084.81$-$01.09 overlaid on a
contour map of the integrated intensity. Right panel: Color-coded map of the
local standard of rest velocity (Vlsr) overlaid on a contour map of the
integrated intensity. Only spectra with signal greater than 5$\sigma$ (FWHM
map) and 3$\sigma$ (Vlsr map) are shown. The circle in the bottom right shows
the beamsize.
Figure 5: Spectra of G084.81$-$01.09 toward the six peak positions (labeled as
P1–P6). The first two rows show NH3 (1,1) and (2,2) spectra. The NH3 (2,2)
lines are moved upward in each panel for clarity. The main (1,1) line feature
of P6 was extracted and drawn beside the spectra. The next two rows show the
13CO (1$-$0) and C18O (1$-$0) spectra. In all cases, the stronger line belongs
to 13CO. The peak positions are referred to in Table 2.
Figure 6: Left: HCO+ (solid) and C18O (dashed) spectra at the positions of the
4 NH3 cores and the 4 HCO+ peaks (indicated by red characters in Fig. 2) from
north to south. The vertical dashed lines indicate the Vlsr of the NH3 (1,1)
line, obtained from a Gaussian fit. Right: Map of $\delta V$, the velocity
difference between the peak of C18O and HCO+ overlaid on a C18O contour map of
the region. Figure 7: NH3 (1,1) channel maps for G084.81$-$01.09. The central
velocity of each channel, in km s-1, is marked on the top-left corner for each
map. The reference position is at 20h57m00.9s,+43°44′15.4″.
Figure 8: 13CO(1$-$0) channel maps (upper panels) for G084.81$-$01.09 and
C18O(1$-$0) channel maps (lower panels) for G084.81$-$01.09. The central
velocity of each channel, in km s-1, is marked on the top-left corner for each
map, and the plus signs indicate the positions of 6 cores identified from NH3
(1,1) map (see Fig. 1). The reference position is the same as in Fig. 7.
Figure 9: The position-velocity map of 13CO (top), C18O (second row), HCO+
(third row) and NH3 (1,1) main component (bottom) along the Declination axis
for the MSX dark cloud G084.81$-$01.09, at Right Ascension of 20h56m47.1s
(left panel with offset $\Delta\delta$=$-$150″) and Right Ascension of
20h57m03.7s (right panel with offset $\Delta\delta$=30″). The declination
reference coordinate is the same as in Fig. 7. Figure 10: Abundances relative
to 13CO toward peak positions (Table 2). The relative abundances were shown in
different grey scale.
Figure 11: Relation between the NH3 (1,1) and C18O (1-0) integrated intensity
and the 1.1 mm flux density. Continuum data were smoothed to corresponding
resolution. The uncertanties of integrated intensity are the rms noise of the
spectra and the error in dust flux is calculated following Rosolowsky et al.
(2009). The solid line represents the least-square fit of the data weighted in
both coordinates.
Table 1: Observation parameters
Line | $\nu_{0}$ | Telescope | During | HPBW | $\delta\nu$ | $\delta v$ | $T_{\rm sys}$
---|---|---|---|---|---|---|---
| (GHz) | | | (″) | (KHz) | (km s-1) | (K)
NH3 ($J,K$=1, 1) | 23.694496 | Effelsberg 100m | 2007 Dec. | 40 | 19.5 | 0.24 | 30-150
| | | 2008 Feb. | 40 | 30.5 | 0.38 | 30-45
NH3 ($J,K$=2, 2) | 23.722633 | Effelsberg 100m | 2007 Dec. | 40 | 19.5 | 0.24 | 30-140
| | | 2008 Feb. | 40 | 30.5 | 0.38 | 30-45
NH3 ($J,K$=3, 3) | 23.870130 | Effelsberg 100m | 2007 Dec. | 40 | 19.5 | 0.24 | 30-140
| | | 2008 Feb. | 40 | 30.5 | 0.38 | 30-45
NH3 ($J,K$=4, 4) | 24.139417 | Effelsberg 100m | 2007 Dec. | 40 | 19.5 | 0.24 | 30-160
12CO ($J$=1-0) | 115.271204 | DLH 13.7m | 2008 Mar.-Apr. | 60 | 142.1 | 0.37 | 200-400
13CO ($J$=1-0) | 110.201353 | DLH 13.7m | 2008 Mar.-Apr. | 64 | 41.7 | 0.11 | 200-400
C18O ($J$=1-0) | 109.782183 | DLH 13.7m | 2008 Mar.-Apr. | 64 | 42.1 | 0.11 | 200-400
HCO+ ($J$=1-0) | 89.188530 | DLH 13.7m | 2008 May.-Jun. | $\sim$78 | 42.1 | 0.14 | 270-420
Note. — $\nu_{0}$ is the rest frequency of the line, HPBW is the half power
beam width of the telescope, $\delta\nu$ and $\delta v$ represent the
frequency and velocity resolutions
Table 2: Cores detected in the NH3 observations of G084.81$-$01.09
Peak | $\alpha$ | $\delta$ | Transition | $V_{\rm LSR}$ | $T_{\rm MB}$ | $FWHM$ | $\tau_{\rm main}$ | $\int{T_{A}^{*}dv}$
---|---|---|---|---|---|---|---|---
| (J2000) | (J2000) | | (km s-1) | (K) | (km s-1) | | (K km s-1)
1 …… | 20 56 47.1 | +43 44 45 | 1-1 | 1.25$\pm$0.01 | 3.84$\pm$0.24 | 1.38$\pm$0.03 | 3.63$\pm$0.20 | 8.73$\pm$0.20
| | | 2-2 | 1.30$\pm$0.04 | 1.14$\pm$0.19 | 0.96$\pm$0.14 | | 1.39$\pm$0.19
2 …… | 20 56 47.1 | +43 43 15 | 1-1 | 1.23$\pm$0.01 | 4.98$\pm$0.37 | 0.75$\pm$0.02 | 3.61$\pm$0.24 | 7.26$\pm$0.19
| | | 2-2 | 1.20$\pm$0.03 | 1.51$\pm$0.17 | 0.88$\pm$0.09 | | 1.61$\pm$0.19
3 …… | 20 57 03.7 | +43 40 45 | 1-1 | 1.22$\pm$0.01 | 3.67$\pm$0.36 | 0.89$\pm$0.04 | 3.25$\pm$0.29 | 6.03$\pm$0.19
| | | 2-2 | 1.20$\pm$0.04 | 1.07$\pm$0.19 | 0.63$\pm$0.08 | | 1.09$\pm$0.19
4 …… | 20 57 00.9 | +43 38 45 | 1-1 | 0.99$\pm$0.01 | 3.71$\pm$0.31 | 1.00$\pm$0.03 | 2.22$\pm$0.20 | 5.91$\pm$0.18
| | | 2-2 | 0.84$\pm$0.10 | 0.58$\pm$0.14 | 1.40$\pm$0.28 | | 0.78$\pm$0.18
5 …… | 20 57 03.7 | +43 37 15 | 1-1 | 1.20$\pm$0.01 | 3.52$\pm$0.25 | 0.97$\pm$0.03 | 2.32$\pm$0.17 | 5.53$\pm$0.10aaPeak 5 was observed in FFT mode with lower noise compared to Peak 1-4 and 6 in frequency switch mode.
| | | 2-2 | 1.21$\pm$0.06 | 0.72$\pm$0.09 | 1.32$\pm$0.21 | | 1.02$\pm$0.11
6 …… | 20 56 55.4 | +43 47 15 | 1-1 | 2.01$\pm$0.02 | 2.51$\pm$0.23 | 1.40$\pm$0.04 | 2.08$\pm$0.21 | 5.02$\pm$0.18
| | | 2-2 | 2.07$\pm$0.14 | 0.53$\pm$0.12 | 1.98$\pm$0.34 | | 1.39$\pm$0.19
Note. — Columns are peak number, offset position from map center, NH3 (J,K)
transition, the local standard rest velocity, main beam brightness
temperature, full width at half maximum, optical depth and integrated
intensity of the main line. Values in column (4), (5), (6) and (7) are the HFS
or GAUSS fitting results and errors estimated in CLASS. Integrated intensities
are calculated from $-$2.5 to 4.5 km s-1 with errors derived from rms.
Table 3: Physical properties of the NH3 cores
Number | $T_{\rm ex}$ | $T_{\rm rot}$aaNH3 rotational temperature given by Ho & Townes (1983) as
$T_{\rm rot}=-41.5\div\ln\left\\{{-0.282\over\tau_{m}(1,1)}\ln\left[1-{T_{R}^{*}(2,2,m)\over T_{R}^{*}(1,1,m)}\times(1-e^{-\tau_{m}(1,1)})\right]\right\\}.$ | $T_{\rm kin}$bbNH3 kinetic temperature $T_{\rm kin}={T_{\rm rot}\over 1-{T_{\rm rot}\over 42}\ln[1+1.1\exp(-16/T_{\rm rot})]}$. | $N(\rm NH_{3})$ccNH3 column density derived from its optical depth via the relation given by Bachiller et al. (1987) as
$N({\rm NH_{3}})=2.784\times 10^{13}\tau J(T_{\rm ex})\Delta V\times Q/Q_{1},$ where $Q$ is the partition function and $Q/Q_{1}={1\over 3}e^{23.4/T_{\rm rot}}+1+{5\over 3}e^{-41.5/T_{\rm rot}}+{14\over 3}e^{-101.5/T_{\rm rot}}+...$. | $\Delta v_{1}$ddThe mean line width is the fitting error-weighted average line width over the core regions. | $\Delta v_{2}$eeThe line width of the mean spectrum is the line width of mean spectrum of the spectra inside the core regions.
---|---|---|---|---|---|---
| (K) | (K) | (K) | ($10^{14}\rm cm^{-2}$) | (km s-1) | (km s-1)
1 …… | 6.7$\pm$0.5 | 11.4$\pm$0.6 | 12.2$\pm$0.7 | 33.5$\pm$2.9 | 1.30$\pm$0.01 | 1.56$\pm$0.01
2 …… | 7.8$\pm$0.8 | 11.5$\pm$0.5 | 12.3$\pm$0.6 | 21.1$\pm$2.2 | 0.95$\pm$0.01 | 1.21$\pm$0.01
3 …… | 6.5$\pm$0.8 | 11.7$\pm$0.7 | 12.5$\pm$0.9 | 18.3$\pm$2.5 | 0.87$\pm$0.01 | 0.98$\pm$0.02
4 …… | 6.9$\pm$0.8 | 10.5$\pm$0.7 | 11.1$\pm$0.9 | 17.5$\pm$2.2 | 1.06$\pm$0.01 | 1.19$\pm$0.02
5 …… | 6.6$\pm$0.6 | 11.2$\pm$0.5 | 11.9$\pm$0.6 | 15.3$\pm$1.6 | 1.01$\pm$0.01 | 1.05$\pm$0.02
6 …… | 5.6$\pm$0.8 | 11.5$\pm$0.8 | 12.4$\pm$1.0 | 16.1$\pm$2.2 | 1.21$\pm$0.02 | 1.49$\pm$0.03
Note. — Columns are peak number, excitation temperature, NH3 rotational
temperature, kinetic temperature, NH3 column density derived from the optical
depth, the mean line width and the line width of the mean spectrum of the
regions.
Table 4: CO and HCO+ line parameters at the positions of the NH3 cores.
Peak | Molecule | $V_{\rm LSR}$ | $T^{*}_{\rm R}$ | $\Delta V$ | $\int{T_{R}^{*}dV}$ | $\sigma$
---|---|---|---|---|---|---
| | (km s-1) | (K) | (km s-1) | (K km s-1) | (K)
1 …… | 12CO (1$-$0) | 2.31 | 10.76 | 6.38 | 89.21$\pm$0.44 | 0.20
| 13CO (1$-$0) | 0.96 | 8.00 | 3.19 | 31.44$\pm$0.22 | 0.22
| C18O (1$-$0) | 0.83 | 3.04 | 2.17 | 6.41$\pm$0.10 | 0.17
| HCO+ (1$-$0) | 1.44 | 1.00 | 4.76 | 4.99$\pm$0.25 | 0.21
2 …… | 12CO (1$-$0) | 2.67 | 11.02 | 6.47 | 89.02$\pm$0.72 | 0.33
| 13CO (1$-$0) | 1.03 | 7.48 | 3.08 | 30.84$\pm$0.22 | 0.22
| C18O (1$-$0) | 1.04 | 2.75 | 1.83 | 5.21$\pm$0.13 | 0.21
| HCO+ (1$-$0) | 1.41 | 1.61 | 3.11 | 5.83$\pm$0.20 | 0.17
3 …… | 12CO (1$-$0) | 2.21 | 9.86 | 7.24 | 95.16$\pm$0.51 | 0.23
| 13CO (1$-$0) | 0.90 | 5.87 | 4.05 | 28.53$\pm$0.25 | 0.25
| C18O (1$-$0) | 1.08 | 2.51 | 1.72 | 4.49$\pm$0.15 | 0.24
| HCO+ (1$-$0) | $-$0.52 | 1.08 | 2.11 | 2.44$\pm$0.24 | 0.20
4 …… | 12CO (1$-$0) | 2.64 | 9.79 | 6.69 | 85.58$\pm$0.44 | 0.20
| 13CO (1$-$0) | 1.11 | 5.71 | 3.30 | 25.15$\pm$0.21 | 0.20
| C18O (1$-$0) | 1.04 | 2.83 | 1.62 | 4.80$\pm$0.10 | 0.15
5 …… | 12CO (1$-$0) | 2.80 | 9.76 | 6.34 | 77.12$\pm$0.43 | 0.20
| 13CO (1$-$0) | 1.32 | 5.57 | 3.14 | 23.28$\pm$0.21 | 0.21
| C18O (1$-$0) | 1.24 | 2.47 | 1.50 | 3.99$\pm$0.11 | 0.17
6 …… | 12CO (1$-$0) | 2.47 | 10.89 | 5.85 | 77.47$\pm$0.45 | 0.20
| 13CO (1$-$0) | 1.67 | 8.34 | 2.70 | 28.82$\pm$0.23 | 0.23
| C18O (1$-$0) | 1.57 | 2.62 | 2.08 | 5.18$\pm$0.10 | 0.16
| HCO+ (1$-$0) | 2.08 | 1.35 | 3.33 | 4.91$\pm$0.19 | 0.16
Note. — 12CO, 13CO, C18O and HCO+ line parameters are calculated within the
velocity range from -3.5 to 6.8 km s-1, -2.7 to 7.7 km s-1, -1.0 to 3.0 km s-1
and 0.0 to 9.0 km s-1, respectively. The HCO+ line was not observed towards
cores P4 and P5 due to low signal-to-noise ratio.
Table 5: Physical properties derived from CO.
| $\rm{}^{12}CO$ | | $\rm{}^{13}CO$ | | $\rm C^{18}O$
---|---|---|---|---|---
Peak | $T_{\rm ex}$ | | $\tau$ | $N$ | | $\tau$ | $N$
| (K) | | | ($10^{15}$cm-2) | | | ($10^{15}$cm-2)
1 …… | 14.17$\pm$0.20 | | 1.26$\pm$0.09 | 58.36$\pm$2.12 | | 0.35$\pm$0.03 | 8.91$\pm$0.31
2 …… | 14.43$\pm$0.33 | | 1.18$\pm$0.09 | 56.27$\pm$2.32 | | 0.31$\pm$0.03 | 6.35$\pm$0.33
3 …… | 13.26$\pm$0.22 | | 0.90$\pm$0.07 | 45.31$\pm$1.55 | | 0.34$\pm$0.03 | 6.37$\pm$0.32
4 …… | 12.90$\pm$0.19 | | 0.91$\pm$0.06 | 39.87$\pm$1.18 | | 0.39$\pm$0.03 | 6.63$\pm$0.30
5 …… | 12.25$\pm$0.20 | | 1.01$\pm$0.08 | 37.02$\pm$1.29 | | 0.33$\pm$0.03 | 4.70$\pm$0.26
6 …… | 14.30$\pm$0.21 | | 1.35$\pm$0.10 | 54.75$\pm$2.19 | | 0.30$\pm$0.02 | 7.88$\pm$0.29
Note. — First column is core number. Column densities are calculated under the
assumption of local thermodynamic equilibrium (LTE), in which 13CO and C18O
have the same excitation temperature as that of optically thick 12CO.
Table 6: Velocity gradients detected in the cores
Core | Molecule | $v_{0}\pm\sigma_{v_{0}}$ | $\mathscr{G}\pm\sigma_{\mathscr{G}}$ | $\mathscr{G}$ at 580 pc | $\Theta_{\mathscr{G}}\pm\sigma_{\Theta_{\mathscr{G}}}$ | $\beta$
---|---|---|---|---|---|---
| | (km s-1) | (m s-1 arcsec-1) | (km s-1 pc-1) | (deg E of N) |
1 …… | 13CO (1$-$0) | 1.01$\pm$0.003 | 04.25$\pm$0.096 | 1.51 | $-$8.3$\pm$0.8 |
| C18O (1$-$0) | 0.91$\pm$0.007 | 03.69$\pm$0.205 | 1.31 | $-$12.1$\pm$2.0 |
| NH3 (1,1) | 1.15$\pm$0.014 | 10.76$\pm$0.339 | 3.83 | $-$37.2$\pm$1.2 | 1.3E-2
| NH3 (2,2) | 1.28$\pm$0.024 | 08.76$\pm$0.927 | 3.12 | $-$50.5$\pm$4.4 |
3 …… | NH3 (1,1) | 1.19$\pm$0.012 | 03.69$\pm$0.337 | 1.31 | 165.0$\pm$4.0 | 2.3E-3
4 …… | NH3 (1,1) | 1.01$\pm$0.011 | 04.32$\pm$0.419 | 1.54 | $-$64.5$\pm$4.0 | 3.6E-3
5 …… | NH3 (1,1) | 1.12$\pm$0.013 | 02.31$\pm$0.497 | 0.82 | $-$123.8$\pm$8.5 | 6.2E-4
6 …… | 13CO (1$-$0) | 1.66$\pm$0.004 | 02.87$\pm$0.188 | 1.02 | 16.5$\pm$2.2 |
| C18O (1$-$0) | 1.54$\pm$0.010 | 04.19$\pm$0.507 | 1.49 | 12.6$\pm$4.5 |
| NH3 (1,1) | 1.81$\pm$0.018 | 09.45$\pm$0.772 | 3.36 | 90.2$\pm$3.9 | 1.5E-2
| NH3 (2,2) | 1.88$\pm$0.048 | 14.92$\pm$2.772 | 5.31 | 45.6$\pm$7.6 |
Note. — Columns are core number, molecules fitted, systemic velocity,
magnitude of the velocity gradient, $\mathscr{G}$ at cloud distance, direction
of increasing velocity (measured east of north), and parameter $\beta$ given
in §4.4. Errors quoted are $1~{}\sigma$ uncertainties.
Table 7: NH3 abundance in the cores
Peak | $N(\rm H_{2})_{C^{18}O}$ | $\chi_{\rm{NH}_{3}}$ | $N(\rm H_{2})_{1.12mm}$ | $\chi^{\prime}_{\rm{NH}_{3}}$ | $\chi~{}{\rm in~{}core}$
---|---|---|---|---|---
| ($10^{22}\rm cm^{-2}$) | ($10^{-8}$) | ($10^{22}\rm cm^{-2}$) | ($10^{-8}$) | ($10^{-8}$)
1 …… | 6.23 | 5.4 | 5.36 | 6.3 | 2.41
2 …… | 4.45 | 4.7 | 5.54 | 3.8 | 2.74
3 …… | 4.46 | 4.1 | 2.85 | 6.4 | 2.46
4 …… | 4.64 | 3.8 | 2.16 | 8.1 | 2.24
5 …… | 3.29 | 4.7 | 0.96 | 15.9 | 1.88
6 …… | 5.52 | 2.9 | 1.25 | 12.9 | 2.04
Note. — The columns show the core number, H2 column density and NH3 abundance
$\chi$ based on C18O observations (column 2 and 3) and 1.1 mm dust flux
(column 4 and 5), and the average NH3 abundance over the cores.
Table 8: Masses of the NH3 cores
Region | R | $n({\rm H_{2}})$ | $M_{\rm Jeans}$ | $M_{\rm Virial}$ | $M_{\rm Molecular}$ | $\alpha$ | $v_{\rm escape}$
---|---|---|---|---|---|---|---
| (arcsec) | ($10^{4}\rm cm^{-3}$) | ($M_{\sun}$) | ($M_{\sun}$) | ($M_{\sun}$) | | (km s-1)
1 …… | 69.0 | 13.8 | 198 | 206 | 208 | 0.99 | 3.03
2 …… | 74.9 | 13.2 | 205 | 121 | 255 | 0.41 | 3.22
3 …… | 69.0 | 8.7 | 259 | 92 | 130 | 0.71 | 2.40
4 …… | 60.1 | 10.4 | 198 | 119 | 104 | 1.15 | 2.30
5 …… | 46.7 | 16.6 | 174 | 85 | 77 | 1.10 | 2.24
6 …… | 49.7 | 11.0 | 228 | 129 | 62 | 2.08 | 1.95
Note. — Columns are core number, size of the cores, H2 densities derived from
C18O, Jeans mass, virial masses from NH3 and gas mass, virial parameter
$\alpha$, and escape velocity.
|
arxiv-papers
| 2011-01-21T05:31:11 |
2024-09-04T02:49:16.559837
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. B. Zhang, J. Yang, Y. Xu, J. D. Pandian, K. M. Menten, C. Henkel",
"submitter": "Shaobo Zhang",
"url": "https://arxiv.org/abs/1101.4072"
}
|
1101.4161
|
# Parameter rigid actions of simply connected nilpotent Lie groups
Hirokazu Maruhashi 111Research Fellow of the Japan Society for the Promotion
of Science
Department of Mathematics, Kyoto University
###### Abstract
We show that for a locally free $C^{\infty}$-action of a connected and simply
connected nilpotent Lie group on a compact manifold, if every real valued
cocycle is cohomologous to a constant cocycle, then the action is parameter
rigid. The converse is true if the action has a dense orbit. Using this, we
construct parameter rigid actions of simply connected nilpotent Lie groups
whose Lie algebras admit rational structures with graduations. This
generalizes the results of dos Santos [8] concerning the Heisenberg groups.
## 1 Introduction
Let $G$ be a connected Lie group with Lie algebra ${\mathfrak{g}}$ and $M$ a
$C^{\infty}$-manifold without boundary. Let $\rho:M\times G\to M$ be a
$C^{\infty}$ right action. We call $\rho$ locally free if every isotropy
subgroup of $\rho$ is discrete in $G$. Assume that $\rho$ is locally free.
Then we have the orbit foliation ${\mathcal{F}}$ of $\rho$ whose tangent
bundle $T{\mathcal{F}}$ is naturally isomorphic to a trivial bundle
$M\times{\mathfrak{g}}$.
The action $\rho$ is parameter rigid if any action $\rho^{\prime}$ of $G$ on
$M$ with the same orbit foliation ${\mathcal{F}}$ is $C^{\infty}$-conjugate to
$\rho$, more precisely, there exist an automorphism $\Phi$ of $G$ and a
$C^{\infty}$-diffeomorphism $F$ of $M$ which preserves each leaf of
${\mathcal{F}}$ and homotopic to identity through $C^{\infty}$-maps preserving
each leaf of ${\mathcal{F}}$ such that
$F(\rho(x,g))=\rho^{\prime}(F(x),\Phi(g))$
for all $x\in M$ and $g\in G$.
Parameter rigidity of actions has been studied by several authors, for
instance, Katok and Spatzier [3], Matsumoto and Mitsumatsu [4], Mieczkowski
[5], dos Santos [8] and Ramírez [7]. Most of known examples of parameter rigid
actions are those of abelian groups and nonabelian actions have not been
considered so much.
Parameter rigidity is closely related to cocycles over actions. Now suppose
$G$ is contractible and $M$ is compact. Let $H$ be a Lie group. A
$C^{\infty}$-map $c:M\times G\to H$ is called a $H$-valued cocycle over $\rho$
if $c$ satisfies
$c(x,gg^{\prime})=c(x,g)c(\rho(x,g),g^{\prime})$
for all $x\in M$ and $g,g^{\prime}\in G$.
A cocycle $c$ is constant if $c(x,g)$ is independent of $x$. A constant
cocycle is just a homomorphism $G\to H$.
$H$-valued cocycles $c,c^{\prime}$ are cohomologous if there exists a
$C^{\infty}$-map $P:M\to H$ such that
$c(x,g)=P(x)^{-1}c^{\prime}(x,g)P(\rho(x,g))$
for all $x\in M$ and $g\in G$.
The action $\rho$ is $H$-valued cocycle rigid if every $H$-valued cocycle over
$\rho$ is cohomologous to a constant cocycle.
###### Proposition 1 ([4]).
If $\rho$ is $G$-valued cocycle rigid, then it is parameter rigid.
###### Remark.
In [4] Matsumoto and Mitsumatsu assume that $\rho$ has at least one trivial
isotropy subgroup, but this assumption is not necessary.
###### Proposition 2 ([4]).
When $G={\mathbb{R}}^{n}$, the following are equivalent:
1. 1.
$\rho$ is ${\mathbb{R}}$-valued cocycle rigid.
2. 2.
$\rho$ is ${\mathbb{R}}^{n}$-valued cocycle rigid.
3. 3.
$\rho$ is parameter rigid.
###### Remark.
The equivalence of the first two conditions is obvious.
In this paper we consider actions of simply connected nilpotent Lie groups. In
[8], dos Santos proved that for actions of a Heisenberg group $H_{n}$,
${\mathbb{R}}$-valued cocycle rigidity implies $H_{n}$-valued cocycle rigidity
and using this, he constructed parameter rigid actions of Heisenberg groups.
To the best of my knowledge these are the only known nontrivial parameter
rigid actions of nonabelian nilpotent Lie groups. We prove the following.
###### Theorem 1.
Let $N$ be a connected and simply connected nilpotent Lie group, $M$ a compact
manifold and $\rho$ a locally free $C^{\infty}$-action of $N$ on $M$. Then,
the following are equivalent:
1. 1.
$\rho$ is ${\mathbb{R}}$-valued cocycle rigid.
2. 2.
$\rho$ is $N$-valued cocycle rigid.
3. 3.
$\rho$ is parameter rigid and every orbitwise constant real valued
$C^{\infty}$-function of $\rho$ on $M$ is constant on $M$.
This theorem enables us to construct parameter rigid actions of nilpotent Lie
groups. The most interesting one is the following.
###### Theorem 2 ([7]).
Let $N$ denote the group of all upper triangular real matrices with $1$ on the
diagonal, $\Gamma$ a cocompact lattice of ${\rm SL}(n,{\mathbb{R}})$ and
$\rho$ the action of $N$ on $\Gamma\backslash{\rm SL}(n,{\mathbb{R}})$ by
right multiplication. If $n\geq 4$, $\rho$ is ${\mathbb{R}}$-valued cocycle
rigid.
###### Remark.
In [7], Ramírez proved more general theorems.
###### Corollary.
The above action $\rho$ is parameter rigid.
In Section 4 we construct parameter rigid actions of nilpotent groups using
Theorem 1. It is a generalization of dos Santos’ example. Let $N$ be a
connected and simply connected nilpotent Lie group and $\Gamma$, $\Lambda$ be
lattices in $N$. Consider the action of $\Lambda$ on ${\Gamma\backslash N}$ by
right multiplication and let $\tilde{\rho}$ be its suspended action of $N$.
###### Theorem 3.
If $\Lambda$ is Diophantine with respect to $\Gamma$, then the action
$\tilde{\rho}$ of $N$ is parameter rigid.
For the definition of Diophantine lattices, see Section 4.
## 2 Preliminaries
Let $G$ be a contractible Lie group with Lie algebra ${\mathfrak{g}}$, $M$ a
compact manifold and $\rho$ a locally free action of $G$ on $M$ with orbit
foliation ${\mathcal{F}}$. Let $H$ be a Lie group with Lie algebra
${\mathfrak{h}}$. We denote by $\Omega^{p}({\mathcal{F}},{\mathfrak{h}})$ the
set of all $C^{\infty}$-sections of
$\mathop{\mathrm{Hom}}\nolimits(\bigwedge^{p}T{\mathcal{F}},{\mathfrak{h}})$.
The exterior derivative
${d_{\mathcal{F}}}:\Omega^{p}({\mathcal{F}},{\mathfrak{h}})\to\Omega^{p+1}({\mathcal{F}},{\mathfrak{h}})$
is defined since $T{\mathcal{F}}$ is integrable.
By differentiating, $H$-valued cocycles over $\rho$ are in one-to-one
correspondence with ${\mathfrak{h}}$-valued leafwise one forms
$\omega\in\Omega^{1}({\mathcal{F}},{\mathfrak{h}})$ such that
${d_{\mathcal{F}}}\omega+[\omega,\omega]=0.$
###### Proposition 3.
Let $c_{1},c_{2}$ be $H$-valued cocycles over $\rho$ and let
$\omega_{1},\omega_{2}$ be corresponding differential forms. For a
$C^{\infty}$-map $P:M\to H$, the following are equivalent:
1. 1.
$c_{1}(x,g)=P(x)^{-1}c_{2}(x,g)P(\rho(x,g))$ for all $x\in M$ and $g\in G$.
2. 2.
$\omega_{1}=\mathop{\mathrm{Ad}}\nolimits(P^{-1})\omega_{2}+P^{*}\theta$ where
$\theta\in\Omega^{1}(H,{\mathfrak{h}})$ is the left Maurer-Cartan form on $H$.
###### Corollary ([4]).
The following are equivalent:
1. 1.
$\rho$ is $G$-valued cocycle rigid.
2. 2.
For each $\omega\in\Omega^{1}({\mathcal{F}},{\mathfrak{g}})$ such that
${d_{\mathcal{F}}}\omega+[\omega,\omega]=0$, there exist a endomorphism
$\Phi:{\mathfrak{g}}\to{\mathfrak{g}}$ of Lie algebra and a $C^{\infty}$-map
$P:M\to G$ such that
$\omega=\mathop{\mathrm{Ad}}\nolimits(P^{-1})\Phi+P^{*}\theta.$
Proposition 3 is obtained by examining the proof of Corollary Corollary in
[4]. In this paper, we will identify a cocycle with its corresponding
differential form.
Let us consider real valued cocycles. A real valued cocycle over $\rho$ is
given by $\omega\in\Omega^{1}({\mathcal{F}},{\mathbb{R}})$ satisfying
${d_{\mathcal{F}}}\omega=0$. Two real valued cocycles $\omega_{1},\omega_{2}$
are cohomologous if and only if $\omega_{1}=\omega_{2}+{d_{\mathcal{F}}}P$ for
some $C^{\infty}$-function $P:M\to{\mathbb{R}}$. Leafwise cohomology
$H^{*}({\mathcal{F}})$ of ${\mathcal{F}}$ is the cohomology of the cochain
complex $(\Omega^{*}({\mathcal{F}},{\mathbb{R}}),{d_{\mathcal{F}}})$. Thus
$H^{1}({\mathcal{F}})$ is the set of all equivalence classes of real valued
cocycles.
The identification $T{\mathcal{F}}\simeq M\times{\mathfrak{g}}$ induces a map
$H^{*}({\mathfrak{g}})\to H^{*}({\mathcal{F}})$ where $H^{*}({\mathfrak{g}})$
is the cohomology of the Lie algebra ${\mathfrak{g}}$. By the compactness of
$M$, this map is injective on $H^{1}({\mathfrak{g}})$. Hence we identify
$H^{1}({\mathfrak{g}})$ with its image. Note that $H^{1}({\mathfrak{g}})$ is
the set of all equivalence classes of constant real valued cocycles. Thus real
valued cocycle rigidity is equivalent to
$H^{1}({\mathcal{F}})=H^{1}({\mathfrak{g}})$.
Notice that $H^{0}({\mathcal{F}})$ is the set of leafwise constant real valued
$C^{\infty}$-functions of ${\mathcal{F}}$ on $M$ and $H^{0}({\mathfrak{g}})$
consists of constant functions on $M$. Therefore the equivalence of 1 and 3 in
Theorem 1 can be stated as follows:
$H^{1}({\mathcal{F}})=H^{1}({\mathfrak{n}})$ if and only if $\rho$ is
parameter rigid and $H^{0}({\mathcal{F}})=H^{0}({\mathfrak{n}})$.
## 3 Proof of Theorem 1
Let $N$ be a simply connected nilpotent Lie group with Lie algebra
${\mathfrak{n}}$, $M$ a compact manifold and $\rho$ a locally free action of
$N$ on $M$ with orbit foliation ${\mathcal{F}}$.
We first prove that $N$-valued cocycle rigidity implies real valued cocycle
rigidity. There exist closed subgroups $N^{\prime}$ and $A$ of $N$ such that
$N^{\prime}\triangleleft N$, $N=N^{\prime}\rtimes A$ and
$A\simeq{\mathbb{R}}$. Let $c$ be any real valued cocycle over $\rho$. We
regard $c$ as a $N$-valued cocycle over $\rho$ via the inclusion
${\mathbb{R}}\simeq A\hookrightarrow N$. By $N$-valued cocycle rigidity, there
exist an endomorphism $\Phi$ of $N$ and a $C^{\infty}$-map $P:M\to N$ such
that $c(x,g)=P(x)^{-1}\Phi(g)P(\rho(x,g))$ for all $x\in M$ and $g\in N$.
Applying the natural projection $\pi:N\to A\simeq{\mathbb{R}}$, we obtain
$c(x,g)=(\pi\circ P)(x)^{-1}(\pi\circ\Phi)(g)(\pi\circ P)(\rho(x,g))$. Thus
$c$ is cohomologous to a constant cocycle $\pi\circ\Phi$.
Next we assume $H^{1}({\mathcal{F}})=H^{1}({\mathfrak{n}})$ and prove
$N$-valued cocycle rigidity. We need the following two lemmata.
###### Lemma 1.
Let $V$ be a finite dimensional real vector space. Assume that
$\omega\in\Omega^{1}({\mathcal{F}},V)$ satisfies the equation
${d_{\mathcal{F}}}\omega=\varphi$, where
$\varphi\in\mathop{\mathrm{Hom}}\nolimits(\bigwedge^{2}{\mathfrak{n}},V)$ is a
constant leafwise two form. Then there exists a constant leafwise one form
$\psi\in\mathop{\mathrm{Hom}}\nolimits({\mathfrak{n}},V)$ with
$\varphi={d_{\mathcal{F}}}\psi$.
###### Proof.
Since $N$ is amenable, there exists a $N$-invariant Borel probability measure
$\mu$ on $M$. Define $\psi\in\mathop{\mathrm{Hom}}\nolimits({\mathfrak{n}},V)$
by
$\psi(X)=\int_{M}\omega(X)d\mu$
where $X\in{\mathfrak{n}}$. Since
$\varphi(X,Y)=X\omega(Y)-Y\omega(X)-\omega([X,Y])$ for all
$X,Y\in{\mathfrak{n}}$, we obtain
$\varphi(X,Y)=-\int_{M}\omega([X,Y])d\mu.$
Thus
${d_{\mathcal{F}}}\psi(X,Y)=-\psi([X,Y])=-\int_{M}\omega([X,Y])d\mu=\varphi(X,Y),$
hence ${d_{\mathcal{F}}}\psi=\varphi$. ∎
Set ${\mathfrak{n}}^{1}={\mathfrak{n}}$,
${\mathfrak{n}}^{i}=[{\mathfrak{n}},{\mathfrak{n}}^{i-1}]$. Then
${\mathfrak{n}}^{s}\neq 0$, ${\mathfrak{n}}^{s+1}=0$ for some $s$. For each
$1\leq i\leq s$, choose a subspace $V_{i}$ with
${\mathfrak{n}}^{i}=V_{i}\oplus{\mathfrak{n}}^{i+1}$, so that
${\mathfrak{n}}=\bigoplus_{i=1}^{s}V_{i}$.
###### Lemma 2.
Let $\omega\in\Omega^{1}({\mathcal{F}},{\mathfrak{n}})$ be such that
${d_{\mathcal{F}}}\omega+[\omega,\omega]=0$. Decompose $\omega$ as
$\omega=\xi+\omega_{k}+\omega_{k+1}$
where $\xi\in\Omega^{1}({\mathcal{F}},\bigoplus_{i=1}^{k-1}V_{i})$,
$\omega_{k}\in\Omega^{1}({\mathcal{F}},V_{k})$ and
$\omega_{k+1}\in\Omega^{1}({\mathcal{F}},{\mathfrak{n}}^{k+1})$. If $\xi$ is
constant, then there exists
$\omega^{\prime}\in\Omega^{1}({\mathcal{F}},{\mathfrak{n}})$ with
${d_{\mathcal{F}}}\omega^{\prime}+[\omega^{\prime},\omega^{\prime}]=0$ which
is cohomologous to $\omega$ and such that
$\omega^{\prime}=\xi^{\prime}+\omega_{k+1}^{\prime}$
where $\xi^{\prime}\in\Omega^{1}({\mathcal{F}},\bigoplus_{i=1}^{k}V_{i})$ is
constant and
$\omega_{k+1}^{\prime}\in\Omega({\mathcal{F}},{\mathfrak{n}}^{k+1})$.
###### Proof.
By cocycle equation,
$0={d_{\mathcal{F}}}\xi+{d_{\mathcal{F}}}\omega_{k}+{d_{\mathcal{F}}}\omega_{k+1}+[\xi,\xi]+\mbox{an
element of}\ \Omega^{2}({\mathcal{F}},{\mathfrak{n}}^{k+1}).$
Comparing $V_{k}$ component, we see that ${d_{\mathcal{F}}}\omega_{k}$ is
constant. Hence by Lemma 1,
${d_{\mathcal{F}}}\omega_{k}={d_{\mathcal{F}}}\psi$ for some
$\psi\in\mathop{\mathrm{Hom}}\nolimits({\mathfrak{n}},V_{k})$. Since we are
assuming that $H^{1}({\mathcal{F}})=H^{1}({\mathfrak{n}})$, there exists
$\psi^{\prime}\in\mathop{\mathrm{Hom}}\nolimits({\mathfrak{n}},V_{k})$ and
$C^{\infty}$-map $h:M\to V_{k}$ such that
$\omega_{k}=\psi+\psi^{\prime}+{d_{\mathcal{F}}}h.$
Put $P=e^{h}:M\to N$. Let $x\in M$ and $X\in T_{x}{\mathcal{F}}$. Choose a
path $x(t)$ such that $X=\frac{d}{dt}x(t)\big{|}_{t=0}$. Let
$\theta\in\Omega^{1}(N,{\mathfrak{n}})$ be the left Maurer-Cartan form on $N$.
Then
$\displaystyle P^{*}\theta(X)$
$\displaystyle=\frac{d}{dt}P(x)^{-1}P(x(t))\Big{|}_{t=0}=\frac{d}{dt}e^{-h(x)}e^{h(x(t))}\Big{|}_{t=0}$
$\displaystyle=\frac{d}{dt}\exp(-h(x)+h(x(t))+\mbox{an element of
}{\mathfrak{n}}^{k+1})\Big{|}_{t=0}$
$\displaystyle={d_{\mathcal{F}}}h(X)+\mbox{an element of
}{\mathfrak{n}}^{k+1}.$
Thus $P^{*}\theta={d_{\mathcal{F}}}h+\mbox{an element of
}\Omega^{1}({\mathcal{F}},{\mathfrak{n}}^{k+1})$. Note that
$\mathop{\mathrm{Ad}}\nolimits(P^{-1})=\exp\mathop{\mathrm{ad}}\nolimits(-h)$
is identity on $\bigoplus_{i=1}^{k}V_{i}$ and preserves
${\mathfrak{n}}^{k+1}$. Hence
$\displaystyle\omega-P^{*}\theta$
$\displaystyle=\xi+\psi+\psi^{\prime}+\mbox{an element of
}\Omega^{1}({\mathcal{F}},{\mathfrak{n}}^{k+1})$
$\displaystyle=\mathop{\mathrm{Ad}}\nolimits(P^{-1})(\xi+\psi+\psi^{\prime}+\mbox{an
element of }\Omega^{1}({\mathcal{F}},{\mathfrak{n}}^{k+1})).$
∎
Let $\omega$ be any $N$-valued cocycle. Using Lemma 2, we can exchange
$\omega$ for a cohomologous cocycle whose $V_{1}$-component is constant.
Applying Lemma 2 repeatedly, we eventually get a constant cocycle cohomologous
to $\omega$. This proves $N$-valued cocycle rigidity.
Next we assume that $\rho$ is parameter rigid and
$H^{0}({\mathcal{F}})=H^{0}({\mathfrak{n}})$. Let ${\mathfrak{n}}^{i}$ and
$V_{i}$ be as above. Note that ${\mathfrak{n}}^{s}$ is central in
${\mathfrak{n}}$. Fix a nonzero $Z\in{\mathfrak{n}}^{s}$.
Let $[\omega]\in H^{1}({\mathcal{F}})$. Let $\omega_{0}$ be the $N$-valued
cocycle over $\rho$ corresponding to the constant cocycle $\mathrm{id}:N\to
N$. We call $\omega_{0}$ the canonical $1$-form of $\rho$. Fix a $\epsilon>0$
and put $\eta:=\omega_{0}+\epsilon\omega Z$. $\eta$ is an $N$-valued cocycle
over $\rho$ since
${d_{\mathcal{F}}}\eta+[\eta,\eta]={d_{\mathcal{F}}}\omega_{0}+\epsilon({d_{\mathcal{F}}}\omega)Z+[\omega_{0},\omega_{0}]=0.$
Since $M$ is compact, we can assume
$\eta_{x}:T_{x}{\mathcal{F}}\to{\mathfrak{n}}$ is bijective for all $x\in M$
by choosing $\epsilon>0$ small. There exists a unique action $\rho^{\prime}$
of $N$ on $M$ whose orbit foliation is ${\mathcal{F}}$ and canonical $1$-form
is $\eta$. See [1]. By parameter rigidity $\rho^{\prime}$ is conjugate to
$\rho$. Thus there exist a $C^{\infty}$-map $P:M\to N$ and an automorphism
$\Phi$ of $N$ satisfying
$\omega_{0}+\epsilon\omega
Z=\mathop{\mathrm{Ad}}\nolimits(P^{-1})\Phi_{*}\omega_{0}+P^{*}\theta.$ (1)
Note that $\mathop{\mathrm{log}}\nolimits:N\to{\mathfrak{n}}$ is defined since
$N$ is simply connected and nilpotent. Let us decompose
$\omega_{0}=\sum_{i=1}^{s}\omega_{0i}$,
$\Phi_{*}\omega_{0}=\sum_{i=1}^{s}\omega_{0i}^{\prime}$ and
$\mathop{\mathrm{log}}\nolimits P=\sum_{i=1}^{s}P_{i}$ according to the
decomposition ${\mathfrak{n}}=\bigoplus_{i=1}^{s}V_{i}$.
###### Lemma 3.
Assume that $P_{1}=\dotsm=P_{k-1}=0$ i.e. $\mathop{\mathrm{log}}\nolimits
P\in{\mathfrak{n}}^{k}$.
1. 1.
If $k<s$, then there exist a $C^{\infty}$-map $Q:M\to N$ and an automorphism
$\Psi$ of $N$ such that
$\omega_{0}+\epsilon\omega
Z=\mathop{\mathrm{Ad}}\nolimits(Q^{-1})\Psi_{*}\omega_{0}+Q^{*}\theta$
and $Q_{1}=\dotsm=Q_{k}=0$ where $\mathop{\mathrm{log}}\nolimits
Q=\sum_{i=1}^{s}Q_{i}$.
2. 2.
If $k=s$, then $\omega$ is cohomologous to a constant cocycle.
###### Proof.
For all $X=\frac{d}{dt}x(t)\big{|}_{t=0}\in T_{x}{\mathcal{F}}$,
$\displaystyle P^{*}\theta(X)$
$\displaystyle=\frac{d}{dt}P(x)^{-1}P(x(t))\Big{|}_{t=0}=\frac{d}{dt}\exp\left(-\sum_{i=k}^{s}P_{i}(x)\right)\exp\left(\sum_{i=k}^{s}P_{i}(x(t))\right)\Big{|}_{t=0}$
$\displaystyle=\frac{d}{dt}\exp\left\\{\sum_{i=k}^{s}\left(P_{i}(x(t))-P_{i}(x)\right)+\mbox{an
element of }{\mathfrak{n}}^{k+1}\right\\}\Big{|}_{t=0}$
$\displaystyle=\frac{d}{dt}\exp\left(P_{k}(x(t))-P_{k}(x)+\mbox{an element of
}{\mathfrak{n}}^{k+1}\right)\Big{|}_{t=0}$
$\displaystyle={d_{\mathcal{F}}}P_{k}(X)+\mbox{an element of
}{\mathfrak{n}}^{k+1}.$
We have
$\displaystyle\mathop{\mathrm{Ad}}\nolimits(P^{-1})\Phi_{*}\omega_{0}$
$\displaystyle=\exp\left(\mathop{\mathrm{ad}}\nolimits\left(-\sum_{i=k}^{s}P_{i}\right)\right)\sum_{i=1}^{s}\omega_{0i}^{\prime}$
$\displaystyle=\sum_{i=1}^{s}\omega_{0i}^{\prime}+\mbox{an element of
}{\mathfrak{n}}^{k+1}.$
Comparing the $V_{k}$-component of (1) we get
$\omega_{0k}+\delta_{ks}\epsilon\omega
Z=\omega_{0k}^{\prime}+{d_{\mathcal{F}}}P_{k}.$
When $k=s$ the equation
$\omega
Z=\epsilon^{-1}(\omega_{0s}^{\prime}-\omega_{0s})+{d_{\mathcal{F}}}(\epsilon^{-1}P_{s})$
shows that $\omega$ is cohomologous to a constant cocycle.
If $k<s$, then ${d_{\mathcal{F}}}P_{k}=\phi\circ\omega_{0}$ for some linear
map $\phi:{\mathfrak{n}}\to V_{k}$. For any $X\in{\mathfrak{n}}$, let
$\tilde{X}$ denote the vector field on $M$ determined by $X$ via $\rho$. We
have $\tilde{X}P_{k}=\phi(X)$ and by integrating over an integral curve
$\gamma$ of $\tilde{X}$ we get $P_{k}(\gamma(T))-P_{k}(\gamma(0))=\phi(X)T$
for all $T>0$. Since $M$ is compact, $\phi(X)=0$. Therefore
${d_{\mathcal{F}}}P_{k}=0$, so that $P_{k}$ is constant on each leaf of
${\mathcal{F}}$. Thus $P_{k}$ is constant on $M$ by our assumption. Put
$g:=\exp(-P_{k})$ and $Q:=gP=\exp\left(\sum_{i=k+1}^{s}P_{i}+\mbox{an element
of }{\mathfrak{n}}^{k+1}\right)$. Then
$\displaystyle\omega_{0}+\epsilon\omega Z$
$\displaystyle=\mathop{\mathrm{Ad}}\nolimits(Q^{-1}g)\Phi_{*}\omega_{0}+(L_{g^{-1}}\circ
Q)^{*}\theta$
$\displaystyle=\mathop{\mathrm{Ad}}\nolimits(Q^{-1})\Psi_{*}\omega_{0}+Q^{*}\theta$
where $\Psi_{*}:=\mathop{\mathrm{Ad}}\nolimits(g)\Phi_{*}$. ∎
Applying Lemma 3 repeatedly, we see that $\omega$ is cohomologous to a
constant cocycle.
Finally we assume $H^{1}({\mathcal{F}})=H^{1}({\mathfrak{n}})$ and prove that
$\rho$ is parameter rigid and $H^{0}({\mathcal{F}})=H^{0}({\mathfrak{n}})$.
Parameter rigidity of $\rho$ follows from Proposition 1. Let $f\in
H^{0}({\mathcal{F}})$. Fix a nonzero $\phi\in H^{1}({\mathfrak{n}})$ and
denote the corresponding leafwise $1$-form on $M$ by $\tilde{\phi}$. Then
$f\tilde{\phi}\in H^{1}({\mathcal{F}})=H^{1}({\mathfrak{n}})$. Thus there
exist $\psi\in H^{1}({\mathfrak{n}})$ and a $C^{\infty}$-function
$g:M\to{\mathbb{R}}$ such that $f\tilde{\phi}-\tilde{\psi}={d_{\mathcal{F}}}g$
where $\tilde{\psi}$ is the leafwise $1$-form corresponding to $\psi$. Choose
$X\in{\mathfrak{n}}$ satisfying $\phi(X)\neq 0$. Let $x(t)$ be an integral
curve of $\tilde{X}$ where $\tilde{X}$ is the vector field corresponding to
$X$. We have $f(x(t))\phi(X)-\psi(X)=\tilde{X}_{x(t)}g=\frac{d}{dt}g(x(t))$.
By integrating over $[0,T]$, we get
$T(f(x(0))\phi(X)-\psi(X))=g(x(T))-g(x(0))$. Since $g$ is bounded,
$f(x(0))\phi(X)-\psi(X)$ must be zero. Then $f(x(0))=\frac{\psi(X)}{\phi(X)}$
and $f$ is constant on $M$.
This completes the proof of Theorem 1.
## 4 A construction of parameter rigid actions
Let us now construct real valued cocycle rigid actions of nilpotent groups.
For the structure theory of nilpotent Lie groups, see [2]. A basis
$X_{1},\dots,X_{n}$ of ${\mathfrak{n}}$ is called a strong Malcev basis if
$\mathop{\mathrm{span}}\nolimits_{{\mathbb{R}}}\\{X_{1},\dots,X_{i}\\}$ is an
ideal of ${\mathfrak{n}}$ for each $i$. If $\Gamma$ is a lattice in $N$, there
exists a strong Malcev basis $X_{1},\dots,X_{n}$ of ${\mathfrak{n}}$ such that
$\Gamma=e^{{\mathbb{Z}}X_{1}}\cdots e^{{\mathbb{Z}}X_{n}}$. Such a basis is
called a strong Malcev basis strongly based on $\Gamma$. Let $\Gamma$ and
$\Lambda$ be lattices in $N$.
###### Definition 1.
$\Lambda$ is Diophantine with respect to $\Gamma$ if there exists a strong
Malcev basis $X_{1},\dots,X_{n}$ of ${\mathfrak{n}}$ strongly based on
$\Gamma$ and a strong Malcev basis $Y_{1},\dots,Y_{n}$ of ${\mathfrak{n}}$
strongly based on $\Lambda$ such that $Y_{i}=\sum_{j=1}^{i}a_{ij}X_{j}$ for
every $1\leq i\leq n$, where $a_{ii}$ is Diophantine.
Let $\rho$ be the action of $\Lambda$ on ${\Gamma\backslash N}$ by right
multiplication. First we will prove the following.
###### Theorem 4.
If $\Lambda$ is Diophantine with respect to $\Gamma$, then every real valued
$C^{\infty}$ cocycle $c:{\Gamma\backslash N}\times\Lambda\to{\mathbb{R}}$ over
$\rho$ is cohomologous to a constant cocycle.
Note that $X_{1}$ is in the center of ${\mathfrak{n}}$. Let
$\pi:N\to\bar{N}:={e^{{\mathbb{R}}X_{1}}}\backslash N$ be the projection.
Since $\Gamma\cap{e^{{\mathbb{R}}X_{1}}}=e^{{\mathbb{Z}}X_{1}}$ is a cocompact
lattice in ${e^{{\mathbb{R}}X_{1}}}$,
$\bar{\Gamma}:=\pi(\Gamma)={e^{{\mathbb{R}}X_{1}}}\backslash\Gamma{e^{{\mathbb{R}}X_{1}}}$
is a cocompact lattice in $\bar{N}$. Let
$\bar{{\mathfrak{n}}}={\mathbb{R}}X_{1}\backslash{\mathfrak{n}}$, then
$\bar{X_{2}},\dots,\bar{X_{n}}$ is a strong Malcev basis of
$\bar{{\mathfrak{n}}}$ strongly based on $\bar{\Gamma}$.
We will see that the naturally induced map $\bar{\pi}:{\Gamma\backslash
N}\to{\bar{\Gamma}\backslash\bar{N}}$ is a principal $S^{1}$-bundle. Indeed,
$\Gamma\backslash\Gamma{e^{{\mathbb{R}}X_{1}}}\hookrightarrow{\Gamma\backslash
N}\twoheadrightarrow\Gamma{e^{{\mathbb{R}}X_{1}}}\backslash N$
is a principal $\Gamma\backslash\Gamma{e^{{\mathbb{R}}X_{1}}}$-bundle and we
have
$\Gamma\backslash\Gamma{e^{{\mathbb{R}}X_{1}}}\simeq\Gamma\cap{e^{{\mathbb{R}}X_{1}}}\backslash{e^{{\mathbb{R}}X_{1}}}=e^{{\mathbb{Z}}X_{1}}\backslash{e^{{\mathbb{R}}X_{1}}}\simeq{\mathbb{Z}}\backslash{\mathbb{R}}$
and
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Since $\Lambda\cap{e^{{\mathbb{R}}X_{1}}}=\Lambda\cap
e^{{\mathbb{R}}Y_{1}}=e^{{\mathbb{Z}}Y_{1}}$ is a cocompact lattice in
$e^{{\mathbb{R}}X_{1}}$, $\bar{\Lambda}:=\pi(\Lambda)$ is a cocompact lattice
in $\bar{N}$. $\bar{Y_{2}},\dots,\bar{Y_{n}}$ is a strong Malcev basis of
$\bar{{\mathfrak{n}}}$ strongly based on $\bar{\Lambda}$ and
$\bar{Y_{i}}=\sum_{j=2}^{i}a_{ij}\bar{X_{j}}$ where $a_{ii}$ is Diophantine.
Therefore $\bar{\Lambda}$ is Diophantine with respect to $\bar{\Gamma}$.
Since $\bar{\pi}$ is $\Lambda$-equivariant, the action $\rho$ of $\Lambda$
when restricted to $e^{{\mathbb{Z}}Y_{1}}$, preserves fibers of $\bar{\pi}$.
Let $z\in{\bar{\Gamma}\backslash\bar{N}}$. Choose a point $\Gamma x$ in
${\bar{\pi}^{-1}(z)}$. Then we have a trivialization
$\iota_{\Gamma x}:{\mathbb{Z}}\backslash{\mathbb{R}}\simeq{\bar{\pi}^{-1}(z)}$
of ${\bar{\pi}^{-1}(z)}$ given by $\iota_{\Gamma x}(s)=\Gamma e^{sX_{1}}x$.
Note that if we take another point $\Gamma y\in{\bar{\pi}^{-1}(z)}$,
$\iota_{\Gamma y}^{-1}\circ\iota_{\Gamma
x}:{\mathbb{Z}}\backslash{\mathbb{R}}\to{\mathbb{Z}}\backslash{\mathbb{R}}$ is
a rotation.
Let $Y_{1}=aX_{1}$ where $a$ is Diophantine. If we identify
${\bar{\pi}^{-1}(z)}$ with ${\mathbb{Z}}\backslash{\mathbb{R}}$ by
$\iota_{\Gamma x}$, then the action of $e^{Y_{1}}$ on
${\mathbb{Z}}\backslash{\mathbb{R}}$ is $s\mapsto s+a$.
Let $\mu_{z}$ be the normalized Haar measure naturally defined on
$\bar{\pi}^{-1}(z)$, $\mu$ the $N$-invariant probability measure on
${\Gamma\backslash N}$ and $\nu$ the $\bar{N}$-invariant probability measure
on ${\bar{\Gamma}\backslash\bar{N}}$. For any $f\in C({\Gamma\backslash N})$,
$\int_{{\Gamma\backslash
N}}fd\mu=\int_{{\bar{\Gamma}\backslash\bar{N}}}\int_{\bar{\pi}^{-1}(z)}fd\mu_{z}d\nu.$
(2)
###### Lemma 4.
$\rho$ is ergodic with respect to $\mu$.
###### Proof.
We use induction on $n$. If $n=1$, $\rho$ is an irrational rotation on
${\mathbb{Z}}\backslash{\mathbb{R}}$, hence the result is well known. In
general, Let $f:{\Gamma\backslash N}\to{\mathbb{C}}$ be a $\Lambda$-invariant
$L^{2}$-function with $\int_{{\Gamma\backslash N}}fd\mu=0$. Since the action
of $e^{{\mathbb{Z}}Y_{1}}$ on ${\bar{\pi}^{-1}(z)}$ is ergodic,
$f|_{{\bar{\pi}^{-1}(z)}}$ is constant $\mu_{z}$-almost everywhere. We denote
this constant by $g(z)$. Then
$g:{\bar{\Gamma}\backslash\bar{N}}\to{\mathbb{C}}$ is
$\bar{\Lambda}$-invariant measurable function. By induction, $g$ is constant
$\nu$-almost everywhere. By (2), this constant must be zero. Therefore $f$ is
zero $\mu$-almost everywhere. ∎
Let $c:{\Gamma\backslash N}\times\Lambda\to{\mathbb{R}}$ be a
$C^{\infty}$-cocycle over $\rho$. We must show that $c$ is cohomologous to a
constant cocycle $c_{0}:\Lambda\to{\mathbb{R}}$ where
$c_{0}(\lambda):=\int_{{\Gamma\backslash N}}c(x,\lambda)d\mu(x)$. Therefore we
may assume that $\int_{{\Gamma\backslash N}}c(x,\lambda)d\mu(x)=0$ for all
$\lambda\in\Lambda$, and we will show that $c$ is a coboundary. We prove this
by induction on $n$. When $n=1$, $\rho$ is a Diophantine rotation on
${\mathbb{Z}}\backslash{\mathbb{R}}$, hence the result is well known.
###### Lemma 5.
For all $m\in{\mathbb{Z}}$,
$\int_{{\bar{\pi}^{-1}(z)}}c(s,e^{mY_{1}})d\mu_{z}(s)=0.$
###### Proof.
Fix $m$ and put $g(z)=\int_{{\bar{\pi}^{-1}(z)}}c(s,e^{mY_{1}})d\mu_{z}(s)$.
For any $\lambda\in\Lambda$, cocycle equation gives
$c(x,\lambda)+c(x\lambda,e^{mY_{1}})=c(x,e^{mY_{1}})+c(xe^{mY_{1}},\lambda)$.
By integrating this equation on ${\bar{\pi}^{-1}(z)}$, we get
$g(z\pi(\lambda))=g(z)$. Since the action of $\bar{\Lambda}$ on
${\bar{\Gamma}\backslash\bar{N}}$ is ergodic, $g$ is constant. By (2), $g$
must be zero. ∎
Let $f:{\mathbb{Z}}\backslash{\mathbb{R}}\xrightarrow{\iota_{\Gamma
x}}{\bar{\pi}^{-1}(z)}\xrightarrow{c(\cdot,e^{Y_{1}})}{\mathbb{R}}$. We define
$h_{z}(\iota_{\Gamma
x}(s))=\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}\frac{\hat{f}(k)}{-1+e^{2\pi
ika}}e^{2\pi iks}.$
Then $h_{z}:{\bar{\pi}^{-1}(z)}\to{\mathbb{R}}$ is $C^{\infty}$, since $f$ is
$C^{\infty}$ and $a$ is Diophantine. By Lemma 5, we have
$c(\iota_{\Gamma x}(s),e^{Y_{1}})=-h_{z}(\iota_{\Gamma
x}(s))+h_{z}(\iota_{\Gamma x}e^{Y_{1}}).$
If we choose another point $\Gamma e^{s_{0}X_{1}}x\in{\bar{\pi}^{-1}(z)}$ to
define $h_{z}$,
$\displaystyle h_{z}(\iota_{\Gamma x}(s))$ $\displaystyle=h_{z}(\Gamma
e^{sX_{1}}x)=h_{z}(\iota_{\Gamma e^{s_{0}X_{1}}x}(s-s_{0}))$
$\displaystyle=\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}\frac{1}{-1+e^{2\pi
ika}}\int_{0}^{1}c(\Gamma e^{(u+s_{0})X_{1}}x,e^{Y_{1}})e^{-2\pi iku}du\
e^{2\pi ik(s-s_{0})}$
$\displaystyle=\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}\frac{1}{-1+e^{2\pi
ika}}\int_{0}^{1}f(u+s_{0})e^{-2\pi iku}du\ e^{2\pi ik(s-s_{0})}$
$\displaystyle=\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}\frac{\hat{f}(k)}{-1+e^{2\pi
ika}}e^{2\pi iks},$
so that $h_{z}$ is determined only by $z$. Define $h:{\Gamma\backslash
N}\to{\mathbb{R}}$ by $h|_{{\bar{\pi}^{-1}(z)}}=h_{z}$. Then for all
$x\in{\Gamma\backslash N}$ and $m\in{\mathbb{Z}}$,
$c(x,e^{mY_{1}})=-h(x)+h(xe^{mY_{1}})$.
Let $U\subset{\bar{\Gamma}\backslash\bar{N}}$ be open and
$\sigma:U\to\bar{\pi}^{-1}(U)$ a section of $\bar{\pi}$. Then we have a
trivialization ${\mathbb{Z}}\backslash{\mathbb{R}}\times
U\simeq\bar{\pi}^{-1}(U)$ which sends $(s,z)$ to $\iota_{\sigma(z)}(s)=\Gamma
e^{sX_{1}}\sigma(z)$. Hence
$h(\iota_{\sigma(z)}(s))=\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}\frac{1}{-1+e^{2\pi
ika}}\int_{0}^{1}c(\iota_{\sigma(z)}(u),e^{Y_{1}})e^{-2\pi iku}du\ e^{2\pi
iks}$
on $\bar{\pi}^{-1}(U)$. The following lemma shows $h$ is $C^{\infty}$ on
${\Gamma\backslash N}$.
###### Lemma 6.
Let $U\subset{\mathbb{R}}^{n}$ be open and let
$f:{\mathbb{Z}}\backslash{\mathbb{R}}\times U\to{\mathbb{R}}$ be a
$C^{\infty}$-function. Define
$h(s,z)=\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}\frac{1}{-1+e^{2\pi
ika}}\widehat{f_{z}}(k)e^{2\pi iks}$
where $f_{z}(u)=f(u,z)$. Then $h:{\mathbb{Z}}\backslash{\mathbb{R}}\times
U\to{\mathbb{R}}$ is $C^{\infty}$.
###### Proof.
Let $V$ be open such that $\bar{V}\subset U$ and $\bar{V}$ is compact. We will
show that $h$ is $C^{\infty}$ on ${\mathbb{Z}}\backslash{\mathbb{R}}\times V$.
Choose constants $C,\alpha>0$ such that $\lvert-1+e^{2\pi ika}\rvert\geq
C\lvert k\rvert^{-\alpha}$ for all $k\in{\mathbb{Z}}\setminus\\{0\\}$.
We will first prove that $h$ is continuous. Since for any
$m\in{\mathbb{Z}}_{>0}$,
$\frac{\partial^{m}f_{z}}{\partial s^{m}}(s)=\sum_{k\in{\mathbb{Z}}}(2\pi
ik)^{m}\widehat{f_{z}}(k)e^{2\pi iks}$
in $L^{2}({\mathbb{Z}}\backslash{\mathbb{R}})$,
$\displaystyle\Bigl{\lVert}\frac{\partial^{m}f_{z}}{\partial
s^{m}}\Bigr{\rVert}_{2}^{2}$ $\displaystyle=\sum_{k\in{\mathbb{Z}}}\lvert(2\pi
ik)^{m}\widehat{f_{z}}(k)\rvert^{2}$ $\displaystyle\geq(2\pi)^{2m}\lvert
k\rvert^{2m}\lvert\widehat{f_{z}}(k)\rvert^{2}\geq\lvert
k\rvert^{2m}\lvert\widehat{f_{z}}(k)\rvert^{2}.$
Since $\Bigl{\lVert}\frac{\partial^{m}f_{z}}{\partial
s^{m}}\Bigr{\rVert}_{2}=\left(\int_{0}^{1}\Bigl{\lvert}\frac{\partial^{m}}{\partial
s^{m}}f(s,z)\Bigr{\rvert}^{2}ds\right)^{\frac{1}{2}}$ is continuous in $z$,
there exists $M>0$ such that $\Bigl{\lVert}\frac{\partial^{m}f_{z}}{\partial
s^{m}}\Bigr{\rVert}_{2}<M$ for every $z\in\bar{V}$. Hence for all
$k\in{\mathbb{Z}}$ and $z\in\bar{V}$, $\lvert
k\rvert^{m}\lvert\widehat{f_{z}}(k)\rvert\leq M$. Therefore, for any
$z\in\bar{V}$,
$\displaystyle\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}\Bigl{\lvert}\frac{1}{-1+e^{2\pi
ika}}\widehat{f_{z}}(k)e^{2\pi iks}\Bigr{\rvert}$ $\displaystyle\leq
C^{-1}\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}\frac{1}{\lvert
k\rvert^{2}}\lvert k\rvert^{\alpha+2}\lvert\widehat{f_{z}}(k)\rvert$
$\displaystyle\leq
C^{-1}M\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}\frac{1}{\lvert
k\rvert^{2}}<\infty.$
This implies continuity of $h$ on
${\mathbb{Z}}\backslash{\mathbb{R}}\times\bar{V}$.
We have
$\frac{\partial h}{\partial
s}(s,z)=\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}\frac{2\pi ik}{-1+e^{2\pi
ika}}\widehat{f_{z}}(k)e^{2\pi iks}.$
Thus a similar argument shows that $\frac{\partial h}{\partial s}$ is
continuous.
Let $z=(z_{1},\dots,z_{n})$. For any $z\in\bar{V}$,
$\displaystyle\Bigl{\lvert}\frac{\partial}{\partial
z_{j}}\left(\frac{1}{-1+e^{2\pi ika}}\widehat{f_{z}}(k)e^{2\pi
iks}\right)\Bigr{\rvert}$ $\displaystyle=\Bigl{\lvert}\frac{1}{-1+e^{2\pi
ika}}\widehat{\frac{\partial f}{\partial z_{j}}(\cdot,z)}(k)e^{2\pi
iks}\Bigr{\rvert}$ $\displaystyle\leq C^{-1}\frac{1}{\lvert k\rvert^{2}}\lvert
k\rvert^{\alpha+2}\Bigl{\lvert}\widehat{\frac{\partial f}{\partial
z_{j}}(\cdot,z)}(k)\Bigr{\rvert}$ $\displaystyle\leq
C^{-1}M^{\prime}\frac{1}{\lvert k\rvert^{2}}\ \in
L^{1}({\mathbb{Z}}\setminus\\{0\\}).$
Thus
$\frac{\partial h}{\partial
z_{j}}(s,z)=\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}\frac{1}{-1+e^{2\pi
ika}}\widehat{\frac{\partial f}{\partial z_{j}}(\cdot,z)}(k)e^{2\pi iks}.$
Hence $\frac{\partial h}{\partial z_{j}}$ is continuous by an argument similar
to those above. For higher derivatives of $h$, continue this procedure. ∎
Set $c_{1}(x,\lambda)=c(x,\lambda)+h(x)-h(x\lambda)$. $c_{1}:{\Gamma\backslash
N}\times\Lambda\to{\mathbb{R}}$ is a $C^{\infty}$-cocycle and
$c_{1}(x,e^{mY_{1}})=0$. Thus for any $\lambda\in\Lambda$, cocycle equation
implies $c_{1}(x,\lambda)=c_{1}(xe^{Y_{1}},\lambda)$. Since the action of
$e^{{\mathbb{Z}}Y_{1}}$ on ${\bar{\pi}^{-1}(z)}$ is ergodic,
$c_{1}(x,\lambda)$ is constant on ${\bar{\pi}^{-1}(z)}$. Therefore we can
define a cocycle
$\bar{c}:{\bar{\Gamma}\backslash\bar{N}}\times\bar{\Lambda}\to{\mathbb{R}}$ by
$\bar{c}(\bar{\pi}(x),\pi(\lambda))=c_{1}(x,\lambda)$. Indeed, if
$\bar{\pi}(x)=\bar{\pi}(y)$ and $\pi(\lambda)=\pi(\lambda^{\prime})$, then
there exists a $m\in{\mathbb{Z}}$ with $\lambda=e^{mY_{1}}\lambda^{\prime}$,
so that
$c_{1}(x,\lambda)=c_{1}(x,e^{mY_{1}}\lambda^{\prime})=c_{1}(xe^{mY_{1}},\lambda^{\prime})=c_{1}(y,\lambda^{\prime}).$
Furthermore,
$\displaystyle\int_{{\bar{\Gamma}\backslash\bar{N}}}\bar{c}(x,\pi(\lambda))d\nu(z)$
$\displaystyle=\int_{{\bar{\Gamma}\backslash\bar{N}}}\int_{{\bar{\pi}^{-1}(z)}}c_{1}(s,\lambda)d\mu_{z}(s)d\nu(z)$
$\displaystyle=\int_{{\Gamma\backslash N}}c_{1}(x,\lambda)d\mu(x)=0.$
By induction, there exists a $C^{\infty}$-function
$P:{\bar{\Gamma}\backslash\bar{N}}\to{\mathbb{R}}$ such that
$\bar{c}(z,\pi(\lambda))=-P(z)+P(z\pi(\lambda))$. Put $Q=P\circ\bar{\pi}$.
Then $c_{1}(x,\lambda)=\bar{c}(\bar{\pi}(x),\pi(\lambda))=-Q(x)+Q(x\lambda)$.
This proves Theorem 4.
Let $\tilde{\rho}:M\times N\to M$ be the suspension of $\rho:{\Gamma\backslash
N}\times\Lambda\to{\Gamma\backslash N}$ where $M={\Gamma\backslash
N}\times_{\Lambda}N$ is a compact manifold. Then $\tilde{\rho}$ is locally
free and let ${\mathcal{F}}$ be its orbit foliation. We have
$H^{1}({\mathcal{F}})\simeq H^{1}(\Lambda,C^{\infty}({\Gamma\backslash N}))$
by [6] where the right hand side is the first cohomology of $\Lambda$-module
$C^{\infty}({\Gamma\backslash N})$ obtained by $\rho$. It is easy to prove
that $\mathop{\mathrm{Hom}}\nolimits(\Lambda,{\mathbb{R}})\to
H^{1}(\Lambda,C^{\infty}({\Gamma\backslash N}))$ is injective. By Theorem 4,
$H^{1}(\Lambda,C^{\infty}({\Gamma\backslash
N}))=\mathop{\mathrm{Hom}}\nolimits(\Lambda,{\mathbb{R}}).$
###### Lemma 7.
$\mathop{\mathrm{dim}}\nolimits\mathop{\mathrm{Hom}}\nolimits(\Lambda,{\mathbb{R}})=\mathop{\mathrm{dim}}\nolimits
H^{1}({\mathfrak{n}}).$
###### Proof.
Recall that $[N,N]\backslash\Lambda[N,N]$ is a cocompact lattice in
$[N,N]\backslash N$ and that
$[\Lambda,\Lambda]\backslash\left(\Lambda\cap[N,N]\right)$ is finite. Since
$0\to[\Lambda,\Lambda]\backslash\left(\Lambda\cap[N,N]\right)\to[\Lambda,\Lambda]\backslash\Lambda\to[N,N]\backslash\Lambda[N,N]\to
0$
is exact, we have
$\mathop{\mathrm{rank}}\nolimits[\Lambda,\Lambda]\backslash\Lambda=\mathop{\mathrm{rank}}\nolimits[N,N]\backslash\Lambda[N,N]=\mathop{\mathrm{dim}}\nolimits[N,N]\backslash
N.$
Thus
$\displaystyle\mathop{\mathrm{dim}}\nolimits\mathop{\mathrm{Hom}}\nolimits(\Lambda,{\mathbb{R}})$
$\displaystyle=\mathop{\mathrm{dim}}\nolimits\mathop{\mathrm{Hom}}\nolimits([\Lambda,\Lambda]\backslash\Lambda,{\mathbb{R}})$
$\displaystyle=\mathop{\mathrm{rank}}\nolimits[\Lambda,\Lambda]\backslash\Lambda$
$\displaystyle=\mathop{\mathrm{dim}}\nolimits[N,N]\backslash N$
$\displaystyle=\mathop{\mathrm{dim}}\nolimits\mathop{\mathrm{Hom}}\nolimits_{{\mathbb{R}}}([{\mathfrak{n}},{\mathfrak{n}}]\backslash{\mathfrak{n}},{\mathbb{R}})$
$\displaystyle=\mathop{\mathrm{dim}}\nolimits H^{1}({\mathfrak{n}}).$
∎
Therefore we obtain
$H^{1}({\mathcal{F}})=H^{1}({\mathfrak{n}}).$
This proves Theorem 3.
## 5 Existence of Diophantine lattices
Let ${\mathfrak{n}}_{{\mathbb{Q}}}$ be a rational structure of
${\mathfrak{n}}$. We construct Diophantine lattices when
${\mathfrak{n}}_{{\mathbb{Q}}}$ admits a graduation. Namely, we assume that
${\mathfrak{n}}_{{\mathbb{Q}}}$ has a sequence $V_{i}$ of
${\mathbb{Q}}$-subspaces such that
${\mathfrak{n}}_{{\mathbb{Q}}}=\bigoplus_{i=1}^{k}V_{i}$ and
$[V_{i},V_{j}]\subset V_{i+j}$. Let $X_{1},\dots,X_{n}$ be a
${\mathbb{Q}}$-basis of ${\mathfrak{n}}_{{\mathbb{Q}}}$ such that
$X_{1},\dots,X_{i_{1}}\in V_{k}$, $X_{i_{1}+1},\dots,X_{i_{2}}\in
V_{k-1}$$,\dots$, $X_{i_{k-1}+1},\dots,X_{n}\in V_{1}$. Then
$X_{1},\dots,X_{n}$ is a strong Malcev basis of ${\mathfrak{n}}$ with rational
structure constants. Multiplying $X_{1},\dots,X_{n}$ by a integer if
necessary, we may assume that $\Gamma:=e^{{\mathbb{Z}}X_{1}}\cdots
e^{{\mathbb{Z}}X_{n}}$ is a cocompact lattice in $N$. Let $\alpha$ be a root
of a irreducible polynomial of degree $k+1$ over ${\mathbb{Q}}$. Since
$\alpha,\alpha^{2},\dots,\alpha^{k}$ are irrational algebraic numbers, they
are Diophantine. If we define a linear map
$\varphi:{\mathfrak{n}}\to{\mathfrak{n}}$ by $\varphi(X)=\alpha^{i}X$ for
$X\in V_{i}\otimes{\mathbb{R}}$, then $\varphi$ is an automorphism of Lie
algebra ${\mathfrak{n}}$. Put $Y_{i}=\varphi(X_{i})$. $Y_{1},\dots,Y_{n}$ is a
strong Malcev basis of ${\mathfrak{n}}$ strongly based on
$\Lambda:=e^{{\mathbb{Z}}Y_{1}}\cdots e^{{\mathbb{Z}}Y_{n}}$. Thus $\Lambda$
is Diophantine with respect to $\Gamma$.
## Acknowledgement
The author would like to thank his advisor, Masayuki Asaoka, for helpful
comments.
## References
* [1] M. Asaoka. Deformation of lacally free actions and the leafwise cohomology. arXiv:1012.2946.
* [2] L. Corwin and F. P. Greenleaf. Representations of nilpotent Lie groups and their applications. Part 1:Basic theory and examples. Cambridge studies in advanced mathematics, vol. 18, Cambridge University Press, Cambridge, 1990.
* [3] A. Katok and R. J. Spatzier. First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity. Inst. Hautes Études Sci. Publ. Math. 79(1994), 131–156.
* [4] S. Matsumoto and Y. Mitsumatsu. Leafwise cohomology and rigidity of certain Lie group actions. Ergod. Th. & Dynam. Sys. 23(2003), 1839–1866.
* [5] D. Mieczkowski. The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory. J. Mod. Dyn. 1(2007), 61–92.
* [6] M. S. Pereira and N. M. dos Santos. On the cohomology of foliated bundles. Proyecciones 21(2)(2002), 175–197.
* [7] F. A. Ramírez. Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups. J. Mod. Dyn. 3(2009), 335–357.
* [8] N. M. dos Santos. Parameter rigid actions of the Heisenberg groups. Ergod. Th. & Dynam. Sys. 27(2007), 1719–1735.
|
arxiv-papers
| 2011-01-21T15:38:38 |
2024-09-04T02:49:16.570774
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Hirokazu Maruhashi",
"submitter": "Hirokazu Maruhashi",
"url": "https://arxiv.org/abs/1101.4161"
}
|
1101.4306
|
# A Matrix-Analytic Solution for Randomized Load Balancing Models with Phase-
Type Service Times
Quan-Lin Li1 John C.S. Lui2 Wang Yang3
1 School of Economics and Management Sciences
Yanshan University, Qinhuangdao 066004, China
2 Department of Computer Science & Engineering
The Chinese University of Hong Kong, Shatin, N.T, Hong Kong
3 Institute of Network Computing & Information Systems
Peking University, China
###### Abstract
In this paper, we provide a matrix-analytic solution for randomized load
balancing models (also known as _supermarket models_) with phase-type (PH)
service times. Generalizing the service times to the phase-type distribution
makes the analysis of the supermarket models more difficult and challenging
than that of the exponential service time case which has been extensively
discussed in the literature. We first describe the supermarket model as a
system of differential vector equations, and provide a doubly exponential
solution to the fixed point of the system of differential vector equations.
Then we analyze the exponential convergence of the current location of the
supermarket model to its fixed point. Finally, we present numerical examples
to illustrate our approach and show its effectiveness in analyzing the
randomized load balancing schemes with non-exponential service requirements.
## 1 Introduction
In the past few years, a number of companies (e.g., Amazon, Google, ,…etc) are
offering the _cloud computing_ service to enterprises. Furthermore, many
content publishers and application service providers are increasingly using
_Data Centers_ to host their services. This emerging computing paradigm allows
service providers and enterprises to concentrate on developing and providing
their own services/goods without worrying about computing system maintenance
or upgrade, and thereby significantly reduce their operating cost. For
companies that offer cloud computing service in their data centers, they can
take advantage of the variation of computing workloads from these customers
and achieve the computational multiplexing gain. One of the important
technical challenges that they have to address is how to utilize these
computing resources in the data center efficiently since many of these servers
can be virtualized. There is a growing interest to examine simple and robust
load balancing strategies to efficiently utilize the computing resource of the
server farms.
Distributed load balancing strategies, in which individual job (or customer)
decisions are based on information on a limited number of other processors,
have been studied analytically by Eager, Lazokwska and Zahorjan [4, 5, 6] and
through trace-driven simulations by Zhou [26]. Further, randomized load
balancing is a simple and effective mechanism to fairly utilize computing
resources, and also can deliver surprisingly good performance measures such as
reducing collisions, waiting times, backlogs,… etc. In a supermarket model,
each arriving job randomly picks a small subset of servers and examines their
instantaneous workload, and the job is routed to the least loaded server. When
a job is committed to a server, jockeying is not allowed and each server uses
the first-come-first-service (FCFS) discipline to process all jobs, e.g., see
Mitzenmacher [11, 12]. For the supermarket models, most of recent research
applied density dependent jump Markov processes to deal with the simple case
with Poisson arrival processes and exponential service times, and illustrated
that there exists a fixed point which decreases doubly exponentially. Readers
may refer to, such as, a simple supermarket model by [1, 24, 11, 12]; simple
variations by [19, 13, 14, 17, 23, 18, 25]; load information by [20, 3, 16,
18]; fast Jackson network by Martin and Suhov [10, 9, 21]; and general service
times by Bramson, Lu and Prabhakar [2]. When the arrival processes or the
service times are more general, the available results of the supermarket
models are few up to now. The purpose of this paper is to provide a novel
approach for studying a supermarket model with PH service times, and show that
the fixed point decreases doubly exponentially.
The remainder of this paper is organized as follows. In the next section, we
describe the supermarket model with the PH service times as a system of
differential vector equations based on the density dependent jump Markov
processes. In Section 3, we set up a system of nonlinear equations satisfied
by the fixed point, provide a doubly exponential solution to the system of
nonlinear equations, and compute the expected sojourn time of any arriving
customer. In Section 4, we study the exponential convergence of the current
location of the supermarket model to its fixed point. In Section 5, numerical
examples illustrate that our approach is effective in analyzing the
supermarket models from non-exponential service time requirements. Some
concluding remarks are given in Section 6.
## 2 Supermarket Model
In this section, we describe a supermarket model with the PH service times as
a system of differential vector equations based on the density dependent jump
Markov processes.
Let us formally describe the supermarket model, which is abstracted as a
multi-server multi-queue stochastic system. Customers arrive at a queueing
system of $n>1$ servers as a Poisson process with arrival rate $n\lambda$ for
$\lambda>0$. The service times of these customers are of phase type with
irreducible representation $\left(\alpha,T\right)$ of order $m$. Each arriving
customer chooses $d\geq 1$ servers independently and uniformly at random from
these $n$ servers, and waits for service at the server which currently
contains the fewest number of customers. If there is a tie, servers with the
fewest number of customers will be chosen randomly. All customers in every
server will be served in the FCFS manner. Please see Figure 1 for an
illustration.
Figure 1: The supermarket model: each customer can probe the loading of $d$
servers
For the supermarket models, the PH distribution allows us to model more
realistic systems and understand their performance implication under the
randomized load balancing strategy. As indicated in [7], the process lifetime
of many parallel jobs, in particular, jobs to data centers, tends to be non-
exponential. For the PH service time distribution, we use the following
irreducible representation: $\left(\alpha,T\right)$ of order $m$, the row
vector $\alpha$ is a probability vector whose $j$th entry is the probability
that a service begins in phase $j$ for $1\leq j\leq m$; $T$ is an $m\times m$
matrix whose $\left(i,j\right)^{th}$ entry is denoted by $t_{i,j}$ with
$t_{i,i}<0$ for $1\leq i\leq m$, and $t_{i,j}\geq 0$ for $1\leq i,j\leq m$ and
$i\neq j$. Let $T^{0}=-Te\gvertneqq 0$, where $e$ is a column vector of ones
with a suitable dimension in the context. The expected service time is given
by $1/\mu=-\alpha T^{-1}e$. Unless we state otherwise, we assume that all
random variables defined above are independent, and that the system is
operating in the stable region $\rho=\lambda/\mu<1$.
We define $n_{k}^{\left(i\right)}\left(t\right)$ as the number of queues with
at least $k$ customers and the service time in phase $i$ at time $t\geq 0$.
Clearly, $0\leq n_{k}^{\left(i\right)}\left(t\right)\leq n$ for $k\geq 0$ and
$1\leq i\leq m$. Let
$X_{n}^{\left(0\right)}\left(t\right)=\frac{n}{n}=1,$
and $k\geq 1$
$X_{n}^{\left(k,i\right)}\left(t\right)=\frac{n_{k}^{\left(i\right)}\left(t\right)}{n},$
which is the fraction of queues with at least $k$ customers and the service
time in phase $i$ at time $t\geq 0$. We write
$X_{n}^{\left(k\right)}\left(t\right)=\left(X_{n}^{\left(k,1\right)}\left(t\right),X_{n}^{\left(k,2\right)}\left(t\right),\ldots,X_{n}^{\left(k,m\right)}\left(t\right)\right),\text{
\ }k\geq 1,$
$X_{n}\left(t\right)=\left(X_{n}^{\left(0\right)}\left(t\right),X_{n}^{\left(1\right)}\left(t\right),X_{n}^{\left(2\right)}\left(t\right),\ldots\right).$
The state of the supermarket model may be described by the vector
$X_{n}\left(t\right)$ for $t\geq 0$. Since the arrival process to the queueing
system is Poisson and the service times of each server are of phase type, the
stochastic process $\left\\{X_{n}\left(t\right),t\geq 0\right\\}$ describing
the state of the supermarket model is a Markov process whose state space is
given by
$\displaystyle\Omega_{n}$ $\displaystyle=$
$\displaystyle\\{\left(g_{n}^{\left(0\right)},g_{n}^{\left(1\right)},,g_{n}^{\left(2\right)}\ldots\right):g_{n}^{\left(0\right)}=1,g_{n}^{\left(k-1\right)}\geq
g_{n}^{\left(k\right)}\geq 0,$ $\displaystyle\text{and \ \
}ng_{n}^{\left(k\right)}\text{ \ is a vector of nonnegative integers for
}k\geq 1\\}.$
Let
$s_{0}\left(n,t\right)=E\left[X_{n}^{\left(0\right)}\left(t\right)\right]$
and $k\geq 1$
$s_{k}^{\left(i\right)}\left(n,t\right)=E\left[X_{n}^{\left(k,i\right)}\left(t\right)\right].$
Clearly, $s_{0}\left(n,t\right)=1$. We write
$S_{k}\left(n,t\right)=\left(s_{k}^{\left(1\right)}\left(n,t\right),s_{k}^{\left(2\right)}\left(n,t\right),\ldots,s_{k}^{\left(m\right)}\left(n,t\right)\right),\text{
\ }k\geq 1.$
As shown in Martin and Suhov [10] and Luczak and McDiarmid [8], the Markov
process $\left\\{X_{n}\left(t\right),t\geq 0\right\\}$ is asymptotically
deterministic as $n\rightarrow\infty$. Thus the limits
$\lim_{n\rightarrow\infty}E\left[X_{n}^{\left(0\right)}\left(t\right)\right]$
and $\lim_{n\rightarrow\infty}E\left[X_{n}^{\left(k,i\right)}\right]$ always
exist by means of the law of large numbers. Based on this, we write
$S_{0}\left(t\right)=\lim_{n\rightarrow\infty}s_{0}\left(n,t\right)=1,$
for $k\geq 1$
$s_{k}^{\left(i\right)}\left(t\right)=\lim_{n\rightarrow\infty}s_{k}^{\left(i\right)}\left(n,t\right),$
$S_{k}\left(t\right)=\left(s_{k}^{\left(1\right)}\left(t\right),s_{k}^{\left(2\right)}\left(t\right),\ldots,s_{k}^{\left(m\right)}\left(t\right)\right)$
and
$S\left(t\right)=\left(S_{0}\left(t\right),S_{1}\left(t\right),S_{2}\left(t\right),\ldots\right).$
Let $X\left(t\right)=\lim_{n\rightarrow\infty}X_{n}\left(t\right)$. Then it is
easy to see from the Poisson arrivals and the PH service times that
$\left\\{X\left(t\right),t\geq 0\right\\}$ is also a Markov process whose
state space is given by
$\Omega=\left\\{\left(g^{\left(0\right)},g^{\left(1\right)},g^{\left(2\right)},\ldots\right):g^{\left(0\right)}=1,g^{\left(k-1\right)}\geq
g^{\left(k\right)}\geq 0\right\\}.$
If the initial distribution of the Markov process
$\left\\{X_{n}\left(t\right),t\geq 0\right\\}$ approaches the Dirac delta-
measure concentrated at a point $g\in$ $\Omega$, then its steady-state
distribution is concentrated in the limit on the trajectory
$S_{g}=\left\\{S\left(t\right):t\geq 0\right\\}$. This indicates a law of
large numbers for the time evolution of the fraction of queues of different
lengths. Furthermore, the Markov process $\left\\{X_{n}\left(t\right),t\geq
0\right\\}$ converges weakly to the fraction vector
$S\left(t\right)=\left(S_{0}\left(t\right),S_{1}\left(t\right),S_{2}\left(t\right),\ldots\right)$,
or for a sufficiently small $\varepsilon>0$,
$\lim_{n\rightarrow\infty}P\left\\{||X_{n}\left(t\right)-S\left(t\right)||\geq\varepsilon\right\\}=0,$
where $||a||$ is the $L_{\infty}$-norm of vector $a$.
In what follows we provide a system of differential vector equations in order
to determine fraction vector $S\left(t\right)$. To that end, we introduce the
_Hadamard Product_ of two matrices $A=\left(a_{i,j}\right)$ and
$B=\left(b_{i,j}\right)$ as follows:
$A\odot B=\left(a_{i,j}b_{i,j}\right).$
Specifically, for $k\geq 2$, we have
$A^{\odot k}=\underset{k\text{ matrix }A}{\underbrace{A\odot A\odot\cdots\odot
A}}.$
To determine the fraction vector $S\left(t\right)$, we need to set up a system
of differential vector equations satisfied by $S\left(t\right)$ by means of
the density dependent jump Markov process. To that end, we provide a concrete
example for $k\geq 2$ to indicate how to derive the the system of differential
vector equations.
Consider the supermarket model with $n$ servers, and determine the expected
change in the number of queues with at least $k$ customers over a small time
period of length d$t$. The probability vector that during this time period,
any arriving customer joins a queue of size $k-1$ is given by
$n\left[\lambda S_{k-1}^{\odot d}\left(n,t\right)-\lambda S_{k}^{\odot
d}\left(n,t\right)\right]\text{d}t.$
Similarly, the probability vector that a customer leaves a server queued by
$k$ customers is given by
$n\left[S_{k}\left(n,t\right)T+S_{k+1}\left(n,t\right)T^{0}\alpha\right]\text{d}t.$
Therefore we can obtain
$\displaystyle\text{d}E\left[n_{k}\left(n,t\right)\right]=$ $\displaystyle
n\left[\lambda S_{k-1}^{\odot d}\left(n,t\right)-\lambda S_{k}^{\odot
d}\left(n,t\right)\right]\text{d}t$
$\displaystyle+n\left[S_{k}\left(n,t\right)T+S_{k+1}\left(n,t\right)T^{0}\alpha\right]\text{d}t,$
which leads to
$\frac{\text{d}S_{k}\left(n,t\right)}{\text{d}t}=\lambda S_{k-1}^{\odot
d}\left(n,t\right)-\lambda S_{k}^{\odot
d}\left(n,t\right)+S_{k}\left(n,t\right)T+S_{k+1}\left(n,t\right)T^{0}\alpha.$
Taking $n\rightarrow\infty$ in the both sides of Equation (LABEL:Equ1), we
have
$\frac{\text{d}S_{k}\left(t\right)}{\text{d}t}=\lambda S_{k-1}^{\odot
d}\left(t\right)-\lambda S_{k}^{\odot
d}\left(t\right)+S_{k}\left(t\right)T+S_{k+1}\left(t\right)T^{0}\alpha.$
Using a similar analysis to Equation (LABEL:Equ2), we can obtain a system of
differential vector equations for the fraction vector
$S\left(t\right)=\left(S_{0}\left(t\right),S_{1}\left(t\right),S_{2}\left(t\right),\ldots\right)$
as follows:
$S_{0}\left(t\right)=1,$
$\frac{\mathtt{d}}{\text{d}t}S_{0}\left(t\right)=-\lambda
S_{0}^{d}\left(t\right)+S_{1}\left(t\right)T^{0},$ (1)
$\frac{\mathtt{d}}{\text{d}t}S_{1}\left(t\right)=\lambda\alpha
S_{0}^{d}\left(t\right)-\lambda S_{1}^{\odot
d}\left(t\right)+S_{1}\left(t\right)T+S_{2}\left(t\right)T^{0}\alpha,$ (2)
and for $k\geq 2$,
$\frac{\mathtt{d}}{\text{d}t}S_{k}\left(t\right)=\lambda S_{k-1}^{\odot
d}\left(t\right)-\lambda S_{k}^{\odot
d}\left(t\right)+S_{k}\left(t\right)T+S_{k+1}\left(t\right)T^{0}\alpha.$ (3)
###### Remark 1
Mitzenmacher [11, 12] provided an heuristical and interesting method to
establish such systems of differential equations, but they lack a rigorous
mathematical meaning for understanding the stochastic process
$\left\\{X_{n}\left(t\right),t\geq 0\right\\}$ in which
$X_{n}\left(t\right)=(X_{n}^{\left(0\right)}\left(t\right),X_{n}^{\left(1\right)}\left(t\right),$
$X_{n}^{\left(2\right)}\left(t\right),\ldots)$ and
$X_{n}^{\left(k\right)}\left(t\right)=n_{k}\left(t\right)/n$ for $k\geq 0$.
This section, following Martin and Suhov [10] and Luczak and McDiarmid [8],
gives some necessary mathematical analysis for the stochastic process
$\left\\{X_{n}\left(t\right),t\geq 0\right\\}$ and the system of differential
vector equations (1), (2) and (3).
## 3 A Matrix-Analytic Solution
In this section, we provide a doubly exponential solution to the fixed point
of the system of differential vector equations (1), (2) and (3).
A row vector $\pi=\left(\pi_{0},\pi_{1},\pi_{2},\ldots\right)$ is called a
fixed point of the fraction vector $S\left(t\right)$ if
$\lim_{t\rightarrow+\infty}S\left(t\right)=\pi$. In this case, it is easy to
see that
$\lim_{t\rightarrow+\infty}\left[\frac{\mathtt{d}}{\text{d}t}S\left(t\right)\right]=0.$
Therefore, as $t\rightarrow+\infty$ the system of differential vector
equations (1), (2) and (3) can be simplified as
$-\lambda\pi_{0}^{d}+\pi_{1}T^{0}=0,$ (4)
$\lambda\alpha\pi_{0}^{d}-\lambda\pi_{1}^{\odot
d}+\pi_{1}T+\pi_{2}T^{0}\alpha=0,$ (5)
and for $k\geq 2$,
$\lambda\pi_{k-1}^{\odot d}-\lambda\pi_{k}^{\odot
d}+\pi_{k}T+\pi_{k+1}T^{0}\alpha=0.$ (6)
In general, it is more difficult and challenging to express the fixed point of
the supermarket models with more general arrival processes or service times,
because the systems of nonlinear equations are more complicated for
computation. Fortunately, we can derive a closed-form expression for the fixed
point $\pi=(\pi_{0},\pi_{1},\pi_{2},...)$ for the supermarket model with PH
service times by means of a novel matrix-analytic approach given as follows.
Noting that $S_{0}\left(t\right)=1$ for all $t\geq 0$, it is easy to see that
$\pi_{0}=1$. It follows from Equation (4) that
$\pi_{1}T^{0}=\lambda.$ (7)
To solve Equation (7), we denote by $\omega$ the stationary probability vector
of the irreducible Markov chain $T+T^{0}\alpha$. Obviously, we have
$\omega T^{0}=\mu,$ $\frac{\lambda}{\mu}\omega T^{0}=\lambda.$ (8)
Thus, we obtain $\pi_{1}=\frac{\lambda}{\mu}\omega=\rho\cdot\omega$. Based on
the fact that $\pi_{0}=1$ and $\pi_{1}=\rho\cdot\omega$, it follows from
Equation (5) that
$\lambda\alpha-\lambda\rho^{d}\cdot\omega^{\odot d}+\rho\cdot\omega
T+\pi_{2}T^{0}\alpha=0,$
which leads to
$\lambda-\lambda\rho^{d}\cdot\omega^{\odot d}e+\rho\cdot\omega
Te+\pi_{2}T^{0}=0.$
Note that $\omega Te=-\mu$, we obtain
$\pi_{2}T^{0}=\lambda\rho^{d}\omega^{\odot d}e.$
Let $\theta=\omega^{\odot d}e$. Then it is easy to see that
$\theta\in\left(0,1\right)$, and
$\pi_{2}T^{0}=\lambda\theta\rho^{d}.$
Using a similar analysis to Equation (8), we have
$\pi_{2}=\frac{\lambda\theta\rho^{d}}{\mu}\omega=\theta\rho^{d+1}\cdot\omega.$
(9)
Based on $\pi_{1}=\rho\cdot\omega$ and $\pi_{2}=\theta\rho^{d+1}\cdot\omega$,
it follows from Equation (6) that for $k=2$,
$\lambda\rho^{d}\cdot\omega^{\odot
d}-\lambda\theta^{d}\rho^{d^{2}+d}\cdot\omega^{\odot
d}+\theta\rho^{d+1}\cdot\omega T+\pi_{3}T^{0}\alpha=0,$
which leads to
$\lambda\theta\rho^{d}-\lambda\theta^{d+1}\rho^{d^{2}+d}+\theta\rho^{d+1}\cdot\omega
Te+\pi_{3}T^{0}=0,$
thus we obtain
$\pi_{3}T^{0}=\lambda\theta^{d+1}\rho^{d^{2}+d}.$
Using a similar analysis on Equation (8), we have
$\pi_{3}=\frac{\lambda\theta^{d+1}\rho^{d^{2}+d}}{\mu}\omega=\theta^{d+1}\rho^{d^{2}+d+1}\cdot\omega.$
(10)
Based on Equations (9) and (10), we may infer that there is a structured
expression
$\pi_{k}=\theta^{d^{k-2}+d^{k-3}+\cdots+d+1}\rho^{d^{k-1}+d^{k-2}+\cdots+d+1}\cdot\omega$,
for $k\geq 1$. To that end, the following theorem states this important
result.
###### Theorem 1
The fixed point $\pi=\left(\pi_{0},\pi_{1},\pi_{2},\ldots\right)$ is unique
and is given by
$\pi_{0}=1,\hskip 14.45377pt\pi_{1}=\rho\cdot\omega$
and for $k\geq 2,$
$\pi_{k}=\theta^{d^{k-2}+d^{k-3}+\cdots+1}\rho^{d^{k-1}+d^{k-2}+\cdots+1}\cdot\omega,$
(11)
or
$\displaystyle\pi_{k}$ $\displaystyle=$
$\displaystyle\theta^{\frac{d^{k-1}-1}{d-1}}\rho^{\frac{d^{k}-1}{d-1}}\cdot\omega=\rho^{d^{k-1}}\left(\theta\rho\right)^{\frac{d^{k-1}-1}{d-1}}\cdot\omega.$
Proof: By induction, one can easily derive the above result.
It is clear that Equation (11) is correct for the cases with $l=2,3$ according
to Equations (9) and (10). Now, we assume that Equation (11) is correct for
the cases with $l=k$. Then it follows from Equation (6) that for $l=k+1$, we
have
$\displaystyle\lambda$
$\displaystyle\theta^{d^{k-2}+d^{k-3}+\cdots+d}\rho^{d^{k-1}+d^{k-2}+\cdots+d}\cdot\omega^{\odot
d}-\lambda\theta^{d^{k-1}+d^{k-2}+\cdots+d}\rho^{d^{k}+d^{k-1}+\cdots+d}\cdot\omega^{\odot
d}$
$\displaystyle+\theta^{d^{k-2}+d^{k-3}+\cdots+1}\rho^{d^{k-1}+d^{k-2}+\cdots+1}\cdot\omega
T+\pi_{k+1}T^{0}\alpha=0,$
which leads to
$\displaystyle\lambda$
$\displaystyle\theta^{d^{k-2}+d^{k-3}+\cdots+d+1}\rho^{d^{k-1}+d^{k-2}+\cdots+d}-\lambda\theta^{d^{k-1}+d^{k-2}+\cdots+d+1}\rho^{d^{k}+d^{k-1}+\cdots+d}$
$\displaystyle+\theta^{d^{k-2}+d^{k-3}+\cdots+1}\rho^{d^{k-1}+d^{k-2}+\cdots+1}\cdot\omega
Te+\pi_{k+1}T^{0}=0,$
thus we obtain
$\pi_{k+1}T^{0}=\lambda\theta^{d^{k-1}+d^{k-2}+\cdots+d+1}\rho^{d^{k}+d^{k-1}+\cdots+d}.$
By a similar analysis to (8), we have
$\displaystyle\pi_{k+1}$
$\displaystyle=\frac{\lambda\theta^{d^{k-1}+d^{k-2}+\cdots+d+1}\rho^{d^{k}+d^{k-1}+\cdots+d}}{\mu}\omega$
$\displaystyle=\theta^{d^{k-1}+d^{k-2}+\cdots+d+1}\rho^{d^{k}+d^{k-1}+\cdots+d+1}\cdot\omega.$
This completes the proof.
Now, we compute the expected sojourn time $T_{d}$ that a tagged arriving
customer spends in the supermarket model. For the PH service times, a tagged
arriving customer is the $k$th customer in the corresponding queue with
probability vector $\pi_{k-1}^{\odot d}-\pi_{k}^{\odot d}$. When $k\geq 1$,
the head customer in the queue has been served, and so its service time is
residual and is denoted as $X_{R}$. Let $X$ be of phase type with irreducible
representation $\left(\alpha,T\right)$. Then $X_{R}$ is of phase type with
irreducible representation $\left(\omega,T\right)$. Clearly, we have
$E\left[X\right]=\alpha\left(-T\right)^{-1}e,\text{ \
}E\left[X_{R}\right]=\omega\left(-T\right)^{-1}e.$
Thus it is easy to see that the expected sojourn time of the tagged arriving
customer is given by
$\displaystyle E\left[T_{d}\right]$
$\displaystyle=\left(\pi_{0}^{d}-\pi_{1}^{\odot
d}e\right)E\left[X\right]+\sum_{k=1}^{\infty}\left(\pi_{k}^{\odot
d}-\pi_{k+1}^{\odot
d}\right)e\left\\{E\left[X_{R}\right]+kE\left[X\right]\right\\}$
$\displaystyle=\pi_{1}^{\odot
d}e\left\\{E\left[X_{R}\right]-E\left[X\right]\right\\}+E\left[X\right]\left[1+\sum_{k=1}^{\infty}\pi_{k}^{\odot
d}e\right]$
$\displaystyle=\rho^{d}\theta\left(\omega-\alpha\right)\left(-T\right)^{-1}e+\alpha\left(-T\right)^{-1}e\left(1+\sum_{k=1}^{\infty}\theta^{\frac{d^{k}-1}{d-1}}\rho^{\frac{d^{k+1}-d}{d-1}}\right).$
When the arrival process and the service time distribution are Poisson and
exponential, respectively, it is clear that $\alpha=\omega=\theta=1$ and
$\alpha\left(-T\right)^{-1}e=1/\mu$, thus we have
$E\left[T_{d}\right]=\frac{1}{\mu}\sum_{k=0}^{\infty}\rho^{\frac{d^{k+1}-d}{d-1}},$
which is the same as Corollary 3.8 in Mitzenmacher [12].
In what follows we consider an interesting problem: how many moments of the
service time distribution are needed to obtain a better accuracy for computing
the fixed point or the expected sojourn time. It is well-known from the theory
of probability distributions that the first three moments is basic for
analyzing such an accuracy, and we can construct a PH distribution of order 2
by using the first three moments. Telek and Heindl [22] provided a fitting
procedure for matching a PH distribution of order 2 from the first three
moments exactly. It is necessary to list the fitting procedure as follows:
For a nonnegative random variable $X$, let $m_{n}=E\left[X^{n}\right]$, $n\geq
1$. We take a PH distribution of order 2 with the canonical representation
$\left(\alpha,T\right)$, where $\mathbf{\alpha=}\left(\eta,1-\eta\right)$ and
$T\mathbf{=}\left(\begin{array}[]{cc}-\xi_{1}&\xi_{1}\\\
0&-\xi_{2}\end{array}\right),$
$0\leq\eta\leq 1$ and $0<\xi_{1}\leq\xi_{2}$. Note that the three unknown
parameters $\eta$, $\xi_{1}$ and $\xi_{2}$ can be obtained from the first
three moments $m_{1}$, $m_{2}$ and $m_{3}$ of an arbitrary general
distribution.
Table 1: Specific Bounds of the First Three Moments Moment | Condition | Bounds
---|---|---
$m_{1}$ | | $0<m_{1}<\infty$
$m_{2}$ | | $1.5m_{1}^{2}\leq m_{2}$
$m_{3}$ | $0.5\leq c_{X}^{2}\leq 1$ | $3m_{1}^{3}\left(3c_{X}^{2}-1+\sqrt{2}\left(1-c_{X}^{2}\right)^{\frac{3}{2}}\right)\leq m_{3}\leq 6m_{1}^{3}c_{X}^{2}$
| $1<c_{X}^{2}$ | $\frac{3}{2}m_{1}^{3}\left(1+c_{X}^{2}\right)^{2}<m_{3}<\infty$
In Table 1, $c_{X}^{2}=m_{2}\diagup m_{1}^{2}-1$ is the squared coefficient of
variation. If the moments do not satisfy these conditions in Table 1, then we
may analyze the following four cases:
(a.1) if $m_{2}<1.5m_{1}^{2}$, then we take $m_{2}=1.5m_{1}^{2}$;
(a.2) if $0.5\leq c_{X}^{2}\leq 1$, and
$m_{3}<3m_{1}^{3}\left(3c_{X}^{2}-1+\sqrt{2}\left(1-c_{X}^{2}\right)^{\frac{3}{2}}\right)$,
then we take
$m_{3}=3m_{1}^{3}\left(3c_{X}^{2}-1+\sqrt{2}\left(1-c_{X}^{2}\right)^{\frac{3}{2}}\right)$;
(a.3) if $0.5\leq c_{X}^{2}\leq 1$, and $m_{3}>6m_{1}^{3}c_{X}^{2}$, then we
take $m_{3}=6m_{1}^{3}c_{X}^{2}$; and
(a.4) if $1<c_{X}^{2}$, and
$m_{3}\leq\frac{3}{2}m_{1}^{3}\left(1+c_{X}^{2}\right)^{2}$, then we take
$m_{3}=\frac{3}{2}m_{1}^{3}\left(1+c_{X}^{2}\right)^{2}$.
Let $c=3m_{2}^{2}-2m_{1}m_{3}$, $d=2m_{1}^{2}-m_{2}$, $b=3m_{1}m_{2}-m_{3}$
and $a=b^{2}-6cd$. If the moments respectively satisfy their specific bounds
__ shown in Table 1 or the exceptive four cases, then three unknown parameters
$\eta$, $\xi_{1}$ and $\xi_{2}$ can be computed in the following three cases.
(1) If $c>0$, then
$\eta=\frac{-b+6m_{1}d+\sqrt{a}}{b+\sqrt{a}},\text{
}\xi_{1}=\frac{b-\sqrt{a}}{c},\text{ }\xi_{2}=\frac{b+\sqrt{a}}{c}.$
(2) If $c<0$, then
$\eta=\frac{b-6m_{1}d+\sqrt{a}}{-b+\sqrt{a}},\text{
}\xi_{1}=\frac{b+\sqrt{a}}{c},\text{ }\xi_{2}=\frac{b-\sqrt{a}}{c}.$
(3) If $c=0$, then
$\eta=0,\text{ }\xi_{1}>0,\text{ }\xi_{2}=\frac{1}{m_{1}}.$
From the above discussion, we can always construct a PH distribution of order
2 to approximate an arbitrary general distribution with the same first three
moments. In fact, such an approximation achieves a better accuracy in
computation.
For the PH distribution of order 2, we have
$T+T^{0}\alpha=\left(\begin{array}[]{cc}-\xi_{1}&\xi_{1}\\\
0&-\xi_{2}\end{array}\right)+\left(\begin{array}[]{c}0\\\
\xi_{2}\end{array}\right)\left(\begin{array}[]{cc}\eta&1-\eta\end{array}\right)=\left(\begin{array}[]{cc}-\xi_{1}&\xi_{1}\\\
\xi_{2}\eta&-\xi_{2}\eta\end{array}\right),$
which leads to
$\omega=\left(\frac{\xi_{2}\eta}{\xi_{1}+\xi_{2}\eta},\frac{\xi_{1}}{\xi_{1}+\xi_{2}\eta}\right)$
and
$\theta=\frac{\xi_{1}^{d}+\xi_{2}^{d}\eta^{d}}{\left(\xi_{1}+\xi_{2}\eta\right)^{d}}.$
Thus, the PH distribution of order 2 can effectively approximates an arbitrary
general service time distribution in the supermarket model under the same
first three moments, and specifically, all the computations are very simple to
implement.
###### Remark 2
Bramson, Lu and Prabhakar [2] provided a modularized program based on ansatz
for treating the supermarket model with a general service time distribution.
They organized a functional equation $\pi=F\left(G\left(\pi\right)\right)$ for
analyzing the stationary probability vector $\pi$ in terms of insensitivity
and generalized Fibonacci sequences, although the operators $F$ and $G$ are
not easy to be given for this supermarket model. This paper studies the
supermarket model with a PH service time distribution, provides the doubly
exponential solution to the fixed point, and is specifically related to the
phase type environment by means of the crucial factor $\theta=\omega^{\odot
d}e$. Note that the PH distributions are dense in the set of all nonnegative
random variables, this paper can numerically provide necessary understanding
for the role played by the general service time distribution in performance
analysis of the supermarket model by means of the PH approximation of order 2.
## 4 Exponential convergence to the fixed point
In this section, we study the exponential convergence of the current location
$S\left(t\right)$ of the supermarket model to its fixed point $\pi$.
For the supermarket model, the initial point $S\left(0\right)$ can affect the
current location $S\left(t\right)$ for each $t>0$, since the service process
in the supermarket model is under a unified structure. To that end, we provide
some notation for comparison of two vectors. Let
$a=\left(a_{1},a_{2},a_{3},\ldots\right)$ and
$b=\left(b_{1},b_{2},b_{3},\ldots\right)$. We write $a\prec b$ if
$a_{k}<b_{k}$ for some $k\geq 1$ and $a_{l}\leq b_{l}$ for $l\neq k,l\geq 1$;
and $a\preceq b$ if $a_{k}\leq b_{k}$ for all $k\geq 1$. Now, we can obtain
the following useful proposition whose proof is clear from a sample path
analysis and thus is omitted here.
###### Proposition 1
If $S\left(0\right)\preceq\widetilde{S}\left(0\right)$, then
$S\left(t\right)\preceq\widetilde{S}\left(t\right)$.
Based on Proposition 1, the following theorem shows that the fixed point $\pi$
is an upper bound of the current location $S\left(t\right)$ for all $t\geq 0$.
###### Theorem 2
For the supermarket model, if there exists some $k$ such that
$S_{k}\left(0\right)=0$, then the sequence
$\left\\{S_{k}\left(t\right)\right\\}$ has an upper bound sequence which
decreases doubly exponentially for all $t\geq 0$, that is,
$S\left(t\right)\preceq\pi$ for all $t\geq 0$.
Proof: Let $\widetilde{S}_{k}\left(0\right)=\pi_{k}$ for $k\geq 1$. Then for
each $k\geq 1$,
$\widetilde{S}_{k}\left(t\right)=\widetilde{S}_{k}\left(0\right)=\pi_{k}$ for
all $t\geq 0$, since
$\widetilde{S}\left(0\right)=\left(\widetilde{S}_{1}\left(0\right),\widetilde{S}_{2}\left(0\right),\widetilde{S}_{2}\left(0\right),\ldots\right)$
is a fixed point in the supermarket model. If $S_{k}\left(0\right)=0$ for some
$k$, then $S_{k}\left(0\right)\prec\widetilde{S}_{k}\left(0\right)$ and
$S_{j}\left(0\right)\preceq\widetilde{S}_{j}\left(0\right)$ for $j\neq k,j\geq
1$, thus $S\left(0\right)\preceq\widetilde{S}\left(0\right)$. It is easy to
see from Proposition 1 that
$S_{k}\left(t\right)\preceq\widetilde{S}_{k}\left(t\right)=\pi_{k}$ for all
$k\geq 1$ and $t\geq 0$. Thus we obtain that for all $k\geq 1$ and $t\geq 0$
$S_{k}\left(t\right)\leq\theta^{\frac{d^{k-1}-1}{d-1}}\rho^{\frac{d^{k}-1}{d-1}}\cdot\omega.$
This completes the proof.
To show the exponential convergence, we define a Lyapunov function
$\Phi\left(t\right)$ as
$\Phi\left(t\right)=\sum_{k=1}^{\infty}w_{k}\left[\pi_{k}-S_{k}\left(t\right)\right]e$
in terms of the fact that $S_{k}\left(t\right)\preceq\pi_{k}$ for $k\geq 1$
and $\pi_{0}=S_{0}\left(t\right)=1$, where $\left\\{w_{k}\right\\}$ is a
positive scalar sequence with $w_{k+1}\geq w_{k}\geq w_{1}=1$ for $k\geq 2$.
The following theorem measures the distance $\Phi\left(t\right)$ of the
current location $S\left(t\right)$ for $t\geq 0$ to the fixed point $\pi$, and
illustrates that this distance between the fixed point and the current
location is very close to zero with exponential convergence. This shows that
from a suitable starting point, the supermarket model can be quickly close to
the fixed point.
###### Theorem 3
For $t\geq 0$, $\Phi\left(t\right)\leq c_{0}e^{-\delta t}$, where $c_{0}$ and
$\delta$ are two positive constants. In this case, the potential function
$\Phi\left(t\right)$ is exponentially convergent.
Proof: Note that
$\Phi\left(t\right)=\sum_{k=1}^{\infty}w_{k}\left[\pi_{k}-S_{k}\left(t\right)\right]e,$
we have
$\frac{d}{dt}\Phi\left(t\right)=-\sum_{k=1}^{\infty}w_{k}\frac{d}{dt}S_{k}\left(t\right)e.$
It follows from Equations (1) to (3) that
$\displaystyle\frac{d}{dt}\Phi\left(t\right)=$ $\displaystyle-w_{1}[\lambda
S_{0}^{d}\left(t\right)\alpha-\lambda S_{1}^{\odot
d}\left(t\right)+S_{1}\left(t\right)T+S_{2}\left(t\right)T^{0}\alpha]e$
$\displaystyle-\sum_{k=1}^{\infty}w_{k}[\lambda S_{k-1}^{\odot
d}\left(t\right)-\lambda S_{k}^{\odot
d}\left(t\right)+S_{k}\left(t\right)T+S_{k+1}\left(t\right)T^{0}\alpha]e.$
By means of $S_{0}\left(t\right)=1$ and $Te=-T^{0}$, we can obtain
$\displaystyle\frac{d}{dt}\Phi\left(t\right)=$ $\displaystyle-
w_{1}[\lambda-\lambda S_{1}^{\odot
d}\left(t\right)e-S_{1}\left(t\right)T^{0}+S_{2}\left(t\right)T^{0}]$
$\displaystyle-\sum_{k=2}^{\infty}w_{k}[\lambda S_{k-1}^{\odot
d}\left(t\right)e-\lambda S_{k}^{\odot
d}\left(t\right)e-S_{k}\left(t\right)T^{0}+S_{k+1}\left(t\right)T^{0}].$ (12)
We take some nonnegative constants $c_{k}\left(t\right)$ and
$d_{k}\left(t\right)$ for $k\geq 1$ such that
$\lambda=f_{1}\left(t\right)S_{1}\left(t\right)T^{0},$
for $k\geq 1$
$\lambda S_{k}^{\odot
d}\left(t\right)e=c_{k}\left(t\right)\left[\pi_{k}-S_{k}\left(t\right)\right]e$
and
$S_{k}\left(t\right)T^{0}=d_{k}\left(t\right)\left[\pi_{k}-S_{k}\left(t\right)\right]e.$
Then it follows from (12) that
$\displaystyle\frac{d}{dt}\Phi\left(t\right)$
$\displaystyle=-\left\\{\left[\left(w_{2}-w_{1}\right)\right]c_{1}\left(t\right)+w_{1}\left[f_{1}\left(t\right)-1\right]d_{1}\left(t\right)\right\\}\cdot\left[\pi_{1}-S_{1}\left(t\right)\right]e$
$\displaystyle-\sum_{k=2}^{\infty}\left[\left(w_{k+1}-w_{k}\right)c_{k}\left(t\right)+\left(w_{k-1}-w_{k}\right)d_{k}\left(t\right)\right]\cdot\left[\pi_{k}-S_{k}\left(t\right)\right]e.$
For a constant $\delta>0$, we take
$w_{1}=1,$
$\left[\left(w_{2}-w_{1}\right)\right]c_{1}\left(t\right)+w_{1}\left[f_{1}\left(t\right)-1\right]d_{1}\left(t\right)\geq\delta
w_{1}$
and
$\left(w_{k+1}-w_{k}\right)c_{k}\left(t\right)+\left(w_{k-1}-w_{k}\right)d_{k}\left(t\right)\geq\delta
w_{k}.$
In this case, it is easy to see that
$w_{2}\geq 1+\frac{\delta+1-f_{1}\left(t\right)}{c_{1}\left(t\right)}$
and for $k\geq 2$
$w_{k+1}\geq w_{k}+\frac{\delta
w_{k}}{c_{k}\left(t\right)}+\frac{d_{k}\left(t\right)}{c_{k}\left(t\right)}\left(w_{k}-w_{k-1}\right).$
Thus we have
$\frac{d}{dt}\Phi\left(t\right)\leq-\delta\sum_{k=0}^{\infty}w_{k}\left[\pi_{k}-S_{k}\left(t\right)\right]e=-\delta\Phi\left(t\right),$
which can leads to
$\Phi\left(t\right)\leq c_{0}e^{-\delta t}.$
This completes the proof.
## 5 Numerical examples
In this section, we provide some numerical examples to illustrate that our
approach is effective and efficient in the study of supermarket models with
non-exponential service requirements, including Erlang service time
distributions, hyper-exponential service time distributions and PH service
time distributions.
Example one (Erlang Distribution) We consider an $m$-order Erlang distribution
with the irreducible PH representation $(\alpha,T)$,
where$\alpha=\left(1,0,\ldots,0,0\right)$ and
$T=\left(\begin{array}[]{ccccc}-\eta&\eta&&&\\\ &-\eta&\eta&&\\\
&&\ddots&\ddots&\\\ &&&-\eta&\eta\\\ &&&&-\eta\end{array}\right),\text{ \ \
}T^{0}=\left(\begin{array}[]{c}0\\\ 0\\\ \vdots\\\ 0\\\
\eta\end{array}\right).$
It is clear that
$T+T^{0}\alpha=\left(\begin{array}[]{ccccc}-\eta&\eta&&&\\\ &-\eta&\eta&&\\\
&&\ddots&\ddots&\\\ &&&-\eta&\eta\\\ \eta&&&&-\eta\end{array}\right),$
which leads to the stationary probability vector of the Markov chain
$T+T^{0}\alpha$ as follows:
$\omega=\left(\frac{1}{m},\frac{1}{m},\ldots\frac{1}{m},\frac{1}{m}\right);\hskip
7.22743pt\mu=\omega T^{0}=\frac{\eta}{m};\hskip
7.22743pt\rho=\frac{\lambda}{\mu}=\frac{m\lambda}{\eta};\hskip
7.22743pt\theta=m\left(\frac{1}{m}\right)^{d}=m^{1-d}.$
Thus we obtain
$\displaystyle\pi_{k}$
$\displaystyle=m^{1-d^{k-1}}\left(\frac{m\lambda}{\eta}\right)^{\frac{d^{k}-1}{d-1}}\left(\frac{1}{m},\frac{1}{m},\ldots\frac{1}{m},\frac{1}{m}\right)$
$\displaystyle=m^{\frac{d^{k-1}+d-2}{d-1}}\left(\frac{\lambda}{\eta}\right)^{\frac{d^{k}-1}{d-1}}\left(\frac{1}{m},\frac{1}{m},\ldots\frac{1}{m},\frac{1}{m}\right).$
Let $\lambda=1$. If $\rho=\frac{m\lambda}{\eta}<1$, then this supermarket
model is stable. In the stable case, $\eta>m$. We may consider the following
simple cases:
(a) If $m=2$ and $d=2$, then $\pi_{k}=2^{2^{k-1}}\eta^{1-2^{k}}$.
(b) If $m=3$ and $d=2$, then $\pi_{k}=3^{2^{k-1}}\eta^{1-2^{k}}$.
Based on the two simple examples with $\lambda=1$ and $d=2$, we need to
illustrate how the fixed point depends on the stage number $m$ and the
exponential service rate $\eta$. To that end, we write
$\pi_{k}\left(m,\eta\right)$. It is easy to see that for a given pair
$\left(k,\eta\right)$ for $\eta>m$ and $k=1,2,\ldots,$ we have
$\pi_{k}\left(1,\eta\right)<\pi_{k}\left(2,\eta\right)<\cdots<\pi_{k}\left(m,\eta\right)<\cdots.$
On the other hand, for a given pair $\left(k,m\right)$ for $m,k=1,2,\ldots,$
we can see that $\pi_{k}\left(m,\eta\right)$ is a decreasing function of
$\eta$.
Let us consider the average response time of the supermarket model with an
$m-$stage Erlang distribution. We first consider a parallel system with
$n=100$ servers and the service time distribution is exponential. We normalize
the average service time to unity and vary the arrival rate $\lambda$. For the
$m-$stage Erlang distribution, the bigger the number $m$ is, the bigger its
variance is. Table 2 illustrates the average response time under different
probe size $d$. One can observe that there is a dramatic improvement (or
reduction) in the average response time when increasing the probe size $d$.
Table 2: Average response time for exponential service time number of servers ($n$) | probe size ($d$) | arrival rate ($\lambda$) | response time ($E[\mathcal{T}]$)
---|---|---|---
100 | 2 | 0.500000 | 1.395977
100 | 2 | 0.700000 | 1.768194
100 | 2 | 0.800000 | 2.072020
100 | 2 | 0.900000 | 2.721852
100 | 3 | 0.500000 | 1.395320
100 | 3 | 0.700000 | 1.604113
100 | 3 | 0.800000 | 1.802933
100 | 3 | 0.900000 | 2.209601
100 | 5 | 0.900000 | 1.916280
We further analyze the cases that the service time is either distributed
according to $2$-stage Erlang or $3$-stage Erlang distribution. Similarly, we
normalized the total average service time as unity and we vary the arrival
rate $\lambda$. Tables 3 and 4 illustrate the average response time under
different probe size $d$. One can observe that
* •
Simple probing size $d$ can significantly improve the performance by lowering
the average response time.
* •
When the service time has lower variance, the average response time is lower.
Table 3: Average response time for $2-$stage Erlang service time number of servers ($n$) | probe size ($d$) | arrival rate ($\lambda$) | response time ($E[\mathcal{T}]$)
---|---|---|---
100 | 2 | 0.500000 | 1.353783
100 | 2 | 0.700000 | 1.599851
100 | 2 | 0.800000 | 1.829199
100 | 2 | 0.900000 | 2.298470
100 | 3 | 0.500000 | 1.325610
100 | 3 | 0.700000 | 1.492651
100 | 3 | 0.800000 | 1.639987
100 | 3 | 0.900000 | 1.941196
100 | 5 | 0.900000 | 1.739867
Table 4: Average response time for $3-$stage Erlang service time number of servers ($n$) | probe size ($d$) | arrival rate ($\lambda$) | response time ($E[\mathcal{T}]$)
---|---|---|---
100 | 2 | 0.500000 | 1.322544
100 | 2 | 0.700000 | 1.539621
100 | 2 | 0.800000 | 1.739972
100 | 2 | 0.900000 | 2.148191
100 | 3 | 0.500000 | 1.298863
100 | 3 | 0.700000 | 1.452785
100 | 3 | 0.800000 | 1.581663
100 | 3 | 0.900000 | 1.834704
100 | 5 | 0.900000 | 1.678233
Example two (Hyper-Exponential Distribution) We consider an $m$-order hyper-
exponential distribution
$F\left(x\right)=1-\sum\limits_{k=1}^{m}\alpha_{k}\exp\left\\{-\eta_{k}x\right\\}$,
or the probability density function
$f\left(x\right)=\sum\limits_{k=1}^{m}\alpha_{k}\eta_{k}\exp\left\\{-\eta_{k}x\right\\}$.
It is clear that the hyper-exponential distribution is of phase type with the
irreducible representation $(\alpha,T)$, where
$\alpha=\left(\alpha_{1},\alpha_{2},\ldots,\alpha_{m}\right)$, and
$T=\left(\begin{array}[]{cccc}-\eta_{1}&&&\\\ &-\eta_{2}&&\\\ &&\ddots&\\\
&&&-\eta_{m}\end{array}\right),\text{ \
}T^{0}=\left(\begin{array}[]{c}\eta_{1}\\\ \eta_{2}\\\ \vdots\\\
\eta_{m}\end{array}\right),$
which lead to
$T+T^{0}\alpha=\left(\begin{array}[]{cccc}-\eta_{1}\left(1-\alpha_{1}\right)&\eta_{1}\alpha_{2}&\cdots&\eta_{1}\alpha_{m}\\\
\eta_{2}\alpha_{1}&-\eta_{2}\left(1-\alpha_{2}\right)&\cdots&\eta_{2}\alpha_{m}\\\
\vdots&\vdots&&\vdots\\\
\eta_{m}\alpha_{1}&\eta_{m}\alpha_{2}&\cdots&-\eta_{m}\left(1-\alpha_{m}\right)\end{array}\right).$
In general, the system of equations $\omega\left(T+T^{0}\alpha\right)=0$ and
$\omega e=1$ does not admit a simple analytic solution. For a convenient
description, we only consider a simple one with $m=2$. In this case, we obtain
$\omega=\left(\frac{\alpha_{1}\eta_{2}}{\alpha_{1}\eta_{2}+\alpha_{2}\eta_{1}},\frac{\alpha_{2}\eta_{1}}{\alpha_{1}\eta_{2}+\alpha_{2}\eta_{1}}\right),\hskip
14.45377pt\mu=\frac{\eta_{1}\eta_{2}\left(\alpha_{1}+\alpha_{2}\right)}{\alpha_{1}\eta_{2}+\alpha_{2}\eta_{1}},$
$\rho=\frac{\lambda}{\mu}=\frac{\lambda\left(\alpha_{1}\eta_{2}+\alpha_{2}\eta_{1}\right)}{\eta_{1}\eta_{2}\left(\alpha_{1}+\alpha_{2}\right)},\hskip
7.22743pt\theta=\left(\frac{\alpha_{1}\eta_{2}}{\alpha_{1}\eta_{2}+\alpha_{2}\eta_{1}}\right)^{d}+\left(\frac{\alpha_{2}\eta_{1}}{\alpha_{1}\eta_{2}+\alpha_{2}\eta_{1}}\right)^{d}$
and
$\displaystyle\pi_{k}=$
$\displaystyle\left[\left(\frac{\alpha_{1}\eta_{2}}{\alpha_{1}\eta_{2}+\alpha_{2}\eta_{1}}\right)^{d}+\left(\frac{\alpha_{2}\eta_{1}}{\alpha_{1}\eta_{2}+\alpha_{2}\eta_{1}}\right)^{d}\right]^{\frac{d^{k-1}-1}{d-1}}$
$\displaystyle\cdot\left[\frac{\lambda\left(\alpha_{1}\eta_{2}+\alpha_{2}\eta_{1}\right)}{\eta_{1}\eta_{2}\left(\alpha_{1}+\alpha_{2}\right)}\right]^{\frac{d^{k}-1}{d-1}}\left(\frac{\alpha_{1}\eta_{2}}{\alpha_{1}\eta_{2}+\alpha_{2}\eta_{1}},\frac{\alpha_{2}\eta_{1}}{\alpha_{1}\eta_{2}+\alpha_{2}\eta_{1}}\right).$
Tables 5 and 6 indicate how the doubly exponential solution ($\pi_{1}$ to
$\pi_{5}$) depends on the vectors $\eta=\left(\eta_{1},\eta_{2}\right)$ and
$\alpha=\left(\alpha_{1},\alpha_{2}\right)$, respectively.
Table 5: The doubly exponential solution depends on $\eta$ | $\eta=(3,3)$ | $\eta=(3,10)$ | $\eta=(3,20)$
---|---|---|---
$\pi_{1}$ | (0.1667, 0.1667) | (0.1667, 0.0500) | (0.1667, 0.0250)
$\pi_{2}$ | (0.0093, 0.0093) | (0.0050, 0.0015) | (0.0047, 0.0007)
$\pi_{3}$ | (2.858e-05, 2.858e-05) | (4.626e-06, 1.388e-06) | (3.819e-06, 5.728e-07)
$\pi_{4}$ | (2.722e-10, 2.722e-10) | (3.888e-12, 1.166e-12) | (2.485e-12, 3.728e-13)
$\pi_{5}$ | (2.470e-20, 2.470e-20) | (2.746e-24, 8.238e-25) | (1.053e-24, 1.579e-25)
Table 6: The doubly exponential solution depends on $\alpha$ | $\alpha=(0.5,\;0.5)$ | $\alpha=(0.2,\;0.8)$ | $\alpha=(0.8,\;0.2)$
---|---|---|---
$\pi_{1}$ | (0.1667, 0.1667) | (0.0667, 0.0267) | (0.2667, 0.0067)
$\pi_{2}$ | (0.0047, 0.0005) | (0.0003, 0.0001) | (0.0190, 0.0005)
$\pi_{3}$ | (3.680e-06, 3.680e-07) | (9.136e-09, 3.654e-09) | (9.607e-05, 2.402e-06)
$\pi_{4}$ | (2.280e-12, 2.280e-13) | (6.454e-18, 2.582e-18) | (2.463e-09, 6.157e-11)
$\pi_{5}$ | (8.752e-25, 8.752e-26) | (3.221e-36, 1.289e-36) | (1.618e-18, 4.046e-20)
Let us consider the average response time of the supermarket model with an
$m$-stage hyper-exponential service time distribution. We consider a parallel
system with $n=100$ servers and the probability density function of the
service time of a customer is given by
$f(x)=0.5\times(2\times e^{-2x})+0.25\times(0.5\times
e^{-0.5x})+0.25\times(e^{-x}).$
Note that the total average service time is normalized to unity and we vary
the arrival rate $\lambda$. Table 7 illustrates the average response time
under different probe size $d$. One can observe that there is a dramatic
reduction in the average response time when increasing the probe size.
Furthermore, when the service time has a higher variance (we here compare it
with the exponential distribution or $m-$stage Erlang distribution), the
average service time is much higher. This indicates that we improve the
performance of the supermarket model, one has to increase the probe size $d$.
Table 7: Average response time for $3-$stage Hyper-exponential service time number of servers ($n$) | probe size ($d$) | arrival rate ($\lambda$) | response time ($E[\mathcal{T}]$)
---|---|---|---
100 | 2 | 0.500000 | 1.552282
100 | 2 | 0.700000 | 1.969132
100 | 2 | 0.800000 | 2.360255
100 | 2 | 0.900000 | 3.225117
100 | 3 | 0.500000 | 1.462128
100 | 3 | 0.700000 | 1.723764
100 | 3 | 0.800000 | 1.947548
100 | 3 | 0.900000 | 2.476718
100 | 5 | 0.900000 | 2.066462
Example three (PH Distribution) We consider an $m$-order PH distribution with
irreducible representation $\left(\alpha,T\right)$. For
$m=2,d=2,\alpha=\left(1/2,1/2\right)$ and
$T\left(1\right)=\left(\begin{array}[]{cc}-4&3\\\
2&-7\end{array}\right),T\left(2\right)=\left(\begin{array}[]{cc}-5&3\\\
2&-7\end{array}\right),T\left(3\right)=\left(\begin{array}[]{cc}-4&4\\\
2&-7\end{array}\right),$
Table 8 illustrates how the doubly exponential solution depends on the PH
matrices $T\left(1\right)$, $T\left(2\right)$ and $T\left(3\right)$,
respectively.
Table 8: The doubly exponential solution depends on the PH matrices $T(i)$ | $T(1)$ | $T(2)$ | $T(3)$
---|---|---|---
$\pi_{1}$ | (0.2045, 0.1591) | (0.1410, 0.1026) | (0.3125, 0.2500)
$\pi_{2}$ | (0.0137, 0.0107) | (0.0043, 0.0031) | (0.0500, 0.0400)
$\pi_{3}$ | (6.193e-05, 4.817e-05) | (3.965e-06, 2.884e-06) | (0.0013 , 0.0010)
$\pi_{4}$ | (1.259e-09, 9.793e-10) | (3.390e-12, 2.465e-12) | (8.446e-07, 6.757e-07)
$\pi_{5}$ | (5.204e-19, 4.048e-19) | (2.478e-24, 1.802e-24) | (3.656e-13, 2.925e-13)
To discuss how different caused by a non-exponential distribution versus an
exponentially distributed service time with the same mean, for the above three
PH distributions we take three corresponding exponential distributions with
service rates $\mu(1)=2.7500,\mu(2)=3.4118$ and $\mu(3)=2.3529$, respectively.
Table 9 illustrates how the doubly exponential solution ($\pi_{1}$ to
$\pi_{5}$) depends on the three service rates. Since the exponential
distribution has a lower variance than the PH distribution, it is seen from
Tables 8 and 9 that the service time has lower variance,
$\pi_{k}$(Exp)$<\pi_{k}$(PH)$e$.
Table 9: The doubly exponential solution depends on exponential service rates $\mu(i)$ | $\mu(1)=2.7500$ | $\mu(2)=3.4118$ | $\mu(3)=2.3529$
---|---|---|---
$\pi_{1}$ | 0.3636 | 0.2931 | 0.4250
$\pi_{2}$ | 0.0481 | 0.0252 | 0.0768
$\pi_{3}$ | 8.408e-04 | 1.858e-04 | 0.0025
$\pi_{4}$ | 2.571e-07 | 1.012e-08 | 2.667e-06
$\pi_{5}$ | 2.402e-14 | 3.004e-17 | 3.030e-12
For the PH and exponential service times, the following two figures provides a
comparison for the expected sojourn time. Clearly, the PH service time makes
the lower expected sojourn time.
Figure 2: $E\left[T_{d}\right]$s of the PH and exponential distributions for
$T(1)$ and $T(2)$, respectively
For $m=3,d=5,\alpha\left(1\right)=\left(1/3,1/3,1/3\right)$ and
$\alpha\left(2\right)=\left(1/12,7/12,1/3\right)$,
$T=\left(\begin{array}[]{ccc}-10&2&4\\\ 3&-7&4\\\ 0&2&-5\end{array}\right).$
Table 10 shows how the doubly exponential solution ($\pi_{1}$ to $\pi_{4}$)
depends on the vectors $\alpha\left(1\right)$ and $\alpha\left(2\right)$,
respectively.
Table 10: The doubly exponential solution depends on the vectors $\alpha$ | $\alpha=(\frac{1}{3},\frac{1}{3},\frac{1}{3})$ | $\alpha=(\frac{1}{12},\frac{7}{12},\frac{1}{3})$
---|---|---
$\pi_{1}$ | (0.0741, 0.1358 , 0.2346) | (0.0602, 0.1728, 0.2531)
$\pi_{2}$ | (5.619e-05, 1.030e-05, 1.779e-04 ) | (7.182e-05, 2.063e-04, 3.020e-04)
$\pi_{3}$ | (1.411e-20, 2.587e-20, 4.469e-20) | (1.739e-19, 4.993e-19, 7.311e-19)
$\pi_{4}$ | (1.410e-98, 2.586e-98, 4.466e-98) | (1.444e-92, 4.148e-92, 6.074e-92)
## 6 Concluding remarks
In this paper, we provide a matrix-analytic solution for supermarket models.
We describe the supermarket model with PH service times as a system of
differential vector equations, and provide a doubly exponential solution to
the fixed point of the system of differential vector equations. We also
provide some numerical examples to illustrate that our approach is effective
and efficient in the study of randomized load balancing schemes with non-
exponential service requirements, such as, Erlang service time distributions,
hyper-exponential service time distributions and PH service time
distributions. We expect that this approach will be applicable to study other
randomized load balancing schemes, for example, generalizing the arrival
process to non-Poisson such as renewal process or Markovian arrival process,
generalizing the service times to general probability distributions, and
analyzing retrial and processor-sharing service disciplines.
## References
* [1] Y. Azar, A.Z. Broder, A.R. Karlin and E. Upfal (1999). Balanced allocations. SIAM Journal on Computing 29, 180–200. A preliminary version of this paper appeared in Proceedings of the Twenty-Sixth Annual ACM Symposium on the Theory of Computing, 1994.
* [2] M. Bramson, Y. Lu and B. Prabhakar (2010). Randomized load balancing with general service time distributions. In Proceedings of the ACM SIGMETRICS international conference on Measurement and modeling of computer systems, pages 275–286.
* [3] M. Dahlin (1999). Interpreting stale load information. IEEE Transactions on Parallel and Distributed Systems 11, 1033 - 1047.
* [4] D.L. Eager, E.D. Lazokwska and J. Zahorjan (1986). Adaptive load sharing in homogeneous distributed systems. IEEE Transactions on Software Engineering 12, 662–675.
* [5] D.L. Eager, E.D. Lazokwska and J. Zahorjan (1986). A comparison of receiver-initiated and sender-initiated adaptive load sharing. Performance Evaluation Review 6, 53–68.
* [6] D.L. Eager, E.D. Lazokwska and J. Zahorjan (1988). The limited performance benefits of migrating active processes for load sharing. Performance Evaluation Review 16, 63–72.
* [7] M. Harchol-Balter, A.B. Downey (1997). Exploiting process lifetime distributions for dynamic load balancing. ACM Transactions on Computer Systems 15, 253–285.
* [8] M. Luczak and C. McDiarmid (2006). On the maximum queue length in the supermarket model. The Annals of Probability 34, 493–527.
* [9] J.B. Martin (2001). Point processes in fast Jackson networks. Annals of Applied Probability 11, 650-663.
* [10] J.B. Martin and Y.M Suhov (1999). Fast Jackson networks. Annals of Applied Probability 9, 854–870.
* [11] M.D. Mitzenmacher (1996). Load balancing and density dependent jump Markov processes. In Proceedings of the Thirty-Seventh Annual Symposium on Foundations of Computer Science, pages 213–222.
* [12] M.D. Mitzenmacher (1996). The power of two choices in randomized load balancing. PhD thesis, University of California at Berkeley, Department of Computer Science, Berkeley, CA, 1996.
* [13] M. Mitzenmacher (1998). Analyses of load stealing models using differential equations. In Proceedings of the Tenth ACM Symposium on Parallel Algorithms and Architectures, pages 212–221.
* [14] M. Mitzenmacher (1999). On the analysis of randomized load balancing schemes. Theory of Computing Systems 32, 361–386.
* [15] M. Mitzenmacher (1999). Studying balanced allocations with differential equations. Combinatorics, Probability, and Computing 8, 473–482.
* [16] M. Mitzenmacher (2000). How useful is old information? IEEE Transactions on Parallel and Distributed Systems 11, 6–20.
* [17] M. Mitzenmacher (2001). The power of two choices in randomized load balancing. IEEE Transactions on Parallel and Distributed Computing 12, 1094-1104.
* [18] M. Mitzenmacher, A. Richa, and R. Sitaraman (2001). The power of two random choices: a survey of techniques and results. In Handbook of Randomized Computing: volume 1, edited by P. Pardalos, S. Rajasekaran and J. Rolim, pages 255-312.
* [19] M. Mitzenmacher and B. Vöcking (1998). The asymptotics of selecting the shortest of two, improved. In Proceedings of the 37th Annual Allerton Conference on Communication, Control, and Computing, pages 326–327. A full version is available as Harvard Computer Science TR-08-99.
* [20] R. Mirchandaney, D. Towsley, and J.A. Stankovic (1989). Analysis of the effects of delays on load sharing. IEEE Transactions on Computers 38, 1513–1525.
* [21] Y.M. Suhov and N.D. Vvedenskaya (2002). Fast Jackson Networks with Dynamic Routing. Problems of Information Transmission 38, 136{153.
* [22] M. Telek and A. Heindl (2002). Matching moments for acyclic discrete and continuous phase-type distributions of second order. International Journal of Simulation: Systems, Science & Technology 3, 47–57.
* [23] B. Vöcking (1999). How asymmetry helps load balancing. In Proceedings of the Fortieth Annual Symposium on Foundations of Computer Science, pages 131–140.
* [24] N.D. Vvedenskaya, R.L. Dobrushin and F.I. Karpelevich. (1996). Queueing system with selection of the shortest of two queues: An asymptotic approach. Problems of Information Transmissions 32, 20–34.
* [25] N.D. Vvedenskaya and Y.M. Suhov (1997). Dobrushin’s mean-field approximation for a queue with dynamic routing. Markov Processes and Related Fields 3, 493–526.
* [26] S. Zhou (1988). A trace-driven simulation study of dynamic load balancing. IEEE Transactions on Software Engineering 14, 1327–1341.
|
arxiv-papers
| 2011-01-22T18:35:45 |
2024-09-04T02:49:16.578805
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Quan-Lin Li, John C.S. Lui, Yang Wang",
"submitter": "Quan-Lin Li",
"url": "https://arxiv.org/abs/1101.4306"
}
|
1101.4435
|
# Solutions for the MIMO Gaussian Wiretap Channel with a Cooperative
Jammer††thanks: The authors are with the Dept. of Electrical Engineering and
Computer Science, University of California, Irvine, CA 92697-2625, USA.
e-mail:{afakoori, swindle}@uci.edu††thanks: This work was supported by the
U.S. Army Research Office under the Multi-University Research Initiative
(MURI) grant W911NF-07-1-0318.
S. Ali. A. Fakoorian*, Student Member, IEEE and A. Lee Swindlehurst, Fellow,
IEEE
###### Abstract
We study the Gaussian MIMO wiretap channel with a transmitter, a legitimate
receiver, an eavesdropper and an external helper, each equipped with multiple
antennas. The transmitter sends confidential messages to its intended
receiver, while the helper transmits jamming signals independent of the source
message to confuse the eavesdropper. The jamming signal is assumed to be
treated as noise at both the intended receiver and the eavesdropper. We obtain
a closed-form expression for the structure of the artificial noise covariance
matrix that guarantees no decrease in the secrecy capacity of the wiretap
channel. We also describe how to find specific realizations of this covariance
matrix expression that provide good secrecy rate performance, even when there
is no non-trivial null space between the helper and the intended receiver.
Unlike prior work, our approach considers the general MIMO case, and is not
restricted to SISO or MISO scenarios.
###### Index Terms:
Physical-layer security, interference channel, MIMO wiretap channel,
cooperative jamming.
EDICS: WIN-CONT, WIN-PHYL, WIN-INFO, MSP-CAPC
## I Introduction
Recent information-theoretic research on secure communication has focused on
enhancing security at the physical layer. The wiretap channel, first
introduced and studied by Wyner [1], is the most basic physical layer model
that captures the problem of communication security. This work led to the
development of the notion of perfect secrecy capacity, which quantifies the
maximum rate at which a transmitter can reliably send a secret message to its
intended recipient, without it being decoded by an eavesdropper. The Gaussian
wiretap channel, in which the outputs of the legitimate receiver and the
eavesdropper are corrupted by additive white Gaussian noise, was studied in
[2]. The secrecy capacity of a Gaussian wiretap channel, which is in general a
difficult non-convex optimization problem, has been addressed and solved for
in [3]-[7]. The secrecy capacity under an average power constraint is treated
in [4] and [5], where in [4] a beamforming approach, based on the generalized
singular value decomposition (GSVD), is proposed that achieves the secrecy
capacity in the high SNR regime. In [5], we propose an optimal power
allocation that achieves the secrecy capacity of the GSVD-based multiple-
input, multiple-output (MIMO) Gaussian wiretap channel for any SNR. In [7], a
closed-form expression for the secrecy capacity is derived under a certain
power-covariance constraint.
It was shown in [8] that, for a wiretap channel without feedback, a non-zero
secrecy capacity can only be obtained if the eavesdropper’s channel is of
lower quality than that of the intended recipient. Otherwise, it is infeasible
to establish a secure link under Wyner’s wiretap channel model. In such
situations, one approach is to exploit user cooperation in facilitating the
transmission of confidential messages from the source to the destination. In
[9]-[13], for example, a four-terminal relay-eavesdropper channel is
considered, where a source wishes to send messages to a destination while
leveraging the help of a relay/helper node to hide the messages from the
eavesdropper. While the relay can assist in the transmission of confidential
messages, its computational cost may be prohibitive and there are difficulties
associated with the coding and decoding schemes at both the relay and the
intended receiver. Alternatively, a cooperating node can be used as a helper
that simply transmits jamming signals, independent of the source message, to
confuse the eavesdropper and increase the range of channel conditions under
which secure communications can take place. The strategy of using a helper to
improve the secrecy of the source-destination communication is generally known
as cooperative jamming [9, 11] or noise-forwarding [12] in prior work.
In [9], the scenario where multiple single-antenna users communicate with a
common receiver (i.e., the multiple access channel) in the presence of an
eavesdropper is considered, and the optimal transmit power allocation that
achieves the maximum secrecy sum-rate ia obtained. The work of [9] shows that
any user prevented from transmitting based on the obtained power allocation
can help increase the secrecy rate for other users by transmitting artificial
noise to the eavesdropper (cooperative jamming). In [11], a source-destination
system in the presence of multiple helpers and multiple eavesdroppers is
considered, where the helpers can transmit weighted jamming signals to degrade
the eavesdropper’s ability to decode the source. While the objective is to
select the weights so as to maximize the secrecy rate under a total power
constraint, or to minimize the total power under a secrecy rate constraint,
the results in [11] yield sub-optimal weights for both single and multiple
eavesdroppers, due to the assumption that the jamming signal must be nulled at
the destination. The noise forwarding scheme of [12] requires that the
interferer’s codewords be decoded by the intended receiver. A generalization
of [9, 11] and [12] is proposed in [13], in which the helper’s codewords do
not have to be decoded by the receiver.
The prior work in [9]-[13] assumes single antenna nodes and models single-
input, single-output (SISO) or multiple-input, single-output (MISO) cases. A
more general MIMO case with multiple cooperative jammers was studied in [14],
in which the jammers aligned their interference to lie within a pre-specified
“jamming subspace” at the receiver, but the dimensions of the subspace and the
power allocation were not optimized. In this paper, we also address the
general MIMO case, where the transmitter, legitimate receiver, eavesdropper
and helper are in general all equipped with multiple antennas. The transmitter
sends confidential messages to its intended receiver, while the helper node
assists the transmitter by sending jamming signals independent of the source
message to confuse the eavesdropper. While the previous work on this problem
shows the fundamental role of jamming as a means to increase secrecy rates, it
also emphasizes the fact that that non-carefully designed jamming strategies
can preclude secure communication [15].
In this work, we derive a closed-form expression for the structure of the
artificial noise covariance matrix of a cooperating jammer that guarantees no
decrease in the secrecy capacity of the wiretap channel, assuming the jamming
signal from the helper is treated as noise at both the intended receiver and
the eavesdropper. We describe algorithms for finding specific realizations of
this covariance expression that provide good secrecy rate performance, and
show that even when there is no non-trivial nullspace between the helper and
the intended receiver, the helper can still transmit artificial noise that
does not impact the mutual information between the transmitter and the
intended receiver, while decreasing the mutual information between the
transmitter and the eavesdropper. Hence, the secrecy level of the confidential
message is increased. The situation we consider is different from the one in
[16], where the transmitter itself rather than an external helper broadcasts
artificial noise to degrade the eavesdropper’s channel. However, both
approaches are able to achieve a positive perfect secrecy rate in scenarios
where the secrecy capacity in the absence of jamming is zero.
The remainder of the paper is organized as follows. In Section II, we describe
the system model for the helper-assisted Gaussian MIMO wiretap channel and
formulate the problem to be solved. In Sections III and IV, we derive the
artificial noise covariance matrix that guarantees no decrease in the secrecy
capacity of the wiretap channel. Numerical results in Section V are presented
to illustrate the proposed solution. Finally, Section VI concludes the paper.
Notation: Throughout the paper, we use boldface uppercase letters to denote
matrices. Vector-valued random variables are written with non-boldface
uppercase letters (e.g., $X$), while the corresponding lowercase boldface
letter (${\mathbf{x}}$) denotes a specific realization of the random variable.
Scalar variables are written with non-boldface (lowercase or uppercase)
letters. We use $(.)^{T}$ to represent matrix transposition, $(.)^{H}$ the
Hermitian (i.e., conjugate) transpose, Tr(.) the matrix trace, $E$ the
expectation operator, I the identity matrix, and 0 a matrix or vector with all
zeros. Mutual information between the random variables $A$ and $B$ is denoted
by $I(A;B)$, and $\mathcal{CN}(0,1)$ represents the complex circularly
symmetric Gaussian distribution with zero mean and unit variance.
## II System Model
We consider a MIMO wiretap channel that includes a transmitter, an intended
receiver, a helping interferer and an eavesdropper, with $n_{t}$, $n_{r}$,
$n_{h}$ and $n_{e}$ antennas, respectively. The transmitter sends a
confidential message to the intended receiver with the aid of the helper, in
the presence of an eavesdropper. We assume that the helper does not know the
confidential message and transmits only a Gaussian jamming signal which is not
known at the intended receiver nor the eavesdropper and which is treated as
noise at both receivers. The mathematical model for this scenario is given by:
$\displaystyle{\mathbf{y}}_{1}$ $\displaystyle=$
$\displaystyle{\mathbf{H}}_{1}{\mathbf{x}}_{1}+{\mathbf{G}}_{2}{\mathbf{x}}_{2}+{\mathbf{z}}_{1}$
(1) $\displaystyle{\mathbf{y}}_{2}$ $\displaystyle=$
$\displaystyle{\mathbf{H}}_{2}{\mathbf{x}}_{2}+{\mathbf{G}}_{1}{\mathbf{x}}_{1}+{\mathbf{z}}_{2}\;,$
(2)
where ${\mathbf{x}}_{1}$ is a zero-mean $n_{t}\times 1$ transmitted signal
vector, ${\mathbf{x}}_{2}$ is a zero-mean $n_{h}\times 1$ jamming vector
transmitted by the helper, and ${\mathbf{z}}_{1}\in\mathbb{C}^{n_{r}\times
1}$, ${\mathbf{z}}_{2}\in\mathbb{C}^{n_{e}\times 1}$ are additive white
Gaussian noise (AWGN) vectors at the intended receiver and the eavesdropper,
respectively, with i.i.d. entries distributed as $\mathcal{CN}(0,1)$. The
matrices ${\mathbf{H}}_{1},{\mathbf{G}}_{1}$ represent the channels from the
transmitter to the intended receiver and eavesdropper, respectively, while
${\mathbf{H}}_{2},{\mathbf{G}}_{2}$ are the channels from the helper to the
eavesdropper and intended receiver, respectively. The channels are assumed to
be independent of each other and full rank with arbitrary dimensions. We also
assume that the transmitter has full channel state information and is aware of
the effective noise covariance at both receivers, where the effective noise is
the background noise plus the received artificial noise. Both the helper and
the eavesdropper are also aware of all channel matrices as well.
The jamming signal transmitted by the helper satisfies an average power
constraint:
$\text{Tr}(E\\{X_{2}X_{2}^{H}\\})=\text{Tr}({\mathbf{K}}_{w})\leq P_{h}$ (3)
where $X_{2}$ is the random variable associated with the specific realization
${\mathbf{x}}_{2}$ and ${\mathbf{K}}_{w}$ is the corresponding covariance
matrix. The channel input is subject to a matrix power constraint [7, 17]
$E\\{X_{1}X_{1}^{H}\\}={\mathbf{K}}_{x}\preceq{\mathbf{S}}$ (4)
where ${\mathbf{K}}_{x}$ is the input covariance matrix, ${\mathbf{S}}$ is a
positive semi-definite matrix, and “$\preceq$” denotes that
${\mathbf{S}}-{\mathbf{K}}_{x}$ is positive semi-definite. Note that (4) is a
rather general power constraint that subsumes many other important power
constraints, including the average total and per-antenna power constraints as
special cases. The approach developed in this paper will assume that $P_{h}$
and ${\mathbf{S}}$ (or $\text{Tr}({\mathbf{S}})\leq P_{t}$) are fixed, and
that power is not allocated jointly between the transmitter and helper. The
numerical results presented later, however, will illustrate the trade-off
associated with the power allocation when $P_{h}+P_{t}$ is fixed.
As mentioned before, we assume Gauusian signaling for the helper. Thus the
effective noise at both receivers is Gaussian and consequently the above MIMO
wiretap channel model is Gaussian. For this case, a Gaussian input signal is
the optimal choice [6, 17]. Hence, the general optimization problem is
equivalent to finding the matrices ${\mathbf{K}}_{x}\succeq 0$ and
${\mathbf{K}}_{w}\succeq 0$ that allow the secrecy capacity of the network to
be obtained. A matrix characterization of this optimization problem is given
by:
$\displaystyle C_{sec}$ $\displaystyle=$
$\displaystyle\max_{{\mathbf{K}}_{x}\succeq 0,{\mathbf{K}}_{w}\succeq
0}[I(X_{1};Y_{1})-I(X_{1};Y_{2})]$ (5) $\displaystyle=$
$\displaystyle\max_{{\mathbf{K}}_{x}\succeq 0,{\mathbf{K}}_{w}\succeq
0}\log|{\mathbf{K}}_{x}{\mathbf{H}}_{1}^{H}({\mathbf{G}}_{2}{\mathbf{K}}_{w}{\mathbf{G}}_{2}^{H}+\textbf{I})^{-1}{\mathbf{H}}_{1}+\textbf{I}|$
$\displaystyle\qquad\qquad\quad-\log|{\mathbf{K}}_{x}{\mathbf{G}}_{1}^{H}({\mathbf{H}}_{2}{\mathbf{K}}_{w}{\mathbf{H}}_{2}^{H}+\textbf{I})^{-1}{\mathbf{G}}_{1}+\textbf{I}|\;,$
where the non-convex maximization problem in carried out under the power
constraints given in (3) and (4). Lemma 1: For a given ${\mathbf{K}}_{w}$, the
maximum of (5) is given by
$C_{sec}({\mathbf{S}})=\sum_{i=1}^{\rho}\log\gamma_{i}$ (6)
where $\gamma_{i}$, $i=1,\cdots,\rho$, are the generalized eigenvalues of the
pencil
$({\mathbf{S}}^{\frac{1}{2}}{\mathbf{H}}_{1}^{H}({\mathbf{G}}_{2}{\mathbf{K}}_{w}{\mathbf{G}}_{2}^{H}+\textbf{I})^{-1}{\mathbf{H}}_{1}{\mathbf{S}}^{\frac{1}{2}}+\textbf{I},\quad{\mathbf{S}}^{\frac{1}{2}}{\mathbf{G}}_{1}^{H}({\mathbf{H}}_{2}{\mathbf{K}}_{w}{\mathbf{H}}_{2}^{H}+\textbf{I})^{-1}{\mathbf{G}}_{1}{\mathbf{S}}^{\frac{1}{2}}+\textbf{I})$
(7)
that are greater than 1. Proof: When the optimization problem in (5) is
performed over ${\mathbf{K}}_{x}$ under the matrix power constraint (4) for a
given ${\mathbf{K}}_{w}$, it is equivalent to a simple MIMO Gaussian wiretap
channel without a helper, where the noise covariance matrices at the receiver
and the eavesdropper are
$({\mathbf{G}}_{2}{\mathbf{K}}_{w}{\mathbf{G}}_{2}^{H}+\textbf{I})$ and
$({\mathbf{H}}_{2}{\mathbf{K}}_{w}{\mathbf{H}}_{2}^{H}+\textbf{I})$,
respectively. The above lemma is a natural extension of [7] and [17, Theorem
3] for the standard MIMO Gaussian wiretap channel.
Note that since both elements of the pencil (7) are strictly positive
definite, all of the generalized eigenvalues are real and positive [17, 18].
In (6), a total of $\rho$ of them are assumed to be greater than one. Clearly,
if there are no such eigenvalues, then the information signal received at the
intended receiver is a degraded version of that of the eavesdropper, and in
this case the secrecy capacity is zero. Note also that Lemma 1 only provides
the secrecy capacity for the optimal ${\mathbf{K}}_{x}$, but does not give an
explicit expression for this ${\mathbf{K}}_{x}$. A general expression for the
maximizing ${\mathbf{K}}_{x}$ will be given in the next section.
To solve the general optimization problem in (5), we would need to find the
${\mathbf{K}}_{w}$ that maximizes (6). Unfortunately, this appears to be a
very difficult problem to solve without resorting to some type of ad hoc
search. In the following we obtain a sub-optimal closed-form solution for the
artificial noise covariance matrix ${\mathbf{K}}_{w}$ that guarantees no
decrease in the mutual information between the transmitter and the intended
receiver compared with the case where ${\mathbf{K}}_{w}=\mathbf{0}$, while
maintaining the power constraint in (5). Hence, the new non-zero
${\mathbf{K}}_{w}$ will only interfere with the eavesdropper, and the secrecy
level of the confidential message will be increased. Once such a
${\mathbf{K}}_{w}$ is found, additional improvement in the secrecy rate can be
achieved if the transmitter updates its covariance matrix ${\mathbf{K}}_{x}$
for the obtained ${\mathbf{K}}_{w}$. The final secrecy rate for this method is
obtained by simply computing (6) and (7) for the resulting ${\mathbf{K}}_{w}$.
Note that we will not propose an iterative algorithm that would further
alternate between calculating ${\mathbf{K}}_{x}$ and ${\mathbf{K}}_{w}$. We
will see in the next section that there is no clear way to update
${\mathbf{K}}_{w}$ from a known non-zero value.
## III Analytical Method
We begin with the case where the helper transmits no signal
$({\mathbf{K}}_{w}=0)$. In this case, the communication system is reduced to a
simple MIMO Gaussian wiretap channel without helper. Based on Lemma 1, the
maximum of (5) when ${\mathbf{K}}_{w}=0$ is obtained by applying the
generalized eigenvalue decomposition to the following two Hermitian positive
definite matrices [7, 17]:
${\mathbf{S}}^{\frac{1}{2}}{\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}{\mathbf{S}}^{\frac{1}{2}}+\textbf{I},\quad{\mathbf{S}}^{\frac{1}{2}}{\mathbf{G}}_{1}^{H}{\mathbf{G}}_{1}{\mathbf{S}}^{\frac{1}{2}}+\textbf{I}\;.$
In particular, there exists an invertible generalized eigenvector matrix
${\mathbf{C}}$ such that [18]
${\mathbf{C}}^{H}\left[{\mathbf{S}}^{\frac{1}{2}}{\mathbf{G}}_{1}^{H}{\mathbf{G}}_{1}{\mathbf{S}}^{\frac{1}{2}}+\textbf{I}\right]{\mathbf{C}}=\textbf{I}$
(8)
${\mathbf{C}}^{H}\left[{\mathbf{S}}^{\frac{1}{2}}{\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}{\mathbf{S}}^{\frac{1}{2}}+\textbf{I}\right]{\mathbf{C}}=\mathbf{\Lambda}$
(9)
where $\mathbf{\Lambda}=\text{diag}\\{\lambda_{1},...,\lambda_{n_{t}}\\}$ is a
positive definite diagonal matrix and $\lambda_{1},...,\lambda_{n_{t}}$
represent the generalized eigenvalues. Without loss of generality, we assume
the generalized eigenvalues are ordered as
$\lambda_{1}\geq...\geq\lambda_{b}>1\geq\lambda_{b+1}\geq...\geq\lambda_{n_{t}}>0$
so that a total of $b$ $(0\leq b\leq n_{t})$ are assumed to be greater than 1.
Hence, we can write $\mathbf{\Lambda}$ as
$\mathbf{\Lambda}=\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}&0\\\
0&\mathbf{\Lambda}_{2}\end{array}\right]$ (10)
where $\mathbf{\Lambda}_{1}=\text{diag}\\{\lambda_{1},...,\lambda_{b}\\}$ and
$\mathbf{\Lambda}_{2}=\text{diag}\\{\lambda_{b+1},...,\lambda_{n_{t}}\\}$.
Also, we can write ${\mathbf{C}}$ as
${\mathbf{C}}=[{\mathbf{C}}_{1}\quad{\mathbf{C}}_{2}]$ (11)
where ${\mathbf{C}}_{1}$ is the $n_{t}\times b$ submatrix representing the
generalized eigenvectors corresponding to $\\{\lambda_{1},...,\lambda_{b}\\}$
and ${\mathbf{C}}_{2}$ is the $n_{t}\times(n_{t}-b)$ submatrix representing
the generalized eigenvectors corresponding to
$\\{\lambda_{b+1},...,\lambda_{n_{t}}\\}$.
For the case of ${\mathbf{K}}_{w}=0$, the secrecy capacity of (5) under the
matrix power constraint (4) is given by (Lemma 1 or [17, Theorem 3]):
$C_{sec}=\sum_{i=1}^{b}\log\lambda_{i}=\log|\mathbf{\Lambda}_{1}|$ (12)
and the input covariance matrix ${\mathbf{K}}_{x}^{*}$ that maximizes (5) is
given by ([7, 17]):
${\mathbf{K}}_{x}^{*}={\mathbf{S}}^{\frac{1}{2}}{\mathbf{C}}\left[\begin{array}[]{ccc}({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}&0\\\
0&0\end{array}\right]{\mathbf{C}}^{H}{\mathbf{S}}^{\frac{1}{2}}\;.$ (13)
Note that (13) is a general expression for the ${\mathbf{K}}_{x}$ that
optimizes (5) for a given ${\mathbf{K}}_{w}$ even when ${\mathbf{K}}_{w}\neq
0$, although in this case the ${\mathbf{C}}$ will be the generalized
eigenvector matrix of the pencil (7). From (9) we note that
${\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}$ can be written as
${\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}={\mathbf{S}}^{-1/2}\left[{\mathbf{C}}^{-H}\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}&0\\\
0&\mathbf{\Lambda}_{2}\end{array}\right]{\mathbf{C}}^{-1}-\textbf{I}\right]{\mathbf{S}}^{-1/2}\;.$
(14)
The following lemma gives the mutual information $I(X_{1};Y_{1})$ between the
transmitter and the intended receiver when ${\mathbf{K}}_{w}=0$ and
${\mathbf{K}}_{x}$ is given by (13). Lemma 2: The following equality holds:
$I(X_{1};Y_{1})|_{{\mathbf{K}}_{w}=0,{\mathbf{K}}_{x}={\mathbf{K}}_{x}^{*}}=\log\left|{\mathbf{K}}_{x}^{*}{\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}+\textbf{I}\right|=\log\left|({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}\mathbf{\Lambda}_{1}\right|\;.$
(15)
Proof: Following the same steps as the proof of [7, App. D] and using (13) and
(14), we have
$\displaystyle\left|{\mathbf{K}}_{x}^{*}{\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}+\textbf{I}\right|$
$\displaystyle=$
$\displaystyle\left|{\mathbf{S}}^{\frac{1}{2}}{\mathbf{C}}\left[\begin{array}[]{ccc}({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}&0\\\
0&0\end{array}\right]{\mathbf{C}}^{H}\times\left[{\mathbf{C}}^{-H}\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}&0\\\
0&\mathbf{\Lambda}_{2}\end{array}\right]{\mathbf{C}}^{-1}-\textbf{I}\right]{\mathbf{S}}^{-1/2}+\textbf{I}\right|$
(20) $\displaystyle=$
$\displaystyle\left|\left[\begin{array}[]{ccc}({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}&0\\\
0&0\end{array}\right]\times\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}&0\\\
0&\mathbf{\Lambda}_{2}\end{array}\right]-\left[\begin{array}[]{ccc}({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}&0\\\
0&0\end{array}\right]{\mathbf{C}}^{H}{\mathbf{C}}+\textbf{I}\right|$ (27)
$\displaystyle=$
$\displaystyle\left|\left[\begin{array}[]{ccc}({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}\mathbf{\Lambda}_{1}&0\\\
0&0\end{array}\right]-\left[\begin{array}[]{ccc}\textbf{I}&({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{2}\\\
0&0\end{array}\right]+\textbf{I}\right|$ (32) $\displaystyle=$
$\displaystyle\left|\left[\begin{array}[]{ccc}({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}\mathbf{\Lambda}_{1}&-({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{2}\\\
0&{\mathbf{I}}\end{array}\right]\right|$ (35) $\displaystyle=$
$\displaystyle\left|({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}\mathbf{\Lambda}_{1}\right|$
(36)
where (27) follows from the fact that
$\left|{\mathbf{A}}{\mathbf{B}}+{\mathbf{I}}\right|=\left|{\mathbf{B}}{\mathbf{A}}+{\mathbf{I}}\right|$,
and (32) follows since
${\mathbf{C}}^{H}{\mathbf{C}}=\left[{\mathbf{C}}_{1}\quad{\mathbf{C}}_{2}\right]^{H}\left[{\mathbf{C}}_{1}\quad{\mathbf{C}}_{2}\right]=\left[\begin{array}[]{ccc}{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1}&{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{2}\\\
{\mathbf{C}}_{2}^{H}{\mathbf{C}}_{1}&{\mathbf{C}}_{2}^{H}{\mathbf{C}}_{2}\end{array}\right]\;.$
We now return to the general optimization problem in (5) with non-zero
${\mathbf{K}}_{w}$. As the helper begins to broadcast artificial noise, both
the mutual information between the transmitter and the intended receiver
$I(X_{1};Y_{1})$ and the mutual information between the transmitter and the
eavesdropper $I(X_{1};Y_{2})$ are in general decreased. Both of these
functions are non-increasing in ${\mathbf{K}}_{w}$ since
$\frac{\left|{\mathbf{A}}+{\mathbf{B}}\right|}{\left|{\mathbf{B}}\right|}\geq\frac{\left|{\mathbf{A}}+{\mathbf{B}}+\mathbf{\bigtriangleup}\right|}{\left|{\mathbf{B}}+\mathbf{\bigtriangleup}\right|}$
when ${\mathbf{A}}$, $\mathbf{\bigtriangleup}\succeq 0$ and ${\mathbf{B}}\succ
0$ [20]. A favorable choice for ${\mathbf{K}}_{w}$ would be one that reduces
$I(X_{1};Y_{2})$ more than $I(X_{1};Y_{1})$. Since the optimal solution to (5)
is intractable, we propose a suboptimal approach that introduces an additional
constraint; namely, we search among those ${\mathbf{K}}_{w}$ matrices that
guarantee no decrease in the favorable term $I(X_{1};Y_{1})$ while the power
constraint (3) is satisfied. It should be noted that this approach is more
general than the cooperative jamming schemes proposed in [10, 11] for the MISO
case where the jamming signal is nulled out at the destination. Clearly, such
sub-optimal solutions are restricted to the case where there exists a null
space between the helper and the intended receiver.
In the following, we obtain an expression that represents all
${\mathbf{K}}_{w}\succeq 0$ matrices with the power constraint
$\text{Tr}({\mathbf{K}}_{w})=P_{h}$ that do not impact the mutual information
between the transmitter and the intended receiver; i.e.,
$I(X_{1};Y_{1})|_{{\mathbf{K}}_{w}\succeq
0,{\mathbf{K}}_{x}={\mathbf{K}}_{x}^{*}}=I(X_{1};Y_{1})|_{{\mathbf{K}}_{w}=0,{\mathbf{K}}_{x}={\mathbf{K}}_{x}^{*}}\;,$
or from (15)
$\log\left|{\mathbf{K}}_{x}^{*}{\mathbf{H}}_{1}^{H}({\mathbf{G}}_{2}{\mathbf{K}}_{w}{\mathbf{G}}_{2}^{H}+{\mathbf{I}})^{-1}{\mathbf{H}}_{1}+{\mathbf{I}}\right|=\log\left|{\mathbf{K}}_{x}^{*}{\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}+{\mathbf{I}}\right|=\log\left|({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}\mathbf{\Lambda}_{1}\right|.$
(37)
Note that, without loss of generality, we have used an equality power
constraint $\text{Tr}({\mathbf{K}}_{w})=P_{h}$ since for the desired
${\mathbf{K}}_{w}$ the best performance is in general obtained when helper
transmits at maximum power.
Theorem 1: All ${\mathbf{K}}_{w}\succeq 0$ matrices for which
$\log\left|{\mathbf{K}}_{x}^{*}{\mathbf{H}}_{1}^{H}({\mathbf{G}}_{2}{\mathbf{K}}_{w}{\mathbf{G}}_{2}^{H}+{\mathbf{I}})^{-1}{\mathbf{H}}_{1}+{\mathbf{I}}\right|=\log\left|{\mathbf{K}}_{x}^{*}{\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}+{\mathbf{I}}\right|=\log\left|({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}\mathbf{\Lambda}_{1}\right|$
satisfy the following relation:
${\mathbf{H}}_{1}^{H}({\mathbf{G}}_{2}{\mathbf{K}}_{w}{\mathbf{G}}_{2}^{H}+{\mathbf{I}})^{-1}{\mathbf{H}}_{1}={\mathbf{S}}^{-1/2}\left[{\mathbf{C}}^{-H}\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}&0\\\
0&{\mathbf{N}}\end{array}\right]{\mathbf{C}}^{-1}-\textbf{I}\right]{\mathbf{S}}^{-1/2}$
(38)
where
$\begin{array}[]{c}\mathbf{\Lambda}_{22}\preceq{\mathbf{N}}\preceq\mathbf{\Lambda}_{2}\\\
\mathbf{\Lambda}_{22}={\mathbf{C}}_{2}^{H}{\mathbf{C}}_{2}+{\mathbf{C}}_{2}^{H}{\mathbf{C}}_{1}(\mathbf{\Lambda}_{1}-{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{2}\end{array}$
(39)
and $\mathbf{\Lambda}_{1}$, $\mathbf{\Lambda}_{2}$, ${\mathbf{C}}$,
${\mathbf{C}}_{1}$ and ${\mathbf{C}}_{2}$ are defined in (8)-(11).
Proof: In Appendix A, using similar steps as those used to obtain (36), we
show that all $\mathbf{\Sigma}\succeq 0$ matrices for which
$\log\left|{\mathbf{K}}_{x}^{*}\mathbf{\Sigma}+{\mathbf{I}}\right|=\log\left|({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}\mathbf{\Lambda}_{1}\right|$
must have the following form
$\mathbf{\Sigma}={\mathbf{S}}^{-1/2}\left[{\mathbf{C}}^{-H}\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}&{\mathbf{M}}\\\
{\mathbf{M}}^{H}&{\mathbf{N}}\end{array}\right]{\mathbf{C}}^{-1}-\textbf{I}\right]{\mathbf{S}}^{-1/2}\;.$
(40)
In the following, we obtain matrices ${\mathbf{N}}\succeq 0$ and
${\mathbf{M}}$ and complete the proof by considering the following specific
choice for $\mathbf{\Sigma}$:
$\mathbf{\Sigma}={\mathbf{H}}_{1}^{H}({\mathbf{G}}_{2}{\mathbf{K}}_{w}{\mathbf{G}}_{2}^{H}+{\mathbf{I}})^{-1}{\mathbf{H}}_{1}\;.$
(41)
For the specific $\mathbf{\Sigma}$ in (41), it is evident that
$0\preceq\mathbf{\Sigma}\preceq{\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}.$ (42)
By applying the constraint
$\mathbf{\Sigma}\preceq{\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}$ on (40) and using
(14), it is enough to show that:
$\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}&{\mathbf{M}}\\\
{\mathbf{M}}^{H}&{\mathbf{N}}\end{array}\right]\preceq\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}&0\\\
0&\mathbf{\Lambda}_{2}\end{array}\right]$
or equivalently that
$\left[\begin{array}[]{ccc}0&-{\mathbf{M}}\\\
-{\mathbf{M}}^{H}&\mathbf{\Lambda}_{2}-{\mathbf{N}}\end{array}\right]\succeq
0\;.$
By applying the Schur Complement Lemma [18], the above relationship is true
_iff_ $\mathbf{\Lambda}_{2}-{\mathbf{N}}\succeq 0$ and
$-{\mathbf{M}}(\mathbf{\Lambda}_{2}-{\mathbf{N}})^{-1}{\mathbf{M}}^{H}\succeq
0$, which in turn is true only when
$\displaystyle{\mathbf{M}}$ $\displaystyle=$ $\displaystyle 0$ (43)
$\displaystyle\mathbf{\Lambda}_{2}-{\mathbf{N}}$ $\displaystyle\succeq$
$\displaystyle 0\;.$ (44)
Applying the results of (43) and (44) in (40) for the specific choice of
$\mathbf{\Sigma}={\mathbf{H}}_{1}^{H}({\mathbf{G}}_{2}{\mathbf{K}}_{w}{\mathbf{G}}_{2}^{H}+{\mathbf{I}})^{-1}{\mathbf{H}}_{1}$,
we have:
$\mathbf{\Sigma}={\mathbf{S}}^{-1/2}\left[{\mathbf{C}}^{-H}\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}&0\\\
0&{\mathbf{N}}\end{array}\right]{\mathbf{C}}^{-1}-\textbf{I}\right]{\mathbf{S}}^{-1/2}\;.$
(45)
Based on (42), we also need to show that $\mathbf{\Sigma}\succeq 0$. From
(45), it is enough to show that
$\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}&0\\\
0&{\mathbf{N}}\end{array}\right]-{\mathbf{C}}^{H}{\mathbf{C}}=\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}-{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1}&-{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{2}\\\
-{\mathbf{C}}_{2}^{H}{\mathbf{C}}_{1}&{\mathbf{N}}-{\mathbf{C}}_{2}^{H}{\mathbf{C}}_{2}\end{array}\right]\succeq
0\;.$
By applying the Schur Complement Lemma, the above relationship is true _iff_
$\mathbf{\Lambda}_{1}-{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1}\succeq 0$ and
${\mathbf{N}}-{\mathbf{C}}_{2}^{H}{\mathbf{C}}_{2}-{\mathbf{C}}_{2}^{H}{\mathbf{C}}_{1}(\mathbf{\Lambda}_{1}-{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{2}\succeq
0$. Using Eqs. (8)-(10), it is evident that
$\mathbf{\Lambda}_{1}-{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1}={\mathbf{C}}_{1}^{H}\left[{\mathbf{S}}^{\frac{1}{2}}{\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}{\mathbf{S}}^{\frac{1}{2}}+\textbf{I}\right]{\mathbf{C}}_{1}-{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1}={\mathbf{C}}_{1}^{H}{\mathbf{S}}^{\frac{1}{2}}{\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}{\mathbf{S}}^{\frac{1}{2}}{\mathbf{C}}_{1}\succeq
0$
and finally the lower bound for ${\mathbf{N}}$ is given by
${\mathbf{N}}\succeq{\mathbf{C}}_{2}^{H}{\mathbf{C}}_{2}+{\mathbf{C}}_{2}^{H}{\mathbf{C}}_{1}(\mathbf{\Lambda}_{1}-{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{2}\succ
0\;,$ which completes the proof.
It should be noted that as ${\mathbf{N}}\rightarrow\mathbf{\Lambda}_{22}$, we
have Tr$({\mathbf{K}}_{w})\rightarrow\infty$. Moreover,
Tr$({\mathbf{K}}_{w})=0$ is achieved by ${\mathbf{N}}=\mathbf{\Lambda}_{2}$.
Hence, for each scalar $P_{h}$, there always exists an ${\mathbf{N}}$ in the
range $\mathbf{\Lambda}_{22}\preceq{\mathbf{N}}\preceq\mathbf{\Lambda}_{2}$
that will lead to a ${\mathbf{K}}_{w}$ that satisfies (38) with
Tr$({\mathbf{K}}_{w})=P_{h}$.
Thus far, we have not made any assumption on the number of antennas at each
node. But it is clear from (38) that, for example when ${\mathbf{G}}_{2}$ has
more columns than rows, for a fixed ${\mathbf{N}}$ in the acceptable range
(39) there will be an infinite number of ${\mathbf{K}}_{w}$ matrices that
satisfy (38) and consequently do not decrease $I(X_{1};Y_{1})$. In fact, in
this example, a common policy for the helper is to simply transmit artificial
noise in the null space of ${\mathbf{G}}_{2}$. A more interesting case occurs
when no such null space exists, i.e., when the number of antennas at the
helper is less than or equal to that of the intended receiver ($n_{h}\leq
n_{r}$). The above result demonstrates the non-trivial fact that even when
$n_{h}\leq n_{r}$, it is possible to find a non-zero jamming signal that does
not impact $I(X_{1};Y_{1})$ even when the jamming signal can not be nulled by
the channel. In the next section, we find more constructive expressions for
the ${\mathbf{K}}_{w}$ matrices that satisfy (38) for various combinations of
the number of antennas at different nodes. In particular, we show that when
$n_{h}\leq n_{r}$, a closed-form expression for ${\mathbf{K}}_{w}$ can be
found.
## IV Results for Different Scenarios
In this section, we consider all possible combinations of the number of
antennas at the transmitter, helper and intended receiver, and obtain
constructive methods for computing specific ${\mathbf{K}}_{w}$ matrices that
satisfy (38). Such ${\mathbf{K}}_{w}$ will have no impact on $I(X_{1};Y_{1})$,
but will in general decrease $I(X_{1};Y_{2})$, the mutual information between
the transmitter and the eavesdropper, compared with the case that there is no
helper. Hence, the secrecy level of the confidential message is increased. As
mentioned before, additional improvement in the secrecy rate can be achieved
if the transmitter updates its covariance matrix ${\mathbf{K}}_{x}$ once
${\mathbf{K}}_{w}$ is computed. Note, however, that such an iterative process
will not be pursued beyond updating ${\mathbf{K}}_{x}$; unlike the first step,
where ${\mathbf{K}}_{w}$ was updated from its initial value of zero, there is
no guarantee that finding a new ${\mathbf{K}}_{w}$ will reduce
$I(X_{1};Y_{2})$. Hence, the final secrecy rate for the proposed method is
obtained by simply computing (6) and (7) for the resulting ${\mathbf{K}}_{w}$
matrices derived in this section.
### IV-A Case 1: $n_{h}\leq\min\\{n_{r},n_{t}\\}$
We show here that for the case where $n_{h}\leq\min\\{n_{r},n_{t}\\}$ and for
a fixed ${\mathbf{N}}$ in the acceptable range (39), there is only one
${\mathbf{K}}_{w}$ matrix that satisfies (38) and consequently does not
decrease $I(X_{1};Y_{1})$. Using the matrix inversion lemma, Eq. (38) can be
written as:
$\displaystyle{\mathbf{H}}_{1}^{H}({\mathbf{G}}_{2}{\mathbf{K}}_{w}{\mathbf{G}}_{2}^{H}+{\mathbf{I}})^{-1}{\mathbf{H}}_{1}$
$\displaystyle=$
$\displaystyle{\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}-{\mathbf{H}}_{1}^{H}{\mathbf{G}}_{2}({\mathbf{G}}_{2}^{H}{\mathbf{G}}_{2}+{\mathbf{K}}_{w}^{-1})^{-1}{\mathbf{G}}_{2}^{H}{\mathbf{H}}_{1}$
(48) $\displaystyle=$
$\displaystyle{\mathbf{S}}^{-1/2}\left[{\mathbf{C}}^{-H}\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}&0\\\
0&{\mathbf{N}}\end{array}\right]{\mathbf{C}}^{-1}-\textbf{I}\right]{\mathbf{S}}^{-1/2}\;.$
Replacing ${\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}$ with (14), we have:
${\mathbf{H}}_{1}^{H}{\mathbf{G}}_{2}({\mathbf{G}}_{2}^{H}{\mathbf{G}}_{2}+{\mathbf{K}}_{w}^{-1})^{-1}{\mathbf{G}}_{2}^{H}{\mathbf{H}}_{1}={\mathbf{S}}^{-1/2}{\mathbf{C}}^{-H}\left[\begin{array}[]{ccc}0&0\\\
0&\mathbf{\Lambda}_{2}-{\mathbf{N}}\end{array}\right]{\mathbf{C}}^{-1}{\mathbf{S}}^{-1/2}\;.$
(49)
Since we have assumed that the channels are full rank, in the case of
$n_{h}\leq n_{r}\leq n_{t}$ or $n_{h}\leq n_{t}\leq n_{r}$, it is clear that
rank$({\mathbf{G}}_{2}^{H}{\mathbf{H}}_{1})=n_{h}.$ Thus, from (49) we have:
$({\mathbf{G}}_{2}^{H}{\mathbf{G}}_{2}+{\mathbf{K}}_{w}^{-1})^{-1}={\mathbf{O}}^{H}{\mathbf{S}}^{-1/2}{\mathbf{C}}^{-H}\left[\begin{array}[]{ccc}0&0\\\
0&\mathbf{\Lambda}_{2}-{\mathbf{N}}\end{array}\right]{\mathbf{C}}^{-1}{\mathbf{S}}^{-1/2}{\mathbf{O}}$
(50)
where ${\mathbf{O}}$ is the right inverse of
${\mathbf{G}}_{2}^{H}{\mathbf{H}}_{1}$, which, for example when $n_{h}\leq
n_{r}\leq n_{t}$, can be written as
${\mathbf{O}}={\mathbf{H}}_{1}^{H}({\mathbf{H}}_{1}{\mathbf{H}}_{1}^{H})^{-1}{\mathbf{G}}_{2}({\mathbf{G}}_{2}^{H}{\mathbf{G}}_{2})^{-1}$.
The following lemma is a direct result of Eqs. (49) and (50).
Lemma 3: For the case of $n_{h}\leq\min\\{n_{r},n_{t}\\}$ and for a fixed
${\mathbf{N}}$ in the acceptable range (39), the ${\mathbf{K}}_{w}\succeq 0$
matrix for which (38) is satisfied and $I(X_{1};Y_{1})$ is not decreased is
given by
${\mathbf{K}}_{w}={\mathbf{Q}}-{\mathbf{Q}}{\mathbf{G}}_{2}^{H}({\mathbf{G}}_{2}{\mathbf{Q}}{\mathbf{G}}_{2}^{H}-{\mathbf{I}})^{-1}{\mathbf{G}}_{2}{\mathbf{Q}}$
(51)
where ${\mathbf{Q}}$ is the RHS of (50).
Proof: After applying the matrix inversion lemma on the LHS of (50), a
straightforward computation yields (51).
As is evident from Eqs. (50)-(51), we still have a design parameter,
${\mathbf{N}}$, that should be chosen in its acceptable range
$\mathbf{\Lambda}_{22}\preceq{\mathbf{N}}\preceq\mathbf{\Lambda}_{2}$ such
that the power constraint $\text{Tr}({\mathbf{K}}_{w})=P_{h}$ is satisfied.
Finding the optimal ${\mathbf{N}}$ that minimizes $I(X_{1};Y_{2})$ when
${\mathbf{K}}_{x}$ and ${\mathbf{K}}_{w}$ are given by (13) and (51),
respectively, is as intractable as the general optimization problem in (5).
Instead, we simply restrict the ${\mathbf{N}}$ we consider to those that can
be linearly parameterized within the acceptable range, as follows:
${\mathbf{N}}=\mathbf{\Lambda}_{22}+t\left(\mathbf{\Lambda}_{2}-\mathbf{\Lambda}_{22}\right)\;.$
(52)
Consequently the term $\mathbf{\Lambda}_{2}-{\mathbf{N}}$ in Eq. (51) becomes
$\mathbf{\Lambda}_{2}-{\mathbf{N}}=(1-t)\left(\mathbf{\Lambda}_{2}-\mathbf{\Lambda}_{22}\right)$
where the scalar $0\leq t\leq 1$ is chosen such that the power constraint
$\text{Tr}({\mathbf{K}}_{w})=P_{h}$ is satisfied. Note that as $t\rightarrow
0$ $({\mathbf{N}}\rightarrow\mathbf{\Lambda}_{22})$ then
$\text{Tr}({\mathbf{K}}_{w})\rightarrow\infty$, and as $t\rightarrow 1$
$({\mathbf{N}}\rightarrow\mathbf{\Lambda}_{2})$ then
$\text{Tr}({\mathbf{K}}_{w})\rightarrow 0$. Thus, we are guaranteed that an
acceptable ${\mathbf{N}}$ can be found in this way.
### IV-B Case 2: $n_{h}>\min\\{n_{r},n_{t}\\}$
As mentioned before, for the case of $n_{h}>n_{r}$ and for a fixed
${\mathbf{N}}$ in the acceptable range (39), there are many ${\mathbf{K}}_{w}$
matrices that satisfy (38) and consequently do not decrease $I(X_{1};Y_{1})$.
A common policy for the helper in this case is to transmit artificial noise in
the null space of ${\mathbf{G}}_{2}$. However, as (38) shows, this policy is
sufficient but it is not necessary. In other words, it is possible that the
optimal ${\mathbf{K}}_{w}$ satisfying (38) has elements outside the null space
of ${\mathbf{G}}_{2}$. Because of the non-linear constraint in (38), finding
the optimal ${\mathbf{K}}_{w}$ is intractable. A similar discussion applies
for the case of $n_{t}<n_{h}\leq n_{r}$.
In this section, we present an approach for computing a suitable
${\mathbf{K}}_{w}$. Consider the following jamming signal covariance matrix:
${\mathbf{K}}_{w}=\mathbf{\Gamma}\,\mathbf{\Pi}\,\mathbf{\Gamma}^{H}\;,$ (53)
where $\mathbf{\Pi}$ is a $d\times d$ positive semidefinite matrix, and
$\mathbf{\Gamma}$ is an $n_{h}\times d$ matrix. For the case of
$n_{t}<n_{h}\leq n_{r}$ or $n_{h}>n_{r}$, we can choose $\mathbf{\Gamma}$ such
that ${\mathbf{G}}_{2}\,\mathbf{\Gamma}$ is orthogonal to
${\mathbf{H}}_{1}\,{{\mathbf{K}}_{x}^{*}}^{\frac{1}{2}}$, i.e.,
${{\mathbf{K}}_{x}^{*}}^{\frac{1}{2}}{\mathbf{H}}_{1}^{H}{\mathbf{G}}_{2}\,\mathbf{\Gamma}={\boldsymbol{0}}$.
For example, $\mathbf{\Gamma}$ can be chosen as the $d$ right singular vectors
in the nullspace of
${{\mathbf{K}}_{x}^{*}}^{\frac{1}{2}}{\mathbf{H}}_{1}^{H}{\mathbf{G}}_{2}$.
Since ${\mathbf{K}}_{x}$ will often be rank deficient, the value of $d$ will
typically be larger than $n_{h}-n_{t}$ for the case of $n_{t}<n_{h}\leq
n_{r}$, and larger than $n_{h}-n_{r}$ for the case of $n_{h}>n_{r}$. For this
choice of $\mathbf{\Gamma}$, the resulting ${\mathbf{K}}_{w}$ in (53)
satisfies (38), and doesn’t decrease $I(X_{1};Y_{1})$ for
${\mathbf{N}}=\mathbf{\Lambda}_{2}$, as is clear from (38). Given
$\mathbf{\Gamma}$, the choice of $\mathbf{\Pi}$ can be made to maximize the
transfer of the “information” in the helper’s jamming signal to the
eavesdropper. In particular, note that at the eavesdropper, the covariance of
the helper’s jamming signal will be given by
${\mathbf{H}}_{2}\mathbf{\Gamma\Pi\Gamma}^{H}{\mathbf{H}}_{2}^{H}$. If the
eigenvalue decomposition of
$\mathbf{\Gamma}^{H}{\mathbf{H}}_{2}^{H}{\mathbf{H}}_{2}\mathbf{\Gamma}$ is
written as
$\mathbf{\Gamma}^{H}{\mathbf{H}}_{2}^{H}{\mathbf{H}}_{2}\mathbf{\Gamma}={\mathbf{U}}\,{\mathbf{D}}\,{\mathbf{U}}^{H}$
with ${\mathbf{U}}$ unitary and ${\mathbf{D}}$ square and diagonal, then
$\mathbf{\Pi}$ can be found via waterfilling; i.e.,
$\mathbf{\Pi}={\mathbf{U}}\,\mathbf{\Delta}\,{\mathbf{U}}^{H}\;,$
where $\mathbf{\Delta}=\left[\eta{\mathbf{I}}-{\mathbf{D}}^{-1}\right]^{+}$,
the operation $[{\mathbf{A}}]^{+}$ zeros out any negative elements, and the
water-filling level $\eta$ is chosen such that
$\text{Tr}({\mathbf{K}}_{w})=\text{Tr}(\mathbf{\Delta})=P_{h}$.
## V Numerical Results
In this section, we present numerical results to illustrate our theoretical
findings. In all of the following figures, channels are assumed to be quasi-
static flat Rayleigh fading and independent of each other. The channel
matrices ${\mathbf{H}}_{1}\in\mathbb{C}^{n_{r}\times n_{t}}$ and
${\mathbf{G}}_{2}\in\mathbb{C}^{n_{r}\times n_{h}}$ have i.i.d. entries
distributed as $\mathcal{CN}(0,\sigma_{d}^{2})$, while
${\mathbf{G}}_{1}\in\mathbb{C}^{n_{e}\times n_{t}}$ and
${\mathbf{H}}_{2}\in\mathbb{C}^{n_{e}\times n_{h}}$ have i.i.d. entries
distributed as $\mathcal{CN}(0,\sigma_{c}^{2})$. In each figure, values for
the number of antennas at each node, as well as $\sigma_{d}^{2}$ and
$\sigma_{c}^{2}$, will be depicted. Unless otherwise indicated, results are
calculated based on an average of at least 500 independent channel
realizations.
In the first example, Fig. 1, we randomly generate positive definite matrices
${\mathbf{S}}$ such that $\text{Tr}({\mathbf{S}})\leq P_{t}$. For each
${\mathbf{S}}$, we compute the secrecy capacity of the MIMO Gaussian wiretap
channel without helper (${\mathbf{K}}_{w}={\boldsymbol{0}}$) as given by (12).
Next, using (51), we obtain a ${\mathbf{K}}_{w}$ with the average power
constraint $\text{Tr}({\mathbf{K}}_{w})=P_{h}$ that does not decrease
$I(X_{1};Y_{1})$, and then update ${\mathbf{K}}_{x}$ and compute
$C_{sec}({\mathbf{S}})$, using (6) and (7), accordingly. Fig. 1 compares the
secrecy capacity of the wiretap channel with (solid lines) and without (dotted
lines) the helper. Note that the vertical difference between the solid curves
(about 0.6 bps/channel use) represents the role of the transmit power $P_{t}$
on the secrecy capacity with helper when $P_{t}$ changes from 100 to 150 and
$P_{h}=20$. This relatively small difference indicates that, in this example,
$P_{t}$ does not have a big impact on the secrecy capacity. Its role is even
more negligible when $P_{h}=0$, where only an increase of $0.3$ bps/channel
use is obtained as $P_{t}$ increases from 100 to 150. The role of the helper
on the other hand is significantly more important; increasing $P_{h}$ from 0
to 20 while holding $P_{t}$ fixed results in an increase on the order of 3
bps/channel use. Furthermore, the use of the helper with a total power of only
120 ($P_{t}=100,P_{h}=20$) provides significantly better secrecy performance
than not using the helper and transmitting with total power equal to 150
($P_{t}=150,P_{h}=0$).
In the next examples, we calculate the secrecy capacity of the proposed
algorithms under the assumption of an average power constraint $P_{t}$ at the
transmitter, and under the constraint that the helper does not reduce the
mutual information between the transmitter and receiver. While Eqs. (6) and
(7) provide the performance for a specific ${\mathbf{S}}$, one must solve
[17], [20, Lemma 1]
$C_{sec}(P_{t})=\max_{{\mathbf{S}}\succeq 0,\text{Tr}({\mathbf{S}})\leq
P_{t}}C_{sec}({\mathbf{S}})$ (54)
to find the secrecy capacity over all ${\mathbf{S}}$ that satisfy the average
power constraint. In the examples that follow, we perform a numerical search
to solve (54) and compute the secrecy capacity.
Fig. 2 shows the secrecy capacity versus $P_{h}$ for a fixed total average
power $P_{t}+P_{h}=110$. In this figure, we consider a situation in which
$\sigma_{c}>\sigma_{d}$, or in other words where the channel between the
transmitter and the intended receiver is weaker than the channel between the
transmitter and the eavesdropper, and the channel between the helper and the
intended receiver is weaker than the channel between the helper and the
eavesdropper. The arrow in the figure shows the secrecy capacity without the
helper $(P_{h}=0)$. The figure shows that a helper with just a single antenna
can provide a dramatic improvement in secrecy rate with very little power
allocated to the jamming signal; in fact, the optimal rate is obtained when
$P_{h}$ is less than 2% of the total available transmit power. If the number
of antennas at the helper increases, a much higher secrecy rate can be
obtained, but at the expense of allocating more power to the helper and less
to the signal for the desired user.
In Fig. 3, we consider a situation in which, unlike the above example, we have
$\sigma_{d}>\sigma_{c}$. Thus, the intended receiver, in comparison with the
eavesdropper, receives a weaker information signal and a stronger jamming
signal than the eavesdropper. It might seem that in this situation, the helper
cannot be very useful, but the figure shows that even in this case we can have
a notable improvement in the secrecy rate (about 4 bps/channel use) by
increasing the number of antennas at the helper, and with an appropriate power
assignment between the transmitter and the helper, without requiring extra
total transmit power for the helper node.
In Fig. 4, we consider a specific scenario where the secrecy capacity in the
absence of the helper node is zero. While channel matrices ${\mathbf{H}}_{2}$
and ${\mathbf{G}}_{2}$ are generated randomly with i.i.d. entries distributed
as $\mathcal{CN}(0,\sigma_{c}^{2})$ and $\mathcal{CN}(0,\sigma_{d}^{2})$,
respectively, we assume the following specific choices for ${\mathbf{H}}_{1}$
and ${\mathbf{G}}_{1}$:
${\mathbf{H}}_{1}=\left[\begin{array}[]{ccc}-0.25+0.5i&-0.35&-1.25-0.9i\\\
-0.4+0.1i&-0.2+0.75i&-i\end{array}\right]$
${\mathbf{G}}_{1}=\left[\begin{array}[]{ccc}2+0.25i&1.5+0.5i&2i\\\
0.25+0.25i&-0.7+1.5i&0.5+0.33i\\\ -1.5&-0.5-i&-2.9i\end{array}\right].$
Since
${\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}\preceq{\mathbf{G}}_{1}^{H}{\mathbf{G}}_{1}$,
all the generalized eigenvalues of the pencil
$\left({\mathbf{S}}^{\frac{1}{2}}{\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}{\mathbf{S}}^{\frac{1}{2}}+\textbf{I}\right)-\gamma\left({\mathbf{S}}^{\frac{1}{2}}{\mathbf{G}}_{1}^{H}{\mathbf{G}}_{1}{\mathbf{S}}^{\frac{1}{2}}+\textbf{I}\right)$
are zero for all ${\mathbf{S}}\succeq 0$ and consequently, the secrecy
capacity without helper will be zero. In this example, we also assume that not
only is the total power fixed at $P_{t}+P_{h}=110$, but also the total number
of transmit antennas is fixed at $n_{t}+n_{h}=3$. As in the other examples,
the secrecy rate of the wiretap channel is considerably improved with the
helper. In this case, the best performance is obtained when the helper has
only a single antenna.
Finally, in Fig. 5, we consider the role of number of antennas at the helper,
$n_{h}$, in the secrecy rate for the specific matrix power constraint
${\mathbf{S}}=\frac{P_{t}}{n_{t}}{\mathbf{I}}$. Note that the solution of
Section IV-A applies for $n_{h}\leq 3$, while the solution of Section IV-B
holds for $n_{h}>3$. In all cases, we see that the secrecy rate increases
considerably as $n_{h}$ increases.
## VI Conclusions
In this paper, we have studied the Gaussian MIMO Wiretap channel in the
presence of an external jammer/helper, where the helper node assists the
transmitter by sending artificial noise independent of the source message to
confuse the eavesdropper. The jamming signal from the helper is not required
to be decoded by the intended receiver and is treated as noise at both the
intended receiver and the eavesdropper. We obtained a closed-form relationship
for the structure of the helper’s artificial noise covariance matrix that
guarantees no decrease in the mutual information between the transmitter and
the intended receiver. We showed how to find appropriate solutions within this
covariance matrix framework that provide very good secrecy rate performance,
even when there is no non-trivial null space between the helper and the
intended receiver. The proposed scheme is shown to achieve a notable
improvement in secrecy rate even for a fixed average total power and a fixed
total number of antennas at the transmitter and the helper, without requiring
extra power or antennas to be allocated to the helper node.
## Appendix A
We are interested in finding a relationship that represents all matrices
$\mathbf{\Sigma}\succ 0$ for which
$\log\left|{\mathbf{K}}_{x}^{*}\mathbf{\Sigma}+{\mathbf{I}}\right|=\log\left|({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}\mathbf{\Lambda}_{1}\right|\;,$
(55)
where
${\mathbf{K}}_{x}^{*}={\mathbf{S}}^{\frac{1}{2}}{\mathbf{C}}\left[\begin{array}[]{ccc}({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}&0\\\
0&0\end{array}\right]{\mathbf{C}}^{H}{\mathbf{S}}^{\frac{1}{2}}\;.$ (56)
Using the fact that
$|{\mathbf{A}}{\mathbf{B}}+{\mathbf{I}}|=|{\mathbf{B}}{\mathbf{A}}+{\mathbf{I}}|$,
it is clear that $\mathbf{\Sigma}$ will have the form
$\mathbf{\Sigma}={\mathbf{S}}^{-\frac{1}{2}}{\mathbf{C}}^{-H}{\mathbf{X}}{\mathbf{C}}^{-1}{\mathbf{S}}^{-\frac{1}{2}}$
for some matrix ${\mathbf{X}}={\mathbf{X}}^{H}$. Substituting this expression
for $\mathbf{\Sigma}$ into (55) results in the following equation that must be
solved for ${\mathbf{X}}$:
$\log\left|\left[\begin{array}[]{ccc}({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}&0\\\
0&0\end{array}\right]{\mathbf{X}}+{\mathbf{I}}\right|=\log\left|({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}\mathbf{\Lambda}_{1}\right|\;.$
(57)
Write ${\mathbf{X}}$ as
${\mathbf{X}}=\left[\begin{array}[]{ccc}{\mathbf{X}}_{1}&{\mathbf{X}}_{2}\\\
{\mathbf{X}}_{2}^{H}&{\mathbf{X}}_{3}\end{array}\right]$ so that we have
$\left[\begin{array}[]{ccc}({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}&0\\\
0&0\end{array}\right]{\mathbf{X}}+{\mathbf{I}}=\left[\begin{array}[]{ccc}({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}{\mathbf{X}}_{1}+{\mathbf{I}}&({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}{\mathbf{X}}_{2}\\\
0&{\mathbf{I}}\end{array}\right]\;,$
and note that the determinant of the above matrix is given by
$\left|({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}{\mathbf{X}}_{1}+{\mathbf{I}}\right|$.
By comparing this result with (55), we see that
${\mathbf{X}}_{1}=\mathbf{\Lambda}_{1}-({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})$.
Consequently, we have:
$\mathbf{\Sigma}={\mathbf{S}}^{-\frac{1}{2}}{\mathbf{C}}^{-H}\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}-({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})&{\mathbf{X}}_{2}\\\
{\mathbf{X}}_{2}^{H}&{\mathbf{X}}_{3}\end{array}\right]{\mathbf{C}}^{-1}{\mathbf{S}}^{-\frac{1}{2}}$
(58)
where ${\mathbf{X}}_{2}$ and ${\mathbf{X}}_{3}$ are still unknown and must be
found as described in the text. It is clear that (58) and (40) are equivalent.
## References
* [1] A. Wyner, “The wire-tap channel,” _Bell. Syst. Tech. J._ , vol. 54, no. 8, pp. 1355-1387, Jan. 1975.
* [2] S. K. Leung-Yan-Cheong and M. E. Hellman, “The Gaussian wire-tap channel,” _IEEE Trans. Inf. Theory_ , vol. 24, pp. 451-456, Jul. 1978.
* [3] F. Oggier and B. Hassibi, “The secrecy capacity of the MIMO wiretap channel,” in _Proc. IEEE Int. Symp. Information Theory_ Toronto, ON, Canada, Jul. 2008, pp. 524-528.
* [4] A. Khisti and G. Wornell, “Secure transmission with multiple antennas II: The MIMOME wiretap channel,” to appear, _IEEE Trans. Inf. Theory_ , 2010. Available at: http://allegro.mit.edu/pubs/posted/journal/2008-khisti-wornell-it.pdf
* [5] S. Ali. A. Fakoorian and A. L. Swindlehurst, “Optimal power allocation for the GSVD based MIMO Gaussian wiretap channel,” submitted to _IEEE Trans. Inf. Theory_ , Available: http://arxiv.org/abs/1006.1890
* [6] T. Liu and S. Shamai (Shitz), “A note on secrecy capacity of the multi-antenna wiretap channel,” _IEEE Trans. Inf. Theory_ , vol. 55, no. 6, pp. 2547-2553, 2009.
* [7] R. Bustin, R. Liu, H. V. Poor, and S. Shamai (Shitz), “A MMSE approach to the secrecy capacity of the MIMO Gaussian wiretap channel,” _EURASIP Journal on Wireless Communications and Networking_ , vol. 2009, Article ID 370970, 8 pages, 2009.
* [8] I. Csiszar and J. Korner, “Broadcast channels with confidential messages,” _IEEE Trans. Inf. Theory_ , vol. 24, pp. 339-348, May 1978.
* [9] E. Tekin and A. Yener, “The general Gaussian multiple access and two-way wire-tap channels: Achievable rates and cooperative jamming,” _IEEE Trans. Inf. Theory_ , vol. 54, no. 6, pp. 2735 2751, Jun. 2008.
* [10] L. Dong, Z. Han, A. P. Petropulu, H. V. Poor, “Cooperative jamming for wireless physical layer security”, in Proc. of _IEEE Workshop on Statistical Signal Processing_ , Cardiff, Wales, U.K. 2009
* [11] L. Dong, Z. Han, A. P. Petropulu, and H. V. Poor, “Improving wireless physical layer security via cooperating relays,” _IEEE Trans. Signal Proc._ , vol. 58, NO. 3, pp. 1875-1888, Mar. 2010.
* [12] L. Lai and H. El Gamal, “The relay-eavesdropper channel: Cooperation for secrecy,” _IEEE Trans. Inf. Theory_ , vol. 54, no. 9, pp. 4005 4019, Sep. 2008.
* [13] X. Tang, R. Liu, P. Spasojevic, and H. V. Poor, “The Gaussian wiretap channel with a helping interferer,” in _Proc. IEEE Int. Symp. Inf. Theory_ , Toronto, ON, Canada, Jul. 2008.
* [14] J. Wang and A. Swindlehurst, “Cooperative jamming in MIMO ad hoc networks,” in _Proc. Asilomar Conf. on Signals, Systems and Computers_ , pp. 1719-1723, Nov., 2009.
* [15] E. MolavianJazi, M. Bloch, and J. N. Laneman, “Arbitrary jamming can preclude secure communication,” in _Proc. Allerton Conf._ Communications, Control, and Computing, Monticello, IL, Sept. 2009.
* [16] S. Goel and R. Negi, “Guaranteeing secrecy using artificial noise,” _IEEE Trans. Wireless Commun_., vol. 7, no. 6, pp. 2180-2189, June 2008.
* [17] Ruoheng Liu, Tie Liu, H. Vincent Poor, and Shlomo Shamai (Shitz), “Multiple-input multiple-output Gaussian broadcast channels with confidential messages,” _IEEE Trans. Inf. Theory_ , to appear.
* [18] R. A. Horn and C. R. Johnson, _Matrix Analysis_ , University Press, Cambridge, UK, 1985.
* [19] S. W. Peters and R. W. Heath, Jr., “Interference alignment via alternating minimization,” in Proc. of _IEEE ICASSP_ , April 2009, Taiwan.
* [20] H. Weingarten, Y. Steinberg, and S. Shamai (Shitz), “The capacity region of the Gaussian multiple-input multiple-output broadcast channel,” _IEEE Trans. Inf. Theory_ , vol. 52, no. 9, pp. 3936-3964, 2006
Figure 1: Comparison of secrecy capacity for MIMO Gaussian wiretap channel
with and without helper for different $P_{t}$ and $P_{h}$. Figure 2:
Comparison of the secrecy capacity for the MIMO Gaussian wiretap channel with
and without a helper versus $P_{h}$ for different number of antennas at the
helper, $P_{t}+P_{h}=110$, assuming the eavesdropper’s channels are stronger
than those of the receiver ($\sigma_{d}^{2}=1,\sigma_{c}^{2}=5$). Figure 3:
Comparison of the secrecy capacity for the MIMO Gaussian wiretap channel with
and without a helper versus $P_{h}$ for different number of antennas at the
helper, $P_{t}+P_{h}=110$, assuming the receiver’s channels are stronger than
those of the eavesdropper ($\sigma_{d}^{2}=2,\sigma_{c}^{2}=1$). Figure 4:
Comparison of the secrecy capacity for the MIMO Gaussian wiretap channel with
and without a helper versus $P_{h}$ for different number of antennas at the
helper, $P_{t}+P_{h}=110$, and $n_{t}+n_{h}=3$. Figure 5: Secrecy data rate
versus $n_{h}$ for a specific matrix power constraint
${\mathbf{S}}=\frac{P_{t}}{n_{t}}{\mathbf{I}}$.
|
arxiv-papers
| 2011-01-24T03:09:16 |
2024-09-04T02:49:16.586353
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. Ali. A. Fakoorian and A. Lee Swindlehurst",
"submitter": "Ali Fakoorian",
"url": "https://arxiv.org/abs/1101.4435"
}
|
1101.4458
|
arxiv-papers
| 2011-01-24T07:49:23 |
2024-09-04T02:49:16.592269
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Qun Mo and Yi Shen",
"submitter": "Yi Shen",
"url": "https://arxiv.org/abs/1101.4458"
}
|
|
1101.4464
|
# Spatially Modulated Interaction Induced Bound States and Scattering
Resonances
Ran Qi and Hui Zhai Institute for Advanced Study, Tsinghua University,
Beijing, 100084, China
###### Abstract
We study the two-body problem with a spatially modulated interaction potential
using a two-channel model, in which the inter-channel coupling is provided by
an optical standing wave and its strength modulates periodically in space. As
the modulation amplitudes increases, there will appear a sequence of bound
states. Part of them will cause divergence of the effective scattering length,
defined through the phase shift in the asymptotic behavior of scattering
states. We also discuss how the local scattering length, defined through
short-range behavior of scattering states, modulates spatially in different
regimes. These results provide a theoretical guideline for new control
technique in cold atom toolbox, in particular, for alkali-earth-(like) atoms
where the inelastic loss is small.
Feshbach resonances (FR) and optical lattices (OL) are two major techniques in
cold atom toolbox. FR can be used to control the interactions by tuning a
bound state in the so-called “closed channel” to the scattering threshold via
magnetic field, laser field or external confinement RMP ; OFR ; CIR . At
resonance, the $s$-wave scattering length diverges and the system becomes a
strongly interacting one. OL can strongly modify the single particle spectrum
of atoms, which suppress the kinetic energy so that the interaction effects
are enhanced. With these two methods, many interesting many-body physics, such
as BEC-BCS crossover, superfluid to Mott insulator transition and strongly
correlated quantum fluids in low dimensions, have been studied extensively in
cold atom systems during the last decade review .
In this letter we theoretically study a new control tool for cold atom system.
It is analogous to OL because it also makes use of two counter propagating
laser fields that lead to a periodic modulation of laser intensity in space;
however, its main effect is not acted on single particle, but manifested on
the interaction term when two particles collide with each other. It generates
a spatial modulation of two-body interaction, i.e. the two-body interaction
potential not only depends on the relative coordinate of two particles under
collision, but also depends on their center-of-mass coordinate. As far as we
know, this is a situation not encountered in interacting systems studied
before, ranging from high-energy and nuclear physics to condensed matter
systems. One can expect a spatial modulated interaction potential will result
in many fascinating phenomena.
The explicit model under consideration is schematically shown in Fig. 1. The
open and closed channels are different orbital states and are coupled by laser
field. Such an optical FR has been studied before for uniform laser intensity
OFR , and has been observed experimentally for both alkaline and alkaline-
earth-like Yb atoms exp . In this letter we shall consider the situation that
the laser field is a standing wave whose intensity, and therefore the coupling
strength between open and closed channel, is modulated periodically in space.
Such a setup allows one to control spatial modulation of inter-atomic
interaction on the scale of sub-micron. Recently, this has been realized in
174Yb condensation Kyoto , although the optical standing wave is a pulsed one.
Alkiline-earth(-like) atom like Yb is particular suitable for such an
experiment because the narrow ${}^{1}S_{0}$-${}^{3}P_{1}$ inter-combination
transition line can avoid large inelastic scattering loss. In this experiment,
a spatially modulated mean-field energy has been observed from diffraction
pattern in a time-of-flight imagine Kyoto . However, the theoretical study of
this system is still very limited, and even the two-body problem has not been
studied. In this work we show that surprises indeed arise even in the two-body
problem of this model.
Figure 1: A schematic of the model we studied. See text for detail
description.
Coupled Two-channel Model. We consider a two-body Hamiltonian for a FR in
which the open and closed channels are modeled by two square well potentials
cheng
$\mathcal{H}=-\frac{\hbar^{2}}{4m}\nabla^{2}_{\bf{R}}-\frac{\hbar^{2}}{m}\nabla^{2}_{\bf{r}}+v({\bf
R},{\bf r})$ (1)
where ${\bf R}=({\bf r_{1}}+{\bf r_{2}})/2$ and ${\bf r}={\bf r_{1}}-{\bf
r_{2}}$. For $r<r_{0}$,
$v({\bf R},{\bf r})=\left(\begin{array}[]{cc}-V_{\text{o}}&\hbar\Omega({\bf
R})\\\ \hbar\Omega({\bf R})&-V_{\text{c}}\end{array}\right)$ (2)
and for $r>r_{0}$,
$v({\bf R},{\bf r})=\left(\begin{array}[]{cc}0&0\\\
0&+\infty\end{array}\right).$ (3)
In this model $V_{\text{o}}$ is given by the background scattering length
$a_{\text{bg}}$ as
$\tan(k_{\text{o}}r_{0})/(k_{\text{o}}r_{0})=1-a_{\text{bg}}/r_{0}$ where
$k_{\text{o}}=\sqrt{mV_{\text{o}}}/\hbar$, and $V_{\text{c}}$ is determined by
the binding energy of closed channel molecule $\epsilon_{\text{c}}$ through
$V_{\text{c}}=\hbar^{2}\pi^{2}/(mr^{2}_{0})-\epsilon_{\text{c}}$. The size of
inter-atomic potential $r_{0}$ is much smaller than all the other length
scales. Conventionally, the inter-channel coupling $\Omega({\bf R})$ is a
constant independent of ${\bf R}$. Such a model captures all key features of a
FR RMP ; cheng . A bound state appears at threshold and causes scattering
resonance at $\Omega_{0}=\sqrt{\epsilon_{\text{c}}/|\beta|}$, and the
scattering length is given by cheng
$a_{\text{s}}=a_{\text{bg}}\left(1-\frac{\beta\Omega^{2}}{\epsilon_{\text{c}}+\beta\Omega^{2}}\right)$
(4)
where $\beta=32r_{0}a_{\text{bg}}/(9\pi^{2})$.
Figure 2: First four bound states as a function of the amplitude of coupling
$\Omega$. The parameters for this plot are $q=0$,
$\epsilon_{\text{c}}=0.05E_{\text{R}}$, $Ka_{\text{bg}}=-0.01$ and
$Kr_{0}=10^{-3}$. $E_{\text{R}}=\hbar^{2}K^{2}/m$ is taken as energy unit.
Now consider the situation $\Omega$ depends on ${\bf R}$. Solving the
Schrödinger equation follows two steps: (i) in the regime $r<r_{0}$, because
the $-\hbar^{2}\nabla^{2}_{{\bf r}}/m$ term commutes with the Hamiltonian, we
consider the wave function of following form
$\psi_{\text{o}}=\frac{\sin kr}{r}a({\bf R});\ \ \psi_{\text{c}}=\frac{\sin
kr}{r}b({\bf R})$ (5)
where $a({\bf R})$ and $b({\bf R})$ satisfy a coupled equation
$\displaystyle\left[-\frac{\hbar^{2}}{4m}\nabla^{2}_{\bf
R}-V_{\text{o}}\right]a({\bf R})+\Omega({\bf R})b({\bf R})=\epsilon a({\bf
R})$ (6) $\displaystyle\left[-\frac{\hbar^{2}}{4m}\nabla^{2}_{\bf
R}-V_{\text{c}}\right]b({\bf R})+\Omega({\bf R})a({\bf R})=\epsilon b({\bf
R})$ (7)
where $\epsilon=E-\hbar^{2}k^{2}/m$. There will be a set of eigen-function
$a_{l}({\bf R})$, $b_{l}({\bf R})$ and $k_{l}$ that give rise to the same
energy $E$. The eigen wave function in the regime $r<r_{0}$ should be assumed
as
$\psi({\bf R},{\bf r})=\sum\limits_{l}A_{l}\frac{\sin
k_{l}r}{r}\left(\begin{array}[]{c}a_{l}({\bf R})\\\ b_{l}({\bf
R})\end{array}\right)$ (8)
(ii) The superposition coefficient $A_{l}$, the binding energy $E$ for bound
states, as well as the phase shift $\delta(E)$ for scattering states, are
determined by matching the wave function in the regime of $r>r_{0}$ at
$r=r_{0}$ for any ${\bf R}$.
Hereafter we will consider an explicit situation where $\Omega({\bf
R})=\Omega\cos(Kx)$ ($x$ denotes the $x$-component of ${\bf R}$). Note that
there is still a discrete translation symmetry $x\rightarrow x+2\pi/K$, we can
introduce a good quantum number “crystal momentum” $q$. In the regime
$r>r_{0}$, $\psi_{\text{c}}=0$, and for the bound states whose energy
$E<\hbar^{2}q^{2}/(4m)$, $\psi^{q}_{\text{o}}(x,r)$ can always be expanded as
$\psi^{q}_{\text{o}}(x,r)=e^{iqx}\sum\limits_{n}U^{q}_{n}e^{inKx}\frac{e^{-r\sqrt{(q+nK)^{2}-4mE/\hbar^{2}}}}{r}$
(9)
Eq. (9) can be viewed as the Bloch wave function for molecules. And for the
low energy scattering state whose energy is greater than but close to
$\hbar^{2}q^{2}/(4m)$, we have
$\displaystyle\psi^{q}_{\text{o}}(x,r)$
$\displaystyle=e^{iqx}\left(U_{0}\frac{\sin(kr-\delta)}{r\sin\delta}\right.$
$\displaystyle\left.+\sum\limits_{n\neq
0}U^{q}_{n}e^{inKx}\frac{e^{-r\sqrt{(q+nK)^{2}-4mE/\hbar^{2}}}}{r}\right)$
(10)
where $k=\sqrt{mE/\hbar^{2}-q^{2}/4}$, and $\delta$ is a function of $k$.
Figure 3: $rw(x,r)$ (where $w(x,r)$ is the “wannier wave function”) for the
first four bound states. $a=2\pi/K$ is the “lattice spacing”. The parameters
for this plot are $\epsilon_{\text{c}}=0.05E_{\text{R}}$,
$Ka_{\text{bg}}=-0.01$, $Kr_{0}=10^{-3}$ and $Kr=0.1$.
Results 1– Bound States: In contrast to the uniform case where there is only
one bound state when $\Omega>\Omega_{0}$, in this case we find a sequence of
bound states as $\Omega$ increases, as shown in Fig. 2. This is because the
periodic structure of coupling $\Omega({\bf R})$ leads to a “ band structure ”
for the molecules, and as the coupling strength increases, the molecules with
zero crystal momentum but in different bands touch the scattering threshold
one after the other. We can introduce the “wannier” wave function as
$w(x-x_{0},r)=\int_{-K/2}^{K/2}e^{iqx_{0}}\psi^{q}_{\text{o}}(x,r)dq$ (11)
As shown in Fig. 3, the “wannier” function for the bound states that appear at
larger $\Omega$ have more oscillation, which means that they come from higher
bands. This can also be illustrated from the symmetry of $U_{n}$ in the Bloch
function of Eq. (9), as summarized in the Table 1 for the first four bound
states. The first two bound state has even parity while the other two have odd
parity.
Figure 4: The effective scattering length defined as Eq. (12) $a_{\text{eff}}/|a_{\text{bg}}|$ as a function of $\Omega/\Omega_{0}$. (b) and (c) are enlarged plot around $\Omega/\Omega_{0}=2.64$ (b), $9.20$ (c). The arrows indicate the positions at which we plot the local scattering length $a_{\text{loc}}$ in Fig. 5(a-d). | $U_{-2}$ | $U_{-1}$ | $U_{0}$ | $U_{1}$ | $U_{2}$
---|---|---|---|---|---
1st | 0 | $+$ | 0 | $+$ | 0
2nd | $+$ | 0 | $+$ | 0 | $+$
3rd | $+$ | 0 | 0 | 0 | $-$
4th | 0 | $+$ | 0 | $-$ | 0
Table 1: Symmetry of Bloch wave function for the first four bound states
Results 2 – Effective Scattering Length: For the scattering state wave
function, at large ${\bf r}$ only the first term in Eq. (10) will not
exponentially decay, and the asymptotic behavior of the scattering wave
function is still the same as that in the uniform case. Hence we can introduce
an effective scattering length as
$a_{\text{eff}}=\lim_{k\rightarrow 0}\frac{\tan\delta(k)}{k}.$ (12)
Note that though the interaction is spatially dependent, the effective
scattering length defined as Eq. (12) is a spatial independent one. Among the
first four bound states, $a_{\text{eff}}$ only diverges when the second bound
state appears at threshold, as one can see by comparing Fig. 4(a) with Fig. 2.
This is because the divergence of $a_{\text{eff}}$ implies the first term in
Eq. (10) goes like $1/r$, which should be smoothly connected to a zero-energy
bound state with non-zero $U_{0}$. Therefore, for the other three bound states
whose $U_{0}=0$, their coupling to the low-energy scattering states vanish and
will not cause divergency of $a_{\text{eff}}$. In Fig. 4(c) we show that
$a_{\text{eff}}$ diverges when the sixth bound state (whose $U_{0}\neq 0$)
appears at scattering threshold, but the width of resonance becomes narrower
compared to Fig. 4(b) because this bound state comes from higher band and its
coupling to low-energy scattering state ( i.e. the absolute value of $U_{0}$)
is smaller.
Figure 5: The local scattering length $a_{\text{loc}}$ as a function of
position $x/a$ for $\Omega/\Omega_{0}=0.71,2.55,2.64$ and $2.7$ (a-d). The
solid blue line is calculated results, the black dashed line is the fitting
formula Eq(17) or (18), and the green dash-dotted line in (a) is from simple
replacement formula Eq. (15).
Results 3 – Local Scattering Length: At short distance the wave function Eq.
(10) satisfies the Bethe-Peierls contact condition and display
$1/r-1/a_{\text{loc}}(x)$ behavior, hence we can introduce a local scattering
length as
$a_{\text{loc}}(x)=-\lim\limits_{r\rightarrow
r_{0}}\frac{r\psi_{\text{o}}(x,r)}{\partial_{r}(r\psi_{\text{o}}(x,r))}$ (13)
Unlike in the uniform case, $a_{\text{eff}}$ and $a_{\text{loc}}$ are
different. Similar situation has also been encountered for scattering in
confined geometry CIR , lattices Cui and mixed dimension tan . What is unique
here is that $a_{\text{loc}}$ is spatially dependent. Naively, one may think
that $a_{\text{loc}}(x)$ can be obtained by replacing $\Omega$ in Eq. (4) by
local $\Omega(x)$, i.e.
$\displaystyle a_{\text{loc}}(x)$
$\displaystyle=a_{\text{bg}}\left(1-\frac{\beta\Omega^{2}\cos^{2}(Kx)}{\epsilon_{\text{c}}+\beta\Omega^{2}\cos^{2}(Kx)}\right)$
(14) $\displaystyle\approx
a_{\text{bg}}\left[1-\beta\Omega^{2}\cos^{2}(Kx)/\epsilon_{\text{c}}\right]$
(15)
where the second line is valid for small $\Omega$. This formula in fact
corresponds to an oversimplified approximation in our model that the kinetic
energy term of the center-of-mass motion
($-\hbar^{2}\nabla^{2}_{\bf{R}}/(4m)$) is completely ignored in Eq. (1). In
fact, what we really obtained from the wave function Eq. (10) is
$\displaystyle a_{\text{loc}}(x)$ $\displaystyle=\frac{1-\sum_{m\neq
0}U_{m}\cos(mKx)/U_{0}}{a^{-1}_{\text{eff}}-\sum_{m\neq
0}U_{m}|m|K\cos(mKx)/(2U_{0})}$
$\displaystyle\approx\frac{1-2U_{2}\cos(2Kx)/U_{0}}{a^{-1}_{\text{eff}}-2U_{2}K\cos(2Kx)/U_{0}}$
(16)
The second line is also valid when $\Omega$ is not too large, so the
coefficient $U_{m>2}$ is small enough that can be ignored.
Away from a resonance, $Ka_{\text{eff}}\ll 1$, Eq. (16) can be well
approximated as
$a_{\text{loc}}(x)=a_{\text{eff}}\left[1-\frac{2U_{2}}{U_{0}}\cos(2Kx)\right]$
(17)
In fact, we show in Fig. 5(a), (b) and (d) that the formula Eq. (17) (dashed
black line) is a very good approximation to the actual results (solid blue
line). In Fig. 5(a) we show the simple replacement formula Eq. (14) already
significantly deviates from the actual results in weak coupling regime. From
Fig. 5(b) and (d) one can also see that the mean value of $a_{\text{loc}}(x)$
changes sign as $a_{\text{eff}}$ changes sign. At resonance,
$a^{-1}_{\text{eff}}\rightarrow 0$, Eq. (16) can be approximated as
$a_{\text{loc}}(x)=\frac{1}{K}\left[1-\frac{U_{0}}{2U_{2}\cos(2Kx)}\right]$
(18)
We show in Fig. 5(c) that Eq. (18) is also a very good approximation to actual
$a_{\text{loc}}$ at resonance. Hence, we show that $a_{\text{loc}}$ behaves
very differently in the regime nearby or away from a scattering resonance.
Implications to Many-body Physics: In summary, we have revealed a number of
novel features in the two-body problem with a spatially modulated interaction
potential, which have strong implications for many-body physics and provide
new insights for developing new tools for quantum control in cold atom
systems.
First, when $a_{\text{eff}}$ diverges, the system enters a strongly
interacting regime and is expected to exhibit universal behavior, which can
even be manifested in the high temperature regime high-T . For a two-component
Fermi gas, it provides a new route toward BEC-BCS crossover physics, and
“high-temperature” superfluid may exist in this regime. The periodic structure
will add new ingredient to the crossover physics.
Secondly, for the low-energy states whose energy $|E|\ll E_{\text{R}}$, the
energy dependence of scattering length can be ignored and the many-body system
can be effectively described by a pseudo-potential model:
$\hat{H}=-\sum\limits_{i}\frac{\hbar^{2}\nabla^{2}_{\bf{r}_{i}}}{2m}+\sum\limits_{ij}\frac{4\pi\hbar^{2}a_{\text{loc}}({\bf
R}_{ij})}{m}\delta^{3}(\mathbf{r_{ij}})\frac{\partial}{\partial
r_{ij}}r_{ij},$ (19)
where ${\bf R}_{ij}=({\bf r}_{i}+{\bf r}_{j})/2$ and $r_{ij}={\bf r}_{i}-{\bf
r}_{j}$. It is very important that $a_{\text{loc}}({\bf R})$ in the pseudo-
potential of Eq. (19) is given by Eq. (16) from the two-body calculation, so
that a two-body problem of the Hamiltonian Eq. (19) can produce correct low-
energy eigen-wave function and the effective scattering length as from model
potential.
For bosons, with a mean-field approximation, Eq. (19) implies that the
interaction energy should take the form
$E_{\text{mf}}=\frac{4\pi\hbar^{2}}{m}\int a_{\text{loc}}(x)n^{2}(x)dx$ (20)
which leads to a modulation of condensate density $n(x)$ and self-trapping
nearby the minimum of $a_{\text{loc}}(x)$. It is very likely a strong enough
modulation of condensate density will eventually result in the loss of
superfluidity and the system enters an insulating phase. If so, it provides a
completely different mechanism for superfluid to insulator transition where
the transition is not driven by suppression of kinetic energy as in
conventional OL.
Final Comments: In this work we choose a coupled two square-well model whose
advantage is that the physics can be demonstrated in a simple and transparent
way. However, some more sophisticated effects in real system, such as the
inelastic loss, are ignored. We have also implemented more systematic
scattering theory which includes these effects and found that the physics
discussed here will remain qualitatively unchanged. These results will be
published elsewhere Qiran .
Moreover, the formalism used in this work can be easily generalized to other
realizations of spatial modulation of interactions. For instance, in a
magnetic FR, one can consider the presence of a magnetic field gradient so
that the closed channel molecular energy varies spatially. This effect is
particularly important for a narrow resonance. One can also optically couple
the closed channel molecule to another molecular state via a bound-bound
transition, which leads to a periodic variation of molecule energy Rempe .
Similar effects as discussed in Results 1-3 also present in these cases Qiran
.
Acknowledgements. We thank Xiaoling Cui, Zeng-Qiang Yu, Peng Zhang and Zhenhua
Yu for helpful discussions. This work is supported by Tsinghua University
Initiative Scientific Research Program, NSFC under Grant No. 11004118 and
NKBRSFC under Grant No. 2011CB921500.
## References
* (1) C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev. Mod. Phys. 82, 1225 (2010).
* (2) P. O. Fedichev, Yu. Kagan, G. V. Shlyapnikov, and J. T. M. Walraven, Phys. Rev. Lett. 77, 2913 (1996); J. L. Bohn and P. S. Julienne, Phys. Rev. A 56, 1486 (1997).
* (3) M. Olshanii, Phys. Rev. Lett. 81, 938 (1998); T. Bergeman, M. G. Moore, and M. Olshanii, Phys. Rev. Lett. 91, 163201 (2003).
* (4) I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008).
* (5) F. K. Fatemi, K. M. Jones, and P. D. Lett, Phys. Rev. Lett. 85, 4462 4465 (2000); M. Theis, et al. Phys. Rev. Lett. 93, 123001 (2004) and K. Enomoto, K. Kasa, M. Kitagawa, and Y. Takahashi, Phys. Rev. Lett. 101, 203201 (2008).
* (6) R Yamazaki, S. Taie, S. Sugawa, and Y. Takahashi, Phys. Rev. Lett. 105, 050405 (2010).
* (7) C. Chin, arXiv: 0506313.
* (8) X. Cui, Y. Wang, and F. Zhou, Phys. Rev. Lett. 104, 153201 (2010) and H. P. Büchler, Phys. Rev. Lett. 104, 090402 (2010)
* (9) Y. Nishida and S. Tan, Phys. Rev. Lett. 101, 170401 (2008)
* (10) T. L. Ho and E. J. Mueller, Phys. Rev. Lett. 92, 160404 (2004).
* (11) R. Qi, P. Zhang and H. Zhai in preparation.
* (12) D. M. Bauer, et al. Nature Physics 5, 339 (2009).
Appendix: In this appendix, we present some details of solving the two-body
Schrödinger equation. Using the discrete translation symmetry, we expand
$\displaystyle a^{q}(x)=e^{iqx}\sum_{n}e^{inKx}a^{q}_{n}$ (21) $\displaystyle
b^{q}(x)=e^{iqx}\sum_{n}e^{inKx}b^{q}_{n},$ (22)
$a_{n}$ and $b_{n}$ satisfy coupled matrix equation
$\displaystyle\left(\frac{\hbar^{2}(q+nK)^{2}}{4m}-V_{\text{o}}\right)a^{q}_{n}+\frac{\Omega}{2}\left(b^{q}_{n-1}+b^{q}_{n+1}\right)=\epsilon^{q}a^{q}_{n}$
(23)
$\displaystyle\left(\frac{\hbar^{2}(q+nK)^{2}}{4m}-V_{\text{c}}\right)b^{q}_{n}+\frac{\Omega}{2}\left(a^{q}_{n-1}+a^{q}_{n+1}\right)=\epsilon^{q}b^{q}_{n}$
(24)
This matrix has a set of eigen-values $\epsilon^{q}_{l}$ and their eigen-
vectors $\\{a^{q}_{l,n},b^{q}_{l,n}\\}$. Hence there are a set of wave
functions sharing the same energy $E$, which ar
$\displaystyle\psi^{q}_{\text{o},l}=e^{iqx}\frac{\sin(k^{q}_{l}r)}{r}\sum\limits_{n}e^{inKx}a^{q}_{l,n}$
(25)
$\displaystyle\psi^{q}_{\text{c},l}=e^{iqx}\frac{\sin(k^{q}_{l}r)}{r}\sum\limits_{n}e^{inKx}b^{q}_{l,n}$
(26)
where $k^{q}_{l}=\sqrt{m(E-\epsilon^{q}_{l})}/\hbar$ is a function of $E$. In
general, the eigen-states take the form
$\psi_{q}=\sum\limits_{l}A^{q}_{l}\left(\begin{array}[]{c}\psi^{q}_{\text{o},l}\\\
\psi^{q}_{\text{c},l}\end{array}\right)=e^{iqx}\sum\limits_{n}e^{inKx}\left(\begin{array}[]{c}\varphi^{q}_{\text{o},n}\\\
\varphi^{q}_{\text{c},n}\end{array}\right)$ (27)
where
$\displaystyle\varphi^{q}_{\text{o},n}({\bf
r})=\sum\limits_{l}A^{q}_{l}\frac{\sin(k^{q}_{l}r)}{r}a^{q}_{l,n};\ \
\varphi^{q}_{\text{c},n}({\bf
r})=\sum\limits_{l}A^{q}_{l}\frac{\sin(k^{q}_{l}r)}{r}b^{q}_{l,n}$
For bound states, to match the boundary condition with Eq. (9) at $r_{0}$ in
both open and closed channels, we obtain a matrix equation
$M^{q}_{kl}(E)A^{q}_{l}=0$ where
$\displaystyle M^{q}_{2n+1,l}=\sin(k^{q}_{l}r_{0})b^{q}_{l,n}$ $\displaystyle
M^{q}_{2n,l}=(k^{q}_{l}\cos(k^{q}_{l}r_{0})+\sqrt{(q+nK)^{2}-\frac{4mE}{\hbar^{2}}}\sin(k^{q}_{l}r_{0}))a^{q}_{l,n}$
Therefore for a given $q$ the eigen-energy $E$ is determined by
$\text{Det}(M)=0$ and
$U^{q}_{n}=e^{r_{0}\sqrt{\frac{\hbar^{2}(q+nK)^{2}}{4m}-E}}\sum\limits_{l}A_{l}a^{l}_{n}\sin(k^{l}r_{0})$
(28)
For the scattering states, $M^{q}_{2n+1,l}$ and $M^{q}_{2n,l}$ ($n\neq 0$) are
the same as bound state, while
$\displaystyle
M^{q}_{0,l}=\sin(k^{q}_{l}r_{0})\cos(kr_{0}-\delta)k-\cos(k^{q}_{l}r_{0})\sin(kr_{0}-\delta)a^{l}_{0}$
where $k=\sqrt{mE/\hbar^{2}-q^{2}/4}$. In this case $\text{Det}(M)=0$ gives
rise to the relation between phase shift $\delta$ and energy $E$. $U^{q}_{n}$
($n\neq 0$) is also the same as Eq. (28), while for $n=0$,
$U^{q}_{0}=\frac{\sin\delta}{\sin(kr_{0}-\delta)}\sum\limits_{l}A_{l}\sin(k^{q}_{l}r_{0})a^{l}_{0}$
(29)
|
arxiv-papers
| 2011-01-24T08:36:31 |
2024-09-04T02:49:16.595851
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ran Qi and Hui Zhai",
"submitter": "Ran Qi",
"url": "https://arxiv.org/abs/1101.4464"
}
|
1101.4556
|
# Distinct Fermi Surface Topology and Nodeless Superconducting Gap in
(Tl0.58Rb0.42)Fe1.72Se2 Superconductor
Daixiang Mou1, Shanyu Liu1, Xiaowen Jia1, Junfeng He1, Yingying Peng1, Lin
Zhao1, Li Yu1, Guodong Liu1, Shaolong He1, Xiaoli Dong1, Jun Zhang1, Hangdong
Wang2, Chiheng Dong2, Minghu Fang2, 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, Zhejiang University, Hangzhou 310027, China
3Technical Institute of Physics and Chemistry, Chinese Academy of Sciences,
Beijing 100190, China
(January 24, 2011)
###### Abstract
High resolution angle-resolved photoemission measurements have been carried
out to study the electronic structure and superconducting gap of the
(Tl0.58Rb0.42)Fe1.72Se2 superconductor with a Tc=32 K. The Fermi surface
topology consists of two electron-like Fermi surface sheets around $\Gamma$
point which is distinct from that in all other iron-based compounds reported
so far. The Fermi surface around the M point shows a nearly isotropic
superconducting gap of $\sim$12 meV. The large Fermi surface near the $\Gamma$
point also shows a nearly isotropic superconducting gap of $\sim$15 meV while
no superconducting gap opening is clearly observed for the inner tiny Fermi
surface. Our observed new Fermi surface topology and its associated
superconducting gap will provide key insights and constraints in understanding
superconductivity mechanism in the iron-based superconductors.
###### pacs:
74.70.-b, 74.25.Jb, 79.60.-i, 71.20.-b
The discovery of the Fe-based superconductorsKamihara ; ZARenSm ; RotterSC ;
MKWu11 ; CQJin111 has attracted much attention because it represents the
second class of high temperature superconductors in addition to the copper-
oxide (cuprate) superconductorsBednorz . It is important to explore whether
the high-Tc superconductivity mechanism in this new Fe-based system is
conventional, parallel to that in cuprates, or along a totally new
routeNPReview . Different from the cuprates where the low-energy electronic
structure is dominated by Cu 3d${}_{x^{2}-y^{2}}$ orbital, the electronic
structure of the Fe-based compounds involves all five Fe 3d-orbitals forming
multiple Fermi surface sheets: hole-like Fermi surface sheets around
$\Gamma$(0,0) and electron-like ones around M($\pi$,$\pi$)DJSingh1111 ; Kuroki
. It has been proposed that the interband scattering between the hole-like
bands near $\Gamma$ and electron-like bands near M gives rise to electron
pairing and superconductivityKuroki ; FeSCMagnetic . An alternative picture is
also proposed based on the interaction of local Fe magnetic momentJ1J2Picture
.
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 ; MHFang ;
GFChen is surprising that provides new perspectives in understanding Fe-based
compounds. First, it may involve Fe vacancies in the FeSe layerMHFang ; GFChen
; ZWang . This is against the general belief that a perfect Fe-sublattice is
essential for superconductivity in the Fe-based compounds, similar to that
perfect CuO2 plane is considered essential for cuprate superconductors.
Second, the superconductivity of AxFe2-ySe2 is realized in a close proximity
to an antiferromagnetic semiconducting (insulating) phaseMHFang ; GFChen ;
QMSi . This is in strong contrast to other Fe-based compounds where the parent
compounds are spin-ordered metalsDongSDW ; RotterParent ; PCDai ; BFSNeutron3
. Third, the intercalation of A=(K,Tl,Cs,Rb and etc.) in AxFe2-ySe2 is
expected to introduce large number of electrons into the system; this usually
would lead to suppression or disappearance of superconductivity like in
heavily electron-doped Ba(Fe,Co)2As2 systemCo122 . The existence of
superconductivity in AxFe2-ySe2 at such a high Tc (over 30 K) with so high
electron doping is unexpected. Most interestingly, band structure
calculationsLJZhang ; IRShein ; XWYan and electronic structure
measurementsYZhang ; TQian all suggest that high electron doping in
AxFe2-ySe2 may lead to the disappearance of the hole-like Fermi surface sheets
around $\Gamma$. This would render it impossible for the electron scattering
between the hole-like bands near $\Gamma$ and electron-like bands near M that
is considered to be important for the electron pairing in Fe-based
superconductors by some theoriestsKuroki ; FeSCMagnetic . Therefore,
investigations of the AxFe2-ySe2 system would be important in searching for
common ingredients underlying the physical properties, especially the
superconductivity of the Fe-based superconductors.
Figure 1: Fermi surface of (Tl0.58Rb0.42)Fe1.72Se2 superconductor (Tc=32 K).
(a). Spectral weight distribution integrated within [-20meV,10meV] energy
window near the Fermi level as a function of kx and ky measured using
h$\nu$=21.2 eV light source. Two Fermi surface sheets are observed around
$\Gamma$ point which are marked as $\alpha$ for the inner small sheet and
$\beta$ for the outer large one. Near the M($\pi$,$\pi$) point, one Fermi
surface sheet is clearly observed which is marked as $\gamma$. (b) Three-
dimensional image of Fig. 1a. (c). Fermi surface mapping measured using
h$\nu$=40.8 eV light source Although the signal is relatively weak, one can
see traces of two Fermi surface sheets around $\Gamma$ and one around M.
In this paper, we report observation of a distinct Fermi surface topology and
nearly isotropic nodeless superconducting gap in (Tl0.58Rb0.42)Fe1.72Se2
superconductor (Tc=32 K) from high resolution angle-resolved photoemission
(ARPES) measurements. We have observed an electron-like Fermi surface sheet
near M($\pi$,$\pi$) and two electron-like Fermi surface sheets near
$\Gamma$(0,0). This Fermi surface topology is distinct from the hole-like
Fermi surface sheets near the $\Gamma$ point found in other Fe-based
compoundsDJSingh1111 ; Kuroki or disappearance of hole-like Fermi surface
sheets near $\Gamma$ in AxFe2-ySe2 compoundsLJZhang ; IRShein ; XWYan ; YZhang
; TQian . We observe nearly isotropic superconducting gap around the Fermi
surface sheets near $\Gamma$ ($\sim$15 meV) and M ($\sim$12 meV); no gap node
is observed in both Fermi surface sheets. These rich information on this new
Fe-based superconductor will provide key insights on the superconductivity
mechanism in the Fe-based superconductors.
High resolution angle-resolved photoemission measurements were carried out on
our lab system equipped with a Scienta R4000 electron energy analyzerGDLiu .
We use Helium discharge lamp as the light source which can provide photon
energies of h$\upsilon$= 21.218 eV (Helium I) and 40.8 eV (Helium II). The
energy resolution was set at 10 meV for the Fermi surface mapping (Fig. 1a)
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 (Tl,Rb)Fe2-ySe2 crystals
were grown by the Bridgeman methodMHFang . Their composition determined by
using an Energy Dispersive X-ray Spectrometer (EDXS) measurement is
(Tl0.58Rb0.42)Fe1.72Se2. The crystals have a sharp superconducting transition
at Tc(onset)=32 K with a transition width of $\sim$1 K. The crystal was
cleaved in situ and measured in vacuum with a base pressure better than
5$\times$10-11 Torr.
Figure 2: Band structure and photoemission spectra of (Tl0.58Rb0.42)Fe1.72Se2
measured along two high symmetry cuts. (a). Band structure along the Cut 1
crossing the $\Gamma$ point; the location of the cut is shown on top of Fig.
2a. The $\alpha$ band and two Fermi crossings of the $\beta$ band ($\beta_{L}$
and $\beta_{R}$) are marked. (b). Corresponding EDC second derivative image of
Fig. 2a. (c). Band structure along the Cut 2 crossing M point; the location of
the cut is shown on top of Fig. 2c. The two Fermi crossings of the $\gamma$
band ($\gamma_{L}$ and $\gamma_{R}$) are marked. (d). Corresponding EDC second
derivative image of Fig. 2c. (e). EDCs corresponding to Fig. 2a for the Cut 1.
(f). EDCs corresponding to Fig. 2c for the Cut 2.
Fig. 1 shows Fermi surface mapping of the (Tl0.58Rb0.42)Fe1.72Se2
superconductor covering multiple Brillouin zones. The band structure along two
typical high symmetry cuts are shown in Fig. 2. An electron-like Fermi surface
is clearly observed around M($\pi$,$\pi$) (Fig. 1a, Figs. 2c and 2d). This
Fermi surface (denoted as $\gamma$ hereafter) is nearly circular with a Fermi
momentum (kF) of 0.35 in a unit of $\pi$/a (lattice constant a=3.896 $\AA$).
The Fermi surface near the $\Gamma$ point consists of two sheets. The inner
tiny pocket (denoted as $\alpha$) is electron-like with a band bottom barely
touching the Fermi level ( Figs. 2a and 2b for the Cut 1). The outer larger
Fermi surface sheet around $\Gamma$ (denoted as $\beta$) (Fig. 1a) is
electron-like (Figs. 2a and 2b) with a Fermi momentum of 0.35 $\pi$/a.
The observation of two electron-like Fermi surface sheets, $\alpha$ and
$\beta$, around $\Gamma$ in (Tl0.58Rb0.42)Fe1.72Se2 is distinct from that
observed in other Fe-based compounds where hole-like pockets are expected
around the $\Gamma$ pointDJSingh1111 ; Kuroki . It is also different from the
band structure calculationsLJZhang ; IRShein ; XWYan ; YZhang ; TQian and
previous ARPES measurementsYZhang ; TQian on AxFe2-ySe2 that suggest
disappearance of hole-like Fermi surface sheets near $\Gamma$ because of the
lifted chemical potential due to a large amount of electron doping.
Figure 3: Temperature dependence of energy bands and superconducting gap near
$\Gamma$ and M points. (a). Photoemission images along the Cut B (bottom-right
inset). (b). Photoemission spectra at the Fermi crossing of $\gamma$ Fermi
surface and their corresponding symmetrized spectra (c) measured at different
temperatures. (d). Temperature dependence of the superconducting gap. The
dashed line is a BCS gap form. (e). Photoemission images along the Cut A
(bottom-right inset) at different temperatures. The original EDCs at the
$\Gamma$ point and at the Fermi crossing of $\beta$ Fermi surface measured at
different temperatures are shown in (f) and their corresponding symmetrized
EDCs are shown in (g).
One immediate question is on the origin of the electron-like $\beta$ band
around $\Gamma$. The first possibility is whether it could be a surface state.
While surface state on some Fe-based compounds like the “1111” system was
observed beforeHYLiu , it has not been observed in the “11”-type Fe(Se,Te)
systemFeSTARPES . The second possibility is whether the $\beta$ band can be
caused by the folding of the electron-like $\gamma$ Fermi surface near M. It
is noted that the Fermi surface size, the band dispersion, and the band width
of the $\beta$ band at $\Gamma$ is similar to that of the $\gamma$ band near
M. A band folding picture would give a reasonable account for such a
similarity if there exists a ($\pi$,$\pi$) modulation in the system that can
be either structural or magnetic. An obvious issue with this scenario is that,
in this case, one should also expect the folding of the $\alpha$ band near
$\Gamma$ onto the M point; but such a folding is not observed at the M point
(Fig. 1a and Fig. 2c). The third possibility is whether the measured $\beta$
sheet is a Fermi surface at a special kz cut. Although the Fermi surface at
$\Gamma$ is absent in TlFe2Se2 from the band structure calculationsLJZhang ,
there is a 3-dimensional Fermi pocket that is present near the zone center at
kz=$\pi$$/$c when x is close to 1 in KxFe2Se2IRShein and CsxFe2Se2XWYan . We
note that the electron doping in (Tl0.58Rb0.42)Fe1.72Se2 is lower than that of
(K,Cs)Fe2Se2. Also we observed similar $\beta$ Fermi surface at different
photon energies (Fig. 1a and Fig. 1c) which corresponds to different kz. The
final resolution of this possibility needs further detailed photon energy
dependent measurements.
The clear identification of various Fermi surface sheets makes it possible to
investigate the superconducting gap in this new superconductor. We start first
by examining the superconducting gap near the M point. Fig. 3a shows the
photoemission images along the Cut B near M (its location shown in the bottom-
right inset of Fig. 3) at different temperatures. The corresponding
photoemission spectra (energy distribution curves, EDCs) on the Fermi momentum
at different temperatures are shown in Fig. 3b. To visually inspect possible
gap opening and remove the effect of Fermi distribution function near the
Fermi level, we have symmetrized these original EDCs to get spectra in Fig.
3c, following the procedure that is commonly used in high temperature cuprate
superconductorsMNorman . For the $\gamma$ pocket near M, there is a clear gap
opening at low temperature (15 K), as indicated by an obvious dip at the Fermi
energy in the symmetrized EDCs (Fig. 3c). With increasing temperature, the dip
at EF is gradually filled up and is almost fully filled above Tc=32 K. The gap
size at different temperatures is extracted from the peak position of the
symmetrized EDCs or fitted by the phenomenological formulaMNorman (Fig. 3d);
it is $\sim$11 meV at 15 K. The temperature dependence of the gap size roughly
follows the BCS-type form (Fig. 3d). Similar temperature dependent
measurements of the superconducting gap were also carried out along the
$\Gamma$-M cut near $\Gamma$ (Figs. 3e-g). The Fermi crossing on the $\beta$
Fermi surface also displays a clear superconducting gap in the superconducting
state which is closed above Tc (lower curves in Figs. 3f and 3g). For the
peculiar tiny $\alpha$ pocket near $\Gamma$, we do not find signature of clear
superconducting gap opening below Tc (upper curves in Figs. 3f and 3g).
Now we come to the momentum-dependent measurements of the superconducting gap.
For this purpose we took high resolution Fermi surface mapping (energy
resolution of 4 meV) of the $\gamma$ pocket at M (Fig. 4a) and the $\beta$
pocket at $\Gamma$ (Fig. 4b). Fig. 4c shows photoemission spectra around the
$\gamma$ Fermi surface measured in the superconducting state (T= 15 K); the
corresponding symmetrized photoemission spectra are shown in Fig. 4d. The
extracted superconducting gap (Fig. 4g) is nearly isotropic with a size of
(12$\pm$2) meV. The superconducting gap around the $\beta$ Fermi surface near
$\Gamma$ is also nearly isotropic with a size of (15$\pm$2) meV (Figs. 4e, 4f
and 4g).
Figure 4: Momentum dependent superconducting gap along the $\gamma$ and the
$\beta$ Fermi surface sheets measured at T=15 K. Fermi surface mapping near M
(a) and near $\Gamma$ (b) and the corresponding Fermi crossings marked by red
circles. (c). EDCs along the $\gamma$ Fermi surface and their corresponding
symmetrized EDCs (d). (e). EDCs along the $\beta$ Fermi surface and their
corresponding symmetrized EDCs (f). (g). Momentum dependence of the
superconducting gap along the $\gamma$ Fermi surface sheet (red circles) and
along the $\beta$ Fermi surface sheet (blue circles).
The observation of a distinct Fermi surface topology in
(Tl0.58Rb0.42)Fe1.72Se2 has important implications to the understanding of
superconductivity in Fe-based superconductors. The realization of high Tc in
this new superconductor with a distinct Fermi surface topology is helpful to
sort out key electronic structure ingredient that is responsible for
superconductivity. With the electron-like $\beta$ Fermi surface present in
(Tl0.58Rb0.42)Fe1.72Se2, the possibility of electron scattering between the
$\Gamma$ Fermi surface sheet(s) and the M Fermi surface sheet(s), proposed by
some theories to account for superconductivity in the Fe-based
superconductorKuroki ; FeSCMagnetic , cannot be ruled out. However, the
electron scattering between two electron-like bands may have different effect
on the electron pairing from that between an electron-like band and a hole-
like band. The nearly isotropic superconducting gap on the $\beta$ and
$\gamma$ Fermi surface sheets, together with the absence of gap nodes, appears
to favor s-wave superconducting gap symmetry in (Tl0.58Rb0.42)Fe1.72Se2. This
is similar to that in (Ba0.6K0.4)Fe2As2122Gap and NdFeAsO0.9F0.1Kaminski1111
. The gap size of 12 (for $\gamma$) and 15 meV (for $\beta$) gives a ratio of
2$\Delta$/kTc=9 and 11, respectively, which is significantly larger than the
traditional BCS weak-coupling value of 3.52 for an s-wave gap. This indicates
that this new superconductor is at least in the strong coupling regime in
terms of the BCS picture. These will put strong constraints on various
proposed gap symmetries and the underlying pairing mechanisms for the iron-
based superconductors.
In summary, we have identified a distinct Fermi surface topology in the new
(Tl0.58Rb0.42)Fe1.72Se2 superconductor that is different from all other Fe-
based superconductors reported so far. Near the $\Gamma$ point, two electron-
like Fermi surface sheets are observed that are different from the band
structure calculations and previous ARPES measurement results. We observed
nearly isotropic superconducting gap around the Fermi surface sheets near
$\Gamma$ and M without gap nodes. These rich information will shed more light
on the nature of superconductivity in the Fe-based superconductors.
XJZ and MHF thank the funding support from NSFC (Grant No. 10734120 and
10974175) and the MOST of China (973 program No: 2011CB921703 and
2011CBA00103).
∗Corresponding author: XJZhou@aphy.iphy.ac.cn
## References
* (1) Y. Kamihara et al., J. Am. Chem. Soc. 130, 3296 (2008).
* (2) Z. A. Ren et al., Chin. Phys. Lett. 25, 2215 (2008).
* (3) M. Rotter et al., Phys. Rev. Lett. 101, 107006(2008).
* (4) F. C. Hsu et al., Proc. Natl. Acad. Sci. USA 105, 14262 (2008).
* (5) X. C. Wang et al., Solid State Commun. 148, 538(2008).
* (6) J. G. Bednorz and K. A. Mueller, Z. Phys. B 64, 189 (1986).
* (7) K. Ishida et al., J. Phys. Soc. Japan 78, 062001 (2009); J. Paglione and R. L. Greene, Nature Phys. 6, 645 (2010).
* (8) D. J. Singh and M.-H. Du, Phys. Rev. Lett. 100, 237003 (2008).
* (9) K. Kuroki et al., Phys. Rev. Lett. 101, 087004 (2008).
* (10) 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).
* (11) T. Yildirim, Phys. Rev. Lett. 101, 057010(2008); Q. M. Si and E. Abrahams, Phys. Rev. Lett. 101, 076401(2008); J. S. Wu et al.,, Phys. Rev. Lett. 101, 126401(2008); C. Fang et al., Phys. Rev. B 77, 224509 (2008); C. K. Xu et al., Phys. Rev. B 78, 020501(R) (2008).
* (12) J. G. Guo et al., Phys. Rev. B 82, 180520(R) (2010).
* (13) A. Krzton-Maziopa et al., arXiv:1012.3637.
* (14) M. H. Fang et al., arXiv:1012.5236.
* (15) D. M. Wang et al., arXiv:1101.0789.
* (16) Z. Wang et al., arXiv:1101.2059.
* (17) R. Yu et al., arXiv:1101.3307.
* (18) J. Dong et al., Europhys. Lett. 83, 27006 (2008).
* (19) M. Rotter et al., Phys. Rev. B 78, 020503(R)(2008).
* (20) C. de la Cruz et al., Nature (London) 453, 899 (2008).
* (21) Q. Huang et al., Phys. Rev. Lett. 101, 257003 (2008).
* (22) Y. Laplace et al., Phys. Rev. B 80, 140501 (2008).
* (23) L. J. Zhang and D. J. Singh, Phys. Rev. B 79, 094528 (2009).
* (24) I.R. Shein and A.L. Ivanovskii, arXiv:1012.5164.
* (25) X. W. Yan et al., arXiv:1012.5536.
* (26) Y. Zhang et al., arXiv:1012.5980.
* (27) T. Qian et al., arXiv:1012.6017.
* (28) G. D. Liu et al., Rev. Sci. Instruments 79, 023105 (2008).
* (29) H. Y. Liu et al., Phys. Rev. Lett. 105, 027001. (2010).
* (30) Y. Xia et al., Phys. Rev. Lett. 103 037002 (2009); F. Chen, Phys. Rev. B. 81 014526 (2009); A. Tamai et al., Phys. Rev. Lett. 104 097002 (2010); K. Nakayama et al., Phys. Rev. Lett. 105, 197001 (2010).
* (31) M. R. Norman et al., Phys. Rev. B 57, R11093 (1998).
* (32) L. Zhao et al., Chin. Phys. Lett. 25, 4402(2008); H. Ding et al., Europhys. Lett. 83, 47001 (2008); D. V. Evtushinsky, et al., Phys. Rev. B 79, 054517 (2009); L. Wray et al., Phys. Rev. B 78, 184508 (2008).
* (33) T. Kondo et al., Phys. Rev. Lett. 101 (2008)147003.
|
arxiv-papers
| 2011-01-24T14:56:57 |
2024-09-04T02:49:16.602615
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Daixiang Mou, Shanyu Liu, Xiaowen Jia, Junfeng He, Yingying Peng, Lin\n Zhao, Li Yu, Guodong Liu, Shaolong He, Xiaoli Dong, Jun Zhang, Hangdong Wang,\n Chiheng Dong, Minghu Fang, Xiaoyang Wang, Qinjun Peng, Zhimin Wang, Shenjin\n Zhang, Feng Yang, Zuyan Xu, Chuangtian Chen and X. J. Zhou",
"submitter": "Xingjiang Zhou",
"url": "https://arxiv.org/abs/1101.4556"
}
|
1101.4580
|
# $B$-physics with dynamical domain-wall light quarks and relativistic
$b$-quarks
Ruth S. Van de Water and
Brookhaven National Laboratory, Department of Physics, Upton, NY 11973, USA
E-mail
###### Abstract:
We report on our progress in calculating the $B$-meson decay constants and
$B^{0}$-$\bar{B}^{0}$ mixing parameters using domain-wall light quarks and
relativistic $b$-quarks. We present our computational method and show some
preliminary results obtained on the coarser ($a\approx 0.11$fm) $24^{3}$
lattices. This work is presented on behalf of the RBC and UKQCD
collaborations.
## 1 Introduction
The study of $B$-meson physics on the lattice is of special phenomenological
interest because it allows one to obtain constraints on the CKM unitarity
triangle. In the standard global unitarity triangle fit, the apex of the CKM
unitarity triangle is constrained using lattice input for neutral $B$-meson
mixing. Experimentally, $B_{q}$–$\bar{B}_{q}$ mixing is well measured in terms
of mass differences (oscillation frequencies) $\Delta m_{q}$; in the standard
model it is parameterized by [1]
$\displaystyle\Delta
m_{q}=\frac{G_{F}^{2}m^{2}_{W}}{6\pi^{2}}\eta_{B}S_{0}m_{B_{q}}{f_{B_{q}}^{2}B_{B_{q}}}\lvert
V_{tq}^{*}V_{tb}\rvert^{2},$ (1)
where the index $q$ denotes either a $d$\- or a $s$-quark, $m_{B_{q}}$ is the
mass of the $B_{q}$-meson, and $V_{tq}^{*}$ and $V_{tb}$ are CKM matrix
elements. The Inami-Lim function, $S_{0}$ [2], and the QCD coefficient,
$\eta_{B}$ [1], can be computed perturbatively, whereas the non-perturbative
input is given by $f_{q}^{2}B_{B_{q}}$: the leptonic decay constant
$f_{B_{q}}$ and the $B$-meson bag parameter $B_{B_{q}}$. Computing the ratio
$\Delta m_{s}$/$\Delta m_{d}$ is particularly advantageous because statistical
and systematic uncertainties largely cancel and, moreover, the ratio of CKM
matrix elements becomes accessible
$\displaystyle\frac{\Delta m_{s}}{\Delta
m_{d}}=\frac{m_{B_{s}}}{m_{B_{d}}}\,{\xi^{2}}\,\frac{\lvert
V_{ts}\rvert^{2}}{\lvert V_{td}\rvert^{2}}.$ (2)
The non-perturbative input is now solely contained in the $SU(3)$-breaking
ratio
$\displaystyle\xi$
$\displaystyle=\frac{f_{B_{s}}\sqrt{B_{B_{s}}}}{f_{B_{d}}\sqrt{B_{B_{d}}}}.$
(3)
An alternative way of constraining the CKM triangle has been proposed by
Lunghi and Soni [3] that uses the CKM matrix elements $V_{ub}$ or $V_{cb}$,
thereby avoiding the tension between inclusive and exclusive determinations of
both $V_{ub}$ (3$\sigma$) and $V_{cb}$ (2$\sigma$). This alternative method,
however, requires precise knowledge of the decay constant $f_{B}$ as well as
$BR(B\to\tau\nu)$ and $\Delta m_{s}$. Moreover, Lunghi and Soni point out [4,
5] that already the current data may show signs of physics beyond the standard
model manifesting themselves as deviations of experimental values for
$\sin(2\beta)$ (3.3$\sigma$) and $BR(B\to\tau l\nu)$ (2.8$\sigma$) from the
Standard Model. They argue that the most likely sources for new physics are in
$B_{q}$ mixing and $\sin(2\beta)$.
Therefore, it is timely for the lattice community to determine these
parameters precisely and prove by using statistically and systematically
independent setups that all sources of errors are well under control.
Currently, there are three determinations for $\xi$ using 2+1-flavor gauge
field configurations: an exploratory study by the RBC-UKQCD collaboration [6]
with non-competitive errors and two precise determinations by Fermilab-MILC
[7, 8] and HPQCD [9], which however both rely on the same set of MILC gauge
field configurations. The latter two collaborations also use the same
configurations to obatin the decay constants $f_{B_{d}}$ and $f_{B_{s}}$ and
find good agreement. Here we present the first results of our project to
determine $B$-meson parameters using domain-wall light quarks and relativistic
$b$-quarks.
## 2 Computational setup
Our project is based on using the RBC-UKQCD 2+1-flavor domain wall lattices
with the Iwasaki gauge action for the gluons. Our first results presented in
these proceedings are obtained on the coarser $a\approx 0.11$fm lattices with
a spatial volume of $24^{3}$ in lattice units [10], whereas the full project
will also utilize the finer $a\approx 0.08$fm lattices ($32^{3}$) [11] (for
details see Tab. 1).
| | | | | approx. | # time
---|---|---|---|---|---|---
L | $a$(fm) | $m_{l}$ | $m_{s}$ | $m_{\pi}$(MeV) | # configs. | sources
24 | $\approx$ 0.11 | 0.005 | 0.040 | 331 | 1640 | 1
24 | $\approx$ 0.11 | 0.010 | 0.040 | 419 | 1420 | 1
24 | $\approx$ 0.11 | 0.020 | 0.040 | 558 | 350 | 8
32 | $\approx$ 0.08 | 0.004 | 0.030 | 307 | 600 | 1
32 | $\approx$ 0.08 | 0.006 | 0.030 | 366 | 900 | 1
32 | $\approx$ 0.08 | 0.008 | 0.030 | 418 | 550 | 1
Table 1: Overview of the gauge field ensembles to be used for this project.
For the heavy quarks we use a relativistic formulation derived from the
“Fermilab action” [12] in which we tune the three relevant parameters of the
action non-perturbatively. As demonstraged by Christ, Li and Lin, this
requires only one additional experimental input compared to the formulation of
Fermilab [13]. The relativistic heavy quark (RHQ) action is given by
$\displaystyle
S=\sum_{n,n^{\prime}}\bar{\Psi}_{n}\left\\{\\!m_{0}+\gamma_{0}D_{0}-\\!\frac{aD_{0}^{2}}{2}+\zeta\left[\vec{\gamma}\cdot\vec{D}-\frac{a\left(\vec{D}\right)^{2}}{2}\right]\\!-a\sum_{\mu\nu}\frac{ic_{P}}{4}\sigma_{\mu\nu}F_{\mu\nu}\\!\right\\}_{\\!\\!n,n^{\prime}}\\!\\!\\!\\!\Psi_{n^{\prime}},$
(4)
where the covariant derivative is denoted by $D$, $F_{\mu\nu}$ is the field
strength tensor and we need to tune the three parameters $m_{0}a$, $c_{P}$ and
$\zeta$. Exploratory studies on how to tune these parameters have been
performed by Li and Peng [14, 15, 16]. The idea is to use experimental values
for the spin averaged meson mass ($\overline{m}=(m_{B_{s}}+3m_{B_{s}^{*}})/4$)
and the hyperfine splitting ($\Delta_{m}=m_{B^{*}_{s}}-m_{B_{s}}$) as inputs
together with the constraint from the dispersion relation that the rest mass
$m_{1}$ equal the kinetic mass $m_{2}$. These three quantities are computed
for a set of seven trial parameters determined by making an initial guess for
$m_{0}a$, $c_{P}$, and $\zeta$ and then varying it by a chosen uncertainty
$\pm\sigma_{\\{m_{0}a,c_{P},\zeta\\}}$:
$\displaystyle\left[\\!\\!\begin{array}[]{c}m_{0}a\\\ c_{P}\\\ \zeta\\\
\end{array}\right],\left[\\!\\!\begin{array}[]{c}m_{0}a-\sigma_{m_{0}a}\\\
c_{P}\\\ \zeta\\\
\end{array}\\!\\!\right],\;\left[\\!\\!\begin{array}[]{c}m_{0}a+\sigma_{m_{0}a}\\\
c_{P}\\\ \zeta\\\
\end{array}\\!\\!\right],\;\left[\\!\\!\begin{array}[]{c}m_{0}a\\\
c_{P}-\sigma_{c_{P}}\\\ \zeta\\\
\end{array}\\!\\!\right],\;\left[\\!\\!\begin{array}[]{c}m_{0}a\\\
c_{P}+\sigma_{c_{P}}\\\ \zeta\\\
\end{array}\\!\\!\right],\;\left[\\!\\!\begin{array}[]{c}m_{0}a\\\ c_{P}\\\
\zeta-\sigma_{\zeta}\\\
\end{array}\\!\\!\right],\;\left[\\!\\!\begin{array}[]{c}m_{0}a\\\ c_{P}\\\
\zeta+\sigma_{\zeta}\\\ \end{array}\\!\\!\right]$ (26)
(see Fig. 1). We iterate over the parameters $\\{m_{0}a,\,c_{P},\,\zeta\\}$
until we determine the values that reproduce the known experimental
measurements for $\overline{m},\,\Delta_{m},\,m_{1}/m_{2}$:
$\displaystyle\left[\begin{array}[]{c}m_{0}a\\\ c_{P}\\\
\zeta\end{array}\right]^{\text{RHQ}}=J^{-1}\times\left(\left[\begin{array}[]{c}\overline{m}\\\
\Delta_{m}\\\ \frac{m_{1}}{m_{2}}\end{array}\right]^{\text{PDG}}-A\right)$
(33) with $\displaystyle
J=\left[\frac{Y_{3}-Y_{2}}{2\sigma_{m_{0}a}},\,\frac{Y_{5}-Y_{4}}{2\sigma_{c_{P}}},\,\frac{Y_{7}-Y_{6}}{2\sigma_{\zeta}}\right]\qquad\text{and}\quad
A=Y_{1}-J\times\left[m_{0}a,\,c_{P},\,\zeta\right]^{t}.$ (34)
In (34) we use the vectors $Y_{i}$ as shorthand notation for
$[\overline{m},\,\Delta_{m},\,m_{1}/m_{2}]^{t}_{i}$ obtained for the input
parameters as given in Eq. (26), labeled 1-7 from left to right. Eq. (33)
assumes a linear dependence of the meson masses on the parameters of the
action; therefore we must be in a linear regime to reliably extract
$\\{m_{0}a,\,c_{P},\,\zeta\\}$. We stop the iteration once the tuned values
lie within our variation range.
Figure 1: Initial guess for the RHQ parameters and their uncertainties.
In order to finally compute $B$-meson decay constants and mixing parameters as
precisely as possible we repeated the original tuning presented at Lattice
2008 [14] with increased statistics and optimized smeared wavefunction
parameters used for generating the heavy quark propagators. Moreover,
performing the tuning on the same configurations we intend to use for
computing weak matrix elements such as $f_{B}$ will allow for an improved
error analysis in which the correlations among the three RHQ parameters can be
fully taken into account.
Our method for computing the $\Delta B=2$ four-quark operators requires the
light quarks to be generated with a point source and sink, but for the heavy
quarks we are free to explore different smearing choices. In addition to point
sources and sinks, we tried Gaussian smeared sources/sinks and also varied the
radius of the Gaussian smearing. As a first guess we chose the radius of the
Gaussian source to be the rms radius of the $b\bar{b}$\- and $c\bar{c}$ states
[17]. Later we extended the radius and found that $r_{\text{rms}}=0.634$fm
gives the best signal as can be seen in Fig. 2.
Figure 2: $B_{l}$-meson effective masses for different $b$-quark spatial
wavefunctions; in each case the light quark has a point source.
Using this setup we obtain the RHQ parameters on the three $24^{3}$ ensembles
as given in Table 2. In contrast to earlier work [14] the determination uses
only quantities from the heavy-light system. We expect these values to be
close to our final determination. Moreover, we observe that within statistical
uncertainties there is no dependence on the light sea quark mass
($m_{\text{sea}}^{l}$).
$m_{\text{sea}}^{l}$ | $m_{0}a$ | $c_{P}$ | $\zeta$
---|---|---|---
0.005 | 8.41(9) | 5.7(2) | 3.1(2)
0.010 | 8.4(1) | 5.8(2) | 3.1(1)
0.020 | 8.4(1) | 5.6(2) | 3.1(1)
Table 2: Preliminary determination of the RHQ parameters on the three $24^{3}$
ensembles with $a\approx 0.11$fm.
## 3 Computation of decay constants
Using these newly determined RHQ parameters we compute as a first non-trivial
test the decay constants of the unitary ($B_{l}$) and strange ($B_{s}$) mesons
via the relation
$\displaystyle f_{B_{q}}=Z_{\Phi}\;\Phi_{B_{q}}\;a^{-3/2}/\sqrt{m_{B_{q}}},$
(35)
where $\Phi_{B}$ is the lattice decay amplitude, $Z_{\Phi}$ is the
renormalization factor. For the $B_{s}$ meson (domain-wall valence quark has
mass of physical $s$-quark) we generated data for the set of seven different
RHQ input parameters. Therefore we can use similar equations to (33) and (34)
in order to determine the decay amplitude $\Phi_{B_{s}}$ at the tuned RHQ
parameters:
$\displaystyle\Phi^{\text{RHQ}}=J_{\Phi}^{(1\times
3)}\times\left[\begin{array}[]{c}m_{0}a\\\ c_{P}\\\
\zeta\end{array}\right]^{\text{RHQ}}+A_{\Phi}$ (39) with
$\displaystyle\displaystyle
J_{\Phi}=\left[\frac{\Phi_{3}-\Phi_{2}}{2\sigma_{m_{0}a}},\,\frac{\Phi_{5}-\Phi_{4}}{2\sigma_{c_{P}}},\,\frac{\Phi_{7}-\Phi_{6}}{2\sigma_{\zeta}}\right]\qquad\text{and}\qquad
A_{\Phi}=\Phi_{1}-J_{\Phi}\times\left[m_{0}a,\,c_{P},\,\zeta\right]^{t}.$ (40)
In Fig. 3 we show the results for $\Phi_{B_{s}}$ obtained on the ensemble with
$m_{\text{sea}}^{l}=0.005$. To test the linearity with respect to the input
parameters $\\{m_{0}a,\,c_{P},\,\zeta\\}$ we used three different sets of
seven RHQ parameters all centered around the same point. The vertical black
line with the gray error band indicates the tuned value of $m_{0}a$, $c_{P}$
or $\zeta$ and allows for a simple estimate of the error in $\Phi_{B_{s}}$ due
to the uncertainty in each of the three parameters. These plots show that the
effect of the uncertainty in $m_{0}a$ and $c_{P}$ is negligible, whereas
$\zeta$ contributes an error of about 1%. Moreover, we see that $\Phi_{B_{s}}$
depends linearly on $m_{0}a$, $c_{P}$ and $\zeta$ in the range of interest.
Figure 3: Dependence of $\Phi_{B_{s}}$ on the three RHQ parameters
$\\{m_{0}a,\,c_{P},\,\zeta\\}$; results are shown for the lightest sea quark
mass $m_{\text{sea}}^{l}=0.005$ on the $24^{3}$ ensembles.
Alternatively, one may simply use the tuned RHQ values to compute the needed
correlation functions. We followed this procedure to calculate the $B_{l}$
meson (domain-wall valence quark mass equals the light sea quark mass) the
decay amplitude $\Phi_{B_{l}}$.
Atfer renormalizing $\Phi_{B}$ multiplicatively at 1-loop [18] we obtain the
decay constants which are given in Tab. 3 and shown in Fig. LABEL:Fig-fB. As
mentioned above, the values for $f_{B_{l}}$ are obtained by simulating
directly at the tuned RHQ parameters, whereas the $f_{B_{s}}$ values are
extracted using Eq. (39). The statistical errors on $f_{B_{s}}$ are therefore
larger because they take into account the statistical uncertainties in the
three RHQ parameters. For a better comparison, we increase the errors in
$f_{B_{l}}$ in the Fig. LABEL:Fig-fB by the error due to the statistical
uncertainty in the RHQ parameters estimated in Fig. 3. Moreover, we emphasize
that this computation is performed without $O(a)$ improvement.
## 4 Conclusion
We presented our first results computing $B$-meson decay constants using
domain-wall light and relativistic $b$-quarks with all parameters of the RHQ
action tuned non-perturbatively. Currently, our results still need to be
${\cal O}(a)$ improved. For $f_{B_{s}}$ we expect only a mild chiral
extrapolation but of course need to perform an extrapolation to the continuum
and estimate other systematic errors. Despite these caveats the small
statistical errors and the fact that our central values lie in the same
ballpark as results of other collaborations indicates the promise of our
method.
$m_{sea}^{l}$ | $f_{B_{l}}$(MeV) | $f_{B_{s}}$(Mev)
---|---|---
0.005 | 188(2) | 215(3)
0.010 | 194(2) | 214(4)
0.020 | — | 221(2)
Table 3: Preliminary results for the decay constants using 1-loop
multiplicative renormalization, but without $O(a)$ improvement of the axial-
current operator.
## Acknowledgments
We are thankful to all the members of the RBC and UKQCD collaborations.
Numerical computations for this work utilized USQCD resources and were
performed on the kaon and jpsi clusters at FNAL, in part funded by the Office
of Science of the U.S. Department of Energy. This manuscript has been authored
by an employee of Brookhaven Science Associates, LLC under Contract No. DE-
AC02-98CH10886 with the U.S. Department of Energy.
## References
* [1] A. J. Buras, M. Jamin, and P. H. Weisz, Nucl. Phys. B347, 491 (1990)
* [2] T. Inami and C. S. Lim, Prog. Theor. Phys. 65, 297 (1981)
* [3] E. Lunghi and A. Soni, Phys.Rev.Lett. 104, 251802 (2010), arXiv:0912.0002 [hep-ph]
* [4] E. Lunghi and A. Soni, Phys. Lett. B666, 162 (2008), arXiv:0803.4340 [hep-ph]
* [5] E. Lunghi and A. Soni, arXiv:1010.6069 [hep-ph]
* [6] C. Albertus, Y. Aoki, P. Boyle, N. Christ, T. Dumitrescu, _et al._ , Phys.Rev. D82, 014505 (2010), arXiv:1001.2023 [hep-lat]
* [7] R. T. Evans, E. Gamiz, and A. X. El-Khadra, PoS LAT2008, 052 (2008)
* [8] R. T. Evans, E. Gamiz, A. El-Khadra, and A. Kronfeld (Fermilab Lattice and MILC Collaborations), PoS LAT2009, 245 (2009), arXiv:0911.5432 [hep-lat]
* [9] E. Gamiz _et al._ (HPQCD), Phys. Rev. D80, 014503 (2009), arXiv:0902.1815 [hep-lat]
* [10] C. Allton _et al._ (RBC-UKQCD), Phys. Rev. D78, 114509 (2008), arXiv:0804.0473 [hep-lat]
* [11] Y. Aoki _et al._ (RBC and UKQCD Collaborations), arXiv:1011.0892 [hep-lat]
* [12] A. X. El-Khadra, A. S. Kronfeld, and P. B. Mackenzie, Phys. Rev. D55, 3933 (1997), arXiv:hep-lat/9604004
* [13] N. H. Christ, M. Li, and H.-W. Lin, Phys.Rev. D76, 074505 (2007), arXiv:hep-lat/0608006
* [14] M. Li (RBC and UKQCD collaborations), PoS LATTICE2008, 120 (2008), arXiv:0810.0040 [hep-lat]
* [15] H. Peng (RBC and UKQCD collaborations), PoS LATTICE2009, 094 (2009)
* [16] H. Peng (RBC and UKQCD collaborations), PoS LATTICE2010, 107 (2010)
* [17] D. P. Menscher, “Charmonium and Charmed Mesons with Improved Lattice QCD,” (2005), PhD thesis
* [18] N. Yamada, S. Aoki, and Y. Kuramashi, Nucl. Phys. B713, 407 (2005), arXiv:hep-lat/0407031
|
arxiv-papers
| 2011-01-24T16:13:41 |
2024-09-04T02:49:16.607567
|
{
"license": "Public Domain",
"authors": "Ruth S. Van de Water and Oliver Witzel",
"submitter": "Oliver Witzel",
"url": "https://arxiv.org/abs/1101.4580"
}
|
1101.4632
|
# A Secure Web-Based File Exchange Server
Software Requirements Specification Document
CIISE Security Investigation Initiative
Represented by:
Serguei A. Mokhov
Marc-André Laverdière
Ali Benssam
Djamel Benredjem
{mokhov,ma_laver,d_benred,al_ben}@ciise.concordia.ca
Montréal, Québec, Canada
(December 14, 2005)
###### Contents
1. 1 Introduction
1. 1.1 Purpose
2. 1.2 Scope
3. 1.3 Definitions and Acronyms
2. 2 Overall Description
1. 2.1 Product Perspective
1. 2.1.1 System interfaces
2. 2.1.2 User Interfaces
3. 2.1.3 Hardware Interfaces
4. 2.1.4 Software Interfaces
2. 2.2 Product Functions
3. 3 Specific Requirements
1. 3.1 Functional Requirements
1. 3.1.1 Domain Model
2. 3.1.2 Use Case Model
2. 3.2 Software System Attributes
1. 3.2.1 Security
2. 3.2.2 Reliability
3. 3.2.3 Availability
4. 3.2.4 Maintainability
5. 3.2.5 Portability
3. 3.3 Logical Database Requirements
###### List of Figures
1. 3.1 SFS system packages
2. 3.2 Normal user use-case
3. 3.3 Administrator use-case
## Chapter 1 Introduction
Building Trust is the basis of all communication, especially electronic one,
as the identity of the other entity remains concealed. To address problems of
trust, authentication and security over the network, electronic communications
and transactions need a framework that provides security policies, encryption
mechanisms and procedures to generate manage and store keys and certificates.
This software requirements specification (SRS) document demonstrates all the
concerns and specifications of the secure web-based file exchange server
(SFS). SFS is a security architecture that we propose here to provide an
increased level of confidence for exchanging information over increasingly
insecure networks, such as the Internet. SFS is expected to offer users a
secure and trustworthy electronic transaction.
### 1.1 Purpose
The intent of implementation and deployment of SFS facilities is to meet its
basic purpose of providing Trust. Presently, SFS needs to perform the
following security functions:
* •
_Mutual authentication of entities taking part in the communication:_ Only
authenticated principals can access files to which they have privileges.
* •
_Ensure data integrity:_ By issuing digital certificates which guarantee the
integrity of the public key. Only the public key for a certificate that has
been authenticated by a certifying authority should work with the private key
possessed by an entity. This eliminates impersonation and key modification.
* •
_Enforce security:_ Communications are more secure by using SSL to exchange
information over the network.
### 1.2 Scope
SFS is implemented to secure sensitive resources of the organization and avoid
security breaches. The SFS allows trustworthy communication between the
different principals. These principals must be authenticated and the access to
the resources (files) should be secured and regulated. Any principal wants to
access to the database needs to perform the following steps:
* •
_Mutual authentication:_ The Web Server via which the database is contacted
authenticates the principal using its digital certificate and username to
ensure that it is who it claims to be . The principal authenticates also the
server using its certificate information.
* •
_Principal validation:_ To validate the principal, the server looks up
information from an LDAP server which contains the hierarchy of all principals
along with certificates and credentials.
* •
_Enforcing security:_ The security is enforced by using SSL to communicate
between the Web Server and the LDAP server, the Web Server and the database
and between the principal and Web Server.
* •
Principal authentication: Upon successful authentication, the Web Server will
allow the principal to perform actions on the database according to a pre
specified Access Control List.
* •
Kinds of principals: There are two kinds of principals, administrators and
clients: _Clients_ have the ability to upload, download, delete and view
files. _Administrators_ have the ability to: Upload, download, delete and view
files; Add, delete and modify users; Generate user’s certificate, with all
required information; Generate ACL to users; Manage groups, Perform
maintenance.
Finally, this infrastructure allows additional features such as the ability to
assign users to groups in order to provide users with the access to files
prepared by other group members.
### 1.3 Definitions and Acronyms
* •
PKI: Public Key Infrastructure
* •
OpenLDAP : is a free, open source implementation of the Lightweight Directory
Access Protocol (LDAP).
* •
OpenSSL: an open source SSL library and certificate authority
* •
Apache Tomcat: A Java based Web Application container that was created to run
Servlets and JavaServer Pages (JSP) in Web applications
* •
PostgreSQL: An open source object-relational database server
* •
SSL: Secure Socket Layer
* •
JSP: Java Server Pages
* •
JCE: Java Cryptography Extension
* •
API: Application Programming Interface
* •
JDBC: Java Database Connectivity
* •
JNDI: Java Naming and Directory Interface
* •
LDAP: Lightweight Directory Access Protocol
* •
X.509: A standard for defining a Digital Certificate used by SSL
* •
SRS: Specification Request Document
* •
SDD: Specification Design Document
* •
DER: Distinguished Encoding Rules
* •
Mutual Authentication: The process of two principals proving their identities
to each other
* •
SFS: Secure File Exchange Server, this product
* •
COTS: Commercial Off The Shelf, common commercially or freely available
software
The coming sections of the SRS are a description of all the requirements to be
implemented in SFS system. The requirements specifications are organized in
two major sections: Overall Description and Specific Requirements.
## Chapter 2 Overall Description
In this chapter we provide an overall insight of the general factors that
affect the SFS system and its requirements.
### 2.1 Product Perspective
The SFS system is intended to operate in a distributed environment: clients
machines, application server, database server, and LDAP server. SFS is
accessed via secure connections we intend to provide in this work. The
system’s user can be either a normal user or an administrator. A normal user
has the ability to upload, download, delete and view files. An administrator
is able to: Upload, download, delete and view files; add, delete and modify
users; generate user certificates along with all required information;
generate an ACL to each and every user; manage groups; perform various
maintenance actions such as: check log files, delete files, etc.
#### 2.1.1 System interfaces
The various parts of the SFS application will be installed on client machines
and different servers. The client that uses the system has to have the
certificate installed on his machine to provide client authentication. The
servers are responsible of one of the following functions: provide the
database, provide the LDAP server functionalities, and provide the application
server for different clients.
#### 2.1.2 User Interfaces
The user interfaces consist of web-based graphical components that allow the
user to interact with the SFS system. The user will use a web browser to send
and receive data. If the user is the administrator, s/he will have more
options to add, delete, etc. users, generate users certificates, generate ACL
for each user, etc.
#### 2.1.3 Hardware Interfaces
The hardware interfaces will be achieved through the abstraction layer of the
Java Virtual Machine (JVM). The keyboard and the mouse are examples of such
hardware interfaces that allow users to interact with the SFS system.
#### 2.1.4 Software Interfaces
Among the most important software interfaces used in this project, we have:
* •
The SFS system is OS-independent due to the cross platform Java
implementation. It will support web browsers such as Internet Explorer,
Mozilla Fireworks, etc.
* •
Access to databases will be provided by JDBC 3 on both Windows and Linux
environment.
* •
JXplorer? will be used to provide a graphical access to LDAP server.
* •
OpenLDAP? is used to host users’ certificates.
* •
Java 1.5 JDK from Sun.
* •
JRE 1.5 from Sun.
* •
Servlets for client and administrator interfaces.
* •
Apache Tomcat5 server as the web server used in this project.
* •
PostegreSQL? is the database used to host users information, files, etc.
* •
OpenSSL toolkit to generate the certificates for users.
Hereafter, we provide the software and documentation’s locations related to
these interfaces:
* •
OpenLDAP? software and documentation found at: _http://www.openldap.org/_
* •
PostgreSQL database and documentation found at: _http://www.postgresql.org/_
* •
Java development kit 1.5 available at: _http://java.sun.com/j2se/_
* •
JSP documentation found at: _http://java.sun.com/products/jsp/_
* •
OpenSSL Toolkit found at: _http://www.openssl.org/_
* •
Apache Tomcat 5.0 web server found at: _http://tomcat.apache.org/_
### 2.2 Product Functions
The SFS system will implement the following functionalities:
* •
_Server authentication:_ This use case allow the user to authenticate the web
server is connecting to.
* •
_Client authentication:_ This is use case allow the server to authenticate the
user he is trying to connect to.
* •
_Secure communication:_ Between users over the network.
* •
_Files handling:_ Such as downloading, uploading, and deletion files.
* •
Users management: Administrator has the ability to add and delete users, add
and delete groups, and assign users to groups.
## Chapter 3 Specific Requirements
In this section we describe the software requirements to design the SFS
system. The system design should satisfy the following requirements.
### 3.1 Functional Requirements
Hereafter, we express the expectations in terms of system functions and
constraints. This includes the domain model and the most important use case
diagrams of the SFS system.
#### 3.1.1 Domain Model
The SFS system domain model consists of many packages.
* •
_Client authentication module:_ used to provide users the ability to
authenticate the server.
* •
_Server authentication module:_ used to provide the server the ability to
authenticate the clients.
* •
_LDAP connection module:_ used to provide connection to the LDAP server in
order to check clients’ credentials.
* •
_Database connection module:_ used to provide users the ability to connect to
the database server in a secure mode.
Figure 3.1: SFS system packages
#### 3.1.2 Use Case Model
The SFS system consists of a set of use cases manageable by the users of the
application. There are two types of users of the system: normal users and
administrator. Normal can view, delete, download, and upload files. For the
administrator, s/he can: add, delete, etc. users; generate users’
certificates; generate ACL for each user, etc.
The diagram in Figure 3.2 shows the capabilities of a normal user.
Figure 3.2: Normal user use-case
The diagram in Figure 3.3 shows the capabilities of the administrator user.
Figure 3.3: Administrator use-case
### 3.2 Software System Attributes
There are a number of software attributes that can serve as requirements. It
is important that required attributes by specified so that their achievement
can be objectively verified. The following items are some of the most
important ones: security, reliability, availability, maintainability, and
portability.
#### 3.2.1 Security
Security is the most important attribute of the SFS design and implementation.
The mutual authentication between the server and clients is crucial for the
system use. The system should be able to authenticate users and differentiate
among them, either are normal users or administrator. In order to achieve the
security feature expected from the SFS system, the following tasks have to be
realized:
* •
Utilize cryptographic techniques
* •
Check users’ credentials before using the system and accessing the database
* •
Provide secure communications between different parts of the system
#### 3.2.2 Reliability
The basis for the definition of reliability is the probability that a system
will fail during a given period. The reliability of the whole system depends
on the reliability of its components and on the reliability of the
communication between its components. The SFS system is based mainly on some
standard components such as OpenLDAP, OpenSSL, JDBC, PostgreSQL, etc. The
reliability of these service components is already proved. This fact improves
the reliability of the system and restricts the proof work on assuring only of
the reliability of the added components and the communications between the
different components. In addition, the system must ensure the security of the
communications which is the most important issue of the SFS system.
#### 3.2.3 Availability
The SFS system must be able to work continuously in order to provide users
with an access to different server’s parts of the system. However, since this
system depends on distributed information systems and databases, many
constraints should be taken into account such as:
* •
The connection to the web server that provides access to the system
* •
The interconnection between different parts of the system should always be
available; otherwise, the users cannot complete their tasks using the system.
* •
The database should be available in the database server side
* •
The LDAP server should be always available in order to check users’
credentials
* •
The web server should be also available in order to allow users connecting to
the system.
#### 3.2.4 Maintainability
Maintainability is defined as the capacity to undergo repairs and
modifications. The main goal in designing SFS system is to keep it easy to be
modified and extended.
#### 3.2.5 Portability
The portability is one of the main specifications of Java. Since SFS is
implemented using the Java programming language, the portability is
automatically satisfied and the system is able to run on any machine or
operating system which supports the execution of a Java virtual machine.
### 3.3 Logical Database Requirements
The rationale behind SFS system is to provide secure connections for users
accessing databases to view, delete, upload, and download files through a web
server and LDAP server. After analyzing the requirements we propose using a
relational database model to meet our requirements. This database is required
to store information about users, files, groups of users, etc. The database is
expected to work on 24 hours and 7 days in order to provide nonstop access to
the users. Therefore, backup of the database should be taken periodically
(daily or weekly. The relational database itself guarantees the flexibility,
simplicity and elimination of redundancy once designed carefully. The entity
relationship model will be elaborated in detail in the database design of the
design part, in this document.
## Bibliography
* [Apa11] Apache Foundation. Apache Jakarta Tomcat. [online], 1999–2011. http://jakarta.apache.org/tomcat/index.html.
* [CBH03] Suranjan Choudhury, Katrik Bhatnagar, and Wisam Haque. Public Key Infrastructure, Implementation and Design. NIIT Books, 2003.
* [Deb05] Mourad Debbabi. INSE 6120: Cryptographic protocols and network security, lecture notes. Concordia Institute for Information Systems Engineering, Concordia University, Montreal, Canada, 2005. http://users.encs.concordia.ca/~debbabi/inse6120.html.
* [E+11] Eclipse contributors et al. Eclipse Platform. eclipse.org, 2000–2011. http://www.eclipse.org, last viewed February 2010.
* [GB11] Erich Gamma and Kent Beck. JUnit. [online], Object Mentor, Inc., 2001–2011. http://junit.org/.
* [Lar06] Craig Larman. Applying UML and Patterns: An Introduction to Object-Oriented Analysis and Design and Iterative Development. Pearson Education, third edition, April 2006. ISBN: 0131489062.
* [Leu05] Leuvens Universitair. A few frequently used SSL commands. [online], Leuvens Universitair Dienstencentrum voor Informatica en Telematica, 2005. http://shib.kuleuven.be/docs/ssl_commands.shtml.
* [Ope05a] OpenLDAP Community. OpenLDAP: The open source for LDAP software and information. [online], 2005. www.openldap.org.
* [Ope05b] OpenSSL Community. OpenSSL: The open source toolkit for SSL/TLS. [online], 2005. http://www.openssl.org.
* [O’R05a] O’Reilly. Home of com.oreilly.servlet. [online], 2005. http://servlets.com/cos.
* [O’R05b] O’Reilly. JXplorer – a Java LDAP browser. [online], 2005. http://sourceforge.net/projects/jxplorer/.
* [Sun05a] Sun Microsystems, Inc. Java servlet technology. [online], 1994–2005. http://java.sun.com/products/servlets.
* [Sun05b] Sun Microsystems, Inc. JavaServer pages technology. [online], 2001–2005. http://java.sun.com/products/jsp/.
* [The11] The PostgreSQL Global Development Group. PostgreSQL – the world’s most advanced open-source database. [ditigal], 1996–2011. http://www.postgresql.org/, last viewed January 2010.
## Index
* Introduction Chapter 1
|
arxiv-papers
| 2011-01-24T19:49:14 |
2024-09-04T02:49:16.612528
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serguei A. Mokhov and Marc-Andr\\'e Laverdi\\`ere and Ali Benssam and\n Djamel Benredjem",
"submitter": "Serguei Mokhov",
"url": "https://arxiv.org/abs/1101.4632"
}
|
1101.4634
|
# Upper Bound on Fidelity of Classical Sagnac Gyroscope
Thomas B. Bahder Aviation and Missile Research, Development, and Engineering
Center,
US Army RDECOM, Redstone Arsenal, AL 35898, U.S.A.
###### Abstract
Numerous quantum mechanical schemes have been proposed that are intended to
improve the sensitivity to rotation provided by the classical Sagnac effect in
gyroscopes. A general metric is needed that can compare the performance of the
new quantum systems with the classical systems. The fidelity (Shannon mutual
information between the measurement and the rotation rate) is proposed as a
metric that is capable of this comparison. A theoretical upper bound is
derived for the fidelity of an ideal classical Sagnac gyroscope. This upper
bound for the classical Sagnac gyroscope should be used as a benchmark to
compare the performance of proposed enhanced classical and quantum rotation
sensors. In fact, the fidelity is general enough to compare the quality of two
different apparatuses (two different experiments) that attempt to measure the
same quantity.
###### pacs:
PACS number 07.60.Ly, 03.75.Dg, 06.20.Dk, 07.07.Df
The Sagnac effect Sagnac (1913a, b, 1914); Post (1967) is the basis for all
modern rotation sensors Lefevre (1993) and their applications to inertial
navigation systems Titterton and Weston (2004). Besides its practical
applications, the Sagnac effect is being contemplated for studying general
relativistic effects, such as Lense-Thirring frame dragging, detecting
gravitational waves, and testing local Lorentz invariance Chow et al. (1985);
Stedman (1997).
The original experiments of Sagnac consisted of mirrors mounted on a rotating
disk, see Fig. 1. The mirrors define two paths, one clockwise (CW) and the
other counter-clockwise (CCW) on the disk. A source of light at point $S$ was
mounted on the rotating disk, having wavelength $\lambda=2\pi c/\omega$, where
$\omega$ is the angular frequency as measured in an inertial frame and $c$ is
the speed of light in vacuum. The beam is split at the beam splitter $BS$ and
light is propagated along the clockwise and counter-clockwise paths. The two
beams are brought together at the beam splitter $BS$ and observed at point
$O$. When the interferometer is rotated at angular velocity $\Omega$, with
light source and detector mounted on the rotating disk, a fringe shift $\Delta
N$ is observed with respect to the fringe position for the stationary
interferometer, given by Post (1967); Chow et al. (1985)
$\Delta N=\frac{4{\bf A}\cdot{\bf\Omega}}{\lambda\,c}$ (1)
where ${\bf A}$ is a vector perpendicular to the area enclosed by the paths,
having magnitude $A=|{\bf A}|$, and ${\bf\Omega}$ is the vector in the
direction of the angular velocity of rotation, with magnitude
$|{\bf\Omega}|=\Omega$. The fringe shift, $\Delta N=\Delta L/\lambda=c\Delta
t/\lambda$, can be expressed in terms of the path length difference, $\Delta
L$, or time difference, $\Delta t=4{\bf A}\cdot{\bf\Omega}/c^{2}$, for the CW
and CCW paths, as measured in an inertial frame. For typical rotation rates in
the laboratory, the classical Sagnac effect is small, and the effect has to be
enhanced to make a practical rotation sensor.
The classical Sagnac effect is exploited for sensing rotation by either
measuring a frequency shift or a phase shift. In the active ring laser
gyroscope (RLG), where the optical medium is inside the cavity Chow et al.
(1985), or in a resonant fiber-optic gyroscope (R-FOG) Lefevre (1993), a
measurement is made of the frequency shift, $\Delta\omega$, between the CW and
CCW propagating modes Post (1967); Chow et al. (1985)
$\Delta\omega=\frac{4A\,\Omega}{L\,c}\omega=S\,\Omega$ (2)
where $L$ is the length of the perimeter of the path measured in an inertial
frame, and $S$ is commonly called the scale factor.
In a passive fiber ring interferometer (I-FOG) Lefevre (1993), or a passive
ring laser gyroscope with light source outside the medium Chow et al. (1985),
the phase shift $\Delta\phi$, is measured between CW and CCW propagating
beams,
$\Delta\phi=\frac{4A\Omega}{c^{2}}\omega$ (3)
For a fiber-optic gyroscope with phase shift enhanced by $N$ turns, the
frequency shift is $\Delta\phi_{N}=N\Delta\phi$.
Figure 1: Schematic of a Sagnac interferometer is shown, with light from
source $S$ incident on beam splitter $BS$, mirrors at $M_{1}$, $M_{2}$, and
$M_{3}$, and the observer at $O$ that detects the frequency shift
$\Delta\omega$.
Recently, much effort has been expended on experiments with quantum Sagnac
interferometers, using single-photons Bertocchi et al. (2006), using cold
atoms Gustavson et al. (2000); Gilowski et al. (2009) and using Bose-Einstein
condensates(BEC) Gupta et al. (2005); Wang et al. (2005); Tolstikhin et al.
(2005), in efforts to make improvements over the sensitivity to rotation of
the classical Sagnac effect. Schemes have also been proposed to improve the
sensitivity of rotation sensing using multi-photon correlations Kolkiran and
Agarwal (2007) and using entangled particles, which are expected to have
Heisenberg limited precision that scales as $1/N$, where $N$ is the number of
particles Cooper et al. (2010).
The utility of these quantum systems as rotation sensors must be compared with
the classical Sagnac effect using classical light. The metric used to compare
the classical and quantum systems must be sufficiently general to treat both
types of systems on an equal basis. Information measures are examples of such
metrics because they are general enough to compare quantum and classical
systems.
The determination of the rotation rate is a specific example of the more
general problem of parameter estimation, whose goal is to determine one or
more parameters from measurements 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). The Cramér-Rao
theoremCramér (1958); Cover and Thomas (2006) can be applied to get an upper
bound on the variance of an unbiased estimator of a parameter of interest
(here the rotation rate) in terms of the classical and quantum Fisher
information, see Ref. Bahder (2010) and references contained therein. One
potential drawback of this approach is that the Fisher information, and
therefore the upper bound on the variance of the estimator, can depend on the
true value of the parameter to be determined Bahder (2010). Of course, the
true value of the parameter is unknown. Specifically, the Fisher information
can depend on the true angular rotation rate when the Sagnac interferometer is
not unitary, which occurs when scattering or dissipation are present Bahder
(2010). Consequently, I propose to characterize a rotation sensor by its
fidelity, which is the Shannon mutual information, a quantity that does not
depend on the true rotation rate.
In this letter, I calculate a fundamental upper bound on the rotation
sensitivity of a classical Sagnac gyroscope that follows Eq. (2). This upper
bound can be used as a benchmark to compare the performance of rotation
sensors based on newly proposed quantum and classical methodologies, see
earlier discussion. In addition, the fidelity is a useful measure to compare
other proposed gyroscopes based on slow light generated by electromagnetically
induced transparency Leonhardt and Piwnicki (2000) and other classical optical
enhancements Matsko et al. (2004); Smith et al. (2008, 2009).
The fidelity Bahder and Lopata (2006); Bahder (2010) is the Shannon mutual
information Shannon (1948); Cover and Thomas (2006) between the measurement
(the frequency shift), $\Delta\omega$, and the parameter to be measured (the
rotation rate) $\Omega$:
$\displaystyle H$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}d({\Delta\omega})\,\,\int_{-\infty}^{+\infty}d\Omega\,\,p(\Delta\omega|\Omega)\,p(\Omega)\,\,\times\,$
(4)
$\displaystyle\log_{2}\left[\frac{p(\Delta\omega|\Omega)\,}{\int_{-\infty}^{+\infty}\,\,\,p(\Delta\omega|\Omega^{\prime})\,p(\Omega^{\prime})\,\,d\,\Omega^{\prime}}\right].$
where $p(\Delta\omega|\Omega)$ is the conditional probability density of
measuring a frequency shift $\Delta\omega$, given a true rotation rate
$\Omega$. Our prior information on the rotation rate is given by the
probability density $p(\Omega)$. The fidelity, $H$, is the Shannon mutual
information in a communication problem between Alice and Bob, wherein Alice
sends messages to Bob Shannon (1948); Cover and Thomas (2006). The fidelity
$H$ does not depend on the measurements, $\Delta\omega$, or on the parameter,
$\Omega$, because it is an average over all possible measurements and
parameter values. If we are completely ignorant of the rotation rate, we can
take the prior probability as flat distribution, $p(\Omega)=1/(2\pi)$,
indicative of no prior information on our part. In this case, the fidelity $H$
characterizes the quality of the Sagnac interferometer itself, in terms of
mutual information that the measurement of $\Delta\omega$ carries about the
parameter of interest, the rotation rate $\Omega$. In fact, the fidelity $H$
is a specific example of a general way to characterize the quality of all
physical measurements.
The fidelity in Eq.(4) is a completely general way to describe any classical
or quantum measurement experiment. The classical or quantum apparatus is
viewed as a channel through which information flows from the phenomena to be
measured to the measurements. The fundamental quantity that describes this
process is the conditional probability of a measurement and the probability
distribution that describes our prior information, above notated as
$p(\Delta\omega|\Omega)$ and $p(\Omega^{\prime})$, respectively. In the
language of communication, there is mutual information $H$ between the
continuous alphabet of the parameter, $\Omega$, and the continuous alphabet of
the measurements, $\Delta\omega$.
In order to compute the fidelity for the classical Sagnac gyroscope from
Equation (4) a model is needed for the conditional probability density
$p(\Delta\omega|\Omega)$. In the case of a quantum system, these probabilities
are given by a trace:
$p(\Delta\omega|\Omega,\rho)=\rm{tr}\left(\hat{\rho}\,\hat{\Pi}\left(\Delta\omega\right)\right)$
(5)
where the state is specified by the density matrix, $\hat{\rho}$, and the
measurements are defined by the positive-operator valued measure (POVM),
$\hat{\Pi}(\Delta\omega)$.
For a classical Sagnac system, I obtain an upper bound on the fidelity in
Eq.(4). I consider classical light of bandwidth $\Delta\omega$ and center
frequency $\bar{\omega}$, input into a Sagnac gyroscope that satisfies Eq.(2).
Therefore, I define the classical measurement probabilities,
$p(\Delta\omega|\Omega)$, by
$p(\Delta\omega|\Omega)=\sum\limits_{n=0}^{\infty}{p\left({\Delta\omega|\Omega,\omega_{n}}\right)\,P_{in}\left({\omega_{n}}\right)}$
(6)
where $p\left({\Delta\omega|\Omega,\omega_{n}}\right)$ is the probability
density for measuring $\Delta\omega$, given the that the true rotation rate is
$\Omega$, and the input was monochromatic at frequency $\omega_{n}$. In Eq.
(6), for convenience, I assume that the allowed frequency modes are discrete,
$\omega_{n}$, for $n=0,1,\cdots\infty$. The probability
$P_{in}\left(\omega\right)$ gives the distribution of input frequencies, which
has center frequency $\bar{\omega}$ and bandwidth $\Delta\omega$. As an
example, I can take $P_{in}\left(\omega\right)$ to be a Gaussian distribution
of input frequencies with mean $\bar{\omega}$ and standard deviation
$\sigma_{\omega}$
$P_{in}\left({\omega_{n}}\right)=\left({\frac{{\delta\omega}}{{2\pi\bar{\omega}}}}\right)^{1/2}\,\exp\left[{-\frac{{\left({\omega_{n}-\bar{\omega}}\right)^{2}}}{{2\,\delta\omega\,\bar{\omega}}}}\right]$
(7)
where $\delta\omega=\omega_{n+1}-\omega_{n}$ and the variance is given by
$\sigma_{\omega}^{2}=\delta\omega\,\bar{\omega}$. The distribution of
frequencies, $P_{in}\left({\omega_{n}}\right)$, is normalized
$\sum\limits_{n=0}^{\infty}{P_{in}\left({\omega_{n}}\right)=1}$ (8)
in the limit $\bar{\omega}/\delta\omega\gg 1$. The size of bandwidth,
$\sigma_{\omega}$, is due to fundamental physical processes in the experiment.
I want to obtain an upper bound on the fidelity in Eq.(4) for a classical
system. Therefore, I assume that classical measurements are have no noise and
no bias. The classical measurement probability, $p(\Delta\omega|\Omega)$, in
Eq. (6) is obtained from Eq. (2) as
$p\left({\Delta\omega|\Omega,\omega}\right)=\delta\left({\Delta\omega-\frac{{4A\omega}}{{Lc}}\Omega}\right)$
(9)
where $\delta(x)$ is the Dirac delta function. Using Eq. (9) in Eq.(6) gives
the classical probability of measuring $\Delta\omega$ given the true rotation
rate is $\Omega$:
$p(\Delta\omega|\Omega)=\left|{\frac{{Lc}}{{4A\Omega}}}\right|\,P_{in}\left({\frac{{Lc}}{{4A{\kern
1.0pt}\Omega}}\,\Delta\omega}\right)$ (10)
Note that Eq. (10) is valid for an arbitrary input frequency distribution
$P_{in}\left({\omega}\right)$. As an example, for a monochromatic frequency
$\bar{\omega}$ input,
$P_{in}\left(\omega\right)=\delta\left({\omega-\bar{\omega}}\right)$ (11)
Eq.(10) gives the probability of classical measurement as expected:
$p\left({\Delta\omega|\Omega}\right)=\delta\left({\Delta\omega-\frac{{4A{\kern
1.0pt}\Omega}}{{Lc}}\bar{\omega}}\right)$ (12)
For classical light input, with the Gaussian distribution in Eq. (7), Eq.(10)
gives the probability of classical measurement as
$p\left({\Delta\omega|\Omega}\right)=\left({\frac{1}{{2\pi}}}\right)^{1/2}\frac{{Lc}}{{4A\left|\Omega\right|{\kern
1.0pt}\sigma_{\omega}}}\exp\left[{-\frac{{\left({\frac{{Lc}}{{4A{\kern
1.0pt}\Omega}}\Delta\omega-\bar{\omega}}\right)^{2}}}{{2{\kern
1.0pt}\sigma_{\omega}^{2}}}}\right]$ (13)
The conditional probability density in Eq. (13) can be inverted by using
Bayes’ rule
$p\left({\Omega|\Delta\omega}\right)=\frac{{p\left({\Delta\omega|\Omega}\right)p\left(\Omega\right)}}{{\int\limits_{-\infty}^{+\infty}{p\left({\Delta\omega|\Omega^{\prime}}\right)p\left({\Omega^{\prime}}\right)\,d\Omega^{\prime}}}}$
(14)
where $p\left(\Omega\right)$ specifies the prior probability distribution on
rate of rotation, based on our prior information on the rotation rate. With
the probability distribution in Eq. (13), the conditional probability
distribution $p\left({\Omega|\Delta\omega}\right)$ defined by Eq. (14) has a
divergence. However, our prior information on the rotation rate, given by the
distribution $p\left(\Omega\right)$ provides a natural cutoff on the integral
in Eq. (14). We can be reasonably sure that $p(\Omega)\rightarrow 0$ as
$|\Omega|\rightarrow\pm\infty$. For example, we can take
$p\left(\Omega\right)=\left\\{{\begin{array}[]{*{20}c}{\frac{1}{{2\Omega_{\max}}},}&{-\Omega_{\max}<\Omega<+\Omega_{\max}}\\\
{0,}&{{\rm{otherwise}}}\\\ \end{array}}\right.$ (15)
where $\Omega_{\max}$ represents the maximum expected rotation rate on
physical grounds.
For the monochromatic input frequency in Eq. (11), the probability of rotation
is
$p\left(\Omega|\Delta\omega\right)=\delta\left(\Omega-\frac{Lc}{4A}\frac{\Delta\omega}{\bar{\omega}}\right)$
(16)
For the Gaussian frequency distribution in Eq. (7), Eq. (14) gives the
probability of rotation as
$p\left({\Omega|\Delta\omega}\right)=\frac{{\bar{\omega}}}{{\sqrt{2\pi}{\kern
1.0pt}\sigma_{\omega}}}\frac{1}{{\left|\Omega\right|}}\exp\left[{-\frac{1}{{2\sigma_{\omega}^{2}}}\left({\frac{{Lc{\kern
1.0pt}\Delta\omega}}{{4A}}}\right)^{2}\left({\frac{1}{\Omega}-\frac{{4A\bar{\omega}}}{{Lc{\kern
1.0pt}\Delta\omega}}}\right)^{2}}\right]$ (17)
In the limit $\Omega\rightarrow\infty$, the probability distribution for
$\Omega$, defined by Eq.(17) approaches the function
$p\left({\Omega|\Delta\omega}\right)=\frac{{\bar{\omega}}}{{\sqrt{2\pi}{\kern
1.0pt}\sigma_{\omega}}}\frac{1}{{\left|\Omega\right|}}\exp\left[{-\frac{1}{2}\left({\frac{{\bar{\omega}}}{{\sigma_{\omega}}}}\right)^{2}}\right]$
(18)
and hence it is not a normalizable probability distribution because its
integral diverges logarithmically like $\log\Omega$ for
$\Omega\rightarrow\infty$. However, this divergence is multiplied by the
exponentially small factor
$\frac{{\bar{\omega}}}{{\sigma_{\omega}}}\exp\left[{-\frac{1}{2}\left({\frac{{\bar{\omega}}}{{\sigma_{\omega}}}}\right)^{2}}\right]\ll
1$ (19)
since $\bar{\omega}/\sigma_{\omega}\gg 1$. Note that the peak in the
probability distribution for $\Omega$ in Eq. (17) occurs at a value
$\bar{\Omega}=Lc\Delta\omega/(4A\bar{\omega})$, which is consistent with
Eq.(2). The probability distribution for the rotation rate in Eq. (17) is not
a Gaussian distribution. However it is possible to define a width,
$\sigma_{\Omega}$, that depends on $\Omega$:
$\sigma_{\Omega}=\frac{{\sigma_{\omega}}}{{\bar{\omega}}}\Omega$ (20)
Equation (20) gives the uncertainty in the rotation rate, $\sigma_{\Omega}$,
in terms of the true rotation rate, $\Omega$, the bandwidth of the input
classical light, $\sigma_{\omega}$, and the center frequency, $\bar{\omega}$,
used in the classical Sagnac gyroscope. As expected, the uncertainty in the
rotation rate, $\sigma_{\Omega}$, is proportional to the bandwidth of the
input light, $\sigma_{\omega}$. The uncertainty also decreases with higher
input frequency, ${\bar{\omega}}$.
The upper bound on the fidelity (Shannon mutual information), $H_{\max}$, for
the classical Sagnac gyroscope is given by Eq. (4) using Eq. (13) and Eq.
(15):
$H_{\max}=\frac{1}{2}{\kern
1.0pt}\log_{2}\left[{\left({\frac{e}{{2\pi}}}\right)^{1/2}\frac{{\bar{\omega}}}{{\sigma_{\omega}}}}\right]$
(21)
Equation (21) represents a fundamental theoretical upper bound on the
information (in bits) that an ideal classical Sagnac gyroscope can provide to
a user, for each measurement of frequency shift, $\Delta\omega$. The value in
Eq. (21) is an upper bound because we have assumed an ideal classical
measurement that has no associated noise. Therefore, the upper bound in Eq.
(21) for the classical Sagnac gyroscope is a benchmark to which we can compare
rotation sensors based on new quantum technologies, see references above.
In summary, I have proposed the use of a new metric, the Shannon mutual
information (called the fidelity) between the rotation rate and the
measurements (frequency shift) to judge the quality of a rotation sensor. The
fidelity metric is general enough to allow comparison of classical and quantum
rotation sensors. For an ideal classical Sagnac gyroscope, I have computed a
theoretical upper bound on the mutual information that the gyroscope can give
to a user by assuming a classical measurement model with no noise.
Consequently, $H_{\max}$ in Eq. (21) is the Shannon capacity of a classical
Sagnac gyroscope. This upper bound is a benchmark to compare the performance
of new rotation sensors based on improved classical and quantum technologies.
In addition, in Eq. (20) I have derived a relation between the bandwidth of
light input into a classical Sagnac gyroscope, $\sigma_{\omega}$, and an
estimate of the uncertainty in the rotation rate, $\sigma_{\Omega}$.
Finally, the fidelity defined in Eq. (4) is general enough to describe the
quality of any physical measurement. Consequently, the fidelity can be used to
compare the quality of two different apparatuses (two different experiments)
that attempt to measure the same quantity.
## References
* Sagnac (1913a) G. Sagnac, Compt. Rend. 157, 708 (1913a).
* Sagnac (1913b) G. Sagnac, Compt. Rend. 157, 1410 (1913b).
* Sagnac (1914) G. Sagnac, J. Phys. Radium 5th Series 4, 177 (1914).
* Post (1967) E. J. Post, Rev. Mod. Phys. 39, 475 (1967).
* Lefevre (1993) H. Lefevre, _The fiber-optic gyroscope_ (Artech House, Boston, USA, 1993).
* Titterton and Weston (2004) D. H. Titterton and J. Weston, _Strapdown Inertial Navigation Technology_ (The Insitution of Engineering and Technology and The American Institute of Aeronautics, London, U.K. and Reston, Virginia, USA, 2004), second edition ed.
* Chow et al. (1985) W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, Rev. Mod. Phys. 57, 61 (1985).
* Stedman (1997) G. E. Stedman, Rep. Prog. Phys. 60, 615 (1997).
* Bertocchi et al. (2006) G. Bertocchi, O. Alibart, D. B. Ostrowsky, S. Tanzilli, and P. Baldi, J. Phys. B 39, 1011 (2006).
* Gustavson et al. (2000) T. L. Gustavson, A. Landragin, and M. A. Kasevich, Class. Quantum Grav. 17, 2385 (2000).
* Gilowski et al. (2009) M. Gilowski, C. Schubert, T. Wendrich, P. Berg, G. Tackmann, W. Ertmer, and E. M. Rasel, Frequency Control Symposium, 2009 Joint with the 22nd European Frequency and Time forum. IEEE International pp. 1173 – 1175 (2009).
* Gupta et al. (2005) S. Gupta, K. W. Murch, K. L. Moore, T. P. Purdy, and D. M. Stamper-Kurn, Phys. Rev. Lett. 95, 143201 (2005).
* Wang et al. (2005) Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, Phys. Rev. Lett. 94, 090405 (2005).
* Tolstikhin et al. (2005) O. I. Tolstikhin, T. Morishita, and S. Watanabe, Phys. Rev. A 72, 051603(R) (2005).
* Kolkiran and Agarwal (2007) A. Kolkiran and G. S. Agarwal, Optics Express 15, 6798 (2007).
* Cooper et al. (2010) J. J. Cooper, D. W. Hallwood, and J. A. Dunningham, Phys. Rev. A 81, 043624 (2010).
* Cramér (1958) H. Cramér, _Mathematical Methods of Statistics_ (Princeton University Press, Princeton, 1958), eight 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.
* Bahder (2010) T. B. Bahder, submitted to Phys. Rev. A xx, xxx (2010), URL http://arxiv.org/abs/1012.5293.
* Leonhardt and Piwnicki (2000) U. Leonhardt and P. Piwnicki, Phys. Rev. A 62, 055801 (2000).
* Matsko et al. (2004) A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, Opt. Commun. 233, 107 (2004).
* Smith et al. (2008) D. D. Smith, H. Chang, L. Arissian, and J. C. Diels, Phys. Rev. A 78, 053824 (2008).
* Smith et al. (2009) D. D. Smith, K. Myneni, J. A. Odutola, and J. C. Diels, Phys. Rev. A 80, 011809 (2009).
* Bahder and Lopata (2006) T. B. Bahder and P. A. Lopata, Phys. Rev. A 74, 051801R (2006), URL http://arxiv.org/abs/quant-ph/0602123.
* Shannon (1948) C. E. Shannon, The Bell System Technical Journal 27, 379 (1948).
|
arxiv-papers
| 2011-01-24T19:52:52 |
2024-09-04T02:49:16.617103
|
{
"license": "Public Domain",
"authors": "Thomas B. Bahder",
"submitter": "Thomas B. Bahder",
"url": "https://arxiv.org/abs/1101.4634"
}
|
1101.4640
|
# Design and Implementation of a Secure Web-Based File Exchange Server
Specification Design Document
CIISE Security Investigation Initiative
Represented by:
Serguei A. Mokhov
Marc-André Laverdière
Ali Benssam
Djamel Benredjem
{mokhov,ma_laver,al_ben,d_benred}@ciise.concordia.ca
Montréal, Québec, Canada
(December 14, 2005)
###### Contents
1. 1 Introduction
1. 1.1 Purpose
2. 1.2 Scope
3. 1.3 Definitions and Acronyms
2. 2 System Architecture
1. 2.1 Architectural Philosophy
2. 2.2 Components
1. 2.2.1 User Interface
2. 2.2.2 Web Server
3. 2.2.3 Database Server
4. 2.2.4 LDAP Server
5. 2.2.5 Certificate Authority
6. 2.2.6 Logging Engine
3. 2.3 Interactions
4. 2.4 Software Interface Design
1. 2.4.1 Web Server
2. 2.4.2 Database Server
3. 2.4.3 LDAP Server
4. 2.4.4 Certificate Authority
5. 2.4.5 Logging Engine
5. 2.5 Hardware Environment
6. 2.6 Code View
3. 3 Detailed System Design
1. 3.1 Class Diagrams
2. 3.2 User Interface
3. 3.3 Class Diagrams
1. 3.3.1 LDAPConnection
2. 3.3.2 UserCredentials
3. 3.3.3 OptionsFileLoaderSingleton
4. 3.3.4 DatabaseConnection
4. 3.4 Data Storage Format
1. 3.4.1 Entity Relationship Diagram
5. 3.5 Options File
6. 3.6 Directory Configuration
7. 3.7 External System Interfaces
1. 3.7.1 External Systems and Databases
8. 3.8 User Scenarios
1. 3.8.1 Typical Scenario
2. 3.8.2 Variant scenario
3. 3.8.3 Variant scenario
9. 3.9 Administrator Scenarios
1. 3.9.1 Typical Scenario
2. 3.9.2 Variant Scenario
3. 3.9.3 Variant Scenario
## Chapter 1 Introduction
Building Trust is the basis of all communication, especially electronic one,
as the identity of the other entity remains concealed. To address problems of
trust, authentication and security over the network, electronic communications
and transactions need a framework that provides security policies, encryption
mechanisms and procedures to generate manage and store keys and certificates.
The Public Key Infrastructure (PKI) is a security architecture that has been
introduced to provide an increased level of confidence for exchanging
information over increasingly insecure networks, such as the Internet. A PKI
infrastructure is expected to offer its users a secure and trustworthy
electronic transaction.
### 1.1 Purpose
The intent of implementation and deployment of PKI facilities is to meet its
basic purpose of providing Trust. Presently, PKI needs to perform the
following security functions:
* •
_Mutual authentication of entities taking part in the communication:_ Only
authenticated principals can access files to which they have privileges.
* •
_Ensure data integrity:_ By issuing digital certificates which guarantee the
integrity of the public key. Only the public key for a certificate that has
been authenticated by a certifying authority should work with the private key
possessed by an entity. This eliminates impersonation and key modification.
* •
_Enforce security:_ Communications are more secure by using SSL to transmit
information.
### 1.2 Scope
PKI is implemented to secure sensitive resources of the organization and avoid
security breaches. The PKI environment allows trustworthy communication
between the different principals. These principals must be authenticated and
the access to the resources (files) should be secured and regulated. Any
principal wants to access to the database needs to perform the following
steps:
* •
_Mutual authentication:_ The Web Server via which the database is contacted
authenticates the principal using its digital certificate and username to
ensure that it is who it claims to be . The principal authenticates also the
server using its certificate information.
* •
_Principal validation:_ To validate the principal, the server looks up
information from an LDAP server which contains the hierarchy of all principals
along with certificates and credentials. The LDAP service is compliant with
the X.500 database structure.
* •
_Enforcing security:_ The security is enforced by using SSL to communicate
between the Web Server and the LDAP server, the Web Server and the database
and between the principal and Web Server.
* •
_Principal authentication:_ Upon successful authentication, the Web Server
will allow the principal to perform actions on the database according to a pre
specified Access Control List.
* •
_Kinds of users:_ We distinguish between a normal and an administrator. While
a normal user can upload, download, delete and view files; the administrator
has the ability to: upload, download, delete and view files; add, delete and
modify users; generate user’s certificate, with all required information;
generate ACL to users; manage groups, perform maintenance.
Finally, this infrastructure allows additional features such as the ability to
assign users to groups in order to provide users with the access to files
prepared by other group members.
### 1.3 Definitions and Acronyms
* •
PKI: Public Key Infrastructure
* •
OpenLDAP : is a free, open source implementation of the Lightweight Directory
Access Protocol (LDAP).
* •
OpenSSL: an open source SSL library and certificate authority
* •
Apache Tomcat: A Java based Web Application container that was created to run
Servlets and JavaServer Pages (JSP) in Web applications
* •
PostgreSQL: An open source object-relational database server
* •
SSL: Secure Socket Layer
* •
JSP: Java Server Pages
* •
JCE: Java Cryptography Extension
* •
API: Application Programming Interface
* •
JDBC: Java Database Connectivity
* •
JNDI: Java Naming and Directory Interface
* •
LDAP: Lightweight Directory Access Protocol
* •
X.509: A standard for defining a Digital Certificate used by SSL
* •
SRS: Specification request Document
* •
SDD: Specification Design document
* •
DER: Distinguished Encoding Rules
* •
Mutual Authentication: The process of two principals proving their identities
to each other
* •
SFS: Secure File Exchange Server, this product
* •
COTS: Commercial Off The Shelf, common commercially or freely available
software
## Chapter 2 System Architecture
This chapter is intended to provide an overview of the whole system as
proposed in the previous requirements and specification document. It describes
the product’s perspective, interfaces and design constraints as we have
assumed. We will first describe the architectural guidelines for this product,
followed by software interface design design, and hardware environment.
### 2.1 Architectural Philosophy
The SFS technology hereby implemented is running on architecture that provides
a high level of secrecy and integrity for exchanging information. The system
is externally visible only through a web application for normal users, and is
also entirely visible and accessible for administrators in the scope of normal
operations. In addition, the system assumes an internal certificate authority
which is explicitly trusted by all principals using SFS.
For the proposed architecture, it requires mutual authentication between the
user and the web server, an LDAP validation of the user by using digital
certificates, the use of the SSL protocol to enforce the security over the
communication between modules and the preservation of files in a database.
This architecture must respect the following properties:
* •
_Security:_ The confidentiality, integrity and availability of information.
This is to be implemented by supporting the data encryption and certificates
mechanisms for secured communication, as well as specifying an access control
mechanism for the files stored
* •
_Trustworthiness:_ The use of electronic certificates internally generated,
and of specific use for the application, allow an high trust to be given to
the user.
* •
_Scalability:_ SFS can be expanded easily to cope with large loads. Methods
such as load balancing and replication can be easily integrated.
* •
_Openness:_ The proposed architecture can be implemented and deployed using
Java Technologies and open source tools that are well-used and rely on
standards. The SFS itself is an open source product developed to achieve
security objectives.
* •
_Component-Based Software Engineering:_ The SFS framework may be treated as
components (modules) . We have already mentioned the relevant technologies
that can better fit for each of these modules.
* •
_Usability:_ The SFS service must be designed for high usability. All the
required information for a single operation should be grouped in a single
screen, with a minimum number of screens needed for all the application.
This architecture aims at maximizing software reuse by the integration of COTS
applications, portability and interoperability by the use of standards,
security and scalability by the user of a single access point.
### 2.2 Components
The Figure 2.1 describes the overall architecture of the SFS system. We see
four major interacting components: the user interface, the web server, the
database and the LDAP server. Two components are not displayed on this figure
, which are the certificate authority and the logging engine.
Figure 2.1: Main system architecture
#### 2.2.1 User Interface
The user interface constitutes of HTML web pages that the user uses through a
web browser. Those web pages are generated by the web server and interact
exclusively with the SFS web server.
#### 2.2.2 Web Server
The web server is the single access point of the system. It handles
authentication responsibilities, database access and user interaction. This is
to be handled by the Apache Tomcat 5.0 server and custom J2SE 5.0 code.
#### 2.2.3 Database Server
This database server contains all the information about the files and their
access control rights. It contains also a subset of the user information. This
is to be handled by PostgreSQL.
#### 2.2.4 LDAP Server
This specialized server holds the user credentials (notably user name and
password). It could be extended to include user certificates. This module will
be realized by OpenLDAP.
#### 2.2.5 Certificate Authority
This responsibility is manually managed by administrators. Using software
tools, they are able to generate the user and server certificates. In our
case, we use OpenSSL to perform those functions.
#### 2.2.6 Logging Engine
This component is responsible for collecting the audit trails and debug
information from other components and store it locally. We wanted to use
log4j, but we finally opted for the logging mechanisms available in the tools
we are using, notably by using Tomcat’s logging.
### 2.3 Interactions
We will now describe the inter-module interactions by the use of a system
scenario.
The user, with a web browser, connects to the web server using SSL. The web
server, being configured as to require client authentication, both parties
exchange their certificates and validate their peer’s identity. The web server
then prompts the user for a user name and password, thereby enforcing 2-factor
authentication.
Upon receiving this information, the web server queries the LDAP directory
based on the user name and retrieves the user’s password hash and certificate
(if any is defined). The web server then proceeds to hash the plaintext
password (using SHA1) received from the user and compare with the one from the
LDAP server. If that information (as well as the certificates, if any)
matches, the user is logged in the system.
The web server will then query the database server for the access rights of
the user (administrator or normal user) and the list of files the user has
access to. Based on this information, it will display the appropriate user
interface functions and the file list.
On user requests to upload, delete or download files, the web server will
request the database server to perform the needed transactions.
### 2.4 Software Interface Design
This section describes the software interfaces (commonly referred to as APIs)
to be used to communicate between each component.
#### 2.4.1 Web Server
The web server is reached by the client through an HTTPS connection to the
single point of access for the web application, the login screen.
#### 2.4.2 Database Server
The database server is reached using the JDBC API with the official PostgreSQL
JDBC driver.
#### 2.4.3 LDAP Server
The LDAP server is reached using the default Sun LDAP JNDI driver. The LDAP
protocol is used for the queries and is encapsulated by JNDI.
#### 2.4.4 Certificate Authority
This component is not integrated with others and, as such, does not have a
software interface to document.
#### 2.4.5 Logging Engine
On the Web server, this component is called automatically by the use of the
default output and default error streams, which will write the information to
a log instead of a console.
### 2.5 Hardware Environment
The SFS system is expected to work in a networked environment, possibly a LAN,
but not necessarily. We assume that the principals have a network connection
allowing to communicate with each other. Only typical low-end workstation
hardware (such as a Pentium III system with 256+ MB of RAM, 2GB hard drive
with a 10BaseT Ethernet connection) is required to operate all the components
of the system, which may be distributed or centralized as needed. Ordinary P
III with 256+ MB of Ram and 2GB HDD are the minimum requirements needed to
deploy the system.
### 2.6 Code View
We decided to structure our software in a few main packages, as illustrated in
Figure 2.2.
Figure 2.2: Java Packages
Those packages hold as follows:
1. 1.
_securefileserver:_ root of our custom code
2. 2.
_base64:_ Library for encoding and decoding Base64-encoded data
3. 3.
_apps:_ Various applications
4. 4.
_webapps:_ The web application code
5. 5.
_conn:_ Connection abstraction code, such as SSL connections, LDAP connections
and database connections
6. 6.
_cert:_ Certificate authority code
7. 7.
_util:_ Utility classes, such as the configuration file loader
Since we had to deal with a large set of non-code artifacts, we structured our
repository as illustrated in Figure 2.3.s
Figure 2.3: Repository Structure
Those directories hold as follows:
1. 1.
root directory: contains the Ant makefile and all the other subdirectories
2. 2.
_cert:_ certificates generated manually using OpenSSL
3. 3.
_conf:_ configuration files
4. 4.
_CVS:_ repository management code, handled automatically by CVS
5. 5.
_images:_ documentation-related images
6. 6.
_lib:_ libraries in JAR format
7. 7.
_sql:_ database initialization script
8. 8.
_src:_ Java source code
9. 9.
_tex:_ documentation in LaTeX2e format
10. 10.
_txt:_ various notes in text format
11. 11.
_design:_ Rational Rose model of the system
## Chapter 3 Detailed System Design
In this section of the specification document we elaborate the detailed
description of the main modules and subprograms of the SFS system. We provide
the important class diagrams for the different packages of the design phase as
mentioned above.
Please consult Figure 3.1 for a high-level view of the class diagram of our
application. Please note that the servlets login and User also have a fair
amount of business logic integrated in them. This situation is due to the
evolutionary nature of the development method used in this project which,
combined with tight deadlines, did not allow for a proper refactoring of the
classes.
We can also take note the presence of test classes in our class diagram, which
are JUnit test cases that allowed to perform some unit testing. The smallness
of the class diagram is mostly explained by the fact that most of the
functionality was implemented in COTS programs that needed only some
configuration.
### 3.1 Class Diagrams
The following diagrams show some of the important user interfaces of the SFS
software system.
Figure 3.1: SFS main class diagram
Figure 3.2: Conn Package Diagram
Figure 3.3: Application Package Diagram
Figure 3.4: The util Package Diagram
Figure 3.5: The cert Package Diagram
### 3.2 User Interface
The following diagrams show some of the important user interfaces of the SFS
software system. On figure 3.6, we see the interface allowing clients to log
in.
Figure 3.6: User interface Log on
Once the client mutual authentication is achieved and user allowed to use the
system he will get the following screen (3.7).
Figure 3.7: User operations displayed
When the client chooses the file to download, he will be prompted to open the
file or give the path he want to save the file in.
Figure 3.8: User operations displayed
Figure 3.9 shows the upload operation, so the user will be prompted to select
the file he want to upload.
Figure 3.9: User operations displayed
Figure 3.10 shows the administrator capabilities: adding users, remove users,
adding groups, setting rights, etc.
Figure 3.10: User operations displayed
The snapshot of the SFS provides the login interface via which the services of
the SFS system can not be utilized unless the user is already logged in.
### 3.3 Class Diagrams
Figure 3.1 includes most of the classes already present. We will describe a
few classes in detail here. All the details regarding the classes are
available in the Javadoc.
#### 3.3.1 LDAPConnection
This class provides an abstraction of an LDAP JNDI context, as well as pre-
made queries for obtaining user credentials, deleting an user and adding an
user in the database.
It depends on SSLConnectionFactory for ensuring that our SSL Connections are
set with the proper keys. It also depends on UserCredentials, since this is
the data type it returns on a query.
#### 3.3.2 UserCredentials
This class encapsulates a user name, a password, and a certificate. It
integrates the hashing of plaintext passwords, as well as a matching
comparison between two sets of credentials. It depends on TestSSHA for
generating and validating the salted SHA1 hash.
#### 3.3.3 OptionsFileLoaderSingleton
This class is a singleton, meaning that only up to one instance can exist at
any time. It loads and parses a configuration file. It also includes many
default keys of the configuration file as constant strings.
#### 3.3.4 DatabaseConnection
This class provides an abstraction for an SSL-enabled database connection.
Contrary to LDAPConnection, it does not provide high-level methods in it,
leaving to the calling code the responsibility to formulate proper SQL
queries.
### 3.4 Data Storage Format
In this section, we provide a description for the database handling the
security aspect of the system. It consists of the following entity relation
model.
#### 3.4.1 Entity Relationship Diagram
The Groupe entity contains the list of groups a user may belong to. The User
entity contains the list of users having right to use the system. The File
entity contains information about different files a user can upload, download,
delete and view. A user may be an Administrator or normal user. The other
relationships (group_user, group_files) are defined between entities Groupe
and User and Groupe and File to host different information related to both of
them respectively. Hereafter, we provide in Figure 3.11 the Entity
Relationship Model of the Security Database.
Figure 3.11: Entity Relationship Diagram
### 3.5 Options File
A .config file is expected in the execution root in order to read the
configuration options.
Lines can be comments (#), blank, or containing a key=value pair.
The expected configuration options are:
* •
_ca_server:_ host name of the CA server. This option is reserved for future
use.
* •
_db_server:_ host name of the database server.
* •
_keystore_filepath:_ relative or absolute path for the web server’s keys
* •
_keystore_password:_ password for the previously specified keystore
* •
_ca_certificate_filepath:_ path to the CA certificate’s keystore
* •
_ca_certificate_password:_ password for the CA certificate
* •
_ldap_password:_ administrator password for the LDAP server
* •
_ldap_principal:_ administrator user name for the LDAP server
* •
_ldap_server:_ host name of the LDAP server
### 3.6 Directory Configuration
The LDAP directory is to be structured in an abritarily manner, as the DN is
not used in queries. However, the UID parameter is used for querying based on
the user name. The user information is of type inetOrgPerson, with the fields
uid, userPassword and userCertificate for, respectively, the user name, its
password (hashed with salted SHA) and its certificate.
### 3.7 External System Interfaces
The only externally reachable interface to the SFS system is the login page of
the web application. This page should be located at /sfs/login.jsp. It is also
symlinked to it via index.jsp.
#### 3.7.1 External Systems and Databases
The SFS system is not designed for interacting with any other systems than
those described as part of our architecture.
### 3.8 User Scenarios
#### 3.8.1 Typical Scenario
1. 1.
User connects to remote server using a web browser on a secure connection
2. 2.
User is prompted with a logon screen, and provides a username and password
3. 3.
After validation, the user is logged as a normal user and is shown a list of
files to which he or she has access to, as well as their rights
4. 4.
The user clicks to download a file and, if the system validates its rights,
the download begins
#### 3.8.2 Variant scenario
4\. The user chooses to upload a file and the upload begins. The file will be
modifyable by the user and according to the default new file ACL.
#### 3.8.3 Variant scenario
4\. The user chooses to delete a file to which it has delete rights and the
systems perform the deletion
### 3.9 Administrator Scenarios
#### 3.9.1 Typical Scenario
1. 1.
User connects to remote server using a web browser on a secure connection
2. 2.
User is prompted with a logon screen, and provides a username and password
3. 3.
After validation, the user is logged as an administrator and is shown a menu
of options, and chooses to view a list of files in the system
4. 4.
The user clicks on the user edit button and changes the access control list of
the object.
#### 3.9.2 Variant Scenario
1. 1.
User connects to the certificate administration service and generate a
certificate for a given subject through a secure connection
2. 2.
User connects to remote server on a secure connection
3. 3.
User is prompted with a logon screen, and provices username and password
4. 4.
After validation, the user is logged as an administrator and is shown the menu
5. 5.
The user clicks on the certificate edit button and is shown a screen of
certificate maintenance
6. 6.
The user uploads the certificate and the system binds it with the
certificate’s defined principal (or creates the user if none exists already)
#### 3.9.3 Variant Scenario
1. 1.
User connects to remote server using a web browser on a secure connection
2. 2.
User is prompted with a logon screen, and provides a username and password
3. 3.
After validation, the user is logged as an administrator and is shown a menu
of options, and chooses to edit the certificates
4. 4.
The user chooses to remove the certificate from a given principal
5. 5.
The user connects to the certificate administration service and issues a
certificate revocation.
## Bibliography
* [Apa11] Apache Foundation. Apache Jakarta Tomcat. [online], 1999–2011. http://jakarta.apache.org/tomcat/index.html.
* [CBH03] Suranjan Choudhury, Katrik Bhatnagar, and Wisam Haque. Public Key Infrastructure, Implementation and Design. NIIT Books, 2003.
* [E+11] Eclipse contributors et al. Eclipse Platform. eclipse.org, 2000–2011. http://www.eclipse.org, last viewed February 2010.
* [GB11] Erich Gamma and Kent Beck. JUnit. [online], Object Mentor, Inc., 2001–2011. http://junit.org/.
* [Lar06] Craig Larman. Applying UML and Patterns: An Introduction to Object-Oriented Analysis and Design and Iterative Development. Pearson Education, third edition, April 2006. ISBN: 0131489062.
* [Leu05] Leuvens Universitair. A few frequently used SSL commands. [online], Leuvens Universitair Dienstencentrum voor Informatica en Telematica, 2005. http://shib.kuleuven.be/docs/ssl_commands.shtml.
* [Ope05a] OpenLDAP Community. OpenLDAP: The open source for LDAP software and information. [online], 2005. www.openldap.org.
* [Ope05b] OpenSSL Community. OpenSSL: The open source toolkit for SSL/TLS. [online], 2005. http://www.openssl.org.
* [O’R05] O’Reilly. Home of com.oreilly.servlet. [online], 2005. http://servlets.com/cos.
* [Sun05a] Sun Microsystems, Inc. Java servlet technology. [online], 1994–2005. http://java.sun.com/products/servlets.
* [Sun05b] Sun Microsystems, Inc. JavaServer pages technology. [online], 2001–2005. http://java.sun.com/products/jsp/.
* [The11] The PostgreSQL Global Development Group. PostgreSQL – the world’s most advanced open-source database. [ditigal], 1996–2011. http://www.postgresql.org/, last viewed January 2010.
## Index
* Introduction Chapter 1
|
arxiv-papers
| 2011-01-24T20:30:17 |
2024-09-04T02:49:16.622345
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serguei A. Mokhov, Marc-Andr\\'e Laverdi\\`ere, Ali Benssam, Djamel\n Benredjem",
"submitter": "Serguei Mokhov",
"url": "https://arxiv.org/abs/1101.4640"
}
|
1101.4891
|
# The effect of dust cooling on low-metallicity star-forming clouds
Gustavo Dopcke, Simon C. O. Glover, Paul C. Clark and Ralf S. Klessen Zentrum
für Astronomie der Universität Heidelberg, Institut für Theoretische
Astrophysik, Albert-Ueberle-Str. 2, 69120 Heidelberg, Germany
###### Abstract
The theory for the formation of the first population of stars (Pop III)
predicts a IMF composed predominantly of high-mass stars, in contrast to the
present-day IMF, which tends to yield stars with masses less than 1 M⊙. The
leading theory for the transition in the characteristic stellar mass predicts
that the cause is the extra cooling provided by increasing metallicity and in
particular the cooling provided at high densities by dust. The aim of this
work is to test whether dust cooling can lead to fragmentation and be
responsible for this transition. To investigate this, we make use of high-
resolution hydrodynamic simulations. We follow the thermodynamic evolution of
the gas by solving the full thermal energy equation, and also track the
evolution of the dust temperature and the chemical evolution of the gas. We
model clouds with different metallicities, and determine the properties of the
cloud at the point at which it undergoes gravitational fragmentation. We
follow the further collapse to scales of an AU when we replace very dense,
gravitationally bound, and collapsing regions by a simple and nongaseous
object, a sink particle. Our results suggest that for metallicities as small
as 10${}^{-5}\rm Z_{\odot}$, dust cooling produces low-mass fragments and
hence can potentially enable the formation of low mass stars. We conclude that
dust cooling affects the fragmentation of low-metallicity gas clouds and plays
an important role in shaping the stellar IMF even at these very low
metallicities.
###### Subject headings:
early universe — hydrodynamics — methods: numerical — stars: formation —
stars: luminosity function, mass function
## 1\. Introduction
The first burst of star formation in the Universe is thought to give rise to
massive stars, the so-called Population III, with current theory predicting
masses in the range 20-150 M⊙ (Abel et al., 2002; Bromm et al., 2002; O’Shea &
Norman, 2007; Yoshida et al., 2008). This contrasts with present-day star
formation, which tends to yield stars with masses less than 1 M⊙ (Kroupa,
2002; Chabrier, 2003), and so at some point in the evolution of the Universe
there must have been a transition from primordial (Pop. III) star formation to
the mode of star formation we see today (Pop. II/I).
When gas collapses to form stars, gravitational energy is transformed to
thermal energy and unless this can be dissipated in some fashion, it will
eventually halt the collapse. Thermal energy can be dissipated by processes
such as atomic fine structure line emission, molecular rotational or
vibrational line emission, or the heating of dust grains. In some cases, these
processes are able to cool the gas significantly during the collapse. This
temperature drop can promote gravitational fragmentation (Mac Low & Klessen,
2004; Bonnell et al., 2007) by diminishing the Jeans mass, which means that
instead of forming very massive clumps, with fragment masses corresponding to
the initial Jeans mass in the cloud, it can instead form even more fragments
with lower masses.
If the gas is cooled only by molecular hydrogen emission, numerical
simulations show that the stars should be very massive (Abel et al., 2002;
Bromm et al., 2002; O’Shea & Norman, 2007; Yoshida et al., 2008). This happens
because the H2 cooling becomes inefficient for temperatures bellow 200K and
densities above $10^{4}\rm cm^{-3}$. At this temperature and density, the mean
Jeans mass at cloud fragmentation is 1,000 times larger than in present-day
molecular clouds,
$M_{\rm frag}\approx 700M_{\odot}\left(\frac{T_{\rm frag}}{200\rm
K}\right)^{3/2}\left(\frac{n_{\rm frag}}{10^{4}\rm cm^{-3}}\right)^{-1/2},$
(1)
for an atomic gas with temperature $T_{\rm frag}$ and number density $n_{\rm
frag}$.
Metal line cooling and dust cooling are effective at lower temperatures and
larger densities, and so the most widely accepted cause for the transition
from Pop. III to Pop. II is metal enrichment of the interstellar medium by the
previous generations of stars. This suggests that there might be a critical
metallicity Zcrit at which the mode of star formation changes.
The main coolants that have been studied in the literature are CII and OI fine
structure emission (Bromm et al., 2001; Bromm & Loeb, 2003; Santoro & Shull,
2006; Frebel et al., 2007), and dust emission. C and O are identified as the
key species because in the temperature and density conditions that
characterise the early phases of Pop. III star formation, the OI and CII fine-
structure lines dominate over all other metal transitions (Hollenbach & McKee,
1989). By equating the CII or OI fine structure cooling rate to the
compressional heating rate due to free-fall collapse, one can define critical
abundances $[\rm C/H]=-3.5$ and $[\rm O/H]=-3.0$111$[\rm X/\rm
Y]=log_{10}(N_{\rm X}/N_{\rm Y})_{\star}-log_{10}(N_{\rm X}/N_{\rm
Y})_{\odot}$, for elements X and Y, where $\star$ denotes the gas in question,
and where $\rm N_{X}$ and $\rm N_{Y}$ are the mass fractions of the elements X
and Y. for efficient metal line cooling (Bromm & Loeb, 2003). However,
previous works (Jappsen et al., 2009a, b) show that this metallicity threshold
does not represent a critical metallicity: the fact that metal-line cooling
has a larger value than the compressional heating does not necessarily lead to
fragmentation.
Dust-cooling models predict a much lower critical metallicity $(\rm Z_{\rm
crit}\approx 10^{-5}\rm Z_{\odot})$. The conditions for fragmentation in the
low-metallicity dust cooling model are predicted to occur in high density gas,
where the distances between the fragments can be very small (Omukai, 2000;
Omukai et al., 2005; Schneider et al., 2002, 2006; Schneider & Omukai, 2010).
In this regime, interactions between fragments will be common, and analytic
models of fragmentation are unable to predict the mass distribution of the
fragments. A full 3D treatment, following the fragments, is needed.
Initial attempts were made by Tsuribe & Omukai (2006, 2008) and Clark et al.
(2008). However, these treatments used a tabulated equation of state, based on
results from previous one-zone chemical models (Omukai et al., 2005), to
determine the thermal energy. This approximation assumes that the gas
temperature adjusts instantaneously to a new equilibrium temperature whenever
the density changes and hence ignores thermal inertia effects. This may yield
too much fragmentation.
In this work, we improve upon these previous treatments by solving the full
thermal energy equation, and calculating the dust temperature through the
energy equilibrium equation. We assume currently that the only significant
external heat source is the CMB, and include its effects in the calculation of
the dust temperature.
## 2\. Simulations
### 2.1. Numerical method
We model the collapse of a low-metallicity gas cloud using a modified version
of the Gadget 2 (Springel, 2005) smoothed particle hydrodynamics (SPH) code.
To enable us to continue our simulation beyond the formation of the first very
high density protostellar core, we use a sink particle approach (Bate et al.,
1995), based on the implementation of Jappsen et al. (2005). Sink particles
are created once the SPH particles are bound, collapsing, and within an
accretion radius, $h_{acc}$, which is taken to be 1.0 AU. The threshold number
density for sink particle creation is $5.0\times 10^{13}\rm cm^{-3}$. At the
threshold density, the Jeans length at the minimum temperature reached by the
gas is approximately one AU, while at higher densities the gas becomes
optically thick and begins to heat up. Further fragmentation on scales smaller
than the sink particle scale is therefore unlikely to occur. For further
discussion see Clark et al. (2011).
To treat the chemistry and thermal balance of the gas, we use the same
approach as in Clark et al. (2011), with two additions: the inclusion of the
effects of dust cooling, as described below, and formation of H2 on the
surface of dust grains (see Hollenbach & McKee, 1979). The Clark et al. (2011)
chemical network and cooling function were designed for treating primordial
gas and do not include the chemistry of metals such as carbon or oxygen, or
the effects of cooling from these atoms, or molecules containing them such as
CO or H2O. We justify this approximation by noting that previous studies of
very low-metallicity gas (e.g. Omukai et al., 2005, 2010) find that gas-phase
metals have little influence on the thermal state of the gas. Omukai et al.
(2010) showed that H2O and OH are efficient coolants at $10^{8}<n<10^{10}\rm
cm^{-3}$ for their one-zone model. In their hydrodynamical calculations,
however, the collapse is faster, and the effect of H2O and OH is not
perceptible. Therefore we do not expect oxygen-bearing molecules to have a big
effect on the thermal evolution of the gas. For the metallicities and dust-to-
gas ratios considered in this study, the dominant sources of cooling are the
standard primordial coolants (H2 bound-bound emission and collision-induced
emission) and energy transfer from the gas to the dust.
Figure 1.— Results of our low-resolution simulations, showing the dependence
of gas and dust temperatures on gas density for metallicities $10^{-4}$ and
$10^{-5}$ times the solar value. In red, we show the gas temperature, and in
blue the dust temperature for the turbulent and rotating cloud. The simple
core collapse is overploted in dark red and green. The points with thinner
features are from the simulations without rotation or turbulence, while those
showing more scatter come from the simulations with rotation and turbulence.
The dashed lines show constant Jeans mass values.
#### 2.1.1 Dust cooling
Collisions between gas particles and dust grains can transfer energy from the
gas to the dust (if the gas temperature $T$ is greater than the dust
temperature $T_{\rm gr}$), or from the dust to the gas (if $T_{\rm gr}>T$).
The rate at which energy is transferred from gas to dust is given by
(Hollenbach & McKee, 1979)
$\Lambda_{\rm gr}=n_{\rm gr}n\bar{\sigma}_{\rm gr}v_{\rm p}f(2kT-2kT_{\rm
gr})\>\>{\rm erg}\>{\rm s^{-1}}\>{\rm cm^{-3}},$ (2)
where $n_{\rm gr}$ is the number density of dust grains, $n$ is the number
density of hydrogen nuclei, $\bar{\sigma}_{\rm gr}$ is the mean dust grain
cross-section, $v_{\rm p}$ is the thermal speed of the proton, and $f$ is a
factor accounting for the ontribution of species other than protons, as well
as for charge and accommodation effects. We assume that $\bar{\sigma}_{\rm
gr}$ is the same as for Milky Way dust, and that the number density of dust
grains is a factor ${\rm Z}/{\rm Z_{\odot}}$ smaller than the Milky Way value.
To compute the rate at which the dust grains radiate away energy, we use the
approximation (Stamatellos et al., 2007)
$\Lambda_{\rm rad}=4\sigma_{\rm sb}n_{\rm gr}\frac{(T_{\rm gr}^{4}-T_{\rm
cmb}^{4})}{\Sigma^{2}\kappa_{R}+\kappa_{P}^{-1}},$ (3)
where $T_{\rm cmb}$ is the CMB temperature, $\sigma_{\rm sb}$ is the Stefan-
Boltzmann constant, $\kappa_{P}$ and $\kappa_{R}$ are the Planck and Rosseland
mean opacities and $\Sigma$ is the column density of gas measured along a
radial ray from the particle to the edge of the cloud.
As explained by Stamatellos et al. (2007), this expression has the correct
behaviour in the optically thin and optically thick limits, and interpolates
between these two limits in a smooth fashion. In practice, we approximate
further by assuming that the Planck and Rosseland mean opacities are equaland
by using the fact that $\Sigma\sim\rho L_{\rm J}$ for a gravitationally
collapsing gas, where $\rho$ is the mass density of the gas, and $L_{\rm J}$
is the Jeans length, given by $L_{\rm J}=(\pi c_{s}^{2}/G\rho)^{1/2}$, where
$c_{s}$ is the speed of sound in the gas. By approximating $\Sigma$ in this
fashion, we avoid the computational difficulties involved with measuring
column densities directly in the simulation, while still following the
behaviour of the gas reasonably accurately in the optically thick regime. In
any case, most of the interesting behaviour that we find in our simulations
occurs while dust cooling remains in the optically thin regime. To compute the
temperature of the dust grains, we assume that the dust is in thermal
equilibrium, and hence solve the equilibrium equation
$\Lambda_{\rm gr}-\Lambda_{\rm rad}=0.$ (4)
This equation is transcendental, so we solve it numerically.
#### 2.1.2 Dust opacity
We follow the dust opacity model of Goldsmith (2001), and we calculate the
opacity as a function of the dust temperature in the same fashion as in
Banerjee et al. (2006). To convert from the frequency-dependent opacity given
in Goldsmith (2001) to our desired temperature-dependent mean opacity, we
assume that for dust with temperature $T_{\rm gr}$, the dominant contribution
to the mean opacity comes from frequencies close to a frequency $\bar{\nu}$
that is given by $h\bar{\nu}=\alpha kT_{\rm gr}$, where $\alpha=2.70$. At a
reference temperature $T_{0}=6.75$ K, this procedure yields an opacity
$\begin{array}[]{rl}\kappa(T_{0})=&3.3\times 10^{-26}\alpha(\rm n/2\rho_{\rm
gas})\\\ =&2.664\times 10^{-2}/(1+4[\rm He])\end{array}$ (5)
where [He] is the helium abundance, and $n$ is the number density of hydrogen
nuclei. At other temperatures, $\kappa\propto T_{\rm gr}^{2}$, so long as
$T_{\rm gr}<200$ K. For grain temperatures larger than 200 K, it is necessary
to account for the effects of ice-mantle evaporation, while at much higher
grain temperatures, the opacity falls off extremely rapidly due to the melting
of the grains. We account for these effects (see Semenov et al., 2003) and so
our opacity varies with dust temperature following the relationship
$\kappa=\kappa(T_{0})\times\left\\{\begin{array}[]{ll}T^{2}&\hskip
36.135pt\mbox{T $<$ 200K}\\\ T^{0}&\hskip 36.135pt\mbox{200K $<$ T $<$
1500K}\\\ T^{-12}&\hskip 36.135pt\mbox{T $>$ 1500K}\end{array}\right.$ (6)
### 2.2. Setup and Initial conditions
Resolution | Number of | Particle | Turbulence | Angular
---|---|---|---|---
Level | Particles | Mass | | Momentum
| | ($10^{-5}\rm M_{\odot}$) | ($E_{\rm turb}/|E_{\rm grav}|$) | ($E_{\rm rot}/|E_{\rm grav}|$)
High | $40\times 10^{6}$ | $2.5$ | 0.1 | 0.02
Low | $4\times 10^{6}$ | $25.0$ | 0.1 | 0.02
| | | 0.0 | 0.00
Table 1Simulation properties.
We performed three sets of simulations, two at low resolution and one at high
resolution. The details are shown in Table 1. Our low resolution simulations
were performed to explore the thermal evolution of the gas during the
collapse, and had 4 million SPH particles which was insufficient to fully
resolve fragmentation. We used these simulations to model the collapse of an
initially uniform gas cloud with an initial number density of $10^{5}\>{\rm
cm^{-3}}$ and an initial temperature of $300\>{\rm K}$. We modelled two
different metallicities (10${}^{-4}{\rm Z}_{\odot}$ and 10${}^{-5}{\rm
Z}_{\odot}$). The initial cloud mass was $1000\>{\rm M_{\odot}}$, and the mass
resolution was $25\times 10^{-3}\>{\rm M_{\odot}}$. In one set of low-
resolution simulations the gas was initially at rest, while in the other, we
included small amounts of turbulent and rotational energy, with $E_{\rm
turb}/|E_{\rm grav}|=0.1$ and $\beta=E_{\rm rot}/|E_{\rm grav}|=0.02$, where
$E_{\rm grav}$ is the gravitational potential energy, $E_{\rm turb}$ is the
turbulent kinetic energy and $E_{\rm rot}$ is the rotational energy. For our
high resolution simulations, which were designed to investigate whether the
gas would fragment, we employed 40 million SPH particles. We adopted initial
conditions similar to those in the low-resolution run with turbulence and
rotation. Again, we simulated two metallicities, 10${}^{-4}{\rm Z}_{\odot}$
and 10${}^{-5}{\rm Z}_{\odot}$. The mass resolution (taken to be 100 times the
SPH particle mass) was $2.5\times 10^{-3}{\rm M}_{\odot}$.
## 3\. Analysis
### 3.1. Thermodynamical evolution of gas and dust
Figure 2.— Number density maps for a slice through the high density region.
The image shows a sequence of zooms in the density structure in the gas
immediately before the formation of the first protostar.
In Figure 1, we compare the evolution of the dust and gas temperatures in the
low-resolution simulations. The dust temperature, shown in the lower part of
the panels, varies from the CMB temperature in the low density region to the
gas temperature at much higher densities. At densities higher than
$10^{11}$–$10^{12}\>\rm cm^{-3}$, dust cooling starts to be effective and
begins to cool the gas. The gas temperature decreases to roughly 600 K in the
$10^{-5}\>\rm Z_{\odot}$ simulations, and 300 K in the Z $=10^{-4}\rm
Z_{\odot}$ case. This temperature decrease significantly increases the number
of Jeans masses present in the collapsing region, making the gas unstable to
fragmentation. The dust and the gas temperatures couple for densities higher
then $10^{13}\rm cm^{-3}$, when the compressional heating starts to dominate
again over the dust cooling. The subsequent evolution of the gas is close to
adiabatic. If we compare the results of the runs with and without rotation and
turbulence, then the most obvious difference is the much greater scatter in
the $n-T$ diagram in the former case. Variations in the infall velocity lead
to different fluid elements undergoing different amounts of compressional
heating. The overall effect is to reduce both the infall velocity and the
average compressional heating rate. This allows dust cooling to dominate at a
density that is up to five times smaller than in the case without rotation or
turbulence. The gas also reaches a lower temperature, cooling down to
$\approx$ 200K (instead of 300K) for the Z $=10^{-4}\rm Z_{\odot}$ case, and
to $\approx$ 400K (instead of 600K) for the Z $=10^{-5}\rm Z_{\odot}$ case.
This behavior shows that it is essential to use 3D simulations to follow the
evolution of the collapsing gas. A similar effect can be seen in Clark et al.
(2011).
### 3.2. Fragmentation
We follow the thermodynamical evolution of the gas up to very high densities
of order $10^{17}\rm cm^{-3}$, where the Jeans mass is $\approx 10^{-2}\rm
M_{\odot}$, and so we need a high resolution simulation to study the
fragmentation behaviour. The transport of angular momentum to smaller scales
during the collapse leads to the formation of a dense disk-like structure,
supported by rotation which then fragments into several objects. Figure 2
shows the density structure in the gas immediately before the formation of the
first protostar. The top-left panel shows a density slice on a scale
comparable to the size of the initial gas distribution. The structure is very
filamentary and there are two main overdense clumps in the center. If we zoom
in on one of the clumps, we see that its internal structure is also
filamentary. We can follow the collapse down to scales of the order of an AU,
but at this point we reach the limit of our computational approach: as the gas
collapses further, the Courant timestep becomes very small, making it
difficult to follow the further evolution of the cloud. In order to avoid this
difficulty, we replace very dense, gravitationally bound, and collapsing
regions by sink particles. Once the conditions for sink particle creation are
met, they start to form in the highest density regions (Figure 3). Due to
interactions with other sink particles that result in an increase in velocity,
some sink particles can be ejected from the high-density region, but most of
the particles still remain within the dense gas. Within 137 years of the
formation of the first sink particle, 45 sink particles have formed. At this
time, approximately $4.6\rm M_{\odot}$ of gas has been accreted by the sink
particles.
Figure 3.— Number density map showing a slice in the densest clump, and the
sink formation time evolution, for the 40 million particles simulation, and Z
= 10-4Z⊙. The box is 100AU x 100AU and the time is measured from the formation
of the first sink particle.
### 3.3. Properties of the fragments
Figure 4 shows the mass distribution of sink particles when we stop the
calculation. We typically find masses below $1\rm M_{\odot}$, with somewhat
smaller values in the $10^{-4}\rm Z_{\odot}$ case compared to the $10^{-5}\rm
Z_{\odot}$ case. Both histograms have the lowest sink particle mass well above
the resolution limit of $0.0025M_{\odot}$. Note that in both cases, we are
still looking at the very early stages of star cluster evolution. As a
consequence, the sink particle masses in Figure 4 are not the same as the
final protostellar masses – there are many mechanisms that will affect the
mass function, such as continuing accretion, mergers between the newly formed
protostars, feedback from winds, jets and luminosity accretion, etc.
Nevertheless, we can speculate that the typical stellar mass is similar to
what is observed for Pop II stars in the Milky Way. This suggests that the
transition from high-mass primordial stars to Population II stars with mass
function similar to that at the present day occurs early in the metal
evolution history of the universe, at metallicities $\rm
Z_{crit}<10^{-5}Z_{\odot}$. The number of protostars formed by the end of the
simulation, for both $10^{-4}Z_{\odot}$ (45) and $10^{-5}Z_{\odot}$ (19)
cases, is much larger than the initial number of Jeans masses (3) in the
cloud.
Figure 4.— Sink particle mass function at the end of the simulations. High and
low resolution results and corresponding resolution limits are shown. To
resolve the fragmentation, the mass resolution should be smaller than the
Jeans mass at the point in the temperature-density diagram where dust and gas
couple and the compressional heating starts to dominate over the dust cooling.
At the time shown, around 5 M⊙ of gas had been accreted by the sink particles
in each simulation.
## 4\. Conclusions
In this paper we have addressed the question of whether dust cooling can lead
to the fragmentation of low-metallicity star-forming clouds. For this purpose
we performed numerical simulations to follow the thermodynamical and chemical
evolution of collapsing clouds. The chemical model included a primordial
chemical network together with a description of dust evolution, where the dust
temperature was calculated by solving self-consistently the thermal energy
equilibrium equation.
We performed three sets of simulations, two at low resolution and one at high
resolution (Table 1). All simulations had an initial cloud mass of 1000 M⊙,
number density of 105 cm-3, and temperature of 300K. We tested two different
metallicities (10${}^{-4}{\rm Z}_{\odot}$ and 10${}^{-5}{\rm Z}_{\odot}$), and
also the inclusion of small amounts of turbulent and rotational energies.
We found in all simulations that dust can effectively cool the gas, for number
densities higher than $10^{11}\rm cm^{-3}$. An increase in metallicity implies
a higher dust-to-gas ratio, and consequently stronger cooling by dust. This is
reflected in a lower temperature of the dense gas in the higher metallicity
simulation.
For the low resolution case, we tested the effect of adding turbulence and
rotation. These diminish the infall velocity, leading to different fluid
elements undergoing different amounts of compressional heating. This lack of
heating allows the gas to reach a lower temperature.
We found that the transport of angular momentum to smaller scales lead to the
formation of a disk-like structure, which then fragmented into a number of low
mass objects.
We conclude that the dust is already an efficient coolant even at
metallicities as low as 10-5 or 10${}^{-4}Z_{\odot}$, in agreement with
previous works (Clark et al., 2008; Omukai et al., 2010; Schneider et al.,
2002, 2006; Tsuribe & Omukai, 2006, 2008). Our results support the idea that
dust cooling can play an important role in the fragmentation of molecular
clouds and the evolution of the stellar IMF.
We thank Tom Abel, Volker Brom, Kazuyuki Omukai, Raffaella Schneider, Rowan
Smith, and Naoki Yoshida for useful comments. The present work is supported by
the _Landesstiftung Baden Württemberg_ via their program International
Collaboration II (grant P-LS-SPII/18), the German _Bundesministerium für
Bildung und Forschung_ via the ASTRONET project STAR FORMAT (grant 05A09VHA),
a Frontier grant of Heidelberg University sponsored by the German Excellence
Initiative, the International Max Planck Research School for Astronomy and
Cosmic Physics at the University of Heidelberg (IMPRS-HD). All computations
described here were performed at the _Leibniz-Rechenzentrum_ , National
Supercomputer HLRB-II (_Bayerische Akademie der Wissenschaften_), and on the
HPC-GPU Cluster Kolob (University of Heidelberg) .
## References
* Abel et al. (2002) Abel, T., Bryan, G. L., & Norman, M. L. 2002, Science, 295, 93
* Banerjee et al. (2006) Banerjee, R., Pudritz, R. E., & Anderson, D. W. 2006, MNRAS, 373, 1091
* Bate et al. (1995) Bate, M. R., Bonnell, I. A., & Price, N. M. 1995, MNRAS, 277, 362
* Bonnell et al. (2007) Bonnell, I. A., Larson, R. B., & Zinnecker, H. 2007, Protostars and Planets V, 149
* Bromm et al. (2002) Bromm, V., Coppi, P. S., & Larson, R. B. 2002, ApJ, 564, 23
* Bromm et al. (2001) Bromm, V., Ferrara, A., Coppi, P. S., & Larson, R. B. 2001, MNRAS, 328, 969
* Bromm & Loeb (2003) Bromm, V., & Loeb, A. 2003, Nature, 425, 812
* Chabrier (2003) Chabrier, G. 2003, PASP, 115, 763
* Clark et al. (2008) Clark, P. C., Glover, S. C. O., & Klessen, R. S. 2008, ApJ, 672, 757
* Clark et al. (2011) Clark, P. C., Glover, S. C. O., Klessen, R. S., & Bromm, V. 2011, ApJ, 727, 110
* Frebel et al. (2007) Frebel, A., Johnson, J. L., & Bromm, V. 2007, MNRAS, 380, L40
* Goldsmith (2001) Goldsmith, P. F. 2001, ApJ, 557, 736
* Hollenbach & McKee (1979) Hollenbach, D., & McKee, C. F. 1979, ApJS, 41, 555
* Hollenbach & McKee (1989) —. 1989, ApJ, 342, 306
* Jappsen et al. (2009a) Jappsen, A., Klessen, R. S., Glover, S. C. O., & Mac Low, M. 2009a, ApJ, 696, 1065
* Jappsen et al. (2005) Jappsen, A., Klessen, R. S., Larson, R. B., Li, Y., & Mac Low, M. 2005, A&A, 435, 611
* Jappsen et al. (2009b) Jappsen, A., Mac Low, M., Glover, S. C. O., Klessen, R. S., & Kitsionas, S. 2009b, ApJ, 694, 1161
* Kroupa (2002) Kroupa, P. 2002, Science, 295, 82
* Mac Low & Klessen (2004) Mac Low, M., & Klessen, R. S. 2004, Reviews of Modern Physics, 76, 125
* Omukai (2000) Omukai, K. 2000, ApJ, 534, 809
* Omukai et al. (2010) Omukai, K., Hosokawa, T., & Yoshida, N. 2010, ApJ, 722, 1793
* Omukai et al. (2005) Omukai, K., Tsuribe, T., Schneider, R., & Ferrara, A. 2005, ApJ, 626, 627
* O’Shea & Norman (2007) O’Shea, B. W., & Norman, M. L. 2007, ApJ, 654, 66
* Santoro & Shull (2006) Santoro, F., & Shull, J. M. 2006, ApJ, 643, 26
* Schneider et al. (2002) Schneider, R., Ferrara, A., Natarajan, P., & Omukai, K. 2002, ApJ, 571, 30
* Schneider & Omukai (2010) Schneider, R., & Omukai, K. 2010, MNRAS, 402, 429
* Schneider et al. (2006) Schneider, R., Omukai, K., Inoue, A. K., & Ferrara, A. 2006, MNRAS, 369, 1437
* Semenov et al. (2003) Semenov, D., Henning, T., Helling, C., Ilgner, M., & Sedlmayr, E. 2003, A&A, 410, 611
* Springel (2005) Springel, V. 2005, MNRAS, 364, 1105
* Stamatellos et al. (2007) Stamatellos, D., Whitworth, A. P., Bisbas, T., & Goodwin, S. 2007, A&A, 475, 37
* Tsuribe & Omukai (2006) Tsuribe, T., & Omukai, K. 2006, ApJ, 642, L61
* Tsuribe & Omukai (2008) —. 2008, ApJ, 676, L45
* Yoshida et al. (2008) Yoshida, N., Omukai, K., & Hernquist, L. 2008, Science, 321, 669
|
arxiv-papers
| 2011-01-25T18:32:29 |
2024-09-04T02:49:16.632254
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gustavo Dopcke, Simon C. O. Glover, Paul C. Clark and Ralf S. Klessen",
"submitter": "Gustavo Dopcke",
"url": "https://arxiv.org/abs/1101.4891"
}
|
1101.4915
|
National Institute of Standards and Technology, Gaithersburg, MD 20899, U.S.A.
∗Corresponding author: gnave@nist.gov
# Wavelengths of the $\rm\bf
3d^{6}(^{5}D)4s\,a^{6}D-3d^{6}\,(^{5}D)4p\,y^{6}P$ Multiplet of Fe II (UV 8)
Gillian Nave∗, Craig J. Sansonetti
###### Abstract
We investigate the wavenumber scale of Fe I and Fe II lines using new spectra
recorded with Fourier transform spectroscopy and using a re-analysis of
archival spectra. We find that standards in Ar II, Mg I, Mg II and Ge I give a
consistent wavenumber calibration. We use the recalibrated spectra to derive
accurate wavelengths for the a6D-y6P multiplet of Fe II (UV 8) using both
directly measured lines and Ritz wavelengths. Lines from this multiplet are
important for astronomical tests of the invariance of the fine structure
constant on a cosmological time scale. We recommend a wavelength of 1608.45081
Å with a one standard deviation uncertainty of 0.00007 Å for the $\rm
a^{6}D_{9/2}-y^{6}P_{7/2}$ transition.
300.621, 300.6300, 300.6540
## 1 Introduction
The universality and constancy of the laws of nature rely on the invariance of
the fundamental constants. However, some recent measurements of quasar (Quasi-
stellar objects - QSO) absorption line spectra suggest that the fine-structure
constant, $\alpha$, [1] may have had a different value during the early
universe [2]. Other measurements (e.g. [3]) do not show any change. The
attempt to resolve these discrepancies can probe deviations from the standard
model of particle physics and thus provide tests of modern theories of
fundamental interactions that are hard to attain in other ways.
QSO absorption lines are used in these investigations to measure the
wavelength separations of atomic lines in spectra of different elements - the
many-multiplet method [4] \- and compare their values at large redshifts with
their values today. Any difference in the separations would suggest a change
in $\alpha$. Since this method uses many different species in the analysis
that have differing sensitivities to changes in $\alpha$, it can be much more
sensitive than previous methods, such as the alkali-doublet method [5], that
use just one species. However, it requires very accurate laboratory
wavelengths to be used successfully, since the observed changes in $\alpha$
are only a few parts in 105, requiring laboratory wavelengths to 1:107 or
better. This has led to several recent measurements of ultraviolet wavelengths
using both Fourier transform (FT) spectroscopy [6, 7, 8] and frequency comb
metrology [9, 10, 11].
One spectral line of particular interest is the $\rm
3d^{6}(^{5}D)4s\,a^{6}D_{9/2}-3d^{5}\,(^{6}S)4s4p(^{3}P)\,y^{6}P_{7/2}$ line
of Fe II at 1608.45 Å. This line is prominent in many QSO spectra and its
variation with $\alpha$ has the opposite sign from that of other nearby lines
[12]. However, measurement of its wavelength using frequency comb metrology,
which is at present the most accurate method, is extremely difficult due to
its short wavelength. Although this line is strong in many of the FT spectra
of iron-neon hollow cathode lamps recorded at the National Institute of
Standards and Technology (NIST) and Imperial College, London, UK (IC), these
spectra display inconsistencies in the wavelength of the 1608 Å line of around
1.5 parts in 107 \- too great for use in the many-multiplet method to detect
changes in $\alpha$. We discussed some of these discrepancies in our previous
paper [13] presenting reference wavelengths in the spectra of iron, germanium
and platinum around 1935 Å.
Here we present a re-analysis of spectra taken at NIST and IC in order to
resolve these discrepancies and provide a better value for the wavelength of
the 1608.45 Å line of Fe II. The papers involved in this re-analysis are
listed in table 1, together with the proposed corrections to the wavenumber
scale. The proposed corrections are up to three times the previous total
uncertainty, depending on the wavenumber. In section 2 we discuss previous
measurements of the a6D - y6P multiplet. Section 3 describes the archival data
we use to obtain improved wavelengths for this multiplet Additional spectra
taken at NIST in order to re-evaluate the calibration of these archival data
are described in section 4. The accuracy of this calibration in the visible
and ultraviolet wavelength regions is also discussed in section 4. Section 5
describes three different methods for obtaining the wavelengths of the a6D -
y6P multiplet. The first method uses intermediate levels determined using
strong Fe II lines in the visible and ultraviolet in order to obtain the
values of the y6P levels and Ritz wavenumbers for the a6D - y6P multiplet. The
second method uses energy levels optimized by using a large number of spectral
lines to derive Ritz wavenumbers for this multiplet. Although better accuracy
is achieved using this method than the first method because of the increased
redundancy, the way in which the y6P levels are determined is less
transparent. The third method uses experimental wavelengths determined in
spectra that are recalibrated from spectra in which we have re-evaluated the
wavenumber calibration. In section 6 we re-examine the Fe II wavenumbers in
our previous paper [13]. All uncertainties in this paper are reported at the
one standard deviation level.
## 2 Previous measurements of the a6D - y6P multiplet
The region of the a6D - y6P multiplet is shown in figure 1 as observed in a FT
spectrum taken at IC. Nave, Johansson & Thorne [14] report Ritz and
experimental wavelengths for six of the nine lines of the a6D - y6P multiplet.
The Ritz wavelengths are based on energy levels optimized to spectral lines
covering wavelengths from 1500 Å to 5.5 $\rm\mu$m measured with FT
spectroscopy. The estimated uncertainties are about 2x10-4 Å or about 1.2
parts in 107. The published lines do not include the $\rm
a^{6}D_{9/2}-y^{6}P_{7/2}$ line. Johansson [15] contains Ritz wavelengths for
all nine lines, based on unpublished interferometric measurements of Norlén,
with a value of 1608.451 Å for this line. The estimated uncertainty of the
$\rm y^{6}P_{7/2}$ level with respect to the ground state $\rm a^{6}D_{9/2}$
of Fe II is 0.02 cm-1. The uncertainty of the 1608 Å line can be derived
directly from the uncertainty of the $\rm y^{6}P_{7/2}$ level, and corresponds
to a wavelength uncertainty at 1608 Å of 0.0005 Å. Wavelengths for all nine
lines measured using FT spectroscopy are also given in Pickering et al. [16]
in a paper devoted to oscillator strength measurements. No details of the
calibration of these lines or their uncertainties are given. The wavelength
value recommended by Murphy et al. [12] is 1608.45080$\pm$0.00008 Å, with a
reference to Pickering et al. However, this is not the value given in
Pickering et al. and the small uncertainty is improbable without additional
confirmation. The source of this wavelength is unclear.
In addition to these published values, lines from this multiplet are present
in some unpublished archival spectra from IC and NIST. The most important
spectra for the current work are summarized in Table 2. The spectra on which
Nave, Johansson & Thorne [14] is based are part of a much larger set of Fe II
spectra covering all wavelengths from 900 Å to 5.5 $\rm\mu$m. Two of these
spectra cover the region around 1600 Å and contain all nine lines of the
multiplet. The wavelength standards for these spectra are traceable to a set
of Ar II lines between 3729 Å and 5146 Å (see section 4 for details). The
weighted average wavelength for the $\rm a^{6}D_{9/2}-y^{6}P_{7/2}$ line in
these unpublished archival spectra is 1608.45075$\pm$0.00018 Å.
The spectra in Nave & Sansonetti [13] were calibrated with respect to Ge
standards of Kaufman & Andrew [17]. In addition to the spectra used in that
paper, we recorded a spectrum using FT spectroscopy with a pure iron cathode
that covers the wavelength region of the $\rm a^{6}D-y^{6}P$ multiplet (fe1115
in Table 2). It was calibrated with iron lines measured in one of the spectra
used for ref. [13] (lp0301 in Table 2). The resulting value for the wavelength
of the $\rm a^{6}D_{9/2}-y^{6}P_{7/2}$ line was 1608.45050$\pm$0.00004 Å,
1.5x10-7 times smaller than the wavelength obtained from the archival spectra
and outside their joint uncertainty. This inconsistency is also larger than
the uncertainty required for measurements of possible changes in $\alpha$.
## 3 Summary of current experimental data
The spectra we re-analyzed are the same as those used in previous studies of
Fe I and Fe II [18, 19, 14, 13]. Three different spectrometers were used: the
f/60 IR-visible-UV FT spectrometer at the National Solar Observatory, Kitt
Peak, AZ (NSO), the f/25 vacuum ultraviolet (VUV) FT spectrometer at IC [20],
and the f/25 VUV spectrometer at NIST [21]. The light sources for all of the
spectra were high-current hollow cathode lamps containing a cathode of pure
iron run in either neon or argon. Gas pressures of 100 Pa to 500 Pa (0.8 Torr
to 4 Torr) were used with currents from 0.32 A to 1 A. The total number of FT
spectra was 31, covering wavelengths from about 1500 Å to 5 $\rm\mu$m (2000
cm-1 to 66000 cm-1). The wavenumber, intensity and width for all the lines
were obtained with Brault’s decomp program [22] or its modification xgremlin
[23]. Further details of the experiments can be found in [18, 19, 14, 13].
Additional spectra were taken using the NIST 2-m FT spectrometer and are
described in section 4.1.
## 4 Calibration of FT spectra
All of the spectra were calibrated assuming a linear FT wavenumber scale, so
that in principle only one reference line is required to put the measurements
on an absolute scale. In practice, many lines are used. To obtain the absolute
wavenumbers, a multiplicative correction factor, k${}_{\mbox{eff}}$, is
derived from the reference lines and applied to each observed wavenumber
$\sigma_{\mbox{obs}}$ so that
$\sigma_{\mbox{corr}}=(1+k_{\mbox{eff}})\sigma_{\mbox{obs}}$ (1)
where $\sigma_{\mbox{corr}}$ is the corrected wavenumber.
All the spectra in Learner & Thorne (3830 Å to 5760 Å) [18] and Nave et al.
(1830 Å to 3850 Å) [19] trace their calibration to 28 Ar II lines in the
visible. The original calibration in Refs. [18] and [19] used the wavenumbers
of Norlén [24] for these lines. Norlén calibrated these Ar II lines with
respect to 86Kr I lines emitted from an electrodeless microwave discharge lamp
that had in turn been calibrated with respect to an Engelhard lamp, which was
the prescribed source for the primary wavelength standard at the time of his
measurements. The estimated standard uncertainty of Norlén’s Ar II wavenumbers
varies from 0.0007 cm-1 at 19429 cm-1 to 0.001 cm-1 at 22992 cm-1. The Ar II
lines were used to calibrate a ‘master spectrum’ (spectrum k19 in Table 2).
Additional spectra of both Fe-Ne and Fe-Ar hollow cathode lamps covering
wavelengths from 2778 Å to 7387 Å were calibrated from this master spectrum.
The ultraviolet spectra reported in [19] were calibrated with respect to the
results of Learner & Thorne [18] by using a bridging spectrum. This bridging
spectrum used two different detectors, one on each output of the FT
spectrometer. The first overlapped with the visible wavenumbers in Ref. [18]
in order to obtain a wavenumber calibration and the second covered the UV
wavenumbers being measured. Since the two outputs of the FT spectrometer are
not exactly in antiphase, the resulting phase correction has a discontinuity
in the region around 35 000 cm-1 where the two detectors overlap, as shown in
Fig. 1 of [19]. The full procedure is described in detail in [19].
The 28 Ar II lines used as wavenumber standards in Refs. [18, 19] were
subsequently re-measured by Whaling et al. [25] using FT spectroscopy with
molecular CO lines as standards. The uncertainty of these measurements is
0.0002 cm-1. The molecular CO standards used in Ref. [25] were measured using
heterodyne frequency spectroscopy with an uncertainty of around 1:109 and are
ultimately traceable to the cesium primary standard [26]. The wavenumbers of
Whaling et al. [25] are systematically higher than those of Norlén [24] by
6.7$\pm$0.8 parts in 108, corresponding to a wavenumber discrepancy of about
0.0014 cm-1 at 21000 cm-1. Since the results of Whaling et al. [25] are more
accurate and precise than those of Norlén [24], all the wavenumbers in [18],
[19], and Table 3 of [14] have been increased by 6.7 parts in 108 wherever
they are used in the current work.
The spectra in Nave & Sansonetti [13] were calibrated with respect to 29 Ge I
Ritz wavenumbers derived from the energy levels of Kaufman & Andrew [17].
However, the Fe II wavenumbers derived using this calibration were found to be
greater than those in Nave et al. [19] by about 7 parts in 108, even after the
wavenumbers in the latter were adjusted to the wavenumber scale of Whaling et
al. [25].
In order to present accurate wavenumbers for Fe II lines around 1600 Å, it is
necessary first to confirm the accuracy of the iron lines in the visible that
were calibrated with respect to selected lines of Ar II lines [18], to
investigate the accuracy with which this calibration is transferred to the
VUV, and to resolve the discrepancy between iron and germanium standard
wavelengths identified in Ref. [13].
### 4.1 Calibration of the visible-region spectra
In order to confirm the calibration of the master spectrum, k19, used in [18]
and [19], we took additional spectra using the NIST 2 m FT spectrometer [23].
The source was a water-cooled high-current hollow cathode lamp with a current
of 1.5 A and argon at pressures of 130 Pa to 330 Pa (1 Torr to 2.5 Torr). The
spectra covered the region 8500 cm-1 to 37 000 cm-1 with resolutions of either
0.02 cm-1 or 0.03 cm-1. A 1 mm aperture was used in order to minimize possible
illumination effects. The detector was a silicon photodiode detector with a 2
mm $\times~{}$2 mm active area.
The spectrometer was aligned optimally using a diffused, expanded beam from a
helium neon laser, ensuring that the modulation of the laser fringes was
maximized throughout the 2 m scan. Before recording some of the spectra, the
spectrometer was deliberately misaligned and re-aligned in order to test
whether small misalignments that could not be detected using our alignment
procedure affected the wavenumber scale.
The spectra were calibrated using the values of Whaling et al. [25] for Ar II
lines recommended in Ref. [18] that had good signal-to-noise ratio.
Wavenumbers of strong iron lines were then measured and compared with iron
lines taken from Ref. [18] and [19].
Figure 2 shows the calibration of one of our spectra using Ar II and iron
lines from Refs. [25, 18, 19] as standards. The calibration constant
k${}_{\mbox{eff}}$ does not depend on wavenumber and is the same for all three
sets of standards to within 1:108 when the iron lines from the Refs [18, 19]
are adjusted to the wavenumber scale of [25]. The possibility of shifts due to
non-uniform illumination of the aperture were investigated by taking a
spectrum with the 5 mm diameter image of the hollow cathode lamp offset from
the 1 mm aperture by about 2 mm. This spectrum also shows good agreement
between the Ar II and iron calibrations.
Many of the early interferograms from the NSO FT spectrometer were
asymmetrically sampled, with a much larger number of points on one side of
zero optical path difference than the other. A Fourier transform of an
asymmetrically-sampled interferogram gives a spectrum with a large,
antisymmetric imaginary part [27]. A small error in the phase correction
causes a small part of this antisymmetric imaginary part to be rotated into
the real part of the spectrum, distorting the line profiles and causing a
wavenumber shift. The zero optical path difference in spectrum k19 is roughly
1/5 of the way through the interferogram. For a Gaussian profile with a full
width at half maximum of W, this produces a wavenumber shift of roughly 0.3W
per radian of phase error as shown in Fig. 3 of [27].
We decided to re-examine the phase curve for the master spectrum, k19, against
which all the other iron spectra used in [18, 19, 14] were calibrated. The
original interferogram for this spectrum was obtained from the NSO Digital
Archives [28] and re-transformed using Xgremlin. The phase is plotted in Fig.
3. The residual phase error after fitting an 11th order polynomial is less
than 10 mrad for almost all wavenumbers below 35 000 cm-1. This corresponds to
an error of 3.6x10-4 cm-1 for a linewidth of 0.12 cm-1. Above 35 000 cm-1 the
polynomial no longer fits the points adequately and consequently these points
were not used in the comparison. Wavenumbers were measured in the re-
transformed spectrum and calibrated with the 28 Ar II lines recommended in
Ref. [18] using the values of Ref. [25]. Iron lines were then compared with
those from papers [18, 19]. The result is shown in Fig. 4. The two
measurements agree to within 1:108. This confirms that the original phase
correction of k19 was accurate and the wavenumbers in Ref. [18] and Table 3 of
Ref. [19] (2929 Å to 3841 Å) are not affected by phase errors.
We conclude that the wavenumbers measured in the master spectrum, k19, are
accurate. Although results from this spectrum were used in Learner & Thorne
[18] and Table 3 of Nave et al. [19], it did not dominate the weighted average
values reported in these papers.
### 4.2 Calibration of the ultraviolet spectra
Tables 4 and 5 in Nave et al. [19] cover wavenumbers from 33 695 cm-1 to 54
637 cm-1 in Fe I and Fe II respectively. The wavenumbers in these tables were
measured using the vacuum ultraviolet FT spectrometer at IC. The calibration
of these spectra was transferred from the master spectrum (k19 in Table 2)
using a bridging spectrum (i56 in Table 2), as described in section 4. The
principal spectrum covering wavenumbers below 35 000 cm-1 in Table 4 of [19]
is i6 in Table 2. It overlaps with the master spectrum between 33 000 cm-1 and
34 000 cm-1. Figure 5 shows a comparison of wavenumbers in i6 with the master
spectrum k19. The wavenumbers in spectrum i6 are systematically smaller than
in k19 by 3.9$\pm$0.5 parts in 108. Although the region of overlap of i6 with
k19 is small and is thus insensitive to non-linearities in the wavenumber
scale, this result supports our earlier speculation in Nave & Sansonetti [13]
that the calibration of the UV data using the bridging spectrum may be
incorrect. Based on the comparison of Fig 5, we conclude the wavenumbers in
Tables 4 and 5 of Ref. [19] should be increased by 10.6 parts in 108,
consisting of 3.9 parts in 108 to correct the transfer of the calibration to
the ultraviolet and an additional 6.7 parts in 108 to put all the spectra on
the wavenumber scale of Whaling et al. [25].
We compared our corrected values for iron lines in the UV to results of
Aldenius et al. [7, 8], who present wavenumbers of iron lines measured in a
high-current hollow cathode lamp using a UV FT spectrometer similar to the one
used in [19]. Instead of recording a pure iron spectrum, they included small
pieces of Mg, Ti, Cr, Mn and Zn in their Fe cathode. This ensured that
spectral lines due to all of these species were placed on the same wavenumber
scale, which was calibrated using the Ar II lines of Whaling et al. [25].
Table 3 compares the wavenumbers of Ref. [8] with the corrected values of
[19]. Although the wavenumbers of Ref. [8] agree with our revised values
within their joint uncertainties, they are systematically smaller by 3.7 parts
in 108. Although this might suggest that it is incorrect to increase the
wavenumbers of Ref. [19], it might also indicate that the wavenumbers of Ref.
[8] need to be increased.
Fortunately, there are data that allow us to test these alternatives. In
addition to iron lines, the spectra in Ref. [8] contained four lines due to Mg
I and Mg II that have since been measured using frequency comb spectroscopy
[9, 10, 11] with much higher accuracy than achievable using FT spectroscopy.
Table 4 compares the wavenumbers of these four magnesium lines from [8] with
those derived from frequency comb measurements of isotopically pure values.
For this comparison, the results of [8] have been increased by 3.7 parts in
108, as suggested by the comparison of Fe II lines in Table 3. With this
adjustment, the results of Aldenius agree with the frequency comb values
within their joint uncertainties, having a mean deviation of -0.7$\pm$3 parts
in 108. Without the adjustment the mean deviation would be
(-4$\pm$3)$\times$10-8.
We conclude that the wavenumbers in Tables 4 and 5 of Nave et al. [19] should
be increased by 10.6 parts in 108: 3.9 parts in 108 to correct for the
incorrect transfer of the calibration from the master spectrum to the
ultraviolet and 6.7 parts in 108 to put all of the spectra on the scale of
Whaling et al. [25]. We have performed this correction in the following
sections of the current paper. The wavenumbers of Aldenius et al. [8] should
be increased by 3.7 parts in 108 to put them on the same scale. This
adjustment of scale brings the measurements of lines of Mg I and Mg II in [8]
into agreement with the more accurate frequency comb values [9, 10, 11].
## 5 Wavenumbers of a6D - y6P transitions
The wavenumbers of the a6D - y6P transitions can be obtained either from
direct measurements or from energy levels derived from a larger set of
experimental data (Ritz wavenumbers). Direct measurements will have larger
uncertainties due to the cumulative addition of the uncertainties in the
transfer of the calibration from the visible to the UV. Ritz wavenumbers are
more accurate due to the increased redundancy, but use of a large set of
experimental data to derive the energy levels makes it less clear exactly how
the Ritz wavenumbers are derived. We illustrate this process by using a small
subset of the strongest transitions that determine the y6P levels that are
present in the visible and ultraviolet regions of the spectrum where we have
corrected the wavenumber calibration.
The y6P levels can be determined from three sets of lines in the UV and
visible as shown in Fig. 6. The first set of nine lines near 2350 Å determines
the three $\rm 3d^{6}(^{6}D)4p\,z^{6}P$ levels. All nine lines are present in
archival spectra from IC which we have recalibrated using the results of
section 4.2. Two of the nine lines are blended with other lines and a third,
between $\rm a^{6}D_{9/2}$ and $\rm z^{6}P_{7/2}$, is self-absorbed in the IC
spectra. These lines are unsuitable for determining the z6P levels. The
recalibrated values of the remaining six lines are shown in column 4 of Table
5. Each line is observed with a signal-to-noise ratio of over 100 in at least
eight spectra, all of which agree within 0.006 cm-1. The wavenumbers in Table
5 are weighted mean values of the individual measurements and the standard
deviation in the last decimal place is given in parenthesis following the
wavenumber. The lower levels in column 3 are determined from between 10 and 20
different transitions to upper levels and have been optimized to the archival
spectra with the program lopt [29] (described below). The total standard
uncertainty in the upper levels includes the calibration uncertainty of
2.3x10-8 times the level value.
The second set of three transitions around 5000 Å determines the 3d54s2 a6S5/2
level from the three z6P levels. These lines are present in k19 and other
archival spectra taken at NSO that we have recalibrated to correspond to the
wavenumber scale of Whaling et al. [25]. Each line is present in five spectra,
all of which agree within 0.0035 cm-1. Wavenumbers for these transitions are
shown in Table 6 and give a mean value of (23317.6344$\pm$0.0010) cm-1 for the
3d54s2 a6S5/2 level.
Finally, the y6P levels can be determined from the a6S5/2 level from three
lines around 2580 Å, present in the IC spectra. The recalibrated wavenumbers
are shown in Table 7 with the resulting y6P level values. These values were
used to calculate Ritz wavenumbers for the a6D - y6P transitions, as shown in
the third column of Table 8.
Alternate values for the Ritz wavenumbers of the a6D-y6P transitions can be
obtained from energy levels optimized using wavenumbers from the archival Fe
II spectra from NSO and IC corrected according to sections 4.1 and 4.2. The
program lopt [29] was used to derive optimized values for 939 energy levels
from 9567 transitions. Weights were assigned proportional to the inverse of
the estimated variance of the wavenumber. Lines with more than one possible
classification, lines that were blended, or for which the identification was
questionable were assigned a low weight. Two iterations were made. In the
first, lines connecting the lowest a6D term to higher $\rm 3d^{6}\,(^{5}D)4p$
levels were assigned a weight proportional to the inverse of the statistical
variance of the wavenumber, omitting the calibration uncertainty. This was
done to obtain accurate values and uncertainties for the a6D intervals. These
intervals are determined from differences between lines close to one another
in the same spectrum sharing the same calibration factor. Hence the
calibration uncertainty does not contribute to the uncertainty in the relative
values of these energy levels. The values of the a6D levels obtained in this
iteration are given in column 3 of table 5. In the second iteration, the a6D
levels were fixed to the values and uncertainties determined from the first
iteration. The weights of the $\rm a^{6}D-3d^{6}\,(^{5}D)4p$ transitions were
assigned by combining in quadrature the statistical uncertainty in the
measurement of the line position and the calibration uncertainty in order to
obtain accurate uncertainties for the $\rm 3d^{6}\,(^{5}D)4p$ and higher
levels. The values of the y6P levels are given in column 4 of table 7. Ritz
wavenumbers for the $\rm a^{6}D-3d^{6}\,(^{5}D)4p$ transitions based on these
globally optimized level values are presented in column 5 of table 8.
The corrected experimental wavenumbers from the archival spectra are given in
column 4 of table 8. The main contribution to the uncertainty in the
experimental wavenumbers is from the calibration and consists of two
components – the uncertainty in the standards and the uncertainty in
calibrating the spectrum. The calibration uncertainty is common to all lines
in the calibrated spectrum and hence must be added to the uncertainties of
wavenumbers measured using transfer standards, rather than added in quadrature
as would be the case for random errors. Hence the uncertainty in the
wavenumbers increases with each calibration step, resulting in larger
uncertainties at the shortest wavenumbers which are furthest from the
calibration standards. The experimental standard uncertainties in Table 8 are
determined by combining in quadrature the statistical uncertainty in
determining the line position and the calibration uncertainty of 4$\times$10-8
times the wavenumber. The experimental wavenumber and both Ritz wavenumbers
agree within their joint uncertainties. The Ritz wavenumbers determined from
optimized energy levels have the smallest uncertainties. Wavelengths
corresponding to these wavenumbers are given in column 7.
## 6 A re-examination of Fe II wavenumbers from Nave & Sansonetti[13]
The Fe I lines in Nave & Sansonetti [13] were calibrated with respect to lines
of Ge I. Figure 3 of that paper showed that the calibration factor
k${}_{\mbox{eff}}$ derived from Fe I and Fe II lines is smaller than that
derived from Ge I by 6.5 parts in 108. We attributed this to a possible
problem in the transfer of the wavenumber calibration of the Fe I and Fe II
lines from the region of the Ar II wavenumber standards to the vacuum
ultraviolet, thus suggesting that the wavenumber standards in [19] are too
small. In section 4.2 we have confirmed that the wavenumbers in Tables 4 and 5
of Nave et al. [19] should be increased by 3.9 parts in 108 due to the
transfer of the calibration. This reduces but does not fully explain the
calibration discrepancy in [13].
The Ge I lines used to calibrate the spectra in Ref. [13] were measured by
Kaufman and Andrew [17]. The wavenumber standard they used was the 5462 Å line
of 198Hg emitted by an electrodeless discharge lamp maintained at a
temperature of $19\,^{\circ}\rm{C}$, containing Ar at a pressure of 400 Pa (3
Torr). The vacuum wavelength of this line was assumed to be 5462.27075 Å. This
value was based on a vacuum wavelength of 5462.27063 Å measured in the same
lamp at $7\,^{\circ}\rm{C}$ [30], with an adjustment for the different
temperature using the measurements of Emara [31]. The 5462 Å line was
remeasured by Salit et al. [32] using a temperature of $8\,^{\circ}\rm{C}$. A
value of (5462.27085$\pm$0.00007) Å was obtained. More recent work by
Sansonetti & Veza [33] gives the wavelength of this line as 5462.270825(11) Å,
in agreement with [32] but more precise. Adoption of this value for the
wavelength of the 198Hg line implies that all of the Ge I wavenumbers in [17]
should be decreased by 1.4 parts in 108. Figure 7 shows how Fig. 3 in [13]
(Spectrum lp0301 in Table 2) changes with the adjustment of both the iron and
germanium wavenumbers. The calibrations based on Ge and Fe lines now differ by
only 1.5 parts in 108, which is within the joint uncertainties. We thus
conclude that the calibration derived from Fe I and Fe II lines is in
agreement with that derived from Ge I when both sets of standards are adjusted
to correspond with the most recent measurements.
Spectrum lp0301 in Table 2 can be used to calibrate spectrum fe1115 in Table 2
referred to in the last paragraph of section 2. A value of
(62171.634$\pm$0.006) cm-1 is obtained for the wavenumber of the $\rm
a^{6}D_{9/2}-y^{6}P_{7/2}$ line, corresponding to a wavelength of
(1608.45057$\pm$0.00016) Å, This disagrees with the Ritz value by 1.7 times
the joint uncertainty and marginally disagrees with the experimental values of
Table 8. The mean difference in the experimental values for all nine $\rm
a^{6}D-y^{6}P$ lines is (0.008$\pm$0.004) cm-1. We believe this difference is
due to a small slope in the calibration of lp0301 but have been unable to
confirm this with our data. The principal contributors to the uncertainty are
the uncertainty in the iron and germanium standards, the uncertainty in
calibrating the spectrum in ref. [13] from these standards, and the
uncertainty in calibrating spectrum fe1115 from spectrum lp0301.
## 7 Conclusions
We investigated the wavenumber scale of published Fe I and Fe II lines using
new spectra recorded with the NIST 2-m FT spectrometer and a re-analysis of
archival spectra. Our new spectra confirm the wavenumber scale of visible-
region iron lines calibrated using the Ar II wavenumber standards of Whaling
et al. [25].
Having confirmed the wavenumber scale of iron lines in the visible and
ultraviolet regions, we have used lines from these spectra to derive Ritz
values for the wavenumbers and wavelengths of lines in the $\rm a^{6}D-y^{6}P$
multiplet of Fe II (UV 8). Ritz wavenumbers derived using two different
methods agree with one another and with directly measured wavenumbers within
the joint uncertainties. We recommend a value of 1608.45081$\pm$0.00007 Å for
the wavelength of the $\rm a^{6}D_{9/2}-y^{6}P_{7/2}$ line of Fe II, which is
an important line for detection of changes in the fine-structure constant
during the history of the Universe using quasar absorption-line spectra.
We find that the wavenumbers in Learner & Thorne [18] and Table 3 of Nave et
al. [19] should be increased by 6.7 parts in 108 to put them on the scale of
the Ar II lines of Whaling et al. [25]. The wavenumbers in Tables 4 and 5 of
Ref. [19] should be increased by 10.6 parts in 108 to put them on the Ar II
scale of Ref. [25] and to correct for an error in the transfer of this
wavenumber scale to the ultraviolet. The Ge I wavenumbers of Kaufman & Andrew
[17] and all the wavenumbers in Nave & Sansonetti [13] should be decreased by
1.4 parts in 108 to put them on the scale of recent measurements of the 198Hg
line at 5462 Å.
## 8 Acknowledgments
We thank Michael T. Murphy for alerting us to the importance of the Fe II line
at 1608 Å. We also thank Anne P. Thorne and Juliet C. Pickering for helpful
discussions on the calibration of FT spectrometers and the linearity of the FT
wavenumber scale. This work is partially supported by NASA inter-agency
agreement NNH10AN38I.
## References
* [1] P. J. Mohr, B. N. Taylor, and D. B. Newell (2007), “The 2006 CODATA Recommended Values of the Fundamental Physical Constants” (Web Version 5.2). Available: http://physics.nist.gov/constants [2010, May 11]. National Institute of Standards and Technology, Gaithersburg, MD 20899.
* [2] M. T. Murphy, J. K. Webb, V. V. Flambaum, “Further evidence for a variable fine-structure constant from Keck/HIRES QSO absorption spectra,” Mon. Not. R. Astron. Soc. 345, 609–638 (2003).
* [3] H. Chand, R. Srianand, P. Petitjean, B. Aracil, R. Quast, D. Reimers, “Variation of the fine-structure constant: very high resolution spectrum of QSO HE 0515-4414,” Astron. Astrophys., 451, 45–56 (2006).
* [4] V. A. Dzuba, V. V. Flambaum, J. K. Webb, “Space-Time Variation of Physical Constants and Relativistic Corrections in Atoms,” Phys. Rev. Lett. 82, 888–891 (1999).
* [5] J. N. Bahcall, W. L. W. Sargent, M. Schmidt, “An Analysis of the Absorption Spectrum of 3c 191,” Astrophys. J. 149, L11–L15 (1967)
* [6] J. C. Pickering, A. P. Thorne, J. K. Webb, “Precise laboratory wavelengths of the Mg I and Mg II resonance transitions at 2853, 2803 and 2796 Angstroms,” Mon. Not. R. Astron. Soc., 300, 131–134 (1998).
* [7] M. Aldenius, S. Johansson, M. T. Murphy, “Accurate laboratory ultraviolet wavelengths for quasar absorption-line constraints on varying fundamental constants,” Mon. Not. R. Astron. Soc., 370, 444–452 (2006).
* [8] M. Aldenius, “Laboratory wavelengths for cosmological constraints on varying fundamental constants,” Phys. Scr. T134, 014008 (2009).
* [9] E. J. Salumbides, S. Hannemann, K. S. E. Eikema, W. Ubachs, “Isotopically resolved calibration of the 285-nm MgI resonance line for comparison with quasar absorptions,” Mon. Not. R. Astron. Soc., 373, L41–L44 (2006).
* [10] S. Hannemann, E. J. Salumbides, S. Witte, R. Th. Zinkstok, E.-J. van Duijn, K. S. E. Eikema, W. Ubachs, “Frequency metrology on the Mg 3s$\rm{}^{2\,1}S\rightarrow$3s4p 1P line for comparison with quasar data,” Phys. Rev. A, 74 012505 (2006).
* [11] V. Batteiger, S. Knünz, M. Herrmann, G. Saathoff, H. A. Schüssler, B. Bernhardt, T. Wilken, R. Holzwarth, T. W. Hänsch, Th. Udem, “Precision spectroscopy of the 3s-3p fine-structure doublet in Mg+,” Phys. Rev. A 80, 022503 (2009).
* [12] M. T. Murphy, J. K. Webb, V. V. Flambaum, “Further evidence for a variable fine-structure constant from Keck/HIRES QSO absorption spectra,” Mon. Not. R. Astron. Soc. 345, 609–638 (2003).
* [13] G. Nave, C. J. Sansonetti, “Reference wavelengths in the spectra of Fe, Ge, and Pt in the region near 1935 Å,” 21, 442–453 (2004).
* [14] G. Nave, S. Johansson, A. P. Thorne, “Precision vacuum-ultraviolet wavelengths of Fe II measured by Fourier-transform and grating spectrometry,” J. Opt. Soc. Am. B 14, 1035–1042 (1997).
* [15] S. Johansson, “The spectrum and term system of Fe II,” Phys. Scr. 18, 217-265 (1978).
* [16] J. C. Pickering, M. P. Donnelly, H. Nilsson, A. Hibbert, S. Johansson, “The FERRUM Project: Experimental oscillator strengths of the UV 8 multiplet and other UV transitions from the $\mathsf{y^{6}}$P levels of Fe II,” Astron. Astrophys. 396, 715–722 (2002).
* [17] V. Kaufman, K. L. Andrew, “Germanium vacuum ultraviolet Ritz standards,” J. Opt. Soc. Am. 52, 1223–1237 (1962).
* [18] R. C. M. Learner, A. P. Thorne, “Wavelength calibration of Fourier-transform emission spectra with applications to Fe I,” J. Opt. Soc. Am. B 5, 2045–2059 (1988).
* [19] G. Nave, R. C. M. Learner, A. P. Thorne, C. J. Harris, “Precision Fe I and Fe II wavelengths in the ultraviolet spectrum of the iron-neon hollow-cathode lamp,” J. Opt. Soc. Am. B 8, 2028–2041 (1991).
* [20] A. P. Thorne, C. J. Harris, I. Wynne-Jones, R. C. M. Learner, G. Cox, “A Fourier transform spectrometer for the vacuum ultraviolet: design and performance,” J. Phys. E 20, 54–60 (1987).
* [21] U. Griesmann, R. Kling, J. H. Burnett, L. Bratasz, “NIST FT700 vacuum ultraviolet Fourier transform spectrometer: applications in ultraviolet spectrometry and radiometry,” in Ultraviolet Atmospheric and Space Remote Sensing: Methods and Instrumentation II, G. R. Carruthers & K. F. Dymond, eds., Proc. SPIE 3818, 180–188 (1999).
* [22] J. W. Brault, M. C. Abrams, “DECOMP: a Fourier transform spectra decomposition program,” Volume 6 of 1989 OSA Technical Digest Series, 110–112.
* [23] G. Nave, C. J. Sansonetti, U. Griesmann, “Progress on the NIST IR-vis-UV Fourier transform spectrometer,” Volume 3 of 1997 OSA Technical Digest Series, 38–40.
* [24] G. Norlén, “Wavelengths and energy levels of Ar I and Ar II based on new interferometric measurements in the region 3400-9800 Å,” Phys. Scr. 8, 249–268 (1973).
* [25] W. Whaling, W. H. C. Anderson, M. T. Carle, J. W. Brault, H. A. Zarem, “Argon ion linelist and level energies in the hollow-cathode discharge,” J. Quant. Spectrosc. Radiat. Transfer 53, 1–22 (1995).
* [26] A. G. Maki, J. S. Wells, “New wavenumber calibration tables from heterodyne frequency measurements,” J. Res. Natl. Inst. Stand. Tech. 97, 409–470 (1992).
* [27] R. C. M. Learner, A. P. Thorne, I. Wynne-Jones, J. W. Brault, M. C. Abrams, “Phase correction of emission line Fourier transform spectra,” 12, 2165–2171 (1995).
* [28] NSO Digital Library available online at http://diglib.nso.edu/nso$\\_$user.html
* [29] A. E. Kramida, “The program lopt for least-squares optimization of energy levels,” Comp. Phys. Comm. 182, 419-434 (2010).
* [30] V. Kaufman, “Wavelengths, Energy Levels, and Pressure Shifts in Mercury 198,” J. Opt. Soc. Am., 52, 866–870 (1962).
* [31] S. H .Emara, “Wavelength shifts in 198Hg as a function of temperature,” J. Res. Natl. Bur. Standards 65A, 473–474 (1961).
* [32] M. L. Salit, C. J. Sansonetti, D. Veza, and J. C. Travis, “Investigation of Single-Factor Calibration of the Wave-Number Scale in Fourier-Transform Spectroscopy,” J. Opt. Soc. Am B, 21, 1543–1550 (2004).
* [33] C. J. Sansonetti, D. Veza, “Doppler-free measurement of the 546-nm line of mercury,” J. Phys. B 43, 205003 (2010).
Figure 1: The region of the Fe II $\rm a^{6}D-y^{6}P$ transitions. The labeled
lines show the J-values of the lower and upper energy levels respectively.
Figure 2: (Color online) Calibration of wavenumbers in spectrum fe0409.002 in
Table 2 using Ar II standards from [25], and iron standards taken from Learner
& Thorne [18] and Nave et al. [19]. The error bars represent the statistical
uncertainty in the measurement of the wavenumber. Figure 3: Phase in the
master spectrum, k19, used in Learner & Thorne [18]. The insert shows the
residual phase after fitting the points to an 11${}^{\mbox{th}}$ order
polynomial. Figure 4: (Color online) Comparison of wavenumbers in the master
spectrum, k19, calibrated from Ar II standards from [25] with iron standards
taken from [18] and [19] adjusted to the scale of [25]. The error bars
represent the statistical uncertainty in the measurement of the wavenumber.
Figure 5: Comparison of wavenumbers in the master spectrum, k19, with those
in i6, the main spectrum contributing to Table 4 of [19] in this wavelength
region. Figure 6: Partial term diagram of Fe II showing the determination of
the y6P levels using transitions in the UV and visible regions Figure 7:
Figure 3 from Nave & Sansonetti [13], with all of the Ge I wavenumbers reduced
by 1.4 parts in 108 and the Fe I and Fe II wavenumbers increased by 3.9 parts
in 108. The mean value of k${}_{\mbox{eff}}$ for the Ge I wavenumbers is
$(1.221\pm 0.020)\times$10-6, in agreement within the joint uncertainties with
the value of $(1.206\pm 0.020)\times$10-6 from the Fe I and Fe II lines
Table 1: Proposed corrections to previous papers.
Reference | Wavenumber | Previous | Previous | New | Correction to
---|---|---|---|---|---
| range | standard | uncertainty | standard | wavenumber scale
| (cm-1) | | (cm-1) | |
[18] | 17350 - 26140 | Ar II [24] | 0.001 | Ar II [25] | (+6.7$\pm$1.8)x10-8
Table 3 of [19] | 26027 - 34131 | Ar II [24], i56 | 0.002 | Ar II [25] | (+6.7$\pm$1.8)x10-8
Tables 4 & 5 of [19] | 33695 - 54637 | Ar II [24], i56 | 0.002 | Ar II [25], Fig.5 | (+10.6$\pm$2.3)x10-8
[14] | 50128 - 107887 | Ar II [24], i56 | 0.005 | Ar II [25], Fig.5 | (+10.6$\pm$2.3)x10-8
[13] | 51613 - 51692 | Ge I [17] | 0.002 | 198Hg [33] | (-1.4$\pm$2)x10-8
[8] | 38458 - 44233 | Ar II [25] | 0.002 | Table 3,4 | (+3.7$\pm$3)x10-8
[17] (Ge I & Ge II) | 8283 - 100090 | 198Hg,[30, 31] | $<$0.006 | 198Hg [33] | (-1.4$\pm$1.8)x10-8
[24] (Ar II)a | 4348 - 5145 | 86Kr Engelhard lamp | $<$0.001 | Ar II [25] | (+6.7$\pm$0.8)x10-8
11footnotetext: The proposed correction has only been confirmed for the 28 Ar
II lines recommended in [18].
Table 2: Summary of spectra
. Spectrum Instrument Date Wavelength Calibration Comments y/m/d Range (Å)
Spectrum k19 [19] NSO 81/07/22 2800 to 5600 Ar II[25] 810622R0.009 (NSO[28])
A1 in [18] i56 [19] IC 2270 to 4170 k19 i6 [19] IC 89/11/07 2220 to 3030 i56
lp0301 NIST VUV 02/03/01 1830 to 3194 Ge I,II[17] Figs. 3 &4 in [13] fe1115
NIST VUV 02/11/15 1558 to 2689 lp0301 fe0409.002 NIST 2-m 09/04/09 2748 to
5765 Ar II [25], Fe I,II[18, 19]
Table 3: Comparison of wavenumbers of Fe lines in Aldenius et al. [8] and
adjusted wavenumbers of Nave et al. [19]. The wavenumbers of [19] have been
increased by 10.6 parts in 108. The standard uncertainty in the last digits of
the wavenumbers and of the levels is given in parenthesis and is dominated by
the calibration uncertainty.
. Species Nave et al. (cm-1) Aldenius et al. (cm-1) (column 3 / column 2) - 1
[19] [8] Fe II 38458.9912(20) 38458.9908(20) -1.0x10-8 Fe II 38660.0535(20)
38660.0523(20) -3.1x10-8 Fe II 41968.0687(20) 41968.0654(20) -7.7x10-8 Fe II
42114.8374(20) 42114.8365(20) -2.1x10-8 Fe II 42658.2449(20) 42658.2430(20)
-4.5x10-8 Mean (-3.7$\pm$2.6)x10-8
Table 4: Comparison of adjusted wavenumbers of Mg lines in Aldenius et al. [8] with frequency comb measurements (col. 3) taken from paper listed in the reference column. The wavenumbers from [8] have been increased by 3.7 parts in 108. The standard uncertainties in the last digits of the wavenumbers are given in parentheses. Species | Aldenius (cm-1) | Frequency comb (cm-1) | (column 2/column 3) -1 | Reference
---|---|---|---|---
| [8] | | |
Mg I | 35051.2817(20) | 35051.2808(2) | 2.6x10-8 | [9]
Mg II | 35669.3039(20) | 35669.30440(6) | -1.3x10-8 | [11]
Mg II | 35760.8523(20) | 35760.85414(6) | -5.1x10-8 | [11]
Mg I | 49346.7730(30) | 49346.77252(7) | 1.0x10-8 | [10]
| | Mean | (-0.7$\pm$3)x10-8 |
Table 5: Determination of the z6P levels of Fe II from transitions to the ground term around 2350 Å. The statistical uncertainty in the last decimal place of the wavenumbers is given in parenthesis. The total standard uncertainty includes the uncertainty in the calibration. Lower | Upper | lower level value | Wavenumber | Upper level value
---|---|---|---|---
level | level | cm-1 | cm-1 | cm-1
a${}^{6}D_{5/2}$ | z${}^{6}P_{7/2}$ | 667.6829(5) | 41990.5610(3) | 42658.2439(6)
a${}^{6}D_{7/2}$ | z${}^{6}P_{7/2}$ | 384.7872(4) | 42273.4573(4) | 42658.2445(6)
| | | Mean | 42658.2442(5)
| | | Total uncertainty | 0.0011
a${}^{6}D_{5/2}$ | z${}^{6}P_{5/2}$ | 667.6829(5) | 42570.9226(4) | 43238.6055(6)
a${}^{6}D_{7/2}$ | z${}^{6}P_{5/2}$ | 384.7872(4) | 42853.8188(4) | 43238.6060(6)
| | | Mean | 43238.6058(5)
| | | Total uncertainty | 0.0011
a${}^{6}D_{1/2}$ | z${}^{6}P_{3/2}$ | 977.0498(6) | 42643.9332(4) | 43620.9830(7)
a${}^{6}D_{5/2}$ | z${}^{6}P_{3/2}$ | 667.6829(5) | 42953.2994(5) | 43620.9823(7)
| | | Mean | 43620.9827(6)
| | | Total uncertainty | 0.0012
Table 6: Determination of the a6S5/2 level of Fe II using transitions from the z6P levels. The statistical uncertainties in the last digits of the wavenumber and levels are given in parentheses. The total standard uncertainty of the a${}^{6}S$ level includes a contribution of 4$\times 10^{-8}$ times the level uncertainty due to the calibration. Upper level | Upper level value | Wavenumber | a6S level
---|---|---|---
| cm-1 | cm-1 | cm-1
z$\rm{}^{6}P_{7/2}$ | 42658.2442(5) | 19340.6092(2) | 23317.6350(5)
z$\rm{}^{6}P_{5/2}$ | 43238.6058(5) | 19920.9733(2) | 23317.6325(5)
z$\rm{}^{6}P_{3/2}$ | 43620.9827(6) | 20303.3477(3) | 23317.6350(7)
| | Weighted mean | 23317.6340(3)
| | Total uncertainty | 0.0010
Table 7: Determination of the y6P levels of Fe II from transitions to the a6S level. The statistical uncertainty in the last digits of the wavenumbers and levels is given in parentheses. The total standard uncertainty is common to all levels and includes a contribution of 4$\times 10^{-8}\sigma$ due to the calibration. The last column contains the level value and standard uncertainty in parenthesis with respect to the ground level obtained from the lopt program as described in section 5 . Upper level | Wavenumber | Upper level value | Level value from lopt
---|---|---|---
| cm-1 | cm-1 | cm-1
y$\rm{}^{6}P_{3/2}$ | 38657.2997(14) | 61974.9347(14) | 61974.9325(24)
y$\rm{}^{6}P_{5/2}$ | 38731.4041(7) | 62049.0381(8) | 62049.0408(27)
y$\rm{}^{6}P_{7/2}$ | 38853.9885(4) | 62171.6225(5) | 62171.6245(27)
| Total uncertainty | 0.003 |
Table 8: Experimental and Ritz wavenumbers for the a6D-y6P multiplet. The
standard uncertainties in the last digits of the wavenumbers and wavelengths
are given in parentheses.
Lower | Upper | $\sigma_{a6S}^{a}$ | $\sigma_{exp}^{b}$ | $\sigma_{Ritz}^{c}$ | $\lambda_{Ritz}^{d}$
---|---|---|---|---|---
level | level | cm-1 | cm-1 | cm-1 | Å
a6D1/2 | y6P3/2 | 60997.884(3) | 60997.882(3) | 60997.8827(25) | 1639.40117(7)
a6D3/2 | y6P3/2 | 61112.322(3) | 61112.321(3) | 61112.3207(25) | 1636.33125(7)
a6D5/2 | y6P3/2 | 61307.251(3) | 61307.247(3) | 61307.2496(25) | 1631.12847(7)
a6D3/2 | y6P5/2 | 61186.426(3) | 61186.432(4) | 61186.4290(28) | 1634.34934(7)
a6D5/2 | y6P5/2 | 61381.355(3) | 61381.358(3) | 61381.3579(28) | 1629.15914(7)
a6D7/2 | y6P5/2 | 61664.251(3) | 61664.255(3) | 61664.2536(27) | 1621.68508(7)
a6D5/2 | y6P7/2 | 61503.940(3) | 61503.945(8) | 61503.9416(27) | 1625.91205(7)
a6D7/2 | y6P7/2 | 61786.835(3) | 61786.837(3) | 61786.8373(27) | 1618.46769(7)
a6D9/2 | y6P7/2 | 62171.623(3) | 62171.626(4) | 62171.6245(27) | 1608.45081(7)
11footnotetext: Wavenumber calculated using a6S as an intermediate
level22footnotetext: Experimental wavenumber from archival spectra corrected
according to section 4.2.33footnotetext: Ritz wavenumber calculated from all
optimized energy levels44footnotetext: Wavelength calculated from the Ritz
wavenumber in column 5.
|
arxiv-papers
| 2011-01-25T20:15:51 |
2024-09-04T02:49:16.640159
|
{
"license": "Public Domain",
"authors": "Gillian Nave and Craig J. Sansonetti",
"submitter": "Gillian Nave",
"url": "https://arxiv.org/abs/1101.4915"
}
|
1101.5225
|
# Interfacial thermal transport in atomic junctions
Lifa Zhang Department of Physics and Centre for Computational Science and
Engineering, National University of Singapore, Singapore 117542, Republic of
Singapore Pawel Keblinski Department of Materials Science and Engineering,
Rensselaer Polytechnic Institute, New York, 12180, USA. Jian-Sheng Wang
Department of Physics and Centre for Computational Science and Engineering,
National University of Singapore, Singapore 117542, Republic of Singapore
Baowen Li Electronic address: phylibw@nus.edu.sg NUS Graduate School for
Integrative Sciences and Engineering, Singapore 117456, Republic of Singapore
Department of Physics and Centre for Computational Science and Engineering,
National University of Singapore, Singapore 117542, Republic of Singapore
(30 Oct 2010, Revised 10 Jan 2011 )
###### Abstract
We study ballistic interfacial thermal transport across atomic junctions.
Exact expressions for phonon transmission coefficients are derived for thermal
transport in one-junction and two-junction chains, and verified by numerical
calculation based on a nonequilibrium Green’s function method. For a single-
junction case, we find that the phonon transmission coefficient typically
decreases monotonically with increasing freqency. However, in the range
between equal frequency spectrum and equal acoustic impedance, it increases
first then decreases, which explains why the Kapitza resistance calculated
from the acoustic mismatch model is far larger than the experimental values at
low temperatures. The junction thermal conductance reaches a maximum when the
interfacial coupling equals the harmonic average of the spring constants of
the two semi-infinite chains. For three-dimensional junctions, in the weak
coupling limit, we find that the conductance is proportional to the square of
the interfacial coupling, while for intermediate coupling strength the
conductance is approximately proportional to the interfacial coupling
strength. For two-junction chains, the transmission coefficient oscillates
with the frequency due to interference effects. The oscillations between the
two envelop lines can be understood analytically, thus providing guidelines in
designing phonon frequency filters.
###### pacs:
66.70.-f, 05.60.-k, 44.10.+i,
## I Introduction
In the past decade there has been a significant research focus on thermal
transport in micro scaleDhar . Several conceptual thermal devices, such as
thermal rectifiers/diodes, thermal transistors, thermal logical gates, and
thermal memory rectifiers ; transistor ; logicgate ; memory , have been
proposed, which, in principle, make it possible to control heat due to phonons
and process information with phonons. The issue of quantum thermal transport
in nanostructures was also addressed wangjs2008 . In this context, the
critical information is in phonon transmission coefficients that in quasi-one-
dimensional atomic models can be calculated by transfer matrix method tong1999
; macia2000 ; cao2005 ; antonyuk2005 . However, the evaluation of the transfer
matrix may be numerically unstable, particularly when the system size becomes
large. Alternatively, nonequilibrium Green’s function (NEGF) method is an
efficient way to calculate the transmission coefficientnegfref .
Unfortunately, both of these two methods are numerical in nature and do not
give analytical expressions.
For thermal transport and control, the interfacial thermal scattering process
is becoming increasingly important, especially in practical devices. Two
theories, acoustic mismatch model little1959 and the diffuse mismatch model
swartz1989 , have been proposed to study the mechanism of the thermal
interfacial resistance. However, both models offer limited accuracy in
nanoscale interfacial resistance predictions stevens2005 because they neglect
atomic details of actual interfaces. A scattering boundary method within the
lattice dynamic approach was first proposed by Lumpkin and Saslow to study the
Kapitza conductance in a one-dimensional (1D) lattice lumpkin1978 , and was
then applied to calculate the Kapitza resistance in two- and three-dimensional
(3D) lattices paranjape1987 ; young1989 . This method can predict thermal
interfacial conductance between heterogeneous materials with full
consideration of the atomic structures in the interface. Recently, this method
was applied to study the ballistic thermal transport in nanotube
junctionswang2006 , spin chainszhang2008 , and honeycomb lattice ribbons
cuansing2009 .
In this paper we give an explicit analytical expression of transmission
coefficient obtained through the scattering boundary method, and use it to
study the interfacial thermal transport across atomic junctions. First, in
Sec. II, we introduce a model in which two semi-infinite 1D atomic chains are
coupled either via a point junction or an extended junction region. By using
the boundary scattering method we derive the exact expressions for phonon
transmission coefficients for thermal transport in one-junction and two-
junction chains in Sec. III. The role of various parameters on the junction
conductance is analyzed and discussed in Sec. IV. In section IV we also
estimate the interfacial conductance between two 3D solids. In Sec. V, we
introduce briefly the NEGF method, and use it to verify the results from
analytical formulae for the thermal transport in our model. A short summary is
presented in Sec. VI.
Figure 1: (color online) A schematic representation of the 1D atomic chain
model. The size of the center part is $N_{C}=8$. The left and right regions
are two semi-infinite harmonic atomic chains at different temperatures $T_{L}$
and $T_{R}$. The three parts are coupled by harmonic springs with constant
strength $k_{12}$ and $k_{23}$; all of which are harmonic chains with mass and
spring constant as $m_{1},k_{1}$, $m_{2},k_{2}$ and $m_{3},k_{3}$,
respectively.
## II Model
The one-dimensional atomic chain consists of three parts: two semi-infinite
leads and an center region (see Fig. 1). The two leads are in equilibrium at
different temperatures $T_{L}$ and $T_{R}$. The three parts are coupled by
harmonic springs with constant strength $k_{12}$ and $k_{23}$; all of which
are harmonic chains with mass and spring constants $m_{1},k_{1}$,
$m_{2},k_{2}$ and $m_{3},k_{3}$, respectively. So the total Hamiltonian can be
written as
$H=\sum\limits_{\alpha=1,2,3}{H_{\alpha}}+\frac{1}{2}k_{12}(x_{1,1}-x_{2,1})^{2}+\frac{1}{2}k_{23}(x_{2,N_{c}}-x_{3,1})^{2};$
(1)
here,
$H_{\alpha}=\sum\limits_{i=1}^{N_{\alpha}}{\frac{1}{2}m_{\alpha}\dot{x}_{\alpha,i}^{2}+\sum\limits_{i=1}^{N_{\alpha}-1}\frac{1}{2}k_{\alpha}(x_{\alpha,i}-x_{\alpha,i+1})^{2}}.$
(2)
Where $x_{\alpha,i}$ is the relative displacement of i-th atom in $\alpha$-th
part. If there is no center part, that is, the two semi-infinite leads
connected directly by $k_{12}$, then by setting $\alpha=1,2$ and $k_{23}=0$ in
Eq. (1), we can obtain the corresponding Hamiltonian. For the semi-infinite
leads, $N_{\alpha}=\infty$.
## III Analytical Solution from the Scattering Boundary Method
Heat current flowing from left to right through a junction connecting two
leads kept at different equilibrium heat-bath temperatures $T_{L}$ and $T_{R}$
is given by the Landauer formula wangjs2008
$I=\frac{1}{{2\pi}}\int_{0}^{\infty}{\hbar\omega\;\bigl{[}f_{L}(\omega)-f_{R}(\omega)\bigr{]}T[\omega]}d\omega,$
(3)
which allows us to develop the junction conductance formula
$\sigma=\frac{1}{{2\pi}}\int_{0}^{\infty}{d\omega\;\hbar\omega\,T[\omega]\frac{\partial
f(\omega)}{\partial T}},$ (4)
here, $f_{L,R}=\\{\exp[\hbar\omega/(k_{B}T_{L,R})]-1\\}^{-1}$ is the Bose-
Einstein distribution for phonons, and $T[\omega]$ is the frequency dependent
transmission coefficient. Therefore, the key step for the thermal transport
characterization is to calculate the transmission coefficients.
We first consider a point-junction case, that is, two semi-infinite harmonic
chains connected by a spring with constant strength $k_{12}$. We assume a wave
solution transmitting from the left lead to the right lead. We label the atoms
as $-\infty,\cdots,-1,0,1,2,\cdots,+\infty$. Atoms $0$ and $1$ are connected
by $k_{12}$ spring. An incident wave from left is assumed as
$x_{I}=\lambda_{1}^{j}e^{-i\omega t}$. When it arrives at the interface, it
will be partially reflected and partially transmitted. The reflected wave
amplitude is $x_{R}=r_{12}\lambda_{1}^{-j}e^{-i\omega t}$ and the transmission
wave can be written as $x_{T}=t_{12}\lambda_{2}^{j-1}e^{-i\omega t}$. So at
each atom we have
$\cdots,\;x_{-1}=(\lambda_{1}^{-1}+r_{12}\lambda_{1})e^{-i\omega
t},\;x_{0}=(1+r_{12})e^{-i\omega t}$, $x_{1}=t_{12}e^{-i\omega
t},\;x_{2}=t_{12}\lambda_{2}e^{-i\omega t},\;\cdots$. Here,
$\lambda_{j}=e^{iq_{j}a_{j}}$, $q_{j}$ is the wave vector, $a_{j}$ is the
interatomic spacing. For the atom in the $j-th$ part, we can have the equation
of motion as
$m_{j}\frac{d^{2}x_{j,n}}{dt^{2}}=k_{j}(x_{j,n+1}-x_{j,n})+k_{j}(x_{j,n}-x_{j,n-1}),$
(5)
each wave transport separately and satisfies such equation. Thus $\lambda_{j}$
satisfies the dispersion relation of the corresponding lead as
$\omega^{2}m_{j}=-k_{j}\lambda_{j}^{-1}+2k_{j}-k_{j}\lambda_{j}.$ (6)
The quadratic equation has two roots. Which one should we choose? Replacing
$\omega$ with $\omega+i\eta$, $\eta=0^{+}$, none of the eigenvalues $\lambda$
will have modulus exactly 1. We find for the traveling waves velev2004
$|\lambda|=1-\eta\frac{a}{v},$ (7)
thus the forward moving waves with group velocity $v>0$ have $|\lambda|<1$.
Therefore we should take the one with $|\lambda|<1$ of the two roots which are
given as
$\lambda_{j}=\frac{{-h_{j}\pm\sqrt{h_{j}^{2}-4}}}{2},\;\;\;h_{j}=\frac{{m_{j}}}{{k_{j}}}(\omega+i\eta)^{2}-2.$
(8)
From the scattering boundary method, the coefficients $r_{12}$, $t_{12}$ can
be obtained from the continuity condition at the interface as:
$\displaystyle\omega^{2}m_{1}x_{0}=-k_{1}x_{-1}+(k_{1}+k_{12})x_{0}-k_{12}x_{1};$
(9)
$\displaystyle\omega^{2}m_{2}x_{1}=-k_{12}x_{0}+(k_{12}+k_{2})x_{1}-k_{12}x_{2}.$
(10)
Finally we can get the transmission coefficient as
$T[\omega]=1-|r_{12}|^{2}=1-|r_{21}|^{2},$ (11)
here,
$r_{ij}=\frac{{k_{i}(\lambda_{i}-1/\lambda_{i})(k_{j}-k_{ij}-k_{j}/\lambda_{j})}}{{(k_{i}-k_{ij}-k_{i}/\lambda_{i})(k_{j}-k_{ij}-k_{j}/\lambda_{j})-k_{ij}^{2}}}-1.$
(12)
Of course, we can also use $t_{12}$ to express $T[\omega]$ as
$\frac{{m_{2}v_{2}/a_{2}}}{{m_{1}v_{1}/a_{1}}}|t_{12}|^{2}$, here the group
velocity
$v_{i}=\frac{{d\omega}}{{dq_{i}}}=\frac{{a_{i}}}{2}\sqrt{\frac{{4k_{i}}}{{m_{i}}}-\omega^{2}}$,
which is derived from the dispersion relation given by Eq. (6). Thus, the
transmission coefficient can also be expressed as
$T[\omega]=\frac{{\sqrt{4k_{2}m_{2}-\omega^{2}m_{2}^{2}}}}{{\sqrt{4k_{1}m_{1}-\omega^{2}m_{1}^{2}}}}|t_{12}|^{2},$
(13)
here
$t_{ij}=\frac{{-k_{ij}k_{i}(\lambda_{i}-1/\lambda_{i})}}{{(k_{i}-k_{ij}-k_{i}/\lambda_{i})(k_{j}-k_{ij}-k_{j}/\lambda_{j})-k_{ij}^{2}}}.$
(14)
For the long-wave limit, that is, $\omega=0^{+}$, we get
$r_{ij}=\frac{{\sqrt{k_{i}m_{i}}-\sqrt{k_{j}m_{j}}}}{{\sqrt{k_{i}m_{i}}+\sqrt{k_{j}m_{j}}}}$;
and the transmission is
$T[0^{+}]=\frac{{4\sqrt{k_{1}m_{1}k_{2}m_{2}}}}{{(\sqrt{k_{1}m_{1}}+\sqrt{k_{2}m_{2}})^{2}}}.$
(15)
This result is consistent with the one obtained for the acoustic mismatch
model, i.e., $T=\frac{4Z_{1}Z_{2}}{(Z_{1}+Z_{2})^{2}}.$ little1959 Where the
acoustic impedance is $Z_{i}=\rho_{i}v_{i}=(m_{i}/a_{i})v_{i}$, and
$Z_{i}(\omega=0^{+})=\sqrt{k_{i}m_{i}}$. We note that in acoustic mismatch
model the transmission coefficient is frequency independent, and in reality it
only applies in the limit of low frequency/long wavelengths. In this case the
phonon sees the interface only as a discontinuity between two semi-infinite
media and the transmission does not depend on the coupling spring strength
$k_{ij}$. If the two leads have the same acoustic impedance for long wave
limit, then $T[0^{+}]=1$; otherwise $T[0^{+}]<1$.
For a two-junction case, which is shown in Fig. 1, the transmission wave will
be reflected and transmitted by the second boundary, leading to multiple
reflections. Finally the total transmitted wave function is obtained as a
superposition of multiple reflections and transmissions, resulting in the
transmission coefficient through the center part
$T[\omega]=\frac{{(1-|r_{12}|^{2})(1-|r_{23}|^{2})}}{{|1-r_{23}r_{21}\lambda_{2}^{2(N_{C}-1)}|^{2}}},$
(16)
here $r_{ij}$ and $\lambda_{i}$ are determined by Eq. (12) and Eq. (8);
$N_{C}$ is the number of atoms in the center atomic chain. From this
expression, we can find that the transmission coefficient oscillates with
frequency, and is between the envelope lines of maximum and minimum
transmission, which are $T_{{\rm
max}}[\omega]=(1-|r_{12}|^{2})(1-|r_{23}|^{2})/(1-|r_{23}r_{21}|)^{2}$ for
constructive interference and $T_{{\rm
min}}[\omega]=/(1-|r_{12}|^{2})(1-|r_{23}|^{2})/(1+|r_{23}r_{21}|)^{2}$ for
destructive interference.
Figure 2: (color online) The transmission coefficient vs frequency $\omega$
for different interface coupling $k_{12}$ in one-junction chains. (a) shows
the transmission in one junction connected by the same semi-infinite atomic
chains with $k_{1}=k_{2}=1.0,\;m_{1}=m_{2}=1.0$; the solid, dashed, dotted and
dash-dotted lines correspond to $k_{12}=0.1$, 0.5, 1.0 and 2.0, respectively.
(b) shows the transmission in one junction connected by two different semi-
infinite atomic chains with $k_{1}=1.0,\;m_{1}=1.0,\;k_{2}=3.0$ and
$m_{2}=4.0$; the solid, dashed, dotted, dash-dotted and shot-dashed lines
correspond to $k_{12}=0.5$, 1.0, 1.5, 3.0 and 8.0, respectively. Figure 3:
(color online) The thermal conductance vs interface coupling $k_{12}$ in
point-junction model. Here, $k_{1}=1.0,\;m_{1}=1.0$.
## IV Results and Discussions
### IV.1 Thermal transport in 1D one-junction chains
In Sec. III, we have derived the analytical expressions for the phonon
transmission coefficient for point-junction and extended-junction (two point
junction) cases Eq. (11), Eq. (12) and Eq. (16) by using the scattering
boundary method. Using these analytical expressions, we analyze the role of
various parameters on the thermal transport in one- and two- point junctions.
Figure 2 shows the transmission coefficient as a function of frequency for a
different interface spring constant $k_{12}$ for the point-junction model. The
maximum frequency at which the transmission coefficient is above zero is equal
to the minimum of $2\sqrt{k_{1}/m_{1}}$ and $2\sqrt{k_{2}/m_{2}}$. In Fig.
2(a), the two semi-infinite atomic chains have the same mass and spring
constant. When the interface coupling $k_{12}$ equals to that of the chains,
the transmission is equal to one in the whole frequency domain, because of the
homogeneity of the chain structure. If $k_{12}$ increases or decreases, the
transmission coefficient decreases. If we set $k_{1}/m_{1}=k_{2}/m_{2}$, the
transmission coefficient exhibits similar behavior, the only difference is
that the transmission coefficient changes to the value obtained by Eq. (15).
In Fig. 2(b), the two semi-infinite atomic chains have different masses and
spring constants. The transmission decreases with increased frequency for all
the coupling values $k_{12}$. Also, it appears that for a given frequency the
transmission is maximized for a $k_{12}$ value residing between $k_{1}$ and
$k_{2}$. From Eq. (11) and Eq. (12), $T[\omega]=0$, if $k_{12}=0$; and
$T[\omega]$ has definite value
$1-|\frac{{k_{1}(\lambda_{1}-1)-k_{2}(1-\lambda_{2}^{-1})}}{{k_{1}(1-\lambda_{1}^{-1})+k_{2}(1-\lambda_{2}^{-1})}}|^{2}$,
if $k_{12}=\infty$.
Figure 4: (color online) The thermal conductance vs the ratio of
$k_{12}/k_{12m}$ in one-junction atomic chain. Here $k_{12m}$ is the harmonic
average of the spring constants of the two semi-infinite leads. (a)
$k_{1}=1.0$, $m_{1}=m_{2}=1.0$; the solid, dashed, and dotted lines correspond
to $k_{2}=0.1$, 1.0, and 40.0, respectively. (b) $k_{1}=1.0$, $m_{1}=1.0$,
$k_{2}=10.0$; the solid, dashed, and dotted lines correspond to $m_{2}=0.01$,
1.0, and 100.0, respectively.
The maximum transmission concept results in the maximum junction conductance
as shown in Fig. 3. With the increasing of $k_{12}$, we find that the
conductance will first increase, then arrive at maximum value, and then
slightly decrease and at last it will tend to a constant. We find that the
maximum transmission or conductance occurs at $k_{12}$ given by
$k_{12}=k_{12m}=\frac{2k_{1}k_{2}}{k_{1}+k_{2}},$ (17)
i.e., when the coupling spring stiffness is equal to the harmonic average of
spring connecting atoms in the two semi-infinite chains. In Fig. 4, we show
the thermal conductance vs the ratio of $k_{12}$ and $k_{12m}$. For the two
semi-infinite chains with the same mass $m_{1}=m_{2}$, the maximum conductance
occurs exactly at $k_{12m}$. If the two leads have different masses $m_{1}\neq
m_{2}$, the maximum conductance is almost exactly at the $k_{12m}$ point, for
mass ratios ranging from 0.01 to 100.
Figure 5: (color online) The transmission coefficient vs frequency for
different mass ratios $m_{2}/m_{1}$ at the interface coupling $k_{12m}$. Here,
$k_{1}=1.0$, $k_{2}=3.0$, $k_{12}=k_{12m}=1.5$ and $m_{1}=1.0$. Figure 6:
(color online) The transmission coefficient vs frequency for different
interface coupling $k_{12m}$. Here, $k_{1}=1.0,m_{1}=1.0$,
$k_{2}=0.7,m_{2}=0.3$.
In Fig. 5, we show the curves of the transmission as a function of frequency
for interface coupling equal to $k_{12m}$. If $k_{1}/m_{1}=k_{2}/m_{2}$, that
is, when both chains have the same frequency spectrum of
$[0,2\sqrt{k_{1}/m_{1}}]$, the transmission equals to a constant
$T[\omega]=T[0^{+}]$, which can be seen from the solid line in Fig. 5, and
which is consistent with Fig. 2(a). Thus for chains with matched spectra the
transmission is frequency independent. Let us now fix $k_{1},k_{2}$ and
$k_{2}$, and decrease $m_{2}$. In the range between the point of equal-
spectrum ($\omega_{m}=k_{1}/m_{1}=k_{2}/m_{2}$) and the one of equal-impedance
($Z(\omega=0^{+})=k_{1}m_{1}=k_{2}m_{2}$), the transmission will first
increase with frequency and then decrease. Otherwise, there is a monotonic
decrease. The former behavior is quite interesting, as one expects that the
transmission should be the largest in the long wavelength limit. For highly
dissimilar materials, the transmission coefficient in the whole frequency
range is much larger than that in the long wave limit
$T[\omega=0^{+}]=\frac{4Z_{1}Z_{2}}{(Z_{1}+Z_{2})^{2}}$, thus the real
conductance is far larger than that calculated from the acoustic mismatch
model. This result explain why the interfacial resistance calculated from the
acoustic mismatch model is far lager than the experimental value measured at
low temperatures, where the phonon transport can be regarded as ballistic
transport.
Figure 7: (color online)(a) The cutoff frequency vs interface coupling for 1D
one-junction atomic chains. The parameters are: $k_{1}=1.0$, $m_{1}=1.0$. (b)
The transmission as function of interface coupling for 1D one-junction atomic
chains. The parameters are: $k_{1}=1.0$, $m_{1}=1.0$, $k_{2}=0.7$, $m_{2}=0.3$
In many real interfaces, interface coupling is very weak, that is, the
$k_{12}$ is less than $k_{12m}$. So it is desirable to study the thermal
transport in atomic chains in the weak coupling limit. Figure 6 shows the
transmission coefficient as function of interface coupling. In the weak
coupling limit, with the frequency increasing, the transmission decreases
rapidly to zero, so the frequency region where phonons are effectively
transmitted is very narrow. With interface strength increasing, more and more
modes contribute to the transmission and the phonon transmission window
widens. If the interface coupling increases further, that is
$k_{12}/k_{12m}>0.1$, out of the weak interface coupling limit, all the
phonons contribute to the transmission. The only further change with
increasing $k_{12}$ is the actual values of the transmission coefficients
increase. In Fig. 7(a), we show the transmission cutoff frequency as function
of the interface coupling. Here, we define the cutoff frequency $\omega_{\rm
cutoff}$ at which the transmission $T(\omega_{\rm cutoff})=0.1T(0^{+})$. We
find that the cutoff frequency shows linear dependance on interface coupling
in the weak coupling limit $k_{12}<0.1k_{12m}$. If the interface strength
increase further, the cutoff frequency is saturated. In Fig. 7(b), we show the
transmission as function of interface coupling for several different phonons.
We find that in the weak interface coupling region, the transmission is
proportional to the square of the interface coupling, which is consistent with
the formulas Eq. (13) and Eq. (14).
Figure 8: (color online) The thermal conductance vs interface coupling for 1D
point-junction atomic chains. The parameters are: $k_{1}=1.0$, $m_{1}=1.0$.
In the weak interface coupling region, for the 1D atomic one-junction chains,
it is shown that the thermal conductance is linear with the interface coupling
(see Fig. 8). If we strengthen the interface coupling between the two chains,
the conductance will be linearly enhanced. For different mismatched chains,
the absolute values of the conductance are different, but dependence on the
coupling strength is the same.
Figure 9: (color online) The thermal conductance vs interface coupling for 3D
one-junction atomic chains. The parameters are the same with Fig. 8. (a)
Interface coupling is far less than the coupling $k_{12m}$:
$k_{12m}/k_{12}=0.001-0.1$; (b) Interface coupling is in the region of
$0.1k_{12m}\sim 0.9k_{12m}$.
### IV.2 Thermal transport in 3D single-interface structures
The thermal conductance Eq. (4) can also be written as hopkins2009 :
$\sigma=\int_{0}^{\infty}{d\omega\;\hbar\omega\,T[\omega]\frac{\partial
f(\omega)}{\partial T}v(\omega)D(\omega)},$ (18)
because of $v(\omega)=\partial\omega/\partial k$ and phonon density of states
in 1D structure, $D(\omega)=1/(2\pi v)$, we can obtain Eq. (4). In order to
estimate the behavior of the interfacial thermal transport across interfaces
in 3D structures, we only need to change the phonon density of states in the
above equation. Because the density of states for 3D structure within the
Debye approximation is $D(\omega)\sim\omega^{2}$, therefore we can replace
$\omega$ with $\omega^{3}$ in Eq. (4); the thermal conductance as a function
of the coupling strength is shown in Fig. 9. From Fig. 9(a), we find that in
the weak interface limit, conductance is proportional to the square of
interface coupling, which is consistent with the results from other models
scho1980 ; lavr1998 ; prasher2009 , while it is linear dependent on the
interface coupling in 1D junctions. This is due to the fact that in 3D low
frequency region contributes relatively little to the conductance as the
density of states is low there. If the interface coupling increases further,
that is $k_{12}/k_{12m}>0.1$, out of the weak interface coupling limit, all
the modes contribute to the transmittance, the conductance is no longer
proportional to the square of the interface coupling, and the slope
continuously decreases. In some intermediate ranges the conductance is
approximately proportional to the interfacial coupling (see Fig. 9(b)), which
is consistent with the results from molecular simulation approach hu2009 . For
stronger coupling the conductances for the 1D case and 3D one have similar
behaviors, the slope of both cases will decrease continuously to be zero at
point $k_{12m}$, where the conductance will be maximized and then decrease
slightly to a limiting value.
### IV.3 Thermal transport in extended junctions
Figure 10: (color online) The transmission coefficient of the two-junction
atomic chains. Parameters:
$k_{1}=1.0,\,m_{1}=1.0,\,k_{2}=0.9,\,m_{2}=1.6,\,k_{3}=4.5,\,m_{3}=2.0$, The
solid, dotted, dashed and shot dashed lines correspond to maximum
transmission, minimum transmission, $N_{c}=4$ and $N_{c}=9$, respectively. The
interface couplings are different: (a) $k_{12}=0.3,k_{23}=0.7$; (b)
$k_{12}=1.0,k_{23}=4.5$. Figure 11: (color online) The transmission
coefficient of the two-junction atomic chains. Here, $k_{1}=1.0,\,m_{1}=1.0$.
The solid, dotted, dashed and shot dashed lines correspond to maximum
transmission, minimum transmission, $N_{c}=4$ and $N_{c}=9$, respectively. (a)
$k_{2}=3.0,\,m_{2}=5.0,\,k_{3}=1.0,\,m_{3}=1.0,\,k_{12}=k_{23}=1.0$; (b)
$k_{2}=3.0,\,m_{2}=1.0,\,k_{3}=5.0,\,m_{3}=1.0,\,k_{12}=k_{12m}=1.5,\,k_{23}=k_{23m}=3.75$;
(c)
$k_{2}=3.0,\,m_{2}=3.0,\,k_{3}=5.0,\,m_{3}=5.0,\,k_{12}=k_{12m}=1.5,\,k_{23}=k_{23m}=3.75$.
Now we focus on a case where the junction is extended and involves a center
part. The overall behavior of the transmission is the combination of the
transmission behavior in single point-junction case and the oscillatory
behavior due to phonon interferences arising form multiple scattering. We show
the transmission coefficient as a function of frequency of an arbitrary case
in Fig. 10(a). Here, the three chain parts have different masses and spring
constants, and the interface coupling is not special. From the analytical
expression of Eq. (16), we plot curves of the maximum transmission and minimum
transmission, $N_{c}=4$ and $N_{c}=9$. The transmission oscillates between the
envelop lines of maximum and minimum transmission. The maximum transmission
line will increase first, and the minimum transmission line will monotonically
decrease with frequency. However for interface coupling that is the same with
the leads, the two envelop lines will monotonically decrease, which can be
seen in Fig. 10(b).
For some special cases, the transmission coefficient in the frequency domain
has interesting phenomena, which are shown in Fig. 11. In Fig. 11(a), the
transmission for the case of two identical leads is shown. In this case, the
maximum transmission is equal to one, the infinite-long wavelength phonon and
the resonance mode can transmit fully through the center part. The minimum
transmission is very low, indicating efficient destructive interference.
Figure 11(b) shows the transmission when all three parts are different and
connected by interface couplings $k_{12m}$ and $k_{23m}$. We find that overall
trend for the maximum and minimum transmission lines is increasing first, then
decreasing. If, in addition, the ratios of $k_{i}/m_{i}$ are the same for
three parts, then the maximum and minimum transmission are constants in the
whole frequency range, and the transmission coefficient through finite-size
center part oscillate between the two constants, which can be clearly seen in
Fig. 11(c). Therefore, we can use the above properties of transmission to
design the frequency filters. Figure 12 shows the maximum and minimum
transmission coefficient for the filter. If the spring constant of the center
part is very different from the ones of the the two leads, the oscillatory
peak is sharp, and transmission for most of the frequency will tend to zero,
only few resonant frequency can be transmitted. This finding provides
guidelines for the design of selective frequency filters.
Figure 12: (color online) The maximum and minimum transmission coefficient of
the two-junction atomic chains. Here, $k_{1}=1.0,\,m_{1}=1.0$. The solid,
dashed lines correspond to maximum transmission and minimum transmission
$k_{3}=1.0,\,m_{3}=1.0$, respectively; the dotted and dash-dotted lines
correspond to maximum transmission and minimum transmission
$k_{3}=5.0,\,m_{3}=5.0$, respectively. The inset shows the transmission
coefficient with frequency for different $k_{2}$.
$k_{1}=k_{3}=1.0,\,m_{1}=m_{3}=1.0$. The dotted, dashed, and solid lines
correspond to $k_{2}=0.5$, 0.1, and 0.02 respectively. For all the curves,
$m_{2}=k_{2}$ and $k_{12}=k_{12m},k_{23}=k_{23m}$.
## V Verification by Nonequilibrium Green’s Function Method
The NEGF method is an exact approach to study the ballistic thermal transport
through junctions. Following the discussion in Sec. II, if we use a
transformation for the coordinates, $u_{j}=\sqrt{m_{j}}x_{j}$, which is called
the mass-normalized displacement, then the Hamiltonian can be written as
$H=\sum\limits_{\alpha=1,2,3}H_{\alpha}+\sum\limits_{\beta=1,3}{U_{\beta}^{T}V_{\beta,2}U_{2}},$
(19)
where
$H_{\alpha}=\frac{1}{2}\left(P_{\alpha}^{T}P_{\alpha}+U_{\alpha}^{T}K_{\alpha}U_{\alpha}\right)$.
$K_{\alpha}$ is the mass-normalized spring constant matrix, and
$V_{12}=(V_{21})^{T}$ is the coupling matrix of the left lead to the central
region and similarly for $V_{23}$ is the coupling matrix of the right lead to
the central region. As stated in Ref. wangjs2006, the element of the coupling
matrix $V_{\alpha,\beta}^{ij}$ is equal to $-k_{ij}/sqrt{m_{i}m_{j}}$ which
corresponding to the coupling between the $i_{\rm th}$ atom in region $\alpha$
and the $j_{\rm th}$ atom in region $\beta$.
Figure 13: (color online) The comparison of the results from scattering
boundary method and nonequilibrium Green’s function method for the
transmission coefficient in two-junction atomic chains. The square curve and
solid line correspond the parameters:
$N_{c}=6,\,k_{1}=1.0,\,m_{1}=1.0,\,k_{2}=1.5,\,m_{2}=1.3,\,k_{3}=2.0,\,m_{3}=1.7,\,k_{12}=1.3,\,k_{23}=0.8$;
the circle curve and dashed line correspond the parameters:
$N_{c}=13,\,k_{1}=1.0,\,m_{1}=1.0,\,k_{2}=1.5,\,m_{2}=1.3,\,k_{3}=4.0,\,m_{3}=2.7,\,k_{12}=1.3,\,k_{23}=0.8$.
The square and circle curves are the results from nonequilibrium Green’s
function method; The solid and dash lines are the results from scattering
boundary method.
We can use the nonequilibrium Green’s function method wangjs2008 to study the
thermal transport in the atomic chain. We define the contour-ordered Green’s
function as
$G^{\alpha\beta}(\tau,\tau^{\prime})\equiv-\frac{i}{\hbar}\left\langle{\mathcal{T}\,U_{\alpha}(\tau)U_{\beta}(\tau^{\prime})^{T}}\right\rangle,$
(20)
where $\alpha$ and $\beta$ refer to the region that the coordinates belong to
and $\mathcal{T}$ is the contour-ordering operator. Then the equations of
motion of the Green’s function can be derived. In particular, the retarded
Green’s function for the central region in frequency domain is
$G^{r}[\omega]=\Bigl{[}(\omega+i\eta)^{2}-K_{2}-\Sigma^{r}[\omega]\Bigr{]}^{-1}.$
(21)
Here, $\Sigma^{r}=\sum\limits_{\alpha=1,3}{\Sigma_{\alpha}^{r}}$, and
$\Sigma_{\alpha}=V_{2,\alpha}g_{\alpha}V_{\alpha,2}$ is the self-energy due to
interaction with the heat bath,
$g_{\alpha}^{r}=[(\omega+i\eta)^{2}-K_{\alpha}]^{-1}$. And in the advanced
Green’s function $G^{a}=(G^{r})^{\dagger}$, the transmission coefficient can
be calculated by the so-called Caroli formula as
$T_{\beta\alpha}[\omega]={\rm{Tr}}(G^{r}\Gamma_{\beta}G^{a}\Gamma_{\alpha}),$
(22)
where
$\Gamma_{\alpha}=i\bigl{(}\Sigma_{\alpha}^{r}[\omega]-\Sigma_{\alpha}^{a}[\omega]\bigr{)}.$
For single-junction atomic chains, if we regard the two atoms in the interface
(atom 0 and atom 1) as the center part, then we can still use the formulae
above to study the phonon transmission leading to the exact formula yielding
the same result with the one obtained from the scattering boundary method. In
Appendix A, We give the analytical proof of this fact.
For two-junction atomic chains, according to the NEGF formulas, we do the
numerical calculation and plot the curves of the transmission coefficient as a
function of frequency and compare them to the results obtained the scattering
boundary method (see Fig. 13). We find that for any arbitrary case, the
results from the NEGF method and the scattering boundary method are exactly
the same. If there is no many-body interaction, that is, for the ballistic
thermal transport the scattering matrix approach and the Green’s function
method give the same results. These two methods are equivalent, which has been
proved from other points of view in Refs. khomyakov2005 ; harbola2008 .
## VI Conclusion
In this paper, we study the ballistic interfacial thermal transport in atomic
junctions, we give the analytical simple formulae Eq. (11), Eq. (12) and Eq.
(16) for the transmission of one-junction and two-junction cases, which are
consistent with the results from the NEGF method.
For one-junction case, we find the transmission and conductance are maximized
when the interface spring constant equals to the harmonic average of the two
spring constants of the leads. At the point near $k_{12}=k_{12m}$, the
transmission $T[\omega]$ is a constant if $k_{2}/m_{2}=k_{1}/m_{1}$; if not
equal, in the range between $k_{1}/m_{1}=k_{2}/m_{2}$ and
$k_{1}m_{1}=k_{2}m_{2}$, the transmission coefficient increases first then
decreases with the increasing of frequency, otherwise the transmission
monotonically decreases as the frequency increasing. For weak interface
coupling, the cutoff frequency and the interface conductance for 1D chain is
linear dependent with the interface coupling strength.
Because of different density of states, we change the formula of conductance
to mimic the thermal transport in 3D junctions. In weak interface coupling
limit, we find that the conductance is proportional to the square of the
interface coupling, which is consistent with the results from other models.
The slope of the conductance as function of interfacial coupling strength
decreases continuously from two to zero, in certain range of which, the
conductance is linear proportional to the interface coupling, which are
consistent with the results of other molecular simulations.
For two-junction case, the transmission will oscillate with frequency in the
envelop lines of maximum and minimum transmission which are determined by the
one-junction picture. The transmission sometimes oscillates between two
decreasing envelop lines, sometimes between two increasing envelop curves, or
between two constants, etc.
## Acknowledgements
P. K. is supported by the U.S. Air Force Office of Scientific Research Grant
No. MURI FA9550-08-1-0407. J.-S. W. acknowledge support from a NUS research
grant R-144-000-257-112.
## Appendix A Analytical proof of the equality of the two methods for one
junction
In this appendix we give the analytical proof for the equality of the
scattering boundary method and the non-equilibrium Green’s function approach
for the one-junction atomic chains.
From the scattering boundary method, we obtain the transmission Eq. (13) and
Eq. (14), that is
$T[\omega]=\frac{{\sqrt{4k_{2}m_{2}-\omega^{2}m_{2}^{2}}}}{{\sqrt{4k_{1}m_{1}-\omega^{2}m_{1}^{2}}}}\big{|}\frac{{-k_{12}k_{1}(\lambda_{1}-1/\lambda_{1})}}{{(k_{1}-k_{12}-k_{1}/\lambda_{1})(k_{2}-k_{12}-k_{2}/\lambda_{2})-k_{12}^{2}}}\big{|}^{2},$
(23)
From the dispersion relation Eq. (6), we can obtain
$k_{j}-k_{j}/\lambda_{j}=\omega^{2}m_{j}-k_{j}(1-\lambda_{j})$ (24)
; and
$k_{j}^{2}|\lambda_{j}-1/\lambda_{j}|^{2}=\omega^{2}(4k_{j}m_{j}-\omega^{2}m_{j}^{2})$
(25)
, So we can get
$T[\omega]=\frac{k_{12}^{2}\omega^{2}\sqrt{4k_{1}m_{1}-\omega^{2}m_{1}^{2}}\sqrt{4k_{2}m_{2}-\omega^{2}m_{2}^{2}}}{\big{|}[\omega^{2}m_{1}-k_{1}(1-\lambda_{1})-k_{12}][\omega^{2}m_{2}-k_{2}(1-\lambda_{2})-k_{12}]-k_{12}^{2}\big{|}^{2}}.$
(26)
Using the NEGF formulae, we regard the two atoms in the interface (atom 0 and
atom 1) as the center part 0, then the dynamic matrix of the center as
$K_{0}=\left({\begin{array}[]{*{20}c}\frac{k_{1}+k_{12}}{m_{1}}&\frac{-k_{12}}{\sqrt{m_{1}m_{2}}}\\\
\frac{-k_{12}}{\sqrt{m_{1}m_{2}}}&\frac{k_{12}+k_{2}}{m_{1}}\\\
\end{array}}\right).$ (27)
And the coupling matrices between the leads (parts 1 and 2) and the center
(part 0) are $V_{01}=(k_{1}/m_{1}\,,\,0)^{T}$ and
$V_{02}=(0\,,\,k_{2}/m_{2})^{T}$, and according to Ref. wang2007 , we can
obtain the surface Green’s function as
$g_{i}^{r}=-\frac{m_{i}\lambda_{i}}{k_{i}},$ (28)
here, $i=1,2$ corresponds to the left and right lead. Then we can get the self
energy ($\Sigma^{r}=V_{01}g_{1}^{r}V_{10}+V_{02}g_{2}^{r}V_{20}$) as
$\Sigma^{r}=\left({\begin{array}[]{*{20}c}-\frac{k_{1}\lambda_{1}}{m_{1}}&0\\\
0&-\frac{k_{2}\lambda_{2}}{m_{2}}\\\ \end{array}}\right).$ (29)
Thus we can calculate the retarded Green’s function of the center
$G^{r}=(\omega^{2}I-K_{0}-\Sigma^{r})^{-1}$, which reads as
$G^{r}=\left({\begin{array}[]{cc}A_{1}&B\\\ B&A_{2}\\\
\end{array}}\right)^{-1}=\frac{1}{\Delta}\left({\begin{array}[]{cc}A_{2}&-B\\\
-B&A_{1}\\\ \end{array}}\right),$ (30)
here, $I$ is two-dimensional identity matrix and
$\displaystyle
A_{i}=\omega^{2}-\frac{k_{i}}{m_{i}}(1-\lambda_{i})-\frac{k_{12}}{m_{i}};$
(31) $\displaystyle
B=\frac{k_{12}}{\sqrt{m_{1}m_{2}}};\;\Delta=A_{1}A_{2}-B^{2}.$ (32)
The advanced Green’s function $G^{a}$ equals to $(G^{r})^{\dagger}$. And from
the self energy we can get
$\Gamma_{1}=\left({\begin{array}[]{*{20}c}C_{1}&0\\\ 0&0\\\
\end{array}}\right);\;\Gamma_{2}=\left({\begin{array}[]{*{20}c}0&0\\\
0&C_{2}\\\ \end{array}}\right),$ (33)
here, $C_{i}=\frac{\omega}{m_{i}}\sqrt{4k_{i}m_{i}-\omega^{2}m_{i}^{2}}$.
Therefore, we can calculate the transmission coefficient from the Caroli
formula Eq. (22), at last we obtain
$T[\omega]=Tr(G^{r}\Gamma_{1}G^{a}\Gamma_{2})=\frac{B^{2}C_{1}C_{2}}{\Delta\Delta^{*}}=\frac{B^{2}C_{1}C_{2}}{|A_{1}A_{2}-B^{2}|^{2}}$
(34)
Inserting the values of $A_{i},B$ and $C_{i}$, we get exactly the same result
with Eq. (26). Therefore, the results from the scattering boundary method and
non-equilibrium Green’s function approach are equivalent.
## References
* (1) A Dhar, Adv. Phys. 57, 457 (2008).
* (2) M. Terraneo, M. Peyrard, and G. Casati, Phys. Rev. Lett.88, 094302 (2002); B. Li, L. Wang, and G. Casati, Phys. Rev. Lett. 93, 184301 (2004); D. Segal and A. Nitzan, Phys. Rev. Lett. 94, 034301 (2005); C. W. Chang, D. Okawa, A. Majumdar, and A. Zettl, Science 314, 1121 (2006).
* (3) B. Li, L. Wang, and G. Casati, Appl. Phys. Lett. 88, 143501 (2006).
* (4) L. Wang and B. Li, Phys. Rev. Lett. 99, 177208 (2007).
* (5) L. Wang and B. Li, Phys. Rev. Lett. 101, 267203 (2008).
* (6) J.-S. Wang, J. Wang, and J. T. Lü, Eur. Phys. J. B 62, 381 (2008).
* (7) P. Tong, B. Li, and B. Hu, Phys. Rev. B 59, 8639 (1999).
* (8) E. Maciá, Phys. Rev. B 61, 6645 (2000).
* (9) L. S. Cao, R. W. Peng, R. L. Zhang, X. F. Zhang, Mu Wang, X. Q. Huang, A. Hu, and S. S. Jiang, Phys. Rev. B 72, 214301 (2005).
* (10) V. B. Antonyuk, M. Larsson, A. G. Mal shukov, and K. A. Chao, Semicond. Sci. Technol. 20, 347 (2005).
* (11) H. Haug and A. P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors (Springer, 1996); J.-S. Wang, J. Wang, and N. Zeng, Phys. Rev. B 74, 033408 (2006); L. Zhang, J. -S. Wang, and B. Li, New J. Phys. 11, 113038 (2009).
* (12) W. Little, Can. J. Phys. 37, 334 (1959).
* (13) E. Swartz and R. Pohl, Rev. Mod. Phys.61, 605 (1989).
* (14) R. Stevens, A. Smith, and P. Norris,, J. Heat Transfer 127, 315 (2005).
* (15) M. E. Lumpkin and W. M. Saslow, Phys. Rev. B. 17,4295 (1997).
* (16) B. V. Paranjape, N. Arimitsu, and E. S. Krebes, J. Appl. Phys. 61, 888 (1987).
* (17) D. A. Young and H. J. Maris, Phys. Rev. B. 40,3685 (1989).
* (18) J. Wang and J.-S. Wang, Phys. Rev. B. 74,054303 (2006).
* (19) L. Zhang, J. -S. Wang and B. Li, Phys. Rev. B 78, 144416 (2008).
* (20) E. Cuansing and J.-S. Wang, Eur. Phys. J. B 69, 505 (2009).
* (21) J. Velev and W. Butler, J. Phys.: Condens. Matter 16, R637 (2004).
* (22) W. A. Little, Can. J. Phys. 37, 334 (1959).
* (23) P. E. Hopkins, P. M. Norris, M. S. Tsegaye, and A. W. Glosh, J. Appl. Phys. 106, 063503 (2009).
* (24) M. Schoenberg, J. Acoust. Soc. Am. 68, 1516 (1980).
* (25) A. I. Lavrentyev and S. I. Rokhlin, J. Acoust. Soc. Am. 103, 657 (1998).
* (26) R. Prasher, Appl. Phys. Lett. 94, 041905 (2009).
* (27) M. Hu, P. Keblinski, and P. K. Schelling, Phys. Rev. B 79, 104305 (2009).
* (28) P.A. Khomyakov, G. Brocks, V. Karpan, M. Zwierzycki, P.J. Kelly, Phys. Rev. B 72, 035450 (2005)
* (29) U. Harbola, S. Mukamel, Phys. Rep.465, 191 C222 (2008).
* (30) J. -S. Wang, N. Zeng, J. Wang, and C. K. Gan, Phys. Rev. E 75, 061128 (2007)
|
arxiv-papers
| 2011-01-27T08:59:36 |
2024-09-04T02:49:16.650344
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lifa Zhang, Pawel Keblinski, Jian-Sheng Wang, and Baowen Li",
"submitter": "Lifa Zhang",
"url": "https://arxiv.org/abs/1101.5225"
}
|
1101.5229
|
# The phonon Hall effect: theory and application
Lifa Zhang,1 Jie Ren,2,1 Jian-Sheng Wang,1 and Baowen Li 1,2 1 Department of
Physics and Centre for Computational Science and Engineering, National
University of Singapore, Singapore 117542, Republic of Singapore 2 NUS
Graduate School for Integrative Sciences and Engineering, Singapore 117456,
Republic of Singapore
(6 June 2011)
###### Abstract
We present a systematic theory of the phonon Hall effect in a ballistic
crystal lattice system, and apply it on the kagome lattice which is ubiquitous
in various real materials. By proposing a proper second quantization for the
non-Hermite Hamiltonian in the polarization-vector space, we obtain a new heat
current density operator with two separate contributions: the normal velocity
responsible for the longitudinal phonon transport, and the anomalous velocity
manifesting itself as the Hall effect of transverse phonon transport. As
exemplified in kagome lattices, our theory predicts that the direction of Hall
conductivity at low magnetic field can be reversed by tuning temperatures,
which we hope can be verified by experiments in the future. Three phonon-Hall-
conductivity singularities induced by phonon-band-topology change are
discovered as well, which correspond to the degeneracies at three different
symmetric center points, ${\bf\Gamma}$, ${\bf K}$, ${\bf X}$, in the wave-
vector space of the kagome lattice.
###### pacs:
63.22.-m 66.70.-f, 72.20.Pa
## 1 Introduction
In recent years, phononics, the discipline of science and technology in
processing information by phonons and controlling heat flow, becomes more and
more exciting [1, 2]. Various functional thermal devices such as thermal diode
[3], thermal transistor [4], thermal logic gates [5] and thermal memory [6],
etc., have been proposed to manipulate and control phonons, the carrier of
heat energy and information. And very recently, similar to the Hall effect of
electrons, Strohm _et al._ observed the phonon Hall effect (PHE) – the
appearance of a temperature difference in the direction perpendicular to both
the applied magnetic field and the heat current flowing through an ionic
paramagnetic dielectric sample [7], which was confirmed later in Ref. [8].
Such observation of the PHE is really surprising because phonons as charge-
free quasiparticles, different from electrons, cannot directly couple to the
magnetic field through the Lorentz force. Since then, several theoretical
explanations have been proposed [9, 10, 11, 12] to understand this novel
phenomenon. From the work of the PHE in four-terminal nano-junctions and the
phonon Hall conductivity in the two-dimensional periodic crystal lattice, we
know that the PHE can exist even in the ballistic system.
Geometric phase effects [13, 14] are fundamentally important in understanding
electrical transport property in quantum Hall effect [15, 16], anomalous Hall
effect[17, 18], and anomalous thermoelectric transport [19]. It is successful
in characterizing the underlying mechanism of quantum spin Hall effect [20,
21]. Such an elegant connection between mathematics and physics provides a
broad and deep understanding of basic material properties. Although there is a
quite difference between phonons and electrons, we still can use the
topological description to study the underlying properties of the phonon
transport, such as topological phonon modes in dynamic instability of
microtubules [22] and in filamentary structures [23], Berry-phase-induced heat
pumping [24], and the Berry-phase contribution of molecular vibrational
instability [25].
The topological nature of the PHE is recently studied in Ref. [26], where a
general expression for phonon Hall conductivity is obtained in terms of the
Berry curvatures of band structures. In Ref. [26], the authors find a phase
transition in the PHE of the honeycomb lattice, explained from topological
nature and dispersion relations. From the Green-Kubo formula and considering
the contributions from all the phonon bands, the authors obtain the general
formula for the phonon Hall conductivity. Then by looking at the phases of the
polarization vectors of both the displacements and conjugate momenta as a
function of the wave vector, a Berry curvature can be defined uniquely for
each band. Combining the above two steps, at last the phonon Hall conductivity
can be written in terms of Berry curvatures. Such derivation gives us a clear
picture of the contribution to the phonon Hall current from all phonon
branches, as well as the relation between the phonon Hall conductivity and the
geometrical phase of the polarization vectors, which thus helps us to
understand the topological picture of the PHE. However, the process of going
from the Berry phase to the heat flux and the phonon Hall conductivity looks
not very clear and natural.
We know that for the Hall effect of the electrons, in addition to the normal
velocity from usual band dispersion contribution, the Berry curvature induces
an anomalous velocity always transverse to the electric field, which gives
rise to a Hall current, thus the Hall effect occurs [14]. For the magnon Hall
effect [27] recently observed, the authors also found the anomalous velocity
due to the Berry connection which is responsible for the thermal Hall
conductivity. Therefore in this article we will derive the theory of the PHE
in a more natural way where the Berry phase effect inducing the anomalous
velocity contributes to the extra term of the heat current. Thus the Berry
phase effect is straightforward to take the responsibility of the PHE.
A kagome lattice, composed of interlaced triangles whose lattice points each
have four neighboring points [29], becomes popular in the magnetic community
because the unusual magnetic properties of many real magnetic materials are
associated with those characteristic of the kagome lattice [30]. The schematic
figure of kagome lattice is shown in Fig. 1. In this paper we also apply the
PHE theory to the kagome lattice to investigate whether the mechanism of the
phase transition found in Ref. [26] is general and how the phonon Hall
conductivity, Chern numbers and the dispersion relation behave and relate to
each other.
In this paper we organize as follows. In Sec. 2, we give a new systematic
derivation of the theory of the PHE in terms of Berry curvatures. In this
section, we first introduce the Hamiltonian and the modified second
quantization, then derive the heat current density operator which includes
both the normal velocity and the anomalous velocity from the Berry-phase
effect. Using the Green-Kubo formula, the general formula of the phonon Hall
conductivity is obtained. Then we give an application example on the kagome
lattice in Sec. 3. In this section the computation details about the dynamic
matrix, the Chern numbers and the phonon Hall conductivity are given, and the
behaviors and relations between the phonon Hall conductivity, Chern numbers,
and the band structures are discussed. In the end a short conclusion is
presented in Sec. 4.
## 2 The PHE theory
In this section, we give the detailed derivation for the theory of the PHE. We
use the Hamiltonian in Refs. [26] and [31], which is a positive definite
Hamiltonian to describe the ionic crystal lattice with in an applied magnetic
field.
### 2.1 The Hamiltonian and the second quantization
The Hamiltonian for an ionic crystal lattice in a uniform external magnetic
field [26, 28, 31] can be written in a compact form as
$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{2}(p-{\tilde{A}}u)^{T}(p-{\tilde{A}}u)+\frac{1}{2}u^{T}Ku\qquad$
(1) $\displaystyle=$
$\displaystyle\frac{1}{2}p^{T}p+\frac{1}{2}u^{T}(K-{\tilde{A}}^{2})u+u^{T}\\!{\tilde{A}}\,p.$
Here, $u$ is a column vector of displacements from lattice equilibrium
positions for all the degrees of freedom, multiplied by the square root of
mass; $p$ is the conjugate momentum vector, and $K$ is the force constant
matrix. The superscript $T$ stands for the matrix transpose. ${\tilde{A}}$ is
an antisymmetric real matrix, which is block diagonal with elements
$\Lambda=\left(\begin{array}[]{rr}0&h\\\ -h&0\\\ \end{array}\right)$ (in two
dimensions), where $h$ is proportional to the magnetic field, and has the
dimension of frequency. For simplicity, we will call $h$ magnetic field later.
According to [9], $h$ is estimated to be $0.1\,{\rm cm}^{-1}\approx 3\times
10^{9}\,{\rm rad\,Hz}$ at a magnetic field $\textbf{\emph{B}}=1\,{\rm T}$ and
a temperature $T=5.45\,{\rm K}$, which is within the possible range of the
coupling strength in ionic insulators [34, 35]. The on-site term,
$u^{T}{\tilde{A}}p$, can be interpreted as the Raman (or spin-phonon)
interaction. Based on quantum theory and symmetry consideration, the
phenomenological description of the spin-phonon interaction was proposed many
years ago [34, 35, 32, 33, 36, 37, 38, 39]. From the first row of Eq. (1), we
find both of the two terms are positive definite, thus the Hamiltonian (1) is
positive definite. The origin of the Hamiltonian for the PHE is discussed in
detail in the supplementary information of Ref. [26].
The Hamiltonian Eq. (1) is quadratic in $u$ and $p$. We can write the linear
equation of motion as
$\displaystyle\dot{p}$ $\displaystyle=$
$\displaystyle-(K-\tilde{A}^{2})u-\tilde{A}p,$ (2) $\displaystyle\dot{u}$
$\displaystyle=$ $\displaystyle p-\tilde{A}u.$ (3)
The equation of motion for the coordinate is,
$\ddot{u}+2\tilde{A}\dot{u}+Ku=0.$ (4)
Since the lattice is periodic, we can apply the Bloch’s theorem
$u_{l}=\epsilon e^{i({\bf R}_{l}\cdot{\bf k}-\omega t)}$. The polarization
vector $\epsilon$ satisfies
$\bigl{[}(-i\omega+A)^{2}+D\bigr{]}\epsilon=0,$ (5)
where $D({\bf k})=-A^{2}+\sum_{l^{\prime}}K_{ll^{\prime}}e^{i({\bf
R}_{l^{\prime}}-{\bf R}_{l})\cdot{\bf k}}$ denotes the dynamic matrix and $A$
is block diagonal with elements $\Lambda$. $D,K_{l,l^{\prime}},$ and $A$ are
all $nd\times nd$ matrices, where $n$ is the number of particles in one unit
cell and $d$ is the dimension of the vibration.
From Eq. (5), we can require the following relations:
$\epsilon_{-k}^{*}=\epsilon_{k};\;\omega_{-k}=-\omega_{k}.$ (6)
Here, we use the short-hand notation $k=({\bf k},\sigma)$ to specify both the
wavevector and the phonon branch, and $-k$ means $(-{\bf k},-\sigma)$. In
normal lattice dynamic treatment, we usually take $\sigma,\omega\geq 0$ as a
convention, and require $\omega_{\sigma,{\bf k}}=\omega_{\sigma,-{\bf k}}$.
For the current problem, this is not true [26, 31]. It is more convenient to
have the frequency taking both positive and negative values and require the
above equation (6). And from Eq. (3), the momentum and displacement
polarization vectors are related through
$\mu_{k}=-i\omega_{k}\epsilon_{k}+A\epsilon_{k}.$ (7)
Equation (5) is not a standard eigenvalue problem. However, we can describe
the system by the polarization vector $x=(\mu,\epsilon)^{T}$, where $\mu$ and
$\epsilon$ are associated with the momenta and coordinates, respectively.
Using Bloch’s theorem, Eqs. (2) and (3) can be recasted as:
$i\frac{\partial}{{\partial t}}x=H_{\rm eff}x,\;\;\;\;\;\;\;\;\;\;H_{\rm
eff}=i\left(\begin{array}[]{cc}-A&-D\\\ I_{nd}&-A\end{array}\right).$ (8)
Here the $I_{nd}$ is the $nd\times nd$ identity matrix. Therefore, the
eigenvalue problem of the equation of motion (8) reads:
$H_{\rm eff}\,x_{k}=\omega_{k}\,x_{k},\;\;\;\;\tilde{x}_{k}^{T}\,H_{\rm
eff}=\omega_{k}\,\tilde{x}_{k}^{T}.$ (9)
where the right eigenvector $x_{k}=(\mu_{k},\epsilon_{k})^{T}$, the left
eigenvector
${\tilde{x}}_{k}^{T}=(\epsilon^{\dagger}_{k},-\mu^{\dagger}_{k})/(-2i\omega_{k})$,
in such choice the second quantization of the Hamiltonian Eq. (1) holds, which
will be proved later. Because the effective Hamiltonian $H_{\rm eff}$ is not
hermitian, the orthonormal condition then holds between the left and right
eigenvectors, as
${\tilde{x}_{\sigma,{\bf k}}}^{T}\;x_{\sigma^{\prime},{\bf
k}}=\delta_{\sigma\sigma^{\prime}}.$ (10)
We also have the completeness relation as
$\sum_{\sigma}x_{\sigma,{\bf k}}\otimes{\tilde{x}_{\sigma,{\bf
k}}}^{T}=I_{2nd}.$ (11)
The normalization of the eigenmodes is equivalent to [11]
$\epsilon_{k}^{\dagger}\,\epsilon_{k}+\frac{i}{\omega_{k}}\epsilon_{k}^{\dagger}\,A\,\epsilon_{k}=1.$
(12)
From the eigenvalue problem Eq. (9), we know that the completed set contains
the branch of the negative frequency. And from the topological nature of the
PHE [26], the formula of the phonon Hall conductivity can be written in the
form comprises the contribution of all the branches including both positive
and negative frequency branches. In order to simplify the notation, for all
the branches, we define
$a_{-k}=a_{k}^{\dagger}.$ (13)
The time dependence of the operators is given by:
$\displaystyle a_{k}(t)$ $\displaystyle=$ $\displaystyle
a_{k}e^{-i\omega_{k}t},$ (14) $\displaystyle a_{k}^{\dagger}(t)$
$\displaystyle=$ $\displaystyle a_{k}^{\dagger}e^{i\omega_{k}t}.$ (15)
The commutation relation is
$[a_{k},a_{k^{\prime}}^{\dagger}]=\delta_{k,k^{\prime}}{\rm sign}(\sigma).$
(16)
And we can get
$\displaystyle\langle a_{k}^{\dagger}a_{k}\rangle$ $\displaystyle=$
$\displaystyle f(\omega_{k}){\rm sign}(\sigma);$ (17) $\displaystyle\langle
a_{k}a_{k}^{\dagger}\rangle$ $\displaystyle=$
$\displaystyle\bigl{[}1+f(\omega_{k})\bigr{]}{\rm sign}(\sigma).$ (18)
Here $f(\omega_{k})=(e^{\hbar\omega_{k}/(k_{B}T)}-1)^{-1}$ is the Bose
distribution function.
The displacement and momentum operators can be written in the following second
quantization forms
$\displaystyle u_{l}$ $\displaystyle=$
$\displaystyle\sum_{k}\epsilon_{k}e^{i{\bf R}_{l}\cdot{\bf
k}}\sqrt{\frac{\hbar}{2N|\omega_{k}|}}\,a_{k};$ (19) $\displaystyle p_{l}$
$\displaystyle=$ $\displaystyle\sum_{k}\mu_{k}e^{i{\bf R}_{l}\cdot{\bf
k}}\sqrt{\frac{\hbar}{2N|\omega_{k}|}}\,a_{k}.$ (20)
Here, $|\omega_{k}|=\omega_{k}{\rm sign}(\sigma)$. We can verify that the
canonical commutation relations are satisfied:
$[u_{l},p_{l^{\prime}}^{T}]=i\hbar\delta_{ll^{\prime}}I_{nd}$ by using the
completeness Eq. (11) and the commutation relation Eq. (16). The Hamiltonian
Eq. (1) then can be written as [31]
$H=\frac{1}{2}\sum_{l,l^{\prime}}\tilde{\chi}^{T}_{l}\left(\begin{array}[]{cc}A\delta_{l,l^{\prime}}&K_{l,l^{\prime}}-A^{2}\delta_{l,l^{\prime}}\\\
-I_{nd}\delta_{l,l^{\prime}}&A\delta_{l,l^{\prime}}\end{array}\right)\chi_{l^{\prime}}$
(21)
where
$\displaystyle\chi_{l}=\left(\begin{array}[]{rr}p_{l}\\\
u_{l}\end{array}\right)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\hbar}{N}}\sum_{k}x_{k}e^{i{\bf R}_{l}\cdot{\bf
k}}c_{k}\;a_{k};$ (24)
$\displaystyle\tilde{\chi}_{l}=\left(\begin{array}[]{rr}u_{l}\\\
-p_{l}\end{array}\right)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\hbar}{N}}\sum_{k}\tilde{x}_{k}e^{-i{\bf
R}_{l}\cdot{\bf k}}\tilde{c}_{k}\;a_{k}^{\dagger}.$ (27)
Here $c_{k}=\sqrt{\frac{1}{2|\omega_{k}|}}$ and
$\tilde{c}_{k}=(-2i\omega_{k})\sqrt{\frac{1}{2|\omega_{k}|}}$. It is easy to
verify that
$[{\chi}_{l},\tilde{\chi}^{T}_{l^{\prime}}]=-i\hbar\delta_{ll^{\prime}}I_{2nd}$.
Because of $e^{i({\bf R}_{l^{\prime}}\cdot{\bf k^{\prime}}-{\bf
R}_{l}\cdot{\bf k})}=e^{i({\bf R}_{l}\cdot({\bf k^{\prime}}-{\bf k})+({\bf
R}_{l^{\prime}}-{\bf R}_{l})\cdot{\bf k^{\prime}})}$ and the definition of the
dynamic matrix $D$, then the Hamiltonian can be written as
$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{\hbar}{2N}\sum_{k,k^{\prime},l}e^{i{\bf R}_{l}\cdot({\bf
k^{\prime}}-{\bf
k})}\tilde{c}_{k}\,c_{k^{\prime}}\tilde{x}_{k}^{T}\left(\begin{array}[]{cc}A&D({\bf
k^{\prime}})\\\
-I_{nd}&A\end{array}\right)x_{k^{\prime}}a_{k}^{\dagger}a_{k^{\prime}}$
$\displaystyle=$ $\displaystyle\frac{\hbar}{2N}\sum_{k,k^{\prime},l}e^{i{\bf
R}_{l}\cdot({\bf k^{\prime}}-{\bf
k})}\tilde{c}_{k}\,c_{k^{\prime}}\tilde{x}_{k}^{T}iH_{\rm
eff}x_{k^{\prime}}a_{k}^{\dagger}a_{k^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{k}\hbar|\omega_{k}|a_{k}^{\dagger}a_{k},$ (31)
which contains both the positive and negative branches. Here we use the
identity $\sum_{l}e^{i{\bf R}_{l}\cdot({\bf k^{\prime}}-{\bf k})}=N\delta_{\bf
k^{\prime}k}$ and the eigenvalue problem Eq. (9). Using the relations Eqs.
(13) and (16), it is easy to prove that Eq. (2.1) is equivalent to the form
$H=\sum_{\sigma>0,{\bf k}}\hbar\omega_{k}(a_{k}^{\dagger}a_{k}+1/2)$ which
only includes the nonnegative branches.
### 2.2 The heat current operator
The heat current density can be computed as [40]:
${\bf J}=\frac{1}{2V}\sum_{l,l^{\prime}}({\bf R}_{l}\\!-\\!{\bf
R}_{l^{\prime}})u^{T}_{l}K_{ll^{\prime}}\dot{u}_{l^{\prime}},$ (32)
where $V$ is the total volume of $N$ unit cells. Because of the equation of
motion Eq. (3), we can rewrite the heat current as
${\bf J}=\frac{1}{4V}\sum_{l,l^{\prime}}\tilde{\chi}^{T}_{l}{\bf
M}_{l\,l^{\prime}}\chi_{l^{\prime}},$ (33)
where
${\bf M}_{l\,l^{\prime}}=\left(\begin{array}[]{cc}({\bf R}_{l}-{\bf
R}_{l^{\prime}})K_{ll^{\prime}}&-({\bf R}_{l}-{\bf
R}_{l^{\prime}})(K_{ll^{\prime}}A+AK_{ll^{\prime}})\\\ 0&({\bf R}_{l}-{\bf
R}_{l^{\prime}})K_{ll^{\prime}}\end{array}\right)$ (34)
Inserting the Eqs. (24,27), we obtain
${\bf
J}=\frac{\hbar}{4VN}\sum_{k,k^{\prime},l,l^{\prime}}\tilde{c}_{k}c_{k^{\prime}}e^{i({\bf
R}_{l^{\prime}}\cdot{\bf k^{\prime}}-{\bf R}_{l}\cdot{\bf
k})}\tilde{x}^{T}_{k}{\bf
M}_{l\,l^{\prime}}x_{k^{\prime}}a_{k}^{\dagger}a_{k^{\prime}},$ (35)
Because of
$\sum_{l}e^{i{\bf R}_{l}\cdot({\bf k^{\prime}}-{\bf
k})}\sum_{l^{\prime}}e^{i({\bf R}_{l^{\prime}}-{\bf R}_{l})\cdot{\bf
k^{\prime}}}({\bf R}_{l}-{\bf R}_{l^{\prime}})K_{ll^{\prime}}=iN\delta_{\bf
k^{\prime}k}\frac{\partial D}{\partial{\bf k^{\prime}}},$ (36)
the heat current can be written as
${\bf J}=\frac{i\hbar}{4V}\sum_{\sigma,\sigma^{\prime},{\bf
k}}\tilde{c}_{\sigma,{\bf k}}c_{\sigma^{\prime},{\bf
k}}\tilde{x}^{T}_{\sigma,{\bf k}}\frac{\partial H_{\rm eff}^{2}}{\partial{\bf
k}}x_{\sigma^{\prime},{\bf k}}a_{\sigma,{\bf
k}}^{\dagger}a_{\sigma^{\prime},{\bf k}},$ (37)
here we use
$\frac{\partial H_{\rm eff}^{2}}{\partial{\bf
k}}=\left(\begin{array}[]{cc}\frac{\partial D}{\partial{\bf
k}}&-(A\frac{\partial D}{\partial{\bf k}}+\frac{\partial D}{\partial{\bf
k}}A)\\\ 0&\frac{\partial D}{\partial{\bf k}}\end{array}\right)$ (38)
by making the first derivative of the square of the effective Hamiltonian Eq.
(8) with respect to the wave vector ${\bf k}$. From the eigenvalue problem Eq.
(9), we have
$H_{\rm eff}X=X\Omega;\;\;\tilde{X}^{T}H_{\rm eff}=\Omega\tilde{X}^{T}.$ (39)
Where the ${2nd}\times{2nd}$ matrices
$X=(x_{1},x_{2},...,x_{2nd})=\\{x_{\sigma}\\}$ (the system has ${2nd}$ phonon
branches), $\tilde{X}=\\{\tilde{x}_{\sigma}\\}$, and $\Omega={\rm
diag}(\omega_{1},\omega_{2},...,\omega_{2nd})=\\{\omega_{\sigma}\\}$. Because
of the completeness relation Eq. (11), $X\tilde{X}^{T}=I_{2nd}$, we get
$H_{\rm eff}^{2}=X\Omega^{2}\tilde{X}^{T}.$ (40)
By calculating the derivative of the above equation, and using the definition
of Berry connection,
${\bf\mathcal{A}}=\tilde{X}^{T}\frac{\partial X}{\partial{\bf k}}.$ (41)
Taking the first derivative of Eq. (40) with respect to ${\bf k}$, we obtain
$\frac{\partial H_{\rm eff}^{2}}{\partial{\bf
k}}=X\left(\frac{\partial\Omega^{2}}{\partial{\bf
k}}+[{\bf\mathcal{A}},\Omega^{2}]\right)\tilde{X}^{T}.$ (42)
Because of the orthogonality relation between left and right eigenvector Eq.
(10), at last we obtain the heat current as
${\bf J}=\frac{i\hbar}{4V}\sum_{\sigma,\sigma^{\prime},{\bf
k}}\tilde{c}_{\sigma,{\bf k}}c_{\sigma^{\prime},{\bf k}}a_{\sigma,{\bf
k}}^{\dagger}\left(\frac{\partial\Omega^{2}}{\partial{\bf
k}}+[{\bf\mathcal{A}},\Omega^{2}]\right)_{\sigma,\sigma^{\prime}}a_{\sigma^{\prime},{\bf
k}}.$ (43)
The first term $\frac{\partial\Omega^{2}}{\partial{\bf k}}$ in the bracket is
a diagonal one corresponding to
$\omega_{\sigma}\frac{\partial\omega_{\sigma}}{\partial{\bf k}}$ relating the
group velocity. The second term in the bracket $[{\bf\mathcal{A}},\Omega^{2}]$
gives the off-diagonal elements of the heat current density, which can be
regarded as the contribution from anomalous velocities similar to the one in
the intrinsic anomalous Hall effect. The Berry connection ${\bf\mathcal{A}}$,
or we can call it Berry vector potential matrix (the Berry vector potential
defined in Ref. [26], ${\bf A}^{\sigma}({\bf k})$, is equal to
$i{\bf\mathcal{A}}^{\sigma\sigma}=i\tilde{x}_{\sigma}^{T}\frac{\partial
x_{\sigma}}{\partial{\bf k}}$), induces the anomalous velocities to the heat
current, which will take the responsibility of the PHE. Therefore, the Berry
vector potential comes naturally into the heat current and the PHE. Such a
picture is clearer than that in Ref. [26].
### 2.3 The phonon Hall conductivity
Inserting the coefficients $\tilde{c}$ and $c$ to Eq. (43), we get
${\bf J}=\frac{\hbar}{4V}\sum_{\sigma,\sigma^{\prime},{\bf
k}}\frac{\omega_{\sigma,{\bf k}}}{\sqrt{|\omega_{\sigma,{\bf
k}}\omega_{\sigma^{\prime},{\bf k}}|}}a_{\sigma,{\bf
k}}^{\dagger}\left(\frac{\partial\Omega^{2}}{\partial{\bf
k}}+[{\bf\mathcal{A}},\Omega^{2}]\right)_{\sigma,\sigma^{\prime}}a_{\sigma^{\prime},{\bf
k}}.$ (44)
This expression is equivalent to that given in Refs. [26] and [31]. Based on
such expression of heat current, the phonon Hall conductivity can be obtained
through the Green-Kubo formula [41]:
$\kappa_{xy}=\frac{V}{\hbar
T}\int_{0}^{\hbar/(k_{B}T)}\\!\\!\\!\\!d\lambda\int_{0}^{\infty}\\!dt\,\bigl{\langle}J^{x}(-i\lambda)J^{y}(t)\bigr{\rangle}_{\rm
eq},$ (45)
where the average is taken over the equilibrium ensemble with Hamiltonian $H$.
The time dependence of the creation and annihilation operators are given as
Eqs. (14) and (15), which are also true if $t$ is imaginary. From the Wick
theorem, we have
$\displaystyle\langle a_{\sigma,{\bf k}}^{\dagger}a_{\sigma^{\prime},{\bf
k}}a_{\bar{\sigma},{\bf\bar{k}}}^{\dagger}a_{\bar{\sigma^{\prime}},{\bf\bar{k}}}\rangle$
$\displaystyle=$ $\displaystyle\langle a_{\sigma,{\bf
k}}^{\dagger}a_{\sigma^{\prime},{\bf k}}\rangle\langle
a_{\bar{\sigma},{\bf\bar{k}}}^{\dagger}a_{\bar{\sigma^{\prime}},{\bf\bar{k}}}\rangle$
(46) $\displaystyle+$ $\displaystyle\langle a_{\sigma,{\bf
k}}^{\dagger}a_{\bar{\sigma},{\bf\bar{k}}}^{\dagger}\rangle\langle
a_{\sigma^{\prime},{\bf k}}a_{\bar{\sigma^{\prime}},{\bf\bar{k}}}\rangle$
$\displaystyle+$ $\displaystyle\langle a_{\sigma,{\bf
k}}^{\dagger}a_{\bar{\sigma^{\prime}},{\bf\bar{k}}}\rangle\langle
a_{\sigma^{\prime},{\bf k}}a_{\bar{\sigma},{\bf\bar{k}}}^{\dagger}\rangle.$
Using the properties of the operators $a^{\dagger}$ and $a$ as Eq. (17), we
have
$\begin{array}[]{ll}\langle a_{\sigma,{\bf
k}}^{\dagger}a_{\sigma^{\prime},{\bf k}}\rangle\langle
a_{\bar{\sigma},{\bf\bar{k}}}^{\dagger}a_{\bar{\sigma^{\prime}},{\bf\bar{k}}}\rangle&\\\
=f(\omega_{\sigma,{\bf
k}})f(\omega_{\bar{\sigma},{\bf\bar{k}}})\delta_{\sigma\sigma^{\prime}}\delta_{\bar{\sigma}\bar{\sigma^{\prime}}}{\rm
sign}(\sigma){\rm sign}(\bar{\sigma}),&\\\ \langle a_{\sigma,{\bf
k}}^{\dagger}a_{\bar{\sigma},{\bf\bar{k}}}^{\dagger}\rangle\langle
a_{\sigma^{\prime},{\bf k}}a_{\bar{\sigma^{\prime}},{\bf\bar{k}}}\rangle&\\\
=f(\omega_{\sigma,{\bf k}})(f(\omega_{\sigma^{\prime},{\bf
k}})+1)\delta_{\bf\bar{k},-k}\delta_{\sigma,-\bar{\sigma}}\delta_{\sigma^{\prime},-\bar{\sigma^{\prime}}}{\rm
sign}(\sigma){\rm sign}(\sigma^{\prime})&\\\ \langle a_{\sigma,{\bf
k}}^{\dagger}a_{\bar{\sigma^{\prime}},{\bf\bar{k}}}\rangle\langle
a_{\sigma^{\prime},{\bf k}}a_{\bar{\sigma},{\bf\bar{k}}}^{\dagger}\rangle&\\\
=f(\omega_{\sigma,{\bf k}})(f(\omega_{\sigma^{\prime},{\bf
k}})+1)\delta_{\bf\bar{k},k}\delta_{\sigma,\bar{\sigma^{\prime}}}\delta_{\sigma^{\prime},\bar{\sigma}}{\rm
sign}(\sigma){\rm sign}(\sigma^{\prime}).&\end{array}$ (47)
Similar as that in Ref. [26], the diagonal term
$\frac{\partial\Omega^{2}}{\partial{\bf k}}$ in the bracket corresponding to
$\omega_{\sigma}\frac{\partial\omega_{\sigma}}{\partial{\bf k}}$ has no
contribution to the phonon Hall conductivity because which is an odd function
of ${\bf k}$. Because of the off-diagonal term
$[\mathcal{A}_{k_{\alpha}},\Omega^{2}]_{\sigma,\sigma^{\prime}}=(\omega_{\sigma^{\prime}}^{2}-\omega_{\sigma}^{2})\mathcal{A}_{k_{\alpha}}^{\sigma\sigma^{\prime}}$
(48)
and
$\mathcal{A}_{k_{\alpha}}^{\sigma\sigma^{\prime}}=\tilde{x}_{\sigma}^{T}\frac{\partial
x_{\sigma^{\prime}}}{\partial{k_{\alpha}}}$ from the definition, the phonon
Hall conductivity can be written as
$\displaystyle\kappa_{xy}$ $\displaystyle=$
$\displaystyle\frac{\hbar}{{8VT}}\sum_{{\bf
k},\sigma,\sigma^{\prime}\neq\sigma}[f(\omega_{\sigma})-f(\omega_{\sigma^{\prime}})](\omega_{\sigma}+\omega_{\sigma^{\prime}})^{2}$
(49)
$\displaystyle\times\frac{i}{4\omega_{\sigma}\omega_{\sigma^{\prime}}}\frac{{\epsilon_{\sigma}^{\dagger}\frac{{\partial
D}}{{\partial
k_{x}}}\epsilon_{\sigma^{\prime}}\epsilon_{\sigma^{\prime}}^{\dagger}\frac{{\partial
D}}{{\partial
k_{y}}}\epsilon_{\sigma}}}{{(\omega_{\sigma}-\omega_{\sigma^{\prime}})^{2}}}.$
Here we simplify the notation of the subscripts of $\omega,\epsilon$ which
have the same wave vector ${\bf k}$. We can prove $\kappa_{xy}=-\kappa_{yx}$,
such that
$\kappa_{xy}=\frac{\hbar}{{16VT}}\sum_{{\bf
k},\sigma,\sigma^{\prime}\neq\sigma}{[f(\omega_{\sigma})-f(\omega_{\sigma^{\prime}})](\omega_{\sigma}+\omega_{\sigma^{\prime}})^{2}B_{k_{x}k_{y}}^{\sigma\sigma^{\prime}}},$
(50)
here
$\displaystyle B_{k_{x}k_{y}}^{\sigma\sigma^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{i}{{4\omega_{\sigma}\omega_{\sigma^{\prime}}}}\frac{\epsilon_{\sigma}^{\dagger}\frac{{\partial
D}}{{\partial
k_{x}}}\epsilon_{\sigma^{\prime}}\epsilon_{\sigma^{\prime}}^{\dagger}\frac{{\partial
D}}{{\partial k_{y}}}\epsilon_{\sigma}-(k_{x}\leftrightarrow
k_{y})}{(\omega_{\sigma}-\omega_{\sigma^{\prime}})^{2}}.$ (51)
$\displaystyle=$ $\displaystyle i\frac{\tilde{x}_{\sigma}^{T}\frac{{\partial
H_{\rm eff}}}{{\partial
k_{x}}}x_{\sigma^{\prime}}\tilde{x}_{\sigma^{\prime}}^{T}\frac{{\partial
H_{\rm eff}}}{{\partial k_{y}}}x_{\sigma}-(k_{x}\leftrightarrow
k_{y})}{(\omega_{\sigma}-\omega_{\sigma^{\prime}})^{2}}.$ $\displaystyle=$
$\displaystyle-i\left(\mathcal{A}_{k_{x}}^{\sigma\sigma^{\prime}}\mathcal{A}_{k_{y}}^{\sigma^{\prime}\sigma}-(k_{x}\leftrightarrow
k_{y})\right),$
in the last step we use the relation $\tilde{x}_{\sigma}^{T}\frac{{\partial
H_{\rm eff}}}{{\partial
k_{x}}}x_{\sigma^{\prime}}=(\omega_{\sigma^{\prime}}-\omega_{\sigma})\tilde{x}_{\sigma}^{T}\frac{{\partial}}{{\partial
k_{x}}}x_{\sigma^{\prime}}$ and the definition of ${\bf\mathcal{A}}$ in Eq.
(41). And the Berry curvature is
$\displaystyle B_{k_{x}k_{y}}^{\sigma}$ $\displaystyle=$
$\displaystyle\sum_{\sigma^{\prime}\neq\sigma}B_{k_{x}k_{y}}^{\sigma\sigma^{\prime}}$
(52) $\displaystyle=$
$\displaystyle-i\sum_{\sigma^{\prime}}\left(\mathcal{A}_{k_{x}}^{\sigma\sigma^{\prime}}\mathcal{A}_{k_{y}}^{\sigma^{\prime}\sigma}-(k_{x}\leftrightarrow
k_{y})\right)$ $\displaystyle=$ $\displaystyle i\left(\frac{\partial}{\partial
k_{x}}\mathcal{A}_{k_{y}}^{\sigma\sigma}-(k_{x}\leftrightarrow k_{y})\right)$
The definition of Berry curvature here is the same as that of Ref. [26], that
is, $B_{k_{x}k_{y}}^{\sigma}=\frac{\partial}{{\partial k_{x}}}{\bf
A}_{k_{y}}^{\sigma}-\frac{\partial}{{\partial k_{y}}}{\bf
A}_{k_{x}}^{\sigma}$. From the above derivation, we find that a Berry
curvature can be defined uniquely for each band by looking at the phases of
the polarized vectors of both the displacements and conjugate momenta as
functions of the wave vector. If we only look at the polarized vector
$\epsilon$ of the displacement, a Berry curvature cannot properly be defined.
We need both $\epsilon$ and $\mu$. The nontrivial Berry vector potential takes
the responsibility of the PHE. The associated topological Chern number is
obtained through integrating the Berry curvature over the first Brillouin zone
as
$C^{\sigma}=\frac{1}{{2\pi}}\int_{{\rm
BZ}}{dk_{x}dk_{y}B_{k_{x}k_{y}}^{\sigma}}=\frac{{2\pi}}{{L^{2}}}\sum\limits_{\bf
k}{B_{k_{x}k_{y}}^{\sigma}},$ (53)
where, $L$ is the length of the sample.
## 3 Application on the kagome lattice
Figure 1: (color online) The schematic picture of kagome lattice. Each unit
cell has three atoms such as the number shown 1,2,3. The coupling between the
atoms are $K_{01},K_{02},K_{03}$. Each unit cell has six nearest neighbors;
the coupling between the unit cell and the neighbors are
$K_{1},K_{2},...,K_{6}$.
In Ref. [26], we provide a topological understanding of the PHE in dielectrics
with Raman spin-phonon coupling for the honeycomb lattice structure. Because
of the nature of phonons, the phonon Hall conductivity, which is not directly
proportional to the Chern number, is not quantized. We observed a phase
transition in the PHE, which corresponds to the sudden change of band
topology, characterized by the altering of integer Chern numbers. Such PHE can
be explained by touching and splitting of phonon bands. To check whether the
mechanism of the PHE is universal, in the following we apply the theory to the
kagome lattice, which has been used to model many real materials [30].
### 3.1 Calculation of the dynamic matrix $D$
In order to calculate the phonon Hall conductivity, we first need to calculate
the dynamic matrix $D({\bf k})$, for the two-dimensional kagome lattice. As
shown in Fig. 1, each unit cell has three atoms, thus $n=3$. We only consider
the nearest neighbor interaction. The spring constant matrix along $x$
direction is assumed as
$K_{x}=\left(\begin{array}[]{cc}K_{L}&0\\\ 0&K_{T}\\\ \end{array}\right).$
(54)
$K_{L}=0.144\,$eV/(uÅ2) is the longitudinal spring constant and the transverse
one $K_{T}$ is 4 times smaller. The unit cell lattice vectors are $(a,0)$ and
$(a/2,a\sqrt{3}/2)$ with $a=1\,$Å.
To obtain the explicit formula for the dynamic matrix, we first define a
rotation operator in two dimensions as:
$U(\theta)=\left({\begin{array}[]{*{20}c}{\cos\theta}&{-\sin\theta}\\\
{\sin\theta}&{\cos\theta}\\\ \end{array}}\right).$
The three kinds of spring-constant matrices between two atoms are
$K_{01}=K_{x}$ (between atoms 1 and 2 in Fig. 1),
$K_{02}=U(\pi/3)K_{x}U(-\pi/3)$ (between atoms 2 and 3),
$K_{03}=U(-\pi/3)K_{x}U(\pi/3)$ (between atoms 3 and 1), which are $2\times 2$
matrices. Then we can obtain the on-site spring-constant matrix and the six
spring-constant matrices between the unit cell and its nearest neighbors as:
$K_{0}=\left({\begin{array}[]{ccc}2(K_{01}+K_{02})&-K_{01}&-K_{02}\\\
-K_{01}&2(K_{01}+K_{03})&-K_{03}\\\
-K_{02}&-K_{03}&2(K_{02}+K_{03})\end{array}}\right),$ $\displaystyle K_{1}$
$\displaystyle=$ $\displaystyle\left({\begin{array}[]{ccc}0&0&0\\\
-K_{01}&0&0\\\
0&0&0\end{array}}\right),\;K_{2}=\left({\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
-K_{02}&0&0\end{array}}\right),$ (61) $\displaystyle K_{3}$ $\displaystyle=$
$\displaystyle\left({\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
0&-K_{03}&0\end{array}}\right),\;K_{4}=\left({\begin{array}[]{ccc}0&-K_{01}&0\\\
0&0&0\\\ 0&0&0\end{array}}\right),$ (68) $\displaystyle K_{5}$
$\displaystyle=$ $\displaystyle\left({\begin{array}[]{ccc}0&0&-K_{02}\\\
0&0&0\\\ 0&0&0\end{array}}\right),\;K_{6}=\left({\begin{array}[]{ccc}0&0&0\\\
0&0&-K_{03}\\\ 0&0&0\end{array}}\right),$ (75)
which are $6\times 6$ matrices. Finally we can obtain the $6\times 6$ dynamic
matrix $D({\bf k})$ as
$\displaystyle D({\bf k})$ $\displaystyle=$
$\displaystyle-A^{2}+K_{0}+K_{1}e^{ik_{x}}+K_{2}e^{i(\frac{k_{x}}{2}+\frac{\sqrt{3}k_{y}}{2})}$
(76)
$\displaystyle+K_{3}e^{i(-\frac{k_{x}}{2}+\frac{\sqrt{3}k_{y}}{2})}+K_{4}e^{-ik_{x}}$
$\displaystyle+K_{5}e^{i(-\frac{k_{x}}{2}-\frac{\sqrt{3}k_{y}}{2})}+K_{6}e^{i(\frac{k_{x}}{2}-\frac{\sqrt{3}k_{y}}{2})},$
where, $A^{2}=-h^{2}\cdot I_{6}$, here $I_{6}$ is the $6\times 6$ identity
matrix.
Figure 2: (color online) The contour map of dispersion relations for the
positive frequency bands. For all the insets, the horizontal and vertical axes
correspond to wave vector $k_{x}$ and $k_{y}$, respectively. The upper six
insets are the dispersion relations for bands 1 to 6 (from left to right) at
$h=0$, respectively. And $h=10$ rad/ps for the lower ones.
### 3.2 The PHE and the associated phase transition
After we get the expression for the dynamic matrix, we can calculate the
eigenvalues and eigenvectors of the effective Hamiltonian. Inserting the
eigenvalues, eigenvectors and the $D$ matrix to the formula Eq. (50), we are
able to compute the phonon Hall conductivity. As is well known, in quantum
Hall effect for electrons, the Hall conductivity is just the Chern number in
units of $e^{2}/h$ ($h$ is the Planck constant); thus with the varying of
magnetic field, the abrupt change of Chern numbers directly induces the
obvious discontinuity of the Hall conductivity. However, for the PHE, there is
an extra weight of $(\omega_{\sigma}+\omega_{\sigma^{\prime}})^{2}$ in Eq.
(50), which can not be moved out from the summation. As a consequence, the
change of phonon Hall conductivity is smoothened at the critical magnetic
field. However, in the study on the PHE in the honeycomb lattice system [26],
from the first derivative of phonon Hall conductivity with respect to the
magnetic field $h$, at the critical point $h_{c}$, we still can observe the
divergence (singularity) of $d\kappa_{xy}/dh$, where the phase transition
occurs corresponding to the sudden change of the Chern numbers. Can such
mechanism be applied for the kagome lattice system? In the following, we give
a detailed discussion on it.
Inserting the dynamic matrix Eq. (76) to the effective Hamiltonian Eq. (9), we
calculate eigenvalues and eigenvectors of the system, and also get the
dispersion relation of the system. Because each unit cell has three atoms, and
we only consider the two-dimensional motion, we get six phonon branches with
positive frequencies. The branches with negative frequencies have similar
behavior because of $\omega_{-k}=-\omega_{k}$. We show the contour map of the
dispersion relation in Fig. 2. We can see that the dispersion relations have a
6-fold symmetry. For different bands, they are different. With a changing
magnetic field, the dispersion relations vary. The point ${\bf\Gamma}$ (${\bf
k}=(0,0)$) is the 6-fold symmetric center; the point ${\bf K}$ (${\bf
k}=(\frac{4\pi}{3},0)$) is 3-fold symmetric center; and the middle point of
the line between two 6-fold symmetric centers, ${\bf X}$ (${\bf
k}=(\pi,\frac{\sqrt{3}\pi}{3})$) is a 2-fold symmetric center. In the
following discussion, we will see the possible bands touching at these
symmetric centers.
Figure 3: (color online) The phonon Hall conductivity vs magnetic field at
different temperatures. The inset is the zoom-in curve of the phonon Hall
conductivity at weak magnetic field. Here the sample size $N_{L}$=400.
Using the formula Eq. (50), we calculate the phonon Hall conductivity of the
kagome lattice systems, the results are shown in Fig. 3. Similar as shown in
Ref. [26], we find the nontrivial behavior of the phonon Hall conductivity as
a function of the magnetic field. When $h$ is small, $\kappa_{xy}$ is
proportional to $h$, which is shown in the inset of Fig.3; while the
dependence becomes nonlinear when $h$ is large. As $h$ is further increased,
the magnitude of $\kappa_{xy}$ increases before it reaches a maximum magnitude
at certain value of $h$. Then the magnitude of $\kappa_{xy}$ decreases and
goes to zero at very large $h$. The on-site term $\tilde{A}^{2}$ in the
Hamiltonian (1) increases with $h$ quadratically so as to blockade the phonon
transport, which competes with the spin-phonon interaction. Because of the
coefficient of $f(\omega_{\sigma})$ in the summation of the formula Eq. (50),
the sign of the Hall conductivity will change with temperatures, which is
clearly shown in the inset of Fig.3. While the phonon hall conductivity at
weak magnetic field is always positive for the honeycomb lattice, the sign
reverse of the phonon Hall conductivity with temperature for the kagome
lattices is novel and interesting, which could be verified by future
experimental measurements.
Figure 4: (color online) The Chern numbers and the phonon Hall conductivity
vs magnetic field. The dashed line and the dotted line correspond to the Chern
numbers of phonon bands 2 and 3 (left scale). The solid line correspond to the
phonon Hall conductivity (right scale) at $T=50$ K.
We plot the curves of the Chern numbers of bands 2 and 3 as a function of the
magnetic field in Fig. 4. The phonon Hall conductivity at $T=50K$ is also
shown for comparison. To calculate the integer Chern numbers, large number of
${\bf k}$-sampling points $N$ is needed. However there is always a zero
eigenvalue at the ${\bf\Gamma}$ point of the dispersion relation, which
corresponds to a singularity of the Berry curvature. Therefore, we cannot sum
up the Berry curvature very near this point to obtain Chern number of this
band, unless we add a negligible on-site potential
$\frac{1}{2}u^{T}V_{\mathrm{onsite}}u$ to the original Hamiltonian [26], which
will not change the topology of the space of the eigenvectors. In Fig. 4, we
set $V_{\mathrm{onsite}}=10^{-3}K_{L}$. The Chern numbers of bands 2 and 3
have three jumps with the increasing of the magnetic field, although the
phonon Hall conductivity is continuous. For other bands, the Chern numbers
keep constant: $C^{1}=C^{4}=-1$, $C^{5}=0$, and $C^{6}=1$. For the electronic
Hall effect, we know it is quantized because the Hall conductivity is directly
proportional to the quantized Chern numbers. Here we also find the quantized
effect of the Chern numbers from Fig. 3, while there is no quantized effect
for the phonon Hall conductivity. Such difference of the PHE from the
electronic Hall effect comes from the different nature of the phonons
respective to the electrons. In Eq. (50), in the summation, an extra term
$(\omega_{\sigma}+\omega_{\sigma^{\prime}})^{2}$ relating to the phonon energy
which is an analog of the electrical charge term $e^{2}$ in the electron Hall
effect, can not be moved out from the summation. Combining the Bose
distribution, the term
$f(\omega_{\sigma})(\omega_{\sigma}+\omega_{\sigma^{\prime}})^{2}$ make the
phonon Hall conductivity smooth, no discontinuity comes out although the Chern
numbers have some sudden jumps. From the discussion in Ref. [26], the
discontinuity of the Chern numbers corresponds to the phase transitions and
would relate to the divergency of derivative of the phonon Hall conductivity.
Figure 5: (color online) The first derivative of the phonon Hall conductivity
$dk_{xy}/dh$ at $T=50K$ and the Chern numbers of bands 2 and 3 in the vicinity
of the magnetic fields. The solid line correspond to the $dk_{xy}/dh$ at
$T=50$K (left scale); the dashed and dotted lines correspond to the Chern
numbers of bands 2 and 3, respectively (right scale). The inset shows the
second derivative with respective to the magnetic field $dk_{xy}^{2}/dh^{2}$
(vertical axis) vs magnetic field $h$ (horizontal axis) at $T=50$ K.
Figure 6: (color online) The dispersion relations around the critical
magnetic fields. (a), (b), and (c) show the dispersion relations along the
direction from ${\bf\Gamma}$ (${\bf k}$=(0,0)) to ${\bf K}$ (${\bf
k}=(\frac{4\pi}{3},0)$) and to ${\bf X}$ (${\bf
k}=(\pi,\frac{\sqrt{3}\pi}{3})$) at the critical magnetic fields
$h_{c1}=5.07$rad/ps, $h_{c2}=6.75$rad/ps, and $h_{c3}=20.39$rad/ps,
respectively. (d)-(f) show the contour maps of the dispersion relation of band
2 at the three critical magnetic fields. (g)-(i) show the contour maps of the
dispersion relation of band 3 at the three critical magnetic fields. The
squares with number 1, 2, and 3 are marked for the touching points. In (d),
(g) and (e), (h), we only mark one of the six symmetric points by squares of
number 1 and 2 for simplicity.
Figure 5 shows the curves of the derivative of the phonon Hall conductivity
and the Chern numbers at the critical magnetic fields. The first derivative of
phonon Hall conductivity has a minimum or maximum at the magnetic fields
$h_{c1}=5.07,h_{c2}=6.75,{\rm and}h_{c3}=20.39$ rad/ps for the finite-size
sample (the sample has $N=N_{L}^{2}$ unit cells). The first derivative
$d\kappa_{xy}/dh$ at the points $h_{c1},h_{c2},h_{c3}$ diverges when the
system size increases to infinity [26]. At the three critical points the
second derivative $d^{2}\kappa_{xy}/dh^{2}$ is discontinuous, which is shown
in the inset of Fig. 5, across which phase transitions occur. For different
temperatures, the phase transitions occur at exactly the same critical values.
Thus the temperature-independent phase transition does not come from the
thermodynamic effect, but is induced by the topology of the phonon band
structure, which corresponds to the sudden change of the Chern numbers. While
there is one discontinuity of the Chern numbers for the honeycomb lattice
system, for the kagome lattice system, there are three ones corresponding to
the divergency of the derivative of the phonon conductivity, which can be seen
in Fig. 5.
The touching and splitting of the phonon bands near the critical magnetic
field induces the abrupt change of Chern numbers of the phonon band [26]. In
Ref. [26], for the PHE in the honeycomb lattices, we know that band $2$ and
$3$ are going to touch with each other at the ${\bf\Gamma}$ point if the
magnetic field increases to $h_{c}$; at the critical magnetic field, the
degeneracy occurs and the two bands possess the cone shape; above the critical
point $h_{c}$, the two bands split up. Therefore, the difference between the
two bands decreases below and increases above the critical magnetic field, and
is zero at the critical point. The eigenfrequecy difference is in the
denominator of the Berry curvature, thus the variation of the difference
around the critical magnetic field, dramatically affects the Berry curvature
of the corresponding bands. In the kagome lattice systems, we find that the
touching and splitting of the phonon bands not only occurs at the
${\bf\Gamma}$ point, but also occurs at other points, which is shown in Fig.
6. At the first critical points $h_{c1}$, the bands 2 and 3 touch at the point
${\bf K}$ (marked by a square with number 1); at $h_{c2}$ the two bands touch
at ${\bf X}$ ( marked by a square with number 2); while only for the third
critical one $h_{c3}$, band 2 and 3 degenerate at the point ${\bf\Gamma}$
(marked by a square with number 3). From the contour maps of bands 2 and 3, we
clearly see that the critical magnetic fields $h_{c1}$, $h_{c2}$, and
$h_{c3}$, there are local maximum for the band 2 and the local minimum for the
band 3. Thus for all the critical magnetic fields where the Chern numbers have
abrupt changes, in the wave-vector space we can always find the phonon bands
touching and splitting at some symmetric center points.
Therefore, through the study of the PHE in both honeycomb lattices [26] and
kagome lattices, we find discontinuous jumps in Chern numbers, which manifest
themselves as singularities of the first derivative of the phonon Hall
conductivity with respect to the magnetic field. Such associated phase
transition is connected with the crossing of band 2 and band 3, which
corresponds to the touching between a acoustic band and a optical band.
However, we can not observe the similar associated phase transition in
triangular lattices because of no optical bands, where the Chern number of
each band keeps zero while the phonon Hall conductivity is nonzero because of
the nonzero Berry curvatures.
## 4 Conclusion
We present a new systematic theory of the PHE in the ballistic crystal lattice
system, and give an example application of the PHE in the kagome lattice which
is a model structure of the many real magnetic materials. By the proper second
quantization for the Hamiltonian, we obtain the formula for the heat current
density, which considers all the phonon bands including both positive and
negative frequencies. The heat current density can be divided into two parts,
one is the diagonal, another is off-diagonal. The diagonal part corresponds to
the normal velocity; and the off-diagonal part corresponds to the anomalous
velocity which is induced by the Berry vector potential. Such anomalous
velocity induces the PHE in the crystal lattice. Based on such heat current
density we derive the formula of the phonon Hall conductivity which is in
terms of the Berry curvatures. From the application on the kagome lattices, we
find that at weak magnetic field, the phonon Hall conductivity changes sign
with varying temperatures. It is also found that the mechanism on the PHE
about the relation between the phonon Hall conductivity, Chern numbers and the
phonon band structure can be generally appllied for kagome lattices. While
there is only one discontinuity in PHE of the honeycomb lattices, in the
kagome lattices there are three singularities induced by the abrupt change of
the phonon band topology, which correspond to the touching and splitting at
three different symmetric center points in the wave-vector space.
## Acknowledgements
L.Z. thanks Bijay Kumar Agarwalla for fruitful discussions. J.R. acknowledges
the helpful communication with Hosho Katsura. This project is supported in
part by Grants No. R-144-000-257-112 and No. R-144-000-222-646 of NUS.
## References
## References
* [1] Wang L and Li B, Physics World 21, No.3, 27 (2008).
* [2] Wang J-S, Wang J, and Lü J T, Eur. Phys. J. B 62, 381 (2008).
* [3] Li B, Wang L, and Casati G, Phys. Rev. Lett. 93 184301 (2004); Chang C W, Okawa D, Majumdar A, and Zettl A, Science 314, 1121 (2006).
* [4] Li B, Wang L and Casati G, Appl. Phys. Lett. 88, 143501 (2006).
* [5] Wang L and Li B, Phys. Rev. Lett. 99, 177208 (2007).
* [6] Wang L and Li B, Phys. Rev. Lett. 101, 267203 (2008).
* [7] Strohm C, Rikken G L J A, and Wyder P, Phys. Rev. Lett. 95, 155901 (2005).
* [8] Inyushkin A V and Taldenkov A N, JETP Lett. 86, 379 (2007).
* [9] Sheng L, Sheng D N, and Ting C S, Phys. Rev. Lett. 96, 155901 (2006).
* [10] Kagan Y and Maksimov L A, Phys. Rev. Lett. 100, 145902 (2008).
* [11] Wang J-S and Zhang L, Phys. Rev. B 80, 012301 (2009).
* [12] Zhang L, Wang J-S, and Li B, New J. Phys. 11, 113038 (2009).
* [13] Berry M V, Proc. R. Soc. Lond. A 392, 45 (1984).
* [14] Xiao D, Chang M-C, and Niu Q, Rev. Mod. Phys. 82, 1959 (2010).
* [15] Thouless D J, Kohmoto M, Nightingale M P, and den Nijs M, Phys. Rev. Lett. 49, 405 (1982).
* [16] Kohmoto M, Ann. Phys. 160, 343 (1985).
* [17] Nagaosa N, Sinova J, Onoda S, MacDonald A H, and Ong N P, Rev. Mod. Phys. 82, 1539 (2010).
* [18] Fang Z, Nagaosa N, Takahashi K S, Asamitsu A, Mathieu R, Ogasawara T, Yamada H, Kawasaki M, Tokura Y, and Terakura K, Science 302, 92 (2003).
* [19] Xiao D, Yao Y, Fang Z, and Niu Q, Phys. Rev. Lett. 97, 026603 (2006).
* [20] Sheng D N, Weng Z Y, Sheng L, and Haldane F D M, Phys. Rev. Lett. 97, 036808 (2006).
* [21] Koenig M, Buhmann H, Molenkamp L W, Hughes T, Liu C-X, Qi X-L, and Zhang S-C, J. Phys. Soc. Jpn. 77, 031007 (2008).
* [22] Prodan E and Prodan C, Phys. Rev. Lett. 103, 248101 (2009).
* [23] Berg N, Joel K, Koolyk M, and Prodan E, Phys. Rev. E 83, 021913 (2011).
* [24] Ren J, Hänggi P, and Li B, Phys. Rev. Lett. 104, 170601 (2010).
* [25] Lü J-T, Brandbyge M, and Hedegård P, Nano Lett. 10, 1657 (2010).
* [26] Zhang L, Ren J, Wang J-S, and Li B, Phys. Rev. Lett. 105, 225901 (2010).
* [27] Katsura H, Nagaosa N, and Lee P A, Phys. Rev. Lett. 104, 066403 (2010); Onose Y, Ideue T, Katsura H, Shiomi Y, Nagaosa N, Tokura Y, Science 329, 297 (2010).
* [28] Holz A, Il Nuovo Cimento B 9, 83 (1972).
* [29] Mekata M, Physics Today 56, 12(2003).
* [30] Syozi I, Prog. Theor. Phys. 6, 306 (1951); Takano M, Shinjo T, Kiyama M, Takada T, J. Phys. Soc. Jpn. 25, 902 (1968); Wolf M, Schotte K D, J. Phys. A 21, 2195 (1988); Elser V, Phys. Rev. Lett. 62, 2405 (1989); Broholm C, Appli G, Espinosa G P, Cooper A S, Phys. Rev. Lett. 65, 3173 (1990).
* [31] Agarwalla B K, Zhang L, Wang J-S, and Li B, Eur. Phys. J. B 81, 197 (2011).
* [32] Kronig R de L, Physica (Amsterdam) 6, 33 (1939).
* [33] Van Vleck J H, Phys. Rev. 57, 426 (1940).
* [34] Orbach R, Proc. R. Soc. A 264, 458 (1961).
* [35] Spin-Lattice Relaxation in Ionic Solids, edited by Manenkov A A and Orbach R (Harper & Row, New York, 1966).
* [36] Capellmann H and Neumann K U, Z. Phys. B 67, 53 (1987).
* [37] Capellmann H, Lipinski S, and Neumann K U, Z. Phys. B 75, 323 (1989).
* [38] Capellmann H and Lipinski S, Z. Phys. B 83, 199 (1991).
* [39] Ioselevich A S and Capellmann H, Phys. Rev. B 51,11 446 (1995).
* [40] Hardy R J, Phys. Rev. 132, 168 (1963).
* [41] Mahan G D, Many-Particle Physics 3rd ed. (Kluwer Academic, New York, 2000).
|
arxiv-papers
| 2011-01-27T09:09:29 |
2024-09-04T02:49:16.657344
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lifa Zhang, Jie Ren, Jian-Sheng Wang, and Baowen Li",
"submitter": "Lifa Zhang",
"url": "https://arxiv.org/abs/1101.5229"
}
|
1101.5280
|
# Extended Detailed Balance for Systems
with Irreversible Reactions
A. N. Gorban ag153@le.ac.uk Department of Mathematics, University of
Leicester, Leicester, LE1 7RH, UK G. S. Yablonsky Parks College, Department
of Chemistry, Saint Louis University, Saint Louis, MO 63103, USA
###### Abstract
The principle of detailed balance states that in equilibrium each elementary
process is equilibrated by its reverse process. For many real physico-chemical
complex systems (e.g. homogeneous combustion, heterogeneous catalytic
oxidation, most enzyme reactions etc), detailed mechanisms include both
reversible and irreversible reactions. In this case, the principle of detailed
balance cannot be applied directly. We represent irreversible reactions as
limits of reversible steps and obtain the principle of detailed balance for
complex mechanisms with some irreversible elementary processes. We prove two
consequences of the detailed balance for these mechanisms: the structural
condition and the algebraic condition that form together the extended form of
detailed balance. The algebraic condition is the principle of detailed balance
for the reversible part. The structural condition is: the convex hull of the
stoichiometric vectors of the irreversible reactions has empty intersection
with the linear span of the stoichiometric vectors of the reversible
reactions. Physically, this means that the irreversible reactions cannot be
included in oriented cyclic pathways.
The systems with the extended form of detailed balance are also the limits of
the reversible systems with detailed balance when some of the equilibrium
concentrations (or activities) tend to zero. Surprisingly, the structure of
the limit reaction mechanism crucially depends on the relative speeds of this
tendency to zero.
###### keywords:
reaction network , detailed balance , microreversibility , pathway ,
irreversibility , kinetics
###### PACS:
82.40.Qt , 82.20.-w , 82.60.Hc , 87.15.R-
††journal: Chemical Engineering Science
## 1 Introduction
### 1.1 Detailed Balance for Systems with Irreversible Reactions: the Grin of
the Vanishing Cat
The principle of detailed balance was explicitly introduced and effectively
used for collisions by Boltzmann (1964). In 1872, he proved his $H$-theorem
using this principle. In its general form, this principle is formulated for
kinetic systems which are decomposed into elementary processes (collisions, or
steps, or elementary reactions). At equilibrium, each elementary process
should be equilibrated by its reverse process. The arguments in favor of this
property are founded upon microscopic reversibility. The microscopic
“reversing of time” turns at the kinetic level into the “reversing of arrows”:
the elementary processes transform into their reverse processes. For example,
the reaction $\sum_{i}\alpha_{i}A_{i}\to\sum_{j}\beta_{j}B_{j}$ transforms
into $\sum_{j}\beta_{j}B_{j}\to\sum_{i}\alpha_{i}A_{i}$ and conversely. The
equilibrium ensemble should be invariant with respect to this transformation
because of microreversibility and the uniqueness of thermodynamic equilibrium.
This leads us immediately to the concept of detailed balance: each process is
equilibrated by its reverse process.
For a given equilibrium, the principle of detailed balance results in a system
of linear conditions on kinetic constants (or collision kernels). On the
contrary, if we postulate just the existence of an a priori unknown
equilibrium state with the detailed balance property then a system of
nonlinear conditions on kinetic constants appear. These conditions were
introduced in by Wegscheider (1901) and used later by Onsager (1931). They are
known now as the Wegscheider conditions.
For linear kinetics, the Wegscheider conditions have a very simple and
transparent form: for each oriented cycle of elementary processes the product
of kinetic constants is equal to the product of kinetic constants of the
reverse processes.
However, many mechanisms of complex chemical and biochemical reactions, in
particular mechanisms of combustion and enzyme reaction, include some
irreversible (unidirectional) reactions. In many cases, complex mechanisms
consist of some reversible and some irreversible reactions, equilibrium
concentrations and rates of reactions become zeroes, and the standard forms of
the detailed balance do not have a sense.
In physical chemistry, the feasibility of a reaction depends on the energies
and entropies of system states, initial, final, and transition ones.
Nevertheless, some combinations of irreversible reactions are impossible
irrespective of the values of thermodynamic functions. Since Wegscheider’s
time it is known that the cyclic sequence of irreversible reactions (the
completely irreversible cycle) is impossible. It is forbidden by the principle
of detailed balance. In a similar way, the reaction mechanism
$A\rightleftharpoons B$, $A\to C$, $C\to B$ is forbidden as well as
$A\rightleftharpoons B$, $A\rightleftharpoons C$, $C\to B$.
Two fundamental problems can be posed:
(1) Which mechanisms with irreversible steps are allowed, and which such
mechanisms are forbidden by the principle of detailed balance?
In accordance with our knowledge, this question was not answered rigorously
and the general problem was not solved. Beside that, the procedure of
determining the forbidden mechanisms was not described.
(2) Let a mechanism with some irreversible steps be not forbidden. Do we still
have some relationships between kinetic constants of this mechanism?
In our paper, both problems are analyzed based on the same procedure.
Substituting the zero kinetic constants by small, however not zero values we
return to the fully ’reversible case’, in which all steps of the reaction
mechanism are reversible. Then, we analyze a limit case, in which small
kinetic parameters tend to reach 0.
Such an idea was applied previously to several examples. In particular, Chu
(1971) used this idea for a three-step mechanism, demonstrating that the
mechanism $A\rightleftharpoons B$, $A\to C$, $B\to C$ can appear as a limit of
reversible mechanisms which obey the principle of detailed balance, whereas
the system $A\rightleftharpoons B$, $A\to C$, $C\to B$ cannot appear in such a
limit. However, this approach was not applied to the general analysis of
multi-step mechanisms, only to a few systems of low dimensions.
Since Lewis Carroll’s “Alice’s Adventures in Wonderland”, the Cheshire Cat is
well known, in particular its inscrutable grin. Finally this cat disappears
gradually until nothing is left but its grin. Alice makes a remark she has
often seen a cat without a grin but never a grin without a cat.
The detailed balance for systems with irreversible reactions can be compared
with this grin of the Cheshire cat: the whole cat (the reversible system with
detailed balance) vanishes but the grin persists.
### 1.2 Detailed Balance: the Classical Relations
First, let us consider linear systems and write the general first order
kinetic equations:
$\dot{p}_{i}=\sum_{j}(k_{ij}p_{j}-k_{ji}p_{i})\,.$ (1)
Here, $p_{i}$ is the probability of a state $A_{i}$ ($i=1,\ldots,n$) (or, for
monomolecular reactions, the concentration of a reagent $A_{i}$). The kinetic
constant $k_{ij}\geq 0$ ($i\neq j$) is the intensity of the transitions
$A_{j}\to A_{i}$ (i.e., $k_{ij}$ is $k_{i\leftarrow j}$). The rate of the
elementary process $A_{i}\to A_{j}$ is $k_{ji}p_{i}$. The class of equations
(1) includes the Kolmogorov equation for finite Markov chains, the Master
equation in physical kinetics and the chemical kinetics equations for
monomolecular reactions.
Let $p_{i}^{\rm eq}>0$ be a positive equilibrium distribution. According to
the principle of detailed balance, the rate of the elementary process
$A_{i}\to A_{j}$ at equilibrium coincides with the rate of the reverse process
$A_{i}\leftarrow A_{j}$:
$k_{ij}p_{j}^{\rm eq}=k_{ji}p_{i}^{\rm eq}\,.$ (2)
For a given equilibrium, $p_{i}^{\rm eq}$, the principle of detailed balance
is equivalent to this system (2) of linear equalities. To find the conditions
of the existence of such a positive equilibrium that (2) holds, it is
sufficient to write equations (2) in the logarithmic form, $\ln p_{i}^{\rm
eq}-\ln p_{j}^{\rm eq}=\ln k_{ij}-\ln k_{ji}$, to consider this system as a
system of linear equations with respect to the unknown $\ln p_{i}^{\rm eq}$,
and to formulate the standard solvability condition.
After some elementary transformation this condition gives: a positive
equilibrium with detailed balance (2) exists if and only if
1. 1.
If $k_{ij}>0$ then $k_{ji}>0$ (reversibility);
2. 2.
For each oriented cycle of elementary processes, $A_{i_{1}}\to
A_{i_{2}}\to\ldots A_{i_{q}}\to A_{i_{1}}$, the product of the kinetic
constants is equal to the product of the kinetic constants of the reverse
processes:
$\prod_{j=1}^{q}k_{i_{j+1}i_{j}}=\prod_{j=1}^{q}k_{i_{j}i_{j+1}}$ (3)
where the cyclic numeration is used, $i_{q+1}=i_{1}$.
Of course, it is sufficient to use in (3) a basis of independent cycles (see,
for example the review of Schnakenberg (1976)).
Let us introduce the more general Wegscheider conditions for nonlinear
kinetics and the generalized mass action law. (For a more detailed exposition
we refer to the textbook of Yablonskii et al (1991).) The elementary reactions
are given by the stoichiometric equations
$\sum_{i}\alpha_{ri}A_{i}\to\sum_{j}\beta_{rj}A_{j}\;\;(r=1,\ldots,m)\,,$ (4)
where $A_{i}$ are the components and $\alpha_{ri}\geq 0$, $\beta_{rj}\geq 0$
are the stoichiometric coefficients. The reverse reactions with positive
constants are included in the list (4) separately. We need this separation of
direct and reverse reactions to apply later the general formalism to the
systems with some irreversible reactions.
The stoichiometric matrix is $\boldsymbol{\Gamma}=(\gamma_{ri})$,
$\gamma_{ri}=\beta_{ri}-\alpha_{ri}$ (gain minus loss). The stoichiometric
vector $\gamma_{r}$ is the $r$th row of $\boldsymbol{\Gamma}$ with coordinates
$\gamma_{ri}=\beta_{ri}-\alpha_{ri}$.
According to the generalized mass action law, the reaction rate for an
elementary reaction (4) is
$w_{r}=k_{r}\prod_{i=1}^{n}a_{i}^{\alpha_{ri}}\,,$ (5)
where $a_{i}\geq 0$ is the activity of $A_{i}$.
The list (4) includes reactions with the reaction rate constants $k_{r}>0$.
For each $r$ we define $k_{r}^{+}=k_{r}$, $w_{r}^{+}=w_{r}$, $k_{r}^{-}$ is
the reaction rate constant for the reverse reaction if it is on the list (4)
and 0 if it is not, $w_{r}^{-}$ is the reaction rate for the reverse reaction
if it is on the list (4) and 0 if it is not. For a reversible reaction,
$K_{r}=k_{r}^{+}/k_{r}^{-}$
The principle of detailed balance for the generalized mass action law is: For
given values $k_{r}$ there exists a positive equilibrium $a_{i}^{\rm eq}>0$
with detailed balance, $w_{r}^{+}=w_{r}^{-}$. This means that the system of
linear equations
$\sum_{i}\gamma_{ri}x_{i}=\ln k_{r}^{+}-\ln k_{r}^{-}=\ln K_{r}$ (6)
is solvable ($x_{i}=\ln a_{i}^{\rm eq}$). The following classical result gives
the necessary and sufficient conditions for the existence of the positive
equilibrium $a_{i}^{\rm eq}>0$ with detailed balance (see, for example, the
textbook of Yablonskii et al (1991)).
###### Proposition 1.
Two conditions are sufficient and necessary for solvability of (6):
1. 1.
If $k_{r}^{+}>0$ then $k_{r}^{-}>0$ (reversibility);
2. 2.
For any solution $\boldsymbol{\lambda}=(\lambda_{r})$ of the system
$\boldsymbol{\lambda\Gamma}=0\;\;\left(\mbox{i.e.}\;\;\sum_{r}\lambda_{r}\gamma_{ri}=0\;\;\mbox{for
all}\;\;i\right)$ (7)
the Wegscheider identity holds:
$\prod_{r=1}^{m}(k_{r}^{+})^{\lambda_{r}}=\prod_{r=1}^{m}(k_{r}^{-})^{\lambda_{r}}\,.$
(8)
###### Remark 1.
It is sufficient to use in (8) a basis of solutions of the system (7):
$\boldsymbol{\lambda}\in\\{\boldsymbol{\lambda}^{1},\cdots,\boldsymbol{\lambda}^{g}\\}$.
###### Remark 2.
The Wegscheider condition for the linear systems (3) is a particular case of
the general Wegscheider identity (8). Therefore, the solutions
$\boldsymbol{\lambda}$ of equation (7) are generalizations of the (non-
oriented) cycles in the reaction networks. The basis of solutions corresponds
to the basic cycles. This basis is, obviously, not unique.
###### Remark 3.
In equation (6) unknown $x_{i}=\ln a_{i}$ are independent variables and vector
$\boldsymbol{x}$ can take any value in $R^{n}$. In practice, this is not
always true. For example, for heterogeneous systems with solid components some
activities may vary in a narrow interval or may be even constant (see the more
detailed discussion below in Section 3.5). We do not study multiphase
equilibiria in our paper.
###### Remark 4.
All the closed chemical systems have linear conservation laws: conservation of
mass, various sorts of atoms, electric charge and other conserved quantities.
They are linear functions of the amounts $N_{i}$ of chemical components
$A_{i}$. There is a problem of uniqueness and existence of a positive
equilibrium with detailed balance or without it for every set of values of the
independent conservation laws. To solve this problem we need some properties
of the connection between activities and concentrations,
$(a_{i})\leftrightarrow(c_{i})$. We do not assume any hypothesis about this
connection and study just existence of a positive equilibrium with detailed
balance in the space of activities. The Wegscheider identity (8) gives a
necessary and sufficient condition for this existence.
In practice, very often $k_{r}^{-}=0$ for some $r$, whereas $k_{r}^{+}>0$. In
these cases, the standard forms of the detailed balance have no sense. Indeed,
let us consider a linear reversible cycle with an irreversible buffer:
$A_{1}\rightleftharpoons A_{2}\rightleftharpoons\ldots A_{n}\rightleftharpoons
A_{1}\to A_{0}\,.$
This system converges to the state where only $p_{0}>$ and $p_{i}=0$ for
$i>0$. In this state, trivially, $w_{r}^{+}=w_{r}^{-}=0$ and it seems that the
standard principle of detailed balance does not imply any restriction on the
kinetic constants. Of course, this impression is wrong.
Let us consider this system as a limit of the system with a reversible buffer,
$A_{1}\rightleftharpoons A_{0}$ (both reaction rate constants are positive),
when the constant of the reverse reaction is positive but tends to zero:
$k_{1\leftarrow 0}\to 0$, $k_{1\leftarrow 0}>0$. For each positive value
$k_{1\leftarrow 0}>0$ the condition of detailed balance $w_{r}^{+}=w_{r}^{-}$
gives the Wegscheider identity (3) for the cycle $A_{1}\rightleftharpoons
A_{2}\rightleftharpoons\ldots A_{n}\rightleftharpoons A_{1}$: The product of
direct reaction rate constants is equal to the product of the reverse reaction
rate constants. This condition holds also in the limit $k_{1\leftarrow 0}\to
0$. So, any practically negligible but positive value of the reverse kinetic
constant implies the nontrivial Wegscheider condition for the other constants.
If we assume that the negligible values of the constants should not affect the
kinetic systems then this Wegscheider condition should hold for the system
with fully irreversible steps as well. Therefore, the following way for the
formalization of the principle of detailed balance for irreversible reactions
is proposed. We return to reversible reactions in which the principle of
detailed balance is assumed by the introduction of small $k_{r}^{-}>0$. Then
we go to the limit $k_{r}^{-}\to 0$ ($k_{r}^{-}>0$) for these reactions.
It is worth mentioning that the free energy has no limit when some of the
reaction equilibrium constants tend to zero. For example, for the ideal gases
$F=\sum_{i}N_{i}(RT\ln c_{i}+\mu_{i}^{0}-RT)$, where $c_{i}$ is the
concentration, $N_{i}$ is the amount and $\mu_{i}^{0}$ is the standard
chemical potential of the component $A_{i}$. In the irreversible limit some
$\mu_{i}^{0}\to\infty$. On the contrary, the activities
$a_{i}=\exp\left(\frac{\mu_{i}-\mu_{i}^{0}}{RT}\right)\,$ (9)
remain finite (for the ideal gases, for example, $a_{i}=c_{i}$) and the
approach based on the generalized mass action law and the equations
$w_{r}^{+}=w_{r}^{-}$ can be applied in the irreversible limit.
Below, we study systems with irreversible reactions as the limits of the
systems with reversible reactions and detailed balance, when some reaction
rate constants go to zero. We formulate the restrictions on the constants in
this limit and find the finite number of conditions that is necessary and
sufficient to check. First of all we have to discuss the necessary notion of
cycles for general reaction networks.
### 1.3 Main Results
We develop three approaches to the detailed balance conditions for the systems
with some irreversible reactions. The first and the most physical idea is to
consider an irreversible reaction as a limit of a reversible reaction when the
reaction rate constant for a reverse reaction tends to zero. The limits of
systems of reversible reactions with detailed balance conditions cannot be
arbitrary systems with some irreversible reactions and we study the structural
and algebraic restrictions for these systems.
The second approach is based on the technical idea to study the limits of the
Wegscheider identities (8). Here, it is very useful to apply the concept of
the general (nonlinear) irreversible cycles or pathways developed recently far
enough for our purposes by Schuster et al (2000); Gagneur & Klamt (2004) and
other. Let us write all reactions separately (including direct and reverse
reactions) (4). The general oriented cycle or pathway is a relation between
vectors $\gamma_{r}$ with non-negative coefficients :
$\sum_{r}\lambda_{r}\gamma_{r}=0$, $\lambda_{r}\geq 0$ and
$\sum_{r}\lambda_{r}>0$. For each system with all reversible reactions and
detailed balance the Wegscheider identity (8) holds for any oriented cycle.
Therefore, if an oriented cycle persists in the limit with some irreversible
reactions, then, for $\lambda_{r}>0$, the $r$th reaction should remain
reversible and for this cycle the Wegscheider condition persists.
This property motivates the definition of the extended (or weakened, Yablonsky
et al (2010)) form of detailed balance in Section 3.1 through the general
oriented cycles and the Wegscheider conditions. Theorem 1 states that a system
satisfies the extended form of detailed balance if and only if it is a limit
of systems with all reversible reactions and detailed balance. One part of
this theorem (“if”) is proved immediately in Section 3.1, the proof of the
second part (“only if”) exploits the third approach and is postponed till
Section 4.
The third idea is to study the limits when some equilibrium concentrations
(or, more general, activities) tend to zero. For systems with all reversible
reactions, we can explicitly express the constants of the reverse reactions
through the constants of the direct reactions and the equilibrium activities:
just use the detailed balance conditions, $w^{+}(a^{\rm eq})=w^{-}(a^{\rm
eq})$. Here, instead of 2$m$ parameters, $k^{\pm}_{r}$ ($m$ is the number of
reactions) we use $m+n$ parameters: $m$ reaction rate constants $k^{+}_{r}$
and $n$ equilibrium activities $a_{i}^{\rm eq}$. In this description of the
reversible reactions, the principle of detailed balance is trivially
satisfied. Some reactions become irreversible in the limits when some of the
equilibrium activities tends to zero. Surprisingly, the structure of the limit
reaction mechanism crucially depends on the relative speeds of this tendency
to zero.
In Section 4, we assume that $a_{i}^{\rm eq}={\rm
const_{i}}\times\varepsilon^{\delta_{i}}$ and study the limits $\varepsilon\to
0$. The $n$-dimensional space of exponents $\delta=(\delta_{i})$ is split by
$m$ hyperplanes $(\gamma_{r},\delta)=0$ on convex cones. Each of these cones
is given by a set of inequalities $(\gamma_{r},\delta)\lesseqqgtr 0$
($r=1,\ldots,m$). In every such a cone, the limit reaction mechanism for
$\varepsilon\to 0$ is constant.
Using this approach, we prove the second part of Theorem 1 and even more: if a
system satisfies the extended form of detailed balance then it may be obtained
in the limit $\varepsilon\to 0$ from a system with all reversible reactions
with given $k_{r}^{+}$ and $a_{i}^{\rm eq}={\rm
const_{i}}\times\varepsilon^{\delta_{i}}$ for some exponents ${\delta_{i}}$
(Theorem 4). So, all the three approaches to the consequences of the principle
of detailed balance for the systems with some irreversible reactions are
equivalent.
The computational problem associated with the extended form of detailed
balance is nontrivial. For example, the oriented cycles (pathways) form a
convex polyhedral cone and we have to formulate the structural condition of
the extended form of detailed balance for all extreme rays (extreme pathways)
of this cone (Theorem 2): if $\lambda_{r}>0$ for a vector
$\boldsymbol{\lambda}$ from an extreme ray then the $r$th reaction should
remain reversible. Calculation of all these extreme rays is a well known and
computational expensive problem (Fukuda & Prodon, 1996; Papin et al, 2003;
Gagneur & Klamt, 2004). In Theorem 3, we significantly reduce the dimension of
the problem.
Instead of the set of all stoichiometric vectors $\gamma_{r}$ ($r=1,\ldots,m$)
in the whole space of composition $\mathbb{R}^{n}$ ($n$ is the number of
components, $m$ is the number of reactions) it is sufficient to consider the
set of the stoichiometric vectors of the irreversible reactions in the
quotient space $\mathbb{R}^{n}/S$, where $S$ is spanned by the stoichiometric
vectors of all reversible reactions. The simple exclusion of the linear
conservation laws provides additional dimensionality reduction. The
application of reduction methods is demonstrated in the case study in Section
3.5.
In Section 3.3, we formulate the main results for the simple case of linear
(monomolecular) systems. Sections 3.4 and 3.5 are devoted to examples of
nonlinear systems. In Section 3.4, the simple examples with obvious lists of
the extreme pathways are collected. In Section 3.5, we analyze the possible
irreversible limits for a complex reaction of methane reforming with CO2.
## 2 Cycles and Pathways in General Reaction Networks
There exist several graph representations of general reaction networks
(Yablonskii et al, 1991; Temkin et al, 1996) and each of them implies the
correspondent notion of a cycle. For example, each input and output formal sum
in the reaction mechanism (4) can be considered as a vertex (a complex) and
then a reaction with the positive rate constant is an oriented edge. This
graph of the transformation of complexes is convenient for the analysis of the
complex balance condition (Feinberg, 1972).
The bipartite graphs of reactions (Volpert & Khudyaev, 1985) gives us another
example: it includes two types of vertices: components (correspond to $A_{i}$)
and reactions (correspond to elementary reactions from (4)). There is an edge
from the $i$th component to the $s$th reaction if $\alpha_{ri}>0$ and from the
$s$th reaction to the $i$th component if $\beta_{ri}>0$. The correspondent
stoichiometric coefficients are the weights of the edges. This graph is
convenient for the analysis of the system stability, for calculation of
Jacobians for the right hand sides of the kinetic equations and for analysis
of their signs (Ivanova, 1979; Mincheva & Roussel, 2007). For nonlinear
systems, these graphs do not give a simple representation of the detailed
balance conditions.
We need a special notion of a cycle which corresponds to the algebraic
relations between reactions. Let us recall that we include direct and inverse
reactions in the reaction mechanism (4) separately. Each solution of (7) may
be represented as follows:
$\begin{split}+&\left|\underline{\begin{array}[]{l}\lambda_{1}\times\left(\sum_{i}\alpha_{1i}A_{i}\to\sum_{j}\beta_{1j}A_{j}\right)\\\
\lambda_{2}\times\left(\sum_{i}\alpha_{2i}A_{i}\to\sum_{j}\beta_{2j}A_{j}\right)\\\
\cdots\\\
\lambda_{m}\times\left(\sum_{i}\alpha_{mi}A_{i}\to\sum_{j}\beta_{mj}A_{j}\right)\end{array}}\right.\\\
&\;\;\;\;\;\;\;\;\;\;\;=\;\sum_{i}a_{i}A_{i}\to\sum_{j}a_{j}A_{j}\;.\end{split}$
(10)
Here,
$a_{i}=\sum_{s}\lambda_{s}\alpha_{si}\equiv\sum_{s}\lambda_{s}\beta_{si}$.
Therefore, we need the following definition of a cycle.
###### Definition 1.
An oriented cycle is a vector of coefficients $\boldsymbol{\lambda}\neq 0$
with all $\lambda_{i}\geq 0$ that satisfies (10).
###### Remark 5.
Cycles in catalysis and, especially, in biochemistry are called pathways
(Schuster et al, 2000; Papin et al, 2003). An oriented pathway is an oriented
cycle from Definition 1. An extreme (oriented) pathway is a direction vector
of an extreme ray of the cone $\Lambda_{+}$. A solution of equation (7) (a
non-oriented cycle) is a non-oriented pathway.
Qualitatively these concepts have been introduced in the early 1940s by
Horiuti who applied them to heterogeneous catalytic reactions (Horiuti, 1973).
Horiuti used them to eliminate intermediates of the complex catalytic reaction
by adding the steps of the detailed mechanism first multiplied by special
coefficients. As result of such procedure, the chemical equation with no
intermediates is obtained.
All oriented cycles form the cone $\Lambda_{+}$ (without the origin). Extreme
ray of a convex cone is a face that is, at the same time, a ray. Each ray may
be defined by a directional vector $\boldsymbol{\lambda}$ that is an arbitrary
nonzero vector from this ray. The cone $\Lambda_{+}$ is defined by a finite
system of linear equations (7) and inequalities $\lambda_{r}\geq 0$.
Therefore, it has a finite set of extreme rays.
For integer stoichiometric coefficients, $\alpha_{si}$, $\beta_{si}$, any
extreme ray is defined by an uniform linear systems of equations with integer
coefficients supplemented by the conditions $\lambda_{i}\geq 0$ and
$\boldsymbol{\lambda}\neq 0$. Therefore, we can always select a direction
vector with the integer coefficients. For each extreme ray, there exists a
unique direction vector with minimal integer coefficients.
For monomolecular reaction networks, these cycles coincide with the oriented
cycles in the graph of reactions (where vertices are reagents and edges are
reactions).
There exists an oriented cycle of the length two for each pair of mutually
reverse reactions. For these cycles the Wegscheider identities (8) hold
trivially, for any positive values of $k^{\pm}$.
###### Remark 6.
The systems without oriented cycles ($\Lambda_{+}=\\{0\\}$) have a simple
dynamic behavior. First of all, for such a system the convex hull of the
stoichiometric vectors does not include zero: $0\notin{\rm
conv}\\{\gamma_{1},\ldots,\gamma_{m}\\}$. Therefore, there exists a linear
functional $l$ that separates 0 from $\\{\gamma_{1},\ldots,\gamma_{m}\\}$:
$l(\gamma_{s})>0$ for all $s=1,\ldots,m$. This linear function $l(c)$
increases monotonically due to any kinetic equation
$\frac{{\mathrm{d}}c}{{\mathrm{d}}t}=\sum_{s}w_{s}\gamma_{s}$
with reaction rates $w_{s}\geq 0$: ${\mathrm{d}}l(c)/{\mathrm{d}}t>0$ if at
least one reaction rate $w_{s}>0$.
## 3 Extended Form of Detailed Balance
### 3.1 Definition
A practically important reaction mechanism may include reversible and
irreversible steps. However, some mechanisms with irreversible steps may be
wrong because they cannot appear as the limits of reversible mechanisms with
detailed balance. Therefore, the first question is about the mechanism
structure: what is allowed?
The second question is about the constants: let the mechanism not be
forbidden. If it is the limit of a system with detailed balance then the
reaction rate constants may be connected by additional algebraic conditions
like the Wegscheider conditions (3). We should describe all the necessary
conditions. In this Section we answer both questions and formulate both
conditions, structural and algebraic.
We have to study study the identities (8) in the limit when some $k_{r}^{-}\to
0$. First of all, let us consider reversible reactions: if $k_{r}^{+}>0$ then
$k_{r}^{-}>0$. It is sufficient to use in (8) only $\boldsymbol{\lambda}$ with
nonnegative coordinates, $\lambda_{r}\geq 0$. Indeed, the direct and reverse
reactions are included in the list (4) under different numbers. Assume that
$\lambda_{r}<0$ in an identity (8) for some $r$. Let the reverse reaction for
this $r$ have number $r^{\prime}$. Let us substitute
$(k^{+}_{r})^{\lambda_{r}}$ in the left hand side of (8) by
$(k^{+}_{r^{\prime}})^{-\lambda_{r}}$ and $(k^{-}_{r})^{\lambda_{r}}$ in the
right hand side by $(k^{-}_{r^{\prime}})^{-\lambda_{r}}$. The new condition is
equivalent to the previous one. Let us iterate this operation for various $r$.
In the finite number of steps all the powers $\lambda_{r}\geq 0$.
Let us use notation $\Lambda$ for the linear space of solutions of (7) and
$\Lambda_{+}$ for the cone of positive solutions $\boldsymbol{\lambda}$
($\lambda_{r}\geq 0$) of (7).
For reversible reactions, we proved the following proposition. Let the
reactions are reversible and the direct and reverse reactions are included in
the list (4) separately.
###### Proposition 2.
The Wegscheider identity (8) holds for all $\boldsymbol{\lambda}\in\Lambda$ if
and only if it holds for all positive $\boldsymbol{\lambda}\in\Lambda_{+}$.
Elementary linear algebra gives the following corollary for reversible
reactions.
###### Corollary 1.
The solution of the system of linear equations for logarithms of equilibrium
activities (6) exists if and only if for any positive solution
$\boldsymbol{\lambda}$ ($\lambda_{r}\geq 0$) of the system
$\boldsymbol{\lambda}\boldsymbol{\Gamma}=0$ (7) the condition (8) holds.
Let us study identity (8) for a positive $\boldsymbol{\lambda}$ when some of
$k_{r}\to 0$. In this limit, for every $\boldsymbol{\lambda}\in\Lambda_{+}$
Corollary 1 gives two conditions:
###### Corollary 2.
Let $k_{s}>0$, $k_{s}\to k_{s}^{\rm lim}$ and the Wegscheider identity (8)
holds for $k_{s}$. Then
1. 1.
If $\lambda_{s}>0$ and $k_{s}^{+}\to 0$ for some $s$ then for some $q$
$\lambda_{q}>0$ and $k_{q}^{-}\to 0$;
2. 2.
If for all positive components $\lambda_{s}>0$ the limit constants are
positive, $k_{s}^{\rm lim\,\pm}>0$, then the condition (8) holds for
$k_{s}^{\rm lim\,\pm}$.
We can interpret the positive solutions of (7) as oriented cycles (linear or
nonlinear). The first condition means that if a cycle is cut by the limit
$k_{s}^{+}\to 0$ in one direction then it should be also cut by a limit
$k_{q}^{-}\to 0$ in the opposite direction: the irreversible cycle is
forbidden. This remark leads to the definition of the structural condition of
the extended form of detailed balance.
###### Definition 2.
A system of reactions (4) satisfies the structural condition of the extended
form of detailed balance if for every $\boldsymbol{\lambda}\in\Lambda_{+}$ the
reaction which satisfy $\lambda_{s}>0$ are reversible: if $\lambda_{s}>0$ then
$k_{s}^{\pm}>0$.
This condition means that all cycles should be reversible. The second
condition means that for all cycles $\boldsymbol{\lambda}\in\Lambda_{+}$ which
persist in the system with irreversible reactions the correspondent
Wegscheider condition (8) holds. This is the algebraic condition of the
extended form of detailed balance. Now, we are ready to formulate the
definition of the extended form of detailed balance.
###### Definition 3.
The subsystem satisfies the extended form of detailed balance if both the
structural and the algebraic condition hold for all
$\boldsymbol{\lambda}\in\Lambda_{+}$.
The following theorem gives the motivation to this definition.
###### Theorem 1.
A system with irreversible reactions is a limit of systems with reversible
reactions and detailed balance if and only if it satisfies the extended form
of detailed balance.
###### Proof.
Let us prove the direct statement: if a system is a limit of systems with
reversible reactions and detailed balance then it satisfies the extended form
of detailed balance. Indeed, let a system of reactions be a limit of systems
with reversible reactions and detailed balance. This means that for each
$j=1,2,\ldots$ a set of reaction rate constants $k_{s,j}^{\pm}>0$ is given,
$k_{s,j}^{\pm}>0$ satisfy the principle of detailed balance for all $j$ and
$k_{s}^{\pm}=\lim_{j\to\infty}k_{s,j}^{\pm}\,.$
Assume that the structural condition is violated: there exists such a
$\boldsymbol{\lambda}\in\Lambda_{+}$ that $\lambda_{s}>0$ for an irreversible
reaction ($k_{s}^{+}>0$, $k_{s}^{-}=0$). For all $j=1,2,\ldots$ the principle
of detailed balance gives:
$\prod_{r,\,\lambda_{r}>0}(k_{r,j}^{+})^{\lambda_{r}}=\prod_{r,\,\lambda_{r}>0}(k_{r,j}^{-})^{\lambda_{r}}\,.$
(11)
If $\lambda_{r}>0$ then $k_{r}^{+}>0$. Therefore, for these $r$, sufficiently
large $j$ and some $\varepsilon,\delta>0$
$\delta>k_{r,j}^{\pm}>\varepsilon>0$. The left hand side of (11) is separated
from zero. The right hand side of (11) tends to zero because all factors are
bounded and at least one of them tends to zero, $k_{r,j}^{-}\to 0$. This
contradiction proves the structural condition. To prove the algebraic
condition, it is sufficient to notice that the Wegscheider identity for
$k_{s,j}^{\pm}>0$ holds for all $j$, hence, it holds in the limit
$j\to\infty$.
We will prove the converse statement (if a system satisfies the extended form
of detailed balance then it is a limit of systems with reversible reactions
and detailed balance) in Section 4, in the proof of Theorem 4. ∎
### 3.2 Criteria
All $\boldsymbol{\lambda}\in\Lambda_{+}$ participate in the definition of the
extended form of detailed balance. Nevertheless, it is sufficient to use a
finite subset of this cone.
We can check directly that if for a set $\\{\boldsymbol{\lambda}^{s}\\}$ the
structural and the algebraic conditions of the extended form of detailed
balance hold then they hold for any conic combination of
$\\{\boldsymbol{\lambda}^{s}\\}$,
$\boldsymbol{\lambda}=\sum_{s}a_{s}\boldsymbol{\lambda}^{s}$, $a_{s}\geq 0$.
Therefore, it is sufficient to check the conditions for the directional
vectors of the extreme rays of $\Lambda_{+}$.
Let a reaction mechanism satisfy the extended principle of detailed balance.
If we delete from this mechanisms any irreversible elementary reaction or any
couple of mutually reverse elementary reactions, the resulting mechanism
satisfies the extended principle of detailed balance as well.
A cone is pointed if the origin is its extreme point or, which is the same,
this cone does not include a whole straight line. The cone $\Lambda_{+}$ is
pointed because it belongs to the positive orthant $\\{\boldsymbol{\lambda}\
|\ \boldsymbol{\lambda}\geq 0\\}$.
It is a standard task of linear programming and computational convex geometry
to find all the extreme rays of the polyhedral pointed cone $\Lambda_{+}$
(Bertsimas & Tsitsiklis, 1997; Motzkin et al, 1953; Fukuda & Prodon, 1996).
Let the directional vectors of these extreme rays be
$\\{\boldsymbol{\lambda}^{s}\ |\ s=1,\cdots,q\\}$.
###### Theorem 2.
The system satisfies the extended form of detailed balance if and only if the
structural and algebraic conditions hold for the directional vectors
$\\{\boldsymbol{\lambda}^{s}\ |\ s=1,\cdots,q\\}$ of the extreme rays of the
cone $\Lambda_{+}$.
Theorem 2 follows just from the definition of extreme rays and the Minkowski
theorem which states that every pointed cone given by linear inequalities
admits a unique representation as a conic hull of a finite set of extreme
rays.
This criterion can be simplified as well: it is necessary and sufficient to
check the structural conditions for the extreme rays of $\Lambda_{+}$ and then
the algebraic condition for a maximal linear independent subset of
$\\{\boldsymbol{\lambda}^{s}\ |\ s=1,\cdots,q\\}$.
###### Corollary 3.
The system satisfies the extended form of detailed balance if and only if the
structural conditions hold for all directional vectors
$\\{\boldsymbol{\lambda}^{s}\ |\ s=1,\cdots,q\\}$ of the extreme rays of the
cone $\Lambda_{+}$ and the algebraic conditions hold for a maximal linear
independent subset of $\\{\boldsymbol{\lambda}^{s}\ |\ s=1,\cdots,q\\}$.
If, for a given reaction mechanism, the set $\\{\boldsymbol{\lambda}^{s}\ |\
s=1,\cdots,q\\}$ of directional vectors of the extreme rays of $\Lambda_{+}$
is known, then it is easy to check, whether this mechanism satisfies the
structural conditions of the extended form of detailed balance. It is
sufficient to examine for each ${\lambda}^{s}_{r}>0$, whether $k_{r}^{-}>0$.
After these conditions are examined, it is a simple task to extract the
independent set of the Wegscheider identities that should be valid: just
select a maximal linear independent subset from the set of
$\boldsymbol{\lambda}^{s}$ and write the correspondent Wegscheider identities.
It is convenient to use all the extreme pathways especially if we would like
to study all the subsystems of the given system, which satisfy the extended
form of detailed balance. On the other hand, it is computationally expensive
to find the set $\\{\boldsymbol{\lambda}^{s}\ |\ s=1,\cdots,q\\}$ (see, for
example, the paper by Gagneur & Klamt (2004)). The amount of computation could
be significantly reduced because it is not necessary to use all the extreme
pathways.
Let us consider a reaction mechanism, which includes both reversible and
irreversible reactions. For the reversible reactions, let us join the direct
and reverse reactions. Let $\gamma_{1},\ldots,\gamma_{r}$ be the
stoichiometric vectors of the reversible reactions and
$\nu_{1},\ldots,\nu_{s}$ be the stoichiometric vectors of the irreversible
reactions. We use $\boldsymbol{\Gamma}_{r}$ for the stoichiometric matrix of
the reversible reactions and $\Lambda_{r}$ for the solutions of the equations
$\boldsymbol{\lambda}\boldsymbol{\Gamma}_{r}=0$.
The linear subspace $S={\rm
span}\\{\gamma_{1},\ldots,\gamma_{r}\\}\subset\mathbb{R}^{n}$ consists of all
linear combinations of the stoichiometric vectors of the reversible reactions.
Let us consider the quotient space $\mathbb{R}^{n}/S$. We use notation
$\overline{\nu}_{j}$ for the images of $\nu_{j}$ in $\mathbb{R}^{n}/S$.
The following theorem gives the criteria of the extended form of detailed
balance, which are more efficient for computations.
###### Theorem 3.
The system satisfies the extended form of detailed balance if and only if
1. 1.
The convex hull of the stoichiometric vectors of irreversible reactions does
not intersect $S$, i.e.
$0\notin{\rm conv}\\{\overline{\nu}_{1},\ldots,\overline{\nu}_{s}\\}\,;$ (12)
2. 2.
The reversible reactions satisfy the principle of detailed balance.
###### Proof.
Let the condition 1 be violated, i.e. $0\in{\rm
conv}\\{\overline{\nu}_{1},\ldots,\overline{\nu}_{s}\\}$. In this case, there
exist such a nonnegative $\theta_{i}\geq 0$ that $\sum_{j=1}^{s}\theta_{j}=1$
and $\sum_{j=1}^{s}\theta_{j}\nu_{j}\in S$. This means that
$\sum_{j=1}^{s}\theta_{j}\nu_{j}+\sum_{i=1}^{r}\lambda_{i}\gamma_{i}=0$. We
can transform the sum $\sum_{i=1}^{r}\lambda_{i}\gamma_{i}$ in a combination
with positive coefficients if for any negative $\lambda_{i}$ we substitute
$\gamma_{i}$ by the stoichiometric vector of the reverse reaction, that is,
$-\gamma_{i}$. As a result, we get the element of $\Lambda_{+}$, a combination
of the stoichiometric vectors with nonnegative coefficients, which is equal to
zero and includes some stoichiometric vectors of the irreversible reactions
with nonzero coefficients. Therefore, the structural condition of the extended
form of detailed balance is violated.
Let the structural condition be violated. Then there exist a combination
$\sum_{j=1}^{s}\theta_{j}\nu_{j}+\sum_{i=1}^{r}\lambda_{i}\gamma_{i}=0$ with
$\theta_{j}\geq 0$ and $\sum_{j=1}^{s}\theta_{j}>0$. Let us notice that
$\sum_{j=1}^{s}\frac{\theta_{j}}{\sum_{j=1}^{s}\theta_{j}}\nu_{j}=-\sum_{i=1}^{r}\frac{\lambda_{i}}{\sum_{j=1}^{s}\theta_{j}}\gamma_{i}\,,$
and, therefore, $0\in{\rm
conv}\\{\overline{\nu}_{1},\ldots,\overline{\nu}_{s}\\}$. The condition 1 is
violated.
We proved that the condition 1 is equivalent to the structural condition of
the extended form of detailed balance.
If the condition 1 holds then the condition 2 is, exactly, the algebraic
condition of the extended form of detailed balance. ∎
###### Remark 7.
The first condition of Theorem, $0\notin{\rm
conv}\\{\overline{\nu}_{1},\ldots,\overline{\nu}_{s}\\}$, is equivalent to the
existence of such a linear functional $l$ on $\mathbb{R}^{n}$ that
$l(\nu_{j})>0$ for all $j=1,\ldots,s$ and $l(\gamma_{j})=0$ for all
$j=1,\ldots,r$.
### 3.3 Linear Systems
The results of previous Sections for a linear system (1) have a geometrically
clear form (see also the paper by Yablonsky et al (2010)).
###### Proposition 3.
The necessary and sufficient condition for the extended form of detailed
balance is: In any cycle $A_{i_{1}}\to A_{i_{2}}\to\ldots\to A_{i_{q}}\to
A_{i_{1}}$ with the strictly positive constants $k_{i_{j+1}i_{j}}>0$ (here
$i_{q+1}=i_{1}$) all the reactions are reversible ($k_{i_{j}i_{j+1}}>0$) and
the identity (3) holds.
The states (reagents) $A_{q}$ and $A_{r}$ ($q\neq r$) are strongly connected
if there exist oriented paths both from $A_{q}$ to $A_{r}$ and from $A_{r}$ to
$A_{q}$ (each oriented edge corresponds to a reaction with the nonzero
reaction rate constant). From Proposition 3 we get the following statement.
###### Corollary 4.
Let a linear system satisfy the extended form of detailed balance. Then all
reactions in any directed path between strongly connected states are
reversible.
In brief, a linear system with the extended form of detailed balance can be
described as follows: (i) the oriented cycles are reversible and satisfy the
classical condition (3), (ii) the system consists of the reversible parts and
the irreversible transitions between these parts and (iii) the system of these
irreversible transitions is acyclic.
For example, let us analyze subsystems of the simple cycle,
$A_{1}\rightleftharpoons A_{2}\rightleftharpoons A_{3}\rightleftharpoons
A_{1}$.
$\boldsymbol{\Gamma}^{\mathrm{T}}=\left[\begin{array}[]{rrrrrr}-1&0&1&1&0&-1\\\
1&-1&0&-1&1&0\\\ 0&1&-1&0&-1&1\end{array}\right]$ (13)
The cone of nonnegative solutions $\Lambda_{+}$ to the equation
$\boldsymbol{\lambda\Gamma}=0$ has extreme rays with the following direction
vectors: $\boldsymbol{\lambda}^{1}=(1,1,1,0,0,0)$,
$\boldsymbol{\lambda}^{2}=(0,0,0,1,1,1)$,
$\boldsymbol{\lambda}^{3}=(1,0,0,1,0,0)$,
$\boldsymbol{\lambda}^{4}=(0,1,0,0,1,0)$, and
$\boldsymbol{\lambda}^{5}=(0,0,1,0,0,1)$. Vectors $\boldsymbol{\lambda}^{3-5}$
give trivial identities (8) $k_{i}^{+}k_{i}^{-}=k_{i}^{-}k_{i}^{+}$
($i=1,2,3$) and vectors $\boldsymbol{\lambda}^{1,2}$ give the same identity:
$k_{1}^{+}k_{2}^{+}k_{3}^{+}=k_{1}^{-}k_{2}^{-}k_{3}^{-}$.
If we delete one elementary reaction from the simple cycle (i.e. one column
from $\boldsymbol{\Gamma}^{\mathrm{T}}$ (13)) then one of the nonnegative
solutions $\boldsymbol{\lambda}^{1,2}$ persists and, due to the extended
detailed balance principle, all the reactions should be reversible. This means
that the structural condition of extended detailed balance is not satisfied
for the simple reversible cycle without one direct or reverse reaction. If two
reactions are reversible then the third should be reversible or completely
vanish. If we delete one direct reaction (with number 1, 2 or 3) and one
reverse reaction (with number 4, 5 or 6) then there remain no non-trivial
solutions in $\Lambda_{+}$ and, therefore, no non-trivial relations between
the constants persist after deletion of these two reactions.
For the linear systems, the oriented cycles in the graph of reactions (where
vertices are the components and edges are the reactions) give the positive
solutions to the equation (7): for a linear oriented cycle $C$ the sum of the
stoichiometric vectors of its reactions is zero. Moreover, any positive
solution of (7) is a convex combination of such cyclic solutions and,
therefore, the directed vectors of the extreme rays of $\Lambda_{+}$ can be
selected in this form.
### 3.4 Simple Examples of Nonlinear Systems
In this section, we present several elementary examples. For these examples,
the sets of the extreme pathways are obvious.
Let us examine a reaction mechanism with irreversible reactions
$A\xrightarrow{k_{1}}B$ and $2B\xrightarrow{k_{2}}2A$.
$\boldsymbol{\Gamma}^{\mathrm{T}}=\left[\begin{array}[]{rr}-1&2\\\
1&-2\end{array}\right]\,.$ (14)
The cone $\Lambda_{+}$ is a ray with the directional vector
$\boldsymbol{\lambda}=(2,1)$. Both $\lambda_{1,2}>0$, hence, both reactions
should be reversible and the condition holds:
$(k_{1}^{+})^{2}k_{2}^{+}=(k_{1}^{-})^{2}k_{2}^{-}$.
Let us slightly modify this example: $2{\rm H}\to{\rm H}_{2}$, ${\rm H}+{\rm
H_{2}}\to 3{\rm H}$.
$\boldsymbol{\Gamma}^{\mathrm{T}}=\left[\begin{array}[]{rr}-2&2\\\
1&-1\end{array}\right]\,.$ (15)
The cone $\Lambda_{+}$ is a ray with the directional vector
$\boldsymbol{\lambda}=(1,1)$. Both $\lambda_{1,2}>0$, hence, both reactions
should be reversible and the condition holds:
$k_{1}^{+}k_{2}^{+}=k_{1}^{-}k_{2}^{-}$.
If we change the direction of one reaction in the previous example then the
new irreversible systems satisfies the extended form of detailed balance:
$2{\rm H}\to{\rm H}_{2}$, $3{\rm H}\to{\rm H}+{\rm H_{2}}$.
$\boldsymbol{\Gamma}^{\mathrm{T}}=\left[\begin{array}[]{rr}-2&-2\\\
1&1\end{array}\right]\,.$ (16)
The cone $\Lambda_{+}$ is trivial (it includes no rays, just the origin),
hence, the structural condition holds. The algebraic condition trivially
holds, because there is no reversible reaction.
Let us add the forth reversible and nonlinear elementary reaction
$A_{1}+A_{2}\rightleftharpoons 2A_{3}$ (with the constants $k_{4}^{\pm}$) to a
linear reversible cycle. We should add to $\boldsymbol{\Gamma}^{\mathrm{T}}$
(13) two new columns:
$\boldsymbol{\Gamma}^{\mathrm{T}}=\left[\begin{array}[]{rrrrrrrr}-1&0&1&-1&1&0&-1&1\\\
1&-1&0&-1&-1&1&0&1\\\ 0&1&-1&2&0&-1&1&-2\end{array}\right]$ (17)
The extreme rays of $\Lambda_{+}$ include four rays that correspond to pairs
of mutually reverse reactions ($\boldsymbol{\lambda}^{1-4}$), two rays that
correspond to the linear cycle ($\boldsymbol{\lambda}^{5,6}$) and six rays for
three nonlinear cycles ($\boldsymbol{\lambda}^{7-12}$): (i) $A_{1}+A_{2}\to
2A_{3}$, $A_{3}\to A_{2}$, $A_{3}\to A_{1}$; (ii) $A_{1}+A_{2}\to 2A_{3}$,
$A_{3}\to A_{1}$, $A_{1}\to A_{2}$ and (iii) $A_{1}+A_{2}\to 2A_{3}$,
$A_{3}\to A_{2}$, $A_{2}\to A_{1}$:
$\displaystyle\boldsymbol{\lambda}^{5}=(1,1,1,0,0,0,0,0),\;\;$
$\displaystyle\boldsymbol{\lambda}^{6}=(0,0,0,0,1,1,1,0),$
$\displaystyle\boldsymbol{\lambda}^{7}=(0,0,1,1,0,1,0,0),\;\;$
$\displaystyle\boldsymbol{\lambda}^{8}=(0,1,0,0,0,0,1,1),$
$\displaystyle\boldsymbol{\lambda}^{9}=(1,0,2,1,0,0,0,0),\;\;$
$\displaystyle\boldsymbol{\lambda}^{10}=(0,0,0,0,1,0,2,1),$
$\displaystyle\boldsymbol{\lambda}^{11}=(0,0,0,1,1,2,0,0),\;\;$
$\displaystyle\boldsymbol{\lambda}^{12}=(1,2,0,0,0,0,0,1)\,.$
We omit $\boldsymbol{\lambda}^{1-4}$ which do not produce nontrivial
conditions. For the reversible reaction mechanism (when $k^{\pm}_{1-4}>0$),
there are two independent Wegscheider identities (8) that formalize the
classical principle of detailed balance:
$k_{1}^{+}k_{2}^{+}k_{3}^{+}=k_{1}^{-}k_{2}^{-}k_{3}^{-}$ and
$k_{3}^{+}k_{4}^{+}k_{2}^{-}=k_{3}^{-}k_{4}^{-}k_{2}^{+}$. If some of the
elementary reactions are irreversible then the direction vectors
$\boldsymbol{\lambda}^{5-12}$ produce 8 conditions. For
$\boldsymbol{\lambda}^{5,7,9,11}$ these conditions are below.
* 1.
($\boldsymbol{\lambda}^{5}$) If $k_{1,2,3}^{+}>0$ then $k_{1,2,3}^{-}>0$ and
$k_{1}^{+}k_{2}^{+}k_{3}^{+}=k_{1}^{-}k_{2}^{-}k_{3}^{-}$;
* 2.
($\boldsymbol{\lambda}^{7}$) If $k_{3,4}^{+},k_{2}^{-}>0$ then
$k_{3,4}^{-},k_{2}^{+}>0$ and
$k_{3}^{+}k_{4}^{+}k_{2}^{-}=k_{3}^{-}k_{4}^{-}k_{2}^{+}$;
* 3.
($\boldsymbol{\lambda}^{9}$) If $k_{1,3,4}^{+}>0$ then $k_{1,3,4}^{-}>0$ and
$k_{1}^{+}(k_{3}^{+})^{2}k_{4}^{+}=k_{1}^{-}(k_{3}^{-})^{2}k_{4}^{-}$;
* 4.
($\boldsymbol{\lambda}^{11}$) If $k_{4}^{+},k_{1,2}^{-}>0$ then
$k_{4}^{-},k_{1,2}^{+}>0$ and
$k_{4}^{+}k_{1}^{-}(k_{2}^{-})^{2}=k_{4}^{-}k_{1}^{+}(k_{2}^{+})^{2}$.
To obtain the conditions for $\boldsymbol{\lambda}^{6,8,10,12}$ it is
sufficient to change the superscripts + to - and inverse. These 8 conditions
represent the extended form of detailed balance for a given mechanism. To
check, whether a subsystem of this mechanism satisfies the extended form of
detailed balance, it is necessary and sufficient to check these conditions.
### 3.5 Methane Reforming Processes: a Case Study
#### 3.5.1 The System
Methane reforming with CO2 is a complex reaction network (Benson, 1981). The
main reactions in the methane reforming are:
1. 1.
${\rm CO}_{2}+{\rm H_{2}}\rightleftharpoons{\rm CO}+{\rm H}_{2}{\rm O}$ (RWGS,
Reverse water-gas shift);
2. 2.
${\rm CH}_{4}+{\rm CO}_{2}\rightleftharpoons{\rm 2CO}+{\rm 2H_{2}}$ (Dry
reforming);
3. 3.
${\rm CO}_{2}+4{\rm H}_{2}\rightleftharpoons{\rm CH}_{4}+2{\rm H}_{2}{\rm O}$
(Methanation);
4. 4.
${\rm CH}_{4}+{\rm H}_{2}{\rm O}\rightleftharpoons{\rm CO}+3{\rm H}_{2}$
(Steam reforming);
5. 5.
${\rm CH}_{4}\rightleftharpoons{\rm 2H}_{2}+{\rm C}$ (Methane decomposition);
6. 6.
$2{\rm CO}\rightleftharpoons{\rm CO}_{2}+{\rm C}$ (Boudouard reaction);
7. 7.
${\rm C}+{\rm H}_{2}{\rm O}\rightleftharpoons{\rm CO}+{\rm H}_{2}$ (Coal
gasification).
For the reagents, we use the notations $A_{1}={\rm CH}_{4}$, $A_{2}={\rm
CO}_{2}$, $A_{3}={\rm CO}$, $A_{4}={\rm H}_{2}$, $A_{5}={\rm H}_{2}{\rm O}$,
$A_{6}={\rm C}$. Amount of $A_{i}$ is $N_{i}$. There exist three independent
linear conservation laws: $b_{\rm C}=N_{1}+N_{2}+N_{3}+N_{6}$; $b_{\rm
H}=4N_{1}+2N_{4}+2N_{5}$; $b_{\rm O}=2N_{2}+N_{3}+N_{5}$. The number of
degrees of freedom in the closed system is three (six components minus three
independent conservation laws).
This example enriches our discussion because it deviates from the nice
abstract scheme discussed above. First of all, the reactions 1–7 are not
elementary steps. We consider them as overall reactions which have their own
intrinsic and complicated reaction mechanism. This does not cause a serious
problem because the generalized mass action law describes the equilibria of
the complex overall reactions as well as the equilibria of the elementary
ones. Therefore, we can apply the concept of the extended form of detailed
balance and our theorems 1–3 to the process network 1–7 build from the complex
reactions. Rigorously speaking, we deal not with the elementary reaction steps
but with the main equilibria and may discuss, for example, not the “Boudouard
reaction” but the “Boudouard equilibrium”.
The second problem is the heterogeneity of the system: $A_{1},\ldots,A_{5}$
are gases and $A_{6}={\rm C}$ is solid. Some of the reactions go on the
surface of the solid.
If a multiphase system is ideal and the solid components are stoichiometric
ones (i.e. they have a fixed composition) then the free energy has the form
$F=\sum_{A_{i}\;-\;{\rm gas}}N_{i}(RT\ln
c_{i}+\mu_{i}^{0}-RT)+\sum_{A_{i}\;-\;{\rm solid}}N_{i}\mu_{i}^{0}\,.$ (18)
Here, the free energy of solid components differs from the free energy of
gases by the absence of the term $RTN\ln c$. This term corresponds to the
ideal gas pressure $PV=NRT$. In our case,
$F=\sum_{i=1}^{5}N_{i}(RT\ln c_{i}+\mu_{i}^{0}-RT)+N_{6}\mu_{6}^{0}\,.$ (19)
To define the activities, we follow (9). For the ideal gases $a_{i}=c_{i}$ and
for the stoichiometric solids $a_{i}\equiv 1$.
In section 1.2, we studied homogeneous systems and considered $x_{i}=\ln
a_{i}$ as independent unknowns in the detailed balance equations (6):
$\sum_{i}\gamma_{ri}x_{i}=\ln K_{r}\;\;\;(x_{i}=\ln a_{i}^{\rm eq})\,.$ (20)
Therefore, for any solution of this system, the activities $a_{i}=\exp x_{i}$
represented a positive equilibrium.
In a heterogeneous system with the free energy (18) the activities for the
solid components are constant, the correspondent $x_{i}\equiv 0$. Let
$\boldsymbol{x}=(x_{i})$ be a solution to equations (20), $\boldsymbol{\Gamma
x}=\boldsymbol{K}$, where $\boldsymbol{K}$ is the vector of the equilibrium
constants. The vector $\boldsymbol{a}=(a_{i})$, $a_{i}=\ln x_{i}$ is a vector
of equilibrium activities if and only if $x_{i}=0$ for all the solid
components $A_{i}$. Instead of analyzing the solvability of the detailed
balance equations (20) we have to study its solvability under additional
condition: $x_{i}=0$ for all the solid components $A_{i}$.
Let us postpone the discussion of the extended principle of detailed balance
in multiphase systems and consider the system “gaseous mixture + one
stoichiometric solid”. Let $A_{n}$ be solid.
If there is the only solid component then the solvability conditions for the
system (20) and for this system with additional condition $x_{n}=0$ coincide.
Indeed, there exist a positive stoichiometric linear conservation law:
$\sum_{i=1}^{n}\gamma_{ri}b_{i}=0\mbox{ for all }r\mbox{ and }b_{i}>0\mbox{
for all }i\,.$
For example, this may be conservation of mass or of the amount of atoms. Let
$\boldsymbol{b}=(b_{i})$. For any solution of the detailed balance conditions
(20) $\boldsymbol{x}=(x_{i})$, the vector
$\boldsymbol{x}^{\prime}=\boldsymbol{x}-\frac{x_{n}}{b_{n}}\boldsymbol{b}$
is also a solution to (20) with the condition $x^{\prime}_{n}=0$.
So, for our example with seven equilibria 1–7 the conditions of the extended
principle of detailed balance for the heterogeneous system with solid
$A_{6}=$C are described by the theorems 1–3 and we can use the results of the
preceding sections.
#### 3.5.2 The Classical Wegscheider Conditions
To formulate the classical Wegscheider identities, we have to join the direct
and inverse reactions and to find the basic solutions of the system of linear
equations $\boldsymbol{\lambda}\boldsymbol{\Gamma}=0$. The stoichiometric
matrix for this example is:
$\boldsymbol{\Gamma}^{\mathrm{T}}=\left[\begin{array}[]{rrrrrrr}0&-1&1&-1&-1&0&0\\\
-1&-1&-1&0&0&1&0\\\ 1&2&0&1&0&-2&1\\\ -1&2&-4&3&2&0&1\\\ 1&0&2&-1&0&0&-1\\\
0&0&0&0&1&1&-1\end{array}\right]$ (21)
The system of seven equations $\boldsymbol{\lambda}\boldsymbol{\Gamma}=0$ is
redundant. There are only three independent equations (one equation for every
degree of freedom). It is sufficient to take the components of stoichiometric
vectors that correspond to the components $A_{2}$, $A_{4}$ , $A_{6}$. Other
components satisfy the same linear relations as the selected ones. The reduced
matrix $\boldsymbol{\Gamma}_{\rm r}^{\mathrm{T}}$ is
$\boldsymbol{\Gamma}_{\rm
r}^{\mathrm{T}}=\left[\begin{array}[]{rrrrrrr}-1&-1&-1&0&0&1&0\\\
-1&2&-4&3&2&0&1\\\ 0&0&0&0&1&1&-1\end{array}\right]$ (22)
There are four independent solutions of the equations
$\boldsymbol{\lambda}\boldsymbol{\Gamma}=0$ (seven variables minus three
independent equations). For example, we can take the following basis of
solutions: $(-1,1,0,-1,0,0,0)$, $(0,0,0,-1,1,0,1)$, $(1,0,0,0,0,1,1)$,
$(1,0,-1,-1,0,0,0)$.
The correspondent Wegscheider identities are: $K_{2}=K_{1}K_{4}$,
$K_{5}K_{7}=K_{4}$, $K_{1}K_{6}K_{7}=1$, $K_{1}=K_{3}K_{4}$.
#### 3.5.3 Allowed and Forbidden Mechanisms
In general, all the seven reactions can be considered as reversible but under
various conditions some of them are almost irreversible. Let us study which
combinations of irreversible reactions are possible in accordance with the
extended form of detailed balance.
For example, existence of the positive solution $(0,1,0,0,0,1,1)\in\Lambda$
guarantees that the irreversible system ${\rm CO}_{2}+{\rm H_{2}}\to{\rm
CO}+{\rm H}_{2}{\rm O}$, ${\rm CH}_{4}+{\rm CO}_{2}\to{\rm 2CO}+{\rm 2H_{2}}$,
${\rm CO}_{2}+4{\rm H}_{2}\to{\rm CH}_{4}+2{\rm H}_{2}{\rm O}$, ${\rm
CH}_{4}+{\rm H}_{2}{\rm O}\to{\rm CO}+3{\rm H}_{2}$, ${\rm CH}_{4}\to{\rm
2H}_{2}+{\rm C}$, $2{\rm CO}\to{\rm CO}_{2}+{\rm C}$, ${\rm C}+{\rm H}_{2}{\rm
O}\to{\rm CO}+{\rm H}_{2}$ is forbidden by the extended form of detailed
balance. This conclusion is also obvious from the correspondent Wegscheider
condition $K_{2}K_{6}K_{7}=1$. Indeed, if all the $k_{i}^{-}\to 0$ for bounded
from below $k_{i}^{+}>\varepsilon>0$ then all $K_{i}\to\infty$ and
$K_{2}K_{6}K_{7}\to\infty$. This contradicts the Wegscheider condition.
The first reaction (RWGS, Reverse water-gas shift) is reversible in the wide
interval of conditions (Moe, 1962). Let us first study all the reaction
mechanisms with the reversible first reaction and the irreversible reactions
2-7. We find the combinations of the directions of the irreversible reactions
that satisfy the extended form of detailed balance. As a criterion of the
extended form of detailed balance we use Theorem 3. After that, we consider
other reactions as the reversible ones (in addition to RWGS) and study the
corresponding reaction mechanisms.
The space $S$ is a straight line with the directional vector $\gamma_{1}$ with
coordinates $(-1,-1,0)$ in the coordinate system $(N_{2},N_{4},N_{6})$ that
corresponds to the components $A_{2}$, $A_{4}$, $A_{6}$. Let us represent the
quotient space $\mathbb{R}^{3}/S$ in the coordinate system $(N_{2},N_{6})$
that corresponds to the components $A_{2}$, $A_{6}$. For this purpose, we have
to eliminate the coordinate $N_{4}$ using vector $\gamma_{1}$. As a result, we
get the following vectors:
$\begin{split}\overline{\gamma}_{2}=\left(\begin{array}[]{r}-3\\\
0\end{array}\right)\,,\overline{\gamma}_{3}=\left(\begin{array}[]{r}3\\\
0\end{array}\right)\,,\overline{\gamma}_{4}=\left(\begin{array}[]{r}-3\\\
0\end{array}\right)\,,\\\ \overline{\gamma}_{5}=\left(\begin{array}[]{r}-2\\\
1\end{array}\right)\,,\overline{\gamma}_{6}=\left(\begin{array}[]{r}1\\\
1\end{array}\right)\,,\overline{\gamma}_{7}=\left(\begin{array}[]{r}-1\\\
-1\end{array}\right)\,.\end{split}$ (23)
For example, to find $\overline{\gamma}_{2}$, we take $\gamma_{2}$ (the second
column in (22)) and exclude the coordinate $N_{4}$ by adding $2\gamma_{1}$.
The result is a vector $\gamma_{2}+2\gamma_{1}$. In coordinates
$(N_{2},N_{6})$, this vector gives us $\overline{\gamma}_{2}$.
Figure 1: Images of the stoichiometric vectors of irreversible reactions
$\overline{\nu}_{j}=\pm\overline{\gamma}_{j}$ in $\mathbb{R}^{3}/S$ for
various combinations of directions of reactions (24) in coordinates $N_{2}$
(abscissa), $N_{6}$. The configurations with $0\in{\rm
conv}\\{\overline{\nu}_{2},\ldots,\overline{\nu}_{7}\\}$ are outlined. Vectors
$\overline{\nu}_{2}$, $\overline{\nu}_{3}$ and $\overline{\nu}_{4}$ coincide
as well as vectors $\overline{\nu}_{6}$ and $\overline{\nu}_{7}$.
The stoichiometric vectors of irreversible reactions are $+\gamma_{j}$ or
$-\gamma_{j}$ ($j=2,\ldots,7$). Their images in the quotient space
$\mathbb{R}^{3}/S$ are $+\overline{\gamma}_{j}$ or $-\overline{\gamma}_{j}$.
The extended form of detailed balance requires that the convex envelope of
these vectors should not include zero. We have to arrange signs in
$\pm\gamma_{j}$ to provide this property. First of all, we see immediately
from (23) that the second and the forth reaction should have the same
directions and the third reaction should have the opposite direction. The
directions of the sixth and the seventh reactions should be opposite.
Therefore, we have to analyze eight possible reaction mechanisms. Let us
represent them by the directions of reactions:
$\left[\begin{array}[]{c}({\rm a})\\\ \rightleftharpoons\\\ \to\\\
\leftarrow\\\ \to\\\ \to\\\ \to\\\
\leftarrow\end{array}\right];\left[\begin{array}[]{c}({\rm b})\\\
\rightleftharpoons\\\ \leftarrow\\\ \to\\\ \leftarrow\\\ \to\\\ \to\\\
\leftarrow\end{array}\right];\left[\begin{array}[]{c}({\rm c})\\\
\rightleftharpoons\\\ \to\\\ \leftarrow\\\ \to\\\ \leftarrow\\\ \to\\\
\leftarrow\end{array}\right];\left[\begin{array}[]{c}({\rm d})\\\
\rightleftharpoons\\\ \leftarrow\\\ \to\\\ \leftarrow\\\ \leftarrow\\\ \to\\\
\leftarrow\end{array}\right];\left[\begin{array}[]{c}({\rm e})\\\
\rightleftharpoons\\\ \to\\\ \leftarrow\\\ \to\\\ \to\\\ \leftarrow\\\
\to\end{array}\right];\left[\begin{array}[]{c}({\rm f})\\\
\rightleftharpoons\\\ \leftarrow\\\ \to\\\ \leftarrow\\\ \to\\\ \leftarrow\\\
\to\end{array}\right];\left[\begin{array}[]{c}({\rm g})\\\
\rightleftharpoons\\\ \to\\\ \leftarrow\\\ \to\\\ \leftarrow\\\ \leftarrow\\\
\to\end{array}\right];\left[\begin{array}[]{c}({\rm h})\\\
\rightleftharpoons\\\ \leftarrow\\\ \to\\\ \leftarrow\\\ \leftarrow\\\
\leftarrow\\\ \to\end{array}\right].$ (24)
Arrows here correspond to the directions of reactions. For example, the case
(a) corresponds to the reaction mechanism
1. 1.
${\rm CO}_{2}+{\rm H_{2}}\rightleftharpoons{\rm CO}+{\rm H}_{2}{\rm O}$;
2. 2.
${\rm CH}_{4}+{\rm CO}_{2}\to{\rm 2CO}+{\rm 2H_{2}}$;
3. 3.
${\rm CO}_{2}+4{\rm H}_{2}\leftarrow{\rm CH}_{4}+2{\rm H}_{2}{\rm O}$;
4. 4.
${\rm CH}_{4}+{\rm H}_{2}{\rm O}\to{\rm CO}+3{\rm H}_{2}$;
5. 5.
${\rm CH}_{4}\to{\rm 2H}_{2}+{\rm C}$;
6. 6.
$2{\rm CO}\to{\rm CO}_{2}+{\rm C}$;
7. 7.
${\rm C}+{\rm H}_{2}{\rm O}\leftarrow{\rm CO}+{\rm H}_{2}$.
Combinations (c) and (f) contradict the condition 1 from Theorem 3: the origin
belongs to the convex envelope of the vectors $\overline{\nu}_{j}$ of
irreversible reactions (see Fig. 1). Hence, only six combinations of
directions of irreversible reactions satisfy the extended form of detailed
balance (from $2^{6}=64$ possible combinations of directions): (a), (b), (d),
(e), (g) and (h).
Let us extend the list of reversible reactions. If we assume that the second
reaction (dry reforming), is reversible together with the first one (RWGS)
then the third and the forth reactions should be also reversible because
$\gamma_{3}=2\gamma_{1}-\gamma_{2}$ and $\gamma_{4}=\gamma_{2}-\gamma_{1}$,
hence, $\gamma_{3,4}\in{\rm span}\\{\gamma_{1},\gamma_{2}\\}$. According to
the condition 1 from Theorem 3, this contradicts to the extended form of
detailed balance if the first and the second reactions are reversible and the
third and the forth are not.
Analogously, in addition to the reversible reaction RWGS, the Boudouard
equilibrium 6 and coal gasification 7 can be reversible only together because
$\gamma_{7}=-\gamma_{1}-\gamma_{6}$.
We have to consider three possible sets of reversible reactions:
1. 1.
1, 2, 3 and 4;
2. 2.
1 and 5;
3. 3.
1, 6 and 7.
For all three cases, $\dim S=2$ and $\dim(\mathbb{R}^{3}/S)=1$. We will use
for the quotient space the coordinate $N_{6}$ which corresponds to $A_{6}={\rm
C}$.
In the first case, let us exclude the coordinate $N_{4}$ from
$\overline{\gamma}_{5,6,7}$ (23) using vector $\overline{\gamma}_{2}$. We get
one-dimensional vectors
$\overline{\gamma}_{5}=1,\;\overline{\gamma}_{6}=1,\;\overline{\gamma}_{7}=-1\,.$
To satisfy the extended form of detailed balance the directions of the fifth
and the sixth reaction should coincide and the direction of the seventh
reaction should be opposite: there are two possible combinations of arrows in
irreversible reactions 5, 6 and 7 if reactions 1, 2, 3 and 4 are reversible:
$5\to,\,6\to,\,7\leftarrow$ and $5\leftarrow,\,6\leftarrow,\,7\to$.
In the second case, let us exclude the coordinate $N_{4}$ from
$\overline{\gamma}_{2,3,4,6,7}$ (23) using vector $\overline{\gamma}_{5}$. We
get one-dimensional vectors
$\overline{\gamma}_{2}=-2/3,\;\overline{\gamma}_{3}=2/3,\;\overline{\gamma}_{4}=-2/3,\;\overline{\gamma}_{6}=1/2,\;\overline{\gamma}_{7}=-1/2\,.$
Again, according to the extended form of detailed balance, here are two
possibilities of directions of irreversible reactions 2, 3, 4, 6 and 7 if
reactions 1 and 5 are reversible:
$2\to,\,3\leftarrow,\,4\to,\,6\leftarrow,\,7\to$ and
$2\leftarrow,\,3\to,\,4\leftarrow,\,6\to,\,7\leftarrow$.
In the third case, let us exclude the coordinate $N_{4}$ from
$\overline{\gamma}_{2,3,4,5}$ (23) using vector $\overline{\gamma}_{6}$. We
get one-dimensional vectors
$\overline{\gamma}_{2}=3,\;\overline{\gamma}_{3}=-3,\;\overline{\gamma}_{4}=3,\;\overline{\gamma}_{5}=3\,.$
According to the extended form of detailed balance, here are two possibilities
of directions of irreversible reactions 2, 3, 4, and 5 if reactions 1, 6 and 7
are reversible: $2\to,\,3\leftarrow,\,4\to,\,5\to$ and
$2\leftarrow,\,3\to,\,4\leftarrow,\,5\leftarrow$.
In the first and the third cases, there are nontrivial Wegscheider identities
for the reaction equilibrium constants of reversible reactions. If reactions
1, 2, 3 and 4 are reversible (case 1) then $\dim\Lambda=2$ and the basis of
$\Lambda$ is, for example, $\boldsymbol{\lambda}^{1}=(2,-1,-1,0)$
($2\gamma_{1}-\gamma_{2}-\gamma_{3}=0$) and
$\boldsymbol{\lambda}^{2}=(1,-1,0,1)$ ($\gamma_{1}-\gamma_{2}+\gamma_{4}=0$).
The two correspondent Wegscheider identities are: $K_{1}^{2}=K_{2}K_{3}$ and
$K_{1}K_{4}=K_{2}$ (where $K_{i}=k_{i}^{+}/k_{i}^{-}$).
If the reactions 1, 6 and 7 are reversible then $\dim\Lambda=1$ and the basis
of $\Lambda$ consists of one vector $\boldsymbol{\lambda}=(1,1,1)$
($\gamma_{1}+\gamma_{6}+\gamma_{7}=0$). The correspondent Wegscheider identity
is: $K_{1}K_{6}K_{7}=1$.
If we add one more reversible reaction in cases 1-3 then all the reactions 1-7
should be reversible in according to the extended form of detailed balance.
In this case study, we demonstrated also how it is possible to organize
computations and reduce the dimension of the computational problems.
## 4 Multiscale Degenerated Equilibria
Let in a system of reversible reactions with detailed balance some
$k_{s}^{-}\to 0$, when the correspondent $k_{s}^{+}$ remains constant and
separated from zero. In this case, some equilibrium activities also tend to
zero. Indeed, at equilibrium $w_{s}^{+}=w_{s}^{-}$, $w_{s}^{-}\to 0$ because
$k_{s}^{-}\to 0$, hence, $w_{s}^{+}\to 0$ and some of $a_{i}^{\rm eq}$ with
$\alpha_{si}>0$ also tend to zero due to the generalized mass action law (5).
Therefore, the irreversible limits of the reactions with detailed balance are
closely related to the limits when some equilibrium activities tend to zero.
(For the usual mass action law is sufficient to replace the “activity $a_{i}$”
by the “concentration $c_{i}$”.)
In this section we study asymptotics $a_{i}^{\rm eq}={\rm
const}\times\varepsilon^{\delta_{i}}$, $\varepsilon\to 0$ for various values
of non-negative exponents $\delta_{i}\geq 0$ ($i=1,\ldots,n$).
There exists a well known way to satisfy the principle of detailed balance:
just write $k^{-}_{r}=k^{+}_{r}/K_{r}$ where $K_{r}$ is the equilibrium
constant:
$K_{r}=\frac{\prod_{i=1}^{n}(a_{i}^{\rm
eq})^{\beta_{ri}}}{\prod_{i=1}^{n}(a_{i}^{\rm eq})^{\alpha_{ri}}}\,.$
We can define the equilibrium constant through the equilibrium thermodynamics
as well (see, for example, the classical book by Prigogine & Defay (1962)). In
this case, the principle of detailed balance is also satisfied for the mass
action law.
In this approach, we have to group the direct and reverse reactions together.
Therefore, $m$ is here the number of pairs of reactions, direct + inverse
ones. We deal with $m+n$ constants ($m$ rate constants $k_{r}^{+}$ for direct
reactions and $n$ equilibrium data for individual reagents: equilibrium
concentrations or activities) instead of $2m$ constants $k_{r}^{\pm}$. For
these $m+n$ constants, the principle of detailed balance produces no
restrictions (Gorban et al, 1989; Yang, et al, 2006). It holds “by the
construction” for any positive values of these constants if
$k^{-}_{r}=k^{+}_{r}/K_{r}$ and the equilibrium constants are calculated in
accordance with the equilibrium data.
To transform the conditions of $a_{i}^{\rm eq}\to 0$ into irreversibility of
some reactions, it is not sufficient to know which $a_{i}^{\rm eq}\to 0$. We
have to take into account the rates of these convergence to zero for different
$i$. In the simple example, $A_{1}\rightleftharpoons A_{2}\rightleftharpoons
A_{3}\rightleftharpoons A_{1}$, if $a_{1,2}^{\rm eq}\to 0$, $a_{1}/a_{2}\to 0$
then in the limit we get the system $A_{1}\to A_{2}$ (because the
$A_{1}/A_{2}$ equilibrium is shifted to $A_{2}$), $A_{1}\to A_{3}$, $A_{2}\to
A_{3}$. For the inverse relations between $a_{1}$ and $a_{2}$, $a_{2}/a_{1}\to
0$, the limit system is $A_{2}\to A_{1}$ (the $A_{1}/A_{2}$ equilibrium is
shifted to $A_{1}$), $A_{1}\to A_{3}$, $A_{2}\to A_{3}$. For the both limit
systems, the equilibrium activities of $A_{1}$, $A_{2}$ are zero but the
directions of reaction are different.
The limit structure of the reaction mechanism when some of $a_{i}^{\rm eq}\to
0$ depends on the behavior of the ratios $a_{i}^{\rm eq}/a_{i}^{\rm eq}$. To
formalize this dependence, let us introduce a parameter $\varepsilon>0$ and
take $a_{i}^{\rm eq}={\rm const}\times\varepsilon^{\delta_{i}}$. At
equilibrium, each monomial in the generalized mass action law is proportional
to a power of $\varepsilon$:
$w_{r}^{{\rm eq}+}=k_{r}^{+}{\rm
const}\times\varepsilon^{\sum_{i}\alpha_{ri}\delta_{i}}\,,\;\;w_{r}^{{\rm
eq}-}=k_{r}^{-}{\rm const}\times\varepsilon^{\sum_{i}\beta_{ri}\delta_{i}}\,.$
The principle of detailed balance gives: $w_{r}^{{\rm eq}+}=w_{r}^{{\rm
eq}-}$. Therefore,
$\frac{k_{r}^{+}}{k_{r}^{-}}={\rm
const}\times\varepsilon^{(\gamma_{r},\delta)}\,,$ (25)
where $\delta$ is the vector with coordinates $\delta_{i}$.
There are three possibilities for the reversibility of an elementary reaction
in asymptotic $\varepsilon\to 0$:
1. 1.
If $(\gamma_{r},\delta)=0$ then the reaction remains reversible in asymptotic
$\varepsilon\to 0$. This means that $0<\lim(k_{s}^{+}/k_{s}^{-})<\infty$.
Therefore, if one of the reactions persists in the limit then the reverse
reaction also persists.
2. 2.
If $(\gamma_{r},\delta)<0$ then in asymptotic $\varepsilon\to 0$ can remain
only direct reaction. This means that $\lim(k_{s}^{-}/k_{s}^{+})=0$.
3. 3.
If $(\gamma_{r},\delta)>0$ then in asymptotic $\varepsilon\to 0$ can remain
only reverse reaction. This means that $\lim(k_{s}^{+}/k_{s}^{-})=0$.
It is possible that $(\gamma_{r},\delta)=0$ but both $k_{r}^{{\rm lim}\pm}=0$
just because $k^{+}_{r}=0$ and $k^{-}_{r}=0$ and not because of the
equilibrium degeneration. If we delete some irreversible reactions or several
pairs of mutually reverse reaction then the extended form of detailed balance
persists. Therefore, we do not consider these cases separately and always
discuss the limit reaction mechanisms with $\max\\{k_{r}^{{\rm
lim}+},k_{r}^{{\rm lim}-}\\}>0$.
For each stoichiometric vector $\gamma_{r}$ the $n$-dimensional space of
vectors $\delta$ is split in three sets: hyperplane $(\gamma_{r},\delta)=0$
(reaction remains reversible), hemispace $(\gamma_{r},\delta)<0$ (only direct
reaction remains) and hemispace $(\gamma_{r},\delta)>0$ (only reverse reaction
remains). For the reaction mechanism, intersections of these sets for all
$\gamma_{r}$ ($r=1,\ldots,m$) form a tiling of the n-dimensional space of
vectors $\delta$. The intersection of all hyperplanes $(\gamma_{r},\delta)=0$
corresponds to the initial reversible reaction mechanism. Other sets from this
tiling correspond to the reaction mechanisms that are limits of the initial
reaction mechanism when some of the reaction rate constants tend to zero but
the principle of detailed balance is valid. In our study, the exponents
$\delta_{j}$ should be non-negative, hence, we have to study the tiling of the
positive orthant $\delta_{j}\geq 0$ in $\mathbb{R}^{n}$ Description of the
tiling produced by a system of hyperplanes $(\gamma_{r},\delta)=0$ is a
classical problem of combinatorial geometry.
In the usual linear triangle $A_{1}\rightleftharpoons A_{2}\rightleftharpoons
A_{3}\rightleftharpoons A_{1}$ we have to consider three hyperplanes in the
space of exponents $\delta=(\delta_{1},\delta_{2},\delta_{3})$:
$\delta_{1}=\delta_{2}$ ($(\gamma_{1},\delta)=0$), $\delta_{2}=\delta_{3}$
($(\gamma_{2},\delta)=0$) and $\delta_{3}=\delta_{1}$
($(\gamma_{3},\delta)=0$). At least one of the exponents should take zero
value to keep the overall concentration in equilibrium neither zero nor
infinite. Let us take $\delta_{1}=0$. The hyperplanes turn in the straight
lines on the plane $(\delta_{2},\delta_{3})$: $0=\delta_{2}$
($(\gamma_{1},\delta)=0$), $\delta_{2}=\delta_{3}$ ($(\gamma_{2},\delta)=0$)
and $\delta_{3}=0$ ($(\gamma_{3},\delta)=0$). The positive octant on the plane
$(\delta_{2},\delta_{3})$ is split in five sets (A)–(E), that correspond to
the limits with some irreversible reactions, and the origin:
* 1.
(A) $\delta_{2}=0$, $\delta_{3}>0$, $A_{1}\rightleftharpoons A_{2}$, $A_{3}\to
A_{1}$, $A_{3}\to A_{2}$;
* 2.
(B) $\delta_{3}>\delta_{2}>0$, $A_{3}\to A_{2}\to A_{1}$, $A_{3}\to A_{1}$;
* 3.
(C) $\delta_{3}=\delta_{2}>0$, $A_{3}\rightleftharpoons A_{2}$, $A_{2}\to
A_{1}$, $A_{3}\to A_{1}$;
* 4.
(D) $\delta_{2}>\delta_{3}>0$, $A_{2}\to A_{3}\to A_{1}$, $A_{2}\to A_{1}$
(this case differs from (B) by the transposition $2\leftrightarrow 3$);
* 5.
(E) $\delta_{2}>0$, $\delta_{3}=0$ $A_{1}\rightleftharpoons A_{3}$ , $A_{2}\to
A_{1}$, $A_{2}\to A_{3}$ (this case differs from (A) by the transposition
$2\leftrightarrow 3$).
* 6.
The origin corresponds to the fully reversible mechanism.
For a less trivial example, let us analyze the reaction mechanism from Section
3.4: $A_{1}\rightleftharpoons A_{2}\rightleftharpoons A_{3}\rightleftharpoons
A_{1}$, $A_{1}+A_{2}\rightleftharpoons 2A_{3}$. This is a reversible cycle
supplemented by a nonlinear step.
We join the direct and reverse elementary reactions and, therefore,
$\boldsymbol{\Gamma}^{\mathrm{T}}=\left[\begin{array}[]{rrrr}-1&0&1&-1\\\
1&-1&0&-1\\\ 0&1&-1&2\end{array}\right]$ (26)
The columns of this matrix are the stoichiometric vectors $\gamma_{r}$.
Let us study the tiling of the positive orthant in $\mathbb{R}^{3}$ by the
planes $(\gamma_{r},\delta)=0$ ($r=1,\ldots,4$). First of all, it is necessary
and sufficient to study this tiling of the positive octants in three planes:
$\delta_{1}=0$, or $\delta_{2}=0$, or $\delta_{3}=0$ because at least one
equilibrium concentration should not tend to zero and, therefore, has zero
exponent. The symmetry between $A_{1}$ and $A_{2}$ allows us to study two
planes: $\delta_{1}=0$ or $\delta_{3}=0$.
Figure 2: Tiling of the positive octant of the plane
$(\delta_{2},\delta_{3})$ ($\delta_{1}=1$) that corresponds to seven
irreversible limits of the reaction mechanism.
On the plane $\delta_{1}=0$ with coordinates $\delta_{2}$, $\delta_{3}$ we
have four straight lines: $(\gamma_{1},\delta)=0$ ($\delta_{2}=0$),
$(\gamma_{2},\delta)=0$, ($\delta_{2}=\delta_{3}$), $(\gamma_{3},\delta)=0$
($\delta_{3}=0$) and $(\gamma_{4},\delta)=0$ ($\delta_{2}=2\delta_{3}$). These
lines divide the positive octant ($\delta_{2,3}\geq 0$) into seven parts (Fig.
2) and the origin:
1. 1.
(A) $\delta_{2}=0$, $\delta_{3}>0$, $A_{1}\rightleftharpoons A_{2}$, $A_{3}\to
A_{1}$, $A_{3}\to A_{2}$, $2A_{3}\to A_{1}+A_{2}$;
2. 2.
(B) $\delta_{2}>0$, $\delta_{3}>\delta_{2}$, $A_{2}\to A_{1}$, $A_{3}\to
A_{1}$, $A_{3}\to A_{2}$, $2A_{3}\to A_{1}+A_{2}$;
3. 3.
(C) $\delta_{2}=\delta_{3}>0$, $A_{2}\to A_{1}$, $A_{3}\to A_{1}$,
$A_{3}\rightleftharpoons A_{2}$, $2A_{3}\to A_{1}+A_{2}$;
4. 4.
(D) $0<\delta_{3}<\delta_{2}<2\delta_{3}$, $A_{2}\to A_{1}$, $A_{3}\to A_{1}$,
$A_{2}\to A_{3}$, $2A_{3}\to A_{1}+A_{2}$;
5. 5.
(E) $0<\delta_{2}=2\delta_{3}$, $A_{2}\to A_{1}$, $A_{3}\to A_{1}$, $A_{2}\to
A_{3}$, $2A_{3}\rightleftharpoons A_{1}+A_{2}$;
6. 6.
(F) $\delta_{2}>2\delta_{3}>0$, $A_{2}\to A_{1}$, $A_{3}\to A_{1}$, $A_{2}\to
A_{3}$, $A_{1}+A_{2}\to 2A_{3}$;
7. 7.
(G) $\delta_{3}=0$, $\delta_{2}>0$, $A_{2}\to A_{1}$, $A_{1}\rightleftharpoons
A_{3}$, $A_{2}\to A_{3}$, $A_{1}+A_{2}\to 2A_{3}$;
8. 8.
The origin corresponds to the fully reversible mechanism.
The same picture gives us the plane $\delta_{2}=0$ with coordinates
$\delta_{1}$, $\delta_{3}$: we need just to transpose the indexes,
$1\leftrightarrow 2$.
On the plane $\delta_{3}=0$ with coordinates $\delta_{1}$, $\delta_{2}$ the
positive octant is divided into five parts and the origin:
1. 1.
$\delta_{1}=0$, $\delta_{2}>0$, $A_{2}\to A_{1}$, $A_{1}\rightleftharpoons
A_{3}$, $A_{2}\to A_{3}$, $A_{1}+A_{2}\to 2A_{3}$ (this is exactly the case
(G) from Fig. 2);
2. 2.
$0<\delta_{1}<\delta_{2}$, $A_{2}\to A_{1}$, $A_{1}\to A_{3}$, $A_{2}\to
A_{3}$, $A_{1}+A_{2}\to 2A_{3}$;
3. 3.
$0<\delta_{1}=\delta_{2}$, $A_{1}\rightleftharpoons A_{2}$, $A_{1}\to A_{3}$,
$A_{2}\to A_{3}$, $A_{1}+A_{2}\to 2A_{3}$;
4. 4.
$\delta_{1}>\delta_{2}>0$, $A_{1}\to A_{2}$, $A_{1}\to A_{3}$, $A_{2}\to
A_{3}$, $A_{1}+A_{2}\to 2A_{3}$;
5. 5.
$\delta_{2}=0$, $\delta_{1}>0$, $A_{1}\to A_{2}$, $A_{1}\to A_{3}$,
$A_{2}\rightleftharpoons A_{3}$, $A_{1}+A_{2}\to 2A_{3}$;
6. 6.
The origin corresponds to the fully reversible mechanism.
This approach is equivalent to the previous definition of the extended form of
detailed balance based on the pathway analysis. Indeed, if the reaction
mechanism with some irreversible reactions is a limit of the reversible
mechanism with detailed balance then it satisfies the conditions of the
extended form of detailed balance. (This is the direct statement of Theorem 1
proved in Section 3.2.) To prove the converse statement, we have to take a
system that satisfies the extended form of detailed balance and to find such a
set of exponents $\delta_{i}\geq 0$ ($i=1,\ldots,n$) that the system appears
in the limit of a reversible system with detailed balance when $\varepsilon\to
0$ and $a_{i}^{\rm eq}={\rm const}\times\varepsilon^{\delta_{i}}$.
Let a system with some irreversible reactions satisfy the extended form of
detailed balance. We follow the notations of Theorem 3: $\gamma_{j}$
($j=1,\ldots,r$) are the stoichiometric vectors of the reversible reactions
and $\nu_{1},\ldots,\nu_{s}$ are the stoichiometric vectors of the
irreversible reactions. The linear subspace $S={\rm
span}\\{\gamma_{1},\ldots,\gamma_{r}\\}\subset\mathbb{R}^{n}$ consists of all
linear combinations of the stoichiometric vectors of the reversible reactions.
We use notation $\overline{\nu}_{j}$ for the images of $\nu_{j}$ in
$\mathbb{R}^{n}/S$.
Let $k_{j}^{\pm}>0$ ($j=1,\ldots,r$) be the reaction rate constants for the
reversible reactions and $q_{j}=q_{j}^{+}>0$ ($j=1,\ldots,s$) be the reaction
rate constants for the irreversible reactions. We extend the system by adding
the reverse reactions with the constants $q_{j}^{-}>0$. If the extended system
satisfies the principle of detailed balance then
$\frac{k_{j}^{+}}{k_{j}^{-}}=\prod_{i=1}^{n}(a_{i}^{\rm
eq})^{\gamma_{ri}}\;\;\mbox{ and
}\;\;\frac{q_{j}^{+}}{q_{j}^{-}}=\prod_{i=1}^{n}(a_{i}^{\rm
eq})^{\nu_{ri}}\,,$ (27)
where $a_{i}^{\rm eq}$ is a point of detailed balance.
###### Theorem 4.
Let the system satisfy the extended form of detailed balance. Then there
exists a vector of nonnegative exponents $\delta=(\delta_{i})$
($i=1,\ldots,n$) and the family of extended systems with equilibria
$a_{i}^{\rm eq}=a_{i}^{*}\varepsilon^{\delta_{i}}$ such that condition (27)
hold, $k_{j}^{\pm}$ ($j=1,\ldots,r$) and $q_{j}=q_{j}^{+}$ ($j=1,\ldots,s$) do
not depend on $\varepsilon$, and $q_{j}^{-}\to 0$ when $\varepsilon\to 0$.
###### Proof.
If the system satisfies the extended form of detailed balance then the
reversible part satisfies the principle of detailed balance and, hence, there
exists a positive point of detailed balance for the reversible part of the
system (Theorem 3): $a_{i}^{*}>0$ and
$k_{j}^{+}\prod_{i=1}^{n}(a_{i}^{*})^{\alpha_{ri}}=k_{j}^{+}\prod_{i=1}^{n}(a_{i}^{*})^{\beta_{ri}}\,.$
Let us take $a_{i}^{\rm eq}=a_{i}^{*}\varepsilon^{\delta_{i}}$. Due to (27),
$k_{j}^{+}/k_{j}^{-}={\rm const}\times\varepsilon^{(\gamma_{j},\delta)}$. To
keep the $k_{i}^{\pm}$ independent of $\varepsilon$, we have to provide
$(\gamma_{j},\delta)=0$. Analogously, $q_{j}^{+}/q_{j}^{-}={\rm
const}\times\varepsilon^{(\nu_{j},\delta)}$. The rate constant $q_{j}^{+}$
should not depend on $\varepsilon$ and $q_{j}^{-}\to 0$ when $\varepsilon\to
0$. Therefore, $(\nu_{j},\delta)<0$. We came to the system of linear equations
and inequalities with respect to exponents $\delta_{i}$:
$(\gamma_{j},\delta)=0\;(j=1,\ldots,r),\;\;(\nu_{j},\delta)<0\;(j=1,\ldots,s)\,.$
(28)
The solvability of this system is equivalent to the condition 1 of Theorem 3
(see Remark 7). To prove the existence of nonnegative exponents
$\delta_{i}\geq 0$, we have to use existence of positive conservation law:
$b_{i}>0$, $(\gamma_{j},b)=0$, $(\nu_{j},b)=0$. For every solution $\delta$ of
(28) and any number $d$, the vector $\delta+db$ is also a solution of (28).
Therefore, the nonnegative solution exists. We proved the theorem and the
converse statement of Theorem 1. ∎
###### Proposition 4.
Let a system with the stoichiometric vectors $\gamma_{s}$ and the extended
detailed balance be obtained from the reversible systems with detailed balance
in the limit $a_{i}^{\rm eq}={\rm const}\times\varepsilon^{\delta_{i}}$,
$\varepsilon\to 0$. For this system, the linear function $(\delta,c)$ of the
concentrations $c$ monotonically decreases in time due to the kinetic
equations $\frac{{\mathrm{d}}c}{{\mathrm{d}}t}=\sum_{s}w_{s}\gamma_{s}$.
###### Proof.
Indeed,
$\frac{{\mathrm{d}}(\delta,c)}{{\mathrm{d}}t}=\sum_{s}w_{s}(\gamma_{s},\delta)$
(compare to Remark 6). For the reversible reactions, the sign of $w_{s}$ is
indefinite but $(\gamma_{s},\delta)=0$. For the irreversible reactions, we
always can take $w_{s}=w_{s}^{+}\geq 0$ just by the selection of notations. In
this case, only $k^{+}_{s}$ survived in the limit $\varepsilon\to 0$, this
means that $(\gamma_{s},\delta)<0$. Therefore,
$\frac{{\mathrm{d}}(\delta,c)}{{\mathrm{d}}t}\leq 0$ and it is zero if and
only if all the reaction rates of the irreversible reactions vanish. ∎
So, the vector of exponents $\delta$ defines the (partially) irreversible
limit of the reaction mechanism and, at the same time, gives the explicit
construction of the special Lyapunov function for the kinetic equations of the
limit system.
In this Section, we developed the approach to the systems with some
irreversible reactions based on multiscale degeneration of equilibria, when
some $a_{i}\to 0$ as $\varepsilon^{\delta_{i}}$. We proved in Theorem 4 that
this approach is equivalent to the extended form of detailed balance based on
the pathways analysis or on the limits of the systems with detailed balance
when some of the reaction rate constants tend to zero.
## 5 Conclusion
The classical principle of detailed balance operates with mechanisms, which
consist of fully reversible elementary processes (reactions). If such
mechanisms have cycles of reactions, each cycle is characterized by one
Wegscheider relationship (8) between its rate constants. The number of
functionally independent relationships is equal to the number of linearly
independent cycles, linear or nonlinear.
In difference from this classical case, we analyzed mechanisms, which may
include irreversible reactions as well. For such mechanisms we proved an
extended form of detailed balance considering the irreversible reactions as
limits of reversible steps, when the rate constants of the corresponding
reverse reactions approach zero. The novelty of this form is that the extended
detailed balance now is presented as a necessary combination of two
constituents:
* 1.
Structural conditions in accordance to which the irreversible reactions cannot
be included in oriented cyclic pathways.
* 2.
Algebraic conditions which are written for the “reversible part” of the
complex mechanism taken separately, without irreversible reactions, using the
classical Wegscheider relationships.
The computational tools combine linear algebra (some standard tools for
chemical kinetics) with methods of linear programming. The most expensive
computational problem appears when we check the structural condition of the
extended form of detailed balance.
Let $n$ be the number of components, and let $\mathbb{R}^{n}$ be the
composition space. We consider a system with $r$ reversible and $s$
irreversible reactions. Let us use $\gamma_{1},\ldots,\gamma_{r}$ for the
stoichiometric vectors of the reversible reactions, $\nu_{1},\ldots,\nu_{s}$
for the stoichiometric vectors of the irreversible reactions and
$\overline{\nu}_{j}$ for the images of $\nu_{j}$ in the quotient space
$\mathbb{R}^{n}/S$, where $S$ is spanned by the stoichiometric vectors of all
reversible reaction, $S={\rm
span}\\{\gamma_{1},\ldots,\gamma_{r}\\}\subset\mathbb{R}^{n}$. The reaction
mechanism satisfies the structural condition of the extended form of detailed
balance if and only if
$0\notin{\rm conv}\\{\overline{\nu}_{1},\ldots,\overline{\nu}_{s}\\}\,.$
We have to check whether the origin belongs to the convex hull of the vectors
$\overline{\nu}_{1},\ldots,\overline{\nu}_{s}$. In practice, we can always
assume that these vectors have exactly known rational (or even integer)
coordinates.
We combined three approaches to study the restrictions implied by the
principle of detailed balance in the systems with some irreversible reactions:
1. 1.
Analysis of limits of the systems with all reversible reactions and detailed
balance when some of the reaction rate constants tend to zero.
2. 2.
Analysis of the Wegscheider identities for elementary pathways when some of
the reaction rate constants turn into zero.
3. 3.
Analysis of limits of the systems when some equilibrium concentrations (or,
more general, activities) tend to zero.
We proved that these three approaches are equivalent if we take into account
not only which equilibrium concentrations tend to zero, but the speed of this
tendency as well. The various partially or fully irreversible limits of the
reaction mechanisms are, in this sense, multiscale asymptotics of the reaction
networks when some equilibrium concentration tend to zero with different
speed.
## References
* Benson (1981) Benson, H. E. (1981), Processing of Gasification Products, In: Chemistry of Coal Utilization, Elliot, M., ed., New York, USA: John Wiley and Sons, Ch. 25, 1753–1800.
* Bertsimas & Tsitsiklis (1997) Bertsimas, D., Tsitsiklis, J.N. (1997), Introduction to Linear Optimization, Cambridge, MA, USA: Athena Scientific.
* Boltzmann (1964) Boltzmann, L. (1964), Lectures on gas theory, Berkeley, CA, USA: U. of California Press.
* Chu (1971) Chu, Ch. (1971), Gas absorption accompanied by a system of first-order reactions, Chem. Eng. Sci. 26(3), 305–312.
* Feinberg (1972) Feinberg, M. (1972) Complex balancing in general kinetic systems. Arch. Rat. Mechan. Anal. 49 (3), 187–194.
* Fukuda & Prodon (1996) Fukuda, K., Prodon A. (1996), Double description method revisited, In: Combinatorics and Computer Science, Lecture Notes in Computer Science, Volume 1120/1996, 91–111.
* Gagneur & Klamt (2004) Gagneur, J., Klamt, S. (2004), Computation of elementary modes: a unifying framework and the new binary approach, BMC Bioinformatics 5:175.
* Gorban et al (1989) Gorban, A.N., Mirkes, E.M., Bocharov, A.N., Bykov, V.I. (1989), Thermodynamic consistency of kinetic data, Combustion, Explosion, and Shock Waves, 25 (5) , 593–600,
* Horiuti (1973) Horiuti, J. (1973), Theory of reaction rates as based on the stoichiometric number concept, Ann. New York Academy Sci. 213, 5-30
* Ivanova (1979) Ivanova, A.N. (1979), Conditions for uniqueness of stationary state of kinetic systems related to structural scheme of reactions, Kinet. Katal., 20(4), 1019–1023.
* Mincheva & Roussel (2007) Mincheva, M., Roussel, M.R. (2007), Graph-theoretic methods for the analysis of chemical and biochemical networks. I. Multistability and oscillations in ordinary differential equation models, J. Math. Biol., 55, 61–68.
* Moe (1962) Moe, J.M. (1962), Design of water-gas shift reactors, Chem. Eng. Progress 58 (3), 33–36.
* Motzkin et al (1953) Motzkin, T.S., Raiffa, H., Thompson, G.L., Thrall, R.M. (1953), The double description method. In: H.W. Kuhn and A.W.Tucker, eds, Contributions to theory of games, Vol. 2., Princeton, NJ, USA: Princeton University Press, 51–73.
* Onsager (1931) Onsager, L. (1931), Reciprocal relations in irreversible processes. I, Phys. Rev. 37, 405–426; II 38, 2265–2279.
* Papin et al (2003) Papin, J.A., Price, N.D., Wiback, S.J., Fell, D.A., Palsson, B.O. (2003), Metabolic pathways in the post-genome era, Trends in Biochemical Sciences 28 (5), 250–258.
* Prigogine & Defay (1962) Prigogine, I., Defay, R. (1962), Chemical Thermodynamics, New York, USA: Longmans and Green.
* Schnakenberg (1976) Schnakenberg, J. (1976), Network theory of microscopic and macroscopic behavior of master equation systems, Rev. Mod. Phys. 48, 571–585.
* Schuster et al (2000) Schuster, S., Fell, D.A., Dandekar, T. (2000), A general definition of metabolic pathways useful for systematic organization and analysis of complex metabolic networks, Nat. Biotechnol. 18, 326–332.
* Temkin et al (1996) Temkin, O.N., Zeigarnik, A.V., Bonchev, D.G. (1996), Chemical reaction networks: a graph-theoretical approach; Boca Raton, FL, USA: CRC Press.
* Volpert & Khudyaev (1985) Volpert, A.I., Khudyaev, S.I. (1985), Analysis in classes of discontinuous functions and equations of mathematical physics. Dordrecht, The Netherlands: Nijoff.
* Wegscheider (1901) Wegscheider, R. (1911) Über simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reactionskinetik homogener Systeme, Monatshefte für Chemie / Chemical Monthly 32(8), 849–906.
* Yablonskii et al (1991) Yablonskii, G.S., Bykov, V.I., Gorban, A.N., Elokhin, V.I. (1991), Kinetic Models of Catalytic Reactions, Amsterdam, The Netherlands: Elsevier.
* Yablonsky et al (2010) Yablonsky, G.S., Gorban, A.N., Constales, D., Galvita, V.V., Marin, G.B. (2011), Reciprocal relations between kinetic curves, EPL 93, 20004.
* Yang, et al (2006) Yang, J., Bruno, W.J., Hlavacek, W.S., Pearson, J. (2006), On imposing detailed balance in complex reaction mechanisms. Biophys. J. 91, 1136–1141.
|
arxiv-papers
| 2011-01-27T13:24:23 |
2024-09-04T02:49:16.666979
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.N. Gorban, G.S. Yablonsky",
"submitter": "Alexander Gorban",
"url": "https://arxiv.org/abs/1101.5280"
}
|
1101.5373
|
# Beam-beam simulation code BBSIM for particle accelerators
H.J. Kim hjkim@fnal.gov (H.J. Kim) T. Sen Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA
###### Abstract
A highly efficient, fully parallelized, six-dimensional tracking model for
simulating interactions of colliding hadron beams in high energy ring
colliders and simulating schemes for mitigating their effects is described.
The model uses the weak-strong approximation for calculating the head-on
interactions when the test beam has lower intensity than the other beam, a
look-up table for the efficient calculation of long-range beam-beam forces,
and a self-consistent Poisson solver when both beams have comparable
intensities. A performance test of the model in a parallel environment is
presented. The code is used to calculate beam emittance and beam loss in the
Tevatron at Fermilab and compared with measurements. We also present results
from the studies of two schemes proposed to compensate the beam-beam
interactions: a) the compensation of long-range interactions in the
Relativistic Heavy Ion Collider (RHIC) at Brookhaven and the Large Hadron
Collider (LHC) at CERN with a current-carrying wire, b) the use of a low-
energy electron beam to compensate the head-on interactions in RHIC.
###### keywords:
accelerator physics , parallel computing , beam dynamics
###### PACS:
29.27.Bd , 29.27.Fh
††journal: Nucl. Instrum. Methods Phys. Res., Sect. A
## 1 Introduction
In high energy storage-ring colliders, the beam-beam interactions are known to
cause emittance growth and a reduction of beam life time, and to limit the
collider luminosity [1, 2, 3, 4, 5, 6, 7]. It has been a key issue in a high
energy collider to simulate the beam-beam interaction accurately and to
mitigate the interaction effects. A beam-beam simulation code BBSIM has been
developed at Fermilab over the past few years to study the effects of the
machine nonlinearities and the beam-beam interactions [8, 9, 10, 11]. The code
is under continuous development with the emphasis being on including the
important details of an accelerator and the ability to reproduce observations
in diagnostic devices. At present, the code can be used to calculate tune
footprints, dynamic apertures, beam transfer functions, frequency diffusion
maps, action diffusion coefficients, emittance growth, and beam lifetime.
Calculation of the last two quantities over the long time scales of interest
is time consuming even with modern computer technology. In order to run
efficiently on a multiprocessor system, the resulting model was implemented by
using parallel libraries which are MPI (inter-processor Message Passing
Interface standard) [12], state-of-the-art parallel solver libraries
(Portable, Extensible Toolkit for Scientific Calculation, PETSc) [13], and
HDF5 (Hierarchical Data Format) [14].
The organization of the paper is as follows: The physical model used in the
simulation code is described in Section 2. The parallelization algorithm and
performance are described in Section 3. Some applications are presented for
the Tevatron, the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron
Collider (LHC) in Section 4. Section 5 summarizes our results.
## 2 Physical model
In a collider simulation, the two beams moving in opposite direction are
represented by macroparticles. The macroparticles are generated with the same
charge to mass ratio as the particles in the accelerator. The number of
macroparticles chosen is much less than the bunch intensity of the beam
because it becomes prohibitive to follow approximately 1011 particles for even
a few revolutions around the accelerator using modern supercomputers. These
macroparticles are generated and loaded with an initial distribution chosen
for the specific simulation purpose. As an example, a six-dimensional Gaussian
distribution is used for long-term beam evolution. The transverse and
longitudinal motion of particles is calculated by a sequence of linear and
nonlinear transfer maps. During the beam transport, a particle is removed from
the distribution if it reaches a predefined boundary in transverse or
longitudinal direction. In our simulation model, the following effects are
included: head-on and long-range beam-beam interactions, fields of a current-
carrying wire and an electron lens, multipole errors in quadrupole magnets in
interaction regions, sextupoles for chromaticity correction, ac dipole,
resistive wall wake, tune modulation, noise in lattice elements, single and
multiple harmonic rf cavities, and crab cavities. The finite bunch length
effect of the beam-beam interactions is considered by slicing the beam into
several chunks in the longitudinal direction and then applying a synchro-beam
map [15]. Each slice in a beam interacts with slices in the other beam in turn
at a collision point. In the following, linear and nonlinear tracking models
are described in detail.
### 2.1 Transport through an arc
The six-dimensional coordinates of a test particle in the accelerator’s
coordinate frame are:
$\mathbf{x}=\left(x,x^{{}^{\prime}},y,y^{{}^{\prime}},z,\delta\right)^{T}$,
where $x$ and $y$ are horizontal and vertical coordinates, $x^{\prime}$ and
$y^{\prime}$ the trajectory slopes of the coordinates, $z=-c\Delta t$ the
longitudinal distance from the synchronous particle, and $\delta=\Delta
p_{z}/p_{0}$ the relative momentum deviation from the synchronous energy [16].
The transverse linear transformation between two elements denoted by $i$ and
$j$ can be written as
$\mathbf{x}_{j}=\left(\begin{array}[]{cc}\mathcal{M}&\hat{\mathcal{D}}\\\
\hat{\mathcal{A}}&\mathcal{L}\end{array}\right)\mathbf{x}_{i}.$ (1)
Here, $\mathcal{M}$ is a coupled transverse map of _off-momentum_ motion
defined by $\mathcal{M}=\mathcal{R}_{j}\tilde{\mathcal{M}}_{i\rightarrow
j}\mathcal{R}_{i}^{-1}$, where $\tilde{\mathcal{M}}_{i\rightarrow j}$ is the
uncoupled linear map described by Twiss functions at $i$ and $j$ elements, and
the transverse coupling matrix $\mathcal{R}$ is defined as [17]
$\mathcal{R}=\frac{1}{\sqrt{1+\left|C\right|}}\left(\begin{array}[]{cc}I&C^{\dagger}\\\
-C&I\end{array}\right)$ (2)
where $C^{\dagger}$ is the $2\times 2$ matrix and the symplectic conjugate of
the coupling matrix $C$. The $4\times 2$ dispersion matrix is defined by
$\hat{\mathcal{D}}=\left(0,\mathbf{D}\right)$, and the dispersion vector
$\mathbf{D}=\left(D_{x},D_{x}^{{}^{\prime}},D_{y},D_{y}^{{}^{\prime}}\right)^{T}$
is characterized by the transverse dispersion functions and the map
$\mathcal{M}$, i.e., $\mathbf{D}=\mathbf{D}_{j}-\mathcal{M}\mathbf{D}_{i}$
where $\mathbf{D}_{i},\mathbf{D}_{j}$ are the dispersion vectors at $i,j$.
Since the transport matrix has to be symplectic, the matrix
$\hat{\mathcal{A}}$ in Eq. (1) is given by
$\hat{A}=-\hat{\mathcal{D}}^{T}S^{T}\mathcal{M},$ where $S$ is a rearranging
matrix (see subsection 2.7). The longitudinal map $\mathcal{L}$ is given by
$\mathcal{L}=\left(\begin{array}[]{cc}1&-\left(\eta/\beta\right)\Delta s\\\
0&1\end{array}\right)$, where $\eta$ is the slip factor, $\beta=v/c$, and
$\Delta s$ the longitudinal distance between the two elements, i.e., $\Delta
s=s_{j}-s_{i}$. It is noted that $s$ is the axis along the beam direction. The
nonlinearity of synchrotron oscillations is applied by adding the longitudinal
momentum change at a rf cavity:
$\Delta\delta=\frac{eV_{rf}}{\beta^{2}E}\left(\sin
k_{rf}z-\sin\phi_{s}\right)$ (3)
where $V_{rf}$ is the voltage of rf cavity, $\phi_{s}$ the phase angle for a
synchronous particle with respect to the rf wave, and $k_{rf}$ the wave number
of the rf cavity. If there are higher harmonic cavities, their effects are
added to the momentum change.
### 2.2 Beam-beam interactions
In order to achieve high luminosity in a collider one can increase the number
of bunches which reduces the bunch spacing. More bunches can increase the
number of parasitic encounters in the interaction regions. Since the
calculation of beam-beam forces requires large amounts of computational
resources, it has to be executed rapidly and accurately. BBSIM has three
different models for this purpose: a weak-strong model for head-on
interactions, a look-up table model for long-range interactions, and a Poisson
solver model for the head-on interactions when both beams have comparable
intensities (“strong-strong” model).
#### 2.2.1 Weak-strong model
In the weak-strong model we assume that the “weak” beam is affected by the
head-on and long-range interactions while the opposing beam or “strong” beam
is unaffected. The charge distribution of the strong beam is assumed to be
Gaussian:
$\rho\left(x,y,z\right)=\frac{Nq}{\left(2\pi\right)^{3/2}\sigma_{x}\sigma_{y}\sigma_{z}}\exp\left(-\frac{x^{2}}{2\sigma_{x}^{2}}-\frac{y^{2}}{2\sigma_{y}^{2}}-\frac{z^{2}}{2\sigma_{z}^{2}}\right)$
(4)
Here, $N$ is the number of particles per bunch and $q$ is the charge per
particle. Note that the coordinates $\left(x,y,z\right)$ are measured in the
rest frame of the strong beam. The beam-beam force between two beams with
transverse Gaussian distribution $\rho\left(x,y\right)=\int
dz\rho\left(x,y,z\right)$ is well-known [18], and the expression for the slope
change is given by, for elliptical beam with $\sigma_{x}>\sigma_{y}$:
$\left(\begin{array}[]{c}\Delta x^{\prime}\\\ \Delta
y^{\prime}\end{array}\right)=\frac{2Nr_{0}}{\gamma}\frac{\sqrt{\pi}}{\sqrt{2\left(\sigma_{x}^{2}-\sigma_{y}^{2}\right)}}\left(\begin{array}[]{c}\text{Im}\left[F\left(x,y\right)\right]\\\
\text{Re}\left[F\left(x,y\right)\right]\end{array}\right)$ (5)
where
$F\left(x,y\right)=w\left(\frac{x+iy}{\sqrt{2\left(\sigma_{x}^{2}-\sigma_{y}^{2}\right)}}\right)-e^{-\frac{x^{2}}{2\sigma_{x}^{2}}-\frac{y^{2}}{2\sigma_{y}^{2}}}w\left(\frac{\frac{x\sigma_{y}}{\sigma_{x}}+i\frac{y\sigma_{x}}{\sigma_{y}}}{\sqrt{2\left(\sigma_{x}^{2}-\sigma_{y}^{2}\right)}}\right).$
(6)
Here, $w\left(z\right)$ is the complex error function defined by
$w\left(z\right)=e^{-z^{2}}\left(1+\frac{2i}{\sqrt{\pi}}\int_{0}^{z}dt\,e^{t^{2}}\right)$,
and $\gamma$ the Lorentz factor. The constant $r_{0}$ is defined as
$r_{0}\equiv qq_{*}/4\pi\epsilon_{0}m_{0}c^{2}$, where $q_{*}$ is the electric
charge of the test particle, and $m_{0}$ the rest mass of the particle.
#### 2.2.2 Look-up table model
The charge distribution of the strong beam in the weak-strong model is not
varied during the simulations. It is redundant to re-calculate the beam-beam
force at every parasitic location and every turn. A look-up table is one way
to avoid it. The look-up table is used to replace a run time computation with
an array indexing operation. The beam-beam force of a Gaussian beam
distribution is described by the complex error function, as shown in Eq. (6).
The calculation of the complex error function can substantially slow the beam-
beam simulation. However, the look-up table is pre-calculated and stored in a
memory, usually in an array. When the value of the error function is required,
it can be retrieved from the table by an interpolation scheme, instead of
using Eq. (6). The look-up table method can significantly reduce a
computational cost. The property of the complex error functions yields the
symmetry relations of function $F\left(z\right)$ as
$F\left(-z\right)=-F\left(z\right),\;F\left(\bar{z}\right)=-\overline{F\left(z\right)},\;F\left(-\bar{z}\right)=\overline{F\left(z\right)}$
(7)
where $z=x+iy$ is a complex variable. The symmetry conditions of the function
$F\left(z\right)$ can reduce memory space to store the function values. It is
sufficient to build the table for the values of function $F\left(z\right)$ in
the first quadrant of the complex plane, i.e., $\left|x\right|\geq 0$ and
$\left|y\right|\geq 0$.
Interpolation techniques are required to predict a value of a function at a
point inside its domain based upon the known tabulated values. For a given set
of data points $\left(z_{i},f_{i}\right)$, $i=0,\dots,N$, where no two
$z_{i}$’s are the same, the interpolated value $g\left(z\right)$ at a value
$z\neq z_{i}$ is found from
$g\left(z\right)=\sum_{i=0}^{N}f_{i}L_{i}\left(z\right)$ (8)
where the $L_{i}$ is Lagrange’s $N$-th order polynomials
$L_{i}\left(z\right)=\prod_{j=0,j\neq i}^{N}\frac{z-z_{j}}{z_{i}-z_{j}}.$ (9)
In order to save the interpolation time further, one can divide $z$-space and
apply a different degree of the Lagrange polynomial. For an example, we apply
a sixth order polynomial for small amplitudes $\left|z\right|\leq 4\sigma$
while a third order polynomial is applied for $\left|z\right|>4\sigma$,
because the function $F\left(z\right)$ varies more rapidly at small
$\left|z\right|$ and slowly at large $\left|z\right|$ .
#### 2.2.3 Poisson solver model
The weak-strong model is a good approximation when one beam has much smaller
intensity than the other, but it is not valid when the intensities of the two
beams are comparable because each beam’s parameters are changed by the other
beam. One has to solve for the field of each beam self-consistently. The
fields are the solutions of the Poisson equation given by
$\nabla^{2}\phi\left(\mathbf{r}\right)=-4\pi\rho\left(\mathbf{r}\right)$ (10)
where $\phi$ is the electrostatic potential and $\rho$ the density function of
the beam. The solution can be obtained by
$\phi\left(\mathbf{r}\right)=\int
G\left(\mathbf{r},\mathbf{r}_{1}\right)\rho\left(\mathbf{r}_{1}\right)d\mathbf{r}_{1}$
(11)
where $G$ is the Green’s function of the Poisson equation and in two space
dimension, is given by
$G\left(x,y:x_{1},y_{1}\right)=-\frac{1}{4\pi}\ln\left[\left(x-x_{1}\right)^{2}+\left(y-y_{1}\right)^{2}\right].$
(12)
Equation (11) can be efficiently calculated using a convolution theorem and
inverse Fourier transform:
$\phi\left(\mathbf{r}\right)=\mathcal{F}^{-1}\left(\hat{G}\left(\bm{\omega}\right)\hat{\rho}\left(\bm{\omega}\right)\right)$
(13)
where
$\hat{G}\left(\bm{\omega}\right)=\left(\frac{1}{\sqrt{2\pi}}\right)^{2}\int_{\mathbb{R}^{2}}G\left(\mathbf{r}\right)e^{-i\bm{\omega}\cdot\mathbf{r}}d\mathbf{r}$
and
$\hat{\rho}\left(\bm{\omega}\right)=\left(\frac{1}{\sqrt{2\pi}}\right)^{2}\int_{\mathbb{R}^{2}}\rho\left(\mathbf{r}\right)e^{-i\bm{\omega}\cdot\mathbf{r}}d\mathbf{r}$.
It is assumed in Eq. (13) that the density function
$\rho\left(\mathbf{r}\right)$ is periodic in both $x$ and $y$ directions.
However, since the beam has a finite charge distribution surrounded by a
conducting wall in an accelerator system, the transverse beam density does not
meet the periodicity requirement of FFT techniques. In order to apply the
above formalism, the density function should be rewritten by, in the doubled
computational domain [19]:
$\rho_{new}\left(x,y\right)=\begin{cases}\rho\left(x,y\right)&,\;0<x\leq
L_{x},\;0<y\leq L_{y}\\\ 0&,\;L_{x}<x\leq 2L_{x},\;or\;L_{y}<y\leq
2L_{y}.\end{cases}$ (14)
Green’s function is defined in the doubled domain, as follows:
$G_{new}\left(x,y\right)=\begin{cases}G\left(x,y\right)&,\,0<x\leq
L_{x},\;0<y\leq L_{y}\\\ G\left(2L_{x}-x,y\right)&,\,L_{x}<x\leq
2L_{x},\;0<y\leq L_{y}\\\ G\left(x,2L_{y}-y\right)&,\,0<x\leq
L_{x},\;L_{y}<y\leq 2L_{y}\\\ G\left(2L_{x}-x,2L_{y}-y\right)&,\,L_{x}<x\leq
2L_{x},\;L_{y}<y\leq 2L_{y}.\end{cases}$ (15)
Both $\rho_{new}$ and $G_{new}$ are doubly periodic functions with periods
$2L_{x}$ and $2L_{y}$. It is noted that only the potential within a domain
$\left(0,L_{x}\right]\times\left(0,L_{y}\right]$ is valid. The potential
outside the domain is incorrect, but it doesn’t matter because the physical
domain of interest is $\left(0,L_{x}\right]\times\left(0,L_{y}\right]$. When
one beam is separated far from the other, one can apply a shifted Green’s
function approach [20].
#### 2.2.4 Crossing angle
When there exists a finite crossing angle between two colliding beams at an
interaction point, the beam-beam force experienced by a test particle will
have transverse and longitudinal components because the electric field
generated by the opposing beam is not perpendicular to the particle velocity
anymore. The existence of a longitudinal force makes it difficult to apply the
result of previous sections. A transformation can be used to remedy the
difficulty. It transforms a crossing angle collision in the laboratory frame
to a head-on collision in the rotated and boosted frame which is called the
head-on frame [21, 22]. The transformation can be described by a
transformation from the accelerator coordinates to Cartesian coordinates, a
Lorentz boost, and again a backward transformation to the accelerator
coordinates:
$\displaystyle x^{*}$
$\displaystyle=z\cos\alpha\tan\phi+x\left[1+h_{x}^{*}\cos\alpha\sin\phi\right]+yh_{x}^{*}\sin\alpha\sin\phi$
(16) $\displaystyle y^{*}$
$\displaystyle=z\sin\alpha\tan\phi+y\left[1+h_{y}^{*}\sin\alpha\sin\phi\right]+xh_{y}^{*}\cos\alpha\sin\phi$
$\displaystyle z^{*}$
$\displaystyle=\frac{z}{\cos\phi}+h_{z}^{*}\left[x\cos\alpha\sin\phi+y\sin\alpha\sin\phi\right]$
$\displaystyle p_{x}^{*}$
$\displaystyle=\frac{p_{x}}{\cos\phi}-h\cos\alpha\frac{\tan\phi}{\cos\phi}$
$\displaystyle p_{y}^{*}$
$\displaystyle=\frac{p_{y}}{\cos\phi}-h\sin\alpha\frac{\tan\phi}{\cos\phi}$
$\displaystyle p_{z}^{*}$ $\displaystyle=p_{z}-p_{x}\cos\alpha\tan\phi-
p_{y}\sin\alpha\tan\phi+h\tan^{2}\phi$
where a star (*) stands for a dynamical variable in the head-on frame, the
Hamiltonian
$h\left(p_{x},p_{y},p_{z}\right)=p_{z}+1-\sqrt{\left(p_{z}+1\right)^{2}-p_{x}^{2}-p_{y}^{2}}$,
$h_{x}^{*}=\partial h^{*}/\partial p_{x}^{*}$,
$h^{*}\left(p_{x}^{*},p_{y}^{*},p_{z}^{*}\right)=h\left(p_{x}^{*},p_{y}^{*},p_{z}^{*}\right)$,
$\alpha$ the crossing plane angle in the $x-y$ plane, and $\phi$ the half
crossing angle in the $\tilde{x}-s$ plane as shown in Fig. 1.
Figure 1: Definition of crossing angles $\alpha$ and $\phi$: $\alpha$ is the
crossing plane angle in the $x-y$ plane and $\phi$ is the half crossing angle
in the $\tilde{x}-s$ plane. $s$ is the axis along the beam direction when
there is no crossing angle. The $\tilde{x}-s$ plane is the crossing plane
defined by the angle $\alpha$. The beam trajectories, shown by lines with
arrows, lie in the crossing plane.
### 2.3 Finite bunch length
The effects due to the finite (as opposed to infinitesimal) bunch length need
to be considered when the transverse beta functions at the interaction point
are small and comparable to $\sigma_{z}$. The finite longitudinal length is
considered by dividing the beam into longitudinal slices and by a so called
synchro-beam map [15]. We make slices of both beams moving in opposite
directions. Each slice of the strong bunch is integrated over its length, and
has only a transverse charge distribution at its center. We take into account
the collision between a pair of slices: the $i^{th}$ slice of a bunch and the
$j^{th}$ slice of a bunch in the other beam. The collision takes place at
collision point
$S\left(z^{i},z_{*}^{j}\right)=\frac{1}{2}\left(z^{i}-z_{*}^{j}\right)$ which
is usually different from the interaction point. For example, the $i^{th}$
slice of a bunch has successive collisions with slices of a bunch in the other
beam. In addition, the electric field varies along the bunch due to the
inhomogeneity of the charge density in the longitudinal direction, and couples
transverse and longitudinal motions. The coupling can be modeled by the
synchro-beam map which includes beam-beam interactions due to the longitudinal
component of the electric field as well as the transverse components. The
transformation is given by [15]
$\displaystyle x^{new}$
$\displaystyle=x+S\left(z,z_{*}\right)\left.\frac{\partial U}{\partial
x}\right|_{S},\enskip p_{x}^{new}=p_{x}-\left.\frac{\partial U}{\partial
x}\right|_{S}$ (17) $\displaystyle y^{new}$
$\displaystyle=y+S\left(z,z_{*}\right)\left.\frac{\partial U}{\partial
y}\right|_{S},\;p_{y}^{new}=p_{y}-\left.\frac{\partial U}{\partial
y}\right|_{S}$ $\displaystyle z^{new}$
$\displaystyle=z,\>\delta^{new}=\delta-\frac{1}{2}\left.\frac{\partial
U}{\partial x}\right|_{S}\left[p_{x}-\frac{1}{2}\left.\frac{\partial
U}{\partial x}\right|_{S}\right]-\frac{1}{2}\left.\frac{\partial U}{\partial
y}\right|_{S}\left[p_{y}-\frac{1}{2}\left.\frac{\partial U}{\partial
y}\right|_{S}\right]-\frac{1}{2}\left.\frac{\partial U}{\partial
z}\right|_{S}.$
Here, $\left.\right|_{S}$ represents the evaluation at the collision point
$S\left(z,z_{*}\right)$. $U$ is the normalized potential energy
$U=q\Phi/E_{0}$ and is given by
$U\left(x,y;\sigma_{x}\left(s\right),\sigma_{y}\left(s\right)\right)=\frac{N_{*}r_{0}}{\gamma}\int_{0}^{\infty}d\zeta\frac{-1+\exp\left(-\frac{x^{2}}{2\sigma_{x}^{2}+\zeta}-\frac{y^{2}}{2\sigma_{y}^{2}+\zeta}\right)}{\sqrt{\left(2\sigma_{x}^{2}+\zeta\right)\left(2\sigma_{y}^{2}+\zeta\right)}}.$
(18)
The dependence on the bunch length is contained in
$\sigma_{x}(s),\sigma_{y}(s)$. The transverse derivatives of the potential
energy are
$\left.\frac{\partial U}{\partial x}\right|_{S}=-\Delta
x^{\prime}\left(X,Y;S\left(z,z_{*}\right)\right),\;\left.\frac{\partial
U}{\partial y}\right|_{S}=-\Delta
y^{\prime}\left(X,Y;S\left(z,z_{*}\right)\right)$ (19)
where $\left(X,Y\right)$ are the transverse coordinates at
$S\left(z,z_{*}\right)$, and$\Delta x^{\prime}$ and $\Delta y^{\prime}$ are
given by Eq. (5). The longitudinal derivative of the potential energy which is
related to the longitudinal beam-beam kicks is expressed by
$\displaystyle\left.\frac{\partial U}{\partial z}\right|_{S}$
$\displaystyle=\frac{1}{2}\left.\frac{d\sigma_{x}^{2}}{ds}\frac{\partial
U}{\partial\sigma_{x}^{2}}\right|_{s=S\left(z,z_{*}\right)}+\frac{1}{2}\left.\frac{d\sigma_{y}^{2}}{ds}\frac{\partial
U}{\partial\sigma_{y}^{2}}\right|_{s=S\left(z,z_{*}\right)}$ (20)
$\displaystyle\frac{\partial U}{\partial\sigma_{x}^{2}}$
$\displaystyle=\frac{1}{2\left(\sigma_{x}^{2}-\sigma_{y}^{2}\right)}\left[x\Delta
x^{\prime}+y\Delta
y^{\prime}+\frac{2N_{*}r_{0}}{\gamma}\left(\frac{\sigma_{y}}{\sigma_{x}}e^{-\frac{x^{2}}{2\sigma_{x}^{2}}-\frac{y^{2}}{2\sigma_{y}^{2}}}-1\right)\right]$
$\displaystyle\frac{\partial U}{\partial\sigma_{y}^{2}}$
$\displaystyle=\frac{-1}{2\left(\sigma_{x}^{2}-\sigma_{y}^{2}\right)}\left[x\Delta
x^{\prime}+y\Delta
y^{\prime}+\frac{2N_{*}r_{0}}{\gamma}\left(\frac{\sigma_{x}}{\sigma_{y}}e^{-\frac{x^{2}}{2\sigma_{x}^{2}}-\frac{y^{2}}{2\sigma_{y}^{2}}}-1\right)\right].$
Note that $\frac{d\sigma_{x}^{2}}{ds}$ and $\frac{d\sigma_{y}^{2}}{ds}$ have
zero amplitude and change their sign at the interaction point if
$\alpha_{x}=\alpha_{y}=0$. Test particles experience longitudinal acceleration
and deceleration passing through the bunch moving in the opposite direction.
### 2.4 Compensation schemes
In storage-ring colliders, a beam experiences periodic perturbations when it
meets the counter-rotating beam in a common beam pipe. The head-on beam-beam
interactions occur when the beams collide in the detectors while the long-
range interactions occur when the beams are simultaneously present at the same
location but are separated transversely. The nonlinear forces due to these
beam-beam interactions result in a tune spread and can cause emittance growth,
a reduction of beam life time, and therefore reduce the collider luminosity.
The combination of beam-beam and machine nonlinearities excites betatron
resonances which can cause particles to diffuse into the tails of the beam
distribution and even to the physical aperture. Different compensation methods
have been proposed: a current-carrying wire for the effects of the long-range
interactions [23] and an electron lens for the head-on interactions in proton
machines [24, 25, 26]. Beam collisions with a crossing angle at the
interaction point are often necessary in colliders to reduce the effects of
the long-range interactions. The crossing angle reduces the geometrical
overlap of the beams and hence the luminosity. A deflecting mode cavity, also
known as a crab cavity, offers a promising way to compensate the crossing
angle and to realize effective head-on collisions [27, 28]. We now describe
the modeling of these compensation schemes in the program.
#### 2.4.1 Current-carrying wire
When the separations at long-range interactions are large compared to the rms
beam size the strength of these interactions is inversely proportional to the
distance. Its effect on a beam can be compensated by a current-carrying wire
which creates a magnetic field with the same $\frac{1}{r}$ dependence. This
approach is simple and it is possible to deal with all multipole orders at
once. For a finite length $l_{w}$ embedded in the middle of a drift length
$L$, the transfer map of a wire can be obtained by
$\mathcal{M}_{w}^{\left(L\right)}=D_{L/2}\circ\mathcal{M}_{k}^{\left(L\right)}\circ
D_{L/2}$ (21)
where $D_{L/2}$ is the drift map with a length $\frac{L}{2}$, and
$\mathcal{M}_{k}^{\left(L\right)}$ is the wire kick integrated over a drift
length. This kick map $\mathcal{M}_{k}^{\left(L\right)}$ is reproduced by the
following changes in slope [29]
$\left(\begin{array}[]{c}\Delta x^{\prime}\\\ \Delta
y^{\prime}\end{array}\right)=\frac{\mu_{0}}{4\pi}\frac{I_{w}l_{w}}{\left(B\rho\right)}\frac{u-v}{x^{2}+y^{2}}\left(\begin{array}[]{c}x\\\
y\end{array}\right)$ (22)
where $I_{w}$ is the current of the wire ,
$u=\sqrt{\left(\frac{L}{2}+l_{w}\right)^{2}+x^{2}+y^{2}}$ and
$v=\sqrt{\left(\frac{L}{2}-l_{w}\right)^{2}+x^{2}+y^{2}}$. We also take into
account the wire misalignment including pitch and yaw angles
$\left(\theta_{x},\theta_{y}\right)$ respectively as well as lateral shifts
$\left(\Delta x,\Delta y\right)$. The transfer map of a wire can be written as
$\mathcal{M}_{w}=S_{\Delta x,\Delta y}\circ
T_{\theta_{x},\theta_{y}}^{-1}\circ
D_{L/2}\circ\mathcal{M}_{k}^{\left(L\right)}\circ D_{L/2}\circ
T_{\theta_{x},\theta_{y}}$ (23)
where $T_{\theta_{x},\theta_{y}}$ represents the tilt of the coordinate system
by horizontal and vertical angles $\theta_{x},\theta_{y}$ to orient the
coordinate system parallel to the wire, and $S_{\Delta x,\Delta y}$ represents
a shift of the coordinate axes to make the coordinate systems after and before
the wire agree. When the wire is parallel to the beam, Eq. (23) becomes Eq.
(21). For canceling the long-range beam-beam interactions of the round beam
with the wire, one can get the desired wire current and length by equating Eq.
(22) and Eq. (5); the integrated strength of the wire compensator is related
to the integrated current of the beam bunch as $I_{w}l_{w}=cqN$.
#### 2.4.2 Electron lens
For the head-on proton-proton beam collisions, particles of one proton bunch
are focused by a space charge of the counter-rotating proton bunch. The beam-
beam effect on the particles of the proton bunch can be compensated by a
counter-rotating beam of negatively charged particles, for example, a low-
energy electron beam. In order to cancel out the transverse kick by the
counter-rotating proton bunch, the electron beam should have the same
transverse charge profile and current as the proton bunch. The proton bunch
typically exhibits an approximately Gaussian transverse profile. If we choose
a Gaussian distribution of the electron beam, the transverse kick on particles
of the proton bunch from the electron beam is given by
$\left(\begin{array}[]{c}\Delta x^{\prime}\\\ \Delta
y^{\prime}\end{array}\right)=-\frac{2N_{e}r_{0}}{\gamma
r^{2}}\zeta\left(x,y:\sigma_{e}\right)\left(\begin{array}[]{c}x\\\
y\end{array}\right)$ (24)
where $N_{e}$ is the number of electrons of the electron beam adjusted by the
electron beam speed, $r_{0}$ the classic proton radius, $\gamma$ the Lorentz
factor, $r^{2}=x^{2}+y^{2}$, and $\sigma_{e}$ the transverse beam size of the
electron beam. The function $\zeta$ is given by
$\zeta\left(x,y:\sigma_{e}\right)=\left[1-\exp\left(-\frac{x^{2}+y^{2}}{2\sigma_{e}}\right)\right].$
(25)
For a non-Gaussian electron charge distribution we implement a flat top
profile with smooth edges that generates a linear beam-beam force near the
beam center. This flat top beam profile
$\rho_{e}\left(r\right)=\rho_{0}/\left(1+\left(r/\sigma_{e}\right)^{8}\right)$
delivers the transverse kicks given by Eq. (24), but the function $\zeta$ is
as follows:
$\zeta=\frac{\sqrt{2}\tilde{\rho}_{0}}{8}\left[\frac{1}{2}\log\left(\frac{\theta_{+}^{2}+1}{\theta_{-}^{2}+1}\right)+\tan^{-1}\theta_{+}+\tan^{-1}\theta_{-}\right]$
(26)
where $\tilde{\rho}$ is a constant, and
$\theta_{\pm}=\sqrt{2}\left(\frac{r}{\sigma_{e}}\right)^{2}\pm 1$.
#### 2.4.3 Crab cavity
When a particle passes through a crab cavity structure, it experiences a
transverse deflection and a small change in its longitudinal energy. Crab
cavities can compensate for the horizontal or vertical crossing angle at the
interaction point by delivering oppositely directed transverse kicks to the
head and the tail of the bunches. In the case of a horizontal crossing, the
kicks from the crab cavity are given by
$\Delta x^{\prime}=-\frac{qV}{E_{0}}\sin\left(\phi_{s}+\frac{\omega
z}{c}\right),\enskip\Delta\delta=-\frac{qV}{E_{0}}\cos\left(\phi_{s}+\frac{\omega
z}{c}\right)\cdot\frac{\omega}{c}x$ (27)
where $q$ denotes the particle charge, $V$ the voltage of crab cavity, $E_{0}$
the particle energy, $\phi_{s}$ the phase of the synchronous particle with
respect to the crab-cavity rf wave, $\omega$ the angular frequency of the crab
cavity, $c$ the speed of light, $z$ the longitudinal coordinate of the
particle with respect to the bunch center, and $x$ the horizontal coordinate.
In general this is a nonlinear map which introduces synchro-betatron coupling,
but for small $z$, this reduces to a linear map in the horizontal-longitudinal
plane. The crab cavity causes a closed orbit distortion dependent on the
longitudinal position of particles, and the beam envelope is tilted all around
the ring. For a bunch shorter than the rf wavelength of the crab cavity
deflecting mode, the tilt angle of the beam envelope at a location with a beam
position monitor (BPM) is given by
$\tan\theta_{crab}=\frac{qV\omega\sqrt{\beta\beta_{crab}}}{c^{2}p_{0}}\left|\frac{\cos\left(\Delta\varphi-\pi
Q\right)}{2\sin\pi Q}\right|$ (28)
where $\beta$ is the beta function at the BPM position, $\beta_{crab}$ the
beta function at the crab cavity, $\Delta\varphi$ the phase advance between
the crab cavity location and the BPM, and $Q$ the betatron tune. The
simulations of a crab cavity in the SPS accelerator at CERN using BBSIM will
be described in another paper.
### 2.5 Particle distribution
At the beginning of a simulation, the simulation particles are distributed
over the phase space
$\mathbf{x}=\left(x,x^{\prime},y,y^{\prime},z,\delta\right)^{T}$, called the
initial loading. In any simulation the number of particles $N$ is limited by
the computational power. In order to make the best use of a small number of
simulation particles compared to the real number of particles in the
accelerator, the loading should be optimized. Indeed the initial loading is
very important because this choice can reduce the statistical noise in the
physical quantities.
_Gaussian distribution_ : For long-term particle tracking where we calculate
emittance growth, we consider an exponential distribution in action (Gaussian
distribution in coordinates) of the form:
$\rho\left(\mathbf{x}\right)=\rho_{0}\exp\left(-\frac{J_{x}}{2\sigma_{J_{x}}}-\frac{J_{y}}{2\sigma_{J_{y}}}-\frac{J_{z}}{2\sigma_{J_{z}}}\right)$
(29)
where $J_{x}$, $J_{y}$, and $J_{z}$ are the transverse and longitudinal action
variables defined by
$\displaystyle J_{x}$
$\displaystyle=\frac{1}{2\beta_{x}}\left[x^{2}+\left(\beta_{x}x^{{}^{\prime}}+\alpha_{x}x\right)^{2}\right],\;J_{y}=\frac{1}{2\beta_{y}}\left[y^{2}+\left(\beta_{y}y^{{}^{\prime}}+\alpha_{y}y\right)^{2}\right]$
(30) $\displaystyle J_{z}$
$\displaystyle=\frac{8}{\pi}\frac{R\nu_{s}}{h^{2}\left|\eta\right|}\left[E\left(k\right)-\left(1-k^{2}\right)K\left(k\right)\right]$
where $R$ is the radius of the accelerator, $h$ the harmonic number, $\nu_{s}$
the longitudinal tune, $E$ and $K$ the complete elliptical integrals, and
$k^{2}=\frac{1}{4}\frac{h^{2}\eta^{2}}{\nu_{s}^{2}}\left(\frac{\Delta
p}{p}\right)^{2}+\sin^{2}\frac{\phi}{2}.$ (31)
$\sigma_{J_{x}}$, $\sigma_{J_{y}}$, and $\sigma_{J_{z}}$ are the rms sizes of
action variables. The simulation particles are generated by two steps:
1. 1.
The action variables $\left(J_{x},J_{y},J_{z}\right)$ of particles can be
directly generated from the distribution function by the inverse transform
method and the bit-reversed sequence [30].
2. 2.
For example, $x$ and $x^{\prime}$ are correlated and their distribution is
$\hat{\rho}\left(x,x^{\prime}\right)=\hat{\rho}_{0}\exp\left(-\frac{x^{2}+\left(\beta_{x}x^{\prime}+\alpha_{x}x\right)^{2}}{2\sigma_{x}^{2}}\right)$.
Since the horizontal action $J_{x}$ is determined at the first step, the
horizontal coordinates $\left(x,x^{\prime}\right)$ can be obtained from the
random variates:
$x=\sqrt{J_{x}}\cos\theta_{x},\quad
x^{\prime}=\sqrt{J_{x}}\left(\sin\theta_{x}-\alpha_{x}\cos\theta_{x}\right)/\beta_{x}$
where the value of $\theta_{x}$ is randomly distributed within the interval
$0\leq\theta_{x}\leq 2\pi$.
_Hollow Gaussian distribution_ : In most cases of particle tracking, lost
particles are observed only above a certain large transverse action while the
beam core is stable. An example is shown in Section 4.1. A hollow beam is a
beam with zero central intensity along the longitudinal beam axis. For the
generation of a hollow beam, a bunched beam distribution in longitudinal phase
space is a Gaussian, but a distribution in transverse phase space is a hollow
Gaussian. The procedure of generating the hollow distribution is the same as
that for the Gaussian distribution except that the amplitude of transverse
action of a particle should be larger than a minimum value, i.e.,
$J_{x}+J_{y}\geq\sigma_{J}$. Since most of the stable particles are not
included in the tracking simulation, the hollow beam model simulates a large
transverse amplitude Gaussian distribution using a small number of macro-
particles. This distribution is useful when calculating beam lifetimes.
### 2.6 Particle diffusion
Diffusion coefficients can characterize the effects of the nonlinearities
present in an accelerator, and can be used to find numerical solutions of a
diffusion equation for the density [31, 32]. The solutions yield the time
evolution of the beam density distribution function for a given set of machine
and beam parameters. This technique enables us to follow the beam intensity
and emittance growth for the duration of a luminosity store, something that is
not feasible with direct particle tracking. The transverse diffusion
coefficients can be calculated numerically from
$\displaystyle D_{ij}\left(a_{i},a_{j}\right)$
$\displaystyle=\frac{1}{N}\left\langle\left(J_{i}(a_{i},N)-J_{i}(a_{i},0)\right)\left(J_{j}(a_{j},N)-J_{j}(a_{j},0)\right)\right\rangle$
(32)
where $J_{i}\left(a_{i},0\right)$ is the initial action at an amplitude
$a_{i}$, $J_{i}\left(a_{i},N\right)$ the action with initial amplitude $a_{i}$
after $N$ turns, $\left\langle\right\rangle$ the average over simulation
particles, and $(i,j)$ are the horizontal $x$ or the vertical $y$ coordinates.
Equation (29) is averaged over a certain number of turns to eliminate the
fluctuation in action due to the phase space structure, e.g. resonance
islands. These diffusion coefficients can be directly used to compare
amplitude growth under different circumstances, e.g with different tunes.
Emittance growth and beam lifetimes can be calculated when these coefficients
are used in a diffusion equation, as mentioned above.
### 2.7 Symplecticity
In the absence of dissipative effects, particle motion in an accelerator can
be described by Hamilton’s equations of motion. Hamiltonian systems obey the
symplectic condition which guarantees the conservation of phase space volume
as the system evolves, this is also known as Liouville’s theorem. For transfer
maps described in previous subsections the symplectic condition requires
$M^{T}SM=S,\quad S=\left(\begin{array}[]{ccc}s&0&0\\\ 0&s&0\\\
0&0&s\end{array}\right)$ (33)
where $s=\left(\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right)$ is an
antisymmetric $2\times 2$ matrix, and $M$ is a transfer matrix for a linear
system or the Jacobian matrix of a nonlinear map around any particle
trajectory. For a nonlinear map
$\mathcal{M}:\mathbf{x}\longrightarrow\bar{\mathbf{x}}$, the Jacobian matrix
is obtained from first-order partial derivatives of the new coordinates with
respect to the old ones. The elements are defined as
$M_{ij}=\partial\bar{x}_{i}/\partial x_{j}$. During implementation of the maps
for beam dynamics, one should check to ensure that the map is as symplectic as
possible. As a measure of the symplecticity, a matrix norm of
$\left\|M^{T}SM-S\right\|$ is used in BBSIM. The accuracy of the look-up table
model mentioned in subsection 2.2.2, for example, depends on the number of
sample points in a given complex space needed for interpolating the function.
Poor interpolation accuracy may violate the symplecticity, and lead to
emittance blow-up or shrinkage. We use the symplectic norm obtained with the
direct calculation of the complex error function as the benchmark. We find for
example, that in order to maintain the symplectic norm with 70 long-range
beam-beam interactions in the Tevatron, the number of sample points should be
more than 4 points per rms beam size.
### 2.8 Diagnostics
Numerical simulation enables the generation of very large amounts of data. The
BBSIM code monitors physical quantities, for example, particle amplitudes and
saves them into an external file during the simulation. According to a problem
of interest, the quantities to be saved can be chosen in order to extract
valuable information from post-processing. In addition, some diagnostic
functions are calculated in the code as follows:
_Betatron tune distribution_ : The betatron tune in an accelerator is one of
the most important beam parameters. The tune of each particle in the beam
distribution is calculated with a Hanning filter applied to an fast-Fourier
transform of particle coordinates found from tracking [33].
_Beam transfer function_ : The beam transfer function (BTF) is defined as the
beam response to a small external longitudinal or transverse excitation at a
given frequency. BTF diagnostics are widely employed in accelerators due to
its non-destructive nature. A stripline kicker or rf cavity excites betatron
or synchrotron oscillations respectively over the appropriate tune spectrum.
The beam response is observed in a downstream pickup. The fundamental
applications of BTF are to measure the transverse tune and tune distribution
by exciting betatron oscillation, to analyze the beam stability limits, and to
determine the impedance characteristics of the chamber wall, and feedback
system [34]. In the code, we apply a sinusoidal driving force to a beam in a
transverse plane. The driving frequency is swept in equidistant steps over a
continuous frequency range which includes the betatron tune. At each new
frequency there is initially a transient response which must be allowed to
relax before the frequency is extracted from the data. We avoid the issue of
the transients in the simulations by reloading the initial particle
distribution at each new frequency.
_Frequency diffusion_ : We have calculated frequency diffusion maps as another
way to investigate the effects of nonlinear forces. The map represents the
variation of the betatron tunes over two successive sets of the tunes [35]:
The variation can be quantified by
$d=\log\sqrt{\Delta\nu_{x}^{2}+\Delta\nu_{y}^{2}}$, where
($\Delta\nu_{x}=\nu_{x}^{\left(2\right)}-\nu_{x}^{\left(1\right)},\Delta\nu_{y}=\nu_{y}^{\left(2\right)}-\nu_{y}^{\left(1\right)}$)
are the tune variations between the first set and next set of 1024 turns. If
the tunes $\left(\nu_{x}^{\left(1\right)},\nu_{y}^{\left(1\right)}\right)$ are
different from
$\left(\nu_{x}^{\left(2\right)},\nu_{y}^{\left(2\right)}\right)$, the particle
is moving to different amplitudes. A large tune variation is generally an
indicator of fast diffusion and reduced stability.
_Dynamic aperture_ : The dynamic aperture of an accelerator is defined as the
smallest radial amplitude of particles that survive up to a certain time
interval, for example, $10^{6}$ turns. As the number of turns increases, the
dynamic aperture approaches an asymptotic value. Initial particles are
distributed uniformly over the transverse phase space with amplitudes
typically varying between 0-20 $\sigma$, where $\sigma$ is the rms transverse
beam size. The longitudinal amplitude is chosen as largest value within a
bunch.
_Emittance_ : The emittance is defined as the area (or volume) of phase space
enclosed by the ellipse containing all the particles in its interior.
Statistically, the rms beam emittance can be calculated by a determinant of
$\Sigma$-matrix of a beam distribution:
$\epsilon=\left[\det\left(\Sigma\right)\right]^{1/d}$ (34)
where $d$ is the dimension of phase space, the element of $\Sigma$-matrix is
$\Sigma_{ij}=\left\langle\left(\zeta_{i}-\left\langle\zeta_{i}\right\rangle\right)\left(\zeta_{j}-\left\langle\zeta_{j}\right\rangle\right)\right\rangle$,
and $\zeta=\left\\{x,x^{\prime},y,y^{\prime},z,\delta\right\\}$. For example,
horizontal emittance is obtained by
$\epsilon_{x}=\left[\det\left(\begin{array}[]{cc}\Sigma_{xx}&\Sigma_{xx^{\prime}}\\\
\Sigma_{x^{\prime}x}&\Sigma_{x^{\prime}x^{\prime}}\end{array}\right)\right]^{1/2}$.
In addition to the emittance of each degree of freedom, four- and six-
dimensional emittances are calculated to see the correlation and coupling
between the phase space coordinates.
_Beam loss_ : The beam loss is one of the fundamental observables and it can
be directly compared with simulation. During a beam simulation, each particle
is monitored if it reaches a physical aperture transversely or the rf bucket
longitudinally. The particle passing beyond the aperture is considered as a
lost particle. Unlike a real machine, several _virtual_ apertures are placed
inside a beam pipe. The multiple apertures are used to find beam losses at
different apertures.
## 3 Parallelization
Realistic simulations of beam dynamics demand large computational resources.
Calculations on these large number of particles can be distributed over
several processors of a parallel computer to improve performance. Two basic
approaches exist to allocate the calculations to the processors, particle
based and domain (space) based partitions. In the former approach, the
particles are uniformly allocated to the processors. They are not limited to a
certain spatial domain. The completion time of a parallel solution depends on
the processor with the maximum computational workload. The particle
decomposition can distribute the computational load evenly among all
processors while the interaction between particles, for example, intra-beam
scattering needs a very large number of communications between processors
since the interacting particles can be located in a distant processor.
Conversely, in the domain decomposition approach, the spatial domain is
partitioned into elementary regions, and each processor is responsible for one
of these regions. The particles in the accelerator simulation are transported
by the lattice map. The map causes significant particle movement which may
cause the load to become quickly unbalanced. The simulation of colliding beams
has two aspects, i.e., pure particle transport and electromagnetic field
evaluation. The domain deposition approach is an efficient way of
parallelizing the field solver. To achieve the workload balanced, our approach
is to use both decomposition schemes.
We have implemented a parallel calculation in the BBSIM code to perform a
tracking simulation of large numbers of particles. When the weak-strong beam-
beam model is used, only the particle decomposition scheme can be applied for
parallel computation. Its implementation can be made trivially because the
macroparticles are never moved from one processor to another. No inter-
processor communication is necessary while the particle trajectories are being
developed. Most calculations on each node are executed sequentially. In this
model the communication between the parallel processes is only required for
reading input data, generating an initial beam distribution, calculating
diagnostics such as beam emittance, and writing out the diagnostic
information. For the Poisson solver model, however, we have used a particle-
in-cell (PIC) model to update the electromagnetic field. The PIC model
represents the beam as a large number of computational particles moving
according to classical mechanics. The PIC algorithm can be characterized as
follows: (a) integrate over particles to obtain a charge distribution on the
grid point, (b) solve a Poisson equation for the potential, and (c)
interpolate the potential or field onto particles for a small interval of time
to advance the position and velocity of particles. Part (a) requires
$\mathcal{O}\left(N_{g}^{d}\right)$ numeric operations for a FFT Poisson
solver, where $N_{g}$ is the number of grid points per dimension and $d$ is
the number of degrees of freedom. Part (a) and (c) obviously require
$\mathcal{O}\left(N_{p}\right)$ operations, where $N_{p}$ is the number of
computation particles. In general, $N_{p}$ is much larger than $N_{g}$ in that
the number of particles should increase according to the degree of freedom to
maintain the statistical noise to be constant in a higher spatial dimension.
The particle calculations thus dominate the overall computational process,
which suggests a prior parallelization of particle calculation. Master/slave
configuration of computational nodes shown in Fig. 2 is considered due to the
difference of numeric operations between particles and field updates.
Each processor on the master and slave nodes possesses the same number of
particles. All processors are responsible for advancing their particles. On
the contrary, the master node may be a single or many processor(s), depending
on the number of grid points required. The charge density of a beam is
deposited on the computational grids of each processor using standard area
weighting (or higher order) methods [36]. The master node gathers the charge
density from all processors, and solves the Poisson equations in parallel. The
master node broadcasts the solution of the electric field to all processors
such that each processor exerts the electromagnetic force on the particles
owned by the processor.
Figure 2: Master/slave communication diagram.
The performance of the master/slave parallelization approach has been
investigated using a real lattice of the Tevatron which has two head-on beam-
beam collisions and 70 long-range beam-beam interactions. Speedup test has
been performed on the Cray XT5 of the National Energy Research Scientific
Computing Center at Lawrence Berkeley National Laboratory. The system is built
up of 664 nodes with two quad-core AMD 2.4 GHz processors per node. The
speedup of a parallel program is a measure of the utilization of parallel
resources and is simply defined as the ratio between sequential execution time
and parallel execution time [37]:
$S_{p}=\frac{T_{1}}{T_{p}}$ (35)
where $p$ is the number of processors, $T_{1}$ is the execution time of the
sequential algorithm, and $T_{p}$ is the execution time of the parallel
algorithm with $p$ processors. For a fixed number of processors $p$, typically
the speedup is $0<S_{p}\leq p$. Ideally all parallel programs should exhibit a
linear speedup, i.e., $S_{p}=p$, but it is not common because communication
between processors is considerably slower than computation in each processor.
Figure 3 (a) illustrates the resulting speedup as a function of the number of
processors.
(a) (b)
Figure 3: Plots of (b) parallel speedup versus the number of nodes, and (b)
CPU time versus the number of simulation particles. cerf and table represent
the weak-strong model, and look-up table model respectively.
The parallelization speedup based on the total simulation time is compared for
simulations with the weak-strong model and the look-up table model. The
speedup curves are very close to the ideal one below a certain number of
processors, while they are less than optimal when the number of processors
increases above a critical value, for example, $2^{6}$ processors. On large
numbers of processors a relative fraction of the communication time in the
total computing time becomes large. A parallel efficiency, defined as the
speedup factor divided by the number of processors, can be obtained as high as
87% up to the critical number of processors. Though the efficiency falls well
below 38% when the number of processors is beyond $2^{10}$, it runs 367 times
faster than on a single processor. In order to see the scalability of our
parallel code for larger problem sizes, Fig. 3 (b) shows the execution time as
a function of the number of macro-particles. Here the number of processors is
fixed at $2^{6}$ for all cases. It is seen that with increasing the number of
simulation particles, the execution time also increases linearly.
## 4 Applications
In high energy storage-ring colliders, the beam-beam interactions cause
emittance growth, may reduce beam lifetime, and hence limit the collider
luminosity. We have used BBSIM to study beam-beam interactions and their
compensations in the Tevatron, in RHIC and in the LHC.
### 4.1 Tevatron
The luminosity of a collider is found from
$\mathcal{L}=\frac{N_{1}N_{2}fN_{B}}{4\pi\sigma_{x}\sigma_{y}}R$ (36)
where $N_{1}$ and $N_{2}$ are the bunch populations of the colliding beams,
$f$ the revolution frequency, $N_{B}$ the number of bunches in one beam,
$\sigma_{x}$ and $\sigma_{y}$ the horizontal and vertical rms beam sizes at
the collision points respectively, and $R$ the luminosity reduction factor due
to the “hour-glass” effect and due to non-zero crossing angle at the
interaction point. The beam-beam tune shift of beam 1 is proportional to the
factor $N_{2}/\sigma_{x}\sigma_{y}$ and experience from colliders worldwide
has shown that the achievable tune shift (and hence luminosity) is limited by
the dynamics of the beam-beam interaction. In the Tevatron, proton and anti-
proton bunches collide at two detectors called CDF and D0. They share the same
beam pipe. Since the two beams circulate on helical orbits, the optics and
dynamics of the beam-beam interactions are complex. The beam-beam interactions
occur all around the ring and at varying betatron phases. In run II, each beam
has three trains of 12 bunches [38]. Each bunch experiences 72 interactions:
two interactions are the head-on collisions in the detectors. However, the
other 70 interactions are long-range, and are placed at different locations
for each bunch. Consequently the beam separation distances between proton and
anti-proton beams at the long-range locations are different from bunch to
bunch. Figure 4 shows the radial beam separation of three anti-proton bunches
from the proton bunches in units of the rms beam size of the proton beam at
the locations of the beam-beam interactions.
Figure 4: Separation distance between proton and anti-proton beams for anti-
proton bunches #1, #6 and #12. The separation is normalized by proton beam’s
rms size.
The long-range interactions of special importance are those on either side of
the head-on interaction points. These occur at small separations and the beta
functions there are large. It was observed that the emittance growth at the
end bunches of each train is smaller than those in the middle of the train.
Here we choose two end bunches (#1 and #12) and one middle bunch (#6) of the
first train.
Beam emittance growth and loss rate are routinely measured during the Tevatron
operation. They can be directly compared with numerical simulations but only
for relatively short times. Figure 5 (a) shows the time evolution of the four-
dimensional emittance of bunches #1, #6, and #12 for 15 hours of high energy
physics (HEP) run of store # 7650.
(a) (b) (c) (d)
Figure 5: (a) Variation of anti-proton emittance of three bunches, #1, #6, and
#12, of store #7650, (b) non-luminous loss rates of anti-proton during the
first 1 hour of stores #7601-#7650, (c) simulation of anti-proton emittance
growth, and (d) simulation of anti-proton beam loss. Here the emittance is
plotted as $\epsilon_{4d}=\sqrt{\epsilon_{x}\epsilon_{y}}$. In the simulation,
initial anti-proton emittance $\left(\epsilon_{x},\epsilon_{y}\right)$ is
(9.0,7.8) mm-mrad, bunch length 1.5 nsec, and bunch intensity $0.86\times
10^{11}$. Proton’s initial emittance is (18,23) mm-mrad, bunch length 1.7
nsec, bunch intensity $2.64\times 10^{11}$. Nominal tune is (20.571, 20.569).
Revolution frequency is 47.7 kHz.
The emittance is calculated and plotted by
$\epsilon_{4d}=\sqrt{\epsilon_{x}\epsilon_{y}}$. It is observed that during
the HEP run, the emittance growth is nearly linear. The growth rate is
6.7%/hr. Figure 5 (b) shows the measured beam loss rates of anti-proton
bunches during the first 1 hour of store #7601-#7650 at collision energy 960
GeV. In order to see the effects of beam-beam interactions on the beam loss,
the loss rate is obtained by subtracting the particle losses due to luminosity
at the main interaction points from the total beam loss rate. Averaged loss
rates of bunch #1 and #12 are 1.4 %/hr and 1.2 %/hr respectively, while the
loss rate of bunch #6 is 2.3 %/hr. We performed the simulations of emittance
growth and particle loss of anti-proton beam, as shown in Fig. 5 (c)-(d). The
particle tracking is carried out over $10^{7}$ turns corresponding to
approximately 3.5 minutes storage time of the Tevatron. In the simulation,
nominal tune is (20.571, 20.569). Initial transverse (95% normalized)
emittance of anti-protons $\left(\epsilon_{x},\epsilon_{y}\right)$ is set to
be (9.0,7.8) mm-mrad from averaging the measured emittances while proton’s
initial emittance is (18,23) mm-mrad. Bunch intensities of anti-proton and
proton are $0.86\times 10^{11}$ and $2.64\times 10^{11}$ respectively. Figure
5 (c) shows the emittance growth of three bunches during the simulation. The
growth rate is approximately 9 %/hr, which is close to the measured growth
rate 7 %/hr in Fig. 5 (a). The emittance does not vary from bunch to bunch.
However, the beam losses vary considerably from bunch to bunch. As shown in
Fig. 5 (d), bunch #6 loses more particles than bunches #1 and #12, which
agrees well with the observation. For the simulation of beam loss, we used the
hollow Gaussian distribution in transverse action coordinates. Most of the
lost particles have large transverse actions as shown in Fig. 6 (a), while the
lost particles are distributed over the entire range of longitudinal action,
as shown in Fig. 6 (b).
(a) (b)
Figure 6: (a) Scatter plot of lost particles in action space
$\left(\sqrt{J_{x}},\sqrt{J_{y}}\right)$ and (b) plot of lost particles versus
$\sqrt{J_{x}+J_{y}}$ for different longitudinal action. The axis variables are
normalized by rms size of transverse action.
The compensation of long-range effects in the Tevatron with a current-carrying
wire was investigated using an earlier version of the code [8]. It was found
that a single wire was unable to compensate for all the 70 interactions, since
they were all at different betatron phases from the wire.
### 4.2 Relativistic Heavy Ion Collider
We have studied the effects of a current-carrying wire on the beam dynamics in
RHIC [32]. Two current-carrying wires, one for each beam, have been installed
between the magnets $Q3$ and $Q4$ of IP6 in the RHIC tunnel. In the physics
run 9, an attempt was made to compensate the long range beam-beam interaction
which shows the reduction of beam loss [39]. During the physics run 7 and 8,
the impact of current-carrying wires on a beam was measured without an attempt
to compensate the beam-beam interactions. However, the experimental results
help to understand the beam-beam effects because the wire force is similar to
the long-range beam-beam force at large separations. As an example, Fig. 7
plots the beam loss rate due to the wire as a function of beam-wire separation
distance.
(a) (b)
Figure 7: Comparison of the simulated beam loss rates with the measured as a
function of separations. (a) gold beam at collision energy, (b) deuteron beam
at collision energy [32].
The onset of beam losses is observed at 8 $\sigma$ and 9 $\sigma$ for gold and
deuteron beams, respectively. The threshold separation for the onset of sharp
losses observed in the measurements and simulations agree to better than 1
$\sigma$. It is also significant that the simulated loss rates at 7 and 8
$\sigma$ separation for the gold beam and 8 and 9 $\sigma$ for the deuteron
beam are very close to the measured loss rates. At fixed separation, the wire
causes a much higher beam loss with the deuteron beam than with the gold beam.
The loss rate for the gold beam at a 8 $\sigma$ separation is about 10 %/hr
while for the deuteron beam the loss rate is about an order of magnitude
higher both in measurements and simulation. Simulations of the beam loss rate
when the wire is present are in good agreement with the experimental
observations.
In the proton-proton runs of RHIC, the maximum beam-beam parameter reached so
far is about $\xi=0.008$. This tune shift is large enough that the combination
of beam-beam and machine nonlinearities excite betatron resonances which cause
emittance growth and diffuse particles into the tail of beam distribution and
beyond. Consequently RHIC is actively developing an electron lens for
compensating the head-on interactions [40]. In order to seek the electron lens
parameters at which the beam life time is improved, we choose three different
electron beam distribution functions: (a) $1\sigma$ Gaussian distribution with
the same rms beam size as that of the proton beam $\sigma$, (b) $2\sigma$
Gaussian distribution with rms size twice that of the proton beam, and (c)
Smooth-edge-flat-top (SEFT) distribution with an edge around at 4 $\sigma$.
When the electron beam profile matches the proton beam, the full compression
of the tune spread requires the electron beam intensity $N_{e}=4\times
10^{11}$ which is defined as the electron beam intensity required for full
compensation. Table 1 shows the results of particle loss for different
intensities with the three electron beam profiles.
Profile | Intensity $\left(4\times 10^{11}\right)$ | Particle loss†(%)
---|---|---
$1\sigma$ Gaussian | 1 | 635
| 1/2 | 115
| 1/4 | 63
| 1/8 | 30
$2\sigma$ Gaussian | 4 | 93
| 2 | 10
| 1 | 8
| 1/2 | 6
SEFT | 8 | 330
| 4 | 21
| 2 | 22
| 1 | 6
| 1/2 | 6
†relative to that without beam-beam compensation
Table 1: Comparison of particle loss for different electron beam profiles and
intensities.
At an intensity $N_{e}=4\times 10^{11}$, the particle loss is nearly six times
the loss without beam-beam compensation. The beam lifetime at $N_{e}=2\times
10^{11}$ however is comparable with that of no beam-beam compensation. As the
electron beam intensity is decreased, the particle loss decreases
significantly, and is reduced to 30% of that without beam-beam compensation at
$N_{e}=0.5\times 10^{11}$. For the $2\sigma$ Gaussian and SEFT electron beam
profiles, we calculated particle loss for different electron beam intensities.
The upper limits of the electron beam intensity for these two distributions
are chosen so that peak of the electron profile matches that of the full
compensation at $1\sigma$ Gaussian. For the intensities $2\times 10^{11}$ and
$4\times 10^{11}$ of $2\sigma$ Gaussian profile, there is a significant
reduction in beam loss, for example, below 10% of the particle loss without
beam-beam compensation when the electron beam intensity is $2\times 10^{11}$.
A significant improvement of beam lifetime with the SEFT profile is also
observed below $8\times 10^{11}$. There is a threshold electron beam intensity
below which beam life time is increased: $2\times 10^{11}$ for the $1\sigma$
Gaussian, $8\times 10^{11}$ for the $2\sigma$ Gaussian, and $16\times
10^{11}$for the SEFT profile. Particle loss is relatively insensitive to
electron lens current variations below the threshold current with the
$2\sigma$ Gaussian and SEFT profiles. This looser tolerance on the allowed
variations in electron intensity will allow greater intensity fluctuations and
is likely to be beneficial during experiments.
### 4.3 Large Hadron Collider
As mentioned above, long-range beam-beam interactions cause emittance growth
or beam loss in the Tevatron and are expected to deteriorate beam quality in
the LHC. Increasing the crossing angle to reduce their effects has several
undesirable effects, the most important of which is a lower luminosity due to
the smaller geometric overlap. For the LHC, a wire compensation scheme has
been proposed to compensate the long-range interactions [23]. However, several
issues need to be resolved for efficient compensation. With the design bunch
spacing, there are about 30 long-range interactions on both sides of an
interaction point (IP). The beam-beam separation distance varies from 6.3
$\sigma$ to 12.6 $\sigma$. The resulting beam-beam force is not identical to
that generated by a single or multiple wire(s) but can be closely approximated
by the wires. Unlike the Tevatron, the long-range forces in the LHC are all at
nearly the same betatron phase and this makes the compensation scheme
feasible. The wire-beam separation distance is one of the parameters which
determine the performance of a wire compensator. Figure 8 (a) shows the beam-
beam separation distance normalized by the transverse rms bunch size. Two
counter-rotating beams collide at a vertical crossing angle near IP1 while
they collide at a horizontal crossing angle near IP5. The separations are
asymmetric with respect to the interaction points. The reference wire-beam
separation (9 $\sigma$) is chosen as the average of beam-beam separations.
(a) (b)
Figure 8: Plot of (a) beam-beam separation at IP 1 and 5 and (b) particle loss
according to wire separation distance with wire strength 82.8 Am.
Figure 8 (b) shows the results of particle loss for different wire-beam
separations. The particle loss saturates at large separation while there is a
sharp increase of particle loss at small separation. We directly see the
minimum particle loss between 0.9 and 1.0 of the reference separation. It
reveals that the average of beam-beam separations is close to an optimal
separation between the wire and the high energy bunch.
## 5 Summary
In this paper, an efficient parallel beam simulation model for circular
colliders is presented in order to study the effects of beam-beam interactions
and machine nonlinearities, and the effectiveness of beam-beam compensation
schemes. We have included the major nonlinearities present in accelerators in
our program as well as models for several methods to compensate the effects of
beam-beam interactions. A particle-domain decomposition scheme is implemented
with the master/slave configuration to achieve a balanced workload in a
parallel environment. A performance test of beam-beam interactions indicates
that the parallelization scheme scales linearly in both the number of
processors and the number of particles in the beam. We have used the program
to study the emittance growth and beam loss of different bunches due to the
beam-beam interactions in the Tevatron, the compensation of head-on beam-beam
interactions with a low energy electron beam in RHIC, and the long-range beam-
beam compensation using a current-carrying wire in the Tevatron, RHIC and the
LHC. The pattern of beam losses observed in the Tevatron is reproduced in the
simulations. In RHIC, simulations of the beam loss rate when the wire is
present are in good agreement with the experimental observations. We have
several predictions from the results of head-on compensation in RHIC. For
example we find that proton beam life time is increased if the electron beam
intensity is kept below a threshold intensity. An electron beam wider than the
proton beam at the electron lens location is found to increase beam life time.
The results of LHC simulation with the current carrying wire show that the
particle loss is minimized when the beam-wire separation is close to the
average of beam-beam separations.
## 6 Acknowledgments
We thank V. Boocha, B. Erdelyi and V. Ranjbar for their contributions to the
development of BBSIM. This research used resources of the Accelerator Physics
Center at Fermi National Accelerator Laboratory as well as resources of the
National Energy Research Scientific Computing Center at Lawrence Berkeley
National Laboratory, which is supported by the Office of Science of the U.S.
Department of Energy. This work is partially supported by the US Department of
Energy through the US LHC Accelerator Research Program (LARP). Fermi National
Accelerator Laboratory (Fermilab) is operated by Fermi Research Alliance, LLC
under Contract No. DE-AC02-07CH11359 with the United States Department of
Energy.
## References
* [1] J.-Y. Hemery, A. Hofmann, J.-P. Koutchouk, S. Myers, L. Vos, Investigation of the coherent beam-beam effects in the ISR, IEEE T. Nucl. Sci. NS-28 (1981) 2497\.
* [2] L. Evans, J. Gareyte, M. Meddahi, R. Schmidt, Beam-beam effects in the strong-strong regime at the CERN-SPS, in: Proceedings of the 1889 Particle Accelerator Conference, 1989.
* [3] W. Fischer, A. Drees, J. Brennan, R. Connolly, R. Fliller, S. Tepikian, J. van Zeijts, Beam lifetime and emittance growth measurements of gold beams in RHIC at storage, in: Proceedings of the 2001 Particle Accelerator Conference, 2001, p. 21.
* [4] T. Sen, B. Erdelyi, M. Xiao, V. Boocha, Beam-beam effects at the Fermilab Tevatron: Theory, Phys. Rev. Spec. Top. Acceler. Beams 7 (2004) 041001.
* [5] F. Zimmermann, Beam-beam effects in the Large Hadron Collider, in: D. Rice (Ed.), ICFA Beam Dynamics Newsletter, no. 34, 2004, p. 26.
* [6] V. Shiltsev, Y. Alexahin, V. Lebedev, P. Lebrun, R. S. Moore, T. Sen, A. Tollestrup, A. Valishev, X. L. Zhang, Beam-beam effects in the Tevatron, Phys. Rev. Spec. Top. Acceler. Beams 8 (2005) 101001.
* [7] J. Beebe-Wang, S. Y. Zhang, Observation and simulation of beam-beam induced emittance growth in RHIC, in: Proceedings of the 2009 Particle Acclerator Conference, 2009, p. 2622.
* [8] T. Sen, B. Erdelyi, Feasibility study of beam-beam compensation in the Tevatron with wires, in: Proccedings of the 2005 Particle Accelerator Conference, 2005, p. 2645.
* [9] F. Zimmermann, J.-P. Koutchouk, F. Roncarolo, J. Wenninger, T. Sen, V. Shiltsev, Y. Papaphilippou, Experiments on LHC long-range beam-beam compensation and crossing schemes at the CERN SPS in 2004, in: Proceedings of the 2005 Particle Accelerator Conference, 2005, p. 686.
* [10] H. J. Kim, T. Sen, N. P. Abreu, W. Fischer, Simulations of beam-beam and beam-wire interactions in RHIC, Phys. Rev. Spec. Top. Acceler. Beams 12 (2009) 031001.
* [11] http://www-ap.fnal.gov/$\sim$hjkim.
* [12] Message Passing Interface Forum, U. of Tennessee, Knoxbille, TN, MPI-2: Extenstions to the Message-Passing Interface (2003).
* [13] S. Balay, J. Brown, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. Smith, H. Zhang, PETSc Users Manual, Tech. Rep. ANL-95/11 - Revision 3.0.0, Argonne National Laboratory (2008).
* [14] NCSA, UIUC, Urbana, IL, HDF5 User’s Guide (2005).
* [15] K. Hirata, H. Moshammer, F. Ruggiero, A symplectic beam-beam interaction with energy change, Part. Accel. 40 (1993) 205.
* [16] K. Wille, The Physics of Particle Accelerators: an introduction, Oxford University Press, 2000.
* [17] L. C. Teng, Concerning n-dimensional coupled motions, Tech. Rep. FN-229, FNAL (1971).
* [18] M. Bassetti, G. A. Erskine, Closed expression for the electrical field of a two-dimensional Gaussian charge, Tech. Rep. CERN-ISR-TH/80-06, CERN (1980).
* [19] R. W. Hockney, The potential calculation and some applications, Meth. Comput. Phys. 9 (1970) 135–211.
* [20] J. Qiang, M. A. Furman, R. D. Ryne, Strong-strong beam-beam simulation using a green function approach, Phys. Rev. Spec. Top. Acceler. Beams 5 (2002) 104402–1.
* [21] K. Hirata, Don’t be afraid of beam-beam interactions with a large crossing angle, Tech. Rep. SLAC-PUB-637, SLAC (1994).
* [22] L. H. A. Leunissen, F. Schmidt, G.Ripken, Six-dimensional beam-beam kick including coupled motion, Phys. Rev. Spec. Top. Acceler. Beams 3 (2000) 124002\.
* [23] J.-P. Koutchouk, Principle of a correction of the long-range beam-beam effect in LHC using electromagnetic lenses, Tech. Rep. LHC Project Note 223, CERN (2000).
* [24] E. Tsyganov, R. Meinke, W. Nexen, A. Zinchenko, Compensation of the beam-beam effect in proton-proton colliders, Tech. Rep. SSCL-Preprint-519, SSCL (1993).
* [25] V. Shiltsev, V. Danilov, D. Finley, A. Sery, Considerations on compensation of beam-beam effects in the tevatron with electron beams, Phys. Rev. Spec. Top. Acceler. Beams 2 (1999) 071001.
* [26] V. Shiltsev, Y. Alexahin, K. Bishofberger, V. Kamerdzhiev, V. Parkhomchuk, V. Reva, N. Solyak, D. Wildman, X.-L. Zhang, F. Zimmermann, Experimental studies of compensation of beam-beam effects with Tevatron electron lenses, New J. Phys. 10 (2008) 043042.
* [27] R. B. Palmer, Energy scaling, crab crossing and the pair problem, Tech. Rep. SLAC-PUB-4707, SLAC (1988).
* [28] K. Oide, K. Yokoya, Beam-beam collision scheme for storage-ring colliders, Phys. Rev. A: At., Mol., Opt. Phys. 40 (1) (1989) 315–316.
* [29] B. Erdelyi, T. Sen, Compensation of beam-beam effects in the tevatron with wires, Tech. Rep. Fermilab-TM-2268-AD, Fermilab (2004).
* [30] C. K. Birdsall, A. B. Langdon, Plasma Physics via Computer Simulation, McGraw-Hill, New York, 1985.
* [31] T. Sen, J. A. Ellison, Diffusion due to beam-beam interaction and fluctuating fields in hadron colliders, Phys. Rev. Lett. 77 (6) (1996) 1051–1054.
* [32] H. J. Kim, T. Sen, N. P. Abreu, W. Fischer, Studies of wire compensation and beam-beam interaction in rhic, in: Proceedings of the 11th European Particle Accelerator Conference, 2008, p. 3119.
* [33] R. Bartolini, A. Bazzani, M. Giovannozzi, W. Scandale, E. Todesco, Tune evaluation in simulations and experiments, Tech. Rep. SL/95-84, CERN (1995).
* [34] J. Borer, G. Guignard, A. Hofmann, E. Peschardt, F. Sacherer, B. Zotter, Information from beam response to longitudinal and transverse excitation, IEEE T. Nucl. Sci. NS-26 (1979) 3405–3408.
* [35] J. Laskar, Frequency analysis for multi-dimensional systems. global dynamics and diffusion, Physica D 67 (1993) 257–281.
* [36] R. W. Hockney, J. W. Eastwood, Computer Simulation Using Particles, Taylor & Francis, Inc., 1988.
* [37] M. J. Quinn, Parallel Programming in C with MPI and OpenMP, McGraw-Hill, 2004\.
* [38] Run II Handbook, http://www-bd.fnal.gov/runII/index.html.
* [39] R. Calaga, W. Fischer, G. Robert-Demolaize, RHIC BBLR measurement in 2009, in: Proceedings of the 2010 International Particle Accelerator Conference, 2010, p. 510.
* [40] W. Fischer, et al., Status of RHIC head-on beam-beam compensation project, in: Proceedings of the 2010 International Particle Accelerator Conference, 2010, p. 513.
|
arxiv-papers
| 2011-01-27T20:02:23 |
2024-09-04T02:49:16.679495
|
{
"license": "Public Domain",
"authors": "Hyung J. Kim and Tanaji Sen",
"submitter": "Hyung Jin Kim",
"url": "https://arxiv.org/abs/1101.5373"
}
|
1101.5415
|
###### Abstract
Let $M_{R}$ be a module and $\sigma$ an endomorphism of $R$. Let $m\in M$ and
$a\in R$, we say that $M_{R}$ satisfies the condition $\mathcal{C}_{1}$
(respectively, $\mathcal{C}_{2}$), if $ma=0$ implies $m\sigma(a)=0$
(respectively, $m\sigma(a)=0$ implies $ma=0$). We show that if $M_{R}$ is
p.q.-Baer then so is $M[x;\sigma]_{R[x;\sigma]}$ whenever $M_{R}$ satisfies
the condition $\mathcal{C}_{2}$, and the converse holds when $M_{R}$ satisfies
the condition $\mathcal{C}_{1}$. Also, if $M_{R}$ satisfies $\mathcal{C}_{2}$
and $\sigma$-skew Armendariz, then $M_{R}$ is a p.p.-module if and only if
$M[x;\sigma]_{R[x;\sigma]}$ is a p.p.-module if and only if
$M[x,x^{-1};\sigma]_{R[x,x^{-1};\sigma]}$ ($\sigma\in Aut(R)$) is a
p.p.-module. Many generalizations are obtained and more results are found when
$M_{R}$ is a semicommutative module.
On Skew Polynomials over p.q.-Baer
and p.p.-Modules
Mohamed Louzari
Department of mathematics
Abdelmalek Essaadi University
B.P. 2121 Tetouan, Morocco
mlouzari@yahoo.com
This work is dedicated to my Professor El Amin Kaidi Lhachmi from University
of Almería on the occasion of his 62nd birthday.
Mathematics Subject Classification: 16S36, 16D80, 16W80
Keywords: Semicommutative modules, p.q.-Baer modules, p.p.-modules.
000Published in Inter. Math. Forum, Vol. 6, 2011, no. 35, 1739 - 1747
## 1 Introduction
In this paper, $R$ denotes an associative ring with unity and modules are
unitary. We write $M_{R}$ to mean that $M$ is a right module. Throughout,
$\sigma$ is an endomorphism of $R$ (unless specified otherwise), that is,
$\sigma\colon R\rightarrow R$ is a ring homomorphism with $\sigma(1)=1$. The
set of all endomorphisms (respectively, automorphisms) of $R$ is denoted by
$End(R)$ (respectively, Aut(R)). In [10], Kaplansky introduced Baer rings as
rings in which the right (left) annihilator of every nonempty subset is
generated by an idempotent. According to Clark [9], a ring $R$ is said to be
quasi-Baer if the right annihilator of each right ideal of $R$ is generated
(as a right ideal) by an idempotent. These definitions are left-right
symmetric. Recently, Birkenmeier et al. [7] called a ring $R$ a right
$($respectively, left$)$ principally quasi-Baer (or simply right
$($respectively, left$)$ p.q.-Baer) if the right (respectively, left)
annihilator of a principally right (respectively, left) ideal of $R$ is
generated by an idempotent. $R$ is called a p.q.-Baer ring if it is both right
and left p.q.-Baer. A ring $R$ is a right (respectively, left) p.p.-ring if
the right (respectively, left) annihilator of an element of $R$ is generated
by an idempotent. $R$ is called a p.p.-ring if it is both right and left
p.p.-ring.
Lee-Zhou [12] introduced Baer, quasi-Baer and p.p.-modules as follows: $(1)$
$M_{R}$ is called Baer if, for any subset $X$ of $M$, $r_{R}(X)=eR$ where
$e^{2}=e\in R$. $(2)$ $M_{R}$ is called quasi-Baer if, for any submodule $N$
of $M$, $r_{R}(N)=eR$ where $e^{2}=e\in R$. $(3)$ $M_{R}$ is called p.p. if,
for any $m\in M$, $r_{R}(m)=eR$ where $e^{2}=e\in R$.
In [3], a module $M_{R}$ is called principally quasi Baer (p.q.-Baer for
short) if, for any $m\in M$, $r_{R}(mR)=eR$ where $e^{2}=e\in R$. It is clear
that $R$ is a right p.q.-Baer ring if and only if $R_{R}$ is a p.q.-Baer
module. If $R$ is a p.q.-Baer ring, then for any right ideal $I$ of $R$,
$I_{R}$ is a p.q.-Baer module. Every submodule of a p.q.-Baer module is
p.q.-Baer module. Moreover, every quasi-Baer module is p.q.-Baer, and every
Baer module is quasi-Baer module.
A ring $R$ is called semicommutative if for every $a\in R$, $r_{R}(a)$ is an
ideal of $R$ (equivalently, for any $a,b\in R$, $ab=0$ implies $aRb=0$). In
[8], a module $M_{R}$ is semicommutative, if for any $m\in M$ and $a\in R$,
$ma=0$ implies $mRa=0$. Let $\sigma$ an endomorphism of $R$, $M_{R}$ is called
$\sigma$-semicommutative module [13] if, for any $m\in M$ and $a\in R$, $ma=0$
implies $mR\sigma(a)=0$. According to Annin [1], a module $M_{R}$ is
$\sigma$-compatible, if for any $m\in M$ and $a\in R$, $ma=0$ if and only if
$m\sigma(a)=0$.
In [12], Lee-Zhou introduced the following notations. For a module $M_{R}$, we
consider
$M[x;\sigma]:=\left\\{\sum_{i=0}^{s}m_{i}x^{i}:s\geq 0,m_{i}\in M\right\\},$
$M[[x;\sigma]]:=\left\\{\sum_{i=0}^{\infty}m_{i}x^{i}:m_{i}\in M\right\\},$
$M[x,x^{-1};\sigma]:=\left\\{\sum_{i=-s}^{t}m_{i}x^{i}:\;t\geq 0,s\geq
0,m_{i}\in M\right\\},$
$M[[x,x^{-1};\sigma]]:=\left\\{\sum_{i=-s}^{\infty}m_{i}x^{i}:s\geq 0,m_{i}\in
M\right\\}.$
Each of these is an Abelian group under an obvious addition operation.
Moreover $M[x;\sigma]$ becomes a module over $R[x;\sigma]$ under the following
scalar product operation:
For $m(x)=\sum_{i=0}^{n}m_{i}x^{i}\in M[x;\sigma]$ and
$f(x)=\sum_{j=0}^{m}a_{j}x^{j}\in R[x;\sigma]$
$m(x)f(x)=\sum_{k=0}^{n+m}\left(\sum_{k=i+j}m_{i}\sigma^{i}(a_{j})\right)x^{k}$
$None$
Similarly, $M[[x;\sigma]]$ is a module over $R[[x;\sigma]]$. The modules
$M[x;\sigma]$ and $M[[x;\sigma]]$ are called the skew polynomial extension and
the skew power series extension of $M$, respectively. If $\sigma\in Aut(R)$,
then with a scalar product similar to $(*)$ , $M[x,x^{-1};\sigma]$
(respectively, $M[[x,x^{-1};\sigma]]$) becomes a module over
$R[x,x^{-1};\sigma]$ (respectively, $R[[x,x^{-1};\sigma]]$). The modules
$M[x,x^{-1};\sigma]$ and $M[[x,x^{-1};\sigma]]$ are called the skew Laurent
polynomial extension and the skew Laurent power series extension of $M$,
respectively. In [13], a module $M_{R}$ is called $\sigma$-skew Armendariz, if
$m(x)f(x)=0$ where $m(x)=\sum_{i=0}^{n}m_{i}x^{i}\in M[x;\sigma]$ and
$f(x)=\sum_{j=0}^{m}a_{j}x^{j}\in R[x;\sigma]$ implies
$m_{i}\sigma^{i}(a_{j})=0$ for all $i$ and $j$. According to Lee-Zhou [12],
$M_{R}$ is called $\sigma$-Armendariz, if it is $\sigma$-compatible and
$\sigma$-skew Armendariz.
In this paper, we show that if $M_{R}$ is p.q.-Baer then so is
$M[x;\sigma]_{R[x;\sigma]}$ whenever $M_{R}$ satisfies the condition
$\mathcal{C}_{2}$, and the converse holds when $M_{R}$ satisfies the condition
$\mathcal{C}_{1}$ (Proposition 2.3). Also, if $M_{R}$ satisfies
$\mathcal{C}_{2}$ and $\sigma$-skew Armendariz, then $M_{R}$ is a p.p.-module
if and only if $M[x;\sigma]_{R[x;\sigma]}$ is a p.p.-module if and only if
$M[x,x^{-1};\sigma]_{R[x,x^{-1};\sigma]}$ ($\sigma\in Aut(R)$) is a
p.p.-module (Proposition 3.1). As a consequence, if $M_{R}$ is semicommutative
and $\sigma$-compatible then: $M_{R}$ is a p.p.-module $\Leftrightarrow$
$M_{R}$ is a p.q.-Baer module $\Leftrightarrow$ $M[x;\sigma]_{R[x;\sigma]}$ is
a p.p.-module $\Leftrightarrow$ $M[x;\sigma]_{R[x;\sigma]}$ is a p.q.-Baer
module (Theorem 3.6). Moreover, we obtain a generalization of some results in
[3, 4, 6, 12].
## 2 Skew polynomials over p.q.-Baer modules
We start with the next definition.
###### Definition 2.1.
Let $m\in M$ and $a\in R$. We say that $M_{R}$ satisfies the condition
$\mathcal{C}_{1}$ $($respectively, $\mathcal{C}_{2}$$)$, if $ma=0$ implies
$m\sigma(a)=0$ $($respectively, $m\sigma(a)=0$ implies $ma=0$$)$.
Note that $M_{R}$ is $\sigma$-compatible if and only if it satisfies
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$. Let $M_{R}$ be a module and
$\sigma\in End(R)$.
###### Lemma 2.2.
If $M_{R}$ satisfies $\mathcal{C}_{1}$ or $\mathcal{C}_{2}$, then
$me=m\sigma(e)$ for any $m\in M$ and any $e^{2}=e\in R$.
###### Proof.
Suppose $\mathcal{C}_{2}$, from $m\sigma(e)(1-\sigma(e))=0$, we have
$0=m\sigma(e)(1-e)=m\sigma(e)-m\sigma(e)e$, so $m\sigma(e)e=m\sigma(e)$. From
$m(1-\sigma(e))\sigma(e)=0$, we have $0=m(1-\sigma(e))e=me-m\sigma(e)e$, so
$m\sigma(e)=m\sigma(e)e=me$. The same for $\mathcal{C}_{1}$. ∎
###### Proposition 2.3.
Let $M_{R}$ be a module and $\sigma\in End(R)$. $(1)$ If $M_{R}$ is a
p.q.-Baer module then so is $M[x;\sigma]_{R[x;\sigma]}$, whenever $M_{R}$
satisfies the condition $\mathcal{C}_{2}$. $(2)$ If
$M[x;\sigma]_{R[x;\sigma]}$ or $M[[x;\sigma]]_{R[[x;\sigma]]}$ is a p.q.-Baer
module then so is $M_{R}$, whenever $M_{R}$ satisfies the condition
$\mathcal{C}_{1}$.
###### Proof.
$(1)$ Let $m(x)=m_{0}+m_{1}x+\cdots+m_{n}x^{n}\in M[x;\sigma]$. Then
$r_{R}(m_{i}R)=e_{i}R$, for some idempotents $e_{i}\in R\;(0\leq i\leq n)$.
Let $e=e_{0}e_{1}\cdots e_{n}$, then $eR=\cap_{i=0}^{n}r_{R}(m_{i}R)$. We show
that $r_{R[x;\sigma]}(m(x)R[x;\sigma])=eR[x;\sigma]$. Let
$\phi(x)=a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{p}x^{p}\in
r_{R[x;\sigma]}(m(x)R[x;\sigma])$. Since $m(x)R\phi(x)=0$, we have
$m(x)b\phi(x)=0$ for all $b\in R$. Then
$m(x)b\phi(x)=\sum_{\ell=0}^{n+p}\left(\sum_{\ell=i+j}m_{i}\sigma^{i}(ba_{j})\right)x^{\ell}=0.$
* •
$\ell=0$ implies $m_{0}ba_{0}=0$ then $a_{0}\in r_{R}(m_{0}R)=e_{0}R$.
* •
$\ell=1$ implies
$m_{0}ba_{1}+m_{1}\sigma(ba_{0})=0$ $None$
Let $s\in R$ and take $b=se_{0}$, so
$m_{0}se_{0}a_{1}+m_{1}\sigma(se_{0}a_{0})=0$, since $m_{0}se_{0}=0$ we have
$m_{1}\sigma(se_{0}a_{0})=m_{1}\sigma(sa_{0})=0$, so $m_{1}sa_{0}=0$, thus
$a_{0}\in r_{R}(m_{1}R)=e_{1}R$. In equation $(1)$,
$m_{1}\sigma(ba_{0})=m_{1}\sigma(be_{1}a_{0})=m_{1}\sigma(b)e_{1}\sigma(a_{0})=0$,
by Lemma 2.2. Then equation (1) gives $m_{0}ba_{1}=0$, so $a_{1}\in e_{0}R$.
* •
$\ell=2$ implies
$m_{0}ba_{2}+m_{1}\sigma(ba_{1})+m_{2}\sigma^{2}(ba_{0})=0$ $None$
Let $s\in R$ and take $b=se_{0}e_{1}$, so
$m_{0}se_{0}e_{1}a_{2}+m_{1}\sigma(s)e_{0}e_{1}\sigma(a_{1})+m_{2}\sigma^{2}(se_{0}e_{1}a_{0})=0$,
but $m_{0}se_{0}e_{1}a_{2}=m_{1}\sigma(s)e_{0}e_{1}\sigma(a_{1})=0$ we have
$m_{2}\sigma^{2}(se_{0}e_{1}a_{0})=0$, since $e_{0}e_{1}a_{0}=a_{0}$ we have
$m_{2}\sigma^{2}(sa_{0})=0$ and so $m_{2}sa_{0}=0$ for all $s\in R$. Hence
$a_{0}\in e_{2}R$ (thus, $a_{0}\in e_{0}e_{1}e_{2}R$). Equation $(2)$, becomes
$m_{0}ba_{2}+m_{1}\sigma(ba_{1})+m_{2}\sigma^{2}(b)e_{0}e_{1}e_{2}\sigma^{2}(a_{0})=0$,
which gives
$m_{0}ba_{2}+m_{1}\sigma(ba_{1})=0$ $None$
Take $b=se_{0}$ in equation $(2^{\prime})$, we have
$m_{0}se_{0}a_{2}+m_{1}\sigma(se_{0}a_{1})=0$, but $m_{0}se_{0}a_{2}=0$ so
$m_{1}\sigma(se_{0}a_{1})=m_{1}\sigma(sa_{1})=0$ and thus $m_{1}sa_{1}=0$,
hence $a_{1}\in e_{1}R$ (so, $a_{1}\in e_{0}e_{1}R$). Equation $(2^{\prime})$
gives $m_{0}ba_{2}=0$, so $a_{2}\in e_{0}R$.
At this point, we have $a_{0}\in e_{0}e_{1}e_{2}R,\;a_{1}\in e_{1}e_{2}R$ and
$a_{2}\in e_{0}R$. Continuing this procedure yields $a_{i}\in eR$ ($0\leq
i\leq n$). Hence $\phi(x)\in eR[x;\sigma]$. Consequently,
$r_{R[x;\sigma]}(m(x)R[x;\sigma])\subseteq eR[x;\sigma]$. Conversely, let
$\varphi(x)=b_{0}+b_{1}x+b_{2}x^{2}+\cdots+b_{p}x^{p}\in R[x;\sigma]$. Then
$m(x)\varphi(x)e=\sum_{\ell=0}^{n+p}\left(\sum_{\ell=i+j}m_{i}\sigma^{i}(b_{j})\sigma^{\ell}(e)\right)x^{\ell}=\sum_{\ell=0}^{n+p}\left(\sum_{\ell=i+j}m_{i}\sigma^{i}(b_{j})e\right)x^{\ell}.$
Since $e\in\bigcap_{i=0}^{n}r_{R}(m_{i}R)$, then $m_{i}Re=0$ ($0\leq i\leq
n$). Thus $m(x)\varphi(x)e=0$, hence $eR[x;\sigma]\subseteq
r_{R[x;\sigma]}(m(x)R[x;\sigma])$. Thus
$r_{R[x;\sigma]}(m(x)R[x;\sigma])=eR[x;\sigma]$, therefore
$M[x;\sigma]_{R[x;\sigma]}$ is p.q.-Baer.
$(2)$ Let $0\neq m\in M$. We have
$r_{R[x;\sigma]}(mR[x;\sigma])=e{R[x;\sigma]}$ for some idempotent
$e=\sum_{i=0}^{n}e_{i}x^{i}\in R[x;\sigma]$. We have
$r_{R[x;\sigma]}(mR[x;\sigma])\cap R=e_{0}R$. On other hand, we show that
$r_{R[x;\sigma]}(mR[x;\sigma])\cap R=r_{R}(mR)$. Let $a\in r_{R}(mR)$ then
$mRa=0$, so $mR\sigma^{i}(a)=0$ for all $i\geq 1$. So $mR[x;\sigma]a=0$.
Therefore $a\in r_{R[x;\sigma]}(mR[x;\sigma])\cap R$. Conversely, let $a\in
r_{R[x;\sigma]}(mR[x;\sigma])\cap R$, then $mR[x;\sigma]a=0$, in particular
$mRa=0$, so $a\in r_{R}(mR)$. Thus $a\in r_{R}(mR)=e_{0}R$, with
$e_{0}^{2}=e_{0}\in R$. So $M_{R}$ is p.q.-Baer. The same method for
$M[[x;\sigma]]$. ∎
###### Corollary 2.4 ([3, Theorem 11]).
$M_{R}$ is p.q.-Baer if and only if $M[x]_{R[x]}$ is p.q.-Baer.
###### Corollary 2.5 ([6, Theorem 3.1]).
$R$ is right p.q.-Baer if and only if $R[x]$ is right p.q.-Baer.
$M_{R}$ is called $\sigma$-reduced module by Lee-Zhou [12], if for any $m\in
M$ and $a\in R$: $(1)$ $ma=0$ implies $mR\cap Ma=0$, $(2)$ $ma=0$ if and only
if $m\sigma(a)=0$.
###### Corollary 2.6 ([3, Theorem 7(1)]).
Let $M_{R}$ a $\sigma$-compatible module. Then the following hold: $(1)$ If
$M[x;\sigma]_{R[x;\sigma]}$ is a p.q.-Baer module then so is $M_{R}$. The
converse holds if in addition $M_{R}$ is $\sigma$-reduced. $(2)$ If
$M[[x;\sigma]]_{R[[x;\sigma]]}$ is a p.q.-Baer module then so is $M_{R}$.
###### Corollary 2.7 ([4, Corollary 2.6]).
Let $M_{R}$ be a $\sigma$-compatible module. Then $M_{R}$ is p.q.-Baer if and
only if $M[x;\sigma]_{R[x;\sigma]}$ is p.q.-Baer.
## 3 Skew polynomials over p.p.-modules
Let $M_{R}$ be an $\sigma$-Armendariz module, if $me=0$ where $e^{2}=e\in R$
and $m\in M$, then $mfe=0$ for any $f^{2}=f\in R$ (by [12, Lemma 2.10]). This
result still true if we replace the condition “$M_{R}$ is $\sigma$-Armendariz”
by “$M_{R}$ is $\sigma$-skew Armendariz satisfying $\mathcal{C}_{2}$”.
###### Proposition 3.1.
Let $M_{R}$ be a $\sigma$-skew Armendariz module which satisfies the condition
$\mathcal{C}_{2}$. The following statements hold: $(1)$ $M_{R}$ is a
p.p.-module if and only if $M[x;\sigma]_{R[x;\sigma]}$ is a p.p.-module, $(2)$
Let $\sigma\in Aut(R)$, then $M_{R}$ is a p.p.-module if and only if
$M[x,x^{-1};\sigma]_{R[x,x^{-1};\sigma]}$ is a p.p.-module.
###### Proof.
$(1)$$(\Leftarrow)$ Is clear by [12, Theorem 2.11]. $(\Rightarrow)$ Let
$m(x)=m_{0}+m_{1}x+\cdots+m_{n}x^{n}\in M[x;\sigma]$, then
$r_{R}(m_{i})=e_{i}R$, for some idempotents $e_{i}\in R\;(0\leq i\leq n)$. Let
$e=e_{0}e_{1}\cdots e_{n}$, then $m_{i}e=0$ for all $0\leq i\leq n$ ([12,
Lemma 2.10]) and by Lemma 2.2, we have $m_{i}\sigma^{j}(e)=0$ for all $0\leq
i\leq n$ and $j\geq 0$. Therefore $e\in r_{R[x;\sigma]}(m(x))$, so
$eR[x;\sigma]\subseteq r_{R[x;\sigma]}(m(x))$. Conversely, let
$\phi(x)=a_{0}+a_{1}x+\cdots+a_{p}x^{p}\in r_{R[x;\sigma]}(m(x))$, then
$m(x)\phi(x)=0$. Since $M_{R}$ is $\sigma$-skew Armendariz, we have
$m_{i}\sigma^{i}(a_{j})=0$ for all $i,j$ and with the condition
$\mathcal{C}_{2}$ we have $m_{i}a_{j}=0$ for all $i,j$. So $a_{j}\in
r_{R}(m_{i})=e_{i}R$ for all $i,j$. Thus
$a_{j}\in\cap_{i=0}^{n}r_{R}(m_{i})=eR$ for each $j$. Then $\phi(x)\in
e{R[x;\sigma]}$, therefore $r_{R[x;\sigma]}(m(x))=eR[x;\sigma]$. With the same
method, we can prove $(2)$. ∎
###### Corollary 3.2 ([12, Theorem 11(1a,2a)]).
If $M_{R}$ is $\sigma$-Armendariz. Then: $(1)$ $M_{R}$ is a p.p.-module if and
only if $M[x;\sigma]_{R[x;\sigma]}$ is a p.p.-module, $(2)$ Let $\sigma\in
Aut(R)$, then $M_{R}$ is a p.p.-module if and only if
$M[x,x^{-1};\sigma]_{R[x,x^{-1};\sigma]}$ is a p.p.-module.
If $M_{R}$ is a semicommutative module such that, $m\sigma(a)a=0$ implies
$m\sigma(a)=0$ for any $m\in M$ and $a\in R$. Then $M_{R}$ is
$\sigma$-semicommutative and hence it satisfies the condition
$\mathcal{C}_{1}$. To see this, suppose that $ma=0$ then $mRa=0$, in
particular $mr\sigma(a)a=0$ for all $r\in R$. By the above condition,
$mr\sigma(a)=0$ for all $r\in R$. Thus $M_{R}$ is $\sigma$-semicommutative.
###### Lemma 3.3.
If $M_{R}$ is a semicommutative module such that $m\sigma(a)a=0$ implies
$m\sigma(a)=0$ for any $m\in M$ and $a\in R$. Then $M_{R}$ is $\sigma$-skew
Armendariz.
###### Proof.
Let $m(x)=m_{0}+m_{1}x+\cdots+m_{n}x^{n}\in M[x;\sigma]$ and
$f(x)=a_{0}+a_{1}x+\cdots+a_{p}x^{p}\in R[x;\sigma]$. From $m(x)f(x)=0$, we
have $\sum_{i+j=k}m_{i}\sigma^{i}(a_{j})=0$, for $0\leq k\leq n+p$. So,
$m_{0}a_{0}=0$. Assume that $s\geq 0$ and $m_{i}\sigma^{i}(a_{j})=0$ for all
$i,j$ with $i+j\leq s$. Note that for $s+1$, we have
$m_{0}a_{s+1}+m_{1}\sigma(a_{s})+\cdots+m_{s}\sigma^{s}(a_{1})+a_{s+1}\sigma^{s+1}(a_{0})=0$
$None$
Multiplying $(1)$ by $\sigma^{s}(a_{0})$ from the right hand, we obtain
$m_{0}a_{s+1}\sigma^{s}(a_{0})+m_{1}\sigma(a_{s})\sigma^{s}(a_{0})+\cdots+m_{s}\sigma^{s}(a_{1})\sigma^{s}(a_{0})+a_{s+1}\sigma^{s+1}(a_{0})\sigma^{s}(a_{0})=0,$
we have $m_{0}a_{0}=0$, then $m_{0}\sigma^{s}(a_{0})=0$ because $M_{R}$ is
$\sigma$-semicommutative, and so $m_{0}a_{s+1}\sigma^{s}(a_{0})=0$. Also,
$m_{1}\sigma(a_{0})=0$ then $m_{1}\sigma^{s}(a_{0})=0$, thus
$m_{1}\sigma(a_{s})\sigma^{s}(a_{0})=0$. Continuing this process until the
step $s$, $m_{s}\sigma^{s}(a_{0})=0$ then $m_{s}\sigma^{s}(a_{1})$
$\sigma^{s}(a_{0})=0$. Therefore
$m_{s+1}\sigma^{s+1}(a_{0})\sigma^{s}(a_{0})=0$. But
$m_{s+1}\sigma^{s+1}(a_{0})\sigma^{s}(a_{0})=m_{s+1}\sigma[\sigma^{s}(a_{0})]\sigma^{s}(a_{0})=0.$
So $m_{s+1}\sigma^{s+1}(a_{0})=0$ . Therefore, equation $(1)$, becomes
$m_{0}a_{s+1}+m_{1}\sigma(a_{s})+\cdots+m_{s}\sigma^{s}(a_{1})=0$ $None$
Multiplying $(2)$, by $\sigma^{s-1}(a_{1})$ from the right hand to obtain
$m_{s}\sigma^{s}(a_{1})=0$. Continuing this procedure yields
$m_{0}a_{s+1}=m_{1}\sigma(a_{s})=\cdots=m_{s}\sigma^{s}(a_{1})=a_{s+1}\sigma^{s+1}(a_{0})=0.$
A simple induction shows that $m_{i}\sigma^{i}(a_{j})=0$, for all $i,j$. ∎
###### Proposition 3.4.
Let $M_{R}$ be a module such that $m\sigma(a)a=0$ implies $m\sigma(a)=0$ for
any $m\in M$ and $a\in R$. If $M_{R}$ is semicommutative then
$M[x;\sigma]_{R[x;\sigma]}$ and $M[[x;\sigma]]_{R[[x;\sigma]]}$ are
semicommutative.
###### Proof.
Let $m(x)=\sum_{i=0}^{n}m_{i}x^{i}\in M[x;\sigma]$,
$f(x)=\sum_{j=0}^{q}a_{j}x^{j}\in R[x;\sigma]$ and
$\phi(x)=\sum_{k=0}^{p}b_{k}x^{k}\in R[x;\sigma]$. Suppose that $m(x)f(x)=0$.
The coefficients of $m(x)\phi(x)f(x)$ are of the form
$\sum_{u+v=w}m_{u}\sigma^{u}\left(\sum_{i+j=v}b_{i}\sigma^{i}(a_{j})\right)=\sum_{u+v=w}\left(\sum_{i+j=v}m_{u}\sigma^{u}(b_{i})\sigma^{u+i}(a_{j})\right).$
By Lemma 3.3, $m_{u}\sigma^{u}(a_{j})=0$, for all $u,j$ and by
$\mathcal{C}_{1}$, $m_{u}\sigma^{u+i}(a_{j})=0$, for all $i,j,u$. Since
$M_{R}$ is semicommutative then $m_{u}\sigma^{u}(b_{i})\sigma^{u+i}(a_{j})=0$,
therefore
$\sum_{u+v=w}m_{u}\sigma^{u}\left(\sum_{i+j=v}b_{i}\sigma^{i}(a_{j})\right)=0.$
So $m(x)\phi(x)f(x)=0$, then $M[x;\sigma]_{R[x;\sigma]}$ is semicommutative.
The same for $M[[x;\sigma]]_{R[[x;\sigma]]}$. ∎
According to Baser and Harmanci [3], a module $M_{R}$ is reduced if for any
$m\in M$ and $a\in R$, $ma^{2}=0$ implies $mR\cap Ma=0$. By [2, Lemma 2.11],
if $M_{R}$ is semicommutative p.p. or semicommutative p.q.-Baer then it’s
reduced.
###### Corollary 3.5.
Let $M_{R}$ be a semicommutative module satisfying the condition
$\mathcal{C}_{1}$, if $M_{R}$ is p.q.-Baer or p.p. then
$M[x;\sigma]_{R[x;\sigma]}$ and $M[[x;\sigma]]_{R[[x;\sigma]]}$ are
semicommutative.
###### Proof.
Let $a\in R$ and $m\in M$ such that $m\sigma(a)a=0$, then $m(\sigma(a))^{2}=0$
(by $\mathcal{C}_{1}$), since $M_{R}$ is reduced we have $m\sigma(a)=0$. By
Proposition 3.4, $M[x;\sigma]_{R[x;\sigma]}$ and
$M[[x;\sigma]]_{R[[x;\sigma]]}$ are semicommutative. ∎
###### Theorem 3.6.
If $M_{R}$ is semicommutative and $\sigma$-compatible. Then the following are
equivalent: $(1)$ $M_{R}$ is p.p. $(2)$ $M_{R}$ is p.q.-Baer, $(3)$
$M[x;\sigma]_{R[x;\sigma]}$ is p.p., $(4)$ $M[x;\sigma]_{R[x;\sigma]}$ is
p.q.-Baer,
###### Proof.
$(1)\Leftrightarrow(2)$ By [2, Proposition 2.7]. $(2)\Leftrightarrow(4)$ By
Proposition 2.3. $(3)\Rightarrow(4)$ Since $M_{R}$ is a p.p.-module, then
Corollary 3.5 implies that $M[x;\sigma]_{R[x;\sigma]}$ is semicommutative.
Therefore $M[x;\sigma]_{R[x;\sigma]}$ is p.q.-Baer by [2, Proposition 2.7].
$(4)\Rightarrow(3)$ By Proposition 2.3, $M_{R}$ is p.q.-Baer, since $M_{R}$ is
semicommutative then $M[x;\sigma]_{R[x;\sigma]}$ is semicommutative, and so
$M[x;\sigma]_{R[x;\sigma]}$ is a p.p.-module. ∎
###### Corollary 3.7.
Let $M_{R}$ be a semicommutative module. Then the following are equivalent:
$(1)$ $M_{R}$ is p.p. $(2)$ $M_{R}$ is p.q.-Baer, $(3)$ $M[x]_{R[x]}$ is p.p.,
$(4)$ $M[x]_{R[x]}$ is p.q.-Baer,
###### Corollary 3.8 ([5, Theorem 2.8]).
If $M_{R}$ is a reduced module. Then the following are equivalent: $(1)$
$M_{R}$ is p.p. $(2)$ $M_{R}$ is p.q.-Baer, $(3)$ $M[x]_{R[x]}$ is p.p., $(4)$
$M[x]_{R[x]}$ is p.q.-Baer,
###### Proof.
Every reduced module is semicommutative by [12, Lemma 1.2]. ∎
## References
* [1] S. Annin, Associated primes over skew polynomials rings, Comm. Algebra, 30 (2002), 2511-2528
* [2] M. Baser and N. Agayev, On reduced and semicommutative modules, Turk. J. Math., 30 (2006), 285-291.
* [3] M. Baser and A. Harmanci, reduced and p.q.-Baer modules, Taiwanese J. Math., 2 (1) (2007), 267-275.
* [4] M. Baser and A. Harmanci, On quasi-Baer and p.q.-Baer modules, Kyungpook Math. Journal, 49 (2009), 255-263.
* [5] M. Baser and M.T. Kosan, On quasi-Armendariz modules, Taiwanese J. Math., 12 (3) (2008), 573-582.
* [6] G.F. Birkenmeier, J.Y. Kim and J.K. Park, On polynomial extensions of principally quasi-Baer rings, Kyungpook Math. journal, 40 (2) (2000), 247-253.
* [7] G.F. Birkenmeier, J.Y. Kim and J.K. Park, Principally quasi-Baer rings, Comm. Algebra, 29 (2) (2001), 639-660.
* [8] A.M. Buhphang and M.B. Rege, semicommutative modules and Armendariz modules, Arab J. Math. Sciences, 8 (2002), 53-65.
* [9] W.E. Clark, Twisted matrix units semigroup algebras, Duke Math. Soc., 35 (1967), 417-424.
* [10] I. Kaplansky, Rings of operators, Math. Lecture Notes series, Benjamin, New York, 1965.
* [11] T.K. Kwak, Extensions of extended symmetric rings, Bull. Korean Math. Soc., 44 (2007), 777-788.
* [12] T. K. Lee and Y. Lee, Reduced Modules, Rings, modules, algebras and abelian groups, 365-377, Lecture Notes in Pure and App. Math., 236, Dekker, New york, (2004).
* [13] C.P Zhang and J.L. Chen, $\sigma$-skew Armendariz modules and $\sigma$-semicommutative modules, Taiwanese J. Math., 12 (2) (2008), 473-486.
|
arxiv-papers
| 2011-01-27T23:22:31 |
2024-09-04T02:49:16.688833
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mohamed Louzari",
"submitter": "Louzari Mohamed",
"url": "https://arxiv.org/abs/1101.5415"
}
|
1101.5463
|
# Walking on a Graph with a Magnifying Glass
Stratified Sampling via Weighted Random Walks
Maciej Kurant Minas Gjoka Carter T. Butts Athina Markopoulou
University of California, Irvine {mkurant, mgjoka, buttsc, athina}@uci.edu
(2011)
###### Abstract
Our objective is to sample the node set of a large unknown graph via crawling,
to accurately estimate a given metric of interest. We design a random walk on
an appropriately defined weighted graph that achieves high efficiency by
preferentially crawling those nodes and edges that convey greater information
regarding the target metric. Our approach begins by employing the theory of
stratification to find optimal node weights, for a given estimation problem,
under an independence sampler. While optimal under independence sampling,
these weights may be impractical under graph crawling due to constraints
arising from the structure of the graph. Therefore, the edge weights for our
random walk should be chosen so as to lead to an equilibrium distribution that
strikes a balance between approximating the optimal weights under an
independence sampler and achieving fast convergence. We propose a heuristic
approach (stratified weighted random walk, or S-WRW) that achieves this goal,
while using only limited information about the graph structure and the node
properties. We evaluate our technique in simulation, and experimentally, by
collecting a sample of Facebook college users. We show that S-WRW requires
13-15 times fewer samples than the simple re-weighted random walk (RW) to
achieve the same estimation accuracy for a range of metrics.
††conference: SIGMETRICS’11, June 7–11, 2011, San Jose, California,
USA.00footnotetext: * This is an extended version of a paper with the same
title presented at _SIGMETRICS’11_. This work was supported by SNF grant
PBELP2-130871, Switzerland, and by the NSF CDI Award 1028394, USA.
## 1 Introduction
Figure 1: Illustrative example. Our goal is to compare the blue and black
subpopulations (e.g., with respect to their median income) in population (a).
Optimal independence sampler, WIS (b), over-samples the black nodes, under-
samples the blue nodes, and completely skips the white nodes. A naive crawling
approach, RW (c), samples many irrelevant white nodes. WRW that enforces WIS-
optimal probabilities may result in poor or no convergence (d). S-WRW (e)
strikes a balance between the optimality of WIS and fast convergence.
Many types of online networks, such as online social networks (OSNs), Peer-to-
Peer (P2P) networks, or the World Wide Web (WWW), are measured and studied
today via sampling techniques. This is due to several reasons. First, such
graphs are typically too large to measure in their entirety, and it is
desirable to be able to study them based on a small but representative sample.
Second, the information pertaining to these networks is often hard to obtain.
For example, OSN service providers have access to all information in their
user base, but rarely make this information publicly available.
There are many ways a graph can be sampled, e.g., by sampling nodes, edges,
paths, or other substructures [27, 23]. Depending on our measurement goal, the
elements with different properties may have different importance and should be
sampled with a different probability. For example, Fig. 1(a) depicts the
world’s population, with residents of China (1.3B people) represented by blue
nodes, of the Vatican (800 people) by black nodes, and all other nationalities
represented by white nodes. Assume that we want to compare the median income
in China and Vatican. Taking a uniform sample of size 100 from the entire
world’s population is ineffective, because most of the samples will come from
countries other than China and Vatican. Even restricting our sample to the
union of China and Vatican will not help much, as our sample is unlikely to
include any Vatican resident. In contrast, uniformly sampling 50 Chinese and
50 Vaticanese residents would be much more accurate with the same sampling
budget.
This type of problem has been widely studied in the statistical and survey
sampling literature. A commonly used approach is _stratified sampling_ [34,
12, 28], where nodes (e.g., people) are partitioned into a set of non-
overlapping _categories_ (or strata). The objective is then to decide how many
independent draws to take from each category, so as to minimize the
uncertainty of the resulting measurement. This effect can be achieved in
expectation by a weighted independence sampler (WIS) with appropriately chosen
sampling probabilities $\pi^{\scriptscriptstyle\textrm{WIS}}$. In our example,
WIS samples Vatican residents with much higher probabilities than Chinese
ones, and avoids completely the rest of the world, as illustrated in Fig.
1(b).
However, WIS, as every independence sampler, requires a sampling frame, i.e.,
a list of all elements we can sample from (e.g., a list of all Facebook
users). This information is typically not available in today’s online
networks. A feasible alternative is _crawling_ (also known as exploration or
link-trace sampling). It is a graph sampling technique in which we can see the
neighbors of already sampled users and make a decision on which users to visit
next.
In this paper, we study how to perform stratified sampling through graph
crawling. We illustrate the key idea and some of the challenges in Fig. 1.
Fig. 1(c) depicts a social network that connects the world’s population. A
simple random walk (RW) visits every node with frequency proportional to its
degree, which is reflected by the node size. In this particular example, for a
simplicity of illustration, all nodes have the same degree equal to 3. As a
result, RW is equivalent to the uniform sample of the world’s population, and
faces exactly the same problems of wasting resources, by sampling all nodes
with the same probability.
We address these problems by appropriately setting the edge weights and then
performing a random walk on the weighted graph, which we refer to as _weighted
random walk_ (WRW). One goal in setting the weights is to mimic the WIS-
optimal sampling probabilities $\pi^{\scriptscriptstyle\textrm{WIS}}$ shown in
Fig. 1(b). However, such a WRW might perform poorly due to potentially slow
mixing. In our example, it will not even converge because the underlying
weighted graph is disconnected, as shown in Fig. 1(d). Therefore, the edge
weights under WRW (which determine the equilibrium distribution
$\pi^{\scriptscriptstyle\textrm{WRW}}$) should be chosen in a way that strikes
a balance between the optimality of $\pi^{\scriptscriptstyle\textrm{WIS}}$ and
fast convergence.
We propose Stratified Weighted Random Walk (S-WRW), a practical heuristic that
effectively strikes such a balance. We refer to our approach as “walking on
the graph with a magnifying glass”, because S-WRW over-samples more relevant
parts of the graph and under-samples less relevant ones. In our example, S-WRW
results in the graph presented in Fig. 1(e). The only information required by
S-WRW are the categories of neighbors of every visited node, which is
typically available in crawlable online networks, such as Facebook. S-WRW uses
two natural and easy-to-interpret parameters, namely: (i)
$\tilde{f}_{\ominus}$, which controls the fraction of samples from irrelevant
categories and (ii) $\gamma$, which is the maximal resolution of our
magnifying glass, with respect to the largest relevant category.
The main contributions of this paper are the following.
* •
We propose to improve the efficiency of crawling-based graph sampling methods,
by performing a stratified weighted random walk that takes into account not
only the graph structure but also the node properties that are relevant to the
measurement goal.
* •
We design and evaluate S-WRW, a practical heuristic that sets the edge weights
and operates with limited information.
* •
As a case study, we apply S-WRW to sample Facebook and estimate the sizes of
colleges. We show that S-WRW requires 13-15 times fewer samples than a simple
random walk for the same estimation accuracy.
The outline of the rest of the paper is as follows. Section 2 summarizes the
most popular graph sampling techniques, including sampling by exploration.
Section 3 presents classical stratified sampling. Section 4 combines
stratified sampling with graph exploration, presenting a unified WRW approach
that takes into account both network structure and node properties; various
trade-offs and practical issues are discussed and an efficient heuristic
(S-WRW) is proposed based on the insights. Section 5 presents simulation
results. Section 6 presents an implementation of S-WRW for the problem of
estimating the college friendship graph on Facebook. Section 7 presents
related work. Section 8 concludes the paper.
## 2 Sampling techniques
### 2.1 Notation
We consider an undirected, static,111Sampling dynamic graphs is currently an
active research area [40, 35, 42], but out of the scope of this paper. graph
$G=(V,E)$, with $N\\!=\\!|V|$ nodes and $|E|$ edges. For a node $v\in V$,
denote by $\deg(v)$ its degree, and by $\mathcal{N}(v)\subset V$ the list of
neighbors of $v$. A graph $G$ can be weighted. We denote by $\textrm{w}(u,v)$
the weight of edge $\\{u,v\\}\in E$, and by
$\textrm{w}(u)=\sum_{v\in\mathcal{N}(u)}\textrm{w}(u,v)$ (1)
the weight of node $u\in V$. For any set of nodes $A\subseteq V$, we define
its volume $\textrm{vol}(A)$ and weight $\textrm{w}(A)$, respectively, as
$\textrm{vol}(A)=\sum_{v\in A}\deg(v)\quad\textrm{ and
}\quad\textrm{w}(A)=\sum_{v\in A}\textrm{w}(v).$ (2)
We will often use
$f_{A}=\frac{|A|}{|V|}\quad\textrm{ and }\quad
f^{\scriptscriptstyle\textrm{vol}}_{A}=\frac{\textrm{vol}(A)}{\textrm{vol}(V)}$
(3)
to denote the relative size of $A$ in terms of the number of nodes and the
volumes, respectively.
Sampling. We collect a sample $S\subseteq V$ of $n\\!=\\!|S|$ nodes. $S$ may
contain multiple copies of the same node, i.e., the sampling is with
replacement. In this section, we briefly review the techniques for sampling
nodes from graph $G$. We also present the weighted random walk (WRW) which is
the basic building block for our approach.
### 2.2 Independence Sampling
Uniform Independence Sampling (UIS) samples the nodes directly from the set
$V$, with replacements, uniformly and independently at random, i.e., with
probability
$\pi^{\scriptscriptstyle\textrm{UIS}}(v)\ =\ \frac{1}{N}\qquad\textrm{ for
every }v\in V.$ (4)
Weighted Independence Sampling (WIS) is a weighted version of UIS. WIS samples
the nodes directly from the set $V$, with replacements, independently at
random, but with probabilities proportional to node weights $\textrm{w}(v)$:
$\pi^{\scriptscriptstyle\textrm{WIS}}(v)\ =\ \frac{\textrm{w}(v)}{\sum_{u\in
V}\textrm{w}(u)}.$ (5)
In general, UIS and WIS are not possible in online networks because of the
lack of sampling frame. For example, the list of all user IDs may not be
publicly available, or the user ID space may be too sparsely allocated.
Nevertheless, we present them as baseline for comparison with the random
walks.
### 2.3 Sampling via Crawling
In contrast to independence sampling, the crawling techniques are possible in
many online networks, and are therefore the main focus of this paper.
Simple Random Walk (RW) [29] selects the next-hop node $v$ uniformly at random
among the neighbors of the current node $u$. In a connected and aperiodic
graph, the probability of being at the particular node $v$ converges to the
stationary distribution
$\pi^{\scriptscriptstyle\textrm{RW}}(v)\ =\ \frac{\deg(v)}{2\cdot|E|}.$ (6)
Metropolis-Hastings Random Walk (MHRW) is an application of the Metropolis-
Hastings algorithm [30] that modifies the transition probabilities to converge
to a desired stationary distribution. For example, we can achieve the uniform
stationary distribution
$\pi^{\scriptscriptstyle\textrm{MHRW}}(v)\ =\ \frac{1}{N}$ (7)
by randomly selecting a neighbor $v$ of the current node $u$ and moving there
with probability $\min(1,\frac{\deg(u)}{\deg(v)})$. However, it was shown in
[35, 17] that RW (after re-weighting, as in Section 2.4) outperforms MHRW for
most applications. We therefore restrict our attention to comparing against
RW.
Weighted Random Walk (WRW) is RW on a weighted graph [4]. At node $u$, WRW
chooses the edge $\\{u,v\\}$ to follow with probability $P_{u,v}$ proportional
to the weight $\textrm{w}(u,v)\geq 0$ of this edge, i.e.,
$P_{u,v}=\frac{\textrm{w}(u,v)}{\sum_{v^{\prime}\in\mathcal{N}(u)}\textrm{w}(u,v^{\prime})}.$
(8)
The stationary distribution of WRW is:
$\pi^{\scriptscriptstyle\textrm{WRW}}(v)\ =\ \frac{\textrm{w}(v)}{\sum_{u\in
V}\textrm{w}(u)}.$ (9)
WRW is the basic building block of our design. In the next sections, we show
how to choose weights for a specific estimation problem.
Graph Traversals (BFS, DFS, RDS, …) is a family of crawling techniques where
no node is sampled more than once. Because traversals introduce a generally
unknown bias (see Sec. 7), we do not consider them in this paper.
### 2.4 Correcting the bias
RW, WRW, and WIS all produce biased (nonuniform) node samples. But their bias
is known and therefore can be corrected by an appropriate re-weighting of the
measured values. This can be done using the Hansen-Hurwitz estimator [19] as
first shown in [39, 41] for random walks and also used in [35]. Let every node
$v\in V$ carry a value $x(v)$. We can estimate the population total
$x_{\scriptscriptstyle\textrm{tot}}=\sum_{v}x(v)$ by
$\hat{x}_{\scriptscriptstyle\textrm{tot}}=\frac{1}{n}\sum_{v\in
S}\frac{x(v)}{\pi(v)},$ (10)
where $\pi(v)$ is the sampling probability of node $v$ in the stationary
distribution. In practice, we usually know $\pi(v)$, and thus
$\hat{x}_{\scriptscriptstyle\textrm{tot}}$, only up to a constant, i.e., we
know the (non-normalized) weights $\textrm{w}(v)$. This problem disappears
when we estimate the population mean
$x_{\scriptscriptstyle\textrm{av}}=\sum_{v}x(v)/N$ as
$\hat{x}_{\scriptscriptstyle\textrm{av}}\ =\ \frac{\sum_{v\in
S}\frac{x(v)}{\pi(v)}}{\sum_{v\in S}\frac{1}{\pi(v)}}\ =\ \frac{\sum_{v\in
S}\frac{x(v)}{\textrm{w}(v)}}{\sum_{v\in S}\frac{1}{\textrm{w}(v)}}.$ (11)
For example, for $x(v)\\!=\\!1$ if $\deg(v)\\!=\\!k$ (and $x(v)\\!=\\!0$
otherwise), $\hat{x}_{\scriptscriptstyle\textrm{av}}(k)$ estimates the node
degree distribution in $G$.
All the results in this paper are presented _after this re-weighting_ step,
whenever necessary.
## 3 Stratified Sampling
In Sec. 1, we argued that in order to compare the median income of residents
of China and Vatican we should take 50 random samples from each of these two
countries, rather than taking 100 UIS samples from China and Vatican together
(or, even worse, from the world’s population). This problem naturally arises
in the field of survey sampling. The most common solution is _stratified
sampling_ [34, 12, 28], where nodes $V$ are partitioned into a set
$\mathcal{C}$ of non-overlapping node categories (or “strata”), with
$\bigcup_{C\in\mathcal{C}}C=V$. Next, we select uniformly at random $n_{i}$
nodes from category $C_{i}$. We are free to choose the allocation
$(n_{1},n_{2},\ldots,n_{|\mathcal{C}|})$, as long as we respect the total
budget of samples $n\\!=\\!\sum_{i}n_{i}$.
Under _proportional allocation_ [28] (or “prop’) we use $n_{i}\propto|C_{i}|$,
i.e.,
$n_{i}^{\scriptscriptstyle\textrm{prop}}\ =\ |C_{i}|\cdot n/N.$ (12)
Another possibility is to do an optimal allocation (or “opt”) that minimizes
the variance $\mathbb{V}$ of our estimator for the specific problem of
interest. For example, assume that every node $v\in V$ carries a value $x(v)$,
and we may want to estimate the mean of $x$ in various scenarios, as discussed
below.
### 3.1 Examples of Stratified Sampling Problems
#### 3.1.1 Estimating the mean across the entire $V$
A classic application of stratification is to better estimate the population
mean $\mu$, given several groups (strata) of different properties (e.g.,
variances). Given $n_{i}$ samples from category $C_{i}$, we can estimate the
mean $\mu_{i}=\frac{1}{|C_{i}|}\sum_{v\in C_{i}}x(v)$ over category $C_{i}$ by
$\hat{\mu}_{i}=\frac{1}{n_{i}}\sum_{v\in S\cap C_{i}}x(v)\qquad\textrm{ with
}\qquad\mathbb{V}(\hat{\mu}_{i})=\frac{\sigma_{i}^{2}}{n_{i}},$ (13)
where $\mathbb{V}(\hat{\mu}_{i})$ is the variance of this estimator and
$\sigma_{i}^{2}$ is the variance of population $C_{i}$. We can estimate
population mean $\mu$ by a weighted average over all $\hat{\mu}_{i}$s [28],
i.e.,
$\hat{\mu}=\sum_{i}\frac{|C_{i}|}{N}\cdot\hat{\mu}_{i}\qquad\textrm{ with
}\qquad\mathbb{V}(\hat{\mu})=\sum_{i}\frac{(|C_{i}|)^{2}\cdot\sigma_{i}^{2}}{N^{2}\cdot
n_{i}}.$
Under proportional allocation (Eq.(12)), this boils down to
$\mathbb{V}(\hat{\mu}^{\scriptscriptstyle\textrm{prop}})\ =\ \frac{1}{N\cdot
n}\ \sum_{i}|C_{i}|\cdot\sigma_{i}^{2}$. However, we can apply Lagrange
multipliers to find that $\mathbb{V}(\hat{\mu})$ is minimized when
$n_{i}^{\scriptscriptstyle\textrm{opt}}=\frac{|C_{i}|\cdot\sigma_{i}}{\sum_{j}|C_{j}|\cdot\sigma_{j}}\cdot
n.$ (14)
This solution is sometimes called ‘Neyman allocation’ [34]. This gives us the
variance under optimal allocation
$\mathbb{V}(\hat{\mu}^{\scriptscriptstyle\textrm{opt}})\ =\
\frac{1}{N^{2}\cdot n}\ \left(\sum_{i}|C_{i}|\cdot\sigma_{i}\right)^{2}$.
The variances $\mathbb{V}(\hat{\mu}^{\scriptscriptstyle\textrm{prop}})$ and
$\mathbb{V}(\hat{\mu}^{\scriptscriptstyle\textrm{opt}})$ are measures of the
performance of proportional and optimal allocation, respectively. In order to
make their practical interpretation easier, we also show how these variances
translate into sample lengths. We define as _gain_ $\alpha$ of ‘opt’ over
‘prop’ the number of times ‘prop’ must be longer than ‘opt’ in order to
achieve the same variance
$\textrm{gain }\ \alpha\ =\
\frac{n^{\scriptscriptstyle\textrm{prop}}}{n^{\scriptscriptstyle\textrm{opt}}},\
\textrm{ subject to }\
\mathbb{V}^{\scriptscriptstyle\textrm{prop}}\\!=\\!\mathbb{V}^{\scriptscriptstyle\textrm{opt}}.$
In that case, the gain is
$\alpha\ \ =\ \
N\cdot\frac{\sum_{i}|C_{i}|\cdot\sigma_{i}^{2}}{\left(\sum_{i}|C_{i}|\cdot\sigma_{i}\right)^{2}}\qquad(\geq
1).$ (15)
Notice that this gain does not depend on the sample budget $n$. The gain is
one of the main metrics we will use in the evaluation sections to assess the
efficiency of our technique compared to the random walk.
#### 3.1.2 Highest precision for all categories
If we are equally interested in each category, we might want the same (highest
possible) precision of estimating $\mu_{i}$ for all categories $C_{i}$. In
this case, the metric to minimize is $\mathbb{V}_{\max}\ =\
\max_{i}\left\\{\mathbb{V}(\hat{\mu}_{i})\right\\}\
=\max_{i}\left\\{\frac{\sigma_{i}^{2}}{n_{i}}\right\\}.$ Under proportional
allocation, this translates to
$\mathbb{V}_{\max}^{\scriptscriptstyle\textrm{prop}}\
=\frac{N}{n}\max_{i}\frac{\sigma_{i}^{2}}{|C_{i}|}$. But the optimal $n_{i}$,
which makes $\mathbb{V}(\hat{\mu}_{i})$ equal for all $i$, is
$n_{i}^{\scriptscriptstyle\textrm{opt}}=\frac{\sigma^{2}_{i}}{\sum_{j}\sigma^{2}_{j}}\cdot
n.$ (16)
Consequently, $\mathbb{V}_{\max}^{\scriptscriptstyle\textrm{opt}}\ =\
\frac{\sum_{i}\sigma_{i}^{2}}{n},$ which leads to gain
$\alpha=\frac{\max_{i}\left\\{\frac{N}{|C_{i}|}\sigma_{i}^{2}\right\\}}{\sum_{i}\sigma_{i}^{2}}\quad(\geq
1).$ (17)
#### 3.1.3 Smallest sum of variances across categories
Even if we are interested in all categories, an alternative objective is to
maximize the _average_ precision of category pair comparisons (see Sec. 5A.13
in [12]), which is equivalent to minimizing the sum
$\mathbb{V}_{\Sigma}=\sum_{i}\mathbb{V}(\hat{\mu}_{i})=\sum_{i}\frac{\sigma_{i}^{2}}{n_{i}}.$
In this case, proportional allocation achieves
$\mathbb{V}_{\Sigma}^{\scriptscriptstyle\textrm{prop}}=\frac{N}{n}\sum_{i}\frac{\sigma_{i}^{2}}{|C_{i}|}$.
while, using Lagrange multipliers we get
$n_{i}^{\scriptscriptstyle\textrm{opt}}=\frac{\sigma_{i}}{\sum_{j}\sigma_{j}}\cdot
n\qquad\textrm{ and
}\qquad\mathbb{V}_{\Sigma}^{\scriptscriptstyle\textrm{opt}}=\frac{\left(\sum_{i}\sigma_{i}\right)^{2}}{n},$
(18)
which leads to gain
$\alpha\ =\
\frac{\sum_{i}\frac{N}{|C_{i}|}\sigma_{i}^{2}}{\left(\sum_{i}\sigma_{i}\right)^{2}}\quad(\geq
1).$ (19)
#### 3.1.4 Relative sizes of node categories
Stratified sampling assumes that we know the sizes $|C_{i}|$ of node
categories. In some applications, however, these sizes are unknown and among
the values we need to estimate as well (e.g., by using UIS or WIS). We show in
Appendix C (for $|\mathcal{C}|\\!=\\!2$) that the optimal sample allocation
and the corresponding gain $\alpha$ of WIS over UIS are respectively
$n_{i}^{\scriptscriptstyle\textrm{WIS}}=\frac{1}{|\mathcal{C}|}\cdot
n\quad\textrm{ and }\quad\alpha\ =\ \frac{N^{2}}{4|C_{1}|\cdot|C_{2}|}.$ (20)
#### 3.1.5 Irrelevant category $C_{\ominus}$ (aggregated)
In many practical cases, we may want to measure some (but not all) node
categories. E.g., in Fig. 1, we are interested in blue and black nodes, but
not in white ones. Similarly, in our Facebook study in Section 6 we are only
interested in self-declared college students, which accounts for only 3.5% of
all users. We group all categories not covered by our measurement objective as
a single _irrelevant category_ $C_{\ominus}\in\mathcal{C}$, and we set
$n^{\scriptscriptstyle\textrm{opt}}_{\ominus}=0$. In contrast,
$n_{\ominus}^{\scriptscriptstyle\textrm{prop}}\ =\ |C_{\ominus}|\cdot n/N$. As
a result, under ‘opt’ we have $N/(N\\!-\\!|C_{\ominus}|)$ times more useful
samples than under ‘prop’. Now, if we allocate optimally all these useful
samples between the relevant categories
$\mathcal{C}\setminus\\{C_{\ominus}\\}$, the gain $\alpha$ becomes
$\alpha\ \ =\ \ \frac{N}{N-|C_{\ominus}|}\ \cdot\
\alpha(\mathcal{C}\setminus\\{C_{\ominus}\\}),$ (21)
where $\alpha(\mathcal{C}\setminus\\{C_{\ominus}\\})$ is the gain (15), (17),
(19) or (20), depending on the metric, calculated only within categories
$\mathcal{C}\setminus\\{C_{\ominus}\\}$.
In other words, gain $\alpha$ is now composed of two factors: (i) gain in
avoiding irrelevant categories, and (ii) gain in optimal allocation of samples
among the relevant categories.
#### 3.1.6 Practical Guideline
Let us look at the optimal weights in the above scenarios, when all
$\sigma_{i}=\sigma$ are the same. This is a reasonable working assumption in
many practical settings, since we typically do not have prior estimates of
$\sigma_{i}$. With this simplification, Eq.(14) becomes
$n_{i}^{\scriptscriptstyle\textrm{opt}}\ =\ \frac{|C_{i}|}{N}\cdot n\ =\
n_{i}^{\scriptscriptstyle\textrm{prop}}.$
In contrast, Eq.(16), Eq.(18) and Eq.(20) get simplified to
$n_{i}^{\scriptscriptstyle\textrm{opt}}\ =\ \frac{1}{|\mathcal{C}|}\cdot n.$
In conclusion, if we are interested in comparing the node categories with
respect to some properties (e.g., average node degree, category size), rather
than estimating a property across the entire population, we should take an
_equal number of samples from every relevant category_.
## 4 Edge weight setting under WRW
In the previous section, we studied the optimal sample allocation under
(independence) stratified sampling. However, independence node sampling is
typically impossible in large online graphs, while crawling the graph is a
natural, available exploration primitive. In this section, we show how to
perform a weighted random walk (WRW) which approximates the stratified
sampling of the previous section. We can formulate the general problem as
follows:
_Given a measurement objective, error metric and sampling budget
$|S|\\!=\\!n$, set the edge weights in graph $G$ such that the WRW measurement
error is minimized._
Although we are able to solve this problem analytically for some specific and
fully known topologies, it is not obvious how to address it in general,
especially under a limited knowledge of $G$. Instead, in this paper, we
propose S-WRW, a heuristic to set the edge weights. S-WRW starts from a
solution optimal under WIS, and takes into account practical issues that arise
in graph exploration. Once the weights are set, we simply perform WRW as
described in Section 2.3 and collect samples.
### 4.1 Preliminaries
#### 4.1.1 Category-level granularity
One can think of the problem in two levels of granularity: the original graph
$G\\!=\\!(V,E)$ and the category graph $G^{C}\\!=\\!(\mathcal{C},E^{C})$. In
$G^{C}$, nodes represent categories, and every undirected edge
$\\{C_{1},C_{2}\\}\in E^{C}$ represents the corresponding non-empty set of
edges $E_{C_{1},C_{2}}\subset E$ in the original graph $G$, i.e.,
$E_{C_{1},C_{2}}\ =\\{\\{u,v\\}\in E:\ u\in C_{1}\textrm{ and }v\in
C_{2}\\}\neq\emptyset.$
In our approach, we move from the finer granularity of $G$ to the coarser
granularity of $G^{C}$. This means that we are interested in collecting, say,
$n_{i}$ samples from category $C_{i}$, but we do not control how these $n_{i}$
nodes are collected (i.e., with what individual sampling probabilities).
The rationale for that simplification is twofold. From a theoretical point of
view, categories are exactly the properties of interest in the estimation
problems we consider. From a practical point of view, it is relatively easy to
obtain or infer information about categories, as we show e.g., in Sec. 4.2.1.
#### 4.1.2 Stratification in expectation
Ideally, we would like to enforce strictly stratified sampling. However, when
we use crawling instead of independence sampling, sampling exactly $n_{i}$
nodes from category $C_{i}$ (and no other nodes) is possible only by
discarding observations. It is thus more natural to frame the problem in terms
of the probability mass placed on each category in equilibrium. This can be
achieved by making the weight $\textrm{w}(C_{i})$ of each category
proportional to the desired number $n_{i}$ of samples, i.e.,
$\textrm{w}(C_{i})\ \propto\ n_{i}.$ (22)
As a result, we draw $n_{i}$ samples from $C_{i}$ _in expectation_.
#### 4.1.3 Main guideline
As the main guideline, S-WRW tries to realize the category weights
$\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$ that are optimal under
WIS. There are many edge weight settings in $G$ that achieve
$\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$. In our implementation,
we observe that $\textrm{vol}(C_{i})$ counts the number of edges incident on
nodes of $C_{i}$. Consequently, if for every category $C_{i}$ we set in $G$
the weights of all edges incident on nodes in $C_{i}$ to
$\textrm{w}_{e}(C_{i})\ =\
\frac{\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})}{\textrm{vol}(C_{i})}.$
(23)
then weight $\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$ are
achieved.222There exist many other edge weight assignments that lead to
$\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$. Eq.(23) has the
advantage of distributing the weights evenly across all $\textrm{vol}(C_{i})$
edges. This simple observation is central to the S-WRW heuristic.
In order to apply Eq.(23), we first have to calculate or estimate its terms
$\textrm{vol}(C_{i})$ and
$\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$.333In fact, we need to
know $\textrm{w}_{e}(C_{i})$ in Eq.(23) only _up to a constant factor_ ,
because these factors cancel out in the calculation of transition
probabilities of WRW in Eq.(8). Consequently, the same applies to
$\textrm{vol}(C_{i})$ and
$\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$. Below, we show how to do
it in Step 1 and 2, respectively. Next, in Steps 3-5, we show how to modify
these terms to account for practical problems arising mainly from the
underlying graph structure.
Main guideline (to be modified)
Set the edge weights in category $C_{i}$ to
$\textrm{w}^{\textrm{WIS}}(C_{i})\,/\,\textrm{vol}(C_{i}).$
Step 1: Estimation of Category Volumes
Estimate $\textrm{vol}(C_{i})$ with a pilot RW estimator $\hbox
to0.0pt{\raisebox{-1.50694pt}{$\hat{\phantom{\textrm{vol}}}$}\hss}\textrm{vol}(C_{i})$
as in Eq.(35).
Step 2: Category Weights Optimal Under WIS
For given measurement objective, calculate
$\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$ as in Sec. 3.
Step 3: Include Irrelevant Categories
Modify $\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$.
$\tilde{f}_{\ominus}$ \- desired fraction of irrelevant nodes.
Step 4: Tiny and Unknown Categories
Modify $\hbox
to0.0pt{\raisebox{-1.50694pt}{$\hat{\phantom{\textrm{vol}}}$}\hss}\textrm{vol}(C_{i})$.
$\gamma$ \- maximal resolution.
Step 5: Edge Conflict Resolution
Set the weights of inter-category edges to Eq.(28).
WRW sample
Use transition probabilities proportional to edge weights (Sec. 2.3).
Correct for the bias
Apply formulas from Sec. 2.4.
Final result
Figure 2: Overview of our approach.
### 4.2 Our practical solution: S-WRW
#### 4.2.1 Step 1: Estimation of Category Volumes
In general, we have no prior information about $G$ or $G^{C}$. Fortunately, it
is easy and inexpensive estimate the relative category volumes
$f^{\scriptscriptstyle\textrm{vol}}_{i}$ which is the first piece of
information we need in Eq.(23) (see footnote 3). Indeed, it is enough to run a
relatively short pilot RW, and plug the collected sample $S$ in Eq.(35)
derived in Appendix B, as follows
$\widehat{f}^{\scriptscriptstyle\textrm{vol}}_{i}\ =\ \ \frac{1}{n}\sum_{u\in
S}\left(\frac{1}{\deg(u)}\sum_{v\in\mathcal{N}(u)}\\!\\!1_{\\{v\in
C_{i}\\}}\right).$
#### 4.2.2 Step 2: Category Weights Optimal Under WIS
In order to find the optimal WIS category weights
$\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$ in Eq.(23), we first
calculate $n_{i}^{\scriptscriptstyle\textrm{opt}}$ as shown, under various
scenarios, in Sec. 3. Next, we plug the resulting
$n_{i}^{\scriptscriptstyle\textrm{opt}}$ in Eq.(22), e.g., by setting
$\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})=n^{\scriptscriptstyle\textrm{opt}}_{i}$.
#### 4.2.3 Step 3: Irrelevant Categories
Problem: Potentially poor or no convergence. Consider the toy example in Fig.
3(a). We are interested in finding the relative sizes of red (dark) and green
(light) categories. The white node in the middle is irrelevant for our
measurement objective. Due to symmetry, we distinguish between two types of
edges with weights $w_{1}$ and $w_{2}$. Under WIS, Eq.(20) gives us the
optimal weights $w_{1}>0$ and $w_{2}=0$, i.e., WIS samples every non-white
node with the same probability and never samples the white one. However, under
WRW with these weights, relevant nodes get disconnected into two components
and WRW does not converge. We observed a similar problem in Fig. 1.
Guideline: Occasionally visit irrelevant nodes. We show in Appendix D that the
optimal WRW weights in Fig. 3(a) are $w_{1}=0$ and $w_{2}>0$. In that case,
half of the samples are due to visits in the white (irrelevant) node. In other
words, WRW may benefit from allocating small weight
$\textrm{w}(C_{\ominus})\\!>\\!0$ to category $C_{\ominus}$ that groups all
(if any) categories irrelevant to our estimation. The intuition is that
irrelevant nodes may not contribute to estimation but may be needed for
connectivity or fast mixing.
Implementation in S-WRW. In S-WRW, we achieve this goal by replacing
$\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$ with
$\tilde{\textrm{w}}^{\scriptscriptstyle\textrm{WIS}}(C_{i})=\left\\{\begin{array}[]{ll}\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})&\textrm{if
}C_{i}\neq C_{\ominus}\\\ \tilde{f}_{\ominus}\cdot\sum_{C\neq
C_{\ominus}}\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C)&\textrm{if
}C_{i}=C_{\ominus}.\end{array}\right.$ (24)
The parameter $0\leq\tilde{f}_{\ominus}\ll 1$ controls the desired fraction of
visits in $C_{\ominus}$.
Figure 3: Optimal edge weights: WIS vs WRW. The objective is to compare the
sizes of red (dark) and green (light) categories.
#### 4.2.4 Step 4: Tiny and Unknown Categories
Problem: “black holes”. Every optical system has a fundamental magnification
limit due to diffraction and our “graph magnifying glass” is no exception.
Consider the toy graph in Fig. 3(b): it consists of a big clique
$C_{\scriptscriptstyle\textrm{big}}$ of 20 red nodes with edge weights
$w_{2}$, and a green category $C_{\scriptscriptstyle\textrm{tiny}}$ with two
nodes only and edge weights $w_{1}$. In Sec. 3.1.4, we saw that WIS optimally
estimates the relative sizes of red and green categories for
$\textrm{w}(C_{\scriptscriptstyle\textrm{big}})\\!=\\!\textrm{w}(C_{\scriptscriptstyle\textrm{tiny}})$,
i.e., for $w_{1}\\!=\\!190\,w_{2}$. However, for such large values of $w_{1}$,
the two green nodes behave as a sink (or a “black hole”) for a WRW of finite
length, thus increasing the variance of the category size estimation.
Guideline: limit edge weights. In other words, although WIS suggests to over-
sample small categories, WRW should “under-over-sample” very small categories
to avoid black holes. For example, in Fig. 3(b) $w_{1}\simeq 60\,w_{2}\ (\ll
190w_{2})$ is optimal for WRW of length $n\\!=\\!50$ (simulation results).
Implementation in S-WRW. In S-WRW, we achieve this goal by replacing
$\textrm{vol}(C_{i})$ in Eq.(23) with
$\displaystyle\hbox
to0.0pt{\raisebox{-2.15277pt}{$\tilde{\phantom{\textrm{vol}}}$}\hss}\textrm{vol}(C)$
$\displaystyle=$ $\displaystyle\max\Big{\\{}\hbox
to0.0pt{\raisebox{-2.15277pt}{$\hat{\phantom{\textrm{vol}}}$}\hss}\textrm{vol}(C),\
\textrm{vol}_{min}\Big{\\}},\quad\textrm{ where }$ (25)
$\displaystyle\textrm{vol}_{min}$ $\displaystyle=$
$\displaystyle\frac{1}{\gamma}\cdot\max_{C\neq C_{\ominus}}\\{\hbox
to0.0pt{\raisebox{-2.15277pt}{$\hat{\phantom{\textrm{vol}}}$}\hss}\textrm{vol}(C)\\}.$
(26)
Moreover, this formulation takes care of every category $C$ that was not
discovered by the pilot RW in Sec. 4.2.1, by setting $\hbox
to0.0pt{\raisebox{-2.15277pt}{$\tilde{\phantom{\textrm{vol}}}$}\hss}\textrm{vol}(C)\\!=\\!\textrm{vol}_{min}$.
#### 4.2.5 Step 5: Edge Conflict Resolution
Problem: Conflicting desired edge weights. With the above modifications, our
target edge weights defined in Eq.(23) can be rewritten as
$\tilde{\textrm{w}}_{e}(C_{i})\ =\
\frac{\tilde{\textrm{w}}^{\scriptscriptstyle\textrm{WIS}}(C_{i})}{\hbox
to0.0pt{\raisebox{-2.15277pt}{$\tilde{\phantom{\textrm{vol}}}$}\hss}\textrm{vol}(C_{i})}.$
(27)
We can directly set the weight
$\textrm{w}(u,v)\\!=\\!\tilde{\textrm{w}}_{e}(C(u))\\!=\\!\tilde{\textrm{w}}_{e}(C(v))$
for every intra-category edge $\\{u,v\\}$. However, for every inter-category
edge, we usually have “conflicting” weights
$\tilde{\textrm{w}}_{e}(C(u))\neq\tilde{\textrm{w}}_{e}(C(v))$ desired at the
two ends of the edge.
Guideline: prefer inter-category edges. There are several possible edge weight
assignments that achieve the desired category node weights. High weights on
intra-category edges and small weights on inter-category edges result in WRW
staying in small categories $C_{\scriptscriptstyle\textrm{tiny}}$ for a long
time. In order to improve the mixing time, we should do exactly the opposite,
i.e., assign relatively high weights to inter-category edges (connecting
relevant categories). As a result, WRW will enter
$C_{\scriptscriptstyle\textrm{tiny}}$ more often, but will stay there for a
short time. This intuition is motivated by Monte Carlo variance reduction
techniques such as the use of _antithetic variates_ [15], which seek to induce
negative correlation between consecutive draws so as to reduce the variance of
the resulting estimator.
Implementation in S-WRW. We choose to assign an edge weight
$\tilde{\textrm{w}}_{e}$ that is in between these two values
$\tilde{\textrm{w}}_{e}(C(u))$ and $\tilde{\textrm{w}}_{e}(C(v))$. We
considered several candidate such assignments. We may take the arithmetic or
geometric mean of the conflicting weights, which we denote by
$\textrm{w}^{\scriptscriptstyle\textrm{ar}}(u,v)$ and
$\textrm{w}^{\scriptscriptstyle\textrm{ge}}(u,v)$, respectively. We may also
use the maximum of the two values,
$\textrm{w}^{\scriptscriptstyle\textrm{max}}(u,v)$, which should improve
mixing according to the discussion above. However,
$\textrm{w}^{\scriptscriptstyle\textrm{max}}(u,v)$ alone would also add high
weight to irrelevant nodes $C_{\ominus}$ (possibly far beyond
$\tilde{f}_{\ominus}$). To avoid this undesired effect, we distinguish between
the two cases by defining a hybrid solution:
$\textrm{w}^{\scriptscriptstyle\textrm{hy}}(u,v)=\left\\{\begin{array}[]{ll}\textrm{w}^{\scriptscriptstyle\textrm{ge}}(u,v)&\textrm{if
}C_{\ominus}\in\\{C(u),C(v)\\}\\\
\textrm{w}^{\scriptscriptstyle\textrm{max}}(u,v)&\textrm{otherwise.}\end{array}\right.$
(28)
This hybrid edge assignment was the one we found to work best in practice -
see Section 6.
### 4.3 Discussion
#### 4.3.1 Information needed about the neighbors
In the pilot RW (Sec. 4.2.1) as well as in the main WRW, we assume that by
sampling a node $v$ we also learn the category (but not degree) of each of its
neighbors $u\in\mathcal{N}(v)$. Fortunately, such information is often
available in most online graphs at no additional cost, especially when
scraping html pages (as we do). For example, when sampling colleges in
Facebook (Sec. 6), we use the college membership information of all $v$’s
neighbors, which, in Facebook, is available at $v$ together with the friends
list.
#### 4.3.2 Cost of pilot RW
The pilot RW volume estimator described in Sec. 4.2.1 considers the categories
not only of the sampled nodes, but also of their neighbors. As a result, it
achieves high efficiency, as we show in simulations (Sec. 5.3.1) and Facebook
measurements (Sec. 6.1). Given that, and high robustness of S-WRW to
estimation errors (see Sec. 5.3.5), pilot RW should be only a small fraction
of the later WRW (e.g., 6.5% in our Facebook measurements in Sec. 6).
#### 4.3.3 Setting the parameters
S-WRW sets the edge weights trying to achieve roughly
$\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$ as the main goal. We
slightly shape $\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$ to avoid
black holes and improve mixing, which is controlled by two natural and easy-
to-interpret parameters, $\tilde{f}_{\ominus}$ and $\gamma$.
Irrelevant nodes visits $\tilde{f}_{\ominus}$. The parameter
$0\leq\tilde{f}_{\ominus}\ll 1$ controls the desired fraction of visits in
$C_{\ominus}$. When setting $\tilde{f}_{\ominus}$, we should exploit the
information provided by the pilot crawl. If the relevant categories appear
poorly interconnected and often separated by irrelevant nodes, we should set
$\tilde{f}_{\ominus}$ relatively high. We have seen an extreme case in Fig.
3(a), with disconnected relevant categories and optimal
$\tilde{f}_{\ominus}\\!=\\!0.5$. In contrast, when the relevant categories are
strongly interconnected, we should use much smaller $\tilde{f}_{\ominus}$.
However, because we can never be sure that the graph induced on relevant nodes
is connected, we recommend always using $\tilde{f}_{\ominus}>0$. For example,
when measuring Facebook in Sec. 6, we set $\tilde{f}_{\ominus}=1\%$.
Maximal resolution $\gamma$. The parameter $\gamma\geq 1$ can be interpreted
as the maximal resolution of our “graph magnifying glass”, with respect to the
largest relevant category $C_{\scriptscriptstyle\textrm{big}}$. S-WRW will
typically sample well all categories that are less than $\gamma$ times smaller
than $C_{\scriptscriptstyle\textrm{big}}$; all categories smaller than that
are relatively undersampled (see Sec. 6.2.4). In the extreme case, for
$\gamma\rightarrow\infty$, S-WRW tries to cover every category, no matter how
small, which may cause the “black hole” problem discussed in Sec. 4.2.4. In
the other extreme, for $\gamma\\!=\\!1$ (and identical
$\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$ for all categories,
including $C_{\ominus}$), S-WRW reduces to RW. We recommend always setting
$1<\gamma<\infty$. Ideally, we know
$|C_{\scriptscriptstyle\textrm{smallest}}|$ \- the smallest category size that
is still relevant to us. In that case we should set
$\gamma=|C_{\scriptscriptstyle\textrm{big}}|/|C_{\scriptscriptstyle\textrm{smallest}}|$.444Strictly
speaking, $\gamma$ is related to volumes $\textrm{vol}(C_{i})$ rather than
sizes $|C_{i}|$. They are equivalent when category volume is proportional to
its size, which is often the case, and is the central assumption in the
“scale-up method” [9]. For example, in Sec. 6 the categories are US colleges;
we set $\gamma\\!=\\!1000$, because colleges with size smaller than 1/1000th
of the largest one (i.e., with a few tens of students) seem irrelevant to our
measurement. As another rule of thumb, we should try to set smaller $\gamma$
for relatively small sample sizes and in graphs with tight community structure
(see Sec. 5.3.5).
#### 4.3.4 Conservative approach
Note that a reasonable setting of these parameters (i.e.,
$\tilde{f}_{\ominus}>0$ and $1<\gamma<\infty$, and any conflict resolution
discussed in the paper), increases the weights of large categories (including
$C_{\ominus}$) and decreases the weight of small categories, compared to
$\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$. This makes S-WRW
allocate category weights between the two extremes: RW and WIS. Consequently,
S-WRW can be considered _conservative_ (with respect to WIS).
#### 4.3.5 S-WRW is unbiased
It is also important to note that because the collected WRW sample is
eventually corrected with the actual sampling weights as described in Sec.
2.4, S-WRW estimation process is _unbiased_ , regardless of the choice of
weights (so long as convergence is attained). In contrast, suboptimal weights
(e.g., due to estimation error of
$\widehat{f}^{\scriptscriptstyle\textrm{vol}}_{C}$) can increase WRW mixing
time, and/or the _variance_ of the resulting estimator. However, our
simulations and empirical experiments on Facebook (see Sec. 5 and 6) show that
S-WRW is very robust to suboptimal choice of weights.
Figure 4: RW and S-WRW under two scenarios: Random (a-g) and Clustered (h-n).
In (b,i), we show error of two volume estimators: naive Eq.(32) (dotted) and
neighbor-based Eq.(35) (plain). Next, we show error of size estimator as a
function of $n$ (c,j) and $w$ (d,g,k,n); in the latter, UIS and RW correspond
to WIS and S-WRW for $w\\!=\\!1$. In (e,l), we show the empirical probability
that S-WRW visits $C_{\scriptscriptstyle\textrm{tiny}}$ at least once.
Finally, (f,m) is gain $\alpha$ of S-WRW over RW under the optimal choice of
$w$ (plain), and for fixed $\gamma\\!=\\!w\\!=\\!5$ (dashed).
## 5 Simulation results
The gain of our approach compared to RW comes from two main factors. First,
S-WRW avoids, to a large extent or completely, the nodes in $C_{\ominus}$ that
are irrelevant to our measurement. This fact alone can bring an arbitrarily
large improvement ($\frac{N}{N-|C_{\ominus}|}$ under WIS), especially when
$C_{\ominus}$ is large compared to $N$. We demonstrate this in the Facebook
measurements in Section 6. Second, we can better allocate samples among the
relevant categories. This factor is observable in our Facebook measurements as
well, but it is more difficult to evaluate due to the lack of ground-truth
therein. In this section, we evaluate the optimal allocation gain in a
controlled simulation and we demonstrate some key insights.
### 5.1 Setup
We consider a graph $G$ with 101K nodes and 505.5K edges organized in two
densely (and randomly) connected communities555The term “community” refers to
cluster and is defined purely based on topology. The term “category” is a
property of a node and is independent of topology. as shown in Fig. 4(h).
The nodes in $G$ are partitioned into two node categories:
$C_{\scriptscriptstyle\textrm{tiny}}$ with 1K nodes (dark red), and
$C_{\scriptscriptstyle\textrm{big}}$ with 100K nodes (light yellow). We
consider two extreme scenarios of such a partition. The ‘random’ scenario is
purely random, as shown in Fig. 4(a). In contrast, under ‘clustered’,
categories $C_{\scriptscriptstyle\textrm{tiny}}$ and
$C_{\scriptscriptstyle\textrm{big}}$ coincide with the existing communities in
$G$, as shown in Fig. 4(h). It is arguably the worst case scenario for graph
sampling by exploration.
We fix the edge weights of all internal edges in
$C_{\scriptscriptstyle\textrm{big}}$ to 1. All the remaining edges, i.e., all
edges incident on nodes in category $C_{\scriptscriptstyle\textrm{tiny}}$,
have weight $w$ each, where $w\geq 1$ is a parameter. Note that this is
equivalent to setting
$\tilde{\textrm{w}}_{e}(C_{\scriptscriptstyle\textrm{big}})\\!=\\!1$,
$\tilde{\textrm{w}}_{e}(C_{\scriptscriptstyle\textrm{tiny}})\\!=\\!w$, and
‘max’ or ‘hybrid’ conflict resolution.
### 5.2 Measurement objective and error metric
We are mainly interested in measuring the relative sizes
$f_{\scriptscriptstyle\textrm{tiny}}$ and $f_{\scriptscriptstyle\textrm{big}}$
of categories $C_{\scriptscriptstyle\textrm{tiny}}$ and
$C_{\scriptscriptstyle\textrm{big}}$, respectively.
We use Normalized Root Mean Square Error (NRMSE) to assess the estimation
error, defined as [37]:
${\small\textrm{NRMSE}}(\widehat{x})=\frac{\sqrt{\mathbb{E}\big{[}(\widehat{x}-x)^{2}\big{]}}}{x},$
(29)
where $x$ is the real value and $\widehat{x}$ is the estimated one.
### 5.3 Results
#### 5.3.1 Estimating volumes is usually cheap
The first step in S-WRW is obtaining category volume estimates
$\widehat{f}^{\scriptscriptstyle\textrm{vol}}_{i}$. We achieve it by running a
short pilot RW and applying the estimator Eq.(35). We show
${\small\textrm{NRMSE}}(\widehat{f}^{\scriptscriptstyle\textrm{vol}}_{\scriptscriptstyle\textrm{tiny}})$
as plain curves in Fig. 4(b). This estimator takes advantage of the knowledge
of the categories of the neighboring nodes, which makes it much more efficient
than the naive estimator Eq.(32) shown by dashed curves. Moreover, the
advantage of Eq.(35) over Eq.(32) grows with the graph density and the
skewness of its degree distribution (not shown here).
Note that under ‘random’, RW and WIS (with the sampling probabilities of RW)
are almost equally efficient. However, on the other extreme, i.e., under the
‘clustered’ scenario, the performance of RW becomes much worse and the
advantage of Eq.(35) over Eq.(32) diminishes. This is because essentially all
friends of a node from category $C_{i}$ are in $C_{i}$ too, which reduces
formula Eq.(35) to Eq.(32). Nevertheless, we show later in Sec. 5.3.5 that
even severalfold volume estimation errors are likely not to affect
significantly the results.
#### 5.3.2 Visiting the tiny category
Fig. 4(e,l) presents the empirical probability
$\mathbb{P}[C_{\scriptscriptstyle\textrm{tiny}}\textrm{ visited}]$ that our
walk visits at least one node from $C_{\scriptscriptstyle\textrm{tiny}}$. Of
course, this probability grows with the sample length. However, the choice of
weight $w$ also helps in it. Indeed, WRW with $w>1$ is more likely to visit
$C_{\scriptscriptstyle\textrm{tiny}}$ than RW ($w=1$, bottom line). This
demonstrates the first advantage of introducing edge weights and WRW.
#### 5.3.3 Optimal $w$ and $\gamma$
Let us now focus on the estimation error as a function of $w$, shown in Fig.
4(d,k). Interestingly, this error does not drop monotonically with $w$ but
follows a ’U’ shaped function with a clear optimal value
$w^{\scriptscriptstyle\textrm{opt}}$.
Under WIS, we have $w^{\scriptscriptstyle\textrm{opt}}\simeq 100$, which
confirms our findings in Sec. 3.1.4. Indeed, according to Eq.(20), we need the
same number of samples from the two categories, and thus
$\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{\scriptscriptstyle\textrm{tiny}})=\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{\scriptscriptstyle\textrm{big}})$
(by Eq.(22)). By plugging this and
$\textrm{vol}(C_{\scriptscriptstyle\textrm{big}})=100\cdot\textrm{vol}(C_{\scriptscriptstyle\textrm{tiny}})$
to Eq.(23), we finally obtain the WIS-optimal edge weights in
$C_{\scriptscriptstyle\textrm{tiny}}$, i.e.,
$w^{\scriptscriptstyle\textrm{opt}}=\textrm{w}_{e}(C_{\scriptscriptstyle\textrm{tiny}})=100\cdot\textrm{w}_{e}(C_{\scriptscriptstyle\textrm{big}})=100$.666For
simplicity, we ignored in this calculation the conflicts on the 500 edges
between $C_{\scriptscriptstyle\textrm{big}}$ and
$C_{\scriptscriptstyle\textrm{tiny}}$.
In contrast, WRW is optimized for $w<100$. For the sample length $n\\!=\\!500$
as in Fig. 4(d,k), the error is minimized already for
$w^{\scriptscriptstyle\textrm{opt}}\\!\simeq\\!20$ and increases for higher
weights. This demonstrates the “black hole” effect discussed in Sec. 4.2.4. It
is much more pronounced in the ‘clustered’ scenario, confirming our intuition
that black-holes become a problem only in the presence of relatively isolated,
tight communities. Of course, the black hole effect diminishes with the sample
length $n$ (and completely vanishes for $n\\!\rightarrow\\!\infty$), which can
be observed in Fig. 4(g,n), especially in (n).
In other words, the optimal assignment of edge weights (in relevant
categories) under WRW lies somewhere between RW (all weights equal) and WIS.
In S-WRW, we control it by parameter $\gamma$. In this example, we have
$\gamma\equiv w$ for $\gamma\leq 100$. Indeed, by combining Eq.(23), Eq.(25),
Eq.(26),
$\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{\scriptscriptstyle\textrm{tiny}})\\!=\\!\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{\scriptscriptstyle\textrm{big}})$,
we obtain
$\displaystyle w$ $\displaystyle=$ $\displaystyle\frac{w}{1}\ =\
\frac{w_{e}(C_{\scriptscriptstyle\textrm{tiny}})}{w_{e}(C_{\scriptscriptstyle\textrm{big}})}\
=\
\frac{\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{\scriptscriptstyle\textrm{tiny}})/\hbox
to0.0pt{\raisebox{-2.15277pt}{$\tilde{\phantom{\textrm{vol}}}$}\hss}\textrm{vol}(C_{\scriptscriptstyle\textrm{tiny}})}{\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{\scriptscriptstyle\textrm{big}})/\hbox
to0.0pt{\raisebox{-2.15277pt}{$\tilde{\phantom{\textrm{vol}}}$}\hss}\textrm{vol}(C_{\scriptscriptstyle\textrm{big}})}$
$\displaystyle=$ $\displaystyle\frac{\hbox
to0.0pt{\raisebox{-2.15277pt}{$\tilde{\phantom{\textrm{vol}}}$}\hss}\textrm{vol}(C_{\scriptscriptstyle\textrm{big}})}{\hbox
to0.0pt{\raisebox{-2.15277pt}{$\tilde{\phantom{\textrm{vol}}}$}\hss}\textrm{vol}(C_{\scriptscriptstyle\textrm{tiny}})}\
=\
\frac{\textrm{vol}(C_{\scriptscriptstyle\textrm{big}})}{\frac{1}{\gamma}\textrm{vol}(C_{\scriptscriptstyle\textrm{big}})}\
=\ \gamma.$
Consequently, the optimal setting of $\gamma$ is the same as
$w^{\scriptscriptstyle\textrm{opt}}$ discussed above.
#### 5.3.4 Gain $\alpha$
The gain $\alpha$ of WIS over UIS is given by Eq.(20). In this case, we have
$\alpha=(101K)^{2}\cdot(4\cdot 1K\cdot 100K)^{-1}\simeq 25$. Indeed, WIS with
$n\\!=\\!500$ samples shown in Fig. 4(d) achieves
${\small\textrm{NRMSE}}\\!\simeq\\!0.1$, which is the same as UIS of about
$\alpha\\!=\\!25$ times more samples (see Fig. 4(c)).
This gain due to stratification is smaller for sampling by exploration: a
500-hop-long WRW with $w\\!\simeq\\!20$ yields the same error
${\small\textrm{NRMSE}}\\!\simeq\\!0.3$ as a 2000-hop-long RW. This means that
WRW reduces the sampling cost by a factor of $\alpha\simeq 4$. Fig. 4(f) shows
that this gain does not vary much with the sampling length. Under ‘clustered’,
both RW and WRW perform much worse. Nevertheless, Fig. 4(m) shows that also in
this scenario WRW may significantly reduce the sampling cost, especially for
longer samples.
It is worth noting that WRW can sometimes significantly outperform UIS. This
is the case in Fig. 4(d), where UIS is equivalent to WIS with $w\\!=\\!1$.
Because no walk can mix faster than UIS (that is independent and thus has
perfect mixing), improving the mixing time alone [10, 37, 38, 5] cannot
achieve the potential gains of stratification, in general.
So far we focused on the smaller set $C_{\scriptscriptstyle\textrm{tiny}}$
only. When estimating the size of $C_{\scriptscriptstyle\textrm{big}}$, all
errors are much smaller, but we observe similar gain $\alpha$.
#### 5.3.5 Robustness to $\gamma$ and volume estimation
The gain $\alpha$ shown above is calculated for the optimal choice of $w$, or,
equivalently, $\gamma$. Of course, in practice it might be impossible to
obtain this value. Fortunately, S-WRW is relatively robust to the choice of
parameters. The dashed lines in Fig. 4(f,m) are calculated for $\gamma$ fixed
to $\gamma\\!=\\!5$, rather than optimized. Note that this value is often
drastically smaller than the optimal one (e.g.,
$w^{\scriptscriptstyle\textrm{opt}}\\!\simeq\\!50$ for $n\\!=\\!5000$).
Nevertheless, although the performance somewhat drops, S-WRW still reduces the
sampling cost about three-fold.
This observation also addresses potential concerns one might have regarding
the category volume estimation error (see Sec. 4.2.1). Indeed, setting
$\gamma\\!=\\!5$ means that every category $C_{i}$ with volume estimated at
$\hbox
to0.0pt{\raisebox{-2.15277pt}{$\hat{\phantom{\textrm{vol}}}$}\hss}\textrm{vol}(C_{i})\leq\frac{1}{5}\textrm{vol}(C_{\scriptscriptstyle\textrm{big}})$
is treated the same. In Fig. 4(f), the volume of
$C_{\scriptscriptstyle\textrm{tiny}}$ would have to be overestimated by more
than 20 times in order to affect the edge weight setting and thus the results.
We have seen in Sec. 5.3.1 that this is very unlikely, even under smallest
sample lengths and most adversarial scenarios.
### 5.4 Summary
WRW brings two types of benefits (i) avoiding irrelevant nodes $C_{\ominus}$
and (ii) carefully allocating samples between relevant categories of different
sizes. Even when $C_{\ominus}\\!=\\!\emptyset$, WRW can still reduce the
sampling cost by 75%. This second benefit is more difficult to achieve when
the categories form strong and tight communities, which leads to the “black
hole”’ effect. We should then choose smaller, more conservative values of
$\gamma$ in S-WRW, which translate into smaller $w$ in our example. In
contrast, under a looser community structure this problem disappears and WRW
is closer to WIS.
## 6 Implementation in Facebook
Figure 5: 5331 colleges discovered and ranked by RW. (a) Estimated relative
college sizes $\widehat{f}_{i}$. (b) Absolute number of user samples per
college. (c-e) 25 estimates of size $\widehat{f}_{i}$ for three different
colleges and sample lengths $n$. (f) Average NRMSE of college size estimation.
Results in (a,b,f) are binned.
As a concrete application, we apply S-WRW to measure the Facebook social
graph, which is our motivating and canonical example. We also note that it is
an undirected and can also be considered a static graph, for all practical
purposes in this study.777The Facebook characteristics do change but in time
scales much longer than the 3-day duration of our crawls. Websites such as
Facebook statistics, Alexa etc show that the number of Facebook users is
growing with rate 0.1-0.2% per day. In Facebook, every user may declare
herself a member of a college888There also exist categories other than
colleges, namely “work” and “high school”. Facebook requires a valid category-
specific email for verification. he/she attends. This membership information
is publicly available by default and allows us to answer some interesting
questions. For example, how do the college networks (or “colleges” for short)
compare with respect to their sizes? What is the college-to-college friendship
graph? In order to answer these questions, we have to collect many college
user samples, preferably evenly distributed between colleges. This is the main
goal of this section.
### 6.1 Measurement Setup
By default, every Facebook user can see the basic information on any other
user, including the name, photo, and a list of friends together with their
college memberships (if any). We developed a high performance multi-threaded
crawler to explore Facebook’s social graph by scraping this web interface.
To make informed decision for the parameters of S-WRW, we first ran a short
pilot RW (see Sec. 4.2.1) with a total of $65K$ samples (which is only 6.5% of
the length of the main S-WRW sample). Although our pilot walk visited only
2000 colleges, it estimated the relative volumes $f^{\textrm{vol}}_{i}$ for
about 9500 colleges discovered among friends of sampled users, as discussed in
Sec. 4.3.2. In Fig. 6(a), we show that the neighbor-based estimator Eq.(35)
greatly outperforms the naive estimator Eq.(32). These volumes cover several
decades. Because colleges with only a few tens of users are not of our
interest, we set the maximal resolution to $\gamma\\!=\\!1000$ (see the
discussion in Sec. 4.3.3). Finally, because the college students looked very
well interconnected in our pilot RW, we set the desired fraction of irrelevant
nodes to a small number $\tilde{f}_{\ominus}\\!=\\!1\%$.
In the main measurement phase, we collected three S-WRW crawls, each with
different edge weight conflict resolution (hybrid, geometric, and arithmetic),
and one simple RW crawl as a baseline comparison (Table
LABEL:tab:fb_datasets). For each crawl type we collected 1 million _unique_
users. Some of them are sampled multiple times (at no additional bandwidth
cost), which results in higher total number of samples in the second row of
Table LABEL:tab:fb_datasets. Our crawls were performed on Oct. 16-19 2010, and
are available at [1].
### 6.2 Results: RW vs. S-WRW
#### 6.2.1 Avoiding irrelevant categories
Only 9% of the RW’s samples come from colleges, which means that the vast
majority of sampling effort is wasted. In contrast, the S-WRW crawls achieved
6-10 better efficiency, collecting 86% (hybrid), 79% (geometric) and 58%
(arithmetic) samples from colleges. Note that these values are significantly
lower than the target 99% suggested by our choice of
$\tilde{f}_{\ominus}\\!=\\!1\%$, and that S-WRW hybrid reaches the highest
number. This is in agreement with our discussion in Sec. 4.2.5. Finally, we
also note that S-WRW crawls discovered $1.6-1.9$ times more unique colleges
than RW.
It might seem surprising that RW samples colleges in 9% of cases while only
3.5% of Facebook users belong to colleges. This can be explained by looking at
the last rows of Table LABEL:tab:fb_datasets. Indeed, the college users have
on average three times more Facebook friends than average users, and therefore
they attract RW approximately three times more often.
#### 6.2.2 Stratification
The advantage of S-WRW over RW does not lie exclusively in avoiding the nodes
in the irrelevant category $C_{\ominus}$. S-WRW can also over-sample small
categories (here colleges) at the cost of under-sampling large ones (which are
very well sampled anyway). This feature becomes important especially when the
category sizes differ significantly, which is the case in Facebook. Indeed,
Fig. 5(a) shows that college sizes exhibit great heterogeneity. For a fair
comparison, we only include the 5,331 colleges discovered by RW. (In fact,
this filtering actually gives preference to RW. S-WRW crawls discovered many
more colleges that we do not show in this figure.) They span more than two
orders of magnitude and follow a heavily skewed distribution (not shown here).
Fig. 5(b) confirms that S-WRW successfully oversamples the small colleges.
Indeed, the number of S-WRW samples per college is almost constant (roughly
around 100). In contrast, the number of RW samples follows closely the college
size, which results in dramatic 100-fold differences between RW and S-WRW for
smaller colleges.
#### 6.2.3 College size estimation
With more samples per college, we naturally expect a better estimation
accuracy under S-WRW. We demonstrate it for three colleges of different sizes
(in terms of the number of Facebook users): MIT (large), Caltech (medium), and
Eindhoven University of Technology (small). Each boxplot in Fig. 5(c-e) is
generated based on 25 independent college size estimates $\widehat{f}_{i}$
that come from walks of length $n\\!=\\!4$K (left), 20K (middle), and 40K
(right) samples each. For the three studied colleges, RW fails to produce
reliable estimates in all cases except for MIT (largest college) under the two
longest crawls. Similar results hold for the overwhelming majority of middle-
sized and small colleges. The underlying reason is the very small number of
samples collected by RW in these colleges, averaging at below 1 sample per
walk. In contrast, the three S-WRW crawls contain typically 5-50 times more
samples than RW (in agreement with Fig. 5(b)), and produce much more reliable
estimates.
Finally, we aggregate the results over all colleges and compute the gain
$\alpha$ of S-WRW over RW. We calculate the error
${\small\textrm{NRMSE}}(\widehat{f}_{i})$ by taking as our “ground truth”
$f_{i}$ the grand average of $\widehat{f}_{i}$ values over all samples
collected via all full-length walks and crawl types. Fig. 5(f) presents
${\small\textrm{NRMSE}}(\widehat{f}_{i})$ averaged over all 5,331 colleges
discovered by RW, as a function of walk length $n$. As expected, for all crawl
types the error decreases with $n$. However, there is a consistent large gap
between RW and all three versions of S-WRW. RW needs 13-15 times more samples
than S-WRW in order to achieve the same error.
Figure 6: Facebook: Pilot RW and other walks of the same length $n\\!=\\!65K$.
(a) The performance of the neighbor-based volume estimator Eq.(35) (plain
line) and the naive one Eq.(32) (dashed line). As ‘ground-truth’ we used
$f^{\textrm{vol}}_{i}$ calculated for all 4$\times$1M collected samples. (b)
The effect of the choice of $\gamma$.
#### 6.2.4 The effect of the choice of $\gamma$
Recall that in all the S-WRW results described above, we used the resolution
$\gamma\\!=\\!1000$. In order to check how sensitive the results are to the
choice of this parameter, we also tried a (shorter) S-WRW run with
$\gamma\\!=\\!100$, i.e., ten times smaller. In Fig. 6(b), we see that the
number of samples collected in the smallest colleges is smaller under
$\gamma\\!=\\!100$ than under $\gamma\\!=\\!1000$. In fact, the two curves
diverge for colleges about 100 times smaller than the biggest college, i.e.,
exactly at the maximal resolution $\gamma\\!=\\!100$.
In any case, both settings of $\gamma$ perform orders of magnitude better than
RW of the same length.
### 6.3 Summary
Only about 3.5% of 500M Facebook users are college members. There are more
than 10K colleges and they greatly vary in size, ranging from 50 (or fewer) to
50K members (we aggregate students, alumni and staff). In this setting, state-
of-the-art sampling methods such as RW are bound to perform poorly. Indeed,
UIS, i.e., an idealized version of RW, with as many as 1M samples will collect
only one sample from size-500 college, on average. Even if we could magically
sample directly only from colleges, we would typically collect fewer than 30
samples per size-500 college.
S-WRW solves these problems. We showed that S-WRW of the same length collects
typically about 100 samples per size-500 college. As a result, S-WRW
outperforms RW by $\alpha=13-15$ times or $\alpha=12-14$ times if we also
consider the 6.5% overhead from the initial pilot RW. This huge gain can be
decomposed into two factors, say $\alpha=\alpha_{1}\cdot\alpha_{2}$, as we
proposed in Eq.(21). Factor $\alpha_{1}\simeq 8$ can be attributed to a about
8 times higher fraction of college samples in S-WRW compared to RW. Factor
$\alpha_{2}\simeq 1.5$ is due to over-sampling smaller networks, i.e., by
applying stratified sampling.
Another important observation is that S-WRW is robust to the way we resolve
target edge weight conflicts in Sec. 4.2.5. The differences between the three
S-WRW implementations are minor - it is the application of Eq.(27) that brings
most of the benefit.
## 7 Related work
Graph Sampling by Exploration. Early crawling of P2P, OSN and WWW typically
used graph traversals, mainly BFS [3, 32, 31, 43, 33] and its variants.
However, incomplete BFS introduces bias towards high-degree nodes that is
unknown and thus impossible to correct in general graphs [2, 26, 8, 17, 25].
Later studies followed a more principled approach based on random walks (RW)
[29, 4]. The Metropolis-Hasting RW (MHRW) [30, 16] removes the bias during the
walk; it has been used to sample P2P networks [40, 35] and OSNs [17].
Alternatively, we can use RW, whose bias is known and can be corrected for
[20, 39], thus leading to a re-weighted RW [35, 17]. RW was also used to
sample Web [21], P2P networks [40, 35, 18], OSNs [24, 36, 17, 33], and other
large graphs [27]. It was empirically shown in [35, 17] that RW outperforms
MHRW in measurement accuracy. Therefore, RW can be considered as the state-of-
the-art.
Random walks have also been used to sample _dynamic graphs_ [40, 35, 42],
which are outside the scope of this paper.
Fast Mixing Markov Chains. The mixing time of a random walk determines the
efficiency of the sampling. On the practical side, the mixing time of RW in
many OSNs was found larger than commonly believed [33]. Multiple dependent
random walks [37] have been used to sample disconnected and loosely connected
graphs. Random walks with jumps have been used to sample large graphs in [38,
5] and in [27]. All the above methods treat all nodes with equal importance,
which is orthogonal to our technique.
On the theoretical side, in [10], the authors propose a method to set edge
weights that achieve the fastest mixing WRW for a given target stationary
distribution. This technique, although related, is not applicable in our
context. First, [10] requires the knowledge of the graph, which makes it
inapplicable to $G$, yet possibly feasible in $G^{C}$ (after estimating some
limited information about $G^{C}$ as in Sec. 4.2.1). In the latter case,
however, even given a perfect knowledge of $G^{C}$, [10] often assigns weight
0 to some self-loops, which likely makes the underlying graph $G$
disconnected. Finally, and most importantly, [10] takes a target stationary
distribution as input. By taking
$\textrm{w}^{\scriptscriptstyle\textrm{WIS}}$, we will face exactly the same
problems of potentially poor convergence (Sec. 4.2.3) and “black holes” (Sec.
4.2.4) as we addressed by S-WRW.
Stratified Sampling. Our approach builds on _stratified sampling_ [34], a
widely used technique in statistics; see [12, 28] for a good introduction.
A related work in a different networking problem is [14], where threshold
sampling is used to vary sampling probabilities of network traffic flows and
estimate their volume.
Weighted Random Walks for Sampling. Random walks on graphs with weighted
edges, or equivalently reversible Markov chains [29, 4], are well studied and
heavily used in Monte Carlo Markov Chain simulations [16] to sample a state
space with a specified probability distribution. However, to the best of our
knowledge, WRWs have not been designed explicitly for measurements of real
online systems. In the context of sampling OSNs, the closest works are [38,
5]. Technically speaking, they use WRW. But they set as their only objective
the minimization of the mixing time, which makes them orthogonal and
complementary to our approach, as we discussed above.
Very recent applications of weighted random walks in online social networks
include [7, 6]. [7] uses WRW in the context of link prediction. The authors
employ supervised learning techniques to set the edge weights, with the goal
of increasing the probability of visiting nodes that are more likely to
receive new links. [6] introduces WRW-based methods to generate samples of
nodes that are internally well-connected but also approximately uniform over
the population. In both these papers, WRW is used to predict/extract something
from a known graph. In contrast, we use WRW to estimate features of an unknown
graph.
In the context of World Wide Web crawling, _focused crawling_ techniques [11,
13] have been introduced to follow web pages of specified interest and to
avoid the irrelevant pages. This is achieved by performing a BFS type of
sample, except that instead of fifo queue they use a priority queue weighted
by the page relevancy. In our context, such an approach suffers from the same
problems as regular BFS: (i) collected samples strongly depend on the starting
point, and (ii) we are not able to unbias the sample.
## 8 Conclusion
We introduced Stratified Weighted Random Walk (S-WRW) - an efficient way to
sample large, static, undirected graphs via crawling and using minimal
information. S-WRW performs a weighted random walk on the graph with weights
determined by the estimation problem. We apply our approach to measure the
Facebook social graph, and we show that S-WRW greatly outperforms the state-
of-art sampling technique, namely the simple re-weighted random walk.
There are several directions for future work. First, S-WRW is currently an
intuitive and efficient heuristic; in future work, we plan to investigate the
optimal solution to problems identified in this paper and compare against or
improve S-WRW. Second, it may be possible to combine these ideas with existing
orthogonal techniques, some of which have been reviewed in Related Work, to
further improve performance. Finally, we are interested in extending our
techniques to dynamic graphs and non-stratified data.
## References
* [1] Weighted Random Walks of the Facebook social graph: http://odysseas.calit2.uci.edu/research/, 2011.
* [2] D. Achlioptas, A. Clauset, D. Kempe, and C. On the bias of traceroute sampling: or, power-law degree distributions in regular graphs. Journal of the ACM, 2009.
* [3] Y. Ahn, S. Han, H. Kwak, S. Moon, and H. Jeong. Analysis of topological characteristics of huge online social networking services. In WWW, pages 835–844, 2007.
* [4] D. Aldous and J. A. Fill. Reversible Markov Chains and Random Walks on Graphs. In preparation.
* [5] K. Avrachenkov, B. Ribeiro, and D. Towsley. Improving Random Walk Estimation Accuracy with Uniform Restarts. In I7th Workshop on Algorithms and Models for the Web Graph, 2010\.
* [6] L. Backstrom and J. Kleinberg. Network Bucket Testing. In WWW, 2011.
* [7] L. Backstrom and J. Leskovec. Supervised Random Walks: Predicting and Recommending Links in Social Networks. In ACM International Conference on Web Search and Data Minig (WSDM), 2011.
* [8] L. Becchetti, C. Castillo, D. Donato, and A. Fazzone. A comparison of sampling techniques for web graph characterization. In LinkKDD, 2006.
* [9] H. R. Bernard, T. Hallett, A. Iovita, E. C. Johnsen, R. Lyerla, C. McCarty, M. Mahy, M. J. Salganik, T. Saliuk, O. Scutelniciuc, G. a. Shelley, P. Sirinirund, S. Weir, and D. F. Stroup. Counting hard-to-count populations: the network scale-up method for public health. Sexually Transmitted Infections, 86(Suppl 2):ii11–ii15, Nov. 2010\.
* [10] S. Boyd, P. Diaconis, and L. Xiao. Fastest mixing Markov chain on a graph. SIAM review, 46(4):667–689, 2004.
* [11] S. Chakrabarti. Focused crawling: a new approach to topic-specific Web resource discovery. Computer Networks, 31(11-16):1623–1640, May 1999.
* [12] W. G. Cochran. Sampling Techniques, volume 20 of McGraw-Hil Series in Probability and Statistics. Wiley, 1977.
* [13] M. Diligenti, F. Coetzee, S. Lawrence, C. Giles, and M. Gori. Focused crawling using context graphs. In Proceedings of the 26th International Conference on Very Large Data Bases, pages 527–534, 2000.
* [14] N. Duffield, C. Lund, and M. Thorup. Learn more, sample less: control of volume and variance in network measurement. IEEE Transactions on Information Theory, 51(5):1756–1775, May 2005\.
* [15] J. Gentle. Random number generation and Monte Carlo methods. Springer Verlag, 2003.
* [16] W. R. Gilks, S. Richardson, and D. J. Spiegelhalter. Markov Chain Monte Carlo in Practice. Chapman and Hall/CRC, 1996.
* [17] M. Gjoka, M. Kurant, C. T. Butts, and A. Markopoulou. Walking in Facebook: A Case Study of Unbiased Sampling of OSNs. In INFOCOM, 2010.
* [18] C. Gkantsidis, M. Mihail, and A. Saberi. Random walks in peer-to-peer networks. In INFOCOM, 2004.
* [19] M. Hansen and W. Hurwitz. On the Theory of Sampling from Finite Populations. Annals of Mathematical Statistics, 14(3), 1943.
* [20] D. D. Heckathorn. Respondent-Driven Sampling: A New Approach to the Study of Hidden Populations. Social Problems, 44:174–199, 1997.
* [21] M. R. Henzinger, A. Heydon, M. Mitzenmacher, and M. Najork. On near-uniform URL sampling. In WWW, 2000.
* [22] M. H. Kalos and P. A. Whitlock. Monte carlo methods. Volume I: Basics. Wiley, 1986.
* [23] E. D. Kolaczyk. Statistical Analysis of Network Data, volume 69 of Springer Series in Statistics. Springer New York, 2009.
* [24] B. Krishnamurthy, P. Gill, and M. Arlitt. A few chirps about Twitter. In WOSN, 2008.
* [25] M. Kurant, A. Markopoulou, and P. Thiran. On the bias of BFS (Breadth First Search). In ITC, also in arXiv:1004.1729, 2010.
* [26] S. H. Lee, P.-J. Kim, and H. Jeong. Statistical properties of Sampled Networks. Phys. Rev. E, 73:16102, 2006.
* [27] J. Leskovec and C. Faloutsos. Sampling from large graphs. In KDD, pages 631–636, 2006.
* [28] S. Lohr. Sampling: design and analysis. Brooks/Cole, second edition, 2009.
* [29] L. Lovász. Random walks on graphs: A survey. Combinatorics, Paul Erdos is Eighty, 2(1):1–46, 1993.
* [30] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller. Equation of state calculation by fast computing machines. Journal of Chemical Physics, 21:1087–1092, 1953.
* [31] A. Mislove, H. S. Koppula, K. P. Gummadi, P. Druschel, and B. Bhattacharjee. Growth of the Flickr social network. In WOSN, 2008.
* [32] A. Mislove, M. Marcon, K. P. Gummadi, P. Druschel, and B. Bhattacharjee. Measurement and analysis of online social networks. In IMC, pages 29–42, 2007.
* [33] A. Mohaisen, A. Yun, and Y. Kim. Measuring the mixing time of social graphs. IMC, 2010.
* [34] J. Neyman. On the Two Different Aspects of the Representative Method: The Method of Stratified Sampling and the Method of Purposive Selection. Journal of the Royal Statistical Society, 97(4):558, 1934.
* [35] A. Rasti, M. Torkjazi, R. Rejaie, N. Duffield, W. Willinger, and D. Stutzbach. Respondent-driven sampling for characterizing unstructured overlays. In Infocom Mini-conference, pages 2701–2705, 2009.
* [36] A. H. Rasti, M. Torkjazi, R. Rejaie, and D. Stutzbach. Evaluating Sampling Techniques for Large Dynamic Graphs. In Technical Report, volume 1, 2008.
* [37] B. Ribeiro and D. Towsley. Estimating and sampling graphs with multidimensional random walks. In IMC, volume 011, 2010.
* [38] B. Ribeiro, P. Wang, and D. Towsley. On Estimating Degree Distributions of Directed Graphs through Sampling. UMass Technical Report, 2010.
* [39] M. Salganik and D. D. Heckathorn. Sampling and estimation in hidden populations using respondent-driven sampling. Sociological Methodology, 34(1):193–240, 2004.
* [40] D. Stutzbach, R. Rejaie, N. Duffield, S. Sen, and W. Willinger. On unbiased sampling for unstructured peer-to-peer networks. In IMC, 2006.
* [41] E. Volz and D. D. Heckathorn. Probability based estimation theory for respondent driven sampling. Journal of Official Statistics, 24(1):79–97, 2008.
* [42] W. Willinger, R. Rejaie, M. Torkjazi, M. Valafar, and M. Maggioni. OSN Research: Time to Face the Real Challenges. In HotMetrics, 2009.
* [43] C. Wilson, B. Boe, A. Sala, K. P. N. Puttaswamy, and B. Y. Zhao. User interactions in social networks and their implications. In EuroSys, 2009.
## Appendix A: Achieving Arbitrary Node Weights
Achieving arbitrary node weights by setting the edge weights in a graph
$G=(V,E)$ is sometimes impossible. For example, for a graph that is a path
consisting of two nodes ($v_{1}-v_{2}$), it is impossible to achieve
$\textrm{w}(v_{1})\neq\textrm{w}(v_{2})$. However, it is always possible to do
so, if there are self loops in each node.
###### Observation 1
For any undirected graph $G=(V,E)$ with a self-loop $\\{v,v\\}$ at every node
$v\in V$, we can achieve an arbitrary distribution of node weights
$\textrm{w}(v)>0,\ v\in V$, by appropriate choice of edge weights
$\textrm{w}(u,v)\\!>\\!0,\ \\{u,v\\}\\!\in\\!E$.
###### Proof 8.1.
Denote by $\textrm{w}_{\min}$ the smallest of all target node weights
$\textrm{w}(v)$. Set $\textrm{w}(u,v)=\textrm{w}_{\min}/N$ for all non self-
loop edges (i.e., where $u\neq v$). Now, for every self-loop $\\{v,v\\}\in E$
set
$\textrm{w}(v,v)\ \ =\ \
\frac{1}{2}\left(\textrm{w}(v)-\frac{\textrm{w}_{\min}}{N}\cdot(\deg(v)\\!-\\!2)\right).$
It is easy to check that, because there are exactly $\deg(v)\\!-\\!2$ non
self-loop edges incident on $v$, every node $v\in V$ will achieve the target
weight $\textrm{w}(v)$. Moreover, the definition of $\textrm{w}_{\min}$
guarantees that $\textrm{w}(v,v)>0$ for every $v\in V$.
## Appendix B: Estimating Category Volumes
In this section, we derive efficient estimators of the volume ratio
$\widehat{f}^{\scriptscriptstyle\textrm{vol}}_{C}=\frac{\textrm{vol}(C)}{\textrm{vol}(V)}$.
Recall that $S\subset V$ denotes an independent sample of nodes in $G$, with
replacement.
Node sampling
If $S$ is a uniform sample UIS, then we can write
$\widehat{f}^{\scriptscriptstyle\textrm{vol}}_{C}\ =\ \frac{\sum_{v\in
S}\deg(v)\cdot 1_{\\{v\in C\\}}}{\sum_{v\in S}\deg(v)},$ (30)
which is a straightforward application of the classic ratio estimator [28].
In the more general case, when $S$ is selected using WIS, then we have to
correct for the linear bias towards nodes of higher weights $\textrm{w}()$, as
follows:
$\displaystyle\widehat{f}^{\scriptscriptstyle\textrm{vol}}_{C}$
$\displaystyle=$ $\displaystyle\frac{\sum_{v\in S}\deg(v)\cdot 1_{\\{v\in
C\\}}/\textrm{w}(v)}{\sum_{v\in S}\deg(v)/\textrm{w}(v)}.$ (31)
In particular, if $\textrm{w}(v)\sim\deg(v)$, then
$\displaystyle\widehat{f}^{\scriptscriptstyle\textrm{vol}}_{C}$
$\displaystyle=$ $\displaystyle\frac{1}{n}\cdot\sum_{v\in S}1_{\\{v\in C\\}}.$
(32)
Star sampling
Another approach is to focus on the set of all neighbors $\mathcal{N}(S)$ of
sampled nodes (with repetitions) rather than on $S$ itself, i.e., to use ‘star
sampling’ [23]. The probability that a node $v$ is a neighbor of a node
sampled from $V$ by UIS is
$\sum_{u\in V}\frac{1}{N}\cdot 1_{\\{v\in\mathcal{N}(u)\\}}\ \ =\ \
\frac{\deg(v)}{N}.$
Consequently, the nodes in $\mathcal{N}(S)$ are asymptotically equivalent to
nodes drawn with probabilities linearly proportional to node degrees. By
applying Eq.(32) to $\mathcal{N}(S)$, we obtain999As a side note, observe that
formula Eq.(33) generalizes the “scale-up method” [9] used in social sciences
to estimate the size (here $|C|$) of hidden populations (e.g., of drug
addicts). Indeed, if we assume that the average node degree in $V$ is the same
as in $C$, then $\textrm{vol}(C)/\textrm{vol}(V)=|C|/N$, which reduces Eq.(32)
to the core formula of the scale-up method.
$\widehat{f}^{\scriptscriptstyle\textrm{vol}}_{C}\ \ =\ \
\frac{1}{\textrm{vol}(S)}\sum_{u\in
S}\sum_{v\in\mathcal{N}(u)}\\!\\!1_{\\{v\in C\\}},$ (33)
where we used $|\mathcal{N}(S)|=\sum_{u\in S}\deg(u)=\textrm{vol}(S)$.
In the more general case, when $S$ is selected using WIS, then we correct for
the linear bias towards nodes of higher weights $\textrm{w}()$, as follows:
$\widehat{f}^{\scriptscriptstyle\textrm{vol}}_{C}\ \ =\ \
\frac{1}{\displaystyle\sum_{u\in S}\frac{\deg(u)}{\textrm{w}(u)}}\sum_{u\in
S}\left(\frac{1}{\textrm{w}(u)}\sum_{v\in\mathcal{N}(u)}\\!\\!1_{\\{v\in
C\\}}\right).$ (34)
In particular, if $\textrm{w}(v)\sim\deg(v)$, then
$\widehat{f}^{\scriptscriptstyle\textrm{vol}}_{C}\ \ =\ \
\frac{1}{n}\sum_{u\in
S}\left(\frac{1}{\deg(u)}\sum_{v\in\mathcal{N}(u)}\\!\\!1_{\\{v\in
C\\}}\right).$ (35)
Note that for every sampled node $v\in S$, the formulas Eq.(33-35) exploit all
the $\deg(v)$ neighbors of $v$, whereas Eq.(30-32) rely on one node per sample
only. Not surprisingly, Eq.(33-35) performed much better in all our
simulations and implementations.
## Appendix C: Relative sizes of node categories
Consider a scenario with only two node categories, i.e.,
$\mathcal{C}=\\{C_{1},C_{2}\\}$. Denote $f_{1}=|C_{1}|/N$ and
$f_{2}=|C_{2}|/N$. The goal is to estimate $f_{1}$ and $f_{2}$ based on the
collected sample $S$.
##### UIS - Uniform independence sampling
Under UIS, the number $X_{1}$ of times we select a node from $C_{1}$ among $n$
attempts follows the Binomial distribution $X_{1}=Binom(f_{1},n)$. Therefore,
we can estimate $f_{1}$ as
$\hat{f}^{\scriptscriptstyle\textrm{UIS}}_{1}\ =\
\frac{X_{1}}{n}\qquad\textrm{ with
}\qquad\mathbb{V}(\hat{f}^{\scriptscriptstyle\textrm{UIS}}_{1})\ =\
\frac{f_{1}f_{2}}{n}.$ (36)
##### WIS - Weighted independence sampling
In contrast, under WIS, at every iteration the probability $\pi(v)$ of
selecting a node $v$ is:
$\pi(v)=\left\\{\begin{array}[]{rl}\pi_{1}=\frac{1}{N}\cdot\frac{w_{1}}{w_{1}f_{1}+w_{2}f_{2}}&\textrm{
if }v\in C_{1},\textrm{ and }\\\
\pi_{2}=\frac{1}{N}\cdot\frac{w_{2}}{w_{1}f_{1}+w_{2}f_{2}}&\textrm{ if }v\in
C_{2},\end{array}\right.$
where $w_{1}$ and $w_{2}$ are the weights $\textrm{w}(v)$ of nodes in $C_{1}$
and $C_{2}$, respectively.
By applying the Hansen-Hurwitz estimator (separately for nominator and
denominator), we obtain
$\displaystyle\hat{f}^{\scriptscriptstyle\textrm{WIS}}_{1}$ $\displaystyle=$
$\displaystyle\frac{|\hat{C}_{1}|}{\hat{N}}\ =\ \frac{\sum_{v\in S}1_{v\in
C_{1}}\,/\,\pi(v)}{\sum_{v\in S}1\,/\,\pi(v)}$ (37) $\displaystyle=$
$\displaystyle\frac{X_{1}\,/\,\pi_{1}}{X_{1}\,/\,\pi_{1}\ +\
(n-X_{1})\,/\,\pi_{2}}\ $ $\displaystyle=$
$\displaystyle\frac{X_{1}\cdot\pi_{2}}{X_{1}(\pi_{2}-\pi_{1})+n\cdot\pi_{1}}\
$ $\displaystyle=$
$\displaystyle\frac{X_{1}\cdot\textrm{w}_{2}}{X_{1}(\textrm{w}_{2}-\textrm{w}_{1})+n\cdot\textrm{w}_{1}},$
where $X_{1}$ is the number of samples taken from $C_{1}$. Note, that to
calculate $\hat{f}^{\scriptscriptstyle\textrm{WIS}}_{1}$ we only need values
$w_{1}$ and $w_{2}$, which are set by us and thus known.
Computing the variance of $\hat{f}^{\scriptscriptstyle\textrm{WIS}}_{1}$ is a
bit more challenging. We use the second-order Taylor expansions (the ’Delta
method’) to approximate it as follows:
$\displaystyle\frac{\partial\hat{f}^{\scriptscriptstyle\textrm{WIS}}_{1}}{\partial
X_{1}}$ $\displaystyle=$
$\displaystyle\frac{nw_{1}w_{2}}{((w_{2}-w_{1})X_{1}+nw_{1})^{2}},\quad\textrm{
and}$ $\displaystyle\mathbb{V}(\hat{f}^{\scriptscriptstyle\textrm{WIS}}_{1})$
$\displaystyle\cong$
$\displaystyle\left(\frac{\partial\hat{f}^{\scriptscriptstyle\textrm{WIS}}_{1}}{\partial
X_{1}}\big{(}\mathbb{E}(X_{1})\big{)}\right)^{2}\mathbb{V}(X_{1})$ (38)
$\displaystyle=$
$\displaystyle\\!\\!\\!\big{(}\ldots\big{)}=\frac{f_{1}f_{2}}{nw_{1}w_{2}}\cdot(f_{1}w_{1}+f_{2}w_{2})^{2}.$
In the above derivation, we used the fact that
$\mathbb{E}(X_{1})=nNf_{1}\pi_{1}$ and
$\mathbb{V}(X_{1})=nN^{2}f_{1}\pi_{1}f_{2}\pi_{2}$. This comes from the fact
that $X_{1}$ actually follows the binomial distribution
$X_{1}=Binom(Nf_{1}\pi_{1},n).$
For $w_{1}=w_{2}$, we are back in the UIS case. But this is not necessarily
the optimal choice of weights. Indeed, a quick application of Lagrange
multipliers reveals that
$\mathbb{V}(\hat{f}^{\scriptscriptstyle\textrm{WIS}}_{1})$ is minimized when
$w_{1}\,f_{1}=f_{2}\,w_{2}.$ (39)
Moreover, analogous analysis shows that Eq.(39) minimizes
$\mathbb{V}(\hat{f}_{2}^{\scriptscriptstyle\textrm{WIS}})$ as well. In other
words, the estimators of both $f_{1}$ and $f_{2}$ have the lowest variance if
the total weighted mass of $C_{1}$ is equal to that of $C_{2}$. This implies,
in expectation, equal allocation of samples between $C_{1}$ and $C_{2}$, i.e.,
$n^{\scriptscriptstyle\textrm{WIS}}_{i}\ =\ \frac{n}{|\mathcal{C}|}.$
Finally, we can use Eq.(36), Eq.(38) and Eq.(39) to calculate the gain
$\alpha$ of WIS over UIS
$\alpha\ =\ \frac{1}{4f_{1}f_{2}}\quad(\geq 1).$ (40)
Note that we always have $\alpha\geq 1$, and $\alpha$ grows quickly with
growing difference between $f_{1}$ and $f_{2}$.
## Appendix D: Optimal WRW weights in Fig. 3(a)
Every time WRW visits the white node/category in Fig. 3(a), the next node is
chosen uniformly from red and green categories. We stay in this selected
category for $k$ rounds, where $k$ is a geometric random variable with
parameter $p=w_{2}/(w_{1}\\!+\\!w_{2})\in[0,1]$. Next, we come back to the
white category, and reiterate the process. So the number
$n_{\scriptscriptstyle\textrm{red}}$ of times the red category is sampled is
$n_{\scriptscriptstyle\textrm{red}}=\sum_{1}^{Binom(0.5,n_{\scriptscriptstyle\textrm{wh}})}Geom(p),$
where $n_{\scriptscriptstyle\textrm{wh}}$ is the number of visits to the white
category. Because the random variables generated by
$Binom(0.5,n_{\scriptscriptstyle\textrm{wh}})$ and $Geom(p)$ are independent,
we can write
$\displaystyle\mathbb{E}[n_{\scriptscriptstyle\textrm{red}}]$ $\displaystyle=$
$\displaystyle\mathbb{E}[Binom(0.5,n_{\scriptscriptstyle\textrm{wh}})]\cdot\mathbb{E}[Geom(p)]\
=\ 0.5n_{\scriptscriptstyle\textrm{wh}}/p$
$\displaystyle\mathbb{V}[n_{\scriptscriptstyle\textrm{red}}]$ $\displaystyle=$
$\displaystyle\mathbb{E}[Binom()]\mathbb{V}[Geom()]+\mathbb{E}^{2}[Geom()]\mathbb{V}[Binom()]$
$\displaystyle=$
$\displaystyle\frac{n_{\scriptscriptstyle\textrm{wh}}}{4p^{2}}(3-2p).$
A possible unbiased estimator of the relative size
$f_{\scriptscriptstyle\textrm{red}}$ of red category (among relevant
categories) is
$\widehat{f}_{\scriptscriptstyle\textrm{red}}=\frac{n_{\scriptscriptstyle\textrm{red}}}{n_{\scriptscriptstyle\textrm{wh}}/p},$
for which we get
$\displaystyle\mathbb{E}[\widehat{f}_{\scriptscriptstyle\textrm{red}}]$
$\displaystyle=$
$\displaystyle\frac{\mathbb{E}[n_{\scriptscriptstyle\textrm{red}}]}{n_{\scriptscriptstyle\textrm{wh}}/p}\
=\ \frac{1}{2}\quad\textrm{(unbiased)}$
$\displaystyle\mathbb{V}[\widehat{f}_{\scriptscriptstyle\textrm{red}}]$
$\displaystyle=$
$\displaystyle\frac{\mathbb{V}[n_{\scriptscriptstyle\textrm{red}}]}{(n_{\scriptscriptstyle\textrm{wh}}/p)^{2}}\
=\ \frac{3-2p}{4n_{\scriptscriptstyle\textrm{wh}}}.$
This variance is expressed as a function of
$n_{\scriptscriptstyle\textrm{wh}}$, and not of the total sample length $n$.
However, note that $n_{\scriptscriptstyle\textrm{wh}}$ drops with decreasing
$p$. Consequently, the variance
$\mathbb{V}[\widehat{f}_{\scriptscriptstyle\textrm{red}}]$ (expressed as a
function of $n_{\scriptscriptstyle\textrm{wh}}$ or of $n$) is minimized for
$p=1$, i.e., for $w_{1}=0$ and $w_{2}>0$ (and
$n_{\scriptscriptstyle\textrm{wh}}\\!=\\!n/2$).
|
arxiv-papers
| 2011-01-28T06:27:45 |
2024-09-04T02:49:16.694733
|
{
"license": "Public Domain",
"authors": "M. Kurant, M. Gjoka, C. T. Butts, A. Markopoulou",
"submitter": "Maciej Kurant",
"url": "https://arxiv.org/abs/1101.5463"
}
|
1101.5664
|
arxiv-papers
| 2011-01-29T03:51:24 |
2024-09-04T02:49:16.708286
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhancheng Li, Ping Wu, Chenxi Wang, Xiaodong Fan, Wenhua Zhang,\n Xiaofang Zhai, Changgan Zeng, Zhenyu Li, Jinlong Yang, and J. G. Hou",
"submitter": "Changgan Zeng",
"url": "https://arxiv.org/abs/1101.5664"
}
|
|
1101.5700
|
# Infraparticles with superselected direction of motion in two-dimensional
conformal field theory
Wojciech Dybalski111Supported by the DFG grant SP181/25.
Zentrum Mathematik, Technische Universität München,
D-85747 Garching, Germany
E-mail: dybalski@ma.tum.de Yoh Tanimoto222Supported in part by the ERC
Advanced Grant 227458 OACFT “Operator Algebras and Conformal Field Theory”.
Dipartimento di Matematica, Università di Roma “Tor Vergata”
Via della Ricerca Scientifica, 1 - I–00133 Roma, Italy.
E-mail: tanimoto@mat.uniroma2.it
###### Abstract
Particle aspects of two-dimensional conformal field theories are investigated,
using methods from algebraic quantum field theory. The results include
asymptotic completeness in terms of (counterparts of) Wigner particles in any
vacuum representation and the existence of (counterparts of) infraparticles in
any charged irreducible product representation of a given chiral conformal
field theory. Moreover, an interesting interplay between the infraparticle’s
direction of motion and the superselection structure is demonstrated in a
large class of examples. This phenomenon resembles the electron’s momentum
superselection expected in quantum electrodynamics.
## 1 Introduction
Particle aspects and superselection structure of quantum electrodynamics are
plagued by the infrared problem, which has been a subject of study in
mathematical physics for more than four decades [46, 28, 29, 39, 21, 40, 18,
33, 19, 30, 6, 7, 8, 10, 16, 41, 42, 22, 23, 24, 48, 34, 44]. The origin of
this difficulty, inherited from classical electrodynamics, is the emission of
photons which accompanies any change of the electron’s momentum. It has two
important consequences which are closely related: Firstly, the electron is not
a particle in the sense of Wigner [50], but rather an _infraparticle_ [46]
i.e. it does not have a precise mass. Secondly, the electron’s plane wave
configurations of different momenta cannot be superposed into normalizable
wavepackets. In fact, such configurations have different spacelike asymptotic
flux of the electric field, which imposes a superselection rule [7]. The
evidence for this phenomenon of the electron’s _momentum superselection_ comes
from two sources: On the one hand, it appears in models of non-relativistic
QED in the representation structure of the asymptotic electromagnetic field
algebra [18]. On the other hand, it is suggested by structural results in the
general framework of algebraic quantum field theory [7, 8, 16, 41, 42].
However, no examples of local, relativistic theories, describing
infraparticles with superselected momentum, have been given to date. Thus the
logical consistency of this property with the basic postulates of quantum
field theory remains to be settled. As a step in this direction, we
demonstrate in the present paper that a simple variant of this phenomenon -
superselection of direction of motion - occurs in a large class of two-
dimensional conformal field theories.
Conformal field theory has been a subject of intensive research over the last
two decades, both from physical and mathematical viewpoints, motivated, in
particular, by the search for non-trivial quantum field theories. (See e.g.
[3] and references therein). It exhibits particularly interesting properties
in two dimensions, where the symmetry group is infinite dimensional. Since the
seminal work of Buchholz, Mack and Todorov [15] the superselection structure
of these theories has been investigated [31] and deep classification results
have been obtained [36, 37]. It has remained unnoticed, however, that two-
dimensional conformal field theories have also a rich and interesting particle
structure: The concepts of Wigner particles and infraparticles have their
natural counterparts in this setting and both types of excitations appear in
abundance: Any chiral conformal field theory in a vacuum representation has a
complete particle interpretation in terms of Wigner particles. Although such
theories are non-interacting, their (Grosse-Lechner) deformations [14] exhibit
non-trivial scattering and inherit the property of asymptotic completeness as
we show in a companion paper [25]. It is verified in the present work that any
charged irreducible product representation of a chiral conformal field theory
admits infraparticles. In a large class of examples these infraparticles have
superselected direction of motion i.e. their plane wave configurations with
opposite directions of momentum cannot be superposed. Thus subtle particle
phenomena, which are not under control in physical spacetime, can be
investigated in these two-dimensional models.
To keep our analysis general, we rely on the setting of algebraic QFT [32]. We
base our discussion on the concept of a local net of $C^{*}$-algebras on
$\mathbb{R}^{2}$, defined precisely in Subsection 2.1: To any open, bounded
region $\mathcal{O}\subset\mathbb{R}^{2}$ we attach a $C^{*}$-algebra
${\mathfrak{A}}(\mathcal{O})$, acting on a Hilbert space $\mathcal{H}$ of
physical states. This algebra is generated by observables which can be
measured with an experimental device localized in $\mathcal{O}$. It is
contained in the quasilocal algebra ${\mathfrak{A}}$, which is the inductive
limit of the net $\mathcal{O}\to{\mathfrak{A}}(\mathcal{O})$. Moreover, there
acts a unitary representation of translations $\mathbb{R}^{2}\ni x\to U(x)$ on
$\mathcal{H}$, whose adjoint action
$\alpha_{x}(\,\cdot\,)=U(x)\,\cdot\,U(x)^{*}$ shifts the observables in
spacetime. The infinitesimal generators of $U$ are interpreted as the
Hamiltonian $H$ and the momentum operator $\boldsymbol{P}$. Their joint
spectrum is contained in the closed forward lightcone $V_{+}$, to ensure the
positivity of energy. If there exists a cyclic, unit vector
$\Omega\in\mathcal{H}$ which is the unique (up to a phase) joint eigenvector
of $H$ and $\boldsymbol{P}$ with eigenvalue zero, then we say that the theory
is in a vacuum representation. If each of the subspaces
$\mathcal{H}_{\pm}=\ker(H\mp\boldsymbol{P})$ includes some vectors orthogonal
to $\Omega$, then we say that the theory contains Wigner particles. Since we
do not assume that these particles are described by vectors in some
irreducible representation space of the Poincaré group, the present definition
is less restrictive than the conventional one. However, it is better suited
for a description of the dispersionless kinematics of two-dimensional massless
excitations. In particular, it allows us to apply the natural scattering
theory, developed by Buchholz in [5], which we outline in Subsection 2.2
below. We recall that in [5] these excitations are called ‘waves’, to stress
their composite character.
Due to this compound structure of Wigner particles (or ‘waves’), asymptotic
completeness in a vacuum representation is not in conflict with the existence
of charged representations with a non-trivial particle content. However, in
charged representations of massless two-dimensional theories Wigner particles
may be absent, as noticed in [12]. In this case scattering theory from [5]
does not apply and an appropriate framework for the analysis of particle
aspects is the theory of _particle weights_ [10, 47, 16, 35, 41, 42, 23, 24],
developed by Buchholz, Porrmann and Stein, which we revisit in Subsections 2.3
and 2.4. This theory is based on the concept of the asymptotic functional,
given by
$\sigma_{\Psi}^{\mathrm{out}}(C)=\lim_{t\to\infty}\int
d\boldsymbol{x}\,(\Psi|\alpha_{(t,\boldsymbol{x})}(C)\Psi),$ (1.1)
for any vector $\Psi\in\mathcal{H}$ of bounded energy, and suitable
observables $C\in{\mathfrak{A}}$. (In general, some time averaging and
restriction to a subnet may be needed before taking the limit). We remark for
future reference that this functional induces a sesquilinear form
$\psi_{\Psi}^{\mathrm{out}}$ on a certain left ideal of ${\mathfrak{A}}$. We
show in Theorem 2.11 below, that asymptotic functionals are non-zero in
theories of Wigner particles. If non-trivial asymptotic functionals arise in
the absence of Wigner particles, then we say that the theory describes
infraparticles333The conventional definition of infraparticles requires that
both $\mathcal{H}_{+}$ and $\mathcal{H}_{-}$ contain at most multiples of the
vacuum vector. Our (less restrictive) definition imposes this requirement on
one of these subspaces only. Thus theories containing ‘waves’ running to the
right but no ‘waves’ running to the left (or vice versa) describe
infraparticles according to our terminology. Such nomenclature turns out to be
more convenient in the context of two-dimensional, massless theories.. Using
standard decomposition theory, the GNS representation $\pi$ induced by the
sesquilinear form $\psi_{\Psi}^{\mathrm{out}}$ can be decomposed into a direct
integral of irreducible representations
$\pi\simeq\int_{X}d\mu(\xi)\,\pi_{\xi},$ (1.2)
where $(X,d\mu)$ is a Borel space and $\simeq$ denotes unitary equivalence
[49, 42]. Results from [2, 47] suggest that the measurable field of
irreducible representations $\\{\pi_{\xi}\,\\}_{\xi\in X}$ carries information
about all the (infra-)particle types appearing in the theory. In particular,
there exists a field of vectors $\\{\,q_{\xi}\,\\}_{\xi\in X}$ which can be
interpreted as the energy and momentum of plane wave configurations
$\\{\psi_{\xi}\\}_{\xi\in X}$ of the respective (infra-)particles [42]. The
sesquilinear forms $\\{\psi_{\xi}\\}_{\xi\in X}$, called _pure particle
weights_ , induce the representations $\\{\pi_{\xi}\\}_{\xi\in X}$ and satisfy
$\psi_{\Psi}^{\mathrm{out}}=\int_{X}d\mu(\xi)\,\psi_{\xi}.$ (1.3)
The existence of such a decomposition was shown, under certain technical
restrictions, in [41, 42].
The theory of particle weights is sufficiently general to accommodate the
phenomenon of the infraparticle’s momentum superselection, discussed above: In
this case $q_{\xi}\neq q_{\xi^{\prime}}$ should imply that $\pi_{\xi}$ is not
unitarily equivalent to $\pi_{\xi^{\prime}}$ for almost all labels $\xi$,
$\xi^{\prime}$ corresponding to the infraparticle in question. Superselection
of direction of motion is a milder property: It only requires that plane waves
$\psi_{\xi},\psi_{\xi^{\prime}}$, travelling in opposite directions, give rise
to representations $\pi_{\xi},\pi_{\xi^{\prime}}$ which are not unitarily
equivalent. This latter interplay between the infraparticle’s kinematics and
the superselection structure occurs in some two-dimensional conformal field
theories, as we explain below. We state this property precisely in Definitions
2.7 and 2.12, where we restrict attention to representations $\pi$ of (Murray-
von Neumann) type I with atomic center. This is sufficient for our purposes
and allows us to separate our central concept from ambiguities involved in the
general decompositions (1.2), (1.3).
Our discussion of conformal field theory relies on the notion of a local net
of von Neumann algebras on $\mathbb{R}$ which we introduce in Subsection 3.1.
(Such nets arise e.g. by restricting the familiar Möbius covariant nets on the
circle to the real line). With any open bounded region ${\cal
I}\subset\mathbb{R}$ we associate a von Neumann algebra ${\cal A}({\cal I})$,
acting on a Hilbert space $\mathcal{K}$, and denote the quasilocal algebra of
this net by ${\cal A}$. Moreover, the Hilbert space $\mathcal{K}$ carries a
unitary representation of translations $\mathbb{R}\ni s\to V(s)$, whose
spectrum coincides with $\mathbb{R}_{+}$. If there exists a cyclic, unit
vector $\Omega_{0}\in\mathcal{K}$, which is the unique (up to a phase) non-
zero vector invariant under the action of $V$, then we say that the theory is
in a vacuum representation. Given such a net, covariant under the action of
some internal symmetry group, one can proceed to the fixed-point subnet which
has a non-trivial superselection structure. In the simple case, considered in
Subsection 3.4, the action of $\mathbb{Z}_{2}$ is implemented by a unitary
$W\neq I$ on $\mathcal{K}$ s.t. $W^{2}=I$. The fixed-point subnet ${\cal
A}_{\mathrm{ev}}$ consists of all the elements of ${\cal A}$, which commute
with $W$. The subspace $\mathcal{K}_{\mathrm{ev}}=\ker(W-I)$ (resp.
$\mathcal{K}_{\mathrm{odd}}=\ker(W+I)$) is invariant under the action of
${\cal A}_{\mathrm{ev}}$ and gives rise to a vacuum representation (resp. a
charged representation) of the fixed-point theory.
Given two nets of von Neumann algebras on the real line, ${\cal A}_{1}$ and
${\cal A}_{2}$, acting on Hilbert spaces $\mathcal{K}_{1}$ and
$\mathcal{K}_{2}$, one obtains the two-dimensional chiral net
${\mathfrak{A}}$, acting on
$\mathcal{H}=\mathcal{K}_{1}\otimes\mathcal{K}_{2}$, by the standard
construction, recalled in Subsection 3.1: The two real lines are identified
with the lightlines in $\mathbb{R}^{2}$ and for any double cone
$\mathcal{O}={\cal I}\times\mathfrak{J}$ one sets444In the main part of the
paper ${\mathfrak{A}}({\cal I}\times\mathfrak{J})$ denotes a suitable weakly
dense ‘regular subalgebra’ of ${\cal A}_{1}({\cal I})\otimes{\cal
A}_{2}(\mathfrak{J})$. This distinction is not essential for the present
introductory discussion. ${\mathfrak{A}}({\cal I}\times\mathfrak{J})={\cal
A}_{1}({\cal I})\otimes{\cal A}_{2}(\mathfrak{J})$. If the nets ${\cal
A}_{1}$, ${\cal A}_{2}$ are in vacuum representations, with the vacuum vectors
$\Omega_{1}\in\mathcal{K}_{1}$, $\Omega_{2}\in\mathcal{K}_{2}$, then
${\mathfrak{A}}$ is also in a vacuum representation, with the vacuum vector
$\Omega=\Omega_{1}\otimes\Omega_{2}$. In spite of their simple tensor product
structure, chiral nets play a prominent role in conformal field theory. In
fact, with any local conformal net on $\mathbb{R}^{2}$ one can associate a
chiral subnet by restricting the theory to the lightlines. In the important
case of central charge $c<1$ these subnets were instrumental for the
classification results, mentioned above, which clarified the superselection
structure of a large class of models [43, 36]. As we show in the present work,
chiral nets also offer a promising starting point for the analysis of particle
aspects of conformal field theories: Any chiral net in a vacuum representation
is an asymptotically complete theory of Wigner particles. Moreover, any
charged irreducible product representation of such a net contains
infraparticles. With this information at hand, we exhibit examples of
infraparticles with superselected direction of motion. This construction is
summarized briefly in the remaining part of this Introduction.
Let us consider two fixed-point nets ${\cal A}_{1,\mathrm{ev}}$, ${\cal
A}_{2,\mathrm{ev}}$, obtained from ${\cal A}_{1}$ and ${\cal A}_{2}$ with the
help of the unitaries $W_{1}$ and $W_{2}$, implementing the respective actions
of $\mathbb{Z}_{2}$. The resulting chiral net ${\mathfrak{A}}_{\mathrm{ev}}$
acts on the Hilbert space $\mathcal{H}=\mathcal{K}_{1}\otimes\mathcal{K}_{2}$,
which decomposes into four invariant subspaces with different particle
structure:
$\mathcal{H}=(\mathcal{K}_{1,\mathrm{ev}}\otimes\mathcal{K}_{2,\mathrm{ev}})\oplus(\mathcal{K}_{1,\mathrm{odd}}\otimes\mathcal{K}_{2,\mathrm{ev}})\oplus(\mathcal{K}_{1,\mathrm{ev}}\otimes\mathcal{K}_{2,\mathrm{odd}})\oplus(\mathcal{K}_{1,\mathrm{odd}}\otimes\mathcal{K}_{2,\mathrm{odd}}).$
(1.4)
${\mathfrak{A}}_{\mathrm{ev}}$ restricted to
$\mathcal{H}_{0}:=\mathcal{K}_{1,\mathrm{ev}}\otimes\mathcal{K}_{2,\mathrm{ev}}$
is a chiral theory in a vacuum representation. Thus it is an asymptotically
complete theory of Wigner particles, by the result mentioned above.
$\mathcal{H}_{{\rm{R}}}:=\mathcal{K}_{1,\mathrm{odd}}\otimes\mathcal{K}_{2,\mathrm{ev}}$
contains ‘waves’ travelling to the right, but no ‘waves’ travelling to the
left. In
$\mathcal{H}_{{\rm{L}}}:=\mathcal{K}_{1,\mathrm{ev}}\otimes\mathcal{K}_{2,\mathrm{odd}}$
the opposite situation occurs. Thus ${\mathfrak{A}}_{\mathrm{ev}}$ restricted
to $\mathcal{H}_{{\rm{R}}}$ or $\mathcal{H}_{{\rm{L}}}$ describes
infraparticles, according to our terminology. Finally,
${\mathfrak{A}}_{\mathrm{ev}}$ restricted to
$\hat{\mathcal{H}}:=\mathcal{K}_{1,\mathrm{odd}}\otimes\mathcal{K}_{2,\mathrm{odd}}$
is a theory of infraparticles which does not contain ‘waves’. In Theorem 3.10
below, which is our main result, we establish superselection of direction of
motion for infraparticles described by the net
$\hat{\mathfrak{A}}={\mathfrak{A}}_{\mathrm{ev}}|_{\hat{\mathcal{H}}}$. The
argument proceeds as follows: ${\mathfrak{A}}_{\mathrm{ev}}$ is contained in
${\mathfrak{A}}$, which is an asymptotically complete theory of Wigner
particles. Thus we can use the scattering theory from [5] to compute the
asymptotic functionals (1.1) and obtain the decompositions (1.2) of their GNS
representations. Interpreted as a state on ${\mathfrak{A}}$, any vector
$\Psi_{1}\otimes\Psi_{2}\in\hat{\mathcal{H}}$ consists of two ‘waves’ at
asymptotic times: $\Psi_{1}\otimes\Omega_{2}$ travelling to the right and
$\Omega_{1}\otimes\Psi_{2}$ travelling to the left. (Cf. Theorem 3.3 below).
However, these two vectors belong to different invariant subspaces of
${\mathfrak{A}}_{\mathrm{ev}}$, namely to $\mathcal{H}_{{\rm{R}}}$ and
$\mathcal{H}_{{\rm{L}}}$. The corresponding representations of
$\hat{\mathfrak{A}}$ are not unitarily equivalent, since they have different
structure of the energy-momentum spectrum.
Our paper is organized as follows: Section 2, which does not rely on conformal
symmetry, concerns two-dimensional, massless quantum field theories and their
particle aspects: Preliminary Subsection 2.1 introduces the main concepts. In
Subsection 2.2 we recall the scattering theory of two-dimensional, massless
Wigner particles developed in [5]. Subsection 2.3 gives a brief exposition of
the theory of particle weights and introduces our main concept: superselection
of direction of motion. Subsection 2.4 presents our main technical result,
stated in Theorem 2.11, which clarifies the structure of asymptotic
functionals in theories of Wigner particles. Its proof is given in Appendix A.
In Section 3 we apply the concepts and tools presented in Section 2 to chiral
conformal field theories. Our setting, which is slightly more general than the
usual framework of conformal field theory, is presented in Subsection 3.1. In
Subsection 3.2 we show that any chiral theory in a vacuum representation has a
complete particle interpretation in terms of Wigner particles. In Subsection
3.3 we demonstrate that charged irreducible product representations of any
chiral theory describe infraparticles. Subsection 3.4 presents our main
result, that is superselection of the infraparticle’s direction of motion in
chiral theories arising from fixed-point nets of $\mathbb{Z}_{2}$ actions.
Proofs of some auxiliary lemmas are postponed to Appendix B. In Section 4 we
summarize our work and discuss future directions.
## 2 Particle aspects of two-dimensional massless theories
### 2.1 Preliminaries
In this section, which does not rely on conformal symmetry, we present some
general results on particle aspects of massless quantum field theories in two-
dimensional spacetime. We rely on the following variant of the Haag-Kastler
axioms [32]:
###### Definition 2.1.
A local net of $C^{*}$-algebras on $\mathbb{R}^{2}$ is a pair
$({\mathfrak{A}},U)$ consisting of a map
$\mathcal{O}\to{\mathfrak{A}}(\mathcal{O})$ from the family of open, bounded
regions of $\mathbb{R}^{2}$ to the family of $C^{*}$-algebras on a Hilbert
space $\mathcal{H}$, and a strongly continuous unitary representation of
translations $\mathbb{R}^{2}\ni x\to U(x)$ acting on $\mathcal{H}$, which are
subject to the following conditions:
1. 1.
(isotony) If $\mathcal{O}_{1}\subset\mathcal{O}_{2}$, then
${\mathfrak{A}}(\mathcal{O}_{1})\subset{\mathfrak{A}}(\mathcal{O}_{2})$.
2. 2.
(locality) If $\mathcal{O}_{1}\perp\mathcal{O}_{2}$, then
$[{\mathfrak{A}}(\mathcal{O}_{1}),{\mathfrak{A}}(\mathcal{O}_{2})]=0$, where
$\perp$ denotes spacelike separation.
3. 3.
(covariance)
$U(x){\mathfrak{A}}(\mathcal{O})U(x)^{*}={\mathfrak{A}}(\mathcal{O}+x)$ for
any $x\in\mathbb{R}^{2}$.
4. 4.
(positivity of energy) The spectrum of $U$ is contained in the closed forward
lightcone
$V_{+}:=\\{\,(\omega,\boldsymbol{p})\in\mathbb{R}^{2}\,|\,\omega\geq|\boldsymbol{p}|\,\\}$.
5. 5.
(regularity) The group of translation automorphisms
$\alpha_{x}(\,\cdot\,)=U(x)\,\cdot\,U(x)^{*}$ satisfies $\lim_{x\to
0}\|\alpha_{x}(A)-A\|=0$ for any $A\in{\mathfrak{A}}$.
We also introduce the quasilocal $C^{*}$-algebra of this net
${\mathfrak{A}}=\overline{\bigcup_{\mathcal{O}\subset\mathbb{R}^{2}}{\mathfrak{A}}(\mathcal{O})}$.
For any given net $({\mathfrak{A}},U)$ there exists exactly one unitary
representation of translations $U^{\mathrm{can}}$ s.t. $U^{\mathrm{can}}$
implements $\alpha$, all the operators $U^{\mathrm{can}}(x)$,
$x\in\mathbb{R}^{2}$ are contained in ${\mathfrak{A}}^{\prime\prime}$, the
spectrum of $U^{\mathrm{can}}$ is contained in $V_{+}$ and has Lorentz
invariant lower boundary upon restriction to any subspace of $\mathcal{H}$
invariant under the action of ${\mathfrak{A}}^{\prime\prime}$ [4]. We assume
that this _canonical_ representation of translations has been selected above,
i.e. $U=U^{\mathrm{can}}$. We denote by $(H,\boldsymbol{P})$ the corresponding
energy-momentum operators i.e. $U(x)=e^{iHt-i\boldsymbol{P}\boldsymbol{x}}$,
$x=(t,\boldsymbol{x})$. As we are interested in scattering of massless
particles, we introduce the single-particle subspaces ${\cal
H}_{\pm}:=\ker(H\mp\boldsymbol{P})$ and denote the corresponding projections
by $P_{\pm}$. The intersection $\mathcal{H}_{+}\cap\mathcal{H}_{-}$ contains
only translationally invariant vectors. If
$\mathcal{H}_{+}\neq\mathcal{H}_{+}\cap\mathcal{H}_{-}$ and
$\mathcal{H}_{-}\neq\mathcal{H}_{+}\cap\mathcal{H}_{-}$ then we say that the
theory describes Wigner particles. If $U$ has a unique (up to a phase)
invariant unit vector $\Omega\in\mathcal{H}$ and $\Omega$ is cyclic under the
action of ${\mathfrak{A}}$ then we say that the net $({\mathfrak{A}},U)$ is in
a vacuum representation. In this case ${\mathfrak{A}}$ acts irreducibly on
$\mathcal{H}$. (Cf. Theorem 4.6 of [1]). Scattering theory for Wigner
particles in a vacuum representation, developed in [5], will be recalled in
Subsection 2.2.
In the absence of Wigner particles we will apply the theory of particle
weights [10, 16, 41, 42], outlined in Subsection 2.3, to extract the
(infra-)particle content of a given theory. In this context it is necessary to
consider various representations of the net $({\mathfrak{A}},U)$. A
representation of the net $({\mathfrak{A}},U)$ is, by definition, a family of
representations $\\{\pi_{\cal O}\\}$ of local algebras which are consistent in
the sense that if ${\cal O}_{1}\subset{\cal O}_{2}$ then it holds that
$\pi_{{\cal O}_{2}}|_{{\mathfrak{A}}({\cal O}_{1})}=\pi_{{\cal O}_{1}}$. Since
the family of open bounded regions in ${\mathbb{R}}^{2}$ is directed, this
representation uniquely extends to a representation $\pi$ of the quasilocal
$C^{*}$-algebra ${\mathfrak{A}}$. Conversely, a representation of
${\mathfrak{A}}$ induces a consistent family of representations of local
algebras. In the following $\pi$ may refer to a representation of
${\mathfrak{A}}$ or a family of representations. We say that a representation
$\pi:{\mathfrak{A}}\to B(\mathcal{H}_{\pi})$ is covariant, if there exists a
strongly continuous group of unitaries $U_{\pi}$ on $\mathcal{H}_{\pi}$, s.t.
$\pi(\alpha_{x}(A))=U_{\pi}(x)\pi(A)U_{\pi}(x)^{*},\quad
A\in{\mathfrak{A}},\,\,x\in\mathbb{R}^{2}.$ (2.1)
Moreover, we say that this representation has positive energy, if the joint
spectrum of the generators of $U_{\pi}$ is contained in $V_{+}+q$ for some
$q\in\mathbb{R}^{2}$. We denote the corresponding canonical representation of
translations by $U_{\pi}^{\mathrm{can}}$ and note that
$(\pi({\mathfrak{A}}),U^{\mathrm{can}}_{\pi})$ is again a local net of
$C^{*}$-algebras in the sense of Definition 2.1. We say that the net
$(\pi({\mathfrak{A}}),U^{\mathrm{can}}_{\pi})$ is in a charged irreducible
representation, if $\pi({\mathfrak{A}})$ acts irreducibly on a non-trivial
Hilbert space $\mathcal{H}_{\pi}$ which does not contain non-zero invariant
vectors of $U^{\mathrm{can}}_{\pi}$.
We call two representations $(\pi_{1},\mathcal{H}_{\pi_{1}})$ and
$(\pi_{2},\mathcal{H}_{\pi_{2}})$ of $({\mathfrak{A}},U)$ unitarily
equivalent, (in short
$(\pi_{1},\mathcal{H}_{\pi_{1}})\simeq(\pi_{2},\mathcal{H}_{\pi_{2}})$), if
there exists a unitary $W:\mathcal{H}_{\pi_{1}}\to\mathcal{H}_{\pi_{2}}$ s.t.
$\displaystyle W\pi_{1}(A)=\pi_{2}(A)W,\quad\phantom{444}A\in{\mathfrak{A}}.$
(2.2)
If $\pi_{1}$ is a covariant, positive energy representation then so is
$\pi_{2}$ and it is easy to see that
$WU_{\pi_{1}}^{\mathrm{can}}(x)=U_{\pi_{2}}^{\mathrm{can}}(x)W,\quad
x\in\mathbb{R}^{2}.$ (2.3)
###### Remark 2.2.
We note that our (non-standard) Definition 2.1 of the local net neither
imposes the Poincaré covariance nor the existence of the vacuum vector. Thus
it applies both to vacuum representations and charged representations, which
facilitates our discussion. Apart from the physically motivated assumptions,
we adopt the regularity property 5, which can always be assured at the cost of
proceeding to a weakly dense subnet. This property seems indispensable in the
general theory of particle weights [41] e.g. in the proof of Proposition 2.10
stated below. For consistency of the presentation, we proceed to regular
subnets also in our discussion of conformal field theories in Section 3. We
stress, however, that this property is not needed there at the technical
level.
### 2.2 Scattering states
Scattering theory for Wigner particles in a vacuum representation of a two-
dimensional massless theory $({\mathfrak{A}},U)$ was developed in [5]. For the
reader’s convenience we recall here the main steps of this construction.
Following [5], for any $F\in{\mathfrak{A}}$ and $T\geq 1$ we introduce the
approximants:
$\displaystyle F_{\pm}(h_{T})=\int h_{T}(t)F(t,\pm t)dt,$ (2.4)
where $F(x):=\alpha_{x}(F)$,
$h_{T}(t)=|T|^{-\varepsilon}h(|T|^{-\varepsilon}(t-T))$, $0<\varepsilon<1$ and
$h\in C_{0}^{\infty}(\mathbb{R})$ is a non-negative function s.t. $\int
dt\,h(t)=1$. By applying the mean ergodic theorem, one obtains
$\lim_{T\to\infty}F_{\pm}(h_{T})\Omega=P_{\pm}F\Omega.$ (2.5)
Moreover, for $F\in{\mathfrak{A}}(\mathcal{O})$ and sufficiently large $T$ the
operator $F_{+}(h_{T})$ (resp. $F_{-}(h_{T})$) commutes with any observable
localized in the left (resp. right) component of the spacelike complement of
$\mathcal{O}$. Exploiting these two facts, the following result was
established in [5]:
###### Proposition 2.3 ([5]).
Let $F,G\in{\mathfrak{A}}$. Then the limits
$\displaystyle\Phi_{\pm}^{\mathrm{out}}(F):=\underset{T\to\infty}{\mathrm{s}\textrm{-}\lim}\;F_{\pm}(h_{T})\quad$
(2.6)
exist and are called the (outgoing) asymptotic fields. They depend only on the
respective vectors $\Phi_{\pm}^{\mathrm{out}}(F)\Omega=P_{\pm}F\Omega$ and
satisfy:
1. (a)
$\Phi_{+}^{\mathrm{out}}(F)\mathcal{H}_{+}\subset\mathcal{H}_{+},\quad\Phi_{-}^{\mathrm{out}}(G)\mathcal{H}_{-}\subset\mathcal{H}_{-}$.
2. (b)
$\alpha_{x}(\Phi_{+}^{\mathrm{out}}(F))=\Phi_{+}^{\mathrm{out}}(\alpha_{x}(F)),\quad\alpha_{x}(\Phi_{-}^{\mathrm{out}}(G))=\Phi_{-}^{\mathrm{out}}(\alpha_{x}(G))$
for $x\in{\mathbb{R}}^{2}$.
3. (c)
$[\Phi_{+}^{\mathrm{out}}(F),\Phi_{-}^{\mathrm{out}}(G)]=0$.
The incoming asymptotic fields $\Phi_{\pm}^{\mathrm{in}}(F)$ are constructed
analogously, by taking the limit $T\to-\infty$.
With the help of the asymptotic fields one defines the scattering states as
follows: Since ${\mathfrak{A}}$ acts irreducibly on $\mathcal{H}$, for any
$\Psi_{\pm}\in\mathcal{H}_{\pm}$ we can find $F_{\pm}\in{\mathfrak{A}}$ s.t.
$\Psi_{\pm}=F_{\pm}\Omega$ [45]. The vectors
$\displaystyle\Psi_{+}\overset{\mathrm{out}}{\times}\Psi_{-}=\Phi_{+}^{\mathrm{out}}(F_{+})\Phi_{-}^{\mathrm{out}}(F_{-})\Omega$
(2.7)
are called the (outgoing) scattering states. By Proposition 2.3 they do not
depend on the choice of $F_{\pm}$ within the above restrictions. The incoming
scattering states $\Psi_{+}\overset{\mathrm{in}}{\times}\Psi_{-}$ are defined
analogously. The physical interpretation of these vectors, as two independent
excitations travelling in opposite directions at asymptotic times, relies on
the following proposition from [5]:
###### Proposition 2.4 ([5]).
Let $\Psi_{\pm},\Psi_{\pm}^{\prime}\in\mathcal{H}_{\pm}$. Then:
1. (a)
$(\Psi_{+}\overset{\mathrm{out}}{\times}\Psi_{-}|\Psi^{\prime}_{+}\overset{\mathrm{out}}{\times}\Psi^{\prime}_{-})=(\Psi_{+}|\Psi^{\prime}_{+})(\Psi_{-}|\Psi^{\prime}_{-})$,
2. (b)
$U(x)(\Psi_{+}\overset{\mathrm{out}}{\times}\Psi_{-})=(U(x)\Psi_{+})\overset{\mathrm{out}}{\times}(U(x)\Psi_{-})$,
for $x\in\mathbb{R}^{2}$.
Analogous relations hold for the incoming scattering states.
Following [5], we define the subspaces spanned by the respective scattering
states:
$\mathcal{H}^{\mathrm{in}}=\mathcal{H}_{+}\overset{\mathrm{in}}{\times}\mathcal{H}_{-}\,\,\textrm{
and
}\,\,\mathcal{H}^{\mathrm{out}}=\mathcal{H}_{+}\overset{\mathrm{out}}{\times}\mathcal{H}_{-}.$
(2.8)
Next, we introduce the wave operators
$\Omega^{\mathrm{out}}:\mathcal{H}_{+}\otimes\mathcal{H}_{-}\to\mathcal{H}^{\mathrm{out}}$
and
$\Omega^{\mathrm{in}}:\mathcal{H}_{+}\otimes\mathcal{H}_{-}\to\mathcal{H}^{\mathrm{in}}$,
extending by linearity the relations
$\Omega^{\mathrm{out}}(\Psi_{+}\otimes\Psi_{-})=\Psi_{+}\overset{\mathrm{out}}{\times}\Psi_{-}\,\,\textrm{
and
}\,\,\Omega^{\mathrm{in}}(\Psi_{+}\otimes\Psi_{-})=\Psi_{+}\overset{\mathrm{in}}{\times}\Psi_{-}.$
(2.9)
These operators are isometric in view of Proposition 2.4 (a). The scattering
operator $S:\mathcal{H}^{\mathrm{out}}\to\mathcal{H}^{\mathrm{in}}$, given by
$S=\Omega^{\mathrm{in}}(\Omega^{\mathrm{out}})^{*},$ (2.10)
is also an isometry. Now we are ready to introduce two important concepts:
###### Definition 2.5.
(a) If $S=I$ on $\mathcal{H}^{\mathrm{out}}$, then we say that the theory is
non-interacting.
(b) If $\mathcal{H}^{\mathrm{in}}=\mathcal{H}^{\mathrm{out}}=\mathcal{H}$ then
we say that the theory is asymptotically complete (in terms of ‘waves’).
We show in Theorem 3.3 below that any chiral conformal field theory in a
vacuum representation is both non-interacting and asymptotically complete. (We
demonstrated these facts already in [25] in a different context).
To conclude this subsection, we introduce some other useful concepts which are
needed in Theorem 2.11 below: Let us choose some closed subspaces ${\cal
K}_{\pm}\subset\mathcal{H}_{\pm}$, invariant under the action of $U$, and
denote by ${\cal K}_{+}\overset{\mathrm{out}}{\times}{\cal K}_{-}$ the linear
span of the respective scattering states. For any $\Psi\in{\cal
K}_{+}\overset{\mathrm{out}}{\times}{\cal
K}_{-}\subset\mathcal{H}^{\mathrm{out}}$ we introduce the positive functionals
$\rho_{\pm,\Psi}$, given by the relations
$\displaystyle\rho_{+,\Psi}(A)$ $\displaystyle=$
$\displaystyle((\Omega^{\mathrm{out}})^{-1}\Psi|(A\otimes
I)(\Omega^{\mathrm{out}})^{-1}\Psi),$ (2.11) $\displaystyle\rho_{-,\Psi}(A)$
$\displaystyle=$ $\displaystyle((\Omega^{\mathrm{out}})^{-1}\Psi|(I\otimes
A)(\Omega^{\mathrm{out}})^{-1}\Psi),$ (2.12)
where $A\in B(\mathcal{H})$ and the embedding ${\cal K}_{+}\otimes{\cal
K}_{-}\subset\mathcal{H}\otimes\mathcal{H}$ is understood. These functionals
can be expressed as follows
$\rho_{\pm,\Psi}(\,\cdot\,)=\sum_{n\in\mathbb{N}}(\Psi_{\pm,n}|\,\cdot\,\Psi_{\pm,n}),$
(2.13)
where $\Psi_{\pm,n}\in{\cal K}_{\pm}$ and
$\sum_{n\in\mathbb{N}}\|\Psi_{\pm,n}\|^{2}=\|\Psi\|^{2}$. It follows easily
from Lemma A.2, that for $\Psi\in P_{E}({\cal
K}_{+}\overset{\mathrm{out}}{\times}{\cal K}_{-})$, where $P_{E}$ is the
spectral projection on vectors of energy not larger than $E$, one can choose
$\Psi_{\pm,n}\in P_{E}{\cal K}_{\pm}$. We note that for $\|\Psi\|=1$ the
functionals $\rho_{\pm,\Psi}$ are just the familiar reduced density matrices.
### 2.3 Particle weights
Similarly as in the previous subsection we consider a local net of
$C^{*}$-algebras $({\mathfrak{A}},U)$ acting on a Hilbert space $\mathcal{H}$.
However, we do not assume that $\mathcal{H}$ contains the vacuum vector or
non-trivial single-particle subspaces $\mathcal{H}_{\pm}$. To study particle
aspects in this general situation we use the theory of particle weights [10,
16, 41, 42] which we recall in this and the next subsection. With the help of
this theory we formulate in Definitions 2.7 and 2.12 below the central notion
of this paper: superselection of direction of motion.
First, we recall two useful concepts: almost locality and the energy
decreasing property. An observable $B\in{\mathfrak{A}}$ is called almost
local, if there exists a net of operators
$\\{\,B_{r}\in{\mathfrak{A}}(\mathcal{O}_{r})\,|\,r>0\,\\}$, s.t. for any
$k\in\mathbb{N}_{0}$
$\lim_{r\to\infty}r^{k}\|B-B_{r}\|=0,$ (2.14)
where
$\mathcal{O}_{r}=\\{(t,\boldsymbol{x})\in\mathbb{R}^{2}\,|\,|t|+|\boldsymbol{x}|<r\,\\}$.
We say that an operator $B\in{\mathfrak{A}}$ is energy decreasing, if its
energy-momentum transfer is a compact set which does not intersect with the
closed forward lightcone $V_{+}$. We recall that the energy-momentum transfer
(or the Arveson spectrum w.r.t. $\alpha$) of an observable
$B\in{\mathfrak{A}}$ is the closure of the union of supports of the
distributions
$(\Psi_{1}|\tilde{B}(p)\Psi_{2})=(2\pi)^{-1}\int
d^{2}x\,e^{-ipx}(\Psi_{1}|B(x)\Psi_{2})$ (2.15)
over all $\Psi_{1},\Psi_{2}\in\mathcal{H}$, where $p=(\omega,\boldsymbol{p})$,
$x=(t,\boldsymbol{x})$ and $px=\omega t-\boldsymbol{p}\boldsymbol{x}$.
Following [16, 41], we introduce the subspace
$\mathcal{L}_{0}\subset{\mathfrak{A}}$, spanned by operators which are both
almost local and energy decreasing, and the corresponding left ideal in
${\mathfrak{A}}$:
$\mathcal{L}:=\\{\,AB\,|\,A\in{\mathfrak{A}},B\in\mathcal{L}_{0}\,\\}.$ (2.16)
Particle weights form a specific class of sesquilinear forms on $\mathcal{L}$:
###### Definition 2.6.
A particle weight is a non-zero, positive sesquilinear form $\psi$ on the left
ideal $\mathcal{L}$, satisfying the following conditions:
1. 1.
For any $L_{1},L_{2}\in\mathcal{L}$ and $A\in{\mathfrak{A}}$ the relation
$\psi(AL_{1},L_{2})=\psi(L_{1},A^{*}L_{2})$ holds.
2. 2.
For any $L_{1},L_{2}\in\mathcal{L}$ and $x\in\mathbb{R}^{2}$ the relation
$\psi(\alpha_{x}(L_{1}),\alpha_{x}(L_{2}))=\psi(L_{1},L_{2})$ holds.
3. 3.
For any $L_{1},L_{2}\in\mathcal{L}$ the map $\mathbb{R}^{2}\ni
x\to\psi(L_{1},\alpha_{x}(L_{2}))$ is continuous. Its Fourier transform is
supported in a shifted lightcone $V_{+}-q$, where $q\in V_{+}$ does not depend
on $L_{1},L_{2}$.
Let us now summarize the pertinent properties of particle weights established
in [41] (in a slightly different framework). As a consequence of Theorem 2.9,
stated below, particle weights satisfy the following clustering property [41]
$\int d\boldsymbol{x}\,|\psi(L_{1},\alpha_{\boldsymbol{x}}(L_{2}))|<\infty,$
(2.17)
valid for $L_{1}=B_{1}^{*}A_{1}B_{1}^{\prime}$,
$L_{2}=B_{2}^{*}A_{2}B_{2}^{\prime}$, where
$B_{1},B_{1}^{\prime},B_{2},B_{2}^{\prime}\in\mathcal{L}_{0}$ and
$A_{1},A_{2}\in{\mathfrak{A}}$ are almost local. In view of this bound, the
GNS representation $(\pi_{\psi},\mathcal{H}_{\pi_{\psi}})$ induced by a
particle weight $\psi$ is well suited for a description of physical systems
which are localized in space (e.g. configurations of particles). The Hilbert
space $\mathcal{H}_{\pi_{\psi}}$ is given by
$\mathcal{H}_{\pi_{\psi}}=(\,\mathcal{L}/\\{\,L\in\mathcal{L}\,|\,\psi(L,L)=0\\})^{\mathrm{cpl}}$
(2.18)
and the respective equivalence class of an element $L\in\mathcal{L}$ is
denoted by $|L\rangle\in\mathcal{H}_{\pi_{\psi}}$. The completion is taken
w.r.t. the scalar product $\langle L_{1}|L_{2}\rangle:=\psi(L_{1},L_{2})$. The
representation $\pi_{\psi}$ acts on $\mathcal{H}_{\pi_{\psi}}$ as follows
$\pi_{\psi}(A)|L\rangle=|AL\rangle,\quad A\in{\mathfrak{A}}.$ (2.19)
This representation is covariant and the translation automorphisms are
implemented by the strongly continuous group of unitaries $U_{\pi_{\psi}}$,
given by
$U_{\pi_{\psi}}(x)|L\rangle=|\alpha_{x}(L)\rangle,\quad
x\in\mathbb{R}^{2},\,L\in\mathcal{L}$ (2.20)
which is called the standard representation of translations in the
representation $\pi_{\psi}$. By property 3 in Definition 2.6 above, its
spectrum is contained in a shifted closed forward lightcone. The corresponding
canonical representation will be denoted by $U_{\pi_{\psi}}^{\mathrm{can}}$.
(Cf. the discussion below Definition 2.1). We also introduce operators
$(Q^{0},\boldsymbol{Q})$ of _characteristic energy-momentum_ of $\psi$ which
are the generators of the following group of unitaries on
$\mathcal{H}_{\pi_{\psi}}$
$U^{\mathrm{char}}_{\pi_{\psi}}(x)=U^{\mathrm{can}}_{\pi_{\psi}}(x)U_{\pi_{\psi}}(x)^{-1}\in\pi_{\psi}({\mathfrak{A}})^{\prime},$
(2.21)
i.e.
$U^{\mathrm{char}}_{\pi_{\psi}}(x)=e^{iQ^{0}t-i\boldsymbol{Q}\boldsymbol{x}}$.
We call a particle weight _pure_ , if its GNS representation is irreducible.
It follows from definition (2.21) that the operator of characteristic energy-
momentum of such a weight is a vector
$q=(q^{0},\boldsymbol{q})\in\mathbb{R}^{2}$. It can be interpreted as the
energy and momentum of the plane wave configuration of the particle described
by this weight [2, 41].
To extract properties of elementary subsystems (particles) of a physical
system described by a given (possibly non-pure) particle weight, it is natural
to study irreducible subrepresentations of its GNS representation. To ensure
that there are sufficiently many such subrepresentations, we restrict
attention to particle weights $\psi$ whose GNS representations $\pi_{\psi}$
are of type I with atomic center555i.e. whose center is a direct sum of one-
dimensional von Neumann algebras.. (In particular, $\pi_{\psi}$ appearing in
our examples in Subsection 3.4 below belong to this family). Then, by Theorem
1.31 from Chapter V of [49], there exists a unique family of Hilbert spaces
$(\mathfrak{H}_{\alpha},\mathfrak{K}_{\alpha})_{\alpha\in\mathbb{I}}$ and a
unitary
$W:\mathcal{H}_{\pi_{\psi}}\to\bigoplus_{\alpha\in\mathbb{I}}\\{\mathfrak{H}_{\alpha}\otimes\mathfrak{K}_{\alpha}\\}$
s.t.
$\displaystyle W\pi_{\psi}({\mathfrak{A}})^{\prime\prime}W^{-1}$
$\displaystyle=$
$\displaystyle\bigoplus_{\alpha\in\mathbb{I}}\\{B(\mathfrak{H}_{\alpha})\otimes{\mathbb{C}}I\\},$
(2.22) $\displaystyle W\pi_{\psi}({\mathfrak{A}})^{\prime}W^{-1}$
$\displaystyle=$
$\displaystyle\bigoplus_{\alpha\in\mathbb{I}}\\{{\mathbb{C}}I\otimes
B(\mathfrak{K}_{\alpha})\\}.$ (2.23)
We note that a subspace ${\cal K}_{\alpha,e}\subset\mathcal{H}_{\pi_{\psi}}$
carries an irreducible subrepresentation $\pi_{\alpha,e}$ of $\pi_{\psi}$, if
and only if $W{\cal K}_{\alpha,e}=\mathfrak{H}_{\alpha}\otimes{\mathbb{C}}e$
for some $\alpha\in\mathbb{I}$ and $e\in\mathfrak{K}_{\alpha}$. Clearly,
$\pi_{\alpha,e}$ and $\pi_{\alpha,e^{\prime}}$ are unitarily equivalent for
any fixed $\alpha$ and arbitrary vectors
$e,e^{\prime}\in\mathfrak{K}_{\alpha}$. Choosing in any
$\mathfrak{K}_{\alpha}$ an orthonormal basis $B_{\alpha}$, we obtain
$\pi_{\psi}=\bigoplus_{\begin{subarray}{c}\alpha\in\mathbb{I}\\\ e\in
B_{\alpha}\end{subarray}}\pi_{\alpha,e}.$ (2.24)
It is clear from the above discussion that any irreducible subrepresentation
of $\pi_{\psi}$ is unitarily equivalent to some $\pi_{\alpha,e}$ in the
decomposition above.
If all the representations in the decomposition (2.24) are unitarily
equivalent to some fixed vacuum representation, then we call the particle
weight $\psi$ neutral. Otherwise we call $\psi$ charged. In the case of
charged particle weights there may occur an interplay between the
translational and internal degrees of freedom of the system which we call
_superselection of direction of motion_. To introduce this concept, we need
some terminology: Let $\mathcal{H}_{\pi_{\psi},\mathrm{R}}$ (resp.
$\mathcal{H}_{\pi_{\psi},\mathrm{L}}$) be the spectral subspace of the
characteristic momentum operator $\boldsymbol{Q}$ of $\psi$, corresponding to
the interval $[0,\infty)$ (resp. $(-\infty,0)$). Let $\pi$ be an irreducible
subrepresentation of $\pi_{\psi}$, acting on a subspace ${\cal
K}\subset\mathcal{H}_{\pi_{\psi}}$. Then we say that $\pi$ is _right-moving_
(resp. _left-moving_), if ${\cal K}\neq\\{0\\}$ and ${\cal
K}\subset\mathcal{H}_{\pi_{\psi},\mathrm{R}}$ (resp. ${\cal
K}\subset\mathcal{H}_{\pi_{\psi},\mathrm{L}}$). By a suitable choice of the
bases $B_{\alpha}$ one can ensure that each representation $\pi_{\alpha,e}$,
appearing in decomposition (2.24), has one of these properties. (In fact,
exploiting relations (2.21), (2.23), one can choose such basis vectors
$e\in\mathfrak{K}_{\alpha}$ that
$W^{-1}(\mathfrak{H}_{\alpha}\otimes{\mathbb{C}}e)$ belong to
$\mathcal{H}_{\pi_{\psi},\mathrm{R}}$ or
$\mathcal{H}_{\pi_{\psi},\mathrm{L}}$). After this preparation we define the
central concept of the present paper:
###### Definition 2.7.
Let $\mathcal{W}$ be a family of particle weights and assume that their GNS
representations
$\\{\,(\pi_{\psi},\mathcal{H}_{\pi_{\psi}})\,|\,\psi\in\mathcal{W}\,\\}$ are
of type I with atomic centers. Suppose that for any
$\psi,\psi^{\prime}\in\mathcal{W}$ the following properties hold:
1. 1.
$\pi_{\psi}$ has both left-moving and right-moving irreducible
subrepresentations.
2. 2.
No right-moving, irreducible subrepresentation of $\pi_{\psi}$ is unitarily
equivalent to a left-moving irreducible subrepresentation of
$\pi_{\psi^{\prime}}$.
Then we say that this family of particle weights has superselected direction
of motion.
Let us now relate superselection of direction of motion in the above sense to
our discussion of this concept in the Introduction. For this purpose we
consider a particle weight $\psi$, whose GNS representation is of type I with
atomic center and acts on a _separable_ Hilbert space
$\mathcal{H}_{\pi_{\psi}}$. Making use of formula (2.24) and identifying
unitarily each $\pi_{\alpha,e}$, acting on
$W^{-1}(\mathfrak{H}_{\alpha}\otimes{\mathbb{C}}e)$, with
$\pi_{\alpha}:=\pi_{\alpha,e_{0}}$ acting on ${\cal
K}_{\alpha}:=W^{-1}(\mathfrak{H}_{\alpha}\otimes{\mathbb{C}}e_{0})$ for some
chosen $e_{0}\in B_{\alpha}$, we obtain
$\pi_{\psi}({\mathfrak{A}})\simeq\bigoplus_{\alpha\in\mathbb{I}}\\{\pi_{\alpha}({\mathfrak{A}})\otimes{\mathbb{C}}I\\},$
(2.25)
where the r.h.s. acts on $\bigoplus_{\alpha\in\mathbb{I}}\\{{\cal
K}_{\alpha}\otimes\mathfrak{K}_{\alpha}\\}$. In the sense of the same
identification
$\pi_{\psi}({\mathfrak{A}})^{\prime}\simeq\bigoplus_{\alpha\in\mathbb{I}}\\{{\mathbb{C}}I\otimes
B(\mathfrak{K}_{\alpha})\\}.$ (2.26)
Now, following [42], we choose a maximal abelian von Neumann algebra ${\cal
M}$ in $\pi_{\psi}({\mathfrak{A}})^{\prime}$, containing
$\\{\,U^{\mathrm{char}}_{\pi_{\psi}}(x)\,|\,x\in\mathbb{R}^{2}\,\\}$. As a
consequence of formula (2.26)
${\cal M}\simeq\bigoplus_{\alpha\in\mathbb{I}}\\{{\mathbb{C}}I\otimes{\cal
M}_{\alpha}\\},$ (2.27)
where ${\cal M}_{\alpha}\subset B(\mathfrak{K}_{\alpha})$ are maximal abelian
von Neumann subalgebras. For any such ${\cal M}_{\alpha}$ there exists a Borel
space $(Z_{\alpha},d\mu_{\alpha})$ s.t. $({\cal
M}_{\alpha},\mathfrak{K}_{\alpha})\simeq(L^{\infty}(Z_{\alpha},d\mu_{\alpha}),L^{2}(Z_{\alpha},d\mu_{\alpha}))$.
(This fact uses separability of the Hilbert space. See Theorem II.2.2 of
[20]). Adopting this identification in (2.25) and (2.26), we obtain
$U^{\mathrm{char}}_{\pi_{\psi}}\simeq\bigoplus_{\alpha\in\mathbb{I}}\\{I\otimes
U^{\mathrm{char}}_{\alpha}\\},$ (2.28)
where $U^{\mathrm{char}}_{\alpha}(x)\in L^{\infty}(Z_{\alpha},d\mu_{\alpha})$
is the operator of multiplication by (the equivalence class of) the function
$Z_{\alpha}\ni z\to e^{iq_{\alpha,z}x}$, where
$q_{\alpha,z}=(q_{\alpha,z}^{0},\boldsymbol{q}_{\alpha,z})\in\mathbb{R}^{2}$.
Introducing the field of representations
$(\pi_{\alpha,z},\mathfrak{H}_{\alpha,z})_{z\in Z_{\alpha}}$ s.t.
$\pi_{\alpha,z}=\pi_{\alpha}$ and $\mathfrak{H}_{\alpha,z}={\cal K}_{\alpha}$
for all $z\in Z_{\alpha}$, we obtain from relation (2.25) the existence of a
unitary
$\tilde{W}:\mathcal{H}_{\pi_{\psi}}\to\bigoplus_{\alpha\in\mathbb{I}}\int^{\oplus}d\mu_{\alpha}(z)\,\mathfrak{H}_{\alpha,z}$
s.t.
$\tilde{W}\pi_{\psi}(\,\cdot\,)\tilde{W}^{-1}=\bigoplus_{\alpha\in\mathbb{I}}\int^{\oplus}_{Z_{\alpha}}d\mu_{\alpha}(z)\,\pi_{\alpha,z}(\,\cdot\,).$
(2.29)
This is an example of decomposition (1.2), stated in the Introduction.
Moreover, as a consequence of (2.28),
$\tilde{W}U^{\mathrm{char}}_{\pi_{\psi}}(x)\tilde{W}^{-1}=\bigoplus_{\alpha\in\mathbb{I}}\int^{\oplus}_{Z_{\alpha}}d\mu_{\alpha}(z)\,e^{iq_{\alpha,z}x},$
(2.30)
where $\\{q_{\alpha,z}\\}_{z\in Z_{\alpha}}$ is the field of characteristic
energy-momentum vectors666This terminology is consistent with the discussion
after formula (2.21). In fact, under some technical restrictions each
$\pi_{\alpha,z}$ is induced by some pure particle weight $\psi_{\alpha,z}$,
whose characteristic energy-momentum vector is $q_{\alpha,z}$ [41, 42]. Cf.
also formula (1.3). of the representations
$(\pi_{\alpha,z},\mathfrak{H}_{\alpha,z})_{z\in Z_{\alpha}}$. As we required
in the Introduction, for any particle weight with superselected direction of
motion, the relation
$\boldsymbol{q}_{\alpha,z}\cdot\boldsymbol{q}_{\alpha^{\prime},z^{\prime}}\leq
0$ should imply that $\pi_{\alpha,z}$ is not unitarily equivalent to
$\pi_{\alpha^{\prime},z^{\prime}}$ for almost all $z,z^{\prime}$. This is in
fact the case in view of the following proposition.
###### Proposition 2.8.
Suppose that $\psi$ belongs to a family of particle weights which has
superselected direction of motion in the sense of Definition 2.7 and s.t. its
GNS representation acts on a separable Hilbert space. Then
$\\{\pi_{\alpha,z}\\}_{z\in Z_{\alpha}}$, appearing in the decomposition
(2.29) of $\pi_{\psi}$, is a field of right-moving (resp. left-moving)
representations, if and only if $\boldsymbol{q}_{\alpha,z}\geq 0$ (resp.
$\boldsymbol{q}_{\alpha,z}<0$) for almost all $z\in Z_{\alpha}$.
Proof. Suppose that $\pi_{\alpha}$ is a right-moving subrepresentation of
$\pi_{\psi}$ i.e. ${\cal
K}_{\alpha}\subset\mathcal{H}_{\pi_{\psi},\mathrm{R}}$. We recall that
$\pi_{\alpha}$ coincides with $\pi_{\alpha,e_{0}}$ acting on ${\cal
K}_{\alpha,e_{0}}={\cal K}_{\alpha}$. Since every $\pi_{\alpha,e}$,
$e\in\mathfrak{K}$, is unitarily equivalent to $\pi_{\alpha,e_{0}}$, the
property of superselection of direction of motion implies that ${\cal
K}_{\alpha,e}\subset\mathcal{H}_{\pi_{\psi},\mathrm{R}}$ for all
$e\in\mathfrak{K}$. Consequently,
$\mathcal{H}_{\alpha}=W^{-1}(\mathfrak{H}_{\alpha}\otimes\mathfrak{K}_{\alpha})\subset\mathcal{H}_{\pi_{\psi},\mathrm{R}}$.
Since the projection $P_{\alpha}$ on $\mathcal{H}_{\alpha}$ is central, this
subspace is invariant under the action of $U^{\mathrm{char}}_{\pi_{\psi}}$.
Formula (2.28) gives
$U^{\mathrm{char}}_{\pi_{\psi}}(x)P_{\alpha}\simeq I\otimes
U^{\mathrm{char}}_{\alpha}(x),$ (2.31)
thus the spectra of the generators of $\mathbb{R}^{2}\ni x\to
U^{\mathrm{char}}_{\alpha}(x)$ and $\mathbb{R}^{2}\ni x\to
U^{\mathrm{char}}_{\pi_{\psi}}(x)P_{\alpha}$ coincide. In particular the
spectrum of the generator of space translations of
$U^{\mathrm{char}}_{\alpha}$ is contained in $[0,\infty)$. The opposite
implication follows immediately from relation (2.31). $\Box$
### 2.4 Asymptotic functionals
In this subsection we consider a concrete class of particle weights,
introduced in [9, 10, 41], which have applications in scattering theory. Their
construction relies on the following result due to Buchholz (which remains
valid in higher dimensions).
###### Theorem 2.9 ([10]).
Let $({\mathfrak{A}},U)$ be a local net of $C^{*}$-algebras on
$\mathbb{R}^{2}$. Then, for any $E\geq 0$, $L\in\mathcal{L}$,
$\left\|P_{E}\int_{K}d\boldsymbol{x}\,(L^{*}L)(\boldsymbol{x})P_{E}\right\|\leq
c,$ (2.32)
where $P_{E}$ is the spectral projection on vectors of energy not larger than
$E$, $K\subset\mathbb{R}$ is a compact interval, and $c$ is a constant
independent of $K$.
Following [41], we introduce the algebra of detectors
$\mathcal{C}=\mathrm{span}\\{L_{1}^{*}L_{2}:L_{1},L_{2}\in\mathcal{L}\\}$ and
equip it with a locally convex topology, given by the family of seminorms
$p_{E}(C)=\sup\left\\{\int
d\boldsymbol{x}\,|(\Psi|C(\boldsymbol{x})\Psi)|\,\,|\,\,\Psi\in
P_{E}\mathcal{H},\,\|\Psi\|\leq 1\,\right\\},\,\quad C\in\mathcal{C},$ (2.33)
labelled by $E\geq 0$, which are finite by Theorem 2.9. Next, for any
$\Psi\in\mathcal{H}$ of bounded energy, (i.e. belonging to $P_{E}\mathcal{H}$
for some $E\geq 0$), we define a sequence of functionals
$\\{\sigma_{\Psi}^{(T)}\\}_{T\in\mathbb{R}}$ from the topological dual of
$\mathcal{C}$:
$\sigma_{\Psi}^{(T)}(C):=\int dt\,h_{T}(t)\int
d\boldsymbol{x}\,(\Psi|C(t,\boldsymbol{x})\Psi),\quad C\in\mathcal{C}.$ (2.34)
As this sequence is uniformly bounded in $T$ w.r.t. any seminorm $p_{E}$, the
Alaoglu-Bourbaki theorem gives limit points
$\sigma_{\Psi}^{\mathrm{out}}\in\mathcal{C}^{*}$ as $T\to\infty$, which are
called the asymptotic functionals. The following fact was shown in [41]:
###### Proposition 2.10 ([41]).
If $\sigma_{\Psi}^{\mathrm{out}}\neq 0$, then the sesquilinear forms on
$\mathcal{L}$, given by
$\psi_{\Psi}^{\mathrm{out}}(L_{1},L_{2}):=\sigma_{\Psi}^{\mathrm{out}}(L_{1}^{*}L_{2}),$
(2.35)
are particle weights, in the sense of Definition 2.6.
Fundamental results from [2] suggest a physical interpretation of the particle
weights $\psi_{\Psi}^{\mathrm{out}}$ as mixtures of plane wave configurations
of all the particle types described by the theory. (Cf. formulas (1.2),
(1.3)). Accordingly, we say that a given theory has a non-trivial _particle
content_ , if it admits some non-zero asymptotic functionals
$\sigma_{\Psi}^{\mathrm{out}}$. This is the case in any massless two-
dimensional theory of Wigner particles (in a vacuum representation) as a
consequence of the following theorem. A proof of this statement, which is our
main technical result, is given in Appendix A.
###### Theorem 2.11.
Let $({\mathfrak{A}},U)$ be a local net of $C^{*}$-algebras on
$\mathbb{R}^{2}$ in a vacuum representation, acting on a Hilbert space
$\mathcal{H}$. Then, for any $\Psi\in P_{E}\mathcal{H}^{\mathrm{out}}$, $E\geq
0$,
$\displaystyle\psi_{\Psi}^{\mathrm{out}}(L_{1},L_{2})$ $\displaystyle=$
$\displaystyle\lim_{T\to\infty}\int dt\,h_{T}(t)\int
d\boldsymbol{x}\,(\Psi|(L_{1}^{*}L_{2})(t,\boldsymbol{x})\Psi)$ (2.36)
$\displaystyle=$ $\displaystyle\int
d\boldsymbol{x}\,(\rho_{+,\Psi}+\rho_{-,\Psi})\big{(}(L_{1}^{*}L_{2})(\boldsymbol{x})\big{)},$
where the functionals $\rho_{\pm,\Psi}$ are defined by (2.11), (2.12). In
particular, $\psi_{\Psi}^{\mathrm{out}}=0$, if and only if
$\Psi\in{\mathbb{C}}\Omega$.
In a theory of Wigner particles
${\mathbb{C}}\Omega\neq\mathcal{H}_{\pm}\subset\mathcal{H}^{\mathrm{out}}$,
thus the particle content is non-trivial by the above result. However, non-
zero asymptotic functionals may also appear in the absence of Wigner particles
i.e. when one or both of the subspaces $\mathcal{H}_{\pm}$ equal
$\mathcal{H}_{+}\cap\mathcal{H}_{-}$. If this is the case, then we say that
the net $({\mathfrak{A}},U)$ describes _infraparticles_. Theorem 3.6 below
provides a large class of such theories. In Theorem 3.10 we show that some of
these models describe excitations whose direction of motion is superselected
in the following sense:
###### Definition 2.12.
Let $({\mathfrak{A}},U)$ be a net describing infraparticles. We say that the
infraparticles of the net $({\mathfrak{A}},U)$ have superselected direction of
motion, if $\\{\,\psi_{\Psi}^{\mathrm{out}}\,|\,\Psi\neq 0,\,\Psi\in
P_{E}\mathcal{H},\,E\geq 0\,\\}$ is a family of particle weights with
superselected direction of motion in the sense of Definition 2.7.
## 3 Particle aspects of conformal field theories
### 3.1 Preliminaries
In this section we are interested in particle aspects of chiral conformal
field theories. To emphasize the relevant properties of these models, we base
our investigation on the concept of a local net of von Neumann algebras on
${\mathbb{R}}$, defined below. There are many examples of such nets. In
particular, they arise from Möbius covariant nets on $S^{1}$ by means of the
Cayley transform and the subsequent restriction to the real line. The simplest
example is the so-called $U(1)$-current net [15], whose subnets and extensions
are well-studied. For certain classes of nets on $S^{1}$ even classification
results have been obtained [36, 37].
###### Definition 3.1.
A local net of von Neumann algebras on $\mathbb{R}$ is a pair $({\cal A},V)$
consisting of a map ${\cal I}\to{\cal A}({\cal I})$ from the family of open,
bounded subsets of $\mathbb{R}$ to the family of von Neumann algebras on a
Hilbert space $\mathcal{K}$ and a strongly continuous unitary representation
of translations $\mathbb{R}\ni s\to V(s)$, acting on $\mathcal{K}$, which are
subject to the following conditions:
1. 1.
(isotony) If ${\cal I}\subset\mathfrak{J}$, then ${\cal A}({\cal
I})\subset{\cal A}(\mathfrak{J})$.
2. 2.
(locality) If ${\cal I}\cap\mathfrak{J}=\varnothing$, then $[{\cal A}({\cal
I}),{\cal A}(\mathfrak{J})]=0$.
3. 3.
(covariance) $V(s){\cal A}({\cal I})V(s)^{*}={\cal A}({\cal I}+s)$ for any
$s\in\mathbb{R}$.
4. 4.
(positivity of energy) The spectrum of $V$ coincides with $\mathbb{R}_{+}$.
We also denote by ${\cal A}$ the quasilocal $C^{*}$-algebra of this net i.e.
${\cal A}=\overline{\bigcup_{{\cal I}\subset\mathbb{R}}{\cal A}({\cal I})}$.
Since we assumed that ${\cal A}({\cal I})$ are von Neumann algebras, we cannot
demand norm continuity of the functions $s\to\beta_{s}(A)$, $A\in{\cal A}$,
where $\beta_{s}(\,\cdot\,)=V(s)\,\cdot\,V(s)^{*}$. This regularity property
holds, however, on the following weakly dense subnet of $C^{*}$-algebras
${\cal I}\to\bar{\cal A}({\cal I}):=\\{\,A\in{\cal A}({\cal I})\,|\,\lim_{s\to
0}\|\beta_{s}(A)-A\|=0\,\\}.$ (3.1)
The corresponding quasilocal algebra is denoted by $\bar{\cal A}$.
If $V$ has a unique (up to a phase) invariant, unit vector
$\Omega_{0}\in\mathcal{K}$ and $\Omega_{0}$ is cyclic under the action of any
${\cal A}({\cal I})$ (the Reeh-Schlieder property) then we say that the net
$({\cal A},V)$ is in a vacuum representation. In this case ${\cal A}$ acts
irreducibly on $\mathcal{K}$. In the course of our analysis we will also
consider other representations of $({\cal A},V)$. We say that a representation
$\pi:{\cal A}\to B(\mathcal{K}_{\pi})$ is covariant, if there exists a
strongly continuous group of unitaries $V_{\pi}$ on $\mathcal{K}_{\pi}$, s.t.
$\pi(\alpha_{s}(A))=V_{\pi}(s)\pi(A)V_{\pi}(s)^{*},\quad A\in{\cal
A},\,\,s\in\mathbb{R}.$ (3.2)
Moreover, we say that this representation has positive energy, if the spectrum
of $V_{\pi}$ coincides with $\mathbb{R}_{+}$. If $\pi$ is locally normal (i.e.
its restriction to any local algebra ${\cal A}({\cal I})$ is normal) then
$(\pi({\cal A}),V_{\pi})$ is again a net of von Neumann algebras in the sense
of Definition 3.1.
Let $({\cal A}_{1},V_{1})$ and $({\cal A}_{2},V_{2})$ be two nets of von
Neumann algebras on ${\mathbb{R}}$, acting on Hilbert spaces $\mathcal{K}_{1}$
and $\mathcal{K}_{2}$. To construct a local net $({\mathfrak{A}},U)$ on
${\mathbb{R}}^{2}$, acting on the tensor product space ${\cal H}={\cal
K}_{1}\otimes{\cal K}_{2}$, we identify the two real lines with the lightlines
$I_{\pm}=\\{\,(t,\boldsymbol{x})\in\mathbb{R}^{2}\,|\,\boldsymbol{x}\mp
t=0\,\\}$ in ${\mathbb{R}}^{2}$. We first specify the unitary representation
of translations
$U(t,\boldsymbol{x}):=V_{1}\left(\frac{1}{\sqrt{2}}(t-\boldsymbol{x})\right)\otimes
V_{2}\left(\frac{1}{\sqrt{2}}(t+\boldsymbol{x})\right),$ (3.3)
whose spectrum is easily seen to coincide with $V_{+}$ as a consequence of
property 4 from Definition 3.1. We mention for future reference that if
$\alpha_{(t,\boldsymbol{x})}(\,\cdot\,):=U(t,\boldsymbol{x})\,\,\cdot\,\,U(t,\boldsymbol{x})^{*}$
is the corresponding group of translation automorphisms and
$\beta^{(1/2)}_{s}(\,\cdot\,):=V_{1/2}(s)\,\,\cdot\,\,V_{1/2}(s)^{*}$, then
$\alpha_{(t,\boldsymbol{x})}(A_{1}\otimes
A_{2})=\beta^{(1)}_{\frac{1}{\sqrt{2}}(t-\boldsymbol{x})}(A_{1})\otimes\beta^{(2)}_{\frac{1}{\sqrt{2}}(t+\boldsymbol{x})}(A_{2}),\quad
A_{1}\in{\cal A}_{1},\,A_{2}\in{\cal A}_{2}.$ (3.4)
Any double cone $D\subset{\mathbb{R}}^{2}$ can be expressed as a product of
intervals on lightlines $D={\cal I}_{1}\times{\cal I}_{2}$. We define the
corresponding local von Neumann algebra by
${\mathfrak{A}}^{\textrm{vN}}(D):={\cal A}_{1}({\cal I}_{1})\otimes{\cal
A}_{2}({\cal I}_{2})$, and for a general open region $\mathcal{O}$ we put
${\mathfrak{A}}^{\textrm{vN}}(\mathcal{O})=\bigvee_{D\subset\mathcal{O}}{\mathfrak{A}}^{\textrm{vN}}(D)$.
The net of von Neumann algebras $({\mathfrak{A}}^{\textrm{vN}},U)$, which we
call the chiral net, satisfies all the properties from Definition 2.1 except
for the regularity property 5. Therefore, we introduce the following weakly
dense subnet of $C^{*}$-algebras
$\mathcal{O}\to{\mathfrak{A}}(\mathcal{O}):=\\{\,A\in{\mathfrak{A}}^{\textrm{vN}}(\mathcal{O})\,|\,\lim_{x\to
0}\|\alpha_{x}(A)-A\|=0\,\\},$ (3.5)
and denote the corresponding quasilocal algebra by ${\mathfrak{A}}$. Then
$({\mathfrak{A}},U)$ is a local net of $C^{*}$-algebras in the sense of
Definition 2.1. We will call it the regular chiral net and refer to $({\cal
A}_{1},V_{1})$, $({\cal A}_{2},V_{2})$ as its chiral components. We note for
future reference that if ${\mathfrak{A}}$ acts irreducibly on $\mathcal{H}$,
then $U$ is automatically the canonical representation of translations of this
net (cf. Subsection 2.1). Another useful fact is the obvious inclusion
$\bar{\cal A}_{1}\otimes_{\mathrm{alg}}\bar{\cal A}_{2}\subset{\mathfrak{A}},$
(3.6)
where $\otimes_{\mathrm{alg}}$ is the algebraic tensor product.
Let $({\mathfrak{A}}^{\textrm{vN}},U)$ be a chiral net, whose chiral
components are $({\cal A}_{1},V_{1})$ and $({\cal A}_{2},V_{2})$. Let
$\pi_{1}$, $\pi_{2}$ be locally normal, covariant, positive energy
representations of the respective nets on $\mathbb{R}$. Then the chiral net of
$(\pi_{1}({\cal A}_{1}),V_{\pi_{1}})$, $(\pi_{2}({\cal A}_{2}),V_{\pi_{2}})$
is a covariant, positive energy representation of
$({\mathfrak{A}}^{\textrm{vN}},U)$, which will be denoted by
$(\pi({\mathfrak{A}}^{\textrm{vN}}),U_{\pi})$, $\pi=\pi_{1}\otimes\pi_{2}$ and
$\pi$ is called the product representation of $\pi_{1}$ and $\pi_{2}$. We note
that $(\pi({\mathfrak{A}}),U_{\pi})$ is contained in the regular subnet of
$(\pi({\mathfrak{A}}^{\textrm{vN}}),U_{\pi})$. For faithful $\pi$ these two
nets coincide, due to Proposition 2.3.3 (2) of [17]. It is easily seen that
$\pi$ is faithful (resp. irreducible), if $\pi_{1}$ and $\pi_{2}$ are faithful
(resp. irreducible). (Cf. Theorems 5.2 and 5.9 from Chapter IV of [49]).
### 3.2 Vacuum representations and asymptotic completeness
A regular chiral net $({\mathfrak{A}},U)$ is in a vacuum representation, with
the vacuum vector $\Omega\in\mathcal{H}$, if and only if its chiral components
$({\cal A}_{1},V_{1})$, $({\cal A}_{2},V_{2})$ are in vacuum representations
with the respective vacuum vectors $\Omega_{1}\in{\cal K}_{1}$,
$\Omega_{2}\in{\cal K}_{2}$ s.t. $\Omega=\Omega_{1}\otimes\Omega_{2}$. (Cf.
Proposition 3.5 below). In this subsection we show that any such regular
chiral net has a complete particle interpretation in terms of non-interacting
Wigner particles. These facts follow from our results in [25], but the
argument below is more direct.
We start from the observation that the asymptotic fields have a particularly
simple form in chiral theories:
###### Proposition 3.2.
Let $({\cal A}_{1},V_{1})$, $({\cal A}_{2},V_{2})$ be two local nets of von
Neumann algebras in vacuum representations, with the respective vacuum vectors
$\Omega_{1}$, $\Omega_{2}$. Then, for any $A_{1}\in\bar{\cal A}_{1}$,
$A_{2}\in\bar{\cal A}_{2}$
$\displaystyle\Phi_{+}^{\mathrm{out}/\mathrm{in}}(A_{1}\otimes A_{2})$
$\displaystyle=$ $\displaystyle A_{1}\otimes(\Omega_{2}|A_{2}\Omega_{2})I,$
(3.7) $\displaystyle\Phi_{-}^{\mathrm{out}/\mathrm{in}}(A_{1}\otimes A_{2})$
$\displaystyle=$ $\displaystyle(\Omega_{1}|A_{1}\Omega_{1})I\otimes A_{2}.$
(3.8)
Proof. We consider only $\Phi_{+}^{\mathrm{out}}$, as the remaining cases are
analogous. From the defining relation (2.6) and formula (3.4), we obtain
$\Phi_{+}^{\mathrm{out}}(A_{1}\otimes
A_{2})=\underset{T\to\infty}{\mathrm{s}\textrm{-}\lim}\;A_{1}\otimes\int
dt\,h_{T}(t)\beta_{\sqrt{2}t}^{(2)}(A_{2}).$ (3.9)
We set $A_{2}(h_{T}):=\int dt\,h_{T}(t)\beta_{\sqrt{2}t}^{(2)}(A_{2})$. This
sequence has the following properties:
$\displaystyle\lim_{T\to\infty}A_{2}(h_{T})\Omega_{2}$ $\displaystyle=$
$\displaystyle(\Omega_{2}|A_{2}\Omega_{2})\Omega_{2},$ (3.10)
$\displaystyle\lim_{T\to\infty}\|[A_{2}(h_{T}),A]\|$ $\displaystyle=$
$\displaystyle 0,\textrm{ for any }A\in\bar{\cal A}_{2}.$ (3.11)
The first identity above follows from the mean ergodic theorem and the fact
that $\Omega_{2}$ is the only vector invariant under the action of $V_{2}$.
The second equality is a consequence of the locality assumption from
Definition 3.1. Since $\bar{\cal A}_{2}$ acts irreducibly, any
$\Psi\in\mathcal{K}_{2}$ has the form $\Psi=A\Omega_{2}$ for some
$A\in\bar{\cal A}_{2}$ [45]. Thus we obtain from (3.10), (3.11)
$\underset{T\to\infty}{\mathrm{s}\textrm{-}\lim}\;A_{2}(h_{T})=(\Omega_{2}|A_{2}\Omega_{2})I,$
(3.12)
which completes the proof. $\Box$
Now we can easily prove the main result of this subsection:
###### Theorem 3.3.
Any regular chiral net $({\mathfrak{A}},U)$ in a vacuum representation is
asymptotically complete. More precisely:
$\displaystyle\mathcal{H}_{+}=\mathcal{K}_{1}\otimes{\mathbb{C}}\Omega_{2},$
(3.13)
$\displaystyle\mathcal{H}_{-}={\mathbb{C}}\Omega_{1}\otimes\mathcal{K}_{2},$
(3.14)
$\displaystyle\mathcal{H}_{+}\overset{\mathrm{out}}{\times}\mathcal{H}_{-}=\mathcal{H}_{+}\overset{\mathrm{in}}{\times}\mathcal{H}_{-}=\mathcal{H}.$
(3.15)
Moreover, any such theory is non-interacting.
###### Remark 3.4.
This result and Theorem 2.11 imply the convergence of the asymptotic
functional approximants $\\{\sigma_{\Psi}^{(T)}\\}_{T\in\mathbb{R}_{+}}$ for
all $\Psi\in\mathcal{H}$ of bounded energy in any regular chiral net in a
vacuum representation.
Proof. Using formula (2.5) and the cyclicity of the vacuum $\Omega$ under the
action of ${\mathfrak{A}}$, we obtain
$\mathcal{H}_{\pm}=[\,\Phi_{\pm}^{\mathrm{out}}(F)\Omega\,|\,F\in{\mathfrak{A}}\,],$
(3.16)
where $[\,\cdot\,]$ denotes the norm closure. Applying Proposition 3.2 and
exploiting the cyclicity of $\Omega_{1/2}$ under the action of $\bar{\cal
A}_{1/2}$, we obtain (3.13) and (3.14). The asymptotic completeness relation
(3.15) also follows from Proposition 3.2: For any $A_{1}\in\bar{\cal A}_{1}$,
$A_{2}\in\bar{\cal A}_{2}$
$\displaystyle\Phi_{+}^{\mathrm{out}}(A_{1}\otimes
I)\Phi_{-}^{\mathrm{out}}(I\otimes
A_{2})\Omega=\Phi_{+}^{\mathrm{in}}(A_{1}\otimes
I)\Phi_{-}^{\mathrm{in}}(I\otimes A_{2})\Omega=A_{1}\Omega_{1}\otimes
A_{2}\Omega_{2}.$ (3.17)
Exploiting once again cyclicity of $\Omega_{1/2}$, we obtain that scattering
states are dense in the Hilbert space.
Now let us show the lack of interaction: Let $\Psi_{\pm}\in\mathcal{H}_{\pm}$.
Then, by (3.13), (3.14) and the irreducibility of the action of $\bar{\cal
A}_{1/2}$ on $\mathcal{K}_{1/2}$, there exist $A_{1}\in\bar{\cal A}_{1}$,
$A_{2}\in\bar{\cal A}_{2}$ s.t. $\Psi_{+}=A_{1}\Omega_{1}\otimes\Omega_{2}$
and $\Psi_{-}=\Omega_{1}\otimes A_{2}\Omega_{2}$. Then
$\displaystyle\Psi_{+}\overset{\mathrm{out}}{\times}\Psi_{-}=\Phi_{+}^{\mathrm{out}}(A_{1}\otimes
I)\Phi_{-}^{\mathrm{out}}(I\otimes A_{2})\Omega=A_{1}\Omega_{1}\otimes
A_{2}\Omega_{2}$ $\displaystyle=\Phi_{+}^{\mathrm{in}}(A_{1}\otimes
I)\Phi_{-}^{\mathrm{in}}(I\otimes
A_{2})\Omega=\Psi_{+}\overset{\mathrm{in}}{\times}\Psi_{-}.$ (3.18)
Hence the scattering operator, defined in (2.10), equals the identity on
$\mathcal{H}$. $\Box$
### 3.3 Charged representations and infraparticles
It is the goal of this subsection to clarify the particle content of chiral
conformal field theories in charged representations. More detailed particle
properties of such theories, e.g. superselection of direction of motion, will
be studied in the next subsection.
Let us first note the following simple relation between the single-particle
subspaces of a regular chiral net and the invariant vectors of its chiral
components.
###### Proposition 3.5.
Let $({\cal A}_{1},V_{1})$, $({\cal A}_{2},V_{2})$ be local nets of von
Neumann algebras on $\mathbb{R}$. Then $V_{1}$ (resp. $V_{2}$) has a non-
trivial invariant vector, if and only if the single-particle subspace
$\mathcal{H}_{-}$ (resp. $\mathcal{H}_{+}$) of the corresponding regular
chiral net $({\mathfrak{A}},U)$ is non-trivial.
Proof. Suppose there exists a non-zero $\Omega_{1}\in\mathcal{K}_{1}$,
invariant under the action of $V_{1}$. Then, for any
$\Psi_{2}\in\mathcal{K}_{2}$,
$\displaystyle U(t,-t)(\Omega_{1}\otimes\Psi_{2})$ $\displaystyle=$
$\displaystyle\Omega_{1}\otimes\Psi_{2},$ (3.19)
for $t\in\mathbb{R}$. Hence the subspace $\mathcal{H}_{-}$ is non-trivial.
Similarly, the existence of a non-zero $\Omega_{2}\in\mathcal{K}_{2}$,
invariant under the action of $V_{2}$, implies the non-triviality of
$\mathcal{H}_{+}$.
Now suppose $\Psi\in\mathcal{H}_{-}$ and $V_{1}$ has no non-trivial, invariant
vectors. Then, by the mean ergodic theorem,
$\displaystyle\Psi=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}dt\,U(t,-t)\Psi=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}dt\,\big{(}V_{1}(\sqrt{2}t)\otimes
I\big{)}\Psi=0.$ (3.20)
Thus we established that $\mathcal{H}_{-}=\\{0\\}$. Similarly, the absence of
non-trivial, invariant vectors of $V_{2}$ implies that
$\mathcal{H}_{+}=\\{0\\}$. $\Box$
Let $({\mathfrak{A}},U)$ be a regular chiral net in a charged irreducible
(product) representation. That is ${\mathfrak{A}}$ acts irreducibly on a non-
trivial Hilbert space, which has the tensor product structure, by our
definition of chiral nets, and does not contain non-zero invariant vectors of
$U$. The particle structure of such theories is described by the following
theorem.
###### Theorem 3.6.
Let $({\mathfrak{A}},U)$ be a regular chiral net in a charged irreducible
(product) representation acting on a Hilbert space $\mathcal{H}$. Then:
1. (a)
$\mathcal{H}_{+}=\\{0\\}$ or $\mathcal{H}_{-}=\\{0\\}$ i.e. the theory does
not describe Wigner particles.
2. (b)
For any non-zero vector $\Psi\in P_{E}\mathcal{H}$, $E\geq 0$, all the limit
points of the net $\\{\sigma_{\Psi}^{(T)}\\}_{T\in\mathbb{R}_{+}}$, given by
(2.34), are different from zero.
Hence $({\mathfrak{A}},U)$ describes infraparticles.
Proof. Part (a) follows immediately from Proposition 3.5 and the absence of
non-zero invariant vectors of $U$ in $\mathcal{H}$. As for part (b), since
${\mathfrak{A}}$ acts irreducibly on $\mathcal{H}$, its chiral components
$({\cal A}_{1/2},V_{1/2})$ act irreducibly on their respective Hilbert spaces
${\cal K}_{1/2}$. We note that for any non-zero vector $\Psi\in
P_{E}\mathcal{H}$ we can find a sequence of vectors
$\\{\Psi_{n}\\}_{n\in\mathbb{N}}$ from ${\cal K}_{1}$ s.t. $\Psi_{1}\neq 0$
and
$(\Psi|(C\otimes I)\Psi)=\sum_{n\in\mathbb{N}}(\Psi_{n}|C\Psi_{n})$ (3.21)
for all $C\in B({\cal K}_{1})$. Moreover, we can assume without loss of
generality that ${\cal K}_{1}$ does not contain non-trivial invariant vectors
of $V_{1}$. Then we obtain from Lemma A.1 (b) the existence of a local
operator $A\in{\cal A}_{1}$ and $f\in S(\mathbb{R})$ s.t.
$\mathrm{supp}\,\tilde{f}\cap\mathbb{R}_{+}=\varnothing$, which satisfy
$A(f)\Psi_{1}\neq 0$. We note that any $B:=A(f)\otimes I$ is a non-zero
element of ${\mathfrak{A}}$ which is almost local and energy decreasing.
Consequently, $B^{*}B$ belongs to the algebra of detectors $\mathcal{C}$ of
the net $({\mathfrak{A}},U)$. We consider the corresponding asymptotic
functional approximants
$\displaystyle\sigma_{\Psi}^{(T)}(B^{*}B)$ $\displaystyle=$ $\displaystyle\int
dt\,h_{T}(t)\int
d\boldsymbol{x}\,(\Psi|\alpha_{(t,\boldsymbol{x})}(B^{*}B)\Psi)$ (3.22)
$\displaystyle=$ $\displaystyle\int dt\,h_{T}(t)\int
d\boldsymbol{x}\,(\Psi|\beta_{(\sqrt{2})^{-1}(t-\boldsymbol{x})}^{(1)}(A(f)^{*}A(f))\otimes
I)\Psi)$ $\displaystyle\geq$ $\displaystyle\int
d\boldsymbol{x}\,(\Psi_{1}|(\beta^{(1)}_{(\sqrt{2})^{-1}\boldsymbol{x}}(A(f)^{*}A(f)))\Psi_{1})\neq
0,$
where in the last step we made use of (3.21). As the last expression is
independent of $T$, all the limit points of
$\\{\sigma_{\Psi}^{(T)}\\}_{T\in\mathbb{R}_{+}}$ are different from zero.
$\Box$
### 3.4 Infraparticles with superselected direction of motion
In Theorem 3.6 above we have shown that any charged irreducible (product)
representation of a chiral conformal field theory contains infraparticles. In
this subsection we demonstrate that in a large class of examples these
infraparticles have superselected direction of motion in the sense of
Definition 2.12.
Let $({\cal A},V)$ be a local net of von Neumann algebras on $\mathbb{R}$,
acting on a Hilbert space $\mathcal{K}$. We assume that this net is in a
vacuum representation, with the vacuum vector $\Omega_{0}\in\mathcal{K}$. Let
$W$ be a unitary operator on $\mathcal{K}$ which implements a symmetry of this
net i.e.
$\displaystyle W{\cal A}({\cal I})W^{*}\subset{\cal A}({\cal I}),$ (3.23)
$\displaystyle WV(t)W^{*}=V(t),$ (3.24) $\displaystyle
W\Omega_{0}=\Omega_{0},$ (3.25)
for any open, bounded interval ${\cal I}\subset\mathbb{R}$ and any
$t\in\mathbb{R}$. We assume that $W$ gives rise to a non-trivial
representation of the group $\mathbb{Z}_{2}$ i.e. ${\hbox{\rm Ad}}W\neq{\rm
id}$ and $W^{2}=I$. We define the subspaces
$\displaystyle{\cal A}_{\mathrm{ev}}({\cal I})$ $\displaystyle=$
$\displaystyle\\{\,A\in{\cal A}({\cal I})\,|\,WAW^{*}=A\,\\},$ (3.26)
$\displaystyle{\cal A}_{\mathrm{odd}}({\cal I})$ $\displaystyle=$
$\displaystyle\\{\,A\in{\cal A}({\cal I})\,|\,WAW^{*}=-A\,\\}.$ (3.27)
Let ${\cal A}_{\mathrm{ev}}$ (resp. ${\cal A}_{\mathrm{odd}}$) be the norm-
closed linear span of all operators from some ${\cal A}_{\mathrm{ev}}({\cal
I})$ (resp. ${\cal A}_{\mathrm{odd}}({\cal I})$), ${\cal
I}\subset{\mathbb{R}}$. Clearly, $({\cal A}_{\mathrm{ev}},V)$ is again a local
net of von Neumann algebras on the real line acting on $\mathcal{K}$. We
introduce the subspaces $\mathcal{K}_{\mathrm{ev}}=[{\cal
A}_{\mathrm{ev}}\Omega_{0}]$, $\mathcal{K}_{\mathrm{odd}}=[{\cal
A}_{\mathrm{odd}}\Omega_{0}]$, where $[\,\cdot\,]$ denotes the closure, which
are invariant under the action of ${\cal A}_{\mathrm{ev}}$ and $V$, and
satisfy
$\mathcal{K}=\mathcal{K}_{\mathrm{ev}}\oplus\mathcal{K}_{\mathrm{odd}}$.
$\mathcal{K}_{\mathrm{odd}}$ gives rise to the representation
$\displaystyle\pi_{\mathrm{odd}}(A)$ $\displaystyle=$ $\displaystyle
A|_{\mathcal{K}_{\mathrm{odd}}},\quad A\in{\cal A}_{\mathrm{ev}},$ (3.28)
$\displaystyle V_{\mathrm{odd}}(t)$ $\displaystyle=$ $\displaystyle
V(t)|_{\mathcal{K}_{\mathrm{odd}}},\quad t\in\mathbb{R}.$ (3.29)
Its relevant properties are summarized in the following lemma, which we prove
in Appendix B.
###### Lemma 3.7.
$(\pi_{\mathrm{odd}},\mathcal{K}_{\mathrm{odd}})$ is a covariant, positive
energy representation of $({\cal A}_{\mathrm{ev}},V)$, in which the
translation automorphisms are implemented by $V_{\mathrm{odd}}$. Moreover:
1. (a)
$\pi_{\mathrm{odd}}$ is a locally normal, faithful and irreducible
representation of ${\cal A}_{\mathrm{ev}}$.
2. (b)
$V_{\mathrm{odd}}$ does not admit non-trivial invariant vectors.
We set $\hat{\cal A}:=\pi_{\mathrm{odd}}({\cal A}_{\mathrm{ev}})$,
$\hat{V}(t):=V_{\mathrm{odd}}(t)$. By the above lemma $(\hat{\cal A},\hat{V})$
is again a local net of von Neumann algebras on the real line. We define its
representation on $\mathcal{K}_{\mathrm{ev}}$
$\displaystyle\pi_{\mathrm{ev}}(\hat{A})$ $\displaystyle=$
$\displaystyle\pi_{\mathrm{odd}}^{-1}(\hat{A})|_{\mathcal{K}_{\mathrm{ev}}},\quad\hat{A}\in\hat{\cal
A},$ (3.30) $\displaystyle V_{\mathrm{ev}}(t)$ $\displaystyle=$ $\displaystyle
V(t)|_{\mathcal{K}_{\mathrm{ev}}},\quad t\in\mathbb{R}$ (3.31)
and state the following fact, whose proof is given in Appendix B.
###### Lemma 3.8.
$(\pi_{\mathrm{ev}},\mathcal{K}_{\mathrm{ev}})$ is a covariant, positive
energy representation of $(\hat{\cal A},\hat{V})$, in which the translation
automorphisms are implemented by $V_{\mathrm{ev}}$. Moreover
1. (a)
$\pi_{\mathrm{ev}}$ is a locally normal, faithful and irreducible
representation of $\hat{\cal A}$.
2. (b)
$V_{\mathrm{ev}}$ admits a unique (up to a phase) invariant vector, which is
cyclic for any $\pi_{\mathrm{ev}}(\hat{\cal A}({\cal I}))$.
We conclude that $(\pi_{\mathrm{ev}}(\hat{\cal A}),V_{\mathrm{ev}})$ is a
local net of von Neumann algebras in a vacuum representation with the vacuum
vector $\Omega_{0}\in\mathcal{K}_{\mathrm{ev}}$.
We remark that the above abstract construction can be performed in a number of
concrete cases. If a Möbius covariant net ${\cal I}\to{\cal A}({\cal I})$ on
$S^{1}$, in a vacuum representation, admits an automorphism777An automorphism
$\gamma$ of a net ${\cal A}$ is an automorphism of the quasilocal algebra
${\cal A}$ which preserves each local algebra ${\cal A}({\cal I})$. $\gamma$
of order 2 which preserves the vacuum state, then one can define $W$ by
$WA\Omega_{0}=\gamma(A)\Omega_{0},\quad A\in{\cal A}({\cal I}).$ (3.32)
This does not depend on the choice of the interval ${\cal I}$ and defines a
unitary operator thanks to the invariance of the vacuum state. This $W$
automatically commutes with the action of the Möbius group (in particular with
the action of translations) as a consequence of the Bisognano-Wichmann
property [31]. Thus, upon restriction to the real line, we obtain a local net
equipped with a unitary $W$ which satisfies (3.23)-(3.25). Non-trivial
automorphisms $\gamma$ appear, in particular, in the $U(1)$-current net
($\gamma:J(z)\to-J(z)$) [15], in loop group nets of a compact group $G$ with a
${\mathbb{Z}}_{2}$-subgroup in $G$ [51] and in the tensor product net ${\cal
A}\otimes{\cal A}$ for an arbitrary Möbius covariant net ${\cal A}$, where
$\gamma$ is the flip symmetry.
Coming back to the abstract setting, we introduce the class of two-dimensional
theories, we are interested in: Let $({\cal A}_{1},V_{1})$, $({\cal
A}_{2},V_{2})$ be two local nets of von Neumann algebras on $\mathbb{R}$, in
vacuum representations, acting on Hilbert spaces $\mathcal{K}_{1}$,
$\mathcal{K}_{2}$. We denote the respective vacuum vectors by $\Omega_{1}$,
$\Omega_{2}$ and introduce the corresponding regular chiral net
$({\mathfrak{A}},U)$. We assume the existence of unitaries $W_{1}$, $W_{2}$,
which give rise to non-trivial representations of $\mathbb{Z}_{2}$ and
implement symmetries of the respective nets on $\mathbb{R}$ as defined in
(3.23)-(3.25). By the construction described above we obtain the nets
$(\hat{\cal A}_{1},\hat{V}_{1})$, $(\hat{\cal A}_{2},\hat{V}_{2})$, acting on
$\mathcal{K}_{1,\mathrm{odd}}$, $\mathcal{K}_{2,\mathrm{odd}}$. We denote by
$(\hat{\mathfrak{A}}^{\textrm{vN}},\hat{U})$ the corresponding chiral net
acting on
$\hat{\mathcal{H}}=\mathcal{K}_{1,\mathrm{odd}}\otimes\mathcal{K}_{2,\mathrm{odd}}$
and by $(\hat{\mathfrak{A}},\hat{U})$ its regular subnet. Let us summarize its
properties.
###### Proposition 3.9.
The regular chiral net $(\hat{\mathfrak{A}},\hat{U})$, whose chiral components
are $(\hat{\cal A}_{1},\hat{V}_{1})$, $(\hat{\cal A}_{2},\hat{V}_{2})$, has
the following properties:
1. (a)
$\hat{\mathfrak{A}}$ acts irreducibly on $\hat{\mathcal{H}}$.
2. (b)
$(\hat{\mathfrak{A}},\hat{U})$ does not admit Wigner particles
($\hat{\mathcal{H}}_{\pm}=\\{0\\}$), but all the asymptotic functionals of the
form $\\{\,\psi_{\Psi}^{\mathrm{out}}\,|\,\Psi\neq 0,\,\Psi\in
P_{E}\hat{\mathcal{H}},\,E\geq 0\,\\}$ are non-zero.
3. (c)
$\pi_{{\rm{R}}}=\iota_{1}\otimes\pi_{2,\mathrm{ev}}$ and
$\pi_{{\rm{L}}}=\pi_{1,\mathrm{ev}}\otimes\iota_{2}$ are irreducible,
faithful, covariant representations of $(\hat{\mathfrak{A}},\hat{U})$, acting
on
$\mathcal{H}_{\pi_{{\rm{R}}}}:=\mathcal{K}_{1,\mathrm{odd}}\otimes\mathcal{K}_{2,\mathrm{ev}}$
and
$\mathcal{H}_{\pi_{{\rm{L}}}}:=\mathcal{K}_{1,\mathrm{ev}}\otimes\mathcal{K}_{2,\mathrm{odd}}$,
respectively. The respective (canonical) unitary representations of
translations are given by
$U_{\pi_{\rm{R}}}(x):=U(x)|_{\mathcal{H}_{\pi_{{\rm{R}}}}}$ and
$U_{\pi_{\rm{L}}}(x):=U(x)|_{\mathcal{H}_{\pi_{{\rm{L}}}}}$.
4. (d)
$\mathcal{H}_{\pi_{{\rm{R}}},-}=\\{0\\}$ and
$\mathcal{H}_{\pi_{{\rm{R}}},+}\neq\\{0\\}$ while
$\mathcal{H}_{\pi_{{\rm{L}}},-}\neq\\{0\\}$ and
$\mathcal{H}_{\pi_{{\rm{L}}},+}=\\{0\\}$. Consequently, $\pi_{{\rm{R}}}$ is
not unitarily equivalent to $\pi_{{\rm{L}}}$.
In part (c) $\iota_{1/2}$ are the defining representations of $\hat{\cal
A}_{1/2}$. Representations $\pi_{1/2,\mathrm{ev}}$ are defined as in (3.30),
(3.31).
Proof. Part (a) follows from the irreducibility of $\pi_{1/2,\mathrm{odd}}$,
shown in Lemma 3.7. As for part (b), we obtain from Lemma 3.7 (b) and
Proposition 3.5 that $\hat{\mathcal{H}}_{\pm}=\\{0\\}$. On the other hand,
Theorem 3.6 ensures that the relevant asymptotic functionals are non-zero.
Irreducibility and faithfulness of $\pi_{{\rm{R}}/{\rm{L}}}$ in part (c)
follow from Lemma 3.8 (a) and Lemma 3.7 (a). Proceeding to part (d), we note
that, by faithfulness of $\pi_{{\rm{R}}}$, the net
$(\pi_{{\rm{R}}}(\hat{\mathfrak{A}}),U_{\pi_{{\rm{R}}}})$ coincides with the
regular chiral subnet of
$(\pi_{{\rm{R}}}(\hat{\mathfrak{A}}^{\textrm{vN}}),U_{\pi_{{\rm{R}}}})$, whose
chiral components are $(\hat{\cal A}_{1},\hat{V}_{1})$ and
$(\pi_{2,\mathrm{ev}}(\hat{\cal A}_{2}),V_{2,\mathrm{ev}})$. From Lemma 3.7
(b), Lemma 3.8 (b) and Proposition 3.5 we obtain that
$\mathcal{H}_{\pi_{{\rm{R}}},-}=\\{0\\}$ and
$\mathcal{H}_{\pi_{{\rm{R}}},+}\neq\\{0\\}$. An analogous reasoning, applied
to $\pi_{{\rm{L}}}$, shows that $\mathcal{H}_{\pi_{{\rm{L}}},-}\neq\\{0\\}$
and $\mathcal{H}_{\pi_{{\rm{L}}},+}=\\{0\\}$. Hence, due to relation (2.3),
the two nets are not unitarily equivalent. $\Box$
In view of part (b) of the above proposition, the theory
$(\hat{\mathfrak{A}},\hat{U})$ describes infraparticles. In the following
theorem, which is our main result, we show that these infraparticles have
superselected direction of motion, in the sense of Definition 2.12.
###### Theorem 3.10.
Consider the regular chiral net $(\hat{\mathfrak{A}},\hat{U})$, constructed
above. Let $\psi\in\\{\,\psi_{\Psi}^{\mathrm{out}}\,|\,\Psi\neq 0,\,\Psi\in
P_{E}\hat{\mathcal{H}},\,E\geq 0\,\\}$ and let $\pi_{\psi}$ be its GNS
representation. Then $\pi_{\psi}$ is a type I representation with atomic
center. It contains both right-moving and left-moving irreducible
subrepresentations which are unitarily equivalent to $\pi_{{\rm{R}}}$ and
$\pi_{{\rm{L}}}$, respectively. Hence the theory describes infraparticles with
superselected direction of motion.
###### Remark 3.11.
Let us consider the regular chiral net $({\mathfrak{A}},U)$ in the vacuum
representation. Then, similarly as in the theorem above, the GNS
representation $\pi_{\psi}$ induced by any particle weight
$\psi\in\\{\,\psi_{\Psi}^{\mathrm{out}}\,|\,\Psi\notin{\mathbb{C}}\Omega,\Psi\in
P_{E}\mathcal{H},E\geq 0\,\\}$ is of type I with atomic center. However, any
non-trivial irreducible subrepresentation of $\pi_{\psi}$ is unitarily
equivalent to the defining vacuum representation i.e. $\psi$ is neutral. These
facts are easily verified by modifying the proof below.
Proof. Let us first consider the regular chiral net $({\mathfrak{A}},U)$
acting on $\mathcal{H}$. By Theorem 3.3, ${\cal K}_{+}:={\cal
K}_{1,\mathrm{odd}}\otimes{\mathbb{C}}\Omega_{2}\subset\mathcal{H}_{+}$ and
${\cal K}_{-}:={\mathbb{C}}\Omega_{1}\otimes{\cal
K}_{2,\mathrm{odd}}\subset\mathcal{H}_{-}$. Any vector $\Psi\in P_{E}({\cal
K}_{+}\overset{\mathrm{out}}{\times}{\cal K}_{-})$, $E\geq 0$, gives rise to
functionals $\rho_{\Psi,\pm}$, defined by (2.11), (2.12). They have the form
$\rho_{\pm,\Psi}(\,\cdot\,)=\sum_{n\in\mathbb{N}}(\Psi_{\pm,n}|\,\cdot\,\Psi_{\pm,n}),$
(3.33)
where $\Psi_{\pm,n}\in P_{E}{\cal K}_{\pm}$. (Cf. formula (2.13) and the
subsequent discussion). Since $\Psi\neq 0$, we can assume that $\Psi_{+,1}\neq
0$ and $\Psi_{-,1}\neq 0$. We also note for future reference that ${\cal
K}_{+}\subset\mathcal{H}_{\pi_{{\rm{R}}}}$ and ${\cal
K}_{-}\subset\mathcal{H}_{\pi_{{\rm{L}}}}$.
Let us now proceed to the net $(\hat{\mathfrak{A}},\hat{U})$, acting on
$\hat{\mathcal{H}}={\cal K}_{+}\overset{\mathrm{out}}{\times}{\cal
K}_{-}\subset\mathcal{H}$, and let $\hat{\mathcal{L}}$ be the left ideal of
$\hat{\mathfrak{A}}$, given by definition (2.16). For any
${\hat{L}}\in\hat{\mathcal{L}}$ we define
$L=(\pi_{1,\mathrm{odd}}^{-1}\otimes\pi_{2,\mathrm{odd}}^{-1})({\hat{L}})\in\mathcal{L},$
(3.34)
where $\mathcal{L}$ is the corresponding left ideal of ${\mathfrak{A}}$. We
note that such $L$ leaves the subspaces $\mathcal{H}_{\pi_{{\rm{R}}}}$ and
$\mathcal{H}_{\pi_{{\rm{L}}}}$ invariant. Exploiting Theorem 2.11 and formula
(2.13), we obtain
$\displaystyle\psi_{\Psi}^{\mathrm{out}}({\hat{L}}_{1},{\hat{L}}_{2})=\sum_{n\in\mathbb{N}}\int
d\boldsymbol{x}\,\\{(\Psi_{+,n}|(L_{1}^{*}L_{2})(\boldsymbol{x})\Psi_{+,n})+(\Psi_{-,n}|(L_{1}^{*}L_{2})(\boldsymbol{x})\Psi_{-,n})\\}.$
(3.35)
It follows from Theorem 2.9 that for any $L$ given by (3.34) the Fourier
transforms
$\displaystyle L\tilde{\Psi}_{+,n}(\boldsymbol{p})$ $\displaystyle:=$
$\displaystyle(2\pi)^{-1/2}\int
d\boldsymbol{x}\,e^{-i\boldsymbol{p}\boldsymbol{x}}LU_{\pi_{\rm{R}}}(\boldsymbol{x})^{*}\Psi_{+,n},$
(3.36) $\displaystyle L\tilde{\Psi}_{-,n}(\boldsymbol{p})$ $\displaystyle:=$
$\displaystyle(2\pi)^{-1/2}\int
d\boldsymbol{x}\,e^{i\boldsymbol{p}\boldsymbol{x}}LU_{\pi_{\rm{L}}}(\boldsymbol{x})^{*}\Psi_{-,n}$
(3.37)
belong to $\mathcal{H}_{\pi_{{\rm{R}}}}\otimes
L^{2}(\mathbb{R}_{+},d\boldsymbol{p})$ and
$\mathcal{H}_{\pi_{{\rm{L}}}}\otimes L^{2}(\mathbb{R}_{+},d\boldsymbol{p})$
respectively. Since $\pi_{{\rm{R}}}(\hat{\mathfrak{A}})$ acts irreducibly on
$\mathcal{H}_{\pi_{{\rm{R}}}}$ and $U_{\pi_{\rm{R}}}$ does not have non-zero
invariant vectors, we obtain from Lemma A.1 (a) the existence of
${\hat{L}}_{+}\in\hat{\mathcal{L}}$ s.t. $L_{+}\Psi_{+,1}\neq 0$. Since
$L_{+}U_{\pi_{\rm{R}}}(\boldsymbol{x})^{*}\Psi_{+,1}$ is a continuous function
of $\boldsymbol{x}$, it is nonzero as a square-integrable function, hence
$\\{L_{+}\tilde{\Psi}_{+,1}(\boldsymbol{p})\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\neq
0$. Analogously, we can find ${\hat{L}}_{-}\in\hat{\mathcal{L}}$ s.t.
$\\{L_{-}\tilde{\Psi}_{-,1}(\boldsymbol{p})\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\neq
0$. For future reference, we note the equalities
$\displaystyle\alpha_{x}(L)\tilde{\Psi}_{+,n}(\boldsymbol{p})$
$\displaystyle=$ $\displaystyle
e^{-i(\boldsymbol{p},\boldsymbol{p})x}U_{\pi_{\rm{R}}}(x)L\tilde{\Psi}_{+,n}(\boldsymbol{p}),$
(3.38) $\displaystyle\alpha_{x}(L)\tilde{\Psi}_{-,n}(\boldsymbol{p})$
$\displaystyle=$ $\displaystyle
e^{-i(\boldsymbol{p},-\boldsymbol{p})x}U_{\pi_{\rm{L}}}(x)L\tilde{\Psi}_{-,n}(\boldsymbol{p}),$
(3.39)
which hold in the sense of $\mathcal{H}_{\pi_{{\rm{R}}}}\otimes
L^{2}(\mathbb{R}_{+},d\boldsymbol{p})$ and
$\mathcal{H}_{\pi_{{\rm{L}}}}\otimes L^{2}(\mathbb{R}_{+},d\boldsymbol{p})$
respectively. These relations are easily verified for such $\Psi_{\pm,n}\in
P_{E}{\cal K}_{\pm}$ that $\mathbb{R}\ni\boldsymbol{x}\to
LU_{\pi_{{\rm{R}}/{\rm{L}}}}(\boldsymbol{x})^{*}\Psi_{\pm,n}$ decay rapidly in
norm as $|\boldsymbol{x}|\to\infty$, since in this case the Fourier transform
is pointwise defined. The general case follows from the fact that such vectors
form a dense subspace in $P_{E}{\cal K}_{\pm}$ (cf. formula (3.53) below) and
that the maps $P_{E}{\cal
K}_{\pm}\ni\Psi\to\\{L\tilde{\Psi}(\boldsymbol{p})\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\in\mathcal{H}_{\pi_{{\rm{R}}/{\rm{L}}}}\otimes
L^{2}(\mathbb{R}_{+},d\boldsymbol{p})$ are norm-continuous. This latter fact
is a consequence of Theorem 2.9 and the (Hilbert space valued) Plancherel
theorem.
After this preparation we study the structure of the GNS representation
induced by $\psi$. Let us first consider the following auxiliary
representation of $(\hat{\mathfrak{A}},\hat{U})$
$\pi_{1}(\,\cdot\,):=\bigoplus_{n\in\mathbb{N}}\big{(}\\{\pi_{{\rm{R}}}(\,\cdot\,)\otimes
I\\}\oplus\\{\pi_{{\rm{L}}}(\,\cdot\,)\otimes I\\}\big{)},$ (3.40)
acting on
$\mathcal{H}_{\pi_{1}}:=\bigoplus_{n\in\mathbb{N}}\big{(}\\{\mathcal{H}_{\pi_{{\rm{R}}}}\otimes
L^{2}(\mathbb{R}_{+},d\boldsymbol{p})\\}\oplus\\{\mathcal{H}_{\pi_{{\rm{L}}}}\otimes
L^{2}(\mathbb{R}_{+},d\boldsymbol{p})\\}\big{)}$. From definition (3.40) and
relation (2.22) we conclude that $\pi_{1}$ and its subrepresentations are of
type I with atomic center. Moreover, $\pi_{1}$ is covariant and it is easily
seen that the canonical representation of translations is given by
$U^{\mathrm{can}}_{\pi_{1}}(x)=\bigoplus_{n\in\mathbb{N}}\big{(}\big{\\{}U_{\pi_{\rm{R}}}(x)\otimes
I\big{\\}}\oplus\big{\\{}U_{\pi_{\rm{L}}}(x)\otimes I\big{\\}}).$ (3.41)
We note that $\pi_{\psi}$ is unitarily equivalent to a subrepresentation of
$\pi_{1}$. In fact, the map
$W_{1}:\mathcal{H}_{\pi_{\psi}}\to\mathcal{H}_{\pi_{1}}$, given by
$W_{1}:|{\hat{L}}\rangle\to\bigoplus_{n\in\mathbb{N}}\big{(}\\{\,L\tilde{\Psi}_{+,n}(\boldsymbol{p})\,\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\oplus\\{\,L\tilde{\Psi}_{-,n}(\boldsymbol{p})\,\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\big{)},$
(3.42)
intertwines the two representations and is an isometry by formula (3.35) and
the (Hilbert space valued) Plancherel theorem. It is easily checked that the
canonical representation of translations $U_{\pi_{\psi}}^{\mathrm{can}}$ in
the representation $\pi_{\psi}$ is given by the relation
$W_{1}U^{\mathrm{can}}_{\pi_{\psi}}(x)=U^{\mathrm{can}}_{\pi_{1}}(x)W_{1}.$
(3.43)
Recalling that
$U^{\mathrm{char}}_{\pi_{\psi}}(x)=U^{\mathrm{can}}_{\pi_{\psi}}(x)U_{\pi_{\psi}}^{-1}(x)$,
where $U_{\pi_{\psi}}$ is given by (2.20), we obtain
$\displaystyle W_{1}U^{\mathrm{char}}_{\pi_{\psi}}(x)|{\hat{L}}\rangle$
$\displaystyle=$ $\displaystyle
W_{1}U^{\mathrm{can}}_{\pi_{\psi}}(x)|\alpha_{-x}({\hat{L}})\rangle=U^{\mathrm{can}}_{\pi_{1}}(x)W_{1}|\alpha_{-x}({\hat{L}})\rangle$
(3.44) $\displaystyle=$
$\displaystyle\bigoplus_{n\in\mathbb{N}}\big{(}\\{\,U_{\pi_{\rm{R}}}(x)\alpha_{-x}(L)\tilde{\Psi}_{+,n}(\boldsymbol{p})\,\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\oplus\\{\,U_{\pi_{\rm{L}}}(x)\alpha_{-x}(L)\tilde{\Psi}_{-,n}(\boldsymbol{p})\,\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\big{)}$
$\displaystyle=$ $\displaystyle\bigoplus_{n\in\mathbb{N}}\big{(}\\{\,I\otimes
e^{i(\boldsymbol{p},\boldsymbol{p})x}\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\oplus\\{I\otimes
e^{i(\boldsymbol{p},-\boldsymbol{p})x}\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\big{)}W_{1}|{\hat{L}}\rangle,$
where in the last step we made use of relations (3.38), (3.39). Now let
$\boldsymbol{Q}$ be the generator of space translations of
$U^{\mathrm{char}}_{\pi_{\psi}}$ and let $\mathcal{H}_{\pi_{\psi},\mathrm{R}}$
(resp. $\mathcal{H}_{\pi_{\psi},\mathrm{L}}$) be its spectral subspace
corresponding to the interval $[0,\infty)$ (resp. $(-\infty,0)$). Then, by
formula (3.44),
$\displaystyle
W_{1}\mathcal{H}_{\pi_{\psi},\mathrm{R}}=P_{{\rm{R}}}W_{1}\mathcal{H}_{\pi_{\psi}},$
(3.45) $\displaystyle
W_{1}\mathcal{H}_{\pi_{\psi},\mathrm{L}}=P_{{\rm{L}}}W_{1}\mathcal{H}_{\pi_{\psi}},$
(3.46)
where $P_{{\rm{R}}/{\rm{L}}}$ are the projections on the subspaces
$\bigoplus_{n\in\mathbb{N}}\\{\mathcal{H}_{\pi_{{\rm{R}}/{\rm{L}}}}\otimes
L^{2}(\mathbb{R}_{+},d\boldsymbol{p})\\}$ in $\mathcal{H}_{\pi_{1}}$. From
definition (3.42) and the remarks after formula (3.37) we conclude that
$P_{{\rm{R}}}W_{1}\mathcal{H}_{\pi_{\psi}}\neq\\{0\\}$ and
$P_{{\rm{L}}}W_{1}\mathcal{H}_{\pi_{\psi}}\neq\\{0\\}$. Consequently
$\pi_{\psi}$ has both right-moving and left-moving irreducible
subrepresentations.
Let $\pi$ be an irreducible subrepresentation of $\pi_{\psi}$, acting on a
non-trivial subspace ${\cal K}\subset\mathcal{H}_{\pi_{\psi},\mathrm{R}}$
(i.e. a right-moving subrepresentation). Then $W_{1}\pi(\,\cdot\,)W_{1}^{*}$
is an irreducible subrepresentation of $\pi_{\rm{R}}(\,\cdot\,)\otimes I$
acting on
$\mathcal{H}_{\pi_{{\rm{R}}}}\otimes\big{(}\bigoplus_{n\in\mathbb{N}}L^{2}(\mathbb{R}_{+},d\boldsymbol{p})\big{)}$.
By irreducibility, we conclude that $\pi$ is unitarily equivalent to
$(\pi_{{\rm{R}}}(\,\cdot\,)\otimes
I)|_{\mathcal{H}_{\pi_{{\rm{R}}}}\otimes{\mathbb{C}}e}$ for some non-zero
$e\in\bigoplus_{n\in\mathbb{N}}L^{2}(\mathbb{R}_{+},d\boldsymbol{p})$. This
latter representation can be identified with $\pi_{{\rm{R}}}$. An analogous
argument shows that any left-moving irreducible subrepresentation of
$\pi_{\psi}$ is unitarily equivalent to $\pi_{{\rm{L}}}$. Hence, by
Proposition 3.9 (d), $(\hat{\mathfrak{A}},\hat{U})$ describes infraparticles
with superselected direction of motion. $\Box$
Let us assume for a moment that $\hat{\mathcal{H}}$ is separable. Then we
obtain from the above theorem and formula (2.29) that the GNS representation
of any particle weight $\psi^{\mathrm{out}}_{\Psi}$ of the net
$(\hat{\mathfrak{A}},\hat{U})$, where $\Psi\neq 0$ is a vector of bounded
energy, has the form
$\displaystyle\pi_{\psi^{\mathrm{out}}_{\Psi}}\simeq\int^{\oplus}_{Z_{{\rm{R}}}}d\mu_{{\rm{R}}}(z)\,\pi_{{\rm{R}},z}\oplus\int^{\oplus}_{Z_{{\rm{L}}}}d\mu_{{\rm{L}}}(z)\,\pi_{{\rm{L}},z}.$
(3.47)
Here $(Z_{{\rm{R}}/{\rm{L}}},d\mu_{{\rm{R}}/{\rm{L}}})$ are some Borel spaces
and $\pi_{{\rm{R}}/{\rm{L}},z}=\pi_{{\rm{R}}/{\rm{L}}}$ for all $z\in
Z_{{\rm{R}}/{\rm{L}}}$. A decomposition of $\psi^{\mathrm{out}}_{\Psi}$ into
pure particle weights, which induce the irreducible representations appearing
in the decomposition of $\pi_{\psi^{\mathrm{out}}_{\Psi}}$, was obtained by
Porrmann in [41, 42] (cf. formula (1.3) above). However, to apply Porrmann’s
abstract argument, one has to restrict attention to countable (resp.
separable) subsets of all the relevant objects and it is not guaranteed that
the resulting (restricted) pure particle weights extend to the original
domains. It is therefore worth pointing out that the theory
$(\hat{\mathfrak{A}},\hat{U})$ admits a large class of particle weights, whose
decomposition can be performed in the original framework. To our knowledge
this is the first such decomposition in the presence of infraparticles. (See
however [35] for some partial results on the Schroer model). These particle
weights belong to the set
$\\{\,\psi^{\mathrm{out}}_{\Psi}\,|\,\Psi\in\mathcal{D}\,\\}$, where
$\mathcal{D}\subset\hat{\mathcal{H}}$ is a dense domain spanned by vectors of
the form
$\Psi=F_{1}\Omega_{1}\otimes F_{2}\Omega_{2},$ (3.48)
where $F_{1}\in{\cal A}_{1,\mathrm{odd}},F_{2}\in{\cal A}_{2,\mathrm{odd}}$
are s.t. $F_{1}\otimes I,I\otimes F_{2}\in{\mathfrak{A}}$ are almost local and
have compact energy-momentum transfer (see formula (2.15)). The proof of the
following proposition exploits some ideas from [26].
###### Proposition 3.12.
Consider the regular chiral net $(\hat{\mathfrak{A}},\hat{U})$ constructed
above. Denote by $\hat{\mathcal{L}}$ its left ideal, given by definition
(2.16). Then, for any non-zero vector $\Psi\in\mathcal{D}$, there exist
continuous fields of pure particle weights
$\Delta_{{\rm{R}},n}\ni\boldsymbol{p}\to\psi_{{\rm{R}},n,\boldsymbol{p}}(\,\cdot\,,\,\cdot\,)$
and
$\Delta_{{\rm{L}},m}\ni\boldsymbol{p}\to\psi_{{\rm{L}},m,\boldsymbol{p}}(\,\cdot\,,\,\cdot\,)$
s.t. for any ${\hat{L}}_{1},{\hat{L}}_{2}\in\hat{\mathcal{L}}$
$\displaystyle\psi_{\Psi}^{\mathrm{out}}({\hat{L}}_{1},{\hat{L}}_{2})=\sum_{n\in
C_{{\rm{R}}}}\int_{\Delta_{{\rm{R}},n}}d\boldsymbol{p}\,\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}}_{1},{\hat{L}}_{2})+\sum_{m\in
C_{{\rm{L}}}}\int_{\Delta_{{\rm{L}},m}}d\boldsymbol{p}\,\psi_{{\rm{L}},m,\boldsymbol{p}}({\hat{L}}_{1},{\hat{L}}_{2}),$
(3.49)
where $C_{{\rm{R}}},C_{{\rm{L}}}\subset\mathbb{N}$ are non-empty finite
subsets and $\Delta_{{\rm{R}},n},\Delta_{{\rm{L}},m}\subset\mathbb{R}_{+}$ are
non-empty, open subsets for any $n\in C_{{\rm{R}}}$, $m\in C_{{\rm{L}}}$.
Moreover:
1. (a)
The characteristic energy-momentum vectors of the weights
$\psi_{{\rm{R}},n,\boldsymbol{p}}$ (resp. $\psi_{{\rm{L}},m,\boldsymbol{p}}$)
are equal to $q_{{\rm{R}},n,\boldsymbol{p}}=(\boldsymbol{p},\boldsymbol{p})$
(resp. $q_{{\rm{L}},m,\boldsymbol{p}}=(\boldsymbol{p},-\boldsymbol{p})$).
2. (b)
The GNS representation induced by any $\psi_{{\rm{R}},n,\boldsymbol{p}}$
(resp. $\psi_{{\rm{L}},m,\boldsymbol{p}}$) is unitarily equivalent to
$\pi_{{\rm{R}}}$ (resp. $\pi_{{\rm{L}}}$).
The representations $\pi_{{\rm{R}}/{\rm{L}}}$ appeared in Proposition 3.9.
###### Remark 3.13.
Parts (b) and (d) of Proposition 3.9 show that spectral properties of the
energy-momentum operators in the representations induced by the pure particle
weights $\psi_{{\rm{R}}/{\rm{L}},n,\boldsymbol{p}}$ are different from those
in the original representation: In the case of $U_{\pi_{{\rm{R}}}}$ the right
branch of the lightcone contains the singularities characteristic for Wigner
particles, while in the left branch such singularities are absent. (For
$U_{\pi_{{\rm{L}}}}$ the opposite situation occurs). For infraparticles in
physical spacetime a more radical version of this phenomenon may occur: There
one expects isolated singularities at the characteristic energy-momentum
values of the respective pure particle weights. (Cf. Section 2 (iii) of [16]).
Proof. Any vector $\Psi\in\mathcal{D}$ has the form
$\Psi=\sum_{k,l}c_{k,l}F_{{\rm{R}},k}\Omega_{1}\otimes
F_{{\rm{L}},l}\Omega_{2},$ (3.50)
where the sum is finite and $F_{{\rm{R}},k}$, $F_{{\rm{L}},l}$ have properties
specified below formula (3.48). Applying the Gram-Schmidt procedure, we can
ensure that the systems of vectors $\\{F_{{\rm{R}},k}\Omega_{1}\\}_{k=0}^{M}$,
$\\{F_{{\rm{L}},l}\Omega_{2}\\}_{l=0}^{N}$ are orthonormal. Since
$\hat{\mathcal{H}}=\mathcal{K}_{1,\mathrm{odd}}\otimes\mathcal{K}_{2,\mathrm{odd}}\subset\mathcal{K}_{1}\otimes\mathcal{K}_{2}=\mathcal{H}$,
we can write
$\Psi=\sum_{k,l}c_{k,l}\Phi^{\mathrm{out}}_{+}(F_{{\rm{R}},k}\otimes
I)\Phi^{\mathrm{out}}_{-}(I\otimes F_{{\rm{L}},l})\Omega,$ (3.51)
where we made use of Proposition 3.2 applied to the net $({\mathfrak{A}},U)$.
For any ${\hat{L}}\in\hat{\mathcal{L}}$ we define
$L=(\pi_{1,\mathrm{odd}}^{-1}\otimes\pi_{2,\mathrm{odd}}^{-1})({\hat{L}})\in\mathcal{L}$,
where $\mathcal{L}$ is the left ideal of ${\mathfrak{A}}$, given by definition
(2.16). In view of Theorem 2.11, we get
$\displaystyle\psi_{\Psi}^{\mathrm{out}}({\hat{L}}_{1},{\hat{L}}_{2})$
$\displaystyle=$ $\displaystyle\sum_{n\in C_{{\rm{R}}}}\int
d\boldsymbol{x}\,((G_{{\rm{R}},n}\otimes
I)\Omega|(L_{1}^{*}L_{2})(\boldsymbol{x})(G_{{\rm{R}},n}\otimes I)\Omega)$
(3.52) $\displaystyle+$ $\displaystyle\sum_{m\in C_{{\rm{L}}}}\int
d\boldsymbol{x}\,((I\otimes{G}_{{\rm{L}},m})\Omega|(L_{1}^{*}L_{2})(\boldsymbol{x})(I\otimes{G}_{{\rm{L}},m})\Omega),$
where $G_{{\rm{R}},n}=\sum_{k}c_{k,n}F_{{\rm{R}},k}$,
$G_{{\rm{L}},m}=\sum_{l}c_{m,l}F_{{\rm{L}},l}$ and the sets $C_{{\rm{R}}}$ and
$C_{{\rm{L}}}$ are chosen so that $\Psi_{{\rm{R}},n}:=(G_{{\rm{R}},n}\otimes
I)\Omega\neq 0$ and $\Psi_{{\rm{L}},m}:=(I\otimes G_{{\rm{L}},m})\Omega\neq 0$
for $n\in C_{{\rm{R}}}$ and $m\in C_{{\rm{L}}}$. We note that both sets are
non-empty, if $\Psi\neq 0$. (Cf. formula (2.13) and the subsequent remarks).
Let us consider the first sum in (3.52) above: Any $L\in\mathcal{L}$ is a
finite linear combination of operators of the form $AB$, where
$A,B\in{\mathfrak{A}}$ and $B$ is almost local and energy decreasing. Since we
assumed that $F_{{\rm{R}},k}\otimes I$ are almost local, the functions
$\displaystyle\mathbb{R}\ni\boldsymbol{x}\to
ABU_{\pi_{{\rm{R}}}}(\boldsymbol{x})^{*}(G_{{\rm{R}},n}\otimes
I)\Omega=A[B,(G_{{\rm{R}},n}\otimes I)(-\boldsymbol{x})]\Omega$ (3.53)
decrease in norm faster than any inverse power of $|\boldsymbol{x}|$.
Consequently, the Fourier transform
$\displaystyle L\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p}):=(2\pi)^{-1/2}\int
d\boldsymbol{x}\,e^{-i\boldsymbol{p}\boldsymbol{x}}LU_{\pi_{{\rm{R}}}}(\boldsymbol{x})^{*}(G_{{\rm{R}},n}\otimes
I)\Omega$ (3.54)
is a norm-continuous function. It is compactly supported in $\mathbb{R}_{+}$
due to the spectrum condition and the fact that the energy-momentum transfer
of each $G_{{\rm{R}},n}\otimes I$ is bounded. By the (Hilbert space valued)
Plancherel theorem, we can write
$\int d\boldsymbol{x}\,((G_{{\rm{R}},n}\otimes
I)\Omega|(L_{1}^{*}L_{2})(\boldsymbol{x})(G_{{\rm{R}},n}\otimes I)\Omega)=\int
d\boldsymbol{p}\,(L_{1}\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p})|L_{2}\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p})).$
(3.55)
We define
$\displaystyle\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}}_{1},{\hat{L}}_{2}):=(L_{1}\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p})|L_{2}\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p})).$
(3.56)
It is easy to see that non-zero $\psi_{{\rm{R}},n,\boldsymbol{p}}$ are
particle weights in the sense of Definition 2.6: Positivity and property 1 are
obvious. The continuity requirement in property 3 follows from the equality
$\displaystyle\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}}_{1},{\hat{L}}_{2}(y)-{\hat{L}}_{2})$
$\displaystyle=(2\pi)^{-1/2}\int
d\boldsymbol{x}\,e^{-i\boldsymbol{p}\boldsymbol{x}}(L_{1}\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p})|[(L_{2}(y)-L_{2}),(G_{{\rm{R}},n}\otimes
I)(-\boldsymbol{x})]\Omega)$ (3.57)
and from the dominated convergence theorem. Invariance under translations
(property 2) is a straightforward consequence of the formula
$\displaystyle\alpha_{x}(L)\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p})=e^{-i(\boldsymbol{p},\boldsymbol{p})x}U_{\pi_{{\rm{R}}}}(x)L\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p}),\quad
x\in\mathbb{R}^{2}.$ (3.58)
Making use of the above relation and the spectrum condition, it is easy to see
that the distribution
$\displaystyle\mathbb{R}^{2}\ni q\to(2\pi)^{-1}\int
d^{2}x\,e^{-iqx}\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}}_{1},\alpha_{x}({\hat{L}}_{2}))$
(3.59)
is supported in $V_{+}-(\boldsymbol{p},\boldsymbol{p})$.
Now let us show that any function
$\boldsymbol{p}\to\psi_{{\rm{R}},n,\boldsymbol{p}}(\,\cdot\,,\,\cdot\,)$,
$n\in C_{{\rm{R}}}$, is non-zero on a non-empty open set
$\Delta_{{\rm{R}},n}$: Since $(G_{{\rm{R}},n}\otimes
I)\Omega\in\mathcal{H}_{\pi_{{\rm{R}}}}$ is different from zero,
$U_{\pi_{{\rm{R}}}}$ does not have non-zero invariant vectors and
$\pi_{{\rm{R}}}(\hat{\mathfrak{A}})$ acts irreducibly on
$\mathcal{H}_{\pi_{{\rm{R}}}}$, Lemma A.1 ensures the existence of
${\hat{L}}\in\hat{\mathcal{L}}$ s.t. $L(G_{{\rm{R}},n}\otimes I)\Omega\neq 0$.
Consequently, $\mathbb{R}\ni\boldsymbol{x}\to
LU_{\pi_{{\rm{R}}}}(\boldsymbol{x})^{*}(G_{{\rm{R}},n}\otimes I)\Omega$ is a
non-zero function and so is the norm of its Fourier transform
$\mathbb{R}_{+}\ni\boldsymbol{p}\to\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}},{\hat{L}})$.
Since the functions
$\mathbb{R}_{+}\ni\boldsymbol{p}\to\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}}_{1},{\hat{L}}_{2})$
are continuous for all ${\hat{L}}_{1},{\hat{L}}_{2}\in\hat{\mathcal{L}}$, as
we have shown above, the sets
$\Delta_{{\rm{R}},n}:=\bigcup_{{\hat{L}}_{1},{\hat{L}}_{2}\in\hat{\mathcal{L}}}\\{\,\boldsymbol{p}\in\mathbb{R}_{+}\,|\,\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}}_{1},{\hat{L}}_{2})\neq
0\,\\}$ (3.60)
are open and non-empty.
Let us now fix some $n\in C_{{\rm{R}}}$,
$\boldsymbol{p}\in\Delta_{{\rm{R}},n}$ and consider the GNS representation
$\pi$ induced by $\psi_{{\rm{R}},n,\boldsymbol{p}}$, acting on the Hilbert
space
$\mathcal{H}_{\pi}:=(\hat{\mathcal{L}}/\\{\,{\hat{L}}\in\hat{\mathcal{L}}\,|\,\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}},{\hat{L}})=0\,\\})^{\mathrm{cpl}}$.
The equivalence class of ${\hat{L}}\in\hat{\mathcal{L}}$ is denoted by
$|{\hat{L}}\rangle$ and the scalar product is given by
$\langle{\hat{L}}_{1}|{\hat{L}}_{2}\rangle=\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}}_{1},{\hat{L}}_{2})$.
This GNS representation has the form
$\displaystyle\pi(\hat{A})|{\hat{L}}\rangle$ $\displaystyle=$
$\displaystyle|\hat{A}{\hat{L}}\rangle,\quad\,\,\,\,\,\,{\hat{L}}\in\hat{\mathcal{L}},\,\hat{A}\in\hat{\mathfrak{A}},$
(3.61) $\displaystyle U_{\pi}(x)|{\hat{L}}\rangle$ $\displaystyle=$
$\displaystyle|\alpha_{x}({\hat{L}})\rangle,\quad{\hat{L}}\in\hat{\mathcal{L}},\,x\in\mathbb{R}^{2},$
(3.62)
where $U_{\pi}$ is the standard representation of translations. We will show
that $(\pi(\hat{\mathfrak{A}}),U_{\pi})$ is unitarily equivalent to
$(\pi_{{\rm{R}}}(\hat{\mathfrak{A}}),U_{\pi_{{\rm{R}}}})$. To this end, we
introduce the map
$W_{{\rm{R}}}:\mathcal{H}_{\pi}\to\mathcal{H}_{\pi_{{\rm{R}}}}=\mathcal{K}_{1,\mathrm{odd}}\otimes\mathcal{K}_{2,\mathrm{ev}}$
given by
$W_{{\rm{R}}}|{\hat{L}}\rangle=L\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p}),\quad{\hat{L}}\in\hat{\mathcal{L}}.$
(3.63)
This map is clearly an isometry. Since $\pi_{{\rm{R}}}$ acts irreducibly on
$\mathcal{H}_{\pi_{{\rm{R}}}}$, we obtain that $W_{{\rm{R}}}$ has a dense
range and hence it is a unitary operator. From the relation
$W_{{\rm{R}}}\pi(\hat{A})|{\hat{L}}\rangle=\pi_{{\rm{R}}}(\hat{A})L\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p})=\pi_{{\rm{R}}}(\hat{A})W_{{\rm{R}}}|{\hat{L}}\rangle,\quad{\hat{L}}\in\hat{\mathcal{L}},\,\,\hat{A}\in\hat{\mathfrak{A}}$
(3.64)
we conclude that $\pi$ and $\pi_{{\rm{R}}}$ are unitarily equivalent. Next, we
obtain for any ${\hat{L}}\in\hat{\mathcal{L}}$ and $x\in\mathbb{R}^{2}$
$\displaystyle
U_{\pi_{{\rm{R}}}}(x)W_{{\rm{R}}}|{\hat{L}}\rangle=e^{i(\boldsymbol{p},\boldsymbol{p})x}\alpha_{x}(L)\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p})=e^{i(\boldsymbol{p},\boldsymbol{p})x}W_{{\rm{R}}}U_{\pi}(x)|{\hat{L}}\rangle,$
(3.65)
where in the first step we made use of relation (3.58). We recall that the
spectrum of $U_{\pi_{{\rm{R}}}}$ coincides with $V_{+}$ and note that
$\pi(\hat{\mathfrak{A}})$ acts irreducibly on $\mathcal{H}_{\pi}$, by relation
(3.64) and Proposition 3.9 (c). Thus, in view of equality (3.65),
$U^{\mathrm{can}}_{\pi}(x)=e^{i(\boldsymbol{p},\boldsymbol{p})x}U_{\pi}(x),\quad
x\in\mathbb{R}^{2}$ (3.66)
is the canonical representation of translations in the GNS representation of
$\psi_{{\rm{R}},n,\boldsymbol{p}}$. Relation (3.66) shows that
$q_{{\rm{R}},n,\boldsymbol{p}}=(\boldsymbol{p},\boldsymbol{p})$.
The analysis of the second term on the r.h.s. of (3.52) proceeds similarly:
For any $m\in C_{{\rm{L}}}$ and ${\hat{L}}\in\hat{\mathcal{L}}$ one introduces
vectors
$L\tilde{\Psi}_{{\rm{L}},m}(\boldsymbol{p}):=(2\pi)^{-1/2}\int
d\boldsymbol{x}\,e^{i\boldsymbol{p}\boldsymbol{x}}LU(\boldsymbol{x})^{*}(I\otimes
G_{{\rm{L}},m})\Omega$ (3.67)
and functionals
$\psi_{{\rm{L}},m,\boldsymbol{p}}({\hat{L}}_{1},{\hat{L}}_{2})=(L_{1}\tilde{\Psi}_{{\rm{L}},m}(\boldsymbol{p})|L_{2}\tilde{\Psi}_{{\rm{L}},m}(\boldsymbol{p}))$.
By an analogous reasoning as above one shows that for $\boldsymbol{p}$ in some
non-empty, open set $\Delta_{{\rm{L}},m}\subset\mathbb{R}_{+}$ these
functionals are particle weights with characteristic energy-momentum vectors
$q_{{\rm{L}},m,\boldsymbol{p}}=(\boldsymbol{p},-\boldsymbol{p})$. Their GNS
representations are unitarily equivalent to $\pi_{{\rm{L}}}$. $\Box$
## 4 Conclusions and outlook
In this work we carried out a systematic study of particle aspects of two-
dimensional conformal field theories both in vacuum representations and in
charged representations. In the former case we established a complete particle
interpretation in terms of Wigner particles (or ‘waves’ in the terminology of
[5]). In the latter case we proved the existence of infraparticles and
verified superselection of their direction of motion in a large class of
examples. We conclude that conformal field theories provide a valuable testing
ground for fundamental concepts of scattering theory.
An important question which remained outside of the scope of the present work
is the problem of asymptotic completeness in the case of infraparticles. We
remark that the theory of particle weights offers natural formulations of this
property [9, 11] which can be adapted to the case of massless, two-dimensional
theories. We conjecture that any charged representation of a chiral conformal
field theory has a complete particle interpretation in terms of
infraparticles.
A more technical circle of problems concerns the decomposition of particle
weights and their representations stated in formulas (1.2), (1.3). We recall
that the general procedure of [41, 42] is not canonical: Firstly, it involves
a choice of a maximal abelian subalgebra, acting on the representation space
of the original weight. Secondly, it relies on a selection of countable
subsets of all the objects involved. In view of these ambiguities it is not
yet possible to associate a unique family of (infra-)particle types with any
given quantum field theory. We feel that a satisfactory solution of these
problems requires a systematic study of examples. A useful criterion for their
classification is the type of representations induced by particle weights.
Thus in the present paper we focused on representations of type I (with atomic
center) which have a simple decomposition theory. Already in this elementary
case we found a physically interesting phenomenon: superselection of direction
of motion. It is a natural direction of further research to look for theories,
whose asymptotic functionals induce representations which are not of type I
with atomic center. We conjecture that such models exist and some of them
describe infraparticles with superselected momentum, similar to the electron
in QED.
Acknowledgements. The authors would like to thank Prof. D. Buchholz and Prof.
R. Longo for interesting discussions.
## Appendix A Proof of Theorem 2.11
###### Lemma A.1.
(a) Let $({\mathfrak{A}},U)$ be a local net of $C^{*}$-algebras on
$\mathbb{R}^{2}$ in the sense of Definition 2.1, acting irreducibly on a
Hilbert space $\mathcal{H}$ and let $U=U^{\mathrm{can}}$. Let
$\Psi\in\mathcal{H}$ be s.t.
$A(f)\Psi:=\int d^{2}x\,\alpha_{x}(A)f(x)\Psi=0$ (A.1)
for all local operators $A\in{\mathfrak{A}}$ and all $f\in S(\mathbb{R}^{2})$
s.t. $\mathrm{supp}\,\tilde{f}$ is compact and $\mathrm{supp}\,\tilde{f}\cap
V_{+}=\varnothing$. Then $\Psi$ is invariant under the action of $U$. (Here
$\tilde{f}(p):=(2\pi)^{-1}\int d^{2}x\,e^{ipx}f(x)$ ).
(b) Let $({\cal A},V)$ be a local net of von Neumann algebras on $\mathbb{R}$
in the sense of Definition 3.1, acting irreducibly on a Hilbert space ${\cal
K}$. Let $\Psi\in{\cal K}$ be s.t.
$A(f)\Psi:=\int ds\,\beta_{s}(A)f(s)\Psi=0$ (A.2)
for all local operators $A\in{\cal A}$ and all $f\in S(\mathbb{R})$ s.t.
$\mathrm{supp}\,\tilde{f}$ is compact and
$\mathrm{supp}\,\tilde{f}\cap\mathbb{R}_{+}=\varnothing$. Then $\Psi$ is
invariant under the action of $V$. (Here
$\tilde{f}(\omega):=(2\pi)^{-\frac{1}{2}}\int ds\,e^{i\omega s}f(s)$ ).
Proof. The argument exploits some ideas from the proof of Proposition 2.1 of
[13]. As for part (a), suppose that $\Psi$ is not invariant under the action
of $U$. Since the map $B(\mathcal{H})\ni A\to A(f)$ is $\sigma$-weakly
continuous (cf. Lemma 5.3 of [41]) and ${\mathfrak{A}}$ acts irreducibly on
$\mathcal{H}$, condition (A.1) implies that
$P(\Delta_{1})A(f)P(\Delta_{2})\Psi=0,$ (A.3)
where $P(\,\cdot\,)$ is the spectral measure of $U$ and
$\Delta_{1},\Delta_{2}\subset\mathbb{R}^{2}$ are open bounded sets. Since the
spectrum of $U$ has Lorentz invariant lower boundary and $\Psi$ is not
invariant under translations, we can choose $\Delta_{1},\Delta_{2}$ s.t.
$P(\Delta_{1})\neq 0$, $P(\Delta_{2})\Psi\neq 0$ and the closure of
$(\Delta_{1}-\Delta_{2})$ does not intersect with $V_{+}$. Choosing $f\in
S(\mathbb{R}^{2})$ s.t. $\mathrm{supp}\,\tilde{f}\cap V_{+}=\varnothing$ and
$\tilde{f}(p)=1$ for $p$ in the closure of $(\Delta_{1}-\Delta_{2})$, we
obtain that
$P(\Delta_{1})AP(\Delta_{2})\Psi=0$ (A.4)
for any $A\in{\mathfrak{A}}$. Exploiting irreducibility again, we obtain
$P(\Delta_{1})=0$, which is a contradiction. The proof of part (b) is
analogous. $\Box$
###### Lemma A.2.
Let ${\cal K}_{\pm}\subset\mathcal{H}_{\pm}$ be closed subspaces, invariant
under the action of $U$. Let $\\{e_{+,m}\\}_{m\in\mathbb{I}}$ be a complete
orthonormal basis in $(P_{E}{\cal K}_{+})$ and let
$\\{e_{-,n}\\}_{n\in\mathbb{J}}$ be a complete orthonormal basis in
$(P_{E}{\cal K}_{-})$ for some $E\geq 0$. Then any $\Psi\in P_{E}({\cal
K}_{+}\overset{\mathrm{out}}{\times}{\cal K}_{-})$ can be expressed as
$\Psi=\sum_{m,n}c_{m,n}e_{+,m}\overset{\mathrm{out}}{\times}e_{-,n},$ (A.5)
where $\sum_{m,n}|c_{m,n}|^{2}<\infty$.
Proof. First, we define a strongly continuous unitary representation of
translations
$U_{0}(x)(\Psi_{+}\otimes\Psi_{-})=(U(x)\Psi_{+})\otimes(U(x)\Psi_{-}),\quad\Psi_{\pm}\in{\cal
K}_{\pm}$ (A.6)
on ${\cal K}_{+}\otimes{\cal K}_{-}$. Then we obtain from Proposition 2.4
$\Omega^{\mathrm{out}}U_{0}(x)=U(x)\Omega^{\mathrm{out}}.$ (A.7)
For $\Psi^{\prime}=(\Omega^{\mathrm{out}})^{-1}\Psi$ the above relation gives
$P_{0,E}\Psi^{\prime}=\Psi^{\prime}$, where $P_{0,E}$ is the spectral
projection of $U_{0}$ corresponding to the set
$\\{\,(\omega,\boldsymbol{p})\in\mathbb{R}^{2}\,|\,\omega\leq E\,\\}$. By the
functional calculus, we get $P_{0,E}=P_{0,E}(P_{E}\otimes P_{E})$. Hence
$\Psi^{\prime}=(P_{E}\otimes
P_{E})\Psi^{\prime}=\sum_{m,n}c_{m,n}e_{+,m}\otimes e_{-,n}.$ (A.8)
By applying $\Omega^{\mathrm{out}}$, we obtain relation (A.5). $\Box$
###### Lemma A.3.
Let $\Psi^{\prime}\in\mathcal{H}$ be a vector of bounded energy, let
$F_{1},F_{2}\in{\mathfrak{A}}$ be almost local and of compact energy-momentum
transfer, and let $L=\sum_{k=1}^{n}A_{k}B_{k}$, where
$A_{k},B_{k}\in{\mathfrak{A}}$ are almost local and $B_{k}$ are, in addition,
energy decreasing. Then
$\lim_{T\to\infty}(\Psi^{\prime}|[[Q_{T},\Phi^{\mathrm{out}}_{+}(F_{1})],\Phi^{\mathrm{out}}_{-}(F_{2})]\Omega)=0,$
(A.9)
where $Q_{T}=\int dt\,h_{T}(t)\int
d\boldsymbol{x}\,(L^{*}L)(t,\boldsymbol{x})$. (The above sequence is well
defined by Theorem 2.9).
Proof. First, we note that by Proposition 2.3 and Theorem 2.9
$\lim_{T\to\infty}(\Psi^{\prime}|[[Q_{T},\Phi^{\mathrm{out}}_{+}(F_{1})],\Phi^{\mathrm{out}}_{-}(F_{2})]\Omega)=\lim_{T\to\infty}(\Psi^{\prime}|[[Q_{T},F_{1,+}(h_{T})],F_{2,-}(h_{T})]\Omega),$
(A.10)
if the limit on the r.h.s. exists. We introduce the auxiliary operators:
$Q_{\pm,T}:=\int
dt\,h_{T}(t)\int_{\mathbb{R}_{\pm}}d\boldsymbol{x}\,(L^{*}L)(t,\boldsymbol{x}).$
(A.11)
As we show below, they satisfy
$\lim_{T\to\infty}\|P_{E}[Q_{\pm,T},F_{\mp}(h_{T})]P_{E}\|=0$ (A.12)
for any $E\geq 0$ and any $F\in{\mathfrak{A}}$ which is almost local and of
compact energy-momentum transfer. Making use of this relation and the fact
that $Q_{T}=Q_{+,T}+Q_{-,T}$, the proof is completed with the help of the
Jacobi identity and Proposition 2.3 (c).
Let us now verify (A.12). As the two cases are analogous, we focus on one of
them and estimate the corresponding expression as follows.
$\|P_{E}[Q_{-,T},F_{+}(h_{T})]P_{E}\|\leq\int
dtdt_{1}\,h_{T}(t)h_{T}(t_{1})\int_{\mathbb{R}_{-}}d\boldsymbol{x}\|[L^{*}L(t,\boldsymbol{x}),F(t_{1},t_{1})]\|.$
(A.13)
Since $L^{*}L$ and $F$ are almost local, we can find sequences
$C_{r},F_{r}\in{\mathfrak{A}}(\mathcal{O}_{r})$, s.t. for any $n\in\mathbb{N}$
there exist $C_{n},C^{\prime}_{n}$ s.t.
$\displaystyle\|L^{*}L-C_{r}\|\leq\frac{C_{n}}{r^{n}},\quad\|F-F_{r}\|\leq\frac{C^{\prime}_{n}}{r^{n}}.$
(A.14)
We choose $r=(1+\frac{1}{4}|\boldsymbol{x}|)^{\varepsilon}+T^{\varepsilon}$,
where $0<\varepsilon<1$ appeared in the definition of $h_{T}$. We write
$\displaystyle[(L^{*}L)(t,\boldsymbol{x}),F(t_{1},t_{1})]$ $\displaystyle=$
$\displaystyle[(L^{*}L-C_{r})(t,\boldsymbol{x}),F(t_{1},t_{1})]$ (A.15)
$\displaystyle+$
$\displaystyle[C_{r}(t,\boldsymbol{x}),(F-F_{r})(t_{1},t_{1})]$
$\displaystyle+$ $\displaystyle[C_{r}(t,\boldsymbol{x}),F_{r}(t_{1},t_{1})].$
By estimates (A.14), the first two terms on the r.h.s. above give
contributions to (A.13) which tend to zero in the limit $T\to\infty$. The
contribution of the last term can be estimated as follows, exploiting
locality,
$\displaystyle\int
dtdt_{1}\,h_{T}(t)h_{T}(t_{1})\int_{\mathbb{R}_{-}}d\boldsymbol{x}\|[C_{r}(t,\boldsymbol{x}),F_{r}(t_{1},t_{1})]\|$
$\displaystyle\phantom{444444444444}\leq c\int
dtdt_{1}\,h_{T}(t)h_{T}(t_{1})\int_{\mathbb{R}_{-}}d\boldsymbol{x}\chi(|\boldsymbol{x}-t_{1}|\leq|t-t_{1}|+2r),$
(A.16)
where $\chi$ is the characteristic function of the respective set and $c$ is a
constant independent of $T$. Let us now derive some inequalities which hold on
the support of the integrand on the r.h.s. of (A.16). First, we note that
$t,t_{1}\in\mathrm{supp}\,h_{T}$, if and only if $t,t_{1}\in
T^{\varepsilon}\mathrm{supp}\,h+T$, in particular $|t-t_{1}|\leq
c_{1}T^{\varepsilon}$ for some $c_{1}\geq 0$. Exploiting this fact, the
inequality $|\boldsymbol{x}-t_{1}|\leq|t-t_{1}|+2r$ and the relation
$r=(1+\frac{1}{4}|\boldsymbol{x}|)^{\varepsilon}+T^{\varepsilon}$, we find
such $c_{2}\geq 0$ that $|\boldsymbol{x}|\leq c_{2}T$ and $r\leq
c_{2}T^{\varepsilon}$, in particular the r.h.s. of (A.16) is finite for any
$T\geq 1$. Making use of the inequalities $r\leq c_{2}T^{\varepsilon}$ and
$|\boldsymbol{x}-t_{1}|\leq|t-t_{1}|+2r$, and of the fact that
$t,t_{1}\in\mathrm{supp}\,h_{T}$, we obtain that $|\boldsymbol{x}-T|\leq
c_{3}T^{\varepsilon}$ for some $c_{3}\geq 0$ which implies that
$\boldsymbol{x}>0$ for sufficiently large $T$. As the region of integration in
the $\boldsymbol{x}$ variable is restricted to $\mathbb{R}_{-}$, we conclude
that the r.h.s. of (A.16) is zero for such $T$. $\Box$
Proof of Theorem 2.11: Let $Q_{T}=\int dt\,h_{T}(t)\int
d\boldsymbol{x}\,(L^{*}L)(t,\boldsymbol{x})$, where
$L=\sum_{k=1}^{n}A_{k}B_{k}$ is an element of the left ideal $\mathcal{L}$,
$A_{k}$ are almost local and $B_{k}\in\mathcal{L}_{0}$. Moreover, we choose
$\Psi=\Phi^{\mathrm{out}}_{+}(F_{1})\Phi^{\mathrm{out}}_{-}(F_{2})\Omega$ and
$\Psi^{\prime}=\Phi^{\mathrm{out}}_{+}(F_{1}^{\prime})\Phi^{\mathrm{out}}_{-}(F_{2}^{\prime})\Omega$,
where $F_{1/2},F_{1/2}^{\prime}\in{\mathfrak{A}}$ are almost local and have
compact energy-momentum transfer. Since $\Psi$ and $\Psi^{\prime}$ are vectors
of bounded energy, we can write
$\displaystyle(\Psi^{\prime}|Q_{T}\Phi^{\mathrm{out}}_{+}(F_{1})\Phi^{\mathrm{out}}_{-}(F_{2})\Omega)$
$\displaystyle=$
$\displaystyle(\Psi^{\prime}|[[Q_{T},\Phi^{\mathrm{out}}_{+}(F_{1})],\Phi^{\mathrm{out}}_{-}(F_{2})]\Omega)$
(A.17) $\displaystyle+$
$\displaystyle(\Psi^{\prime}|\Phi^{\mathrm{out}}_{-}(F_{2})Q_{T}\Phi^{\mathrm{out}}_{+}(F_{1})\Omega)$
$\displaystyle+$
$\displaystyle(\Psi^{\prime}|\Phi^{\mathrm{out}}_{+}(F_{1})Q_{T}\Phi^{\mathrm{out}}_{-}(F_{2})\Omega).$
The term with the double commutator above vanishes as $T\to\infty$ due to
Lemma A.3. The second term on the r.h.s. of relation (A.17) is treated as
follows:
$\displaystyle\lim_{T\to\infty}(\Psi^{\prime}|\Phi^{\mathrm{out}}_{-}(F_{2})Q_{T}\Phi^{\mathrm{out}}_{+}(F_{1})\Omega)$
$\displaystyle\phantom{44444}=\lim_{T\to\infty}(\Psi^{\prime}|\Phi^{\mathrm{out}}_{-}(F_{2})\int
h_{T}(t)e^{iHt}\int
d\boldsymbol{x}\,(L^{*}L)(\boldsymbol{x})e^{-i\boldsymbol{P}t}\Phi^{\mathrm{out}}_{+}(F_{1})\Omega)$
$\displaystyle\phantom{444444}=\lim_{T\to\infty}(\Psi^{\prime}|\Phi^{\mathrm{out}}_{-}(F_{2})\int
h_{T}(t)e^{i(H-\boldsymbol{P})t}\int
d\boldsymbol{x}\,(L^{*}L)(\boldsymbol{x})\Phi^{\mathrm{out}}_{+}(F_{1})\Omega)$
$\displaystyle\phantom{4444444444444444444444}=(\Psi^{\prime}|\Phi^{\mathrm{out}}_{-}(F_{2})P_{+}\int
d\boldsymbol{x}\,(L^{*}L)(\boldsymbol{x})\Phi^{\mathrm{out}}_{+}(F_{1})\Omega),$
(A.18)
where in the first step we made use of the fact that
$\Phi^{\mathrm{out}}_{+}(F_{1})\Omega=P_{+}\Phi^{\mathrm{out}}_{+}(F_{1})\Omega$,
in the second step we exploited the invariance of $\int
d\boldsymbol{x}\,(L^{*}L)(\boldsymbol{x})$ under translations in space and in
the last step we made use of the mean ergodic theorem as in the proof of Lemma
1 of [5]. Next, we obtain
$\displaystyle(\Phi^{\mathrm{out}}_{+}(F_{1}^{\prime})\Phi^{\mathrm{out}}_{-}(F_{2}^{\prime})\Omega|\Phi^{\mathrm{out}}_{-}(F_{2})P_{+}\int
d\boldsymbol{x}\,(L^{*}L)(\boldsymbol{x})\Phi^{\mathrm{out}}_{+}(F_{1})\Omega)$
$\displaystyle\phantom{44}=(\Phi^{\mathrm{out}}_{-}(F_{2})^{*}\Phi^{\mathrm{out}}_{-}(F_{2}^{\prime})\Omega|\Phi^{\mathrm{out}}_{+}(F_{1}^{\prime})^{*}P_{+}\int
d\boldsymbol{x}\,(L^{*}L)(\boldsymbol{x})\Phi^{\mathrm{out}}_{+}(F_{1})\Omega)$
$\displaystyle\phantom{44444}=(\Omega|\Phi^{\mathrm{out}}_{-}(F_{2}^{\prime})^{*}\Phi^{\mathrm{out}}_{-}(F_{2})\Omega)\,(\Omega|\Phi^{\mathrm{out}}_{+}(F_{1}^{\prime})^{*}\int
d\boldsymbol{x}\,(L^{*}L)(\boldsymbol{x})\Phi^{\mathrm{out}}_{+}(F_{1})\Omega),$
(A.19)
where we made use of the facts that
$[\Phi^{\mathrm{out}}_{+}(F_{1}),\Phi^{\mathrm{out}}_{-}(F_{2})]=0$ and that
$\mathcal{H}_{+}/{\mathbb{C}}\Omega$ is orthogonal to
$\mathcal{H}_{-}/{\mathbb{C}}\Omega$ (as in the proof of Lemma 4 (a) of [5]).
The last term on the r.h.s. of (A.17) is treated analogously.
We note that any $\Psi_{\pm}\in P_{E}\mathcal{H}_{\pm}$ can be approximated by
a sequence of vectors of the form $P_{\pm}F_{n}\Omega$, where
$F_{n}\in{\mathfrak{A}}$ are quasilocal and have energy-momentum transfers in
some fixed compact set. Hence, any
$\Psi=\Psi_{+}\overset{\mathrm{out}}{\times}\Psi_{-}$ has bounded energy. By
the above considerations and Theorem 2.9, we obtain for any
$\Psi=\Psi_{+}\overset{\mathrm{out}}{\times}\Psi_{-}$,
$\Psi^{\prime}=\Psi_{+}^{\prime}\overset{\mathrm{out}}{\times}\Psi_{-}^{\prime}$,
$\Psi_{\pm},\Psi_{\pm}^{\prime}\in P_{E}\mathcal{H}_{\pm}$,
$\displaystyle\lim_{T\to\infty}(\Psi^{\prime}|Q_{T}\Psi)$ $\displaystyle=$
$\displaystyle(\Psi_{+}^{\prime}|\Psi_{+})\int
d\boldsymbol{x}\,(\Psi_{-}^{\prime}|(L^{*}L)(\boldsymbol{x})\Psi_{-})$ (A.20)
$\displaystyle+$ $\displaystyle(\Psi_{-}^{\prime}|\Psi_{-})\int
d\boldsymbol{x}\,(\Psi_{+}^{\prime}|(L^{*}L)(\boldsymbol{x})\Psi_{+}).$
Now in view of Lemma A.2, any $\Psi\in P_{E}\mathcal{H}^{\mathrm{out}}$ has
the form
$\Psi=\sum_{m,n}c_{m,n}e_{+,m}\overset{\mathrm{out}}{\times}e_{-,n},$ (A.21)
where $\\{e_{\pm,m}\\}_{m=0}^{\infty}$ are orthonormal systems in
$\\{P_{E}\mathcal{H}_{\pm}\\}$, which we choose so that $e_{\pm,0}=\Omega$.
Defining
$\displaystyle\Psi_{+,n}=\sum_{m}c_{m,n}e_{+,m},\quad\Psi_{-,n}=\sum_{m}c_{n,m}e_{-,m},$
(A.22)
we obtain
$\rho_{\pm,\Psi}(\,\cdot\,)=\sum_{n}\,(\Psi_{\pm,n}|\,\cdot\,\Psi_{\pm,n})$.
Relation (A.20) gives
$\lim_{T\to\infty}(\Psi|Q_{T}\Psi)=\int
d\boldsymbol{x}\,(\rho_{+,\Psi}+\rho_{-,\Psi})\big{(}(L^{*}L)(\boldsymbol{x})\big{)}.$
(A.23)
Exploiting the Cauchy-Schwarz inequality and the following bounds, valid for
$L=AB$, $A\in\mathfrak{A}$, $B\in\mathcal{L}_{0}$,
$\displaystyle|(\Psi|Q_{T}\Psi)|$ $\displaystyle\leq$
$\displaystyle\|P_{E}\int
d\boldsymbol{x}\,(B^{*}B)(\boldsymbol{x})P_{E}\|\,\|A^{*}A\|,$ (A.24)
$\displaystyle\int
d\boldsymbol{x}\,(\rho_{+,\Psi}+\rho_{-,\Psi})\big{(}(L^{*}L)(\boldsymbol{x})\big{)}$
$\displaystyle\leq$ $\displaystyle 2\|P_{E}\int
d\boldsymbol{x}\,(B^{*}B)(\boldsymbol{x})P_{E}\|\,\|A^{*}A\|,\,\,\,\,$ (A.25)
one extends (A.23) to any $L\in\mathcal{L}$. Now formula (2.36) follows by a
polarization argument.
Let us now show that $\psi_{\Psi}^{\mathrm{out}}=0$ only if
$\Psi\in{\mathbb{C}}\Omega$. By the above considerations we obtain, for any
$B\in\mathcal{L}_{0}$,
$\psi_{\Psi}^{\mathrm{out}}(B,B)=\sum_{n}\int
d\boldsymbol{x}\,\big{\\{}(\Psi_{-,n}|(B^{*}B)(\boldsymbol{x})\Psi_{-,n})+(\Psi_{+,n}|(B^{*}B)(\boldsymbol{x})\Psi_{+,n})\big{\\}}.$
(A.26)
If $\psi_{\Psi}^{\mathrm{out}}=0$, then $B\Psi_{\pm,n}=0$ for each $n$ and any
such $B$. Thus, by Lemma A.1 (a), $\Psi_{\pm,n}$ are proportional to $\Omega$.
Using definitions (A.21), (A.22) and the convention $e_{\pm,0}=\Omega$, it is
easily seen that $\Psi$ is proportional to $\Omega$. $\Box$
## Appendix B Proofs of Lemmas 3.7 and 3.8
Proof of Lemma 3.7: As for the main part of the lemma, it suffices to show
that the spectrum of $V_{\mathrm{odd}}$ coincides with $\mathbb{R}_{+}$. It
follows from the assumption ${\hbox{\rm Ad}}W\neq{\rm id}$ and the Reeh-
Schlieder property of the net $({\cal A},V)$ that ${\cal
A}_{\mathrm{odd}}({\cal I})\neq\\{0\\}$ and ${\cal K}_{\mathrm{odd}}=[{\cal
A}_{\mathrm{odd}}({\cal I})\Omega_{0}]$ for any open, bounded subset ${\cal
I}\subset\mathbb{R}$. Let $P(\,\cdot\,)$ be the spectral measure of $V$ and
suppose that $P(\Delta){\cal K}_{\mathrm{odd}}=\\{0\\}$ for some open subset
$\Delta\subset\mathbb{R}_{+}$. We fix a non-zero $A\in{\cal
A}_{\mathrm{odd}}({\cal I})$. Then, for any $B\in{\cal A}({\cal I})$ the
distribution
$(\Omega_{0}|[B,\widetilde{A}(\omega)]\Omega_{0})=\frac{1}{\sqrt{2\pi}}\int
dt\,e^{-i\omega t}(\Omega_{0}|[B,\beta_{t}(A)]\Omega_{0})$ (B.1)
is supported outside of $\Delta\cup-\Delta$. Since, by locality, this
distribution is a holomorphic function, it must be zero for all
$\omega\in\mathbb{R}$. Thus for any $f\in S(\mathbb{R})$ s.t. $\tilde{f}$ is
supported in the interior of $\mathbb{R}_{+}$ we obtain
$(\Omega_{0}|BA(f)\Omega_{0})=(\Omega_{0}|B\tilde{f}(T)A\Omega_{0})=0$. Here
$T\geq 0$ is the generator of $V$, $A(f)=\int dt\,\beta_{t}(A)f(t)$ and we
made use of the fact that $A(f)^{*}\Omega_{0}=0$, due to the support property
of $\tilde{f}$ and the spectrum condition. Approximating the characteristic
function of the interior of $\mathbb{R}_{+}$ with such $\tilde{f}$ and making
use of the fact that $(\Omega_{0}|A\Omega_{0})=0$, we conclude that
$A\Omega_{0}=0$ and hence, by the Reeh-Schlieder property $A=0$, which
contradicts our assumption. Consequently, $P(\Delta){\cal
K}_{\mathrm{odd}}\neq\\{0\\}$ for any open subset $\Delta$ of
$\mathbb{R}_{+}$, which means that the spectrum of $V_{\mathrm{odd}}$
coincides with $\mathbb{R}_{+}$.
This fact can also be proven as follows: The representation of translations
$V$ can be extended to a representation of the $ax+b$ group thanks to the
Borchers theorem [27]. There is only one non-trivial, irreducible
representation of this group which has positive energy [38] and its spectrum
of translations is ${\mathbb{R}}_{+}$. Since ${\cal K}_{\mathrm{odd}}$ does
not contain non-trivial invariant vectors of $V$, the spectrum of $V|_{{\cal
K}_{\mathrm{odd}}}$ coincides with $\mathbb{R}_{+}$.
Let us now proceed to part (a) of the lemma. To show the irreducibility of
$\pi_{\mathrm{odd}}$, it suffices to check that any vector
$\Psi\in\mathcal{K}_{\mathrm{odd}}$ is cyclic under the action of
$\pi_{\mathrm{odd}}({\cal A}_{\mathrm{ev}})$. By contradiction, we suppose
that there is $\Psi^{\prime}\in\mathcal{K}_{\mathrm{odd}}$ s.t.
$(\Psi^{\prime}|A\Psi)=0$ for any $A\in{\cal A}_{\mathrm{ev}}$. But this
implies that $(\Psi^{\prime}|B\Psi)=0$ for any $B\in{\cal A}$, which
contradicts the irreducibility of the action of ${\cal A}$ on $\mathcal{K}$.
Next we verify the faithfulness of $\pi_{\mathrm{odd}}$ restricted to a local
algebra. Let $A\in{\cal A}_{\mathrm{ev}}({\cal I})$ be a positive local
element which is zero upon restriction to $\mathcal{K}_{\mathrm{odd}}$. For
any local odd element $B\in{\cal A}_{\mathrm{odd}}(\mathfrak{J})$ and for
sufficiently large $s$ we obtain
$0=(\Omega|\beta_{s}(B^{*})A\beta_{s}(B)\Omega)=(\Omega|\beta_{s}(B^{*}B)A\Omega)\to(\Omega|B^{*}B\Omega)\cdot(\Omega|A\Omega),$
(B.2)
where in the last step we took the limit $s\to\infty$. By the Reeh-Schlieder
property it follows that $A=0$. This implies that $\pi_{\mathrm{odd}}$ is
faithful on ${\cal A}_{\mathrm{ev}}({\cal I})$ by Proposition 2.3.3 (3) of
[17]. Now the faithfulness of $\pi_{\mathrm{odd}}$ on the quasilocal algebra
${\cal A}_{\mathrm{ev}}$ follows from Proposition 2.3.3 (2) of [17], which
says that $\pi_{\mathrm{odd}}$ is faithful, if and only if
$\|\pi_{\mathrm{odd}}(A)\|=\|A\|$ for any $A\in{\cal A}_{\mathrm{ev}}$. Local
normality of $\pi_{\mathrm{odd}}$ is obvious, since $\pi_{\mathrm{odd}}$ acts
by the restriction to a subspace. Indeed, making use of Lemma 2.4.19 from [17]
and of the fact that $\pi_{\mathrm{odd}}$ preserves the norm, it is easy to
check that
$\mathrm{l.u.b.}\pi_{\mathrm{odd}}(A_{\alpha})=\pi_{\mathrm{odd}}(\mathrm{l.u.b.}\,A_{\alpha})$,
where l.u.b denotes the least upper bound and
$\\{A_{\alpha}\\}_{\alpha\in\mathbb{I}}$ is a uniformly bounded increasing net
of positive operators from some ${\cal A}_{\mathrm{ev}}({\cal I})$.
Part (b) of the lemma follows from the uniqueness of the invariant vector of
$V$. $\Box$
Proof of Lemma 3.8: We know from Lemma 3.7 that ${\cal
A}_{\mathrm{ev}}\neq{\mathbb{C}}I$, since it can be irreducibly represented on
the infinite dimensional Hilbert space ${\cal K}_{\mathrm{odd}}$.
Consequently, we can find a non-zero $A\in{\cal A}_{\mathrm{ev}}({\cal I})$,
for some open, bounded ${\cal I}$, s.t. $(\Omega_{0}|A\Omega_{0})=0$.
Proceeding identically as in the proof of the main part of Lemma 3.7, we
conclude that the spectrum of $V_{\mathrm{ev}}$ coincides with
$\mathbb{R}_{+}$. Part (b) follows trivially from the fact that the net
$({\cal A},V)$ is in a vacuum representation. Irreducibility in part (a)
follows from part (b). The remaining part of the statement is proven
analogously as the corresponding part of Lemma 3.7. $\Box$
## References
* [1] H. Araki. _Mathematical theory of quantum fields_. Oxford University Press, Oxford, 1999.
* [2] H. Araki and R. Haag. _Collision cross sections in terms of local observables_. Commun. Math. Phys. 4, (1967) 77-91.
* [3] M. Bischoff, D. Meise, K.-H. Rehren and I. Wagner. _Conformal quantum field theory in various dimensions._ Bulg. J. Phys. 36, (2009), no. 3, 170-185.
* [4] H.-J. Borchers and D. Buchholz. _The energy-momentum spectrum in local field theories with broken Lorentz-symmetry_. Commun. Math. Phys. 97, (1985) 169-185.
* [5] D. Buchholz. _Collision theory for waves in two dimensions and a characterization of models with trivial $S$-matrix_. Commun. Math. Phys. 45, (1975) 1-8.
* [6] D. Buchholz. _Collision theory for massless bosons_. Commun. Math. Phys. 52, (1977) 147-173.
* [7] D. Buchholz. _The physical state space of quantum electrodynamics_. Commun. Math. Phys. 85, (1982) 49-71.
* [8] D. Buchholz. _Gauss’ law and the infraparticle problem_. Phys. Lett. B 174, (1986) 331-334.
* [9] D. Buchholz. _Particles, infraparticles and the problem of asymptotic completeness_. In: VIIIth International Congress on Mathematical Physics. Marseille 1986. Singapore: World Scientific 1987.
* [10] D. Buchholz. _Harmonic analysis of local operators_. Commun. Math. Phys. 129, (1990) 631-641.
* [11] D. Buchholz. _On the manifestations of particles_. pp.177-202 in: Mathematical Physics Towards the 21st Century. Proceedings Beer-Sheva 1993, R.N. Sen and A. Gersten Eds., Ben-Gurion University of the Negev Press 1994.
* [12] D. Buchholz. _Quarks, gluons, colour: facts or fiction?_ Nuclear Phys. B 469, (1996) 333-353.
* [13] D. Buchholz and K. Fredenhagen. _Locality and the structure of particle states._ Commun. Math. Phys. 84, (1982) 1-54.
* [14] D. Buchholz, G. Lechner and S. J. Summers. _Warped convolutions, Rieffel deformations and the construction of quantum field theories_. Commun. Math. Phys. 304, (2011) 95-123.
* [15] D. Buchholz, G. Mack and I. Todorov. _The current algebra on the circle as a germ of local field theories._ Nuclear Phys. B Proc. Suppl. 5B, (1988) 20-56.
* [16] D. Buchholz, M. Porrmann and U. Stein. _Dirac versus Wigner. Towards a universal particle concept in local quantum field theory_. Phys. Lett. B 267, (1991) 377-381.
* [17] O. Brattelli and D.W. Robinson. _Operator algebras and quantum statistical mechanics 1_. Springer-Verlag, Berlin Heidelberg New York, 1979.
* [18] T. Chen, J. Fröhlich and A. Pizzo. _Infraparticle scattering states in non-relativistic QED. I. The Bloch-Nordsieck paradigm._ Commun. Math. Phys. 294, (2010) 761-825.
* [19] T. Chen, J. Fröhlich and A. Pizzo. _Infraparticle scattering states in nonrelativistic quantum electrodynamics. II. Mass shell properties._ J. Math. Phys. 50, (2009) 012103.
* [20] K. R. Davidson. _C*-Algebras by Example_. American Mathematical Society, 1996.
* [21] J. Dereziński and C. Gérard. _Scattering theory of infrared divergent Pauli-Fierz Hamiltonians_. Ann. Henri Poincaré 5, (2004) 523-577.
* [22] W. Dybalski. _Haag-Ruelle scattering theory in presence of massless particles_. Lett. Math. Phys. 72, (2005) 27-38.
* [23] W. Dybalski. _Spectral theory of automorphism groups and particle structures in quantum field theory_. Ph.D. Thesis, Universität Göttingen (2008).
http://webdoc.sub.gwdg.de/diss/2009/dybalski/
* [24] W. Dybalski. _Continuous spectrum of automorphism groups and the infraparticle problem_. Commun. Math. Phys. 300, (2010) 273-299.
* [25] W. Dybalski and Y. Tanimoto. _Asymptotic completeness in a class of massless relativistic quantum field theories_. Commun. Math. Phys. 305, (2011) 427-440.
* [26] V. Enss. _Characterization of particles by means of local observables_. Commun. Math. Phys. 45, (1975) 35-52.
* [27] M. Florig. _On Borchers’ theorem._ Lett. Math. Phys. 46, (1998) 289-293.
* [28] J. Fröhlich. _On the infrared problem in a model of scalar electrons and massless, scalar bosons._ Ann. Inst. H. Poincaré Sect. A (N.S.) 19, (1973) 1-103.
* [29] J. Fröhlich. _Existence of dressed one electron states in a class of persistent models_. Fortschr. Phys. 22, (1974) 159-198.
* [30] J. Fröhlich, G. Morchio and F. Strocchi. _Charged sectors and scattering states in quantum electrodynamics._ Ann. Phys. 119, (1979) 241-284.
* [31] F. Gabbiani and J. Fröhlich. _Operator algebras and conformal field theory_. Commun. Math. Phys. 155, (1993) 569-640.
* [32] R. Haag. _Local quantum physics_. Second edition. Springer-Verlag, Berlin, 1996.
* [33] D. Hasler and I. Herbst. _Absence of ground states for a class of translation invariant models of non-relativistic QED._ Commun. Math. Phys. 279, (2008) 769-787.
* [34] A. Herdegen. _Infrared problem and spatially local observables in electrodynamics_. Ann. Henri Poincaré 9, (2008) 373-401.
* [35] K. Johannsen. _Teilchenaspekte im Schroermodell_. Diplomarbeit, Universität Hamburg, 1991.
* [36] Y. Kawahigashi and R. Longo. _Classification of two-dimensional local conformal nets with $c<1$ and 2-cohomology vanishing for tensor categories_. Commun. Math. Phys. 244, (2004) 63-97.
* [37] Y. Kawahigashi and R. Longo. _Classification of local conformal nets. Case $c<1$._ Annals of Mathematics 160, (2004) 493-522.
* [38] R. Longo. _Lectures on Conformal Nets._ _Real Hilbert subspaces, modular theory, ${\rm SL}(2,{\bf R})$ and CFT_. In: Von Neumann algebras in Sibiu: Conference Proceedings, 33–91. Theta, Bucharest, 2008.
* [39] A. Pizzo. _One-particle (improper) states in Nelson’s massless model._ Ann. Henri Poincaré 4, (2003) 439-486.
* [40] A. Pizzo. _Scattering of an infraparticle: the one particle sector in Nelson’s massless model_. Ann. Henri Poincaré 5, (2005) 553-606.
* [41] M. Porrmann. _Particle weights and their disintegration I_. Commun. Math. Phys. 248, (2004) 269-304.
* [42] M. Porrmann. _Particle weights and their disintegration II_. Commun. Math. Phys. 248, (2004) 305-333.
* [43] K.-H. Rehren. _Chiral observables and modular invariants_. Commun. Math. Phys. 208, (2000) 689-712.
* [44] K. Rejzner. _Asymptotic algebra of fields in quantum electrodynamics_. Master’s thesis, University of Cracow, 2009.
* [45] S. Sakai. _$C^{*}$ -algebras and $W^{*}$-algebras_. Berlin, Heidelberg, New York. Springer 1971.
* [46] B. Schroer. _Infrateilchen in der Quantenfeldtheorie_. Fortschr. Phys. 11, (1963) 1-31.
* [47] U. Stein. _Zur Konstruktion von Streuzuständen mit Hilfe lokaler Observabler_. Ph.D. Thesis, Universität Hamburg 1989.
* [48] O. Steinmann. _Perturbative quantum electrodynamics and axiomatic field theory_.
Springer, 2000.
* [49] M. Takesaki. _Theory of operator algebras I_. Berlin, Heidelberg, New York. Springer 1979.
* [50] E. P. Wigner. _On unitary representations of the inhomogeneous Lorentz group_. Ann. Math. 40, (1939) 149-204.
* [51] A. Wassermann. _Operator algebras and conformal field theory. III. Fusion of positive energy representations of LSU(N) using bounded operators._ Invent. Math. 133(3), (1998) 467-538.
|
arxiv-papers
| 2011-01-29T15:36:25 |
2024-09-04T02:49:16.715014
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Wojciech Dybalski, Yoh Tanimoto",
"submitter": "Yoh Tanimoto",
"url": "https://arxiv.org/abs/1101.5700"
}
|
1101.5785
|
# Statistical Compressed Sensing of Gaussian Mixture Models
Guoshen Yu ECE, University of Minnesota, Minneapolis, Minnesota, 55414, USA
Guillermo Sapiro ECE, University of Minnesota, Minneapolis, Minnesota, 55414,
USA
###### Abstract
A novel framework of compressed sensing, namely statistical compressed sensing
(SCS), that aims at efficiently sampling a collection of signals that follow a
statistical distribution, and achieving accurate reconstruction on average, is
introduced. SCS based on Gaussian models is investigated in depth. For signals
that follow a single Gaussian model, with Gaussian or Bernoulli sensing
matrices of $\mathcal{O}(k)$ measurements, considerably smaller than the
$\mathcal{O}(k\log(N/k))$ required by conventional CS based on sparse models,
where $N$ is the signal dimension, and with an optimal decoder implemented via
linear filtering, significantly faster than the pursuit decoders applied in
conventional CS, the error of SCS is shown tightly upper bounded by a constant
times the best $k$-term approximation error, with overwhelming probability.
The failure probability is also significantly smaller than that of
conventional sparsity-oriented CS. Stronger yet simpler results further show
that for any sensing matrix, the error of Gaussian SCS is upper bounded by a
constant times the best $k$-term approximation with probability one, and the
bound constant can be efficiently calculated. For Gaussian mixture models
(GMMs), that assume multiple Gaussian distributions and that each signal
follows one of them with an unknown index, a piecewise linear estimator is
introduced to decode SCS. The accuracy of model selection, at the heart of the
piecewise linear decoder, is analyzed in terms of the properties of the
Gaussian distributions and the number of sensing measurements. A maximum a
posteriori expectation-maximization algorithm that iteratively estimates the
Gaussian models parameters, the signals model selection, and decodes the
signals, is presented for GMM-based SCS. In real image sensing applications,
GMM-based SCS is shown to lead to improved results compared to conventional
CS, at a considerably lower computational cost.
## I Introduction
Compressed sensing (CS) aims at achieving accurate signal reconstruction while
sampling signals at a low sampling rate, typically far smaller than that of
Nyquist/Shannon. Let $\mathbf{x}\in\mathbb{R}^{N}$ be a signal of interest,
$\Phi\in\mathbb{R}^{M\times N}$ a non-adaptive sensing matrix (encoder),
consisting of $M\ll N$ measurements,
$\mathbf{y}=\Phi\mathbf{x}\in\mathbb{R}^{M}$ a measured signal, and $\Delta$ a
decoder used to reconstruct $\mathbf{x}$ from $\Phi\mathbf{x}$. CS develops
encoder-decoder pairs $(\Phi,\Delta)$ such that a small reconstruction error
$\mathbf{x}-\Delta(\Phi\mathbf{x})$ can be achieved.
Reconstructing $\mathbf{x}$ from $\Phi\mathbf{x}$ is an ill-posed problem
whose solution requires some prior information on the signal. Instead of the
frequency band-limit signal model assumed in classic Shannon sampling theory,
conventional CS adopts a sparse signal model, i.e., there exists a dictionary,
typically an orthogonal basis $\Psi\in\mathbb{R}^{N\times N}$, a linear
combination of whose columns generates an accurate approximation of the
signal, $\mathbf{x}\approx\Psi\mathbf{a}$, the coefficients $\mathbf{a}[m]$,
$1\leq m\leq N$, having their amplitude decay fast after being sorted. For
signals following the sparse model, it has been shown that using some random
sensing matrices such as Gaussian and Bernoulli matrices $\Phi$ with
$M=\mathcal{O}(k\log(N/k))$ measurements, and an $l_{1}$ minimization or a
greedy matching pursuit decoder $\Delta$ promoting sparsity, with high
probability CS leads to accurate signal reconstruction: The obtained
approximation error is tightly upper bounded by a constant times the best
$k$-term approximation error, the minimum error that one may achieve by
keeping the $k$ largest coefficients in $\mathbf{a}$ [12, 13, 17, 18].
Redundant and signal adaptive dictionaries that further improve the CS
performance with respect to orthogonal bases have been investigated [11, 19,
31]. In addition to sparse models, manifold models have been considered for CS
as well [6, 16].
The present paper introduces a novel framework of CS, namely statistical
compressed sensing (SCS). As opposed to conventional CS that deals with one
signal at a time, SCS aims at efficiently sampling a collection of signals and
having accurate reconstruction on average. Instead of restricting to sparse
models, SCS works with general Bayesian models. Assuming that the signals
$\mathbf{x}$ follow a distribution with probability density function (pdf)
$f(\mathbf{x})$, SCS designs encoder-decoder pairs $(\Phi,\Delta)$ so that the
average error
$E_{\mathbf{x}}\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{X}=\int\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{X}f(\mathbf{x})d\mathbf{x},\vspace{0ex}$
where $\|\cdot\|_{X}$ is a norm, is small. As an important example, SCS with
Gaussian models is here shown to have improved performance (bounds) relative
to conventional CS, the signal reconstruction calculated with an optimal
decoder $\Delta$ implemented via a fast linear filtering. Moreover, for
Gaussian mixture models (GMMs) that better describe most real signals, SCS
with a piecewise linear decoder is investigated.
The motivation of SCS with Gaussian models is twofold. First, controlling the
average error over a collection of signals is useful in signal acquisition,
not only because one is often interested in acquiring a collection of signals
in real applications, but also because more effective processing of an
individual signal, an image or a sound for example, is usually achieved by
dividing the signal in (often overlapping) local subparts, patches (see Figure
10) or short-time windows for instance, so a signal can be regarded as a
collection of subpart signals [2, 8, 35, 36]. In addition, Gaussian mixture
models (GMMs), which model signals or subpart signals with a collection of
Gaussians, assuming each signal drawn from one of them, have been shown
effective in describing real signals, leading to state-of-the-art results in
image inverse problems [36] and missing data estimation [24].
SCS based on a single Gaussian model is first developed in Section II.
Following a similar mathematical approach as the one adopted in conventional
CS performance analysis [17], it is shown that with the same random matrices
as in conventional CS, but with a considerably reduced number
$M=\mathcal{O}(k)$ of measurements, and with the optimal decoder implemented
via linear filtering, significantly faster than the decoders applied in
conventional CS, the average error of Gaussian SCS is tightly upper bounded by
a constant times the best $k$-term approximation error with overwhelming
probability, the failure probability being orders of magnitude smaller than
that of conventional CS. Moreover, stronger yet simpler results further show
that for any sensing matrix, the average error of Gaussian SCS is upper
bounded by a constant times the best $k$-term approximation with probability
one, and the bound constant can be efficiently calculated.
Section III extends SCS to GMMs. A piecewise linear GMM-based SCS decoder,
which essentially consists of estimating the signal using each Gaussian model
included in the GMM and then selecting the best model, is introduced. The
accuracy of the model selection, at the heart of the scheme, is analyzed in
detail in terms of the properties of the Gaussian distributions and the number
of sensing measurements. These results are then important in the general area
of model selection from compressed measurements.
Following [36], Section IV presents an maximum a posteriori expectation-
maximization (MAP-EM) algorithm that iteratively estimates the Gaussian models
and decodes the signals. GMM-based SCS calculated with the MAP-EM algorithm is
applied in real image sensing, leading to improved results with respect to
conventional CS, at a considerably lower computational cost.
## II Performance Bounds for a Single Gaussian Model
This section analyzes the performance bounds of SCS based on a single Gaussian
model. Perfect reconstruction of degenerated Gaussian signals is briefly
discussed. After reviewing basic properties of linear approximation for
Gaussian signals, the rest of the section shows that for Gaussian signals with
fast eigenvalue decay, the average error of SCS using $k$ measurements and
decoded by a linear estimator is tightly upper bounded by that of best
$k$-term approximation.
Signals $\mathbf{x}\in\mathbb{R}^{N}$ are assumed to follow a Gaussian
distribution $\mathcal{N}(\mu,\Sigma)$ in this section. Principal Component
Analysis (PCA) calculates a basis change
$\mathbf{a}=\mathbf{B}^{T}(\mathbf{x}-\mu)$ of the data $\mathbf{x}$, with
$\mathbf{B}$ the orthonormal PCA basis that diagonalizes the data covariance
matrix
$\Sigma=\mathbf{B}\mathbf{S}\mathbf{B}^{T},$ (1)
where $\mathbf{S}=\mathrm{diag}(\lambda_{1},\ldots,\lambda_{N})$ is a diagonal
matrix whose diagonal elements
$\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{N}$ are the sorted
eigenvalues, and $\mathbf{a}\sim\mathcal{N}(\mathbf{0},\mathbf{S})$ the PCA
coefficient vector [27]. In this section, for most of the time we will assume
without loss of generality that
$\mathbf{x}\sim\mathcal{N}(\mathbf{0},\mathbf{S})$ by looking in the PCA
domain. For Gaussian and Bernoulli matrices that are known to be universal,
analyzing CS in canonical basis or PCA basis is equivalent [4].
### II-A Degenerated Gaussians
Conventional CS is able to perfectly reconstruct $k$-sparse signals, i.e.,
$\mathbf{x}\in\mathbb{R}^{N}$ with at most $k$ non-zero entries (typically
$k\ll N$), with $2k$ measurements [17]. Degenerated Gaussian distributions
$\mathcal{N}(\mathbf{0},\mathbf{S}_{k})$, where
$\mathbf{S}_{k}=\mathrm{diag}(\lambda_{1},\ldots,\lambda_{k},0,\ldots,0)$ with
at most $k$ non-zero eigenvalues, give the counterpart of $k$-sparsity for the
Gaussian signal models considered in this paper. Such signals belong to a
linear subspace $\mathcal{S}_{k}=\\{\mathbf{x}|\mathbf{x}[m]=0,\forall k<m\leq
N\\}$. The next lemma gives a condition for perfect reconstruction of signals
in $\mathcal{S}_{k}$.
###### Lemma 1.
If $\Phi$ is any $M\times N$ matrix and $k$ is a positive integer, then there
is a decoder $\Delta$ such that $\Delta(\Phi\mathbf{x})=\mathbf{x}$, for all
$\mathbf{x}\in\mathcal{S}_{k}$, if and only if
$\mathcal{S}_{k}\cap\mathrm{Null}(\Phi)=\mathbf{0}$.
###### Proof.
Suppose there is a decoder $\Delta$ such that
$\Delta(\Phi\mathbf{x})=\mathbf{x}$, for all $\mathbf{x}\in\mathcal{S}_{k}$.
Let $\mathbf{x}=\mathcal{S}_{k}\cap\mathrm{Null}(\Phi)$. We can write
$\mathbf{x}=\mathbf{x}_{1}-\mathbf{x}_{2}$ where both
$\mathbf{x}_{1},\mathbf{x}_{2}\in\mathcal{S}_{k}$. Since
$\Phi\mathbf{x}=\mathbf{0}$, $\Phi\mathbf{x}_{1}=\Phi\mathbf{x}_{2}$. Plugging
$\Phi\mathbf{x}_{1}$ and $\Phi\mathbf{x}_{2}$ into the decoder $\Delta$, we
obtain $\mathbf{x}_{1}=\mathbf{x}_{2}$ and then
$\mathbf{x}=\mathbf{x}_{1}-\mathbf{x}_{2}=\mathbf{0}$.
Suppose $\mathcal{S}_{k}\cap\mathrm{Null}(\Phi)=\mathbf{0}$. If
$\mathbf{x}_{1},\mathbf{x}_{2}\in\mathcal{S}_{k}$ with
$\Phi\mathbf{x}_{1}=\Phi\mathbf{x}_{2}$, then
$\mathbf{x}_{1}-\mathbf{x}_{2}\in\mathcal{S}_{k}\cap\mathrm{Null}(\Phi)$, so
$\mathbf{x}_{1}=\mathbf{x}_{2}$. $\Phi$ is thus a one-to-one map. Therefore
there must exist a decoder $\Delta$ such that
$\Delta(\Phi\mathbf{x})=\mathbf{x}$. ∎
It is possible to construct matrices of size $M\times N$ with $M=k$ which
satisfies the requirement of the Lemma. A trivial example is
$[\mathbf{I}_{M\times M}|\mathbf{0}_{M\times(N-M)}]$, where
$\mathbf{I}_{M\times M}$ is the identity matrix of size $M\times M$ and
$\mathbf{0}_{M\times(N-M)}$ is a zero matrix of size $M\times(N-M)$. Comparing
with conventional compressed sensing that requires $2k$ measurements for exact
reconstruction of $k$-sparse signals, with only $k$ measurements signals in
$\mathcal{S}_{k}$ can be exactly reconstructed. Indeed, in contrast to the
$k$-sparse signals where the positions of the non-zero entries are unknown
($k$-sparse signals reside in multiple $k$-dimensional subspaces), with the
degenerated Gaussian model $\mathcal{N}(\mathbf{0},\mathbf{S}_{k})$, the
position of the non-zero coefficients are known a priori to be the first $k$
ones. $k$ measurements thus suffice for perfect reconstruction.
In the following, we will concentrate on the more general case of non-
degenerated Gaussian signals (i.e., Gaussians with full-rank covariance
matrices $\Sigma$) with fast eigenvalue decay, in analogy to compressible
signals for conventional CS. As mentioned before and will be further
experimented in this paper, such simple models not only lead to improved
theoretical bounds, but also provide state-of-the-art image reconstruction
results.
### II-B Optimal Decoder
To simplify the notation, we assume without loss of generality that the
Gaussian has zero mean ${\mu}=\mathbf{0}$, as one can always center the signal
with respect to the mean.
It is well-known that the optimal decoders for Gaussian signals are calculated
with linear filtering:
###### Theorem 1.
[23] Let $\mathbf{x}\in\mathbb{R}^{N}$ be a random vector with prior pdf
$\mathcal{N}(\mathbf{0},\Sigma)$, and $\Phi\in\mathbb{R}^{M\times N}$, $M\leq
N$, be a sensing matrix. From the measured signal
$\mathbf{y}=\Phi\mathbf{x}\in\mathbb{R}^{M}$, the optimal decoder $\Delta$
that minimizes the mean square error (MSE)
$E_{\mathbf{x}}[\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{2}^{2}]=\min_{g}E_{\mathbf{x}}[\|\mathbf{x}-g(\Phi\mathbf{x})\|_{2}^{2}],$
as well as the mean absolute error (MAE)
$E_{\mathbf{x}}[\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{1}]=\min_{g}E_{\mathbf{x}}[\|\mathbf{x}-g(\Phi\mathbf{x})\|_{1}],$
where $g:\mathbb{R}^{M}\rightarrow\mathbb{R}^{N}$, is obtained with a linear
MAP estimator,
$\Delta(\Phi\mathbf{x})=\arg\max_{\mathbf{x}}p(\mathbf{x}|\mathbf{y})=\underbrace{\Sigma\Phi^{T}(\Phi\Sigma\Phi^{T})^{-1}}_{\Delta}(\Phi\mathbf{x}),\vspace{0ex}$
(2)
and the resulting error $\eta=\mathbf{x}-\Delta(\Phi\mathbf{x})$ is Gaussian
with mean zero and with covariance matrix
$\Sigma_{\eta}=E_{\mathbf{x}}[\eta\eta^{T}]=\Sigma-\Sigma\Phi^{T}(\Phi\Sigma\Phi^{T})^{-1}\Phi\Sigma,$
whose trace yields the MSE of SCS.
$E_{\mathbf{x}}[\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{2}^{2}]=Tr(\Sigma-\Sigma\Phi^{T}(\Phi\Sigma\Phi^{T})^{-1}\Phi\Sigma).\vspace{0ex}$
(3)
In contrast to conventional CS, for which the $l_{1}$ minimization or greedy
matching pursuit decoders, calculated with iterative procedures, have been
shown optimal under certain conditions on $\Phi$ and the signal sparsity [12,
18], Gaussian SCS enjoys the advantage of having an optimal decoder (2)
calculated fast via a closed-form linear filtering for any $\Phi$.
###### Corollary 1.
If a random matrix $\Phi\in\mathbb{R}^{M\times N}$ is drawn independently to
sense each $\mathbf{x}$, with all the other conditions as in Theorem 1, the
MSE of SCS is
$E_{\mathbf{x},\Phi}[\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{2}^{2}]=E_{\Phi}[Tr(\Sigma-\Sigma\Phi^{T}(\Phi\Sigma\Phi^{T})^{-1}\Phi\Sigma)].\vspace{0ex}$
(4)
Applying an independent random matrix realization to sense each signal has
been considered in [17]. In real applications, these random sensing matrices
need not to be stored, since the decoder can regenerate them itself given the
random seed.
Following a PCA basis change (1), it is equivalent to consider signals
$\mathbf{x}\sim\mathcal{N}(\mu,\Sigma)$ and
$\mathbf{x}\sim\mathcal{N}(\mathbf{0},\mathbf{S})$, where
$\mathbf{S}=\mathrm{diag}(\lambda_{1},\ldots,\lambda_{N})$ is a diagonal
matrix whose diagonal elements
$\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{N}$ are the sorted
eigenvalues. Theorem 1 and Corollary 1 clearly hold for
$\mathbf{x}\sim\mathcal{N}(\mathbf{0},\mathbf{S})$, with (2), (3), and (4)
rewritten as
$\displaystyle\Delta(\Phi\mathbf{x})$ $\displaystyle=$
$\displaystyle\underbrace{\mathbf{S}\Phi^{T}(\Phi\mathbf{S}\Phi^{T})^{-1}}_{\Delta}(\Phi\mathbf{x}),$
(5) $\displaystyle
E_{\mathbf{x}}[\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{2}^{2}]$
$\displaystyle=$ $\displaystyle
Tr(\mathbf{S}-\mathbf{S}\Phi^{T}(\Phi\mathbf{S}\Phi^{T})^{-1}\Phi\mathbf{S}),$
(6) $\displaystyle
E_{\mathbf{x},\Phi}[\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{2}^{2}]$
$\displaystyle=$ $\displaystyle
E_{\Phi}[Tr(\mathbf{S}-\mathbf{S}\Phi^{T}(\Phi\mathbf{S}\Phi^{T})^{-1}\Phi\mathbf{S})].$
(7)
Note that PCA bases, as sparsifying dictionaries, have been applied to do
conventional CS based on sparse models [29], which is fundamentally different
than the Gaussian models and SCS here studied.
### II-C Linear vs Nonlinear Approximation
Before proceeding with the analysis of the SCS performance, let us make some
comments on the relationship between linear and non-linear approximations for
Gaussian signals. In particular, the following is observed via Monte Carlo
simulations:
For Gaussian signals $\mathbf{x}\sim\mathcal{N}(\mathbf{0},\mathbf{S})$, where
$\mathbf{S}=\mathrm{diag}(\lambda_{1},\ldots,\lambda_{N})$ whose eigenvalues
$\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{N}$ decay fast, the best
$k$-term linear approximation
${\mathbf{x}}^{l}_{k}(m)=\left\\{\begin{array}[]{cc}\mathbf{x}(m)&~{}~{}1\leq
m\leq k,\\\ 0&~{}~{}k+1\leq m\leq N,\end{array}\right.$ (8)
and the nonlinear approximation
${\mathbf{x}}^{n}_{k}=T_{k}(\mathbf{x}),$ (9)
where $T_{k}$ is a thresholding operator that keeps the $k$ coefficients of
largest amplitude and setting others to zero, lead to comparable approximation
errors
$\sigma^{l}_{k}(\\{\mathbf{x}\\})_{X}=E_{\mathbf{x}}[\|\mathbf{x}-{\mathbf{x}}^{l}_{k}\|_{X}]~{}~{}~{}\textrm{and}~{}~{}~{}\sigma^{n}_{k}(\\{\mathbf{x}\\})_{X}=E_{\mathbf{x}}[\|\mathbf{x}-{\mathbf{x}}^{n}_{k}\|_{X}].$
(10)
Monte Carlo simulations are performed to test this. Assuming a power decay of
the eigenvalues [27],
$\lambda_{m}=m^{-\alpha},~{}~{}~{}1\leq m\leq N,\vspace{-1.5ex}$ (11)
where $\alpha>0$ is the decay parameter, with $N=64$, Figure 1 plots the MSEs
$\sigma^{l}_{k}(\\{\mathbf{x}\\})_{2}^{2}=E_{\mathbf{x}}[\|\mathbf{x}-{\mathbf{x}}^{l}_{k}\|_{2}^{2}]~{}~{}~{}\textrm{and}~{}~{}~{}\sigma^{n}_{k}(\\{\mathbf{x}\\})_{2}^{2}=E_{\mathbf{x}}[\|\mathbf{x}-{\mathbf{x}}^{n}_{k}\|_{2}^{2}],$
(12)
normalized by the ideal signal energy $\|\mathbf{x}\|_{2}^{2}$, of best
$k$-term linear and nonlinear approximations as a function of $\alpha$, with
typical (for image patches of size $8\times 8$ for example) $k$ values $8$ and
$16$ ($k/N=1/8$ and $1/4$). Both MSEs decrease as $\alpha$ increases, i.e., as
the eigenvalues decay faster. With typical values $\alpha\approx 3$ (similar
to the eigenvalue decay calculated with typical image patches) and $k=8$ or
16, both approximations are accurate and generate small and comparable MSEs,
their difference being about $0.1\%$ of the signal energy and ratio about 2.
| |
---|---|---
(a) | (b) | (c)
Figure 1: (a). MSEs (normalized by the ideal signal energy) of best $k$-term
linear and non-linear approximation, with $k=8$ and $16$ (signal dimension
$N=64$). (b) and (c) Difference and ratio of normalized MSEs of best $k$-term
linear and non-linear approximation shown in (a).
Following this, the error of Gaussian SCS will be compared with that of best
$k$-term linear approximation, which is comparable to that of best $k$-term
nonlinear approximation. For simplicity, the best $k$-term linear
approximation errors will be denoted as
$\sigma_{k}(\\{\mathbf{x}\\})_{X}=\sigma^{l}_{k}(\\{\mathbf{x}\\})_{X}~{}~{}~{}\textrm{and}~{}~{}~{}\sigma_{k}(\\{\mathbf{x}\\})_{2}^{2}=\sigma^{l}_{k}(\\{\mathbf{x}\\})_{2}^{2}.$
(13)
Note that $\sigma_{k}(\\{\mathbf{x}\\})_{2}^{2}=\sum_{m=k+1}^{N}\lambda_{m}$.
### II-D Performance of Gaussian SCS – A Numerical Analysis At First
This section numerically evaluates the MSE of Gaussian SCS, and compares it
with the minimal MSE generated by best $k$-term linear approximation,
proceeding the theoretical bounds later developed.
As before, a power decay of the eigenvalues (11), with $N=64$, is assumed in
the Monte Carlo simulations. An independent random Gaussian matrix realization
$\Phi$ is applied to sense each signal $\mathbf{x}$ [17].
Figures 2 (a) and (c)-top plot the MSE (normalized by the ideal signal energy)
of SCS and that of the best $k$-term linear approximation, as well as their
ratio as a function of $\alpha$, with $k$ fixed at typical values $8$ and $16$
($k/N=1/8$ and $1/4$). As $\alpha$ increases, i.e., as the eigenvalues decay
faster, the MSEs for both methods decrease. Their ratio increases almost
linearly with $\alpha$. The same is plotted in figures 2 (b) and (c)-bottom,
with eigenvalue decay parameter fixed at a typical value $\alpha=3$, and with
$k$ varying from $5$ to $32$ ($k/N$ from $5/64$ to $1/2$). As $k$ increases,
both MSEs decrease, their ratio being almost constant at about $3.7$.
| |
---|---|---
(a) | (b) | (c)
Figure 2: Comparison of the MSE of SCS and that of the best $k$-term linear
approximation for Gaussian signals of dimension $N=64$. (a) and (c)-top. The
MSE (normalized by the ideal signal energy) of SCS and that of best $k$-term
linear approximation, as well as their ratio as a function of $\alpha$, with
$k$ fixed at typical values $8$ and $16$. (b) and (c)-bottom. The same values,
with eigenvalue decay parameter fixed at a typical value $\alpha=3$, and with
$k$ varying from $5$ to $32$.
These results indicate a good performance of Gaussian SCS, its MSE is only a
small number of times larger than that of the best $k$-term linear
approximation. 111Simulations using the same coefficient energy power decay
model (11) show that the ratio between conventional CS based on sparse models,
with $k$ measurements, and that of the best $k$-term nonlinear approximation,
varies as a function of the decay parameter $\alpha$ and $k$. For typical
values $\alpha=3$, the ratio is typically an order of magnitude larger than
that between the MSE of SCS and that of the best $k$-term linear
approximation. The next sections provide mathematical analysis of this
performance.
Let us notice that while the best $k$-term linear approximation decoding is
feasible for signals following a single Gaussian distribution, it is
impractical with GMMs (assuming multiple Gaussians and that each signal is
generated from one of them with an unknown index), since the Gaussian index of
the signal is unknown. SCS with GMMs, which describe real data considerably
better than a single Gaussian model [36], will be described in sections III
and IV.
### II-E Performance Bounds
Following the analysis techniques in [17], this section shows that with
Gaussian and Bernoulli random matrices of $\mathcal{O}(k)$ measurements,
considerably smaller than the $\mathcal{O}(k\log(N/k))$ required by
conventional CS, the average error of Gaussian SCS is tightly upper bounded by
a constant times the best $k$-term linear approximation error with
overwhelming probability, the failure probability being orders of magnitude
smaller than that of conventional CS.
We consider only the encoder-decoder pairs $(\Phi,\Delta)$ that preserve
$\Phi\mathbf{x}$, i.e., $\Phi(\Delta(\Phi\mathbf{x}))=\Phi\mathbf{x}$,
satisfied by the optimal $\Delta$ in (5) for Gaussian signals $\mathbf{x}$,
$\forall\Phi$.
#### II-E1 From Null Space Property to Instance Optimality
The instance optimality in expectation bounds the average error of SCS with a
constant times that of the best $k$-term linear approximation (13), defining
the desired SCS performance:
###### Definition 1.
Let $\mathbf{x}\in\mathbb{R}^{N}$ be a random vector that follows a certain
distribution. Let $K\subset\\{1,\ldots,N\\}$ be any subset of indices. We say
that $(\Phi,\Delta)$ is instance optimal in expectation in $K$ in
$\|\cdot\|_{X}$, with a constant $C_{0}$, if
$E_{\mathbf{x},(\Phi)}[\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{X}]\leq
C_{0}E_{\mathbf{x}}[\|\mathbf{x}-\mathbf{x}_{K}\|_{X}],\vspace{0ex}$ (14)
where $\mathbf{x}_{K}$ is the signal $\mathbf{x}$ restricted to $K$
($\mathbf{x}_{K}[n]=\mathbf{x}[n],~{}\forall~{}n\in K$, and $0$ otherwise),
the expectation on the left side considered with respect to $\mathbf{x}$, and
to $\Phi$ if one random $\Phi$ is drawn independently for each $\mathbf{x}$.
Similarly, the MSE instance optimality in $K$ is defined as
$E_{\mathbf{x},(\Phi)}[\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{2}^{2}]\leq
C_{0}E_{\mathbf{x}}[\|\mathbf{x}-\mathbf{x}_{K}\|_{2}^{2}].\vspace{0ex}$ (15)
In particular, if $K=\\{1,\ldots,k\\}$, then we say that $(\Phi,\Delta)$ is
instance optimal in expectation of order $k$ in $\|\cdot\|_{X}$, with a
constant $C_{0}$, if
$E_{\mathbf{x},(\Phi)}[\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{X}]\leq
C_{0}E_{\mathbf{x}}[\|\mathbf{x}-\mathbf{x}_{K}\|_{X}]=C_{0}\sigma_{k}(\\{\mathbf{x}\\})_{X},\vspace{0ex}$
(16)
and is instance optimal of order $k$ in MSE, with a constant $C_{0}$, if
$E_{\mathbf{x},(\Phi)}[\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{2}^{2}]\leq
C_{0}E_{\mathbf{x}}[\|\mathbf{x}-\mathbf{x}_{K}\|_{2}^{2}]=C_{0}\sigma_{k}(\\{\mathbf{x}\\})_{2}^{2}.\vspace{0ex}$
(17)
The null space property in expectation defined next will be shown equivalent
to the instance optimality in expectation.
###### Definition 2.
Let $\mathbf{x}\in\mathbb{R}^{N}$ be a random vector that follows a certain
distribution. Let $K\subset\\{1,\ldots,N\\}$ be any subset of indices. We say
that $\Phi$ in $(\Phi,\Delta)$ has the null space property in expectation in
$K$ in $\|\cdot\|_{X}$, with constant $C$, if
$E_{\mathbf{x},(\Phi)}[\|\eta\|_{X}]\leq
CE_{\mathbf{x}}[\|\eta-\eta_{K}\|_{X}],~{}\textrm{where}~{}\eta=\mathbf{x}-\Delta(\Phi\mathbf{x}),$
(18)
where $\eta_{K}$ is the signal $\eta$ restricted to $K$
($\eta_{K}[n]=\eta[n],~{}\forall~{}n\in K$, and $0$ otherwise), the
expectation considered on the left side with respect to $\mathbf{x}$, and to
$\Phi$ if one random $\Phi$ is drawn independently for each $\mathbf{x}$. Note
that $\eta\in\mathrm{Null}(\Phi)$. Similarly, the MSE null space property in
$K$ is defined as
$E_{\mathbf{x},(\Phi)}\|\eta\|_{2}^{2}\leq
CE_{\mathbf{x}}[\|\eta-\eta_{K}\|_{2}^{2}],~{}\textrm{where}~{}\eta=\mathbf{x}-\Delta(\Phi\mathbf{x}).$
(19)
In particular, if $K=\\{1,\ldots,k\\}$, with $1\leq k\leq N$, then we say that
$\Phi$ in $(\Phi,\Delta)$ has the null space property in expectation of order
$k$ in $\|\cdot\|_{X}$, with constant $C$, if
$E_{\mathbf{x},(\Phi)}\|\eta\|_{X}\leq
CE_{\mathbf{x}}[\|\eta-\eta_{K}\|_{X}]=C\sigma_{k}(\\{\eta\\})_{X},~{}\textrm{where}~{}\eta=\mathbf{x}-\Delta(\Phi\mathbf{x}),$
(20)
and has the MSE null space property of order $k$, if
$E_{\mathbf{x},(\Phi)}\|\eta\|_{2}^{2}\leq
CE_{\mathbf{x}}[\|\eta-\eta_{K}\|_{2}^{2}]=C\sigma_{k}(\\{\eta\\})_{2}^{2},~{}\textrm{where}~{}\eta=\mathbf{x}-\Delta(\Phi\mathbf{x}).$
(21)
###### Theorem 2.
Let $\mathbf{x}\in\mathbb{R}^{N}$ be a random vector that follows a certain
distribution. Given an $M\times N$ matrix $\Phi$, a norm $\|\cdot\|_{X}$, and
a subset of indices $K\subset\\{1,\ldots,N\\}$, a sufficient condition that
there exists a decoder $\Delta$ such that the instance optimality in
expectation in $K$ in $\|\cdot\|_{X}$ (14) holds with constant $C_{0}$, is
that the null space property in expectation (18) holds with $C=C_{0}/2$ for
this $(\Phi,\Delta)$:
$E_{\mathbf{x},(\Phi)}[\|\eta\|_{X}]\leq\frac{C_{0}}{2}E_{\mathbf{x}}[\|\eta-\eta_{K}\|_{X}],~{}\textrm{where}~{}\eta=\mathbf{x}-\Delta(\Phi\mathbf{x}).$
(22)
A necessary condition is the null space property in expectation (18) with
$C=C_{0}$:
$E_{\mathbf{x},(\Phi)}[\|\eta\|_{X}]\leq
C_{0}E_{\mathbf{x}}[\|\eta-\eta_{K}\|_{X}],~{}\textrm{where}~{}\eta=\mathbf{x}-\Delta(\Phi\mathbf{x}),$
(23)
Similar results hold between the MSE instance optimality in $K$ (15) and the
null space property (19), with the constant $C=C_{0}/4$ in the sufficient
condition.
In particular, if $K=\\{1,\ldots,k\\}$, with $1\leq k\leq N$, the same
equivalence between the instance optimality in expectation of order $k$ in
$\|\cdot\|_{X}$ (16) and the null space property in expectation (20), and that
between the MSE instance optimality of order $k$ (17) and the null space
property (21), hold as well.
###### Proof.
To prove the sufficiency of (22), we consider the decoder $\Delta$ such that
for all $\mathbf{y}=\Phi\mathbf{x}\in\mathbb{R}^{M}$,
$\Delta(\mathbf{y}):=\arg\min_{\mathbf{z}}\|\mathbf{z}-\mathbf{z}_{K}\|_{X}~{}~{}~{}\textrm{s.t.}~{}~{}~{}\Phi\mathbf{z}=\mathbf{y}.$
(24)
By (22), we have
$\displaystyle E\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{X}$ $\displaystyle\leq$
$\displaystyle\frac{C_{0}}{2}E_{\mathbf{x}}[\|(\mathbf{x}-\Delta(\Phi\mathbf{x}))-(\mathbf{x}_{K}-(\Delta\Phi\mathbf{x})_{K})\|_{X}]$
(25) $\displaystyle\leq$
$\displaystyle\frac{C_{0}}{2}(E_{\mathbf{x}}[\|\mathbf{x}-\mathbf{x}_{K}\|_{X}]+E_{\mathbf{x}}[\|\Delta(\Phi\mathbf{x})-(\Delta\Phi\mathbf{x})_{K}\|_{X}])$
$\displaystyle\leq$
$\displaystyle{C_{0}}E_{\mathbf{x}}[\|\mathbf{x}-\mathbf{x}_{K}\|_{X}],$
where the second inequality uses the triangle inequality, and the last
inequality follows from the choice of the decoder (24).
To prove the necessity of (23), let $\Delta$ be any decoder for which (14)
holds. Let $\eta=\mathbf{x}-\Delta(\Phi\mathbf{x})$ and let $\eta_{K}$ be the
linear approximation of $\eta$ in $K$ ($\eta_{K}[n]=\eta[n],~{}\forall~{}n\in
K$, and $0$ otherwise). Let $\eta_{K}=\eta_{1}+\eta_{2}$ be any splitting of
$\eta_{K}$ into two vectors in the linear space
$\mathcal{S}_{K}=\\{\mathbf{x}|\mathbf{x}[m]=0,\forall k\notin K\\}$. We can
write
$\eta=\eta_{1}+\eta_{2}+\eta_{3},$
with $\eta_{3}=\eta-\eta_{K}$. As the right side of (14) is equal to 0 for
$\forall~{}\mathbf{x}\in\mathcal{S}_{K}$, we deduce
$-\eta_{1}=\Delta(\Phi(-\eta_{1}))$. Since $\eta\in\mathrm{Null}(\Phi)$, we
have $\Phi(-\eta_{1})=\Phi(\eta_{2}+\eta_{3})$, so that
$-\eta_{1}=\Delta(\Phi(\eta_{2}+\eta_{3}))$. We derive
$\displaystyle E[\|\eta\|_{X}]$ $\displaystyle=$ $\displaystyle
E[\|\eta_{2}+\eta_{3}-\Delta\Phi(\eta_{2}+\eta_{3})\|_{X}]\leq
C_{0}E_{\mathbf{x}}[\|(\eta_{2}+\eta_{3})-((\eta_{2})_{K}+(\eta_{3})_{K})\|_{X}]$
$\displaystyle=$ $\displaystyle C_{0}E_{\mathbf{x}}[\|\eta-\eta_{K}\|_{X}],$
where the inequality follows from (14), and the second and third equalities
use the fact that $\eta=\eta_{1}+\eta_{2}+\eta_{3}$ and
$\eta_{1}\in\mathcal{S}_{K}$. Thus we have obtained (23).
A similar proof proceeds for MSE instance optimality and null space property.
The second part of the theorem is a direct consequence of the first part. ∎
Comparing to conventional CS that requires the null space property to hold
with the best $2k$-term nonlinear approximation error [17], the requirement
for Gaussian SCS is relaxed to $k$, thanks to the linearity of the best
$k$-term linear approximation for Gaussian signals.
Theorem 2 proves the existence of the decoder $\Delta$ for which the instance
optimality in expectation holds for $(\Phi,\Delta)$, given the null space
property in expectation. However, it does not explain how such decoder is
implemented. The following Corollary, a direct consequence of theorems 1 and
2, shows that for Gaussian signals the optimal decoder (5) leads to the
instance optimality in expectation.
###### Corollary 2.
For Gaussian signals $\mathbf{x}\sim\mathcal{N}(\mathbf{0},\mathbf{S})$, if an
$M\times N$ sensing matrix $\Phi$ satisfies the null space property in
expectation (20) of order $k$ in $\|\cdot\|_{1}$, with constant $C_{0}/2$, or
the MSE null space property (21) of order $k$ with constant $C_{0}/4$, then
the optimal and linear decoder
$\Delta=\mathbf{S}\Phi^{T}(\Phi\mathbf{S}\Phi^{T})^{-1}$ satisfies the
instance optimality in expectation (16) in $\|\cdot\|_{1}$, or the MSE
instance optimality (17).
###### Proof.
It follows from Theorem 1 that the MAP decoder minimizes MAE and MSE among all
the estimators for $\mathbf{x}\sim\mathcal{N}(\mathbf{0},\mathbf{S})$.
Therefore its MAE and MSE are smaller than the ones generated by the decoder
considered in Theorem 2 (24). The latter satisfies the instance optimality, so
is the former. ∎
#### II-E2 From RIP to Null Space Property
The Restricted Isometry Property (RIP) of a matrix measures its ability to
preserve distances, and is related to the null space property in conventional
CS [14, 18]. The new linear RIP of order $k$ restricts the requirement of
conventional RIP of order $k$ to a union of $k$-dimensional linear subspaces
with consecutive supports:
###### Definition 3.
Let $k\leq N$ be a positive integer. Let $\mathcal{K}_{1}$ define a linear
subspace of functions with support in the first $k$ indices in $[1,N]$,
$\mathcal{K}_{2}$ a linear subspace of functions with support in the next $k$
indices, and so on. The functions in the last linear subspace
$\mathcal{K}_{J}$ defined this way may have support with less than $k$
indices. An $M\times N$ matrix $\Phi$ is said to have linear RIP of order $k$
with constant $\delta$ if
$(1-\delta)\|\mathbf{x}\|_{2}\leq\|\Phi\mathbf{x}\|_{2}\leq(1+\delta)\|\mathbf{x}\|_{2},~{}~{}~{}\forall~{}\mathbf{x}\in\cup_{j=1}^{J}\mathcal{K}_{j}.\vspace{0ex}$
(27)
The linear RIP is a special case of the block RIP [21], with block sparsity
one and blocks having consecutive support of the same size.
The following theorem relates the linear RIP (27) of a matrix $\Phi$ to its
null space property in expectation (20).
###### Theorem 3.
Let $\mathbf{x}\in\mathbb{R}^{N}$ be a random vector that follows a certain
distribution. Let $\Phi$ be an $M\times N$ matrix that satisfies the linear
RIP of order $2k$ with $\delta<1$, and let $\Delta$ be a decoder. Let
$\eta=\mathbf{x}-\Delta(\Phi\mathbf{x})$. Assume further that
$E_{\mathbf{x},(\Phi)}|\eta[n]|$ decays in $n$:
$E_{\mathbf{x},(\Phi)}|\eta[n+1]|<E_{\mathbf{x},(\Phi)}|\eta[n]|$, $\forall
n<N-1$. Then $\Phi$ satisfies the null property in expectation of order $k$ in
$\|\cdot\|_{1}$ (20), with constant
$C_{0}=1+k^{1/2}\frac{1+\delta}{1-\delta}$. 222As in [17], the result here is
in the $l_{1}$ norm, while in the next section we will consider a natural
extension of the RIP for SCS which can be studied in the $l_{2}$ norm,
something possible for conventional CS only in a probabilistic setting, with
one random sensing matrix independently drawn for each signal [17].
###### Proof.
Let $K$ denote the set of first $k$ indices of the entries in $\eta$, $K_{1}$
the next $k$ indices, $K_{2}$ the next $k$ indices, etc. We have
$\displaystyle\|\eta_{K}\|_{2}$ $\displaystyle\leq$
$\displaystyle\|\eta_{K\cup K_{1}}\|_{2}\leq(1-\delta)^{-1}\|\Phi\eta_{K\cup
K_{1}}\|_{2}=(1-\delta)^{-1}\|\sum_{j=2}^{J}\Phi\eta_{K_{j}}\|_{2}$
$\displaystyle\leq$
$\displaystyle(1-\delta)^{-1}\sum_{j=2}^{J}\|\Phi\eta_{K_{j}}\|_{2}\leq(1+\delta)(1-\delta)^{-1}\sum_{j=2}^{J}\|\eta_{K_{j}}\|_{2},$
where the second and last inequalities follow the linear RIP property of
$\Phi$, the third inequality follows from the triangle equality, and the
equality holds since $\eta\in\mathrm{Null}(\Phi)$. Hence we have
$E\|\eta_{K}\|_{2}\leq(1+\delta)(1-\delta)^{-1}\sum_{j=2}^{J}E\|\eta_{K_{j}}\|_{2}.$
(28)
Since $E|\eta[n+k]|\leq E|\eta[n]|$, we derive $E\|\eta_{K_{j+1}}\|_{1}\leq
E\|\eta_{K_{j}}\|_{1}$, so that
$E\|\eta_{K_{j+1}}\|_{2}\leq E\|\eta_{K_{j+1}}\|_{1}\leq
E\|\eta_{K_{j}}\|_{1},$ (29)
where the first inequality follows from the fact that
$\|\mathbf{x}\|_{2}\leq\|\mathbf{x}\|_{1},~{}\forall\mathbf{x}$. Inserting
(29) into (28) gives
$E\|\eta_{K}\|_{2}\leq(1+\delta)(1-\delta)^{-1}\sum_{j=1}^{J-1}E\|\eta_{K_{j}}\|_{1}\leq(1+\delta)(1-\delta)^{-1}E\|\eta_{{K}^{C}}\|_{1}.$
(30)
By the Cauchy-Schwartz inequality $\|\eta_{K}\|_{1}\leq
k^{1/2}\|\eta_{K}\|_{2}$, we therefore obtain
$E\|\eta\|_{1}=E\|\eta_{K}\|_{1}+E\|\eta_{K^{C}}\|_{1}\leq\left(1+k^{1/2}\frac{1+\delta}{1-\delta}\right)E\|\eta_{K^{C}}\|_{1},$
(31)
which verifies the null space property with constant $C_{0}$. ∎
For Gaussian signals $\mathbf{x}\in\mathcal{N}(\mathbf{0},\mathbf{S})$, with
$\Phi$ Gaussian or Bernoulli matrices, one realization drawn independently for
each $\mathbf{x}$, and with $\Delta$ the optimal decoder (5), the decay of
$E_{\mathbf{x},\Phi}|\eta[n]|$ assumed in Theorem 3 is verified through Monte
Carlo simulations.
#### II-E3 From Random Matrices to Linear RIP
The next Theorem shows that Gaussian and Bernoulli matrices satisfy the
conventional RIP for one subspace with overwhelming probability. The linear
RIP will be addressed after it.
###### Theorem 4.
[1, 4] Let $\Phi$ be a random matrix of size $M\times N$ drawn according to
any distribution that satisfies the concentration inequality
$\textrm{Pr}(|\|\Phi\mathbf{x}\|_{2}^{2}-\|\mathbf{x}\|_{2}^{2}|\geq\epsilon\|\mathbf{x}\|_{2}^{2})\leq
2e^{-Mc_{0}(\delta/2)},~{}~{}~{}\forall~{}\mathbf{x}\in\mathbb{R}^{N},\vspace{0ex}$
(32)
where $0<\delta<1$, and $c_{0}(\delta/2)>0$ is a constant depending only on
$\delta/2$. Then for any set $K\subset\\{1,\ldots,N\\}$ with $|K|=k<M$, we
have the conventional RIP condition
$(1-\delta)\|\mathbf{x}\|_{2}\leq\|\Phi\mathbf{x}\|_{2}\leq(1+\delta)\|\mathbf{x}\|_{2},~{}~{}~{}\forall~{}\mathbf{x}\in\mathcal{X}_{K},\vspace{0ex}$
(33)
where $\mathcal{X}_{K}$ is the set of all vectors in $\mathbb{R}^{N}$ that are
zero outside of $K$, with probability greater than or equal to
$1-2(12/\delta)^{k}e^{-c_{0}(\delta/2)M}.$ Gaussian and Bernoulli matrices
satisfy the concentration inequality (32).
The linear RIP of order $k$ (27) requires that (33) holds for $N/k\leq N$
subspaces. The next Theorem follows from Theorem 4 by simply multiplying by
$N$ the probability that the RIP fails to hold for one subspace.
###### Theorem 5.
Suppose that $M$, $N$ and $0<\delta<1$ are given. Let $\Phi$ be a random
matrix of size $M\times N$ drawn according to any distribution that satisfies
the concentration inequality (32). Then there exist constants $c_{1},c_{2}>0$
depending only on $\delta$ such that the linear RIP of order $k$ (27) holds
with probability greater than or equal to $1-2Ne^{-c_{2}M}$ for $\Phi$ with
the prescribed $\delta$ and $k\leq c_{1}M$.
###### Proof.
Following Theorem 4, for a $k$-dimensional linear space $\mathcal{X}_{K}$, the
matrix $\Phi$ will fail to satisfy (33) with probability $\leq
2(12/\delta)^{k}e^{-c_{0}(\delta/2)n}$.
The linear RIP requires that (33) holds for at most $N$ such subspaces. Hence
(33) will fail to hold with probability
$\leq
2N(12/\delta)^{k}e^{-c_{0}(\delta/2)M}=2Ne^{-c_{0}(\delta/2)M+k\log(12/\delta)}.$
(34)
Thus for a fixed $c_{1}>0$, whenever $k\leq c_{1}M$, the exponent in the
exponential on the right side of (34) is $\leq c_{2}M$ provided that
$c_{2}\leq c_{0}(\delta/2)-c_{1}(1+\log(12/\delta))$. We can always choose
$c_{1}>0$ small enough to ensure $c_{2}>0$. This proves that with a
probability $1-2Ne^{-c_{2}M}$, the matrix $\Phi$ will satisfy the linear RIP
(27). ∎
Comparing with conventional CS, where the null space property requires that
the RIP (33) holds for $\binom{N}{k}$ subspaces [4, 14, 18], the number of
subspaces in the linear RIP (27) is sharply reduced to $N/k$ for Gaussian SCS,
thanks to the coefficients pre-ordering and the linear estimation in
consequence. Therefore with the same number of measurements $M$, the
probability that a Gaussian or Bernoulli matrix $\Phi$ satisfies the linear
RIP is substantially higher than that for the conventional RIP. Equivalently,
given the same probability that $\Phi$ satisfies the linear RIP or the
conventional RIP of order $k$, the required number of measurements for the
linear RIP is $M\sim\mathcal{O}(k)$, substantially smaller than the
$M\sim\mathcal{O}(k\log(N/k))$ required for the conventional RIP. Similar
improvements have been obtained with model-based CS that assumes structured
sparsity on the signals [5].
With the results above, we have shown that for Gaussian signals, with sensing
matrices satisfying the linear RIP (27) of order $2k$, for example Gaussian or
Bernoulli matrices with $\mathcal{O}(k)$ rows, with overwhelming probability,
and with the optimal and linear decoder (5), SCS leads to the instance
optimality in expectation of order $k$ in $\|\cdot\|_{1}$ (16), with constant
$C_{0}=2(1+k^{1/2}\frac{1+\delta}{1-\delta})$. $k^{1/2}$ is typically small by
the definition of CS.
### II-F Performance Bounds with RIP in Expectation
This section shows that with an RIP in expectation, a matrix isometry property
more adapted to SCS, the Gaussian SCS MSE instance optimality (17) of order
$k$ and constant $C_{0}$, holds in the $l_{2}$ norm with probability one for
any matrix. $C_{0}$ has a closed-form and can be easily computed numerically.
###### Definition 4.
Let $\mathbf{x}\in\mathbb{R}^{N}$ be a random vector that follows a certain
distribution. Let $\Phi$ be an $M\times N$ sensing matrix and let $\Delta$ be
a decoder. Let $\eta=\mathbf{x}-\Delta(\Phi\mathbf{x})$. $\Phi$ in
$(\Phi,\Delta)$ is said to have RIP in expectation in $K$ with constant
$c_{K}$ if
${E_{\mathbf{x},(\Phi)}\|\Phi\eta_{K}\|_{2}^{2}}=c_{K}{E_{\mathbf{x},(\Phi)}\|\eta_{K}\|_{2}^{2}},~{}\textrm{where}~{}\eta=\mathbf{x}-\Delta(\Phi\mathbf{x}),\vspace{0ex}$
(35)
where $K\subset\\{1,\ldots,N\\}$, $\eta_{K}\in\mathbb{R}^{N}$ is the signal
$\eta$ restricted to $K$ ($\eta_{K}[n]=\eta[n],~{}\forall~{}n\in K$, and $0$
otherwise), and the expectation is with respect to $\mathbf{x}$, and to $\Phi$
if one random $\Phi$ is drawn independently for each $\mathbf{x}$.
The conventional RIP is known to be satisfied only by some random matrices,
Gaussian and Bernoulli matrices for example, with high probability. For a
given matrix, checking the RIP property is however NP-hard [4]. By contrast,
the constant of the RIP in expectation (35) can be measured for any matrix via
a fast Monte Carlo simulation, the quick convergence guaranteed by the
concentration of measure [33]. The next proposition, directly following from
(6) and (7), further shows that for Gaussian signals, the RIP in expectation
has its constant in a closed form.
###### Proposition 1.
Assume $\mathbf{x}\sim\mathcal{N}(\mathbf{0},\mathbf{S})$, $\Phi$ is an
$M\times N$ sensing matrix and $\Delta$ is the optimal and linear decoder (5).
Then $\Phi$ in $(\Phi,\Delta)$ satisfies the RIP in expectation in $K$,
${(E_{\Phi})\left[Tr\left(\Phi\mathbf{R}_{K}\mathbf{S}\mathbf{R}_{K}^{T}\Phi^{T}-\Phi\mathbf{R}_{K}\mathbf{S}\Phi^{T}(\Phi\mathbf{S}\Phi^{T})^{-1}\Phi\mathbf{S}\mathbf{R}_{K}^{T}\Phi^{T}\right)\right]}\vspace{-1ex}$
$=c_{K}{(E_{\Phi})\left[Tr\left(\mathbf{R}_{K}\mathbf{S}\mathbf{R}_{K}^{T}-\mathbf{R}_{K}\mathbf{S}\Phi^{T}(\Phi\mathbf{S}\Phi^{T})^{-1}\Phi\mathbf{S}\mathbf{R}_{K}^{T}\right)\right]}\vspace{0ex},$
(36)
where $\mathbf{R}_{K}$ is an $N\times N$ extraction matrix giving
$\eta_{K}=\mathbf{R}_{K}\eta$, i.e., $\mathbf{R}_{K}(i,i)=1$, $\forall i\in
K$, all the other entries being zero. The expectation with respect to $\Phi$
is calculated if one random $\Phi$ is drawn independently for each
$\mathbf{x}$.
###### Proof.
Let
$\eta=\mathbf{x}-\Delta\Phi\mathbf{x}=\mathbf{x}-\mathbf{S}\Phi^{T}(\Phi\mathbf{S}\Phi^{T})^{-1}\Phi\mathbf{x}$,
which follows from the MAP estimation (5). (1) is derived by calculating the
covariance matrices
$\Sigma_{\Phi\eta_{K}}=E\left[\Phi\mathbf{R}_{K}\eta(\Phi\mathbf{R}_{K}\eta)^{T}\right]$
of $\Phi\eta_{K}=\Phi\mathbf{R}_{K}\eta$, and
$\Sigma_{\eta_{K}}=E\left[\mathbf{R}_{K}\eta(\mathbf{R}_{K}\eta)^{T}\right]$
of $\eta_{K}=\mathbf{R}_{K}\eta$, and using the fact that the trace of a
covariance matrix yields the average energy of the underlying random vector. ∎
The next Theorem shows that the RIP in expectation leads to the MSE null space
property holding in equality.
###### Theorem 6.
Let $\mathbf{x}\in\mathbb{R}^{N}$ be a random vector that follows a certain
distribution, $\Phi$ an $M\times N$ sensing matrix, and $\Delta$ a decoder.
Assume ${E_{\mathbf{x},(\Phi)}\|\eta_{K}\|_{2}^{2}}\neq 0$ and
${E_{\mathbf{x},(\Phi)}\|\eta_{K^{C}}\|_{2}^{2}}\neq 0$, for some
$K\subset\\{1,\ldots,N\\}$. Assume that $\Phi$ in $(\Phi,\Delta)$ has the RIP
in expectation in $K$ with constant $a_{K}>0$, and in
$K^{C}=\\{1,\ldots,N\\}\backslash K$ with constant $b_{K}>0$:
$\frac{E_{\mathbf{x},(\Phi)}\|\Phi\eta_{K}\|_{2}^{2}}{E_{\mathbf{x},(\Phi)}\|\eta_{K}\|_{2}^{2}}=a_{K},~{}~{}~{}\frac{E_{\mathbf{x},(\Phi)}\|\Phi\eta_{K^{C}}\|_{2}^{2}}{E_{\mathbf{x},(\Phi)}\|\eta_{K^{C}}\|_{2}^{2}}=b_{K},~{}\textrm{where}~{}\eta=\mathbf{x}-\Delta\Phi\mathbf{x},$
(37)
where $K\subset\\{1,\ldots,N\\}$, and $\eta_{K}\in\mathbb{R}^{N}$ is the
signal $\eta$ restricted to $K$ ($\eta_{K}[n]=\eta[n],~{}\forall~{}n\in K$,
and $0$ otherwise). Then $\Phi$ satisfies
$E_{\mathbf{x},(\Phi)}\|\eta\|_{2}^{2}=C_{0}E_{\mathbf{x},(\Phi)}\|\eta_{K^{C}}\|_{2}^{2},\vspace{0ex}$
(38)
where $C_{0}=1+{b_{K}}/{a_{K}}$. In particular, if $K=\\{1,\ldots,k\\}$, with
$1\leq k\leq N$, then $\Phi$ satisfies the MSE null space property of order
$k$, which holds with equality,
$E_{\mathbf{x},(\Phi)}\|\eta\|_{2}^{2}=C_{0}\sigma_{k}(\\{\eta\\})_{2}^{2}.\vspace{0ex}$
(39)
###### Proof.
We derive (38) by
$\frac{E_{\mathbf{x},(\Phi)}\|\eta\|_{2}^{2}}{E_{\mathbf{x},(\Phi)}\|\eta_{K^{C}}\|_{2}^{2}}=1+\frac{E_{\mathbf{x},(\Phi)}\|\eta_{K}\|_{2}^{2}}{E_{\mathbf{x},(\Phi)}\|\eta_{K^{C}}\|_{2}^{2}}=1+\frac{E_{\mathbf{x},(\Phi)}\|\Phi\eta_{K}\|_{2}^{2}/a_{k}}{E_{\mathbf{x},(\Phi)}\|\Phi\eta_{K^{C}}\|_{2}^{2}/b_{k}}=1+\frac{b_{k}}{a_{k}},$
where the second equality follows from the RIP in expectation (37) and the
last equality holds because $\Phi\eta_{K}=\Phi\eta_{K^{C}}$ since
$\eta=\eta_{K}+\eta_{K^{C}}\in\mathrm{Null}(\Phi)$. (39) is obtained by
inserting (13) in (38).∎
Following Corollary 2, the MSE null space property constant $C_{0}$ indicates
the upper bound of the SCS reconstruction error relative to the best $k$-term
linear approximation. Let us check $C_{0}$ of different sensing matrices in
SCS for Gaussian signals
$\mathbf{x}\in\mathbb{R}^{N}\sim\mathcal{N}(\mathbf{0},\mathbf{S})$, assuming
that the eigenvalues of $\mathbf{S}$ follow a power decay (11) with typical
values $\alpha=3$ and $N=64$. Gaussian, Bernoulli and random subsampling
matrices $\Phi$ of size $M\times N$ are considered, and the optimal and linear
decoder $\Delta$ (5) is applied to reconstruct the signals. For each matrix
distribution, a different random matrix realization $\Phi$ is applied to sense
each signal $\mathbf{x}$. Note that since the random subsampling matrix
$\Phi$, each row containing one entry with value 1 at a random position and 0
otherwise, has the maximal coherence with the canonical basis, this matrix is
not suitable for directly sensing $\mathbf{x}$ [10], and is replaced by
$\Phi\Psi$ in the simulation, with $\Psi$ a DCT basis having low coherence
with $\Phi$.
Monte Carlo simulations are performed to calculate the RIP constants $a_{K}$
and $b_{K}$ (37). Figure 3 (a) plots $C_{0}=1+{b_{K}}/{a_{K}}$, with a typical
value $k=10$ ($k/N=5/32$), for different values of $M$. When the number $M$ of
SCS measurements increases, the reconstruction error of SCS decreases,
resulting in a smaller ratio over the best $k$-term linear approximation error
with a fixed $k$. Gaussian and Bernoulli matrices lead to similar $C_{0}$
values, slightly smaller than that of random subsampling matrices. Figure 3
(b) plots $C_{0}$, as a function of $k$, with $M=k$. Gaussian and Bernoulli
matrices lead to similar $C_{0}\approx 4.5$ that varies little with $k$, in
line with the results obtained in Section II-D (Figure 2-(c)). For random
subsampling matrices $C_{0}$ slowly increases, almost linearly, and is equal
to $5.5$ for a typical value $k=10$, about 20% larger than that of Gaussian
and Bernoulli matrices. The small $C_{0}$ values indicate that the SCS
reconstruction error is tightly upper bounded by a constant times the best
$k$-term approximation error.
|
---|---
(a) | (b)
Figure 3: The MSE null space property constant $C_{0}$ (39) of Gaussian,
Bernoulli, and random subsampling matrices, as a function of $M$, with a fixed
$k=10$ (left), and of $k$ with $M=k$ (right). The signal dimension is $N=64$.
From Corollary 2 and Theorem 6, we obtain the next concluding Theorem, which
shows that for any sensing matrix, the error of Gaussian SCS is upper bounded
by a constant times the best $k$-term linear approximation with probability
one, and the bound constant can be efficiently calculated.
###### Theorem 7.
Assume $\mathbf{x}\sim\mathcal{N}(\mathbf{0},\mathbf{S})$. Let $\Phi$ be an
$M\times N$ sensing matrix and $\Delta$ the optimal and linear decoder (5).
Then $\Phi$ satisfies the MSE instance optimality of order $k$ (17) with
constant $C_{0}=4(1+{b_{K}}/{a_{K}})$, $a_{K}$ and $b_{K}$ given in (37), and
$K=\\{1,\ldots,k\\}$.
Theorem 7, together with the performance comparison of linear and nonlinear
approximation for Gaussian signals described in Section II-C, show that for
signals following a Gaussian distribution with fast eigenvalue decay, the
average error of SCS using $k$ measurements is tightly upper bounded by that
of the best $k$-term approximation.
## III Compressed Sensing Model Selection with GMMs
Section II shows tight error bounds of SCS for signals following a Gaussian
distribution with fast eigenvalue decay. A single Gaussian distribution,
however, is too simplistic for modeling most real signals. Assuming multiple
Gaussian distributions and that each signal follows one of them, Gaussian
mixture models (GMMs) provide more precise signal descriptions. It has been
shown that algorithms based on GMMs lead to results in the ballpark of the
state-of-the-art in various signal inverse problems, for different types of
real data including images and ranking score matrices [24, 36]. GMMs have also
been used to model color distributions [32] and for clustering [20], among
many satisfactory applications with these models.
This section first introduces a piecewise linear decoder for GMM-based SCS,
which essentially consists of estimating a signal using each Gaussian model
included in the GMM and then selecting the best model. At the heart of the
GMM-based SCS decoder is the model selection. The rest of the section analyzes
the accuracy of the model selection in terms of the GMM properties and the
number of the measurements. As correct Gaussian models are selected, the SCS
performance bounds described in Section II apply.
### III-A Piecewise Linear Decoder
GMMs describe signals with a mixture of Gaussian distributions. Assume there
exist $J$ Gaussian distributions $\\{\mathcal{N}(\mu_{j},\Sigma_{j})\\}_{1\leq
j\leq J}$, parametrized by their means $\mu_{j}$ and covariances $\Sigma_{j}$.
To simplify the notation, we assume without loss of generality that the
Gaussians have zero means ${\mu}_{j}=\mathbf{0}$, $\forall j$, as one can
always center the signals with respect to the means. GMM assumes that each
signal $\mathbf{x}\in\mathbb{R}^{N}$ is independently drawn from one of these
Gaussians with an unknown index $j\in[1,J]$, whose probability density
function is
$f(\mathbf{x})=\frac{1}{(2\pi)^{N/2}|\Sigma_{j}|^{1/2}}\exp\left({-\frac{1}{2}\mathbf{x}^{T}\Sigma_{j}^{-1}\mathbf{x}}\right).$
(40)
To decode a measured signal $\mathbf{y}=\Phi\mathbf{x}$, the GMM-based SCS
decoder estimates the signal $\tilde{\mathbf{x}}$ and selects the Gaussian
model $\tilde{j}$ by maximizing the log a-posteriori probability
$(\tilde{\mathbf{x}},\tilde{j})=\arg\max_{\mathbf{x},j}\log
f(\mathbf{x}|\mathbf{y},\Sigma_{j}).$ (41)
(41) is calculated by first computing the linear MAP decoder (2) using each of
the Gaussian models,
$\tilde{\mathbf{x}}_{j}=\Delta_{j}(\Phi\mathbf{x})=\underbrace{\Sigma_{j}\Phi^{T}(\Phi\Sigma_{j}\Phi^{T})^{-1}}_{\Delta_{j}}(\Phi\mathbf{x}),~{}~{}~{}\forall
1\leq j\leq J,\vspace{0ex}$ (42)
and then selecting a best model $\tilde{j}$ that maximizes the log
a-posteriori probability among all the models [36]
$\tilde{j}=\arg\max_{1\leq j\leq
J}-\frac{1}{2}\left(\log|\Sigma_{j}|+\tilde{\mathbf{x}}_{j}^{T}\Sigma_{j}^{-1}\tilde{\mathbf{x}}_{j}\right),$
(43)
whose corresponding decoder $\Delta_{\tilde{j}}$ implements a piecewise linear
estimate:
$\tilde{\mathbf{x}}=\tilde{\mathbf{x}}_{\tilde{j}}=\Delta_{\tilde{j}}(\Phi\mathbf{x}).$
(44)
The model selection (43) is at the heart of the GMM-based SCS.333Correct
model/class selection from compressed measurements is at the core of numerous
applications beyond signal reconstruction, see for example [15] and references
therein. To better understand it, we concentrate next in a simple case, where
the GMM involves $J=2$ Gaussian distributions
$\mathcal{N}(\mathbf{0},\Sigma_{1})$ and $\mathcal{N}(\mathbf{0},\Sigma_{2})$
that have the same “shape” and “size”, but different “orientation,” i.e., the
two covariance matrices have the same eigenvalues, but different PCA bases:
$\Sigma_{1}=\mathbf{B}_{1}\mathbf{S}\mathbf{B}_{1}^{T}~{}~{}~{}\textrm{and}~{}~{}~{}\Sigma_{2}=\mathbf{B}_{2}\mathbf{S}\mathbf{B}_{2}^{T},$
(45)
with $\mathbf{B}_{1}$ and $\mathbf{B}_{2}$ the PCA bases of the two Gaussian
distributions, and $\mathbf{S}=\mathrm{diag}(\lambda_{1},\ldots,\lambda_{N})$
a diagonal matrix, whose diagonal elements
$\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{N}$ are the sorted
eigenvalues. It follows directly that $|\Sigma_{1}|=|\Sigma_{2}|$. This will
be used next.
### III-B Oracle Model Selection
Let us first study the model selection in an oracle situation, where the
underlying signals $\mathbf{x}$ are assumed to be known and, without loss of
generality, to follow the first Gaussian distribution
$\mathbf{x}\sim\mathcal{N}(\mathbf{0},\Sigma_{1})$. Recall that
$|\Sigma_{1}|=|\Sigma_{2}|$ is assumed. The probability of correct oracle
model selection (43) that assigns $\mathbf{x}$ to the first Gaussian
distribution $\mathcal{N}(\mathbf{0},\Sigma_{1})$,
$P_{c}^{o}=\int_{\mathbf{x}^{T}\Sigma_{1}^{-1}\mathbf{x}<\mathbf{x}^{T}\Sigma_{2}^{-1}\mathbf{x}}f_{1}(\mathbf{x})d\mathbf{x}=\int\textrm{sign}\left(\mathbf{x}^{T}\Sigma_{2}^{-1}\mathbf{x}-\mathbf{x}^{T}\Sigma_{1}^{-1}\mathbf{x}\right)f_{1}(\mathbf{x})d\mathbf{x},$
(46)
where
$f_{1}(\mathbf{x})=\frac{1}{(2\pi)^{N/2}|\Sigma_{1}|^{1/2}}\exp\left({-\frac{1}{2}\mathbf{x}^{T}\Sigma_{1}^{-1}\mathbf{x}}\right)$,
will be studied as a function of the relationship between $\mathbf{B}_{1}$ and
$\mathbf{B}_{2}$, the decay rate of the eigenvalues, and the signal dimension
$N$.
#### III-B1 KL Divergence
To better understand (46), let us first check the Kullback-Leibler (KL)
divergence from the first Gaussian distribution to the second
$\displaystyle D_{KL}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int\left(\mathbf{x}^{T}\Sigma_{2}^{-1}\mathbf{x}-\mathbf{x}^{T}\Sigma_{1}^{-1}\mathbf{x}\right)f_{1}(\mathbf{x})d\mathbf{x}$
(47) $\displaystyle=$
$\displaystyle\frac{1}{2}\textrm{Tr}(\Sigma_{2}^{-1}\Sigma_{1}-\mathbf{I}_{N})=\frac{1}{2}(\textrm{Tr}(\Sigma_{2}^{-1}\Sigma_{1})-N),$
(48)
where $\mathbf{I}_{N}$ denotes the $N\times N$ identify matrix, and the second
equality holds since
$E[\mathbf{x}^{T}\mathbf{A}\mathbf{x}]=\textrm{Tr}(\mathbf{A}\Sigma)$ if
$\mathbf{x}\sim\mathcal{N}(\mathbf{0},\Sigma)$ [30]. Comparing (47) and (46),
we observe that $D_{KL}$ is monotonic relative to $P_{c}^{o}$. Analyzing the
behavior of $D_{KL}$ as a function of the two Gaussians thus helps to
understand that of $P_{c}^{o}$.
Inserting (45) into (48) leads to
$D_{KL}=\frac{1}{2}(\textrm{Tr}(\mathbf{B}_{2}\mathbf{S}^{-1}\mathbf{B}_{2}^{T}\mathbf{B}_{1}\mathbf{S}\mathbf{B}_{1}^{T})-N)=\frac{1}{2}(\textrm{Tr}(\mathbf{C}\mathbf{S}\mathbf{C}^{T}\mathbf{S}^{-1})-N),$
(49)
where $\mathbf{C}=\mathbf{B}_{2}^{T}\mathbf{B}_{1}$, and the second equality
follows from the cyclic permutation invariance property of the trace
$\textrm{Tr}(\mathbf{A}\mathbf{B}\mathbf{C})=\textrm{Tr}(\mathbf{C}\mathbf{B}\mathbf{A})$.
Note that $\mathbf{C}$ is an orthogonal matrix:
$\mathbf{C}^{T}\mathbf{C}=\mathbf{I}_{N}$. Maximizing $D_{KL}$ with respect to
$\mathbf{B}_{1}$ and $\mathbf{B}_{2}$ is therefore equivalent to maximizing
$\textrm{Tr}(\mathbf{C}\mathbf{S}\mathbf{C}^{T}\mathbf{S}^{-1})$ with respect
to $\mathbf{C}$. The following lemma shows that in dimension two, $D_{KL}$ is
maximized when the first principal directions of the two Gaussians are
orthogonal, and moreover, the maximum divergence increases as the Gaussians
become more anisotropic.
###### Lemma 2.
Let $\mathbf{B}_{1}$ and $\mathbf{B}_{2}$ be respectively the PCA bases
($\Sigma_{1}=\mathbf{B}_{1}\mathbf{S}\mathbf{B}_{1}^{T}$ and
$\Sigma_{2}=\mathbf{B}_{2}\mathbf{S}\mathbf{B}_{2}^{T}$) of two centered 2D
Gaussian distributions $\mathcal{N}(\mathbf{0},\Sigma_{1})$ and
$\mathcal{N}(\mathbf{0},\Sigma_{2})$, and
$\mathbf{S}=\left[\begin{array}[]{cc}\lambda_{1}&0\\\ 0&\lambda_{2}\\\
\end{array}\right]$, with $\lambda_{1}>0$ and $\lambda_{2}>0$ their common
eigenvalues. The KL divergence from the first Gaussian distribution to the
second (47) has a maximum value
$D_{KL}^{\max}=\max_{\mathbf{B}_{1},\mathbf{B}_{2}}D_{KL}=\frac{1}{2}\left(\frac{\lambda_{2}}{\lambda_{1}}+\frac{\lambda_{1}}{\lambda_{2}}\right),$
(50)
which is obtained when
$\mathbf{B}_{2}^{T}\mathbf{B}_{1}=\left[\begin{array}[]{cc}0&1\\\ 1&0\\\
\end{array}\right]$.
Let the determinant of the covariance matrices
$|\Sigma_{1}|=|\Sigma_{2}|=\lambda_{1}\lambda_{2}$ further be assumed given.
Then $D_{KL}^{\max}$ is minimized as $\lambda_{1}=\lambda_{2}$, and it
increases as the ratio between $\lambda_{1}$ and $\lambda_{2}$ increases.
###### Proof.
The first part of the lemma can be easily checked by maximizing $D_{KL}$ in
(49) with respect to the 2D orthogonal matrix
$\mathbf{C}=\mathbf{B}_{2}^{T}\mathbf{B}_{1}$ and writing
$\mathbf{C}=\left[\begin{array}[]{cc}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\\\ \end{array}\right]$. The second part is verified via
a direct observation of (50). ∎
Figure 4-(a) plots $D_{KL}$ as a function of the angle $\theta$ between the
first principal components of the two 2D Gaussians going from $5^{\circ}$ to
$90^{\circ}$, with different eigenvalue ratios $\lambda_{1}/\lambda_{2}$ from
5 to 100. As indicated by Lemma 2, given $\lambda_{1}/\lambda_{2}$, $D_{KL}$
increases as $\theta$ increases. At a given $\theta$, larger
$\lambda_{1}/\lambda_{2}$ leads to larger $D_{KL}$.
The analysis in higher dimension is more difficult, however, one can check via
a greedy optimization that
$\mathbf{C}=\mathbf{B}_{2}^{T}\mathbf{B}_{1}=\begin{bmatrix}0&\cdots&\cdots&0&1\\\
\vdots&\cdots&\iddots&1&0\\\ \vdots&\iddots&\iddots&\iddots&\vdots\\\
0&1&\iddots&\cdots&\vdots\\\ 1&0&\cdots&\cdots&0\\\ \end{bmatrix},$ (51)
with ones along the anti-diagonal, and zeros elsewhere, gives a local maximum
of (49). In other words, the two Gaussians being “orthogonal” one another,
i.e., the alignment of the first principal component of one Gaussian to the
last principal component of the other, the second principal component of the
former to the second to last principal component of the latter, and so on,
leads to a local maximization of (49). This can be observed by inserting
$\mathbf{C}=\begin{bmatrix}C_{11}&\ldots&C_{1N}\\\ \vdots&\ddots&\vdots\\\
C_{N1}&\ldots&C_{NN}\\\ \end{bmatrix}$
in (49), which gives
$D_{KL}=\frac{1}{2}(\sum_{m=1}^{N}\frac{1}{\lambda_{m}}\sum_{n=1}^{N}\lambda_{n}C_{mn}^{2}-N).$
(52)
A greedy maximization of (52) with respect to $\mathbf{C}$ is calculated by
scanning $\mathbf{C}$ row by row from bottom to top, observing that
$1/\lambda_{m}$ decreases as $m$ goes from $N$ to $1$, and at each $m$-th row
scanning $C_{mn}$ from left to right, observing that $\lambda_{n}$ increases
as $n$ goes from $1$ to $N$, taking into account the constraint
$\mathbf{C}^{T}\mathbf{C}=\mathbf{I}_{N}$. A similar observation of (52) shows
that when $D_{KL}$ is at the local maximum with $\mathbf{C}$ equal to (51),
its value increases as the eigenvalues decay faster from $\lambda_{1}$ to
$\lambda_{N}$.
#### III-B2 Correct Model Seletion Probability
The probability of correct oracle model selection $P_{c}^{o}$ (46) is now
evaluated via Monte Carlo simulations. Figure 4-(b) plots $P_{c}^{o}$ as a
function the angle $\theta$ between the first principal components of the two
2D Gaussians going from $5^{\circ}$ to $90^{\circ}$, with different eigenvalue
ratios $\lambda_{1}/\lambda_{2}$ from 5 to 100. As illustrated in Figure 4,
$P_{c}^{o}$ shows a behavior similar to the KL-divergence $D_{KL}$ as a
function of $\theta$ and of $\lambda_{1}/\lambda_{2}$: Given
$\lambda_{1}/\lambda_{2}$, $P_{c}^{o}$ increases as $\theta$ increases; at a
given $\theta$, larger $\lambda_{1}/\lambda_{2}$ leads to larger $P_{c}^{o}$.
In contrast to $D_{KL}$, whose value is roughly proportional to
$\lambda_{1}/\lambda_{2}$ (as $\lambda_{1}\gg\lambda_{2}$), $P_{c}^{o}$
presents a saturation effect: $\lambda_{1}/\lambda_{2}$ values larger than
about 40 lead to comparable $P_{c}^{o}$ that increases rapidly as a function
of $\theta$, converging to a high value around 0.9; for
$\lambda_{1}/\lambda_{2}$ smaller than about 40, on the other hand,
$P_{c}^{o}$ reduces quickly as $\lambda_{1}/\lambda_{2}$ shrinks towards 1.
|
---|---
(a) | (b)
Figure 4: (a) The KL-divergence (47) between two 2D Gaussians, as a function
the angle $\theta$ between the first principal components of the two Gaussians
going from $5^{\circ}$ to $90^{\circ}$, with different eigenvalue ratios
$\lambda_{1}/\lambda_{2}$ from 5 to 100. (b) The same for the probability of
correct oracle model selection $P_{c}^{o}$ (46).
Figure 5 shows the probability of correct oracle model selection $P_{c}^{o}$
(46) in higher dimensions, under the condition that (51) holds, i.e., the two
Gaussians are “orthogonal.” A power decay of the eigenvalues (11) is assumed
in the Monte Carlo simulations. In different signal dimensions $N$ from $2$ to
$20$, $P_{c}^{o}$ as a function of the eigenvalue decay parameter $\alpha$ is
plotted. For a given dimension, $P_{c}^{o}$ increases as $\alpha$ increases,
i.e., as the eigenvalues decay faster so that the Gaussians are more
anisotropic. It is important to remark that, with the same $\alpha$,
$P_{c}^{o}$ rapidly increases as the signal dimension $N$ increases, which
shows that anisotropic Gaussians with their energy concentrated in the first
few dimensions are more separate in higher dimension.
---
Figure 5: The probability of correct oracle model selection $P_{c}^{o}$ (46)
between two Gaussians, as a function of the eigenvalue decay parameter
$\alpha$ from 1 to 5, for different signal dimensions $N$ from 2 to 20. The
two Gaussians satisfy (51).
### III-C Model Selection and Signal Reconstruction
In SCS, the model selection (43) is calculated with the decoded signals (42)
and not the ideal ones. Assume without loss of generality that the signals
follow the first Gaussian distribution
$\mathbf{x}~{}\sim\mathcal{N}(\mathbf{0},\Sigma_{1})$. This section checks via
Monte Carlo simulations the probability of correct model selection (43)
calculated with the decoded signals
$\tilde{\mathbf{x}}_{1}=\Delta_{1}\Phi\mathbf{x}$ and
$\tilde{\mathbf{x}}_{2}=\Delta_{2}\Phi\mathbf{x}$,
$P_{c}=(E_{\Phi})\left(\int_{\tilde{\mathbf{x}}_{1}^{T}\Sigma_{1}^{-1}\tilde{\mathbf{x}}_{1}<\tilde{\mathbf{x}}_{2}^{T}\Sigma_{2}^{-1}\tilde{\mathbf{x}}_{2}}f_{1}(\mathbf{x})d\mathbf{x}\right)=(E_{\Phi})\left(\int\textrm{sign}\left(\tilde{\mathbf{x}}_{2}^{T}\Sigma_{2}^{-1}\tilde{\mathbf{x}}_{2}-\tilde{\mathbf{x}}_{1}^{T}\Sigma_{1}^{-1}\tilde{\mathbf{x}}_{1}\right)f_{1}(\mathbf{x})d\mathbf{x}\right),$
(53)
where the expectation is with respect to $\Phi$ if one random $\Phi$ is
independently drawn for each $\mathbf{x}$. We also investigate the MSE of the
resulting signal reconstruction,
$E_{\mathbf{x},(\Phi)}\|\mathbf{x}-\tilde{\mathbf{x}}\|^{2}_{2}=(E_{\Phi})\left(\int_{\tilde{\mathbf{x}}_{1}^{T}\Sigma_{1}^{-1}\tilde{\mathbf{x}}_{1}<\tilde{\mathbf{x}}_{2}^{T}\Sigma_{2}^{-1}\tilde{\mathbf{x}}_{2}}\|\mathbf{x}-\tilde{\mathbf{x}}_{1}\|^{2}_{2}f_{1}(\mathbf{x})d\mathbf{x}+\int_{\tilde{\mathbf{x}}_{1}^{T}\Sigma_{1}^{-1}\tilde{\mathbf{x}}_{1}\geq\tilde{\mathbf{x}}_{2}^{T}\Sigma_{2}^{-1}\tilde{\mathbf{x}}_{2}}\|\mathbf{x}-\tilde{\mathbf{x}}_{2}\|^{2}_{2}f_{1}(\mathbf{x})d\mathbf{x}\right),$
(54)
as a function of the number of sensing measurements $M$ and the properties of
the Gaussian distributions.
Figure 6 shows the probability of correct model selection $P_{c}$ (53) and the
MSE of signal reconstruction (54) as a function of the number of measurements
$M$ and the signal dimension $N$. Figure 6-(a) plots $P_{c}$ as a function of
$M$ going from $1$ to $N$, with different $N$ values from 2 to 15, assuming
that (51) holds, i.e., the two Gaussians are “orthogonal.” A power decay model
of the eigenvalues (11) with a typical decay parameter $\alpha=3$ is assumed
in the simulations. A random Gaussian matrix realization $\Phi$ is drawn
independently to sense each signal. As expected, $P_{c}$ increases as $M$ goes
from 1 to $N$, i.e., as more measurements are dedicated. The signal dimension
$N$ plays an important role. With only $M=1$ measurement, the model selection
is uniformly random ($P_{c}\approx 0.5$), independent of the signal dimensions
$N$. At an extremely low dimension $N=2$, even with $M=N$ measurements (which
leads to perfect signal reconstruction, as if in the “oracle” case described
in Section III-B), $P_{c}$ remains lower than 0.8. 444We observe that a
mistake in the model selection will not necessarily lead to a mistake in the
reconstruction, e.g., flat image patches can often be recovered by multiple
different models. When $N$ goes higher, $P_{c}$ rapidly increases converging
towards 1 as $M$ increases. After $N$ stands above a certain value (about 10
in this example, note that for the image examples in the next section $N=64$),
$P_{c}$ converges very close to 1 as far as $M$ reaches a fixed value (about
8) independent of $N$. This indicates that accurate model selection can be
achieved with very low sampling rates $M/N$, given that the energy of the
signals is concentrated in the first few principal dimensions. In signal
sampling, one is more interested in the signal reconstruction error than model
selection. Figure 6-(b) similarly shows the MSE of the decoded signals (54)
(normalized by the ideal signal energy). The MSE decreases as $M$ increases,
and it goes to 0 as $M=N$. At high dimensions $N$ (over about 10), almost
perfect signal reconstruction is obtained as far as $M$ reaches a fixed value
(about 8).
|
---|---
(a) | (b)
Figure 6: (a) The probability of correct model selection (53) as a function
the of the number of measurements $M$ from $1$ to the signal dimension $N$,
with $N$ going from 2 to 15. (b) The same for MSE (54) (normalized by the
ideal signal energy) of the decoded signals.
Similarly, Figure 7 plots the probability of correct model selection $P_{c}$
(53) as well as the MSE of the decoded signals (54) (normalized by the ideal
signal energy), as a function of the measurements $M$ going from 1 to the
signal dimension $N=10$, with different eigenvalue decay parameter $\alpha$
from 1 to 5. As $\alpha$ increases, i.e., as the eigenvalues decay faster,
$P_{c}$ and MSE respectively converge to 1 and 0 at a faster rate as $M$ goes
from 1 to $N$.
|
---|---
(a) | (b)
Figure 7: (a) The probability of correct model selection (53), as a function
of the number of measurements $M$ from $1$ to the signal dimension $N=10$,
with different eigenvalue decay parameter $\alpha$ from 1 to 5. (b) The same
for MSE (54) (normalized by the ideal signal energy) of the decoded signals.
In summary, this section shows that the accuracy of the Gaussian model
selection (43) in GMM-based SCS is influenced by a number of factors including
the geometry of the Gaussian distributions in the GMM, the signal dimension,
and the number of sensing measurements. More accurate model selection is
obtained as the Gaussians distributions are more “orthogonal” one another, as
each of the Gaussians is more anisotropic, as the signals are in a higher
dimension given that the energy of the signals are concentrated in the first
few dimensions, and as the number of sensing measurements increases.
## IV SCS with GMM – Algorithm and Experiments
The GMM-based SCS decoder described in Section III-A assumes that the means
and the covariances of the Gaussian distributions
$\\{\mathcal{N}(\mu_{j},\Sigma_{j})\\}_{1\leq j\leq J}$ in the GMMs are known.
However, in real sensing applications, these parameters are unavailable.
Following [36], this ection presents a maximum a posteriori expectation-
maximization (MAP-EM) algorithm [3] that iteratively estimates the Gaussian
parameters and decodes the signals. GMM-based SCS calculated with the MAP-EM
algorithm is applied in real signal sensing, and is compared with conventional
CS based on sparse models.
### IV-A MAP-EM Algorithm
The MAP-EM algorithm is an iterative procedure that alternates between two
steps:
#### IV-A1 E-step
Assuming that the estimates of the Gaussian parameters
$\\{(\tilde{\mu}_{j},\tilde{\Sigma}_{j})\\}_{1\leq j\leq J}$ are known
(following the previous M-step), the E-step calculates the MAP signal
estimation and model selection for all the signals, following (41)–(44) .
#### IV-A2 M-step
Assuming that the Gaussian model selection $\tilde{j}$ and the signal estimate
$\tilde{\mathbf{x}}$ are known for all the signals (following the previous
E-step), the M-step estimates (updates) the Gaussian models
$\\{(\tilde{\mu}_{j},\tilde{\Sigma}_{j})\\}_{1\leq j\leq J}$.
Let $\mathbf{x}_{i}$, $\mathbf{y}_{i}$, $\tilde{\mathbf{x}}_{i}$ and
$\tilde{j}_{i}$ respectively denote the $i$-th signal in the collection, its
coded version, its estimate, and its estimated Gaussian model index, $1\leq
i\leq I$. Let $\mathcal{C}_{j}$ be the ensemble of the signal indices $i$ that
are assigned to the $k$-th Gaussian model, i.e.,
$\mathcal{C}_{j}=\\{i:\tilde{j}_{i}=j\\}$, and let $|\mathcal{C}_{j}|$ be its
cardinality. The parameters of each Gaussian model are estimated with the
maximum likelihood estimate using all the signals assigned to that Gaussian
model,
$(\tilde{\mu}_{j},\tilde{\Sigma}_{j})=\arg\max_{\mu_{j},\Sigma_{j}}\log
f(\\{\tilde{\mathbf{x}}_{i}\\}_{i\in\mathcal{C}_{j}}|\mu_{j},\Sigma_{j}).$
(55)
With the Gaussian model (40) , it is well-known that the resulting estimate is
the empirical estimate
$\tilde{\mu}_{j}=\frac{1}{|\mathcal{C}_{j}|}\sum_{i\in\mathcal{C}_{j}}\tilde{\mathbf{x}}_{i}~{}~{}\textrm{and}~{}~{}\tilde{\Sigma}_{j}=\frac{1}{|\mathcal{C}_{j}|}\sum_{i\in\mathcal{C}_{j}}(\tilde{\mathbf{x}}_{i}-\tilde{\mu}_{j})(\tilde{\mathbf{x}}_{i}-\tilde{\mu}_{j})^{T}.$
(56)
The computational complexity of the MAP-EM algorithm is dominated by the
matrix inversion $(\Phi\Sigma_{j}\Phi^{T})^{-1}$ in (42) in the E-step. It can
be implemented with $M^{3}/3$ flops through a Cholesky factorization [7]. With
$J$ Gaussian models, the complexity per iteration is therefore dominated by
$JM^{3}/3$ flops.
As the MAP-EM algorithm described above iterates, the MAP probability of the
observed signals $f(\\{\tilde{\mathbf{x}}_{i}\\}_{1\leq i\leq
I}|\\{\mathbf{y}_{i}\\}_{1\leq i\leq
I},\\{\tilde{\mu}_{j},\tilde{\Sigma}_{j}\\}_{1\leq j\leq J})$ always
increases. This can be observed by interpreting the E- and M-steps as a
coordinate descent optimization [22].
The algorithm initialization and the number $J$ of Gaussians in GMM can be
selected according to the type of signals of interest. For sensing natural
images, a geometry-motivated initialization as detailed in [36] will be
applied in the experiments.
### IV-B Experiments
The GMM-based SCS is applied in real image sensing, and is compared with
conventional CS based on sparse models. Following standard practice, an image
is decomposed into $\sqrt{N}\times\sqrt{N}=8\times 8$ local patches
$\\{\mathbf{x}_{i}\\}_{1\leq i\leq I}$ (an image patch is reshaped to and
considered as a vector) [2, 26, 36], which are assumed to follow a GMM [36].
SCS samples each patch $\mathbf{y}_{i}=\Phi_{i}\mathbf{x}_{i}$, with a
possibly different $\Phi_{i}$ for each $\mathbf{x}_{i}$. The decoder is
implemented with the MAP-EM algorithm, initialized with $J=19$ geometry-
motivated Gaussian models, each capturing a local direction [36]. The
algorithm typically converges within 3 iterations. No database is used, and
all the parameters and reconstruction are learned from the compressed sensed
image alone.
The dictionary for conventional CS is learned with K-SVD [2] from 720,000
image patches, extracted from the entire standard Berkeley segmentation
database containing 300 natural images [28]. In image estimation and sensing,
learned dictionaries have been shown to produce better results than off-the-
shelf ones [2, 19, 26]. The decoder is calculated with the $l_{1}$
minimization [34] implemented in [25]. Three standard images Lena ($512\times
512$), House ($256\times 256$), and Peppers ($512\times 512$), as illustrated
in Figure 8, are used in the experiments.
---
Figure 8: From left to right. Three standard images used for the experiments: Lena, House, and Peppers. |
---|---
(a) | (b)
Figure 9: (a) PSNR (dB) vs sampling rate for SCS and CS using Gaussian and
random subsampling sensing matrices on image patches extracted from Lena. (b)
PSNR (dB) vs sampling rate for SCS and CS using Gaussian sensing matrices on
image patches extracted from House and Peppers.
Figure 9 (a) shows the sensing performance on about 260,000 (sliding) patches,
regarded as signals $\mathbf{x}_{i}$, extracted from Lena. The PSNRs generated
by SCS and CS using Gaussian and random subsampling sensing matrices, one
independent realization for each patch, are plotted as a function of the
sampling rate $M/N$. At the same sampling rate, SCS outperforms SC. The gain
increases from about 0.5 dB at very low sampling rates ($M/N\approx 0.1$),
learning a GMM from the poor-quality measured data being more challenging, to
more than 3.5 dB at high sampling rates ($M/N\approx 0.5$). (SC using an
“oracle” dictionary learned from the ideal Lena itself, undoable in practice,
improves its performance from 0.2 dB at low sampling rates to 1.3 dB at high
sample rates, still lower than SCS.) For both SCS and CS, Gaussian and random
subsampling matrices lead to similar PSNRs at low sampling rates ($M/N<0.25$),
and at higher sampling rates Gaussian sensing gains by about 0.5 dB. Recall
that SCS is not just more accurate and significantly faster, but also uses
only the compressed image, while conventional CS uses a pre-learned dictionary
from a large database.
Figure 9 (b) further compares SCS with CS on sliding patches, regarded as
signals, extracted from Peppers (260,000 patches) and House (62,000 patches).
One independent Gaussian matrix realization is applied to sense each patch.
Similar results as on the patches from Lena are observed. At the same sampling
rate, SCS outperforms SC. The gain is smaller (about 1 dB) at very low
sampling rates ($M/N\approx 0.1$), and becomes substantial (about 3 dB) at
high sampling rates ($M/N\approx 0.5$).
Figure 10 illustrates some typical patches with geometry. The ground-truth
patches are shown in the first row, and the patches reconstructed by
conventional CS and SCS, all sensed with Gaussian matrices at a sampling rate
$M/N=1/4$, are respectively illustrated in the second and the third row. Both
CS and SCS lead to accurate reconstruction in uniform regions. SCS outperforms
CS on the more geometrical parts, and the improvement is significant on the
fine contours (the 2nd, 3rd and 7th patches).
---
Figure 10: Some typical $8\times 8$ patches with geometry. First row: ground-
truth patches. Second and third rows: patches reconstructed by conventional CS
and SCS respectively, all sensed at a sampling rate $M/N=1/4$ with Gaussian
matrices.
In most image sensing applications, one is interested in reconstructing whole
images instead of individual patches. Aggregating non-overlapped patches to a
whole images produces block artifacts, as illustrated in Figure 11. It is well
known that averaging overlapped reconstructed patches not only removes the
block artifacts, but also considerably improves the image estimation [2, 26,
36]. However, compressed sensing only allows sensing non-overlapping patches,
since sensing overlapping patches would dramatically increase the sampling
rate. Nevertheless, overlapped reconstructed patches are computable if the
sensing operators, performed on non-overlapped patches, are random subsampling
matrices, which are diagonal operators (one non-zero entry per row). (The
reconstruction is then equivalent to solving an inpainting problem [2, 36].)
Figure 11 shows some typical regions in Lena. The overlapped reconstruction,
which further supports the search for performance on average as in the
proposed SCS, removes the block artifacts and significantly improves the
reconstructed image. Figure 12 plots the PSNRs on the whole image Lena
generated by SCS using random subsampling matrices and overlapped
reconstruction are plotted, in comparison with those obtained using Gaussian
sensing matrices and non-overlapped reconstruction, at different sampling
rates. The former improves from about 3.5 dB, at low sampling rates, to 1.5
dB, at high sampling rates, at a cost of $N=64$ times computation.
| |
---|---|---
Ground truth | No.-ovl. rec. 30.82 dB | Ovl. rec. 34.02 dB
| |
Ground truth | No.-ovl. rec. 24.72 dB | Ovl. rec. 27.87 dB
Figure 11: From left to right. Zoomed crops from Lena, reconstructed images by
SCS using Gaussian sensing matrices and non-overlapping reconstruction, and by
SCS using subsampling random matrices and overlapping reconstruction. The
image is sensed on non-overlapped patches at a sampling rate of $M/N=0.25$.
Local PSNRs are reported.
---
Figure 12: PSNR (dB) vs sampling rate (on the whole image Lena), for SCS using
Gaussian sensing matrices with non-overlapping reconstruction, and subsampling
random matrices with overlapping reconstruction.
## V Conclusion
Statistical compressed sensing (SCS) based on statistical signal models has
been introduced. As opposed to conventional compressed sensing that aims at
efficiently sensing and accurately reconstructing one signal at a time, SCS
deals simultaneously with a collection of signals. While CS assumes signal
sparse models, SCS is based on a more general Bayesian assumption that signals
follow a statistical distribution. SCS based on Gaussian models has been
investigated in depth. It has been shown that based on a single Gaussian
model, with Gaussian or Bernoulli sensing matrices of $\mathcal{O}(k)$
measurements, considerably smaller than the $\mathcal{O}(k\log(N/k))$ required
by conventional CS, where $N$ is the signal dimension, and with an optimal
decoder implemented with linear filtering, significantly faster than the
pursuit decoders applied in conventional CS, the error of SCS is tightly upper
bounded by a constant times the best $k$-term approximation error, with
overwhelming probability. The failure probability is also significantly
smaller than that of conventional CS. Stronger yet simpler results, derived
from a new RIP in expectation property further show that for any sensing
matrix, the error of Gaussian SCS is upper bounded by a constant times the
best $k$-term approximation with probability one, and the bound constant can
be efficiently calculated. For Gaussian mixture models (GMMs) that assume
multiple Gaussian distributions, and that each signal follows one of them with
an unknown index, a piecewise linear estimator is introduced to decode SCS.
The accuracy of model selection, which is at the heart of the piecewise linear
decoder, is analyzed in terms of the properties of the Gaussian distributions
and the number of the sensing measurements. A MAP-EM algorithm that
iteratively estimates the Gaussian models and decodes the compressed signals
is presented for GMM-based SCS. Applications of GMM-based SCS in real image
sensing has been shown. Comparing with conventional CS, SCS leads to improved
results, at a considerably lower computational cost.
This line of research opens numerous new questions in compressed sensing, from
the formal development of bounds in the compressed domain model selection (see
also [9]), to the study of model parameters estimation in the compressed
domain and the extension of the results here reported to non-Gaussian
distributions. The work here reported also shows that compressed sensing is
significant beyond sparse signal models, generating the natural question of
what type of models can benefit from such sensing scenario.
Acknowledgements: Work partially supported by NSF, ONR, NGA, ARO, DARPA, and
NSSEFF. The authors thank very much Stéphane Mallat for co-developing the GMM
framework reported in [36] for solving inverse problems.
## References
* [1] D. Achlioptas. Database-friendly random projections: Johnson-Lindenstrauss with binary coins. Journal of Computer and System Sciences, 66(4):671–687, 2003.
* [2] M. Aharon, M. Elad, and A. Bruckstein. K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation. IEEE Trans. on Signal Proc., 54(11):4311, 2006.
* [3] S. Allassonniere, Y. Amit, and A. Trouvé. Towards a coherent statistical framework for dense deformable template estimation. J.R. Statist. Soc. B, 69(1):3–29, 2007.
* [4] R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin. A simple proof of the restricted isometry property for random matrices. Constructive Approximation, 28(3):253–263, 2008.
* [5] R.G. Baraniuk, V. Cevher, M.F. Duarte, and C. Hegde. Model-based compressive sensing. IEEE Trans. on Info. Theo., 56(4):1982–2001, 2010.
* [6] R.G. Baraniuk and M.B. Wakin. Random projections of smooth manifolds. Foundations of Comp. Math., 9(1):51–77, 2009.
* [7] S.P. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, 2004.
* [8] A. Buades, B. Coll, and J.M. Morel. A review of image denoising algorithms, with a new one. Multiscale Modeling and Simulation, 4(2):490–530, 2006.
* [9] R. Calderbank, S. Jafarpour, and Kent. J. Finding needles in compressed haystacks. in Compressed Sensing, Y. Eldar and G. Kutynok, Eds., Cambridge University Press, to appear, 2011.
* [10] E. Candès and J. Romberg. Sparsity and incoherence in compressive sampling. Inverse Problems, 23:969, 2007.
* [11] E.J. Candes, Y.C. Eldar, D. Needell, , and P. Randall. Compressed sensing with coherent and redundant dictionaries. In Press, Applied and Computational Harmonic Analysis, 2011.
* [12] E.J. Candès, J. Romberg, and T. Tao. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans. on Info. Theo., 52(2):489–509, 2006.
* [13] E.J. Candès and T. Tao. Decoding by linear programming. IEEE Trans. on Info. Theo., 51(12):4203–4215, 2005.
* [14] E.J. Candès and T. Tao. Near-optimal signal recovery from random projections: Universal encoding strategies? IEEE Trans. on Info. Theo., 52(12):5406–5425, 2006.
* [15] G. Chechik, G. Heitz, G. Elidan, P. Abbeel, and D. Koller. Max-margin classification of incomplete data. In Proceedings of the Advances in Neural Information Processing Systems, page 233, 2007.
* [16] M. Chen, J. Silva, J. Paisley, C. Wanng, D. Dunson, and L. Carin. Compressive sensing on manifolds using a nonparametric mixture of factor analyzers: Algorithm and performance bounds. IEEE Trans. on Signal Proc., 58(12):6140–6155, 2010.
* [17] A. Cohen, W. Dahmen, and R. DeVore. Compressed sensing and best k-term approximation. J. of the Am, Math. Soc., 22(1):211–231, 2009.
* [18] D.L. Donoho. Compressed sensing. IEEE Trans. on Info. Theo., 52, 2006.
* [19] J.M. Duarte-Carvajalino and G. Sapiro. Learning to sense sparse signals: Simultaneous sensing matrix and sparsifying dictionary optimization. IEEE Transactions on Image Processing, 18(7):1395–1408, 2009.
* [20] R.O. Duda, P.E. Hart, and D.G. Stork. Pattern Classification. Wiley-Interscience, 2000.
* [21] Y.C. Eldar and M. Mishali. Robust recovery of signals from a structured union of subspaces. IEEE Trans. on Info. Theo., 55(11):5302–5316, 2009.
* [22] R.J. Hathaway. Another interpretation of the EM algorithm for mixture distributions. Statistics & Probability Letters, 4(2):53–56, 1986.
* [23] S.M. Kay. Fundamentals of Statistical Signal Processing, Volume 1: Estimation Theory. Prentice Hall, 1998.
* [24] F. Léger, G. Yu, and G Sapiro. Efficient matrix completion with Gaussian models. Submitted, arxiv.org/abs/1010.4050, Oct., 2010.
* [25] J. Mairal, F. Bach, J. Ponce, and G. Sapiro. Online dictionary learning for sparse coding. In Proceedings of the 26th Annual International Conference on Machine Learning, pages 689–696. ACM, 2009.
* [26] J. Mairal, M. Elad, and G. Sapiro. Sparse representation for color image restoration. IEEE Trans. on Image Proc., 17, 2008.
* [27] S. Mallat. A Wavelet Tour of Signal Processing: The Sparse Way, 3rd Ddition. Academic Press, 2008.
* [28] D. Martin, C. Fowlkes, D. Tal, and J. Malik. A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics. In Proc. ICCV, 2001.
* [29] R. Masiero, G. Quer, D. Munaretto, M. Rossi, J. Widmer, and M. Zorzi. Data acquisition through joint compressive sensing and principal component analysis. In Global Telecommunications Conference, 2009. GLOBECOM 2009. IEEE, pages 1–6. IEEE, 2009.
* [30] K.B. Petersen and M.S. Pedersen. The matrix cookbook. Technical University of Denmark, 2006.
* [31] G. Peyré. Best basis compressed sensing. IEEE Transactions on Signal Processing, 58(5):2613–2622, 2010.
* [32] C. Rother, V. Kolmogorov, and A. Blake. Grabcut: Interactive foreground extraction using iterated graph cuts. ACM Transactions on Graphics (TOG), 23(3):309–314, 2004.
* [33] M. Talagrand. A new look at independence. Ann. Prob., 24:1, 1996.
* [34] R. Tibshirani. Regression shrinkage and selection via the lasso. J. of the Royal Stat. Society, pages 267–288, 1996.
* [35] G. Yu, S. Mallat, and E. Bacry. Audio denoising by time-frequency block thresholding. IEEE Trans. on Signal Proc., 56(5):1830, 2008.
* [36] G. Yu, G. Sapiro, and S. Mallat. Solving inverse problems with piecewise linear estimators: from Gaussian mixture models to structured sparsity. Submitted, arxiv.org/abs/1006.3056, June, 2010.
|
arxiv-papers
| 2011-01-30T17:16:55 |
2024-09-04T02:49:16.728863
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guoshen Yu and Guillermo Sapiro",
"submitter": "Guoshen Yu",
"url": "https://arxiv.org/abs/1101.5785"
}
|
1101.5980
|
# Notes on a particular Weyl Algebra
Giuseppe Iurato
###### Abstract
By means of the notions of cross product algebras of the theory of quantum
groups, in the context of classical Hopf algebra structures, we deduce some
known structures of Weyl algebras type (as the Drinfeld quantum double, the
restricted Heisenberg double, the generalized Schrödinger representation, and
so on) that may be considered as a non-trivial examples of quantum groups
having physical meaning, starting from a particular example of groupoid
motivated by elementary quantum mechanics.
1\. Introduction
In the paper [Iu], following a suggestion of Alain Connes (see [Co], I.1), it
has been introduced a particular, simple groupoid, the so-called Heisenberg-
Born-Jordan EBB-groupoid (or HBJ EBB-groupoid), whose physical motivations
were, mainly, of spectroscopical nature.
An E-groupoid (in the notations of [Iu]) is an algebraic system of the type
$(G,G^{(0)},r,s,\star)$, with $G,G^{(0)}$ non-void sets, $G^{(0)}\subseteq G$,
$G^{(0)}$ set of unities, $r,s:G\rightarrow G^{(0)}$ and
$\star:G^{(2)}\rightarrow G$ partial groupoid law defined on
$G^{(2)}=\\{(g_{1},g_{2})\in G\times G,s(g_{1})=r(g_{2})\\}$, satisfying the
set of axioms described in [Iu], § 1.
The HBJ EBB-groupoid is a particular E-groupoid that has been denoted with
$\mathcal{G}_{HBJ}(\mathcal{F}_{I})=(\Delta\mathcal{F}_{I},\mathcal{F}_{I},r,s,\tilde{+})$,
where $\mathcal{F}_{I}=\\{\nu_{i};\nu_{i}\in\mathbb{R}^{+},i\in
I\subseteq\mathbb{N}\\}$ is the set of energy levels of a certain
spectroscopic physical system,
$\Delta\mathcal{F}_{I}=\\{\nu_{ij};\nu_{ij}=\nu_{i}-\nu_{j},i,j\in
I\subseteq\mathbb{N}\\}$, $r:(i,j)\rightarrow i$ and $s:(i,j)\rightarrow j$
are the range and source maps, respectively, and
$\nu_{ij}\tilde{+}\nu_{jk}=\nu_{ik}$ is the (partial) groupoid law as
algebraic result of the Ritz-Rydberg combination principle.
In [Iu], it has been only considered the structure of a no finitely generated
groupoid algebra on $\mathcal{G}_{HBJ}(\mathcal{F}_{I})$, say
$\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))=(\langle\Delta(\mathcal{F}_{I})\rangle,+,\cdot,\ast)$,
respect to an arbitrary commutative field $\mathbb{K}$ and a non-commutative
convolution product $\ast$; subsequently, it has been built up a (trivial)
structure of braided non-commutative Hopf algebra on it.
We claim that this last (albeit trivial) Hopf structure is the first and most
natural possible one, on such a groupoid algebra, because of the no
(algebraic) finiteness of this generated algebra (since card $I=\infty$, in
general).
Therefore, the main interest of the paper [Iu], must be searched in the
physical construction of the EBJ EBB-groupoid.
In this paper, we’ll try to build up other (less trivial) structures on this
special HBJ EBB-groupoid, through adapted methods and tools of the theory of
quantum groups, relative both to the infinite-dimensional case and finite-
dimensional case111The most interesting case, from the physical view-point, is
that finite-dimensional corresponding to card $I<\infty$, since any physical
spectroscopic system has a finite number of energy levels..
Furthermore, these structures will be introduced taking into account eventual
physical motivations.
The (above mentioned) natural structure of Hopf algebra on
$\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$ is given as
follows: coproduct $\Delta(x)=x\otimes x$, counit $\varepsilon(x)=1$, and
antipode the extended inversion map.
As already said, the first, natural structure of a braided (or
quasitriangular) non-commutative Hopf algebra on such an algebra, is trivially
given by the universal R-matrix $R=1\otimes 1$, whence a (trivial) example of
quantum group if one assume a braided (or quasitriangular) non-commutative
Hopf algebra as definition of quantum group. Instead, a non-trivial example of
quasitriangular Hopf algebra arises from Drinfeld quantum double constructions
(see § 6.).
There exists other definitions of a quantum group structure: for instance, if
we consider a non-commutative and non-cocommutative Hopf algebra as quantum
group, then a cross (or bicross, or double cross) product construction may
provide examples of such a quantum group, whereas, if we consider as special
’quantum objects’ the result of a non-degenerate dual pairing of Hopf
algebras, then a Heisenberg double may be taken as an example of quantum
group.
If one want to determine examples of these last structures starting from
$\mathcal{G}_{HBJ}(\mathcal{F}_{I})$, it is necessary, at first, examines the
possible dual structures of
$\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$, taking into
account the existence of some problems for this particular case study.
The first problem (that we’ll sketch at the paragraph 3.) is related to
dualization in the infinite-dimensional case, whereas the second problem222In
a certain sense, preliminary to the first one., because of the infinity of
$\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$, is due to the tentative of giving a
Hopf algebra structure to the $\mathbb{K}$-algebra of $\mathbb{K}$-valued
functions defined on the HBJ EBB-groupoid
$\mathcal{G}_{HBJ}(\mathcal{F}_{I})$, say
$\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$: in fact, on
this algebra (that is strictly correlated to the first problem of dualization
of $\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$, and
viceversa) it is a problematic question to define the right comultiplication
and counit, for the following reasons.
For a group $(\mathcal{G},\cdot)$, the comultiplication question do not
subsist in the finite-dimensional case, because of the natural identification
$\mathcal{F}_{\mathbb{K}}(\mathcal{G})\otimes\mathcal{F}_{\mathbb{K}}(\mathcal{G})\cong\mathcal{F}_{\mathbb{K}}(\mathcal{G}\times\mathcal{G});$
in such a case, a natural structure of Hopf algebra on
$\mathcal{F}_{\mathbb{K}}(\mathcal{G})$, is given by the following data :
1. 1.
coproduct
$\Delta:\mathcal{F}_{\mathbb{K}}(\mathcal{G})\rightarrow\mathcal{F}_{\mathbb{K}}(\mathcal{G}\times\mathcal{G})$,
given by $\Delta(f)(g_{1},g_{2})=f(g_{1}\cdot g_{2})$, for all
$g_{1},g_{2}\in\mathcal{G}$;
2. 2.
counit
$\varepsilon:\mathcal{F}_{\mathbb{K}}(\mathcal{G})\rightarrow\mathbb{K}$, with
$\varepsilon(f)=1$;
3. 3.
antipode
$S:\mathcal{F}_{\mathbb{K}}(\mathcal{G})\rightarrow\mathcal{F}_{\mathbb{K}}(\mathcal{G})$,
defined as $S(f)(g)=f(g^{-1})$ for all $g\in\mathcal{G}$,
where the functional laws in 1. and 2. are well-defined since, respectively,
the group law is totally defined in $\mathcal{G}$, and there exists a unique
unit.
Instead, if we consider a generic groupoid, these two questions remains
unsolved, in the finite-dimensional case too, both for the partial definition
of the groupoid law and for the existence of many unities: for these reasons,
the initial definitions 1. and 2. of above, are ill-posed in this case.
Nevertheless, it is possible to solve these last problems with some extensions
in the above definitions, remaining in the context of classical Hopf algebra
theory, but with minor usefulness of results.
Instead, in the new realm of the extended Hopf algebra structures, this
problem may be clarified and solved with fruitfulness, at least for the dual
$\mathcal{F}_{\mathbb{K}}^{\ast}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\subseteq\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$.
2\. Cross product algebras
The notions of cross product and bicrossproduct are important tools for an
algebraic setting of some fundamental structures of Quantum Mechanics. In this
paper, we’ll to do only with the notion of cross (or smash, or semidirect)
product.
Let $V_{\mathbb{K}}$ be a $\mathbb{K}$-linear space, $A$ a
$\mathbb{K}$-algebra and $\psi:A\otimes V\rightarrow V$ a $\mathbb{K}$-linear
map; if we pose $\psi(h\otimes v)=\psi_{h}(v)$, and if
$\psi_{ab}(v)=\psi_{a}(\psi_{b}(v)),\ \ \psi_{1}(v)=v,\ \ \forall a,b\in
A,\forall v\in V$, then $(A,V_{\mathbb{K}},\psi)$ is a left $A$-module on
$V_{\mathbb{K}}$; we say that $(A,V_{\mathbb{K}},\psi)$ is a left action of
$A$ on $V_{\mathbb{K}}$, or that $V_{\mathbb{K}}$ is a left $A$-module .
Usually, we write $a\rhd v$ instead of $\psi_{a}(v)$, so that the action
axioms are write as $(ab)\rhd v=a\rhd(b\rhd v)$ and $1\rhd v=v$.
If $A$ is a Hopf algebra, $V_{\mathbb{K}}$ is an $A$-module algebra
[coalgebra], and $a\rhd(vw)=(a_{(1)}\rhd v)(a_{(2)}\rhd w)$, $a\rhd
1_{V}=\varepsilon(h)1_{V}$ [$\Delta(a\rhd v)=(a_{(1)}\rhd
v_{(1)})\otimes(a_{(2)}\rhd v_{(2)})$ (that is to say $\Delta(a\rhd
v)=\Delta_{A}(a)\rhd\Delta(v)$), $\varepsilon(a\rhd
v)=\varepsilon(a)\varepsilon(v)$] for all $v,w\in V_{\mathbb{K}}$ and $a\in
A$, then $V_{\mathbb{K}}$ is said a left $A$-module algebra [coalgebra].
If $V$ is a Hopf algebra [bialgebra], then there exists the following two
natural left actions on itself: the left regular action $L$, given by
$L_{v}(w)=vw$, and the left adjoint action $Ad$, given by
$Ad_{v}(w)=v_{(1)}wS(v_{(2)})$, for all $v,w\in V$.
The left coregular action $R^{\ast}$ of a finite-dimensional Hopf algebra
[bialgebra] $V$ on the dual $V^{\ast}$, is given by
$R^{\ast}_{v}(\phi)=\phi_{(1)}\langle v,\phi_{(2)}\rangle$, whereas, in the
infinite-dimensional case, we set $\langle
R^{\ast}_{v}(\phi),w\rangle=\langle\phi,vw\rangle$, for all $v,w\in V$ and
$\phi\in V^{\ast}$, being $\langle\ ,\ \rangle$ the dual pairing between $V$
and $V^{\ast}$ ($V^{o}$ in the infinite-dimensional case); furthermore,
$R^{\ast}$ makes $V^{\ast}$ [$V^{o}$] into a $V$-module algebra.
The left coadjoint action of a finite-dimensional Hopf algebra [bialgebra] $V$
on the dual $V^{\ast}$, is given by $Ad_{v}^{\ast}(\phi)=\phi_{(2)}\langle
v,(S\phi_{(1)})\phi_{(2)}\rangle$, whereas, in the infinite-dimensional case,
we put $\langle
Ad_{v}^{\ast}(\phi),w\rangle=\langle\phi,(Sv_{(1)})wv_{(2)}\rangle$, for all
$v,w\in V$ and $\phi\in V^{\ast}$, being $\langle\ ,\ \rangle$ the dual
pairing between $V$ and $V^{\ast}$ ($V^{o}$ in the infinite-dimensional case);
furthermore, $Ad^{\ast}$ makes $V^{\ast}$ [$V^{o}$] into a $V$-module
coalgebra.
The concept of $A$-module algebra generalizes333Because the structure of
$A$-module (from commutative algebra) generalize the notion of representation.
the notion of $G$-covariant algebra of the Physics: if $G$ is a symmetry
group, given a $G$-covariant $\mathbb{K}$-algebra $V$, we construct the group
algebra $\mathbb{K}G$ generated by $G$; then, the algebra generated by
$\mathbb{K}G$ and $V$, with commutation relations given by $uv=(u\rhd v)u\ \
\forall v\in V,u\in G$, give rise to a semidirect, or cross, product algebra.
Therefore, we have the following general structure.
Given a Hopf algebra [bialgebra] $A$ and a left $A$-module algebra on $V$,
then there exists a left cross product algebra on $V\otimes A$, with product
$(v\otimes a)(w\otimes b)=v(a_{(1)}\rhd w)\otimes a_{(2)}b,\qquad v,w\in V,\
a,b\in A$
and unit element $1\otimes 1$. This algebra is denoted with444Or with
$V\rtimes_{\psi}A$, if one want to specify the underling left action $\psi$.
$V\rtimes A$.
With obvious modifications, it is possible to have right actions, as follows.
A right action of an algebra $A$ on the $\mathbb{K}$-linear space
$V_{\mathbb{K}}$, is a linear map $V\otimes A\rightarrow V$, denoted by
$v\otimes a\rightarrow v\lhd a$, such that $v\lhd(ab)=(v\lhd a)\lhd b$ and
$v\lhd 1=v$ for all $a,b\in A,\ v\in V$. When $A$ is a Hopf algebra that acts
at right on an algebra $V$, and $(ab)\lhd v=(a\lhd v_{(1)})(b\lhd v_{(2)})$,
$1_{V}\lhd v=1_{V}\varepsilon(v)$, for all $a,b\in A$ and $v\in V$, then we
say that $V$ is a right $A$-module algebra, and we write $(V,A,\lhd)$.
A right $A$-coaction on a $\mathbb{K}$-linear space $V_{\mathbb{K}}$, is a
linear map $\varphi:V\rightarrow V\otimes A$ such that
$(\varphi\otimes\mbox{\rm id})\circ\varphi=(\mbox{\rm
id}\otimes\Delta)\circ\varphi$ and $\mbox{\rm id}=(\mbox{\rm
id}\otimes\varepsilon)\circ\varphi$; we say, also, that $V_{\mathbb{K}}$ is a
right $A$-comodule.
If we set $\varphi(v)=v^{(\bar{1})}\otimes v^{(\bar{2})}$ (in $V\otimes A$),
then a coalgebra $V_{\mathbb{K}}$ is a right $A$-comodule coalgebra if
$V_{\mathbb{K}}$ is a right $A$-comodule and
$v_{(1)}^{(\bar{1})}\otimes v_{(2)}^{(\bar{1})}\otimes
v^{(\bar{2})}=v_{(1)}^{(\bar{1})}\otimes v_{(2)}^{(\bar{1})}\otimes
v_{(1)}^{(\bar{2})}v_{(2)}^{(\bar{2})},\qquad\varepsilon(v^{(\bar{1})})v^{(\bar{2})}=\varepsilon(v).$
If $V$ is a Hopf algebra [bialgebra], there are two natural right actions on
itself: the right regular action R, given by $R_{v}(w)=wv$, and the right
adjoint action Ad, given by $Ad_{v}(w)=(Sv_{(1)})wv_{(2)}$, for all $v,w\in
V$.
If $A$ is a Hopf algebra [bialgebra] and $V$ is a right $A$-comodule
coalgebra, then there exists a right cross coproduct coalgebra structure on
$A\otimes V$, given by
$\Delta(a\otimes v)=a_{(1)}\otimes v_{(1)}^{(\bar{1})}\otimes
a_{(2)}v_{(1)}^{(\bar{2})}\otimes v_{(2)},\quad\varepsilon(a\otimes
v)=\varepsilon_{A}(a)\varepsilon(c),$
for all $a\in A,\ v\in V$; such a coalgebra, is denoted with555Or with
$V\ltimes_{\psi}A$, if one want to specify the underling right action $\psi$.
$A\ltimes V$.
We do not discuss the notion of bicrossproduct algebra, introduced by S. Majid
in connections with an important tentative of unifying Quantum mechanics and
Gravity (see [Ma], Chap. 6), but we give only a simple example of what a
bicrossproduct Hopf algebra is: let $G,M$ two subgroups that factorizes a
given group, so that $G$ acts on $M$, and viceversa (for instance, $M$ may be
the position space, while $G$ may be the momentum group); let $\mathbb{K}(M)$
the algebra of $\mathbb{K}$-valued functions on $M$, and let $\mathbb{K}G$ be
the free algebra on $G$. Then, the following Hopf algebra
$\mathbb{K}(M)\bowtie\mathbb{K}G=\left\\{\begin{array}[]{l}\mathbb{K}(M)\rtimes\mathbb{K}G\qquad\mbox{as\
algebra,}\\\ \mathbb{K}(M)\ltimes\mathbb{K}G\qquad\mbox{as\
coalgebra}\end{array}\right.$
is a first example of bicrossproduct algebra, whose dual Hopf algebra is
$\mathbb{K}M\bowtie\mathbb{K}(G)$.
Another notion strictly correlated to that of bicrossproduct, is the notion of
double cross product (see [Ma]).
The cross, bicross and double cross product constructions, provides a large
class of quantum groups.
3\. The restricted Hopf algebra structure on
$\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$
There exists various methods to define a classical Hopf algebra structure on
$\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$, recalling that
this algebra is infinite-dimensional.
The main method, proceed as follows.
In the case of the groupoid $\mathcal{G}_{HBJ}(\mathcal{F}_{I})$, that we
recall to be a particular example of the general type $(G,G^{(0)},r,s,\star)$,
the most natural modifications to the functional laws on the points 1. and 2.
of the § 1, are the following (see [Va], § 2.2):
1’. coproduct: $\Delta(f)(g_{1},g_{2})=f(g_{1}\star g_{2})$ if
$(g_{1},g_{2})\in G^{(2)}$, and $=0$ otherwise;
2’. counit: $\varepsilon(f)=\sum_{e\in G^{(0)}}f(e)$.
The antipode definition 3., is the same also in this case.
Besides the question relative to the functional laws, there exists the
question related to their definition sets.
Since, in the infinite-dimensional case we have
$\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\otimes\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\subseteq\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I})\times\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$
with
$\Delta:\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\rightarrow\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I})\times\mathcal{G}_{HBJ}(\mathcal{F}_{I})),$
let
$\mathcal{F}^{o}=\Delta^{-1}(\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\otimes\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I})))\subseteq\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$;
then, $\mathcal{F}^{o}$ is a Hopf algebra with the coalgebra structure given
by 1’., 2’. and 3., although it is difficult to determine exactly its set-
theoretic specificity.
It is called the restricted Hopf algebra of
$\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$, and is denoted
with $\mathcal{F}_{\mathbb{K}}^{o}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$.
For our purpose, in the finite-dimensional case, there exists the isomorphism
(see [Ks], III.1; [Ma], Example 1.5.4)
$\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\cong\mathcal{A}_{\mathbb{K}}^{\ast}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$,
so that we may construct a Hopf algebra structure on
$\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$ via
$\mathcal{A}_{\mathbb{K}}^{\ast}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$ by dual
pairing; unfortunately, this isomorphism do not subsists in the infinite-
dimensional case, and such a question will be at the basis of the discussion
of § 6.
Other methods for dualization (as Konstant duality, Cartier duality, Tannaka-
Krein duality, Takeuki duality, Kadison-Szlachányi dual pairing, the weak
antipode plus convolution-inverse method, Pontriyagin duality, and so on), may
be found, for instance, in [Sw], [Sch1], [Sch2], [Sch3], [Ma].
However, in the context of the classical Hopf algebra structures, some of
these methods do not lead to an explicit solution of the problem, while others
provides complicated structures unadapted to the physical applications.
But there exists different generalizations of the structure of Hopf algebra
(for a recent survey of these, see [Ka]) as, for instance, the notions of weak
Hopf algebra (or quantum groupoid) and Hopf algebroid (see [BNS], [NV]),
through which it is possible to solve, more explicitly, the above problem, at
least for the dual
$\mathcal{A}_{\mathbb{K}}^{\ast}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$, in the
context of weak Hopf algebras, and with more possibilities on the side of
physical applications.
Such a question, we’ll be the matter of a further paper.
4\. The restricted Heisenberg double
$\mathcal{H}^{o}_{\mathcal{A}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$
Hence, as regard what has been said above, we may consider the following dual
pairing
$\langle\ ,\
\rangle:\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\times\mathcal{F}_{\mathbb{K}}^{o}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\rightarrow\mathbb{K}$
such that
$\langle a_{1}\otimes a_{2},\Delta_{\mathcal{F}^{o}}(f)\rangle=\langle
a_{1}a_{2},f\rangle,\qquad\langle\Delta_{\mathcal{A}}(a),f_{1}\otimes
f_{2}\rangle=\langle a,f_{1}f_{2}\rangle$ $\langle
1_{\mathcal{A}},f\rangle=\varepsilon_{\mathcal{F}^{o}}(f),\qquad\ \langle
1_{\mathcal{F}^{o}},a\rangle=\varepsilon_{\mathcal{A}}(a)$
for all $f,f_{1},f_{2}\in\mathcal{F}^{o},$ and $a,a_{1},a_{2}\in\mathcal{A}$.
It is know that it is always possible to consider, eventually quotienting, a
non-degenerate dual pairing of this type. Therefore, if we consider the action
$(b,a)\rightarrow b\rhd a=\langle b,a_{(1)}\rangle a_{(2)}\qquad\forall
a\in\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I})),\ \forall
b\in\mathcal{F}_{\mathbb{K}}^{o}(\mathcal{G}_{HBJ}(\mathcal{F}_{I})),$
it follows that it is possible to define the left cross product algebra
$\mathcal{H}_{\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I})),\mathcal{F}_{\mathbb{K}}^{o}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))}=\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\rtimes\mathcal{F}_{\mathbb{K}}^{o}(\mathcal{G}_{HBJ}(\mathcal{F}_{I})),$
called the Heisenberg double of the pair
$\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}),\mathcal{F}_{\mathbb{K}}^{o}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$.
This last construction may be repeated for the restricted dual
$\mathcal{A}_{\mathbb{K}}^{o}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))(\subseteq\mathcal{A}_{\mathbb{K}}^{*}(\mathcal{G}_{HBJ}(\mathcal{F}_{I})))$
of $\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$, obtaining
the so-called Heisenberg double of
$\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$, that we
denote, for simplicity, with
$\mathcal{H}_{\mathcal{A}}^{o}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$.
5\. The restricted Weyl algebra
The notion of cross product lead to an algebraic formulation of some aspects
of quantization.
Let $V$ be a $A$-module algebra, with $A$ a Hopf algebra, and let $V\rtimes A$
be the corresponding left cross product. Hence, there exists a canonical
representation on $V$ itself, given by $(v\otimes a)\rhd w=v(a\rhd w)$, called
the generalized Schrödinger representation of $V$.
The physical motivations to this terminology arise from the quantum meaning
that such a representation has when applied to the bicrossproduct algebra
$\mathbb{K}(M)\rtimes\mathbb{K}G$ of the end of paragraph 2 (see also [Ma],
Chap. 6).
Our interest is on infinite-dimensional case666But not only; for example, in
the finite-dimensional case, we have $V^{o}=V^{\ast}$, and what follows holds
also in this case, with obvious modifications., so let $V$ be a infinite-
dimensional Hopf algebra, with restricted dual $V^{o}$; then, by the left
coregular representation $R^{\ast}$ of $V$ on $V^{o}$ (that does holds also in
the infinite-dimensional case, as seen at § 2.), $V^{o}$ is a $V$-module
algebra, so that we may consider the left cross product algebra $V^{o}\rtimes
V$.
Nevertheless, we are interested to another type of left cross product algebra,
built up as follows (see [Ma], § 6.1, for details).
We consider the following action $\phi\rhd
v=v_{(1)}\langle\phi,v_{(2)}\rangle$ for all $v\in V,\phi\in V^{o}$, making
$V$ into a $V^{o}$-module algebra and that gives rise to the following product
on $V\otimes V^{o}$
$(v\otimes\phi)(w\otimes\psi)=vw_{(1)}\otimes\langle
w_{(2)},\phi_{(1)}\rangle\phi_{(2)}\psi,$
whence a structure of left cross product algebra on $V\otimes V^{o}$, namely
$V\rtimes V^{o}$.
Then, it is possible to prove that the related Schrödinger representation give
rise to an isomorphism (of algebras) $\chi:V\rtimes V^{o}\rightarrow Lin(V)$,
where $Lin(V)$ is the algebra of $\mathbb{K}$-endomorphisms of $V$, given by
$\chi(v\otimes\psi)w=vw_{(1)}\langle\phi,w_{(2)}\rangle$.
Therefore, we have the algebra isomorphism $\mathcal{W}(V)=V\rtimes V^{o}\cong
Lin(V)$; we call $\mathcal{W}(V)$ the restricted Weyl algebra of the Hopf
algebra $V$.
This last construction is an algebraic generalization of the usual Weyl
algebras of Quantum Mechanics on a group, whose finite-dimensional prototype
is as follows.
We consider the strict dual pair given by the $\mathbb{K}$-valued functions on
$G$, say $\mathbb{K}(G)$, and the free algebra on $G$, say $\mathbb{K}G$;
then, the left cross product algebra $\mathbb{K}(G)\rtimes\mathbb{K}G$ is
given by the right action of $G$ on itself, namely $\psi_{u}(s)=su$, that
induces a left regular representation of $G$ on $\mathbb{K}G$, hence a
Schrödinger representation generated by this and by the action of
$\mathbb{K}G$ on itself by pointwise product. Whence, if $V=\mathbb{K}(G)$, we
obtain the left cross product algebra $\mathbb{K}(G)\rtimes\mathbb{K}G$,
isomorphic to $Lin(\mathbb{K}(G))$ by Schrödinger representation, that
formalizes the algebraic quantization of a particle moving on $G$ by
translations.
We may apply these well-known considerations (see [Ma]) to
$\mathcal{G}_{HBJ}(\mathcal{F}_{I})$ when card $I<\infty$ (finite number of
energy levels), taking into account that (in the finite-dimensional case)
$\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))=\mathcal{F}_{\mathbb{K}}^{\ast}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$,
in such a way that the Weyl algebra
$\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\rtimes\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))=\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\rtimes\mathcal{F}_{\mathbb{K}}^{\ast}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))=$
$=\mathcal{W}(\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I})))\cong
Lin(\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I})))$
represents the algebraic quantization of a particle moving on the groupoid
$\mathcal{G}_{HBJ}(\mathcal{F}_{I})$ by translations777This remark may be
think as the starting point for a quantum mechanics on a groupoid..
Instead, for the infinite-dimensional HBJ EBB-groupoid
$\mathcal{G}_{HBJ}(\mathcal{F}_{I})$, we obtain a particular restricted Weyl
algebra of the following type
$\mathcal{W}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))=\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\rtimes\mathcal{F}_{\mathbb{K}}^{o}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$
with
$\mathcal{F}_{\mathbb{K}}^{o}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\neq\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$
because of the no finitely generation of
$\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$; therefore, the
above physical interpretation of the restricted Weyl algebra
$\mathbb{K}(G)\rtimes\mathbb{K}G$, is no longer valid for
$\mathcal{W}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$.
However, as we’ll see in another place, the last lack of physical
interpretation can be restored in the context of extended Hopf algebra
structures.
6\. The Drinfeld quantum double
If $V$ is a Hopf algebra, then through the left adjoint action (in infinite-
dimensional setting) $Ad$ on itself, we have that $V$ is a $V$-module algebra,
so that we may build the left cross product algebra $V\rtimes_{Ad}V$.
We consider the right adjoint action of $\mathcal{G}_{HBJ}(\mathcal{F}_{I})$
on itself, given by $\psi_{g}(h)=g^{-1}\star h\star g$ if exists, $=0$
otherwise; such an action makes
$\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$ into a
$\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$-module algebra.
In the finite-dimensional case we have
$\mathcal{A}_{\mathbb{K}}^{\ast}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\cong\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$,
hence
$\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))=\mathcal{A}_{\mathbb{K}}^{\ast\ast}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\cong\mathcal{F}_{\mathbb{K}}^{\ast}(\mathcal{G}_{HBJ}(\mathcal{F}_{I})),$
so that $\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$ is also
a $\mathcal{F}_{\mathbb{K}}^{\ast}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$-module
algebra, whence the left cross product algebra
$\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\rtimes\mathcal{F}_{\mathbb{K}}^{\ast}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\cong\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\rtimes\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$
that the tensor product coalgebra makes into a Hopf algebra, called the
quantum double of $\mathcal{G}_{HBJ}(\mathcal{F}_{I})$ and denoted with
$D(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$; even in the finite-dimensional case,
it represent the algebraic quantization of a particle constrained to move on
conjugacy classes of $\mathcal{G}_{HBJ}(\mathcal{F}_{I})$ (quantization on
homogeneous spaces over a groupoid).
Besides, it was proved, for a finite group $G$, that this (Drinfeld) quantum
double $D(G)$ has a quasitriangular structure (see [Ma], Chap. 6), given by
$(\delta_{s}\otimes u)(\delta_{t}\otimes
v)=\delta_{u^{-1}su,t}\delta_{t}\otimes uv,\qquad\Delta(\delta_{s}\otimes
u)=\sum_{ab=s}\delta_{a}\otimes u\delta_{b}\otimes u,$
$\varepsilon(\delta_{s}\otimes u)=\delta_{s,e},\qquad S(\delta_{s}\otimes
u)=\delta_{u^{-1}s^{-1}u}\otimes u^{-1},$ $R=\sum_{u\in G}\delta_{u}\otimes
e\otimes 1\otimes u,$
where we have identifies the dual of $\mathbb{K}G$ with $\mathbb{K}(G)$ via
the idempotents $p_{g},\ g\in G$ such that $p_{g}p_{h}=\delta_{g,h}p_{g}$ (see
[NV], 2.5. and [Ma], 1.5.4); such a quantum double represents the algebra of
quantum observables of a certain physical system.
Hence, it is a natural question to ask if such structures may be extended to
$D(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$, that is when we have a quantum double
built on a groupoid.
7\. Conclusions.
From what has been said above (and in [Iu]), it can be meaningful to study as
the common known structures, just described above, may be extended when we
consider a groupoid888Finite or not. instead of a finite group, both in the
classical theory of Hopf algebras and in the new realm of the extended Hopf
structures.
Subsequently, the resulting structures must be interpreted from the physical
view point, with a critical comparison respect to the physical meaning of the
classical Hopf structures just seen in this paper.
These questions are not trivial, because there are recent papers on a
classical Physics on a groupoid (see, for instance, [CDMMM]), and many
important works on the role of Hopf algebras in High Energy Physics999Perhaps,
it should be interesting to apply the extended Hopf algebra structures to this
context. (see, for instance, [Kr] and references therein).
References.
[BNS] G. Böhm, F. Nill, K. Szlachányi, Weak Hopf Algebra I. Integral Theory
and $C^{\ast}$-structure, J. Algebra, 221 (1999) pp. 385-438.
[Co] A. Connes, Noncommutative Geometry, Academic Press, New York, 1994.
[CDMMM] J. Cortes, M. De Leon, J.C. Marrero, D. Martín de Diego, E. Martínez,
A Survey of Lagrangian Mechanics and Control on Lie Algebroids and Groupoids,
preprint e-arXiv: math.ph/0511009v1.
[Iu] G. Iurato, A possible quantic motivation of the structure of quantum
group, JP Journal of Algebra, Number Theory and Applications, 2010 (to
appear).
[Ka] G. Karaali, On the Hopf algebras and their generalizations, preprint
e-arXiv: math.QA/0703441v2.
[Ks] C. Kassel, Quantum Groups, Springer-Verlag, New York, 1995.
[Kr] D. Kreimer, Algebraic structures in Quantum Field Theory, preprint
e-arXiv: hep-th/1007.0341v1.
[Ma] S. Majid, Foundations of quantum group theory, Cambridge University
Press, Cambridge, 1995.
[NV] D. Nikshych, L. Vainerman, Finite quantum groupoids and their
applications, in: New directions in Hopf algebras, S. Montgomery, H.J.
Schneider eds., MSRI Publications, Vol. 43, (2002), pp. 211-262.
[Sch1] P. Schauenburg, Tannaka Duality for arbitrary Hopf Algebras, Verlag
Reinhard Fischer, Munich, 1992.
[Sch2] P. Schauenburg, Duals and doubles of quantum groupoids
($\times_{R}$-bialgebras), in: New Trends in Hopf Algebra Theory, N.
Andruskiewitsch, W.R. Ferrer Santos, and H.J. Schneider, Eds., vol. 267, Amer.
Math. Soc., New York, 2000\.
[Sch3] P. Schauenburg, Weak Hopf Algebras and Quantum Groupoids, preprint
e-arXiv: math.QA/0204180v1.
[Sw] M. Sweedler, Hopf Algebras, W.A. Benjamin Inc., New York, 1969.
[Va] J.M. Vallin, Relative matched pairs of finite groups from depth two
inclusions of Von Neumann algebras to quantum groupoid, preprint e-arXiv:
math.OA/0703886v1.
|
arxiv-papers
| 2011-01-31T14:27:00 |
2024-09-04T02:49:16.739162
|
{
"license": "Public Domain",
"authors": "Giuseppe Iurato",
"submitter": "Giuseppe Iurato",
"url": "https://arxiv.org/abs/1101.5980"
}
|
1101.6040
|
# Generating GHZ state in $2m$-qubit spin network
M. A. Jafarizadeha,d , R. Sufiania , F. Eghbalifama, M. Azimia,
S. F. Taghavic and E. Baratie,
aDepartment of Theoretical Physics and Astrophysics, University of Tabriz,
Tabriz 51664, Iran.
bCenter of excellence for photonic, University of Tabriz, Tabriz 51664, Iran.
cInstitute for Studies in Theoretical Physics and Mathematics, Tehran
19395-1795, Iran.
dResearch Institute for Fundamental Sciences, Tabriz 51664, Iran.
eInstitute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52,
01-224 Warszawa, Poland. E-mail:jafarizadeh@tabrizu.ac.irE-
mail:sofiani@tabrizu.ac.ir
###### Abstract
We consider a pure $2m$-qubit initial state to evolve under a particular
quantum mechanical spin Hamiltonian, which can be written in terms of the
adjacency matrix of the Johnson network $J(2m,m)$. Then, by using some
techniques such as spectral distribution and stratification associated with
the graphs, employed in [1, 2], a maximally entangled $GHZ$ state is generated
between the antipodes of the network. In fact, an explicit formula is given
for the suitable coupling strengths of the hamiltonian, so that a maximally
entangled state can be generated between antipodes of the network. By using
some known multipartite entanglement measures, the amount of the entanglement
of the final evolved state is calculated, and finally two examples of four
qubit and six qubit states are considered in details.
Keywords: maximal entanglement , GHZ states, Johnson network, Stratification,
Spectral distribution
PACs Index: 01.55.+b, 02.10.Yn
## 1 Introduction
The idea to use quantum spin chains for short distance quantum communication
was put forward by Bose [3]. After the work of Bose, the use of spin chains
[4]-[18] and harmonic chains [19] as quantum wires have been proposed. In the
previous work [1], the so called distance-regular graphs have been considered
as spin networks (in the sense that with each vertex of a distance-regular
graph, a qubit or a spin was associated) and perfect state transfer (PST) of a
single qubit state over antipodes of these networks has been investigated. In
that work, a procedure for finding suitable coupling constants in some
particular spin Hamiltonians has been given so that perfect and optimal
transfer of a quantum state between antipodes of the corresponding networks
can be achieved, respectively.
Entanglement is one of the other important tasks in quantum communication.
Quantum entanglement in spin systems is an extensively-studied field in recent
years [20,21,22,23], in the advent of growing realization that entanglement
can be a resource for quantum information processing. Within this general
field, entanglement of spin$1/2$ degrees of freedom, qubits, has been in focus
for an obvious reason of their paramount importance for quantum computers, not
to mention their well-known applicability in various condensed-matter systems,
optics and other branches of physics. In [24], authors attempted to generate a
Bell state between distant vertices in a permanently coupled spin network
interacting via invariant stratification graphs (ISGs). At the first step,
they established an upper bound over achievable entanglement between the
reference site and the other vertices. Due to this upper bound they found that
creation of a Bell state between the reference site and a vertex is possible
if the stratum of that vertex is a single element, e.g. antipodal ISGs. The
present work focuses on the provision of $GHZ$ state, by using a $2m$-qubit
initial product state. To this end, we will consider the Johnson networks
$J(2m,m)$ (which are distance-regular) as spin networks. Then, we use the
algebraic properties of these networks in order to find suitable coupling
constants in some particular spin Hamiltonians so that $2m$-qubit $GHZ$ state
can be achieved.
The organization of the paper is as follows: In section 2, we review some
preliminary facts about graphs and their adjacency matrices, spectral
distribution associated with them; In particular, some properties of the
networks derived from symmetric group $S_{n}$ called also Johnson networks are
reviewed. Section 3 is devoted to $2m$-qubit $GHZ$ state provision by using
algebraic properties of Johnson network $J(2m,m)$, where a method for finding
suitable coupling constants in particular spin Hamiltonians is given so that
maximal entanglement in final state is possible. The paper is ended with a
brief conclusion and two appendices.
## 2 preliminaries
### 2.1 Graphs and their adjacency matrices
A graph is a pair $\Gamma=(V,E)$, where $V$ is a non-empty set called the
vertex set and $E$ is a subset of $\\{(x,y):x,y\in V,x\neq y\\}$ called the
edge set of the graph. Two vertices $x,y\in V$ are called adjacent if
$(x,y)\in E$, and in that case we write $x\sim y$. For a graph $\Gamma=(V,E)$,
the adjacency matrix $A$ is defined as
$\bigl{(}A)_{\alpha,\beta}\;=\;\cases{1&if $\;\alpha\sim\beta$\cr
0&\mbox{otherwise}\cr}.$ (2-1)
Conversely, for a non-empty set $V$, a graph structure is uniquely determined
by such a matrix indexed by $V$. The degree or valency of a vertex $x\in V$ is
defined by
$\kappa(x)=|\\{y\in V:y\sim x\\}|$ (2-2)
where, $|\cdot|$ denotes the cardinality. The graph is called regular if the
degree of all of the vertices be the same. In this paper, we will assume that
graphs under discussion are regular. A finite sequence
$x_{0},x_{1},...,x_{n}\in V$ is called a walk of length $n$ (or of $n$ steps)
if $x_{i-1}\sim x_{i}$ for all $i=1,2,...,n$. Let $l^{2}(V)$ denote the
Hilbert space of $C$-valued square-summable functions on $V$. With each
$\beta\in V$ we associate a vector $|\beta\rangle$ such that the $\beta$-th
entry of it is $1$ and all of the other entries of it are zero. Then
$\\{|\beta\rangle:\beta\in V\\}$ becomes a complete orthonormal basis of
$l^{2}(V)$. The adjacency matrix is considered as an operator acting in
$l^{2}(V)$ in such a way that
$A|\beta\rangle=\sum_{\alpha\sim\beta}|\alpha\rangle.$ (2-3)
### 2.2 Spectral distribution associated with the graphs
Now, we recall some preliminary facts about spectral techniques used in the
paper, where more details have been given in Refs. [26,27,28,29]
Actually the spectral analysis of operators is an important issue in quantum
mechanics, operator theory and mathematical physics [30,31]. As an example
$\mu(dx)=|\psi(x)|^{2}dx$ ($\mu(dp)=|\widetilde{\psi}(p)|^{2}dp$) is a
spectral distribution which is assigned to the position (momentum) operator
$\hat{X}(\hat{P})$. The mathematical techniques such as Hilbert space of the
stratification and spectral techniques have been employed in [32,33] for
investigating continuous time quantum walk on graphs. Moreover, in general
quasi-distributions are the assigned spectral distributions of two hermitian
non-commuting operators with a prescribed ordering. For example the Wigner
distribution in phase space is the assigned spectral distribution for two non-
commuting operators $\hat{X}$ (shift operator) and $\hat{P}$ (momentum
operator) with Wyle-ordering among them [34, 35]. It is well known that, for
any pair $(A,|\phi_{0}\rangle)$ of a matrix $A$ and a vector
$|\phi_{0}\rangle$, one can assign a measure $\mu$ as follows
$\mu(x)=\langle\phi_{0}|E(x)|\phi_{0}\rangle,$ (2-4)
where $E(x)=\sum_{i}|u_{i}\rangle\langle u_{i}|$ is the operator of projection
onto the eigenspace of $A$ corresponding to eigenvalue $x$, i.e.,
$A=\int xE(x)dx.$ (2-5)
Then, for any polynomial $P(A)$ we have
$P(A)=\int P(x)E(x)dx,$ (2-6)
where for discrete spectrum the above integrals are replaced by summation.
Therefore, using the relations (2-4) and (2-6), the expectation value of
powers of adjacency matrix $A$ over reference vector $|\phi_{0}\rangle$ can be
written as
$\langle\phi_{0}|A^{m}|\phi_{0}\rangle=\int_{R}x^{m}\mu(dx),\;\;\;\;\
m=0,1,2,....$ (2-7)
Obviously, the relation (2-7) implies an isomorphism from the Hilbert space of
the stratification onto the closed linear span of the orthogonal polynomials
with respect to the measure $\mu$.
### 2.3 Underlying networks derived from symmetric group $S_{n}$
Let $\lambda=(\lambda_{1},...,\lambda_{m})$ be a partition of $n$, i.e.,
$\lambda_{1}+...+\lambda_{m}=n$. We consider the subgroup $S_{m}\otimes
S_{n-m}$ of $S_{n}$ with $m\leq[\frac{n}{2}]$. Then we assume the finite set
$M^{\lambda}=\frac{S_{n}}{S_{m}\otimes S_{n-m}}$ with
$|M^{\lambda}|=\frac{n!}{m!(n-m)!}$ as vertex set. In fact, $M^{\lambda}$ is
the set of $(m-1)$-faces of $(n-1)$-simplex (recall that, the graph of an
$(n-1)$-simplex is the complete graph with $n$ vertices denoted by $K_{n}$).
If we denote the vertex $i$ by $m$-tuple $(i_{1},i_{2},...,i_{m})$, then the
adjacency matrices $A_{k}$, $k=0,1,...,m$ are defined as
$\bigl{(}A_{k})_{i,j}\;=\left\\{\begin{array}[]{c}1\quad\mathrm{if}\;\;\
\partial(i,j)=k,\\\ 0\quad\quad\mathrm{otherwise}\quad\quad\quad(i,j\in
M^{\lambda})\\\ \end{array}\right.,\;\;\ k=0,1,...,m.$ (2-8)
where, we mean by $\partial(i,j)$ the number of components that
$i=(i_{1},...,i_{m})$ and $j=(j_{1},...,j_{m})$ are different (this is the
same as Hamming distance which is defined in coding theory). The network with
adjacency matrices defined by (2-8) is known also as the Johnson network
$J(n,m)$ and has $m+1$ strata such that
$\kappa_{0}=1,\;\ \kappa_{l}=\left(\begin{array}[]{c}m\\\
m-l\end{array}\right)\left(\begin{array}[]{c}n-m\\\ l\end{array}\right),\;\;\
l=1,2,...,m.$ (2-9)
One should notice that for the purpose of maximal entanglement provision, we
must have $\kappa_{m}=1$ which is fulfilled if $n=2m$, so we will consider the
network $J(2m,m)$ (hereafter we will take $n=2m$ so that we have
$\kappa_{m}=1$). If we stratify the network $J(2m,m)$ with respect to a given
reference node $|\phi_{0}\rangle=|i_{1},i_{2},...,i_{m}\rangle$, where
$|i_{1},i_{2},...,i_{m}\rangle\equiv|0...0\underbrace{1}_{i_{1}}0...0\underbrace{1}_{i_{2}}0...0\underbrace{1}_{i_{m}}0\rangle$
and $i_{1}\neq i_{2}\neq...\neq i_{m}$. The unit vectors $|\phi_{i}\rangle$,
$i=1,...,m$ are defined as
$|\phi_{1}\rangle=\frac{1}{\sqrt{\kappa_{1}}}(\sum_{i^{\prime}_{1}\neq
i_{1}}|i^{\prime}_{1},i_{2},...,i_{m}\rangle+\sum_{i^{\prime}_{2}\neq
i_{2}}|i_{1},i^{\prime}_{2},i_{3},...,i_{m}\rangle+...+\sum_{i^{\prime}_{m}\neq
i_{m}}|i_{1},...,i_{m-1},i^{\prime}_{m}\rangle),$
$|\phi_{2}\rangle=\frac{1}{\sqrt{\kappa_{2}}}\sum_{k\neq
l=1}^{m}\sum_{i^{\prime}_{l}\neq i_{l},i^{\prime}_{k}\neq
i_{k}}|i_{1},...i_{l-1},i^{\prime}_{l},i_{l+1},...,i_{k-1},i^{\prime}_{k},i_{k+1}...,i_{m}\rangle,$
$\vdots$ $\hskip
14.22636pt|\phi_{j}\rangle=\frac{1}{\sqrt{\kappa_{j}}}\sum_{k_{1}\neq
k_{2}\neq...\neq k_{j}=1}^{m}\sum_{i^{\prime}_{k_{1}}\neq
i_{k_{1}},...,i^{\prime}_{k_{j}}\neq
i_{k_{j}}}|i_{1},...,i_{k_{1}-1},i^{\prime}_{k_{1}},i_{k_{1}+1},...,i_{k-1},i^{\prime}_{k_{j}},i_{k_{j}+1}...,i_{m}\rangle,$
$\vdots$ $|\phi_{m}\rangle=\frac{1}{\sqrt{\kappa_{m}}}\sum_{i^{\prime}_{1}\neq
i_{1},...,i^{\prime}_{m}\neq
i_{m}}|i^{\prime}_{1},i^{\prime}_{2},...,i^{\prime}_{m}\rangle.$ (2-10)
Since the network $J(2m,m)$ is distance-regular, the above stratification is
independent of the choice of reference node. The intersection array of the
network is given by
$b_{l}=(m-l)^{2}\;\ ;\;\;\ c_{l}=l^{2}.$ (2-11)
Then, by using the Eq. (B-49), the QD parameters $\alpha_{i}$ and $\omega_{i}$
are obtained as follows
$\alpha_{l}=2l(m-l),\;\ l=0,1,...,m;\;\;\ \omega_{l}=l^{2}(m-l+1)^{2},\;\
l=1,2,...,m.$ (2-12)
Then, one can show that [27]
$A|\phi_{l}\rangle=(l+1)(m-l)|\phi_{l+1}\rangle+2l(m-l)|\phi_{l}\rangle+l(m-l+1)|\phi_{l-1}\rangle.$
(2-13)
## 3 $GHZ$ state generation by using quantum mechanical Hamiltonian in the
network $J(2m,m)$
The model we consider is the distance-regular Johnson network $J(2m,m)$
consisting of $N=C^{2m}_{m}=\frac{(2m)!}{m!m!}$ sites labeled by
$\\{1,2,...,N\\}$ and diameter $m$. Then, we stratify the network with respect
to a chosen reference site, say 1 (the discussion about stratification has
been given in appendix A; In these particular networks, the first and the last
strata possess only one node, i.e., $|\phi_{0}\rangle=|1\rangle$ and
$|\phi_{m}\rangle=|N\rangle$). At time $t=0$, a $2m$-qubit state is prepared
in the first (reference) site of the network. We wish to provide a maximal
quantum entanglement between the state of this site and the state of the
$N$-th site after a well-defined period of time, in which the corresponding
network is evolved under a particular Hamiltonian.
If the network be assumed as a spin network, in which a spin-$1/2$ particle is
attached to each vertex (node) of the network, the Hilbert space associated
with the network is given by ${\mathcal{H}}=(C^{2})^{\otimes 2m}$. The
standard basis for an individual qubit is chosen to be
${|0\rangle=|\downarrow\rangle,|1\rangle=|\uparrow\rangle}$. Then we consider
the Hamiltonian
$H_{s}=\frac{1}{2}\sum\limits_{1\leq i<j\leq 2m}H_{ij}$ (3-14)
where, $H_{ij}=\sigma_{i}\cdot\sigma_{j}$ and $\sigma_{i}$ is a vector with
familiar Pauli matrices $\sigma_{i}^{x},\sigma_{i}^{y}$ and $\sigma_{i}^{z}$.
One can easily see that, the Hamiltoniaan (3.14) commutes with the total $z$
component of the spin, i.e., $[\sigma^{z}_{total},H_{s}]=0$, hence the Hilbert
space ${\mathcal{H}}$ decompose into invariant subspaces, each of which is a
distinct eigenspace of the operator $\sigma^{z}_{total}$. So the total number
of up and down spins are invariant under action of Hamiltonian or time
evolution operator. Now, we recall that the kets
$|i_{1},i_{2},\ldots,i_{2m}\rangle$ with
$i_{1},\ldots,i_{2m}\in\\{\uparrow,\downarrow\\}$ form an orthonormal basis
for Hilbert space ${\mathcal{H}}$. Then, one can easily obtain
$H_{ij}|...\underbrace{\uparrow}_{i}...\underbrace{\uparrow}_{j}...\rangle=|...\underbrace{\uparrow}_{i}...\underbrace{\uparrow}_{j}...\rangle$
and
$H_{ij}|...\underbrace{\uparrow}_{i}...\underbrace{\downarrow}_{j}...\rangle=-|...\underbrace{\uparrow}_{i}...\underbrace{\downarrow}_{j}...\rangle+2|...\underbrace{\downarrow}_{i}...\underbrace{\uparrow}_{j}...\rangle.$
(3-15)
Equation (3.15) implies that the action of $H_{ij}$ on the basis vectors is
equivalent to the action of the operator $2P_{ij}-I$, i.e. we have
$H_{ij}=2P_{ij}-I$ (3-16)
where $P_{ij}$ is the permutation operator acting on sites $i$ and $j$. So
$\frac{1}{2}\sum\limits_{1\leq i<j\leq
2m}\sigma_{i}\cdot\sigma_{j}=\sum\limits_{1\leq i<j\leq
2m}P_{ij}-\frac{1}{2}\left(\begin{array}[]{c}2m\\\ 2\\\ \end{array}\right)I,$
(3-17)
In fact restriction of the operator $\sum_{1\leq i<j\leq 2m}P_{ij}$ on the
$m$-particle subspace (subspace spanned by the states with $m$ spin up) which
has dimension $C^{2m}_{m}$, is written as the adjacency matrix $A$ of the
Johnson network $J(2m,m)$, as
$\sum_{1\leq i<j\leq 2m}P_{ij}=A+m(m-1)I.$ (3-18)
For more details see Ref.[1]. Then we stratify the network with respect to a
chosen reference site, say $|\phi_{0}\rangle$. At time $t=0$, the state is
prepared in the $2m$-qubit state $|\psi(t=0)\rangle=|\underbrace{11\ldots
1}_{m}\rangle|\underbrace{00\ldots 0}_{m}\rangle$. Now, we consider the
dynamics of the system to be governed by the Hamiltonian
$H=\sum_{k=0}^{m}J_{k}P_{k}(1/2\sum_{{}_{1\leq i<j\leq
2m}}{\mathbf{\sigma}}_{i}\cdot{\mathbf{\sigma}}_{j}+\frac{m}{2}I),$ (3-19)
Then, by using (3.17)-(3.20), the Hamiltonian can be written as
$H=\sum_{k=0}^{m}J_{k}P_{k}(A)$ (3-20)
$J_{k}$ is the coupling strength between the reference site $|\phi_{0}\rangle$
and all of the sites belonging to the $k$-th stratum with respect to
$|\phi_{0}\rangle$, and $P_{k}(A)$ are polynomials in terms of adjacency
matrix of the Johnson network. Then, the total system is evolved under unitary
evolution operator $U(t)=e^{-iHt}$ for a fixed time interval, say $t$. The
final state becomes
$|\psi(t)\rangle=\sum_{j=1}^{N}f_{jA}(t)|j\rangle$ (3-21)
where, $N$ is the number of vertices, $|j\rangle$s have $2m$ entries inclusive
$m$ entries equal to 1 and the other entries are 0 and
$|A\rangle=|\underbrace{11\ldots 1}_{m}\underbrace{00\ldots 0}_{m}\rangle$ so
that $f_{jA}(t):=\langle j|e^{-iHt}|A\rangle$.
The evolution with the adjacency matrix $H=A\equiv A_{1}$ for distance-regular
networks (see Appendix B) starting in $|\phi_{0}\rangle$, always remains in
the stratification space. For distance-regular network $J(2m,m)$ for which the
last stratum, i.e., $|\phi_{m}\rangle$ contains only one site, then maximal
entanglement between the starting site $|\phi_{0}\rangle\equiv|A\rangle$ and
the last stratum $|\phi_{m}\rangle$ (the corresponding antipodal node) is
generated, by choosing suitable coupling constants $J_{k}$. In fact, for the
purpose of a maximally entangled $GHZ$ state generation between the first and
the last stratum of the network, we impose the constraints that the amplitudes
$\langle\phi_{i}|e^{-iHt}|\phi_{0}\rangle$ be zero for all $i=1,...,m-1$ and
$\langle\phi_{0}|e^{-iHt}|\phi_{0}\rangle=f$,
$\langle\phi_{m}|e^{-iHt}|\phi_{0}\rangle=f^{\prime}$. Therefore, these
amplitudes must be evaluated. To do so, we use the stratification and spectral
distribution associated with the network $J(2m,m)$ to write
$\langle\phi_{i}|e^{-iHt}|\phi_{0}\rangle=\langle\phi_{i}|e^{-it\sum_{l=0}^{m}J_{l}P_{l}(A)}|\phi_{0}\rangle=\frac{1}{\sqrt{\kappa_{i}}}\langle\phi_{0}|A_{i}e^{-it\sum_{l=0}^{m}J_{l}P_{l}(A)}|\phi_{0}\rangle$
Let the spectral distribution of the graph is
$\mu(x)=\sum_{k=0}^{m}\gamma_{k}\delta(x-x_{k})$ (see Eq. (B-53)). The Johnson
network is a kind of network with a highly regular structure that has a nice
algebraic description; For example, the eigenvalues of this network can be
computed exactly (see for example the notes by Chris Godsil on association
schemes [39] for the details of this calculation). Indeed, the eigenvalues of
the adjacency matrix of the network $J(2m,m)$ (that is $x_{k}$’s in $\mu(x)$)
are given by
$x_{k}=m^{2}-k(2m+1-k),\;\;\ k=0,1,\ldots,m.$ (3-22)
Now, from the fact that for distance-regular graphs we have
$A_{i}=\sqrt{\kappa_{i}}P_{i}(A)$ [27],
$\langle\phi_{i}|e^{-iHt}|\phi_{0}\rangle=0$ implies that
$\sum_{k=0}^{m}\gamma_{k}P_{i}(x_{k})e^{-it\sum_{l=0}^{m}J_{l}P_{l}(x_{k})}=0,\;\;\
i=1,...,m-1$
Denoting $e^{-it\sum_{l=0}^{m}J_{l}P_{l}(x_{k})}$ by $\eta_{k}$, the above
constraints are rewritten as follows
$\sum_{k=0}^{m}P_{i}(x_{k})\eta_{k}\gamma_{k}=0,\;\;\ i=1,...,m-1,$
$\sum_{k=0}^{m}P_{0}(x_{k})\eta_{k}\gamma_{k}=f$
$\sum_{k=0}^{m}P_{m}(x_{k})\eta_{k}\gamma_{k}=f^{\prime}.$ (3-23)
From invertibility of the matrix ${\mathrm{P}}_{ik}=P_{i}(x_{k})$ (see Ref.
[2]) one can rewrite the Eq. (3-23) as
$\left(\begin{array}[]{c}\eta_{0}\gamma_{0}\\\ \eta_{1}\gamma_{1}\\\ \vdots\\\
\eta_{d-1}\gamma_{d-1}\\\
\eta_{d}\gamma_{d}\end{array}\right)=P^{-1}\left(\begin{array}[]{c}f\\\ 0\\\
\vdots\\\ 0\\\ f^{\prime}\end{array}\right).$ (3-24)
The above equation implies that $\eta_{k}\gamma_{k}$ for $k=0,1,...,m$ are the
same as the entries in the first column of the matrix
${\mathrm{P}}^{-1}=WP^{t}$ multiplied with $f$ and the entries in the last
column multiplied with $f^{\prime}$, i.e., the following equations must be
satisfied
$\eta_{k}\gamma_{k}=\gamma_{k}e^{-it\sum_{l=0}^{m}J_{l}P_{l}(x_{k})}={(W{\mathrm{P}}^{t})}_{k0}f+{(W{\mathrm{P}}^{t})}_{km}f^{\prime}\;\
,\;\;\ \mbox{for}\;\ k=0,1,...,m,$ (3-25)
with $W:=diag(\gamma_{0},\gamma_{1},\ldots,\gamma_{m})$. By using the fact
that $\gamma_{k}$ and ${(W{\mathrm{P}}^{t})}_{km}$ are real for
$k=0,1,\ldots,m$, and so we have $\gamma_{k}=|{(W{\mathrm{P}}^{t})}_{km}|$ and
$\gamma_{k}={(W{\mathrm{P}}^{t})}_{k0}$. The Eq. (3.26) can be rewritten as
$\eta_{k}=e^{-it\sum_{l=0}^{m}J_{l}P_{l}(x_{k})}=f+\sigma(k)f^{\prime}$ (3-26)
where $\sigma(k)$ is defined as
$\sigma(k)=\;\cases{-1&for odd $k$\cr 1&\mbox{otherwise}\cr}.$ (3-27)
Assuming $f=|f|e^{i\theta}$ and
$f^{\prime}=|f^{\prime}|e^{i{\theta}^{\prime}}$, it should be considered
${\theta}^{\prime}=\theta\pm\frac{\pi}{2}$ then
$e^{-it\sum_{l=0}^{m}J_{l}P_{l}(x_{k})}=e^{i\theta}(|f|\pm
i\sigma(k)|f^{\prime}|)=e^{i(\theta\pm\arctan{(\frac{\sigma(k)|f^{\prime}|}{|f|})}+2c_{k}\pi)};\;\;\;c_{k}\in\mathcal{Z}$
(3-28)
One should notice that, the Eq. (3.29) can be rewritten as
$(J_{0},J_{1},\ldots,J_{m})=-\frac{1}{t}[\theta+2c_{0}\pi\pm\arctan{(\frac{\sigma(k)|f^{\prime}|}{|f|})},\theta+2c_{1}\pi\pm\arctan{(\frac{\sigma(k)|f^{\prime}|}{|f|})},$
$,\ldots,\theta+2c_{m}\pi\pm\arctan{(\frac{\sigma(k)|f^{\prime}|}{|f|})}](W{\mathrm{P}}^{t})$
(3-29)
or
$J_{k}=-\frac{1}{t}\sum_{j=0}^{m}[\theta+2c_{j}\pi\pm\arctan{(\frac{\sigma(k)|f^{\prime}|}{|f|})}](W{\mathrm{P}}^{t})_{jk}$
(3-30)
where $c_{j}$ for $j=0,1,\ldots,m$ are integers. The result (3-30) gives an
explicit formula for suitable coupling constants so that GHZ state in the
final state can be achieved. The final state is as the form
$|\psi(t)\rangle=f|11\ldots 100\ldots 0\rangle+f^{\prime}|00\ldots 011\ldots
1\rangle$ (3-31)
One attempt to provide a computationally feasible and scalable quantification
of entanglement in multipartite systems was made in Refs. [40,41,42]. For a
pure $n$-qubit state $|\psi\rangle$, the so-called global entanglement is
defined as
$Q(|\psi\rangle)=2(1-\frac{1}{N}\sum\limits_{i=0}^{N-1}Tr[\rho_{i}^{2}])$
(3-32)
where $\rho_{i}$ represents the density matrix of $i$th qubit after tracing
out all other qubits. As seen from this definition, the global entanglement
can be interpreted as the average over the (bipartite) entanglements of each
qubit with the rest of the system. The global entanglement for state in Eq.
(3.32) will be
$Q(|\psi\rangle)=4|f|^{2}|f^{\prime}|^{2}$ (3-33)
Also we introduce a simple multiqubit entanglement quantifier based on the
idea of bipartition and the measure negativity (which is two times the
absolute value of the sum of the negative eigenvalues of the corresponding
partially transposed matrix of a state $\rho$) [43]. For an arbitrary
$N$-qubit state $\rho_{s_{1}s_{2}...s_{N}}$ , a multiqubit entanglement
measure can be formulated as [44]
$\overline{\varrho}=\frac{N}{2}\sum\limits_{1}^{\frac{N}{2}}\varrho_{k|N-k}(\rho_{s_{1}s_{2}...s_{N}})$
(3-34)
where $N$ is assumed even, otherwise $\frac{N}{2}$ should be replaced by
$\frac{N-1}{2}$, and $\varrho_{k|N-k}(\rho_{s_{1}s_{2}...s_{N}})$ is the
entanglement in terms of negativity between two blocks of a bipartition
$k|N-k$ of the state $\rho_{s_{1}s_{2}...s_{N}}$. We can define the following
partition-dependent residual entanglements (PREs)
$\Pi_{q_{1}...q_{m}q_{m+1}...q_{k}|q_{k+1}...q_{n}q_{n+1}...q_{N}}={\varrho}^{2}_{q_{1}...q_{m}q_{m+1}...q_{k}|q_{k+1}...q_{n}q_{n+1}...q_{N}}$
$-\varrho^{2}_{q_{1}...q_{m}|q_{k+1}...q_{n}}-\varrho^{2}_{q_{1}...q_{m}|q_{n+1}...q_{N}}-\varrho^{2}_{q_{m+1}...q_{k}|q_{k+1}...q_{n}}-\varrho^{2}_{q_{m+1}...q_{k}|q_{n+1}...q_{N}}$
(3-35)
and
$\Pi^{\prime}_{q_{1}...q_{k}|q_{k+1}...q_{N}}=\varrho^{2}_{q_{1}...q_{k}|q_{k+1}...q_{N}}-\sum\limits_{i-1}^{k}\sum\limits_{j=k+1}^{N}\varrho^{2}_{q_{i}q_{j}}$
(3-36)
For the state in Eq.(3.32), we have
$\Pi_{q_{1}...q_{m}q_{m+1}...q_{k}|q_{k+1}...q_{n}q_{n+1}...q_{N}}=\Pi^{\prime}_{q_{1}...q_{k}|q_{k+1}...q_{N}}=\varrho^{2}_{q_{1}...q_{k}|q_{k+1}...q_{N}}=4|f|^{2}|f^{\prime}|^{2}$
(3-37)
Another useful entanglement measure was introduced in Refs.[45,46] for
$n$-qubit state $|\psi\rangle=\sum_{i=0}^{2^{n}-1}a_{i}|i\rangle$ with even
$n$, as
$\tau(\psi)=2|{\chi}^{*}(a,n)|$ (3-38)
where
${\chi}^{*}(a,n)=\sum\limits_{i=0}^{2^{n-2}-1}sgn^{*}(n,i)(a_{2i}a_{(2^{n-1}-1)-2i}-a_{2i+1}a_{(2^{n-2}-2)-2i}),$
(3-39) $sgn^{*}(n,i)=\;\cases{{(-1)^{N(i)}}&$0\leq
i\leq{2^{n-3}-1}$\cr{(-1)^{N(i)+n}}&$2^{n-3}\leq i\leq{2^{n-2}-1}$\cr}$ (3-40)
where, $N(i)$ is the number of the occurrences of 1 in the $n$-bit binary
representation of $i$ as $i_{n-1}...i_{1}i_{0}$ ( in binary representation,
$i$ is written as $i=i_{n-1}2^{n-1}+...+i_{1}2^{1}+i_{0}2^{0}$). For the state
Eq.(3.32), one can see that
$\tau(\psi)=2|{\chi}^{*}(a,n)|=2|a_{2^{m}-1}a_{2^{2m}-2^{m}}|=2|ff^{\prime}|.$
(3-41)
In order to achieve maximal entanglement ($GHZ$ state), we should have
$|f|=|f^{\prime}|=\frac{1}{\sqrt{2}}$ (3-42)
Then
$Q(|\psi\rangle)=\Pi_{q_{1}...q_{m}q_{m+1}...q_{k}|q_{k+1}...q_{n}q_{n+1}...q_{N}}=\Pi^{\prime}_{q_{1}...q_{k}|q_{k+1}...q_{N}}=\tau(\psi)=1$.
In the following we consider the four qubit state (the case $m=2$)
$|\psi(t=0)\rangle=|1100\rangle$ and the six qubit state (the case $m=3$)
$|\psi(t=0)\rangle=|111000\rangle$ in details: From Eq. (2-12), for $m=2$, the
QD parameters are given by
$\alpha_{1}=2,\;\ \alpha_{2}=0;\;\;\ \omega_{1}=\omega_{2}=4,$
Then by using the recursion relations (B-48) and (B-51), we obtain
$Q_{2}^{(1)}(x)=x^{2}-2x-4,\;\;\ Q_{3}(x)=x(x-4)(x+2),$
so that the stieltjes function is given by
$G_{\mu}(x)=\frac{Q_{2}^{(1)}(x)}{Q_{3}(x)}=\frac{x^{2}-2x-4}{x(x-4)(x+2)}.$
Then the corresponding spectral distribution is given by
$\mu(x)=\sum_{l=0}^{2}\gamma_{l}\delta(x-x_{l})=\frac{1}{6}\\{3\delta(x)+\delta(x-4)+2\delta(x+2)\\},$
which indicates that
$W=\left(\begin{array}[]{ccc}\gamma_{0}&0&0\\\ 0&\gamma_{1}&0\\\
0&0&\gamma_{2}\\\
\end{array}\right)=\frac{1}{6}\left(\begin{array}[]{ccc}1&0&0\\\ 0&3&0\\\
0&0&2\\\ \end{array}\right).$
In order to obtain the suitable coupling constants, we need also the
eigenvalue matrix $P$ with entries
$P_{ij}=P_{i}(x_{j})=\frac{1}{\sqrt{\omega_{1}\ldots\omega_{i}}}Q_{i}(x_{j})$.
By using the recursion relations (B-48), one can obtain $P_{0}(x)=1,\;\
P_{1}(x)=\frac{x}{2}$ and $P_{2}(x)=\frac{1}{4}(x^{2}-2x-4)$, so that
$P=\left(\begin{array}[]{ccc}1&1&1\\\ 2&0&-1\\\ 1&-1&1\\\ \end{array}\right).$
Then, Eq. (3-30) leads to
$-t(J_{0}+2J_{1}+J_{2})=\theta\pm\frac{\pi}{4}\pm 2c_{0}\pi,$
$-t(J_{0}-J_{2})=\theta\mp\frac{\pi}{4}\pm 2c_{1}\pi,$
$-t(J_{0}-J_{1}+J_{2})=\theta\pm\frac{\pi}{4}\pm 2c_{2}\pi.$
Now, by considering $c_{0}=c_{1}=c_{2}=0$ we obtain
$J_{0}=-\frac{\theta}{t},\;\ J_{1}=0,\;\ J_{2}=\mp\frac{\pi}{4t}.$
Also by considering $c_{0}=0,c_{1}=c_{2}=1$ the coupling constants will be
$J_{0}=-\frac{3\theta\pm 5\pi}{3t},\;\ J_{1}=\pm\frac{2\pi}{3t},\;\
J_{2}=\pm\frac{\pi}{12t}$
and by considering $c_{0}=1,c_{1}=c_{2}=0$
$J_{0}=-\frac{3\theta\pm\pi}{3t},\;\ J_{1}=\mp\frac{2\pi}{3t},\;\
J_{2}=\mp\frac{7\pi}{12t}$
From Eq. (2-12), for $m=3$, the QD parameters are given by
$\alpha_{1}=4,\;\ \alpha_{2}=4,\;\ \alpha_{3}=0;\;\;\
\omega_{1}=\omega_{3}=9,\;\omega_{2}=16$
Then by using the recursion relations (B-48) and (B-51), we obtain
$Q_{3}^{(1)}(x)=x^{3}-8x^{2}-9x+36,\;\;\ Q_{4}(x)=(x^{2}-9)(x-9)(x+1),$
so that the stieltjes function is given by
$G_{\mu}(x)=\frac{Q_{3}^{(1)}(x)}{Q_{4}(x)}=\frac{x^{3}-8x^{2}-9x+36}{(x^{2}-9)(x-9)(x+1)}.$
Then the corresponding spectral distribution is given by
$\mu(x)=\sum_{l=0}^{3}\gamma_{l}\delta(x-x_{l})=\frac{1}{20}\\{\delta(x-9)+5\delta(x-3)+9\delta(x+1)+5\delta(x+3)\\},$
which indicates that
$W=\left(\begin{array}[]{cccc}\gamma_{0}&0&0&0\\\ 0&\gamma_{1}&0&0\\\
0&0&\gamma_{2}&0\\\ 0&0&0&\gamma_{3}\\\
\end{array}\right)=\frac{1}{20}\left(\begin{array}[]{cccc}1&0&0&0\\\
0&5&0&0\\\ 0&0&9&0\\\ 0&0&0&5\\\ \end{array}\right).$
By using the recursion relations (B-48), one can obtain $P_{0}(x)=1,\;\
P_{1}(x)=\frac{x}{3}$, $P_{2}(x)=\frac{1}{12}(x^{2}-4x-9)$ and
$P_{3}(x)=\frac{1}{36}(x^{3}-8x^{2}-9x+36)$, so that
$P=\left(\begin{array}[]{cccc}1&1&1&1\\\ 3&1&-\frac{1}{3}&-1\\\
3&-1&-\frac{1}{3}&1\\\ 1&-1&1&-1\\\ \end{array}\right).$
Then, Eq. (3-30) gives
$-t(J_{0}+3J_{1}+3J_{2}+J_{3})=\theta\pm\frac{\pi}{4}\pm 2c_{0}\pi,$
$-t(J_{0}+J_{1}-J_{2}-J_{3})=\theta\mp\frac{\pi}{4}\mp 2c_{1}\pi,$
$-t(J_{0}-\frac{1}{3}J_{1}-\frac{1}{3}J_{2}+J_{3})=\theta\pm\frac{\pi}{4}\pm
2c_{2}\pi,$ $-t(J_{0}-J_{1}+J_{2}-J_{3})=\theta\mp\frac{\pi}{4}\pm 2c_{3}\pi.$
Again, by considering $c_{0}=c_{1}=c_{2}=c_{3}=0$ we obtain
$J_{0}=-\frac{\theta}{t},\;\ J_{1}=J_{2}=0,\;\ J_{3}=\frac{\mp\pi}{4t}.$
## 4 Conclusion
A $2m$-qubit initial state was prepared to evolve under a particular spin
Hamiltonian, which could be written in terms of the adjacency matrix of the
Johnson graph $J(2m,m)$. By using spectral analysis methods and employing
algebraic structures of the Johnson networks, such as distance-regularity and
stratification, a method for finding a suitable set of coupling constants in
the Hamiltonians associated with the networks was given so that in the final
state, the maximal entanglement of the form $GHZ$ state, could be generated.
In this work we imposed a constraint so that all amplitudes in the final state
were equal to zero except to two amplitudes corresponding to the first and the
final strata (any pair of antinodes of the network), where for $J(2m,m)$ these
strata contain only one vertex, then $GHZ$ state was generated. We hope to
generalize this method to arbitrary Johnson networks $J(n,m)$ and other
various graphs, in order to investigate the entanglement of such systems by
using some multipartite entanglement measures.
## Appendix
## Appendix A Stratification technique
In this section, we recall the notion of stratification for a given graph
$\Gamma$. To this end, let $\partial(x,y)$ be the length of the shortest walk
connecting $x$ and $y$ for $x\neq y$. By definition $\partial(x,x)=0$ for all
$x\in V$. The graph becomes a metric space with the distance function
$\partial$. Note that $\partial(x,y)=1$ if and only if $x\sim y$. We fix a
vertex $o\in V$ as an origin of the graph, called the reference vertex. Then,
the graph $\Gamma$ is stratified into a disjoint union of strata (with respect
to the reference vertex $o$) as
$V=\bigcup_{i=0}^{\infty}\Gamma_{i}(o),\;\ \Gamma_{i}(o):=\\{\alpha\in
V:\partial(\alpha,o)=i\\}$ (A-43)
Note that $\Gamma_{i}(o)=\emptyset$ may occur for some $i\geq 1$. In that case
we have $\Gamma_{i}(o)=\Gamma_{i+1}(o)=...=\emptyset$. With each stratum
$\Gamma_{i}(o)$ we associate a unit vector in $l^{2}(V)$ defined by
$|\phi_{i}\rangle=\frac{1}{\sqrt{\kappa_{i}}}\sum_{\alpha\in\Gamma_{i}(o)}|\alpha\rangle,$
(A-44)
where, $\kappa_{i}=|\Gamma_{i}(o)|$ is called the $i$-th valency of the graph
($\kappa_{i}:=|\\{\gamma:\partial(o,\gamma)=i\\}|=|\Gamma_{i}(o)|$).
One should notice that, for distance regular graphs, the above stratification
is independent of the choice of reference vertex and the vectors
$|\phi_{i}\rangle,i=0,1,...,d-1$ form an orthonormal basis for the so called
Krylov subspace $K_{d}(|\phi_{0}\rangle,A)$ defined as
$K_{d}(|\phi_{0}\rangle,A)=\mathrm{span}\\{|\phi_{0}\rangle,A|\phi_{0}\rangle,\cdots,A^{d-1}|\phi_{0}\rangle\\}.$
(A-45)
Then it can be shown that [25], the orthonormal basis $|\phi_{i}\rangle$ are
written as
$|\phi_{i}\rangle=P_{i}(A)|\phi_{0}\rangle,$ (A-46)
where $P_{i}=a_{0}+a_{1}A+...+a_{i}A^{i}$ is a polynomial of degree $i$ in
indeterminate $A$ (for more details see for example [25,26]).
## Appendix B Spectral distribution associated with the graphs
In this section we recall some facts about spectral techniques used in the
paper. From orthonormality of the unit vectors $|\phi_{i}\rangle$ given in
Eq.(A-44) (with $|\phi_{0}\rangle$ as unit vector assigned to the reference
node) we have
$\delta_{ij}=\langle\phi_{i}|\phi_{j}\rangle=\int_{R}P_{i}(x)P_{j}(x)\mu(dx).$
(B-47)
By rescaling $P_{k}$ as $Q_{k}=\sqrt{\omega_{1}\ldots\omega_{k}}P_{k}$, the
spectral distribution $\mu$ under question will be characterized by the
property of orthonormal polynomials $\\{Q_{k}\\}$ defined recurrently by
$Q_{0}(x)=1,\;\;\;\;\;\ Q_{1}(x)=x,$
$xQ_{k}(x)=Q_{k+1}(x)+\alpha_{k}Q_{k}(x)+\omega_{k}Q_{k-1}(x),\;\;\ k\geq 1.$
(B-48)
The parameters $\alpha_{k}$ and $\omega_{k}$ appearing in (B-48) are defined
by
$\alpha_{0}=0,\;\;\ \alpha_{k}=\kappa-b_{k}-c_{k},\;\;\;\;\
\omega_{k}\equiv\beta^{2}_{k}=b_{k-1}c_{k},\;\;\ k=1,...,d,$ (B-49)
where, $\kappa\equiv\kappa_{1}$ is the degree of the networks and $b_{i}$’s
and $c_{i}$’s are the corresponding intersection numbers. Following Refs.
[34], we will refer to the parameters $\alpha_{k}$ and $\omega_{k}$ as $QD$
(Quantum Decomposition) parameters (see Refs. [26,27,28,34] for more details).
If such a spectral distribution is unique, the spectral distribution $\mu$ is
determined by the identity
$G_{\mu}(x)=\int_{R}\frac{\mu(dy)}{x-y}=\frac{1}{x-\alpha_{0}-\frac{\omega_{1}}{x-\alpha_{1}-\frac{\omega_{2}}{x-\alpha_{2}-\frac{\omega_{3}}{x-\alpha_{3}-\cdots}}}}=\frac{Q_{d}^{(1)}(x)}{Q_{d+1}(x)}=\sum_{l=0}^{d}\frac{\gamma_{l}}{x-x_{l}},$
(B-50)
where, $x_{l}$ are the roots of the polynomial $Q_{d+1}(x)$. $G_{\mu}(x)$ is
called the Stieltjes/Hilbert transform of spectral distribution $\mu$ and
polynomials $\\{Q_{k}^{(1)}\\}$ are defined recurrently as
$Q_{0}^{(1)}(x)=1,\;\;\;\;\;\ Q_{1}^{(1)}(x)=x-\alpha_{1},$
$xQ_{k}^{(1)}(x)=Q_{k+1}^{(1)}(x)+\alpha_{k+1}Q_{k}^{(1)}(x)+\omega_{k+1}Q_{k-1}^{(1)}(x),\;\;\
k\geq 1,$ (B-51)
respectively. The coefficients $\gamma_{l}$ appearing in (B-50) are calculated
as
$\gamma_{l}:=\lim_{x\rightarrow x_{l}}[(x-x_{l})G_{\mu}(x)]$ (B-52)
Now let $G_{\mu}(z)$ is known, then the spectral distribution $\mu$ can be
determined in terms of $x_{l},l=1,2,...$ and Gauss quadrature constants
$\gamma_{l},l=1,2,...$ as
$\mu=\sum_{l=0}^{d}\gamma_{l}\delta(x-x_{l})$ (B-53)
(for more details see Refs. [35,36,37,38]).
## References
* [1] M. A. Jafarizadeh and R. Sufiani, (2008), Phys. Rev. A 77, 022315\.
* [2] M. A. Jafarizadeh, R. Sufiani, S. F. Taghavi and E. Barati, (2008), J. Phys. A: Math. Theor. 41, 475302.
* [3] Sougato Bose, Phys. Rev. Lett. 91, 207901 (2003).
* [4] V. Subrahmanyam, Phys. Rev. A 69, 034304 (2004).
* [5] M. Christandl, N. Datta, A. Ekert and A. J. Landahl, Phys. Rev. Lett. 92, 187902 (2004).
* [6] M. Christandl, N. Datta, T. C. Dorlas, A. Ekert, A. Kay and A. J. Landahl, (2005), Phys. Rev. A 71, 032312.
* [7] C. Albanese, M. Christandl, N. Datta and A. Ekert, Phys. Rev. Lett. 93, 230502 (2004).
* [8] T. J. Osborne and N. Linden, Phys. Rev. A 69, 052315 (2004).
* [9] F. Verstraete, M. A. Martin-Delgado and J. I. Cirac, Phys. Rev. Lett. 92, 087201 (2004).
* [10] F. Verstraete, M. Popp and J. I. Cirac, Phys. Rev. Lett. 92, 027901.
* [11] B.Q. Jin and V. E. Korepin, Phys. Rev. A 69, 062314 (2004).
* [12] M. H Yung, D. W Leung and S. Bose, Quant. Inf. and Comp. 4, 174 (2004).
* [13] L. Amico, A. Osterloh, F. Plastina, R. Fazio and G. M. Palma, Phys. Rev. A 69, 022304 (2004).
* [14] V. Giovannetti and R. Fazio, Phys. Rev. A 71, 032314 (2005).
* [15] D. Burgarth and S. Bose, (2005), Phys. Rev. A 71, 052315.
* [16] D. Burgarth and S. Bose, (2005), New J. Phys. 7, 135.
* [17] A. Chan and R. Hosoya, (2004), J. Algebraic Combinatorics 20, 341-351.
* [18] A. Chan and C. D. Godsil, (2004), J. Combin. Th. Ser. A 106, 165-191.
* [19] M.B. Plenio, J. Hartley and J. Eisert, New J. Phys. 6, 36 (2004).
* [20] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wooters, Phys. Rev. Lett. 70, 1895(1993).
* [21] J. I. Cirac, A. K. Ekert, S. F. Huelga, and C. Macchiavello, Phys. Rev. A 59, 4249 (1999).
* [22] C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. 69, 2881 (1992).
* [23] M. Ghojavand, M. A. Jafarizadeh, and S. Rouhani, J. Stat. Mech. (2010) P03001.
* [24] M. Ghojavand, M. A. Jafarizadeh, and S. Rouhani, J. Stat. Mech. (2010) P12023.
* [25] M. A. Jafarizadeh and S. Salimi, (2006), J. Phys. A : Math. Gen. 39, 1-29.
* [26] M. A. Jafarizadeh, R. Sufiani and S. Jafarizadeh, (2007), J. Phys. A: Math. Theor. 40, 4949-4972.
* [27] M. A. Jafarizadeh, R. Sufiani and S. Jafarizadeh, (2008), Journal of Mathematical Physics 49, 073303.
* [28] M. A. Jafarizadeh, S. Salimi, (2007), Annals of physics 322, 1005-1033.
* [29] M. A. Jafarizadeh, R. Sufiani, S. Salimi and S. Jafarizadeh, Eur. Phys. J. B 59, 199-216.
* [30] M. A. Jafarizadeh, R. Sufiani, (2007), Physica A, 381, 116-142.
* [31] M. A. Jafarizadeh and R. Sufiani, (2007), International Journal of Quantum Information Vol. 5, No. 4, 575-596.
* [32] H. Cycon, R. Forese, W. Kirsch and B. Simon Schrodinger operators (Springer-Verlag, 1987).
* [33] P. D. Hislop and I. M. Sigal, Introduction to spectral theory: With applications to schrodinger operators (1995).
* [34] Y. S. Kim, Phase space picture of quantum mechanics:group theoretical approach, (Science, 1991).
* [35] H. W. Lee, Physics. Report, 259, 147 (1995).
* [36] N. Obata, (2004), Quantum Probabilistic Approach to Spectral Analysis of Star Graphs, Interdisciplinary Information Sciences, Vol. 10, 41-52.
* [37] A. Hora, and N. Obata, (2003), Fundamental Problems in Quantum Physics, World Scientific, 284.
* [38] J. A. Shohat, and J. D. Tamarkin, (1943), The Problem of Moments, American Mathematical Society, Providence, RI.
* [39] C. Godsil, (2005), Association schemes, Combinatorics and Optimization, University of Waterloo.
* [40] D. A. Meyer, and N. R. Wallach, J. Math. Phys. 43 (2002) 4273.
* [41] G. K. Brennen, Quantum Inf. Comput. 3 (2003) 619.
* [42] T. Radtke, S. Fritzsche, Computer Physics Communications. 175 (2006) 145 166.
* [43] G. Vidal, and R. F. Werner, (2002), Phys. Rev. A. 65 032314.
* [44] Z. X. Man, Y. J. Xia and N. B. An, New Journal of Physics. 12 (2010) 033020\.
* [45] D. Li, X. Li, H. Huang and X. Li, Phys. Rev. A. 77 (2008) 056302.
* [46] D. Li, X. Li, H. Huang and X. Li, J. Math. Phys. 50 (2009) 012104.
|
arxiv-papers
| 2011-01-31T17:54:06 |
2024-09-04T02:49:16.745396
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. A. Jafarizadeh, R. Sufiani, F. Eghbalifam, M. Azimi, S. F. Taghavi\n and E. Barati",
"submitter": "Mohamad Ali Jafarizadeh",
"url": "https://arxiv.org/abs/1101.6040"
}
|
1102.0048
|
∎
11institutetext: S. Mottelet 22institutetext: Laboratoire de Mathématiques
Appliquées
Université de Technologie de Compiègne
60205 Compiègne France
22email: stephane.mottelet@utc.fr 33institutetext: L. de Saint Germain, O.
Mondin44institutetext: Luxilon
21, rue du Calvaire
92210 Saint-Cloud France
44email: lsg@lsg-studio.com
# Smart depth of field optimization applied to a robotised view camera
Stéphane Mottelet Luc de Saint Germain Olivier Mondin
###### Abstract
The great flexibility of a view camera allows the acquisition of high quality
images that would not be possible any other way. Bringing a given object into
focus is however a long and tedious task, although the underlying optical laws
are known. A fundamental parameter is the aperture of the lens entrance pupil
because it directly affects the depth of field. The smaller the aperture, the
larger the depth of field. However a too small aperture destroys the sharpness
of the image because of diffraction on the pupil edges. Hence, the desired
optimal configuration of the camera is such that the object has the sharpest
image with the greatest possible lens aperture. In this paper, we show that
when the object is a convex polyhedron, an elegant solution to this problem
can be found. The problem takes the form of a constrained optimization
problem, for which theoretical and numerical results are given.
###### Keywords:
Large format photography Computational photography Scheimpflug principle
## 1 Introduction
---
(a)
(b)
Figure 1: The Sinar e (a) and its metering back (b).
Since the registration of Theodor Scheimpflug’s patent in 1904 (see
Scheimpflug ), and the book of Larmore in 1965 where a proof of the so-called
Scheimpflug principle can be found (see (larmore, , p. 171-173)), very little
has been written about the mathematical concepts used in modern view cameras,
until the development of the Sinar e in 1988 (see Figure 1). A short
description of this camera is given in (Tillmans, , p. 23):
The Sinar e features an integrated electronic computer, and in the studio
offers a maximum of convenience and optimum computerized image setting. The
user-friendly software guides the photographer through the shot without
technical confusion. The photographer selects the perspective (camera
viewpoint) and the lens, and chooses the areas in the subject that are to be
shown sharp with a probe. From these scattered points the Sinar e calculates
the optimum position of the plane of focus, the working aperture needed, and
informs the photographer of the settings needed
Sinar sold a few models of this camera and discontinued its development in the
early nineties. Surprisingly, there has been very little published user
feedback about the camera itself. However many authors started to study (in
fact, re-discover) the underlying mathematics (see e.g. Merklinger and the
references therein). The most crucial aspect is the consideration of depth of
field and the mathematical aspects of this precise point are now well
understood. When the geometrical configuration of the view camera is precisely
known, then the depth of field region (the region of space where objects have
a sharp image) can be determined by using the laws of geometric optics.
Unfortunately, these laws can only be used as a rule of thumb when operating
by hand on a classical view camera. Moreover, the photographer is rather
interested in the inverse problem: given an object which has to be rendered
sharply, what is the optimal configuration of the view camera? A fundamental
parameter of this configuration is the aperture of the camera lens. Decreasing
the lens aperture diameter increases the depth of field but also increases the
diffraction of light by the lens entrance pupil. Since diffraction decreases
the sharpness of the image, the optimal configuration should be such that the
object fits the depth of field region with the greatest aperture.
This paper presents the mathematical tools used in the software of a computer
controlled view camera solving this problem. Thanks to the high precision
machining of its components, and to the known optical parameters of the lens
and digital sensor, a reliable mathematical model of the view camera has been
developed. This model allows the acquisition of 3D coordinates of the object
to be photographed, as explained in Section 2. In Section 3 we study the depth
of field optimization problem from a theoretical and numerical point of view.
We conclude and briefly describe the architecture of the software in Section
4.
## 2 Basic mathematical modeling
### 2.1 Geometrical model of the View Camera
---
(a)
(b)
Figure 2: Geometrical model (a) and robotised view camera (b).
We consider the robotised view camera depicted in Figure 2(a) and its
geometrical model in Figure 2(b). We use a global Euclidean coordinate system
$(\mathbf{O},X_{1},X_{2},X_{3})$ attached to the camera’s tripod. The front
standard, symbolized by its frame with center $L$ of global coordinates
$\mathbf{L}=(L_{1},L_{2},L_{3})^{\top}$, can rotate along its tilt and swing
axes with angles $\theta_{L}$ and $\phi_{L}$. Most camera lenses are in fact
thick lenses and nodal points $H^{\prime}$ and $H$ have to be considered (see
Ray p. 43-46). The rear nodal plane, which is parallel and rigidly fixed to
the front standard, passes through the rear nodal point $H^{\prime}$. Since
$L$ and $H^{\prime}$ do not necessarily coincide, the translation between
these two points is denoted $\mathbf{t^{L}}$. The vector
$\overrightarrow{HH^{\prime}}$ is supposed to be orthogonal to the rear nodal
plane.
The rear standard is symbolized by its frame with center $S$, whose global
coordinates are given by $\mathbf{S}=(S_{1},S_{2},S_{3})^{\top}$. It can
rotate along its tilt and swing axes with angles $\theta_{S}$ and $\phi_{S}$.
The sensor plane is parallel and rigidly fixed to the rear standard. The
eventual translation between $S$ and the center of the sensor is denoted by
$\mathbf{t^{S}}$.
The rear standard center $S$ can move in the three $X_{1},X_{2}$ and $X_{3}$
directions but the front standard center $L$ is fixed. The rotation matrices
associated with the front and rear standard alt-azimuth mounts are
respectively given by $\mathbf{R^{L}}=\mathbf{R}(\theta_{L},\phi_{L})$ and
$\mathbf{R^{S}}=\mathbf{R}(\theta_{S},\phi_{S})$ where
$\mathbf{R}(\theta,\phi)=\left(\begin{array}[]{ccc}\cos\phi&-\sin\phi\sin\theta&-\sin\phi\cos\theta\\\
0&\cos\theta&-\sin\theta\\\
-\sin\phi&\cos\phi\sin\theta&\cos\phi\cos\theta\end{array}\right).$
The intrinsic parameters of the camera (focal length $f$, positions of the
nodal points $H$, $H^{\prime}$, translations $\mathbf{t^{S}}$,
$\mathbf{t^{L}}$, image sensor characteristics) are given by their respective
manufacturers data-sheets. The extrinsic parameters of the camera are
$\mathbf{S}$, $\mathbf{L}$, the global coordinate vectors of $S$ and $L$, and
the four rotation angles $\theta_{S}$, $\phi_{S}$, $\theta_{L}$, $\phi_{L}$.
The precise knowledge of the extrinsic parameters is possible thanks to the
computer-aided design model used for manufacturing the camera components. In
addition, translations and rotations of the rear and front standards are
controlled by stepper motors whose positions can be precisely known. In the
following, we will see that this precise geometrical model of the view camera
allows one to solve various photographic problems. The first problem is the
determination of coordinates of selected points of the object to be
photographed.
In the sequel, for sake of simplicity, we will give all algebraic details of
the computations for a thin lens, i.e. when the two nodal points $H$,
$H^{\prime}$ coincide. In this case, the nodal planes are coincident in a so-
called lens plane. We will also consider that $\mathbf{t^{L}}=(0,0,0)^{\top}$
so that $L$ is the optical center of the lens. Finally, we also consider that
$\mathbf{t}^{S}=(0,0,0)^{\top}$ so that $S$ coincides with the center of the
sensor.
### 2.2 Acquisition of object points coordinates
---
Figure 3: Graphical construction of the image $A$ of an object point $X$ in
the optical coordinate system. Green rays and blue rays respectively lie in
the $(L,x_{3},x_{2})$ and $(L,x_{3},x_{1})$ planes and $F$ is the focal point.
---
Figure 4: Image formation when considering a lens with a pupil.
Let us consider a point $X$ with global coordinates given by
$\mathbf{X}=(X_{1},X_{2},X_{3})^{\top}$. The geometrical construction of the
image $A$ of $X$ through the lens is depicted in Figure 3. We have considered
a local optical coordinate system attached to the lens plane with origin $L$.
The local coordinates of $X$ are given by
$\mathbf{x}=(\mathbf{R^{L}})^{-1}(\mathbf{X}-\mathbf{L})$ and the focal point
$F$ has local coordinates $(0,0,f)^{\top}$. Elementary geometrical optics (see
Ray p. 35-42) allows one to conclude that if the local coordinates of $A$ are
given by $\mathbf{a}=(a_{1},a_{2},a_{3})^{\top}$, then $a_{3}$, $x_{3}$ and
$f$ are linked by the thin lens equation given in its Gaussian form by
$-\frac{1}{a_{3}}+\frac{1}{x_{3}}=\frac{1}{f}.$
Since $A$ lies on the $(XL)$ line, the other coordinates are obtained by
straightforward computations and we have the conjugate formulas
$\displaystyle\mathbf{a}$ $\displaystyle=\frac{f}{f-x_{3}}\mathbf{x},$ (1)
$\displaystyle\mathbf{x}$ $\displaystyle=\frac{f}{f+a_{3}}\mathbf{a}.$ (2)
Bringing an object into focus is one of the main tasks of a photographer but
it can also be used to calculate the coordinates of an object point. It is
important to remember that all light rays emanating from $X$ converge to $A$
but pass through a pupil (or diaphragm) assumed to be circular, as depicted in
Figure 4. Since all rays lie within the oblique circular cone of vertex $A$
and whose base is the pupil, the image of $X$ on the sensor will be in focus
only if the sensor plane passes through $A$, otherwise its extended image will
be a blur spot. By using the full aperture of the lens, the image will rapidly
go out of focus if the sensor plane is not correctly placed, e.g. by
translating $S$ into the $x_{3}$ direction. This is why auto-focus systems on
classical cameras only work at near full aperture: the distance to an object
is better determined when the depth of field is minimal.
The uncertainty on the position of $S$ giving the best focus is related to the
diameter of the so-called “circle of confusion”, i.e. the maximum diameter of
a blur spot that is indistinguishable from a point. Hence, everything depends
on the size of photosites on the sensor and on the precision of the focusing
system (either manual or automatic). This uncertainty is acceptable and should
be negligible compared to the uncertainty of intrinsic and extrinsic camera
parameters.
The previous analysis shows that the global coordinates of $X$ can be
computed, given the position $(u,v)^{\top}$ of its image $A$ on the sensor
plane. This idea has been already used on the Sinar e, where the acquisition
of $(u,v)^{\top}$ was done by using a mechanical metering unit (see Figure 1
(b)). In the system we have developed, a mouse click in the live video window
of the sensor is enough to indicate these coordinates. Once $(u,v)^{\top}$ is
known, the coordinates of $A$ in the global coordinate system are given by
$\mathbf{A}=\mathbf{S}+\mathbf{R^{S}}\left(\begin{array}[]{c}u\\\ v\\\
0\end{array}\right),$
and its coordinates in the optical system by
$\mathbf{a}=(\mathbf{R^{L}})^{-1}(\mathbf{A}-\mathbf{L}).$
Then the local coordinate vector $\mathbf{x}$ of the reciprocal image is
computed with (2) and the global coordinate vector $\mathbf{X}$ is obtained by
$\mathbf{X}=\mathbf{L}+\mathbf{R^{L}}\mathbf{x}.$
By iteratively focusing on different parts of the object, the photographer can
obtain a set of points $\mathcal{X}=\\{{X}^{1},\dots,{X}^{n}\\}$, with $n\geq
3$, which can be used to determine the best configuration of the view camera,
i.e. the positions of front and rear standards and their two rotations, in
order to satisfy focus requirements.
## 3 Focus and depth of field optimization
In classical digital single-lens reflex (DLSR) cameras, the sensor plane is
always parallel to the lens plane and to the plane of focus. For example,
bringing into focus a long and flat object which is not parallel to the sensor
needs to decrease the aperture of the lens in order to extend the depth of
field. On the contrary, view cameras with tilts and swings (or DLSR with a
tilt/shift lens) allow to skew away the plane of focus from the parallel in
any direction. Hence, bringing into focus the same long and flat object with a
view camera can be done at full aperture. This focusing process is
unfortunately very tedious. However, if a geometric model of the camera and
the object are available, the adequate rotations can be estimated precisely.
In the next sections, we will explain how to compute the rear standard
position and the tilt and swing angles of both standards to solve two
different problems:
1. 1.
when the focus zone is roughly flat, and depth of field is not a critical
issue, then the object plane is computed from the set of object points
$\mathcal{X}$. If $n=3$ and the points are not aligned then this plane is
uniquely defined. If $n>3$ and at least $3$ points are not aligned, we compute
the best fitting plane minimizing the sum of squared orthogonal distances to
points of $\mathcal{X}$. Then, we are able to bring this plane into sharp
focus by acting on:
1. (a)
the angles $\theta_{L}$ and $\phi_{L}$ of the front standard and the position
of the rear standard, for arbitrary rotation angles $\theta_{S}$, $\phi_{S}$.
2. (b)
the angles $\theta_{S}$, $\phi_{S}$ and position of the rear standard, for
arbitrary rotation angles $\theta_{L}$, $\phi_{L}$ (in this case there is a
perspective distortion).
2. 2.
when the focus zone is not flat, then the tridimensional shape of the object
has to be taken into account.
The computations in case 1a are detailed in Section 3.1. In Section 3.3 a
general algorithm is described that allows the computation of angles
$\theta_{L}$ and $\phi_{L}$ of the front standard and the position of the rear
standard such that all the object points are in the depth of field region with
a maximum aperture. We give a theoretical result showing that the
determination of the solution amounts to enumerate a finite number of
configurations.
### 3.1 Placement of the plane of sharp focus by using tilt and swing angles
Figure 5: Illustration of the Scheimpflug rule. Figure 6: Illustration of the
Hinge rule.
In this section we study the problem of computing the tilt and swing angles of
front standard and the position of the rear standard for a given sharp focus
plane. Although the underlying laws are well-known and are widely described
(see Merklinger ; Wheeler ; Evens ), the detail of the computations is always
done for the particular case where only the tilt angle $\theta$ is considered.
Since we aim to consider the more general case where tilt and swing angles are
used, we will describe the various objects (lines, planes) and the associated
computations by using linear algebra tools.
#### 3.1.1 The Scheimpflug and the Hinge rules
In order to explain the Scheimpflug rule, we will refer to the diagram
depicted in Figure 5. The Plane of sharp focus (abbreviated $\mathrm{SFP}$) is
determined by a normal vector $\mathbf{n^{SF}}$ and a point $Y$. The position
of the optical center $L$ and a vector $\mathbf{n^{S}}$ normal to the sensor
plane (abbreviated $\mathrm{SP}$) are known. The unknowns are the position of
the sensor center $S$ and a vector $\mathbf{n^{L}}$ normal to the lens plane
(abbreviated $\mathrm{LP}$).
The Scheimplug rule stipulates that if $\mathrm{SFP}$ is into focus, then
$\mathrm{SP}$, $\mathrm{LP}$ and $\mathrm{SFP}$ necessarily intersect on a
common line called the ”Scheimpflug Line” (abbreviated $\mathrm{SL}$). The
diagram of Figure 5a should help the reader to see that this rule is not
sufficient to uniquely determine $\mathbf{n^{L}}$ and $\mathrm{SP}$, as this
plane can be translated toward $\mathbf{n^{S}}$ if $\mathbf{n^{L}}$ is changed
accordingly.
The missing constraints are provided by the Hinge rule, which is illustrated
in Figure 6. This rule considers two complimentary planes: the front focal
plane (abbreviated $\mathrm{FFP}$), which is parallel to $\mathrm{LP}$ and
passes through the focal point $F$, and the parallel to sensor lens plane
(abbreviated $\mathrm{PSLP}$), which is parallel to $\mathrm{SP}$ and passes
through the optical center $L$. The Hinge Rule stipulates that $\mathrm{FFP}$,
$\mathrm{PSLP}$ and $\mathrm{SFP}$ must intersect along a common line called
the Hinge Line (abbreviated $\mathrm{HL}$). Since $\mathrm{HL}$ is uniquely
determined as the intersection of $\mathrm{SFP}$ and $\mathrm{PSLP}$, this
allows one to determine $\mathbf{n^{L}}$, or equivalently the tilt and swing
angles, such that $\mathrm{FFP}$ passes through $\mathrm{HL}$ and $F$. Then
$\mathrm{SL}$ is uniquely defined as the intersection of $\mathrm{LP}$ and
$\mathrm{SFP}$ by the Scheimpflug rule (note that $\mathrm{SL}$ and
$\mathrm{HL}$ are parallel by construction). Since $\mathbf{n^{S}}$ is already
known, any point belonging to $\mathrm{SL}$ is sufficient to uniquely define
$\mathrm{SP}$. Hence, the determination of tilt and swing angles and position
of the rear standard can be summarized as follows:
1. 1.
determination of $\mathrm{HL}$, intersection of $\mathrm{FFP}$ and
$\mathrm{SFP}$,
2. 2.
determination of tilt and swing angles such that $\mathrm{HL}$ belongs to
$\mathrm{FFP}$,
3. 3.
determination of $\mathrm{SL}$, intersection of $\mathrm{LP}$ and
$\mathrm{SFP}$,
4. 4.
translation of $S$ such that $\mathrm{SL}$ belongs to $\mathrm{SP}$.
#### 3.1.2 Algebraic details of the computations
In this section the origin of the coordinate system is the optical center $L$
and the inner product of two vectors $\mathbf{X}$ and $\mathbf{Y}$ is
expressed by using the matrix notation ${\mathbf{X}}^{\top}{\mathbf{Y}}$. All
planes are defined by a unit normal vector and a point in the plane as
follows:
$\displaystyle\mathrm{SP}$
$\displaystyle=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{(\mathbf{X-S})}^{\top}{\mathbf{n^{S}}}=0\right\\}$
$\displaystyle\mathrm{PSLP}$
$\displaystyle=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{\mathbf{X}}^{\top}{\mathbf{n^{S}}}=0\right\\},$
$\displaystyle\mathrm{LP}$
$\displaystyle=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{\mathbf{X}}^{\top}{\mathbf{n^{L}}}=0\right\\},$
$\displaystyle\mathrm{FFP}$
$\displaystyle=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{\mathbf{X}}^{\top}{\mathbf{n^{L}}}-f=0\right\\},$
$\displaystyle\mathrm{SFP}$
$\displaystyle=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{(\mathbf{X}-\mathbf{Y})}^{\top}{\mathbf{n^{SF}}}=0\right\\},$
where the equation of $\mathrm{FFP}$ takes this particular form because the
distance between $L$ and $F$ is equal to the focal length $f$ and we have
imposed that $n^{L}_{3}>0$. The computations are detailed in the following
algorithm:
###### Algorithm 1
* Step 1 : compute the Hinge Line by considering its parametric equation
$\displaystyle\mathrm{HL}$
$\displaystyle=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;\exists\,t\in\mathbb{R},\;\mathbf{X}=\mathbf{W}+t\mathbf{V}\right\\},$
where $\mathbf{V}$ is a direction vector and $\mathbf{W}$ is the coordinate
vector of an arbitrary point of $\mathrm{HL}$. Since this line is the
intersection of $\mathrm{PSLP}$ and $\mathrm{SFP}$, $\mathbf{V}$ is orthogonal
to $\mathbf{n^{L}}$ and $\mathbf{n^{SF}}$. Hence, we can take $\mathbf{V}$ as
the cross product
$\mathbf{V}=\mathbf{n^{SF}}\times\mathbf{n^{S}}$
and $\mathbf{W}$ as a particular solution (e.g. the solution of minimum norm)
of the overdetermined system of equations
$\displaystyle{\mathbf{W}}^{\top}{\mathbf{n^{S}}}$ $\displaystyle=0,$
$\displaystyle{\mathbf{W}}^{\top}{\mathbf{n^{SF}}}$
$\displaystyle={\mathbf{Y}}^{\top}{\mathbf{n^{SF}}}.$
* Step 2 : since $\mathrm{HL}$ belongs to $\mathrm{FFP}$ we have
${(\mathbf{W}+t\mathbf{V})}^{\top}{\mathbf{n^{L}}}-f=0,\quad\forall~{}t\in\mathbb{R},$
hence $\mathbf{n^{L}}$ verifies the overdetermined system of equations
$\displaystyle{\mathbf{W}}^{\top}{\mathbf{n^{L}}}$ $\displaystyle=f,$ (3)
$\displaystyle{\mathbf{V}}^{\top}{\mathbf{n^{L}}}$ $\displaystyle=0.$ (4)
with the constraint $\|\mathbf{n^{L}}\|^{2}=1$. The computation of
$\mathbf{n^{L}}$ can be done by the following two steps:
1. 1.
compute $\widetilde{\mathbf{W}}=\mathbf{V}\times\mathbf{W}$ and
$\widetilde{\mathbf{V}}$ the minimum norm solution of system (3)-(4), which
gives a parametrization
$\mathbf{n^{L}}=\widetilde{\mathbf{V}}+t\widetilde{\mathbf{W}},$
of all its solutions, where $t$ is an arbitrary real.
2. 2.
determination of $t$ such that $\|\mathbf{n^{L}}\|^{2}=1$: this is done by
taking the solution $t$ of the second degree equation
${\widetilde{\mathbf{W}}}^{\top}{\widetilde{\mathbf{W}}}t^{2}+2{\widetilde{\mathbf{W}}}^{\top}{\widetilde{\mathbf{V}}}t+{\widetilde{\mathbf{V}}}^{\top}{\widetilde{\mathbf{V}}}-1=0,$
such that $n^{L}_{3}>0$. The tilt and swing angles are then obtained as
$\theta_{L}=-\arcsin
n_{2}^{L},\quad\phi_{L}=-\arcsin\frac{n_{1}^{L}}{\cos\theta_{L}}.$
* Step 3 : since $\mathrm{SL}$ is the intersection of $\mathrm{LP}$ and $\mathrm{SFP}$, the coordinate vector $\mathbf{U}$ of a particular point $U$ on $\mathrm{SL}$ is obtained as the minimum norm solution of the system
$\displaystyle{\mathbf{U}}^{\top}{\mathbf{n^{L}}}$ $\displaystyle=0,$
$\displaystyle{\mathbf{U}}^{\top}{\mathbf{n^{SF}}}$
$\displaystyle={\mathbf{W}}^{\top}{\mathbf{n^{SF}}},$
where we have used the fact that $W\in\mathrm{SFP}$.
* Step 4 : the translation of $S$ can be computed such that $U$ belongs to $\mathrm{SP}$, i.e.
${(\mathbf{U}-\mathbf{S})}^{\top}{\mathbf{n^{S}}}=0.$
If we only act on the third coordinate of $S$ and leave the two others
unchanged, then $S_{3}$ can be computed as
$S_{3}=\frac{{\mathbf{U}}^{\top}{\mathbf{n^{S}}}-S_{1}n^{S}_{1}-S_{2}n^{S}_{2}}{n^{S}_{3}}.$
###### Remark 1
When we consider a true camera lens, the nodal points $H,H^{\prime}$ and the
front standard center $L$ do not coincide. Hence, the tilt and swing rotations
of the front standard modify the actual position of the $\mathrm{PSLP}$ plane.
In this case, we use the following iterative fixed point scheme:
1. 1.
The angles $\phi_{L}$ and $\theta_{L}$ are initialized with starting values
$\phi_{L}^{0}$ and $\theta_{L}^{0}$.
2. 2.
At iteration $k$,
1. (a)
the position of $\mathrm{PSLP}$ is computed considering $\phi_{L}^{k}$ and
$\theta_{L}^{k}$,
2. (b)
the resulting Hinge Line is computed, then the position of $\mathrm{FFP}$ and
the new values $\phi_{L}^{k+1}$ and $\theta_{L}^{k+1}$ are computed.
Point 2 is repeated until convergence of $\phi_{L}^{k}$ and $\theta_{L}^{k}$
up to a given tolerance. Generally 3 iterations are sufficient to reach the
machine precision.
### 3.2 Characterization of the depth of field region
Figure 7: Position of planes $\mathrm{SFP_{1}}$ and $\mathrm{SFP_{2}}$
delimiting the depth of field region.
As in the previous section, we consider that $L$ and the nodal points $H$ and
$H^{\prime}$ coincide. Moreover, $L$ will be the origin of the global
coordinates system.
We consider the configuration depicted in Figure 7 where the sharp focus plane
$\mathrm{SFP}$, the lens plane $\mathrm{LP}$ and the sensor plane
$\mathrm{SP}$ are tied by the Scheimpflug and the Hinge rule. The depth of
field can be defined as follows:
###### Definition 1
Let $X$ be a 3D point and $A$ its image through the lens. Let $\mathcal{C}$ be
the disk in $\mathrm{LP}$ of center $L$ and diameter $f/N$, where $N$ is
called the f-number. Let $K$ be the cone of base $\mathcal{C}$ and vertex $A$.
The point $X$ is said to lie in the depth of field region if the diameter of
the intersection of $\mathrm{SP}$ and $K$ is lower that $c$, the diameter of
the so-called circle of confusion.
The common values of $c$, which depend on the magnification from the sensor
image to the final image and on its viewing conditions, lie typically between
0.2 mm and 0.01 mm. In the following the value of $c$ is not a degree of
freedom but a given input.
If the ellipticity of extended images is neglected, the depth of field region
can be shown to be equal to the unbounded wedge delimited by
$\mathrm{SFP_{1}}$ and $\mathrm{SFP_{2}}$ intersecting at $\mathrm{HL}$, where
the corresponding sensor planes $\mathrm{SP_{1}}$ and $\mathrm{SP_{2}}$ are
tied to $\mathrm{SFP_{1}}$ and $\mathrm{SFP_{2}}$ by the Scheimpflug rule. By
mentally rotating $\mathrm{SFP}$ around $\mathrm{HL}$, it is easy to see that
$\mathrm{SP}$ is translated through $\mathbf{n^{S}}$ and spans the region
between $\mathrm{SP_{1}}$ and $\mathrm{SP_{2}}$. The position of
$\mathrm{SP_{1}}$ and $\mathrm{SP_{2}}$, the f-number $N$ and the diameter of
the circle of confusion $c$ are related by the formula
$\frac{Nc}{f}=\frac{p_{1}-p_{2}}{p_{1}+p_{2}},$ (5)
where $p_{1}$, respectively $p_{2}$, are the distances between the optical
center $L$ and $\mathrm{SP_{1}}$, respectively $\mathrm{SP_{2}}$, both
measured orthogonally to the optical plane. The distance $p$ between
$\mathrm{SP}$ and L can be shown to be equal to
$p=\frac{2p_{1}p_{2}}{p_{1}+p_{2}},$ (6)
the harmonic mean of $p_{1}$ and $p_{2}$. This approximate definition of the
depth of field region has been proposed by various authors (see Wheeler ;
Bigler ) but when the ellipticity of images is taken into account a complete
study can be found in Evens . For sake of completeness, we give the
justification of formulas (5) and (6) in Appendix A.1. In most practical
situations the necessary angle between $\mathrm{SP}$ and $\mathrm{LP}$ is
small (less that 10 degrees), so that this approximation is correct.
###### Remark 2
The analysis in Appendix A.1 shows that the ratio $\frac{Nc}{f}$ in equation
(5) does not depend on the direction used for measuring the distance between
$\mathrm{SP}$, $\mathrm{SP_{1}}$, $\mathrm{SP_{2}}$ and $L$. The only
condition, in order to take into account the case where $\mathrm{SP}$ and
$\mathrm{LP}$ are parallel, is that this direction is not orthogonal to
$\mathbf{n^{S}}$. Hence, by taking the direction given by $\mathbf{n^{S}}$, we
can obtain an equivalent formula to (5). To compute the distances, we need the
coordinate vector of two points $U_{1}$ and $U_{2}$ on $\mathrm{SP_{1}}$ and
$\mathrm{SP_{2}}$ respectively. To this purpose we consider Step 3 of
Algorithm 1 in section 3.1: if $\mathbf{W}$ is the coordinate vector of any
point $W$ of $\mathrm{HL}$, each vector $\mathbf{U}^{i}$ can be obtained as a
particular solution of the system
$\displaystyle{\mathbf{U}^{i}}^{\top}{\mathbf{n^{L}}}$ $\displaystyle=0,$
$\displaystyle{\mathbf{U}^{i}}^{\top}{\mathbf{n}^{i}}$
$\displaystyle={\mathbf{W}}^{\top}{\mathbf{n}^{i}}.$
Since $\mathrm{SP_{i}}$ can be defined as
$\mathrm{SP_{i}}=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{(\mathbf{X}-\mathbf{U}^{i})}^{\top}{\mathbf{n^{S}}}=0\right\\},$
and $\|\mathbf{n^{S}}\|=1$, the signed distance from $L$ to $\mathrm{SP_{i}}$
is equal to
$d(L,\mathrm{SP_{i}})={\mathbf{U_{i}}}^{\top}{\mathbf{n^{S}}}.$
So the equivalent formula giving the ratio $\frac{Nc}{f}$ is given by
$\displaystyle\frac{Nc}{f}$
$\displaystyle=\left|\frac{d(L,\mathrm{SP_{1}})-d(L,\mathrm{SP_{2}})}{d(L,\mathrm{SP_{1}})+d(L,\mathrm{SP_{2}})}\right|,$
$\displaystyle=\left|\frac{{(\mathbf{U_{1}}-\mathbf{U_{2}})}^{\top}{\mathbf{n^{S}}}}{{(\mathbf{U_{1}}+\mathbf{U_{2}})}^{\top}{\mathbf{n^{S}}}}\right|.$
(7)
The above considerations show that for a given orientation of the rear
standard given by $\mathbf{n^{S}}$, if the depth of field wedge is given, then
the needed f-number, the tilt and swing angles of the front standard and the
translation of the sensor plane, can be determined. The related question that
will be addressed in the following is the question: given a set of points
$\mathcal{X}=\\{{X}^{1},\dots,{X}^{n}\\}$, how can we minimize the f-number
such that all points of $\mathcal{X}$ lie in the depth of field region?
### 3.3 Depth of field optimization with respect to tilt angle
We first study the depth of field optimization in two dimensions, because in
this particular case all computations can be carried explicitly and a closed
form expression is obtained, giving the f-number as a function of front
standard tilt angle and of the slope of limiting planes. First, notice that
$N$ has a natural upper bound, since (5) implies that
$N\leq\frac{f}{c}.$
#### 3.3.1 Computation of f-number with respect to tilt angle and limiting
planes
Figure 8: The depth of field region when only a tilt angle $\theta$ is used.
Without loss of generality, we consider that the sensor plane has the normal
$\mathbf{n^{S}}=(0,0,1)^{\top}$. Let us denote by $\theta$ the tilt angle of
the front standard and consider that the swing angle $\phi$ is zero. The lens
plane is given by
$\mathrm{LP}=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{\mathbf{X}}^{\top}{\mathbf{n^{L}}}=0\right\\},$
where
$\mathbf{n^{L}}=(0,-\sin\theta,\cos\theta)^{\top},$
and any collinear vector to $\mathbf{n^{L}}\times\mathbf{n^{S}}$ is a
direction vector of the Hinge Line. Hence we can take, independently of
$\theta$,
$\mathbf{V}=(1,0,0)^{\top}$
and a parametric equation of $\mathrm{HL}$ is thus given by
$\displaystyle\mathrm{HL}$
$\displaystyle=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;\exists\,t\in\mathbb{R},\;\mathbf{X}=\mathbf{W}(\theta)+t\mathbf{V}\right\\},$
where $\mathbf{W}(\theta)$ is the coordinate vector of a particular point
${W}(\theta)$ on $\mathrm{HL}$, obtained as the minimum norm solution of
$\displaystyle{\mathbf{W}(\theta)}^{\top}{\mathbf{n^{S}}}$ $\displaystyle=0,$
$\displaystyle{\mathbf{W}(\theta)}^{\top}{\mathbf{n^{L}}}$ $\displaystyle=f.$
Straightforward computations show that for $\theta\neq 0$
$\mathbf{W}(\theta)=\left(\begin{array}[]{c}0\\\ -\frac{f}{\sin\theta}\\\
0\end{array}\right).$
Consider, as depicted in Figure 8, the two sharp focus planes
$\mathrm{SFP_{1}}$ and $\mathrm{SFP_{2}}$ passing through $\mathrm{HL}$, with
normals $\mathbf{n^{1}}=(0,-1,a_{1})^{\top}$ and
$\mathbf{n^{2}}=(0,-1,a_{2})^{\top}$,
$\displaystyle\mathrm{SFP_{1}}$
$\displaystyle=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{(\mathbf{X}-\mathbf{W}(\theta))}^{\top}{\mathbf{n^{1}}}=0\right\\},$
$\displaystyle\mathrm{SFP_{2}}$
$\displaystyle=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{(\mathbf{X}-\mathbf{W}(\theta))}^{\top}{\mathbf{n^{2}}}=0\right\\}.$
The two corresponding sensor planes $\mathrm{SP_{i}}$ are given by following
Steps 3-4 in Algorithm 1 by
$\displaystyle\mathrm{SFP_{i}}=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;X_{3}=t_{i}\right\\},\;i=1,2,$
where
$t_{i}=\frac{f}{a_{i}\sin\theta-\cos\theta},\;i=1,2.$
Using equation (7) the corresponding f-number is equal to
$N(\theta,\mathbf{a})=\left|\frac{t_{1}-t_{2}}{t_{1}+t_{2}}\right|\left(\frac{f}{c}\right),$
(8)
where we have used the notation $\mathbf{a}=(a_{1},a_{2})$. Finally we have,
for $\theta\neq 0$
$N(\theta,\mathbf{a})=\operatorname{sign}\theta\frac{(a_{1}-a_{2})\sin\theta}{2\cos\theta-(a_{1}+a_{2})\sin\theta}\left(\frac{f}{c}\right).$
(9)
###### Remark 3
When $\theta=0$, then $\mathrm{SFP}$ does not intersect $\mathrm{PSLP}$ and
the depth of field region is included between two parallel planes
$\mathrm{SFP_{i}}$ given by
$\displaystyle\mathrm{SFP_{i}}=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;X_{3}=z_{i}\right\\},\;i=1,2,$
where $z_{1}$ and $z_{2}$ depend on the f-number and on the position of
$\mathrm{SFP}$. One can show by using the thin lens equation and equation (5)
that the corresponding f-number is equal to
$N_{0}(z_{1},z_{2})=\frac{\left|\frac{1}{z_{2}}-\frac{1}{z_{1}}\right|}{\frac{1}{z_{1}}+\frac{1}{z_{2}}-\frac{2}{f}}\left(\frac{f}{c}\right).$
(10)
#### 3.3.2 Theoretical results for the optimization problem
Without loss of generality, we will consider a set of only three non-aligned
points $\mathcal{X}=\\{{X}^{1},{X}^{2},{X}^{3}\\}$, which have to be within
the depth of field region with minimal f-number and we denote by
$\mathbf{X}^{1},\mathbf{X}^{2},\mathbf{X}^{3}$ their respective coordinate
vectors.
The corresponding optimization problem can be stated as follows: find
$(\theta^{*},\mathbf{a}^{*})=\arg\min_{\begin{array}[]{c}\theta\in\mathbb{R}\\\
a\in\mathcal{A(\theta)}\end{array}}N(\theta,\mathbf{a}),$ (11)
where for a given $\theta$ the set $\mathcal{A(\theta)}$ is defined by the
inequalities
$\displaystyle{(\mathbf{X}^{i}-\mathbf{W}(\theta))}^{\top}{\mathbf{n^{1}}}$
$\displaystyle\geq 0,~{}i=1,2,3,$ (12)
$\displaystyle{(\mathbf{X}^{i}-\mathbf{W}(\theta))}^{\top}{\mathbf{n^{2}}}$
$\displaystyle\leq 0,~{}i=1,2,3,$ (13)
meaning that ${X}^{1},{X}^{2},{X}^{3}$ are respectively under
$\mathrm{SFP_{1}}$ and above $\mathrm{SFP_{2}}$, and by the inequalities
$\displaystyle-{(\mathbf{X}^{i}-\mathbf{W}(\theta))}^{\top}{\mathbf{n^{L}}}+f$
$\displaystyle\leq 0,~{}i=1,2,3,$ (14)
meaning that ${X}^{1},{X}^{2},{X}^{3}$ are in front of $\mathrm{FFP}$.
###### Remark 4
Points behind the focal plane cannot be in focus, and the constraints (14) are
just expressing this practical impossibility. However, we have to notice that
when one of these constraints is active, we can show that $a_{1}=\cot\theta$
or $a_{2}=\cot\theta$ so that $N(\mathbf{a},\theta)$ reaches its upper bound
$\frac{f}{c}$. Moreover, we have to eliminate the degenerate case where the
points ${X}^{i}$ are such that there exists two active constraints in (14): in
this case, there exists a unique admissible pair $(\mathbf{a},\theta)$ and the
problem has no practical interest. To this purpose, we can suppose that
$X^{i}_{3}>f$ for $i=1,2,3$.
For $\theta\neq 0$ the gradient of $N(\mathbf{a},\theta)$ is equal to
$\mathbf{\nabla}(\mathbf{a},\theta)=\frac{2\operatorname{sign}\theta}{(2\cot\theta-(a_{1}+a_{2}))^{2}}\left(\begin{array}[]{c}\cot\theta-
a_{2}\\\ -\cot\theta+a_{1}\\\
\frac{a_{1}-a_{2}}{\sin^{2}\theta}\end{array}\right)$
and cannot vanish since $a_{1}=a_{2}$ is not possible because it would mean
that ${X}^{1},{X}^{2},{X}^{3}$ are aligned. This implies that $\mathbf{a}$
lies on the boundary of $\mathcal{A}(\theta)$ and we have the following
intuitive result (the proof is given in Appendix A.2):
###### Proposition 1
Suppose that $X^{i}_{3}>f$, for $i=1,2,3$. Then when $N(\mathbf{a},\theta)$
reaches its minimum, there exists $i_{1}$, $i_{2}$ with $i_{1}\neq i_{2}$ such
that
$\displaystyle{(\mathbf{X}^{i_{1}}-\mathbf{W}(\theta))}^{\top}{\mathbf{n^{1}}}$
$\displaystyle=0,$
$\displaystyle{(\mathbf{X}^{i_{2}}-\mathbf{W}(\theta))}^{\top}{\mathbf{n^{2}}}$
$\displaystyle=0.$
###### Remark 5
The above result shows that at least two points touch the depth of field
limiting planes $\mathrm{SFP_{1}}$ and $\mathrm{SFP_{2}}$ when the f-number is
minimal. In the following, we will show that the three points
${X}^{1},{X}^{2}$ and ${X}^{3}$ are necessarily in contact with one of the
limiting planes (the proof is given in Appendix A.3):
###### Proposition 2
Suppose that the vertices $\\{{X}^{i}\\}_{i=1\dots 3}$ verify the condition
$\frac{\|\mathbf{X}^{i}\times\mathbf{X}^{j}\|}{\|\mathbf{X}^{i}-\mathbf{X}^{j}\|}>f,~{}~{}i\neq
j.$ (15)
Then $N(\mathbf{a},\theta)$ reaches its minimum when all vertices are in
contact with the limiting planes.
###### Remark 6
If $\theta$ is small, then $N(\theta,\mathbf{a})$ in (9) can be approximated
by
$\tilde{N}(\theta,\mathbf{a})=\operatorname{sign}\theta{(a_{1}-a_{2})\sin\theta}\left(\frac{f}{2c}\right),$
(16)
and the proof of Proposition 2 is considerably simplified: the same result
holds with the weaker condition
$\|\mathbf{X}^{i}\times\mathbf{X}^{j}\|>0.$ (17)
In fact, an approximate way of specifying the depth of field region using the
hyperfocal distance, proposed in Merklinger , leads to the same approximation
of $N(\theta,\mathbf{a})$, under the a priori hypothesis of small $\theta$ and
distant objects, i.e. $X^{i}_{3}\gg f$. This remark is clarified in Appendix
A.4.
We will illustrate the theoretical result by considering sets of 3 points. For
a set with more than 3 vertices (but being equal to the vertices of the convex
hull of $\mathcal{X}$), the determination of the optimal solution is purely
combinatorial, since it is enough to enumerate all admissible situations where
two points are in contact with one plane, and a third one with the other. The
value $\theta=0$ also has to be considered because it can be a critical value
if the object has a vertical edge. We will also give an Example which violates
condition (15) and where $N(\mathbf{a},\theta)$ reaches its minimum when only
two vertices are in contact with the limiting planes.
#### 3.3.3 Numerical results
In this section, we will consider the following function, defined for
$\theta\neq 0$
$n(\theta)=\min_{a\in\mathcal{A}(\theta)}N(\mathbf{a},\theta).$
Finding the minimum of this function allows one to solve the original
constrained optimization problem, but considering the results of the previous
section, $n(\theta)$ is non-differentiable. In fact, the values of $\theta$
for which $n(\theta)$ is not differentiable correspond to the situations where
3 points are in contact with the limiting planes. We extend $n(\theta)$ by
continuity for $\theta=0$ by defining
$n(0)=\frac{\frac{1}{z_{2}}-\frac{1}{z_{1}}}{\frac{1}{z_{1}}+\frac{1}{z_{2}}-\frac{2}{f}}\left(\frac{f}{c}\right),$
where
$z_{1}=\max_{i=1,2,3}X^{i}_{3},~{}~{}z_{2}=\min_{i=1,2,3}X^{i}_{3}.$
This formula can be directly obtained by using conjugation formulas or by
taking $\theta=0$ in equation (24).
###### Example 1
Figure 9: Enumeration of the 3 candidates configurations for Example 1.
---
(a)
(b)
Figure 10: Example 1 graph of $n(\theta)$ for $\theta\in[-0.6,0.6]$ (a) and
$\theta\in[0,0.06]$ (b). Labels give the different types of contact.
We consider a lens with focal length $f=5.10^{-2}\mathrm{m}$ and a confusion
circle $c=3.10^{-5}m$ (commonly used value for 24x36 cameras). The vertices
have coordinates
$\mathbf{X}^{1}=\left(\begin{array}[]{r}0\\\ -1\\\
1\end{array}\right),~{}\mathbf{X}^{2}=\left(\begin{array}[]{r}0\\\ 3\\\
1\end{array}\right),~{}\mathbf{X}^{3}=\left(\begin{array}[]{r}0\\\ 0\\\
1.5\end{array}\right).$
Figure 9 shows the desired depth of field region (dashed zone) and the three
candidates hinge lines corresponding to contacts $E_{ij}V_{k}$:
* •
$E_{12}V_{3}$, obtained when the edge $[{X}^{1},{X}^{2}]$ is in contact with
$\mathrm{SFP_{1}}$ and vertex ${X}^{3}$ with $\mathrm{SFP_{2}}$.
* •
$E_{13}V_{2}$, obtained when the edge $[{X}^{1},{X}^{3}]$ is in contact with
$\mathrm{SFP_{2}}$ and vertex ${X}^{2}$ with $\mathrm{SFP_{1}}$.
* •
$E_{23}V_{1}$, obtained when the edge $[{X}^{2},{X}^{3}]$ is in contact with
$\mathrm{SFP_{1}}$ and vertex ${X}^{1}$ with $\mathrm{SFP_{2}}$.
The associated values of $\theta$ and $n(\theta)$ are given in Table 1,
Contact | $\theta$ | $n(\theta)$
---|---|---
$E_{12}V_{3}$ | 0.0166674 | 9.52
$E_{13}V_{2}$ | 0.025002 | 7.16
$E_{23}V_{1}$ | 0.0500209 | 28.28
Table 1: Value of $\theta$ for each possible optimal contact and corresponding
f-number $n(\theta)$ for Example 1.
which shows that contact $E_{13}V_{2}$ seems to give the minimum f-number.
Condition (15) is verified, since
$\frac{\|\mathbf{X}^{1}\times\mathbf{X}^{2}\|}{\|\mathbf{X}^{1}-\mathbf{X}^{2}\|}=1,~{}\frac{\|\mathbf{X}^{1}\times\mathbf{X}^{3}\|}{\|\mathbf{X}^{1}-\mathbf{X}^{3}\|}=1.34,$
$\frac{\|\mathbf{X}^{2}\times\mathbf{X}^{3}\|}{\|\mathbf{X}^{2}-\mathbf{X}^{3}\|}=1.48.$
Hence, the derivative of $n(\theta)$ cannot vanish and the minimum f-number is
necessarily reached for $\theta^{*}=0.025002$.
We can confirm this by considering the graph of $n(\theta)$ depicted on Figure
10. In the zone of interest, the function $n(\theta)$ is almost piecewise
affine. For each point in the interior of curved segments of the graph, the
value of $\theta$ is such that the contact of $\mathcal{X}$ with the limiting
planes is of type $V_{i}V_{j}$. Clearly, the derivative of $n(\theta)$ does
not vanish. The possible optimal values of $\theta$, corresponding to contacts
of type $E_{i}jV_{k}$, are the abscissa of angular points of the graph, marked
with red dots. The graph confirms that the minimal value of $n(\theta)$ is
reached for contact $E_{12}V_{3}$.
The minimal f-number is equal to $n(\theta^{*})=7.16$. By comparison, the
f-number without tilt optimization is $n(0)=28.74$. This example highlights
the important gain in terms of f-number reduction with the optimized tilt
angle.
###### Example 2
Contact | $\theta$ | $n(\theta)$
---|---|---
$E_{12}V_{3}$ | 0.185269 | 29.49
$E_{23}V_{1}$ | 0.100419 | 83.67
$E_{13}V_{2}$ | 0.235825 | 47.04
Table 2: Value of $\theta$ for each possible optimal contact and corresponding
f-number $n(\theta)$ for Example 2.
---
(a)
(b)
Figure 11: Example 2 graph of $n(\theta)$ for $\theta\in[-0.6,0.6]$ (a) and
$\theta\in[0,0.3]$ (b). Labels give the different types of contact.
We consider the same lens and confusion circle as in Example 1
($f=5.10^{-2}\mathrm{m}$, $c=3.10^{-5}m$) but the vertices have coordinates
$\mathbf{X}^{1}=\left(\begin{array}[]{r}0\\\ -0.1\\\
0.12\end{array}\right),~{}\mathbf{X}^{2}=\left(\begin{array}[]{r}0\\\ 0\\\
0.19\end{array}\right),~{}\mathbf{X}^{3}=\left(\begin{array}[]{r}0\\\
-0.0525\\\ 0.17\end{array}\right).$
The object is almost ten times smaller than the object of the previous example
(it has the size of a small pen), but it is also ten times closer: this is a
typical close up configuration. The values of $\theta$ and $n(\theta)$
associated to contacts of type $E_{ij}V_{k}$ are given in Table 2 which shows
that contact $E_{13}V_{2}$ seems to give the minimum f-number. We have
$\frac{\|\mathbf{X}^{1}\times\mathbf{X}^{2}\|}{\|\mathbf{X}^{1}-\mathbf{X}^{2}\|}=0.16,~{}\frac{\|\mathbf{X}^{1}\times\mathbf{X}^{3}\|}{\|\mathbf{X}^{1}-\mathbf{X}^{3}\|}=0.16,$
$\frac{\|\mathbf{X}^{2}\times\mathbf{X}^{3}\|}{\|\mathbf{X}^{2}-\mathbf{X}^{3}\|}=0.1775527,$
showing that condition (15) is still verified, even if the values are smaller
than the values of Example 1. Hence, the derivative of $n(\theta)$ cannot
vanish and the minimum f-number is reached for $\theta^{*}=0.185269$.
We can confirm this by considering the graph of $n(\theta)$ depicted on Figure
11. As in Example 1, the derivative of $n(\theta)$ does not vanish and the
graph confirms that the minimal value of $n(\theta)$ is reached for contact
$E_{12}V_{3}$.
The minimal f-number is equal to $n(\theta^{*})=29.49$. By comparison, the
f-number without tilt optimization is $n(0)=193.79$. Such a large value gives
an aperture of diameter $0.26$mm, almost equivalent to a pin hole ! Since the
maximum f-number of view camera lenses is never larger than 64, the object
cannot be in focus without using tilt.
This example also shows that Proposition 2 is still valid even if the object
is close to the lens and the obtained optimal tilt angle $0.185269$ (10.61
degrees) is large.
###### Example 3
---
(a)
(b)
Figure 12: Example 3 graphs of $n(\theta)$ for $\theta\in[-1,1]$.
In this example we consider an extremely flat triangle which is almost aligned
with the optical center. From the photographer’s point of view, this is an
unrealistic case. We consider the same optical parameters as before and
consider the following vertices
$\mathbf{X}^{1}=\left(\begin{array}[]{r}0\\\ 0\\\
1\end{array}\right),~{}\mathbf{X}^{2}=\left(\begin{array}[]{r}0\\\ h\\\
1.5\end{array}\right),~{}\mathbf{X}^{3}=\left(\begin{array}[]{r}0\\\ -h\\\
2\end{array}\right),$
for $h=0.01$ and we have
$\frac{\|\mathbf{X}^{1}\times\mathbf{X}^{2}\|}{\|\mathbf{X}^{1}-\mathbf{X}^{2}\|}=0.02,~{}\frac{\|\mathbf{X}^{1}\times\mathbf{X}^{3}\|}{\|\mathbf{X}^{1}-\mathbf{X}^{3}\|}=0.01,$
$\frac{\|\mathbf{X}^{2}\times\mathbf{X}^{3}\|}{\|\mathbf{X}^{2}-\mathbf{X}^{3}\|}=0.07,$
hence condition (15) is violated in configurations $V_{1}V_{2}$ and
$V_{1}V_{3}$. Since this condition is sufficient, the minimum value of
$n(\theta)$ could still occur for a contact of type $E_{ij}V_{k}$. However, we
can see in Figure 12a that $n(\theta)$ has a minimum at a differentiable point
in configuration $V_{1}V_{3}$.
###### Remark 7
Condition (15) is not necessary: in the proof of Proposition 2 (given in
Appendix A.3), it can be seen that (15) is a condition ensuring that the
polynomial $p(\theta)$ defined by equation (25) has no root in
$[-\pi/2,\pi/2]$. However, the relevant interval is smaller because
${X}^{i_{1}}$ and ${X}^{i_{2}}$ do not stay in contact with the limiting
planes for all values of $\theta$ in $[-\pi/2,\pi/2]$, and because all
vertices must be in front of the focal plane. Anyway, it is always possible to
construct absolutely unrealistic configurations. For example, when we consider
the above vertices with $h=0.005$, then no edge is in contact with the
limiting planes and vertices ${X}^{1}$ and ${X}^{3}$ stay in contact with the
limiting planes for all admissible values of $\theta$. The corresponding graph
of $n(\theta)$ is given in Figure 12b.
### 3.4 Depth of field optimization with respect to tilt and swing angles
As in the previous section, we consider the case of a thin lens and where the
optical and sensor centers coincide respectively with the front and rear
standard rotation centers. Without loss of generality, we consider that the
sensor plane has the normal $\mathbf{n^{S}}=(0,0,1)^{\top}$. The lens plane is
given by
$\mathrm{LP}=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{\mathbf{X}}^{\top}{\mathbf{n^{L}}}=0\right\\},$
where
$\mathbf{n^{L}}=(-\sin\phi\cos\theta,-\sin\theta,\cos\phi\cos\theta)^{\top}.$
A parametric equation of $\mathrm{HL}$ is given by
$\displaystyle\mathrm{HL}$
$\displaystyle=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;\exists\,t\in\mathbb{R},\;\mathbf{X}=\mathbf{W}(\theta,\phi)+t\mathbf{V}(\theta,\phi)\right\\},$
where the direction vector is given by
$\mathbf{V}(\theta,\phi)=\mathbf{n^{L}}\times\mathbf{n^{S}}=(-\sin\theta,\sin\phi\cos\theta,0)^{\top},$
and $\mathbf{W}(\theta,\phi)$ is the coordinate vector of a particular point
${W}(\theta,\phi)$ on $\mathrm{HL}$, obtained as the minimum norm solution of
$\displaystyle{\mathbf{W}(\theta)}^{\top}{\mathbf{n^{S}}}$ $\displaystyle=0,$
$\displaystyle{\mathbf{W}(\theta)}^{\top}{\mathbf{n^{L}}}$ $\displaystyle=f.$
Consider, as depicted in Figure 7, the two planes of sharp focus
$\mathrm{SFP_{1}}$ and $\mathrm{SFP_{2}}$ intersecting at HL, with normals
$\mathbf{n^{1}}$ and $\mathbf{n^{1}}$ respectively. The point $W(\theta,\phi)$
belongs to $\mathrm{SFP_{1}}$ and $\mathrm{SFP_{2}}$ and any direction vector
of $\mathrm{SFP_{1}}\cap\mathrm{SFP_{2}}$ is collinear to
$\mathbf{V}(\theta,\phi)$. Hence, we have
$\displaystyle\mathrm{SFP_{1}}$
$\displaystyle=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{(\mathbf{X}-\mathbf{W}(\theta,\phi))}^{\top}{\mathbf{n^{1}}}=0\right\\},$
$\displaystyle\mathrm{SFP_{2}}$
$\displaystyle=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{(\mathbf{X}-\mathbf{W}(\theta,\phi))}^{\top}{\mathbf{n^{2}}}=0\right\\},$
and
$(\mathbf{n^{1}}\times\mathbf{n^{2}})\times\mathbf{V}(\theta,\phi)=0.$ (18)
Using equation (7) the f-number is equal to
$N(\theta,\phi,\mathbf{n^{1}},\mathbf{n^{2}})=\frac{f}{c}\left|\frac{{(\mathbf{U}^{1}-\mathbf{U}^{2})}^{\top}{\mathbf{n^{S}}}}{{(\mathbf{U}^{1}+\mathbf{U}^{2})}^{\top}{\mathbf{n^{S}}}}\right|,$
(19)
where each coordinate vector $\mathbf{U}^{i}$ of point $U^{i}$, for $i=1,2$,
is obtained as a particular solution of the following system:
$\displaystyle{\mathbf{U}^{i}}^{\top}{\mathbf{n^{L}}}$ $\displaystyle=0,$
$\displaystyle{\mathbf{U}^{i}}^{\top}{\mathbf{n}^{i}}$
$\displaystyle={\mathbf{W}(\theta,\phi)}^{\top}{\mathbf{n}^{i}}.$
###### Remark 8
Figure 13: Position of the hinge line when both tilt and swing are used and
$\mathbf{n^{S}}=(0,0,1)^{\top}$. The LP and FFP planes have not been
represented for reasons of readability.
When $\mathbf{n^{S}}=(0,0,1)^{\top}$ the intersections of $\mathrm{HL}$ with
the $(L,X_{1})$ and the $(L,X_{2})$ axes can be determined, as depicted in
Figure 13. In this case, it is easy to show that the coordinates of the
minimum norm $\mathbf{W}(\theta,\phi)$ are given by
$\mathbf{W}(\theta,\phi)=-\frac{f}{\sin^{2}\phi\cos^{2}\theta+\sin^{2}\theta}(\sin\phi\cos\theta,\sin\theta,0)^{\top}.$
We note that up to a rotation of axis $(0,0,1)^{\top}$ and angle $\alpha$, we
recover a configuration where only a tilt angle $\psi$ is used, where these
two angles are respectively defined by
$\displaystyle\sin\psi$
$\displaystyle=\operatorname{sign}\theta\sqrt{\sin^{2}\phi\cos^{2}\theta+\sin^{2}\theta},$
$\displaystyle\sin\alpha$ $\displaystyle=\frac{\sin\phi\cos\theta}{\sin\psi}.$
#### 3.4.1 Optimization problem
Consider a set 4 non coplanar points $\mathcal{X}=\\{{X}^{i}\\}_{i=1\dots 4}$,
which have to be within the depth of field region with minimal f-number and
denote by $\\{\mathbf{X}^{i}\\}_{i=1\dots 4}$ their respective coordinate
vectors. The optimization problem can be stated as follows: find
$(\theta^{*},\phi^{*},\mathbf{n^{1}}^{*},\mathbf{n^{2}}^{*})=\arg\min_{\theta,\phi,\mathbf{n^{1}},\mathbf{n^{2}}}N(\theta,\phi,\mathbf{n^{1}},\mathbf{n^{2}}),$
(20)
where the minimum is taken for $\theta,\phi,\mathbf{n^{1}}$ and
$\mathbf{n^{2}}$ such that equation (18) is verified and such that the points
$\\{{X}^{i}\\}_{i=1\dots 4}$ lie between $\mathrm{SFP_{1}}$ and
$\mathrm{SFP_{2}}$. This last set of constraints can be expressed in a way
similar to equations (12)-(13).
#### 3.4.2 Analysis of configurations
Consider the function $n(\theta,\phi)$ defined by
$n(\theta,\phi)=\min_{\mathbf{n^{1}},\mathbf{n^{2}}}N(\theta,\phi,\mathbf{n^{1}},\mathbf{n^{2}}),$
(21)
where $\mathbf{n^{1}}$ and $\mathbf{n^{2}}$ are constrained as in the previous
section. Consider the two limiting planes $\mathrm{SFP_{1}}$ and
$\mathrm{SFP_{2}}$ with normals $\mathbf{n^{1}}$ and $\mathbf{n^{2}}$
satisfying the minimum in equation (21). Using the rotation argument of Remark
8 we can easily show that $\mathrm{SFP_{1}}$ and $\mathrm{SFP_{2}}$ are
necessary in contact with at least two vertices. However, for each value of
the pair $(\theta,\phi)$, we have three types of possible contact between the
tetrahedron formed by points $\\{{X}^{i}\\}_{i=1\dots 4}$ and the limiting
planes: vertex-vertex, edge-vertex, edge-edge or face-vertex. These
configurations can be analyzed as follows: let us consider a pair
$(\theta_{0},\phi_{0})$ and the corresponding type of contact:
* •
Vertex-vertex: each limiting plane is in contact with only one vertex,
respectively $V_{i}$ and $V_{j}$. In this case, $n(\theta,\phi)$ is
differentiable at $(\theta_{0},\phi_{0})$ and there exists a curve
$\gamma_{vv}$ defined by
$\gamma_{vv}(t)=(\theta(t),\phi(t)),~{}\gamma_{vv}(0)=(\theta_{0},\phi_{0}),$
such that $\frac{d}{dt}n(\theta(t),\phi(t))$ exists and does not vanish for
$t=0$. This can be proved using again the rotation argument of Remark 8.
Hence, $n(\theta_{0},\phi_{0})$ cannot be minimal.
* •
Edge-vertex: one of the two planes is in contact with edge $E_{ij}$ and the
other one is in contact with vertex $V_{k}$. In this case there is still a
degree of freedom since the plane in contact with $E_{ij}$ can rotate around
this edge in either directions while keeping the other plane in contact with
$V_{k}$ only. If the plane in contact with $E_{ij}$ is $\mathrm{SFP_{1}}$, its
normal $\mathbf{n^{1}}$ can be parameterized by using a single scalar
parameter $t$ and we obtain a family of planes defined by
$\mathrm{SFP_{1}}(t)=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{\mathbf{n^{1}}(t)}^{\top}{(\mathbf{X}-\mathbf{X}^{i})}=0\right\\}.$
For each value of $t$, the intersection of $\mathrm{SFP_{1}}(t)$ with
$\mathrm{PSLP}$ defines a Hinge Line and thus a pair $(\theta(t),\phi(t))$ of
tilt and swing angles. Hence, there exists a parametric curve
$\gamma_{ev}(t)=(\theta(t),\phi(t)),~{}\gamma_{ev}(0)=(\theta_{0},\phi_{0}),$
(22)
along which $n(\theta,\phi)$ is differentiable. As we will see in the
numerical results, the curve $\gamma_{ev}(t)$ is almost a straight line when
$\theta$ and $\phi$ are small, and $\frac{d}{dt}n(\theta,\phi(t))$ does not
vanish for $t=0$.
* •
Edge-edge: the limiting planes are respectively in contact with edges
$E_{ij}$, $E_{kl}$ connecting, respectively, vertices $V_{i},V_{j}$ and
vertices $V_{k},V_{l}$. There is no degree of freedom left since these edges
cannot be parallel (otherwise all points would be coplanar). Hence,
$n(\theta,\phi)$ is not differentiable at ($\theta_{0},\phi_{0}$).
* •
Face-vertex: the limiting planes are respectively in contact with vertex
$V_{l}$ and with the face $F_{ijk}$ connecting vertices $V_{i},V_{j},V_{k}$.
As in the previous case, there is no degree of freedom left and
$n(\theta,\phi)$ is not differentiable at ($\theta_{0},\phi_{0}$).
We can already speculate that the first two configurations are necessary
suboptimal. Consequently we just have to compute the f-number associated with
each one of the 7 possible configurations of type edge-edge of face-vertex.
#### 3.4.3 Numerical results
We have considered the flat object of Example 1, translated in plane
$X_{1}=-0.5$, and a complimentary point in order to form a tetrahedron. The
vertices have coordinates
$\mathbf{X}^{1}=\left(\begin{array}[]{r}-0.5\\\ -1\\\
1\end{array}\right),~{}\mathbf{X}^{2}=\left(\begin{array}[]{r}-0.5\\\ 3\\\
1\end{array}\right),$ $\mathbf{X}^{3}=\left(\begin{array}[]{r}-0.5\\\ 0\\\
1.5\end{array}\right),\mathbf{X}^{4}=\left(\begin{array}[]{r}1\\\ 1\\\
1.5\end{array}\right).$
Contact | $\theta$ | $\phi$ | $n(\theta,\phi)$
---|---|---|---
$E_{12}E_{34}$ | 0.021430 | -0.028582 | 12.35
$E_{23}E_{14}$ | 0.150568 | -0.203694 | 90.75
$E_{13}E_{24}$ | 0.030005 | -0.020010 | 8.64
$F_{123}V_{4}$ | 0 | 0.100167 | 88.18
$F_{243}V_{1}$ | 0.075070 | -0.050162 | 42.91
$F_{134}V_{2}$ | 0.033340 | -0.033358 | 9.64
$F_{124}V_{3}$ | 0.018751 | -0.012503 | 10.76
Table 3: Value of $\theta$ and $\phi$ for each possible optimal contact and
corresponding f-number $n(\theta,\phi)$.
All configurations of type edge-edge and face-vertex have been considered and
the corresponding values of $\theta,\phi$ and $n(\theta,\phi)$ are given in
Table 3. The $E_{13}E_{24}$ contact seems to give the minimum f-number.
Figure 14: Graph of $n(\theta,\phi)$ for
$(\theta,\phi)\in[-0.175,0.225]\times[-0.3225,0.2775]$. Figure 15: Level
curves of $n(\theta,\phi)$ and types of contact for
$(\theta,\phi)\in[-0.175,0.225]\times[-0.3225,0.2775]$. Figure 16: Level
curves and types of of $n(\theta,\phi)$ contact for
$(\theta,\phi)\in[-0.015,0.035]\times[-0.0375,-0.0075]$.
Figure 14 gives the graph of $n(\theta,\phi)$ and its level curves in the
vicinity of the minimum are depicted in Figures 15 and 16. In the interior of
each different shaded region, the $(\theta,\phi)$ pair is such that the
contact of $\mathcal{X}$ with the limiting planes is of type $V_{i}V_{j}$. The
possible optimal $(\theta,\phi)$ pairs, corresponding to contacts of type
$E_{ij}E_{kl}$ of $F_{ijk}V_{l}$, are marked with red dots. Notice that the
graph of $n(\theta,\phi)$ is almost polyhedral, i.e. in the interior of
regions of type $V_{i}V_{j}$, the gradient is almost constant and does not
vanish, as seen on the level curves. If confirms that the minimum cannot occur
in these regions, as announced in Section 3.4.2.
The frontiers between regions of type $V_{i}V_{j}$ are curves corresponding to
contacts of type $E_{ij}V_{k}$ and defined by Equation (22). The extremities
of these curves are $(\theta,\phi)$ pairs corresponding to contacts of type
$E_{ij}E_{kl}$ or $F_{ijk}V_{l}$. For example, in Figure 16, the
$(\theta,\phi)$ pairs on the curve separating $V_{2}V_{3}$ and $V_{3}V_{4}$
regions correspond to the $E_{24}V_{3}$ contact. The extremities of this curve
are the two $(\theta,\phi)$ pairs corresponding to contacts $F_{124}V_{3}$ and
$E_{13}E_{24}$. Along this curve, $n(\theta,\phi)$ is strictly monotone as
shown by its level curves.
Finally, the convergence of its level curves in Figure 16 confirms that the
minimum of $n(\theta,\phi)$ is reached for the $E_{13}E_{24}$ contact. Hence,
the optimal angles are $(\theta^{*},\phi^{*})=(0.030005,-0.020010)$ and the
minimal f-number is equal to $n(\theta^{*},\phi^{*})=8.64$. By comparison, the
f-number without tilt and swing optimization is $n(0,0)=28.74$. This example
highlights again the important gain in terms of f-number reduction with the
optimized tilt and swing angles. In our experience, the optimal configuration
for general polyhedrons can be of type edge-edge or face-vertex.
## 4 Trends and conclusion
In this paper, we have given the optimal solution of the most challenging
issue in view camera photography: bring an object of arbitrary shape into
focus and at the same time minimize the f-number. This problem takes the form
of a continuous optimization problem where the objective function (the
f-number) and the constraints are non-linear with respect to the design
variables. When the object is a convex polyhedron, we have shown that this
optimization problem does not need to be solved by classical methods. Under
realistic hypotheses, the optimal solution always occurs when the maximum
number of constraints are saturated. Such a situation corresponds to a small
number of configurations (seven when the object is a tetrahedron). Hence, the
exact solution is found by comparing the value of the f-number for each
configuration.
The linear algebra framework allowed us to efficiently implement the
algorithms in a numerical computer algebra software. The camera software is
able to interact with a robotised view camera prototype, which is actually
used by our partner photographer. With the robotised camera, the time elapsed
in the focusing process is often orders of magnitude smaller than the
systematic trial and error technique.
The client/server architecture of the software allows us to rapidly develop
new problem solvers by validating them first on a virtual camera before
implementing them on the prototype. We are currently working on the fine
calibration of some extrinsic parameters of the camera, in order to improve
the precision of the acquisition of 3D points of the object.
###### Acknowledgements.
This work has been partly funded by the Innovation and Technological Transfer
Center of Région Ile de France.
## Appendix A Appendix
### A.1 Computation of the depth of field region
Figure 17: Construction of four approximate intersections of cones with
$\mathrm{SP}$
(a)
(b)
Figure 18: (a) close-up of a particular intersection exhibiting the vertices
of cones for a given directrix $\mathcal{D}$. (b) geometrical construction
allowing to derive the depth of field formula.
In order to explain the kind of approximation used, we have represented in
Figure 17 the geometric construction of the image space limits corresponding
to the depth of field region. Let us consider the cones whose base is the
pupil and having an intersection with $\mathrm{SP}$ of diameter $c$. The image
space limits are the locus of the vertex of such cones. The key point,
suggested in Bigler and Evens , is the way the diameter of the intersection
is measured.
For a given line $\mathcal{D}$ passing through $L$ and a circle $\mathcal{C}$
of center $L$ in $\mathrm{LP}$ let us call
$\mathcal{K}(\mathcal{C},\mathcal{D})$ the set of cones with directrix
$\mathcal{D}$ and base $\mathcal{C}$. For a given directrix $\mathcal{D}$ let
us call $A$ its intersection with $SP$, as depicted in Figure 18a. Instead of
considering the intersection of cones of directrix $\mathcal{D}$ with
$\mathrm{SP}$, we consider their intersections with the plane passing through
$A$ and parallel to $\mathrm{LP}$. By construction, all intersections are
circles, and there exists only two cones $K_{1}$ and $K_{2}$ in
$\mathcal{K}(\mathcal{C},\mathcal{D})$ such that this intersection has a
diameter equal to $c$, with their respective vertices $A_{1},A_{2}$ on each
side of $\mathrm{SP}$, respectively marked in Figure 18a by a red and a green
spot. Moreover, for all cones in $\mathcal{K}(\mathcal{C},\mathcal{D})$ only
those with a vertex lying on the segment $[A_{1},A_{2}]$ have an ”approximate”
intersection of diameter less that $c$.
The classical laws of homothety show that for any directrix $\mathcal{D}$, the
locus of the vertices of cones $K_{1}$ and $K_{2}$ will be on two parallel
planes located in front of and behind $\mathrm{SP}$, as illustrated by a red
and a green frame in Figure 17a. Hence, the depth of field region in the
object space is the reciprocal image of the region between parallel planes
$\mathrm{SP_{1}}$ and $\mathrm{SP_{2}}$ as depicted in Figure 7.
Formulas (5) and (6) are obtained by considering the directrix that is
orthogonal to $\mathrm{LP}$, as depicted in Figure 18b. If we note $p=AL$,
$p_{1}=A_{1}L$, $p_{2}=A_{2}L$, by considering similar triangles, we have
$\displaystyle\frac{p_{1}-p}{p_{1}}=\frac{p-p_{2}}{p_{2}}=\frac{Nc}{f},$ (23)
which gives immediately
$p=\frac{2p_{1}p_{2}}{p_{1}+p_{2}},$
and by substituting $p$ in (23), we obtain
$\frac{p_{1}-p_{2}}{p_{1}+p_{2}}=\frac{Nc}{f},$
which allows to obtain (5).
### A.2 Proof of Proposition 1
Without loss of generality, we consider that the optimal $\theta$ is positive.
Suppose now that only one constraint is active in (12). Then there exists
$i_{1}$ such that ${(\mathbf{X}^{i_{1}}-\mathbf{W})}^{\top}{\mathbf{n^{1}}}=0$
and the first order optimality condition is verified: if we define
$g(\mathbf{a},\theta)=-{(\mathbf{X}^{i}-\mathbf{W}(\theta))}^{\top}{\mathbf{n^{1}}}=X^{i_{1}}_{2}-a_{1}X^{i_{1}}_{3}+\frac{f}{\sin\theta},$
there exists $\lambda_{1}\geq 0$ such that the Kuhn and Tucker condition
$\nabla N(\mathbf{a},\theta)+\lambda_{1}\nabla g(\mathbf{a},\theta)=0,$
is verified. Hence, we have
$\frac{2}{(2\cot\theta-(a_{1}+a_{2}))^{2}}\left(\begin{array}[]{c}\cot\theta-
a_{2}\\\ -\cot\theta+a_{1}\\\
\frac{a_{1}-a_{2}}{\sin^{2}\theta}\end{array}\right)+\lambda_{1}\left(\begin{array}[]{c}-X^{i_{1}}_{3}\\\
0\\\ -\frac{\cos\theta}{\sin^{2}\theta}\end{array}\right),$
and necessarily, $a_{1}=\cot\theta$ so that $N(a,\theta)$ reaches its upper
bound and thus is not minimal. We obtain the same contradiction when only a
constraint in (13) is active, or only two constraints in (12), or only two
constraints in (13).∎
### A.3 Proof of Proposition 2
Without loss of generality we suppose that $\theta\geq 0$. Suppose that the
minimum of $N(\mathbf{a},\theta)$ is reached with only vertices $i_{1}$ and
$i_{2}$ respectively in contact with limiting planes $\mathrm{SFP_{1}}$ and
$\mathrm{SFP_{2}}$. The values of $a_{1}$ and $a_{2}$ can be determined as the
following functions of $\theta$
$a_{1}(\theta)=\frac{X_{2}^{i_{1}}+\frac{f}{\sin\theta}}{X_{3}^{i_{1}}},~{}a_{1}(\theta)=\frac{X_{2}^{i_{2}}+\frac{f}{\sin\theta}}{X_{3}^{i_{2}}},$
and straightforward computations give
$N(\mathbf{a}(\theta),\theta)=\frac{\left(\frac{X^{i_{1}}_{2}}{X^{i_{1}}_{3}}-\frac{X^{i_{2}}_{2}}{X^{i_{2}}_{3}}\right)\sin\theta+\left(\frac{f}{X^{i_{1}}_{3}}-\frac{f}{X^{i_{2}}_{3}}\right)}{2\cos\theta-\left(\frac{X^{i_{1}}_{2}}{X^{i_{1}}_{3}}+\frac{X^{i_{2}}_{2}}{X^{i_{2}}_{3}}\right)\sin\theta-\left(\frac{f}{X^{i_{1}}_{3}}+\frac{f}{X^{i_{2}}_{3}}\right)}\left(\frac{f}{c}\right).$
(24)
In order to prove the result, we just have to check that the derivative of
$N(\mathbf{a}(\theta),\theta)$ with respect to $\theta$ cannot vanish for
$\theta\in[0,\frac{\pi}{2}]$. The total derivative of
$N(\mathbf{a}(\theta),\theta))$ with respect to $\theta$ is given by
$\frac{d}{d\theta}N(\mathbf{a}(\theta),\theta)=\\\
2\frac{\left(\frac{X^{i_{1}}_{2}}{X^{i_{1}}_{3}}-\frac{X^{i_{2}}_{2}}{X^{i_{2}}_{3}}\right)-f\left(\frac{X^{i_{1}}_{2}-X^{i_{2}}_{2}}{X^{i_{1}}_{3}X^{i_{2}}_{3}}\right)\cos\theta+\left(\frac{f}{X^{i_{1}}_{3}}-\frac{f}{X^{i_{2}}_{3}}\right)\sin\theta}{\left(2\cos\theta-\left(\frac{X^{i_{1}}_{2}}{X^{i_{1}}_{3}}+\frac{X^{i_{2}}_{2}}{X^{i_{2}}_{3}}\right)\sin\theta-\left(\frac{f}{X^{i_{1}}_{3}}+\frac{f}{X^{i_{2}}_{3}}\right)\right)^{2}}\left(\frac{f}{c}\right)$
and its numerator is proportional to the trigonometrical polynomial
$p(\theta)=b_{0}+b_{1}\cos\theta+b_{2}\sin\theta,$ (25)
where $b_{0}=X^{i_{1}}_{2}X^{i_{2}}_{3}-X^{i_{2}}_{2}X^{i_{1}}_{3}$,
$b_{1}=-f\left(X^{i_{2}}_{2}-X^{i_{1}}_{2}\right)$,
$b_{2}=f(X^{i_{2}}_{3}-X^{i_{1}}_{3})$. It can be easily shown by using the
Schwartz inequality that $p(\theta)$ does not vanish provided that
$b_{0}^{2}>b_{1}^{2}+b_{2}^{2}.$ (26)
Since $X_{1}^{i_{2}}=X_{1}^{i_{1}}=0$, whe have
$b_{0}^{2}=\|X_{1}^{i_{1}}\times X_{1}^{i_{1}}\|^{2}$ and
$b_{1}^{2}+b_{2}^{2}=f^{2}\|X_{1}^{i_{1}}-X_{1}^{i_{1}}\|^{2}$. Hence (26) is
equivalent to condition (15), this ends the proof.∎
### A.4 Depth of field region approximation used by A. Merklinger
Figure 19: Depth of field region approximation using the hyperfocal.
In his book (Merklinger , Chapter 7) A. Merklinger has proposed the following
approximation based on the assumption of distant objects and small tilt
angles: if $h$ is the distance from $\mathrm{SFP_{1}}$ to $\mathrm{SFP_{2}}$,
measured in a direction parallel to the sensor plane at distance $z$ from the
lens plane, as depicted in Figure 19, we have
$\frac{h}{2z}\approx\frac{f}{H\sin\theta},$ (27)
where $H$ is the hyperfocal distance (for a definition see Ray p. 221),
related to the f-number $N$ by the formula
$H=\frac{f^{2}}{Nc},$
and $c$ is the diameter of the circle of confusion. Using (27), the f-number
$N$ can be approximated by
$\tilde{N}=\sin\theta\frac{h}{2z}\frac{f}{c}.$
Since the slopes of $\mathrm{SFP_{1}}$ and $\mathrm{SFP_{2}}$ are respectively
given by $a_{1}$ and $a_{2}$, we have
$\frac{h}{z}={a_{1}-a_{2}},$
and we obtain immediately
$\tilde{N}=(a_{1}-a_{2})\sin\theta\left(\frac{f}{2c}\right),$
which is the same as (16).
## References
* (1) E. Bigler. Depth of field and Scheimpflug rule : a minimalist geometrical approach. http://www.galerie-photo.com/profondeur-de-champ-scheimpflug-english.html, 2002.
* (2) L. Evens. View camera geometry. http://www.math.northwestern.edu/~len/photos/pages/vc.pdf, 2008\.
* (3) L. Larmore. Introduction to Photographic Principles. Dover Publication Inc., New York, 1965.
* (4) A. Merklinger. Focusing the view camera. Bedford, Nova Scotia, 1996. http://www.trenholm.org/hmmerk/FVC161.pdf.
* (5) S. F. Ray. Applied Photographic Optics. Focal Press, 2002. third edition.
* (6) T. Scheimpflug. Improved Method and Apparatus for the Systematic Alteration or Distortion of Plane Pictures and Images by Means of Lenses and Mirrors for Photography and for other purposes. GB Patent No. 1196, 1904.
* (7) U. Tillmans. Creative Large Format: Basics and Applications. Sinar AG. Feuerthalen, Switzerland, 1997.
* (8) R. Wheeler. Notes on view camera geometry, 2003. http://www.bobwheeler.com/photo/ViewCam.pdf.
|
arxiv-papers
| 2011-02-01T01:05:10 |
2024-09-04T02:49:16.755599
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "St\\'ephane Mottelet and Luc de Saint Germain and Olivier Mondin",
"submitter": "St\\'ephane Mottelet",
"url": "https://arxiv.org/abs/1102.0048"
}
|
1102.0097
|
Hamiltonian formulation for the theory of gravity
and canonical transformations in extended phase space
T. P. Shestakova
Department of Theoretical and Computational Physics, Southern Federal
University,
Sorge St. 5, Rostov-on-Don 344090, Russia
E-mail: shestakova@sfedu.ru
Abstract
A starting point for the present work was the statement recently discussed in
the literature that two Hamiltonian formulations for the theory of gravity,
the one proposed by Dirac and the other by Arnowitt – Deser – Misner, may not
be related by a canonical transformation. In its turn, it raises a question
about the equivalence of these two Hamiltonian formulations and their
equivalence to the original formulation of General Relativity. We argue that,
since the transformation from components of metric tensor to the ADM variables
touches gauge degrees of freedom, which are non-canonical from the point of
view of Dirac, the problem cannot be resolved in the limits of the Dirac
approach. The proposed solution requires the extension of phase space by
treating gauge degrees of freedom on an equal footing with other variables and
introducing missing velocities into the Lagrangian by means of gauge
conditions in differential form. We illustrate with a simple cosmological
model the features of Hamiltonian dynamics in extended phase space. Then, we
give a clear proof for the full gravitational theory that the ADM-like
transformation is canonical in extended phase space in a wide enough class of
possible parametrizations.
## 1\. Introduction
It is generally accepted that the problem of formulating Hamiltonian dynamics
for systems with constraints has been solved by Dirac in his seminal papers
[1, 2]. It was Dirac who pointed to the importance of Hamiltonian formulation
for any dynamical theory before its quantization [3]. Other approaches, such
as the Batalin – Fradkin – Vilkovisky (BFV) path integral approach [4, 5, 6]
follow the Dirac one in what concerns the rule of constructing a Hamiltonian
and the role of constraints as generators of transformations in phase space.
It is believed that Dirac generalized Hamiltonian dynamics is equivalent to
Lagrangian dynamics of original theory. However, even for electrodynamics the
constraints do not generate a correct transformation for zero component of
vector potential, $A_{0}$. The same situation we face in General Relativity,
since gravitational constraints cannot produce correct transformations for
$g_{00}$, $g_{0\mu}$ components of metric tensor. In fact, it means that the
group of transformations generated by constraints differs from the group of
gauge transformations of the original theory. Some authors have tried to
remedy this shortcoming by modifying the Dirac approach and proposing some
special prescriptions how the generator should be constructed (see, for
example, [7, 8]). Until now this problem has not attracted much attention
mainly because that it touches only transformations of gauge variables which,
according to conventional viewpoint, are redundant and must not affect the
physical content of the theory. It will be demonstrated in this paper that the
role of gauge degrees of freedom may be more significant that it is usually
thought, and the difference in the groups of transformations is the first
indication to the inconsistence of the theory.
Historically, while constructing Hamiltonian dynamics for gravitational field
theorists used various parametrizations of gravitational variables. Dirac
dealt with original variables, which are components of metric tensor [3],
whereas the most famous parametrization is probably that of Arnowitt – Deser –
Misner (ADM) [9], who expressed $g_{00}$, $g_{0\mu}$ through the lapse and
shift functions. To give another example, let us mention the work by Faddeev
[10] where quite specific variables $\lambda^{0}=1/h^{00}+1$,
$\lambda^{i}=h^{0i}/h^{00}$, $q^{ij}=h^{0i}h^{0j}-h^{00}h^{ij}$,
$h^{\mu\nu}=\sqrt{-g}g^{\mu\nu}$ were introduced. From the point of view of
Lagrangian formalism, all the parametrizations are rightful, and the
correspondent formulations are equivalent. Meanwhile, it has been shown in
[11] that components of metric tensor and the ADM variables are not related by
a canonical transformation. In other words, it implies that the Dirac
Hamiltonian formulation for gravitation and the ADM one are not equivalent,
though it is believed that each of them is equivalent to the Einstein
(Lagrangian) formulation. There exists the contradiction that again witnesses
about the incompleteness of the theoretical foundation.
The purpose of the present paper is to demonstrate that this contradiction can
be resolved if one treats gauge gravitational degrees of freedom on an equal
footing with physical variables in extended phase space. The idea of extended
phase space was put forward by Batalin, Fradkin and Vilkovisky [4, 5, 6] who
included integration over gauge and ghost degrees of freedom in their
definition of path integral. However, in their approach gauge variables were
still considered as non-physical, secondary degrees of freedom playing just an
auxiliary role in the theory. To construct Hamiltonian dynamics for a
constrained system which would be completely equivalent to Lagrangian
formulation, we need to take yet another step: we should introduce into the
Lagrangian missing velocities corresponding to gauge variables by means of
special (differential) gauge conditions. It actually extends the phase space
of physical degrees of freedom.
In Section 2 a mathematical formulation of the problem will be given. We shall
see that non-equivalence of Hamiltonian formulations for different
parametrizations prevents from constructing a generator of transformation in
phase space which would produce correct transformations for any
parametrizations. These ideas will be illustrated in Section 3 for a simple
model with finite number of degrees of freedom. The mentioned above algorithms
[7, 8] work correctly only for some parametrizations. One possible point of
view (advocated, in particular, in [11]) is that only these parametrizations
should be allowed while all other, not related with the first ones by
canonical transformations, should be prohibited, including the ADM
parametrization. However, imposing any limitations on admissible
parametrizations or transformations does not seem to be a true solution to the
problem. In Section 4 the outline of Hamiltonian dynamics in extended phase
space will be presented, and in Section 5 it will be demonstrated for the full
gravitational theory that different parametrizations from a wide enough class
are related by canonical transformations. In particular, it will restore a
legitimate status of the ADM parametrization. We shall discuss the results and
future problems in Section 6.
## 2\. Canonical transformations in phase space
It is generally known that for a system without constraints Lagrangian as well
as Hamiltonian equations maintain their form under transformations to a new
set of generalized coordinates
$q^{a}=v^{a}(Q),$ (1)
where $v^{a}(Q)$ are invertible functions of their arguments. It is easy to
see that any transformation (2.1) correspond to a canonical transformation in
phase space. Indeed, consider a quadratic in velocities Lagrangian
$L=\frac{1}{2}\;\Gamma_{ab}(q)\dot{q}^{a}\dot{q}^{b}-U(q).$ (2)
After the transformation (2.1) the Lagrangian (2.2) would read
$L=\frac{1}{2}\;\Gamma_{cd}(Q)\frac{\partial v^{c}}{\partial
Q^{a}}\frac{\partial v^{d}}{\partial Q^{b}}\dot{Q}^{a}\dot{Q}^{b}-U(Q).$ (3)
New momenta $\\{P_{a}\\}$ are expressed through old momenta $\\{p_{a}\\}$ by
relations
$P_{a}=p_{b}\frac{\partial v^{b}}{\partial Q^{a}}.$ (4)
The transformation (2.1), (2.4) is canonical with the generating function
which depends on new coordinates and old momenta,
$\Phi(Q,\,p)=-p_{a}v^{a}(Q).$ (5)
The equations
$q^{a}=-\frac{\partial\Phi}{\partial p_{a}};\qquad
P^{a}=-\frac{\partial\Phi}{\partial Q^{a}}$ (6)
reproduce exactly the transformation (2.1), (2.4). It is also easy to check
that the transformation (2.1), (2.4) maintains the Poisson brackets
$\\{Q^{a},\,Q^{b}\\}=0,\qquad\\{P_{a},\,P_{b}\\}=0,\qquad\\{Q^{a},\,P_{b}\\}=\delta^{a}_{b}.$
(7)
For a system with constraints, gauge variables (i.e. the variables whose
velocities cannot be expressed in terms of conjugate momenta) do not enter
into the Lagrangian quadratically, and a general transformation like (2.1) may
not be canonical. An example can be found in the theory of gravity by the
transformation from components of metric tensor to the ADM variables,
$g_{00}=\gamma_{ij}N^{i}N^{j}-N^{2},\qquad g_{0i}=\gamma_{ij}N^{j},\qquad
g_{ij}=\gamma_{ij}.$ (8)
This transformation concerns gauge degrees of freedom which, from the
viewpoint of Dirac, are not canonical variables at all. To pose the question,
if the transformation (2.8) is canonical, one should formally extend the
original phase space including into it gauge degrees of freedom and their
momenta. In order to prove non-canonicity of (2.8) it is enough to check that
some of the relations (2.7) are broken. Using the transformation inverted to
(2.8), one can see that $\\{N,\,\Pi^{ij}\\}\neq 0$, where $\Pi^{ij}$ are the
momenta conjugate to $\gamma_{ij}$ (see Equation (152) in [11]). More
generally, let us consider the ADM-like transformation
$N_{\mu}=V_{\mu}(g_{0\nu},\,g_{ij}),\qquad\gamma_{ij}=g_{ij}.$ (9)
Here $V_{\mu}$ are some functions of components of metric tensor (but
$N_{\mu}$ ought not to form 4-vector). A feature of this transformation is
that space components of metric tensor remain unchanged, and so do their
conjugate momenta: $\Pi^{ij}=p^{ij}$. Then
$\left.\\{N_{\mu},\,\Pi^{ij}\\}\right|_{g_{\nu\lambda},p^{\rho\sigma}}=\frac{\partial
V_{\mu}}{\partial g_{ij}}.$ (10)
It is equal to zero if only the functions $V_{\mu}$ do not depend on $g_{ij}$.
This is quite a trivial case when old gauge variables are expressed through
some new gauge variables only, and the ADM transformation (2.8) does not
belong to this class.
One could pose the question: Is it worth considering the equivalence of
different formulations in extended phase space? Would not it better to
restrict ourself by transformations in phase space of original canonical
variables in the sense of Dirac? In the second case, we can prove the
equivalence of equation of motion in Lagrangian and Hamiltonian formalism,
however, we have to fix a form of gravitational constraints by forbidding any
reparametrizations of gauge variables. Determination of the constraints’ form
is of importance for a subsequent procedure of quantization which gives rise
to the problem of parametrization noninvariance (see, for example, [12]). From
the viewpoint of subsequent quantization, the ADM parametrization is more
preferable, since the constraints do not depend on gauge variables in this
case. I would like to emphasize that there are no solid grounds for fixing the
form of the constraints, and, as we shall see in this paper, extension of
phase space enables us to solve the problem of equivalence of Lagrangian and
Hamiltonian formalism for gravity without any restriction on parametrizations.
As it has been already mentioned, the constraints, being considered as
generators of transformations in phase space, do not produce correct
transformation for all gravitational variables. To ensure the full equivalence
of two formulations one has to modify the Dirac prescription, according to
which the generator must be a linear combination of constraints, and replace
it by a more sophisticated algorithm. The known algorithms, firstly, are
relied upon algebra of constraints and, secondly, require extension of phase
space. Indeed, a transformation for a variable $q^{a}$ produced by any
generator $G$ in phase space reads
$\delta q^{a}=\\{q^{a},\,G\\}.$ (11)
So, to generate correct transformations for gauge variables the Poisson
brackets should be defined in extended phase space. Again, the dependence of
the algorithm on the algebra of constraints together with non-canonicity of
the transformations like (2.9) leads to the fact that the algorithm would work
only for a limited class of parametrizations. Thus, non-equivalence of
Hamiltonian formulations for different parametrizations, resulting in
different algebra of constraints, prevents from constructing the generator
which would produce correct transformations for any parametrizations. In the
next section we shall illustrate it making use of the algorithm [7], for a
simple model with finite number of degrees of freedom.
## 3\. The generator of gauge transformation: a simple example
Now we shall consider a closed isotropic cosmological model with the
Lagrangian
$L_{1}=-\frac{1}{2}\frac{a\dot{a}^{2}}{N}+\frac{1}{2}Na.$ (1)
This model is traditionally described in the ADM variables ($N$ is the lapse
function, $a$ is the scale factor). For our purpose, it is more convenient to
go to a new variable $\mu=N^{2}$ which corresponds to $g_{00}$. So the
Lagrangian is
$L_{2}=-\frac{1}{2}\frac{a\dot{a}^{2}}{\sqrt{\mu}}+\frac{1}{2}\sqrt{\mu}\,a.$
(2)
The canonical Hamiltonian constructed according to the rule
$H=p_{a}\dot{q}^{a}-L$, where $\\{p_{a},\;q^{a}\\}$ are pairs of variables
called canonical in the sense that all the velocities $\dot{q}^{a}$ can be
expressed through conjugate momenta, for our model is
$H_{C}=p\dot{a}-L_{2}=-\frac{1}{2}\frac{\sqrt{\mu}}{a}\;p^{2}-\frac{1}{2}\sqrt{\mu}\,a$
(3)
($p$ is the momentum conjugate to the scale factor). However, some authors
include into the form $p_{a}\dot{q}^{a}$ also gauge variables and their
momenta which are non-canonical variables in the above sense. Then we have the
so-called total Hamiltonian which for our model takes the form
$H_{T}=\pi\dot{\mu}+p\dot{a}-L_{2}=\pi\dot{\mu}-\frac{1}{2}\frac{\sqrt{\mu}}{a}\;p^{2}-\frac{1}{2}\sqrt{\mu}\,a$
(4)
($\pi$ is the momentum conjugate to the gauge variable $\mu$). Making use of
the total Hamiltonian implies a mixed formalism in which the Hamiltonian is
written in terms of canonical coordinates and momenta but as well of
velocities that cannot be expressed through the momenta. Nevertheless, this
very Hamiltonian plays the central role in the algorithm suggested in [7]
while the canonical Hamiltonian (3.3) will not lead to the correct result.
In [7] the generator of gauge transformations is sought in the form
$G=\sum\limits_{n}\theta_{\mu}^{(n)}G_{n}^{\mu},$ (5)
where $G_{n}^{\mu}$ are first class constraints, $\theta_{\mu}^{(n)}$ are the
$n$th order time derivatives of the gauge parameters $\theta_{\mu}$. In the
theory of gravity the variations of $g_{\mu\nu}$ involve first order
derivatives of gauge parameters, thus the generator is
$G=\theta_{\mu}G_{0}^{\mu}+\dot{\theta}_{\mu}G_{1}^{\mu}.$ (6)
$G_{n}^{\mu}$ satisfy the following conditions that were derived from the
requirement of invariance of motion equations under transformations in phase
space:
$G_{1}^{\mu}\quad{\rm are\;primary\;constraints};$ (7)
$G_{0}^{\mu}+\left\\{G_{1}^{\mu},\;H\right\\}\quad{\rm
are\;primary\;constraints};$ (8) $\left\\{G_{0}^{\mu},\;H\right\\}\quad{\rm
are\;primary\;constraints}.$ (9)
In our case $\pi=0$ is the only primary constraint of the model, so that
$G_{1}=\pi$. The secondary constraint is
$\dot{\pi}=\left\\{\pi,\;H_{T}\right\\}=-\frac{\partial
H_{T}}{\partial\mu}=\frac{1}{4}\frac{1}{a\sqrt{\mu}}\;p^{2}+\frac{1}{4}\frac{a}{\sqrt{\mu}}=T.$
(10)
The canonical Hamiltonian (3.3) appears to be proportional to the secondary
constraint $T$, $H_{C}=-2\mu T$.
The condition (3.8) becomes
$G_{0}+\left\\{\pi,\;H_{T}\right\\}=\alpha\pi;$ (11) $G_{0}=-T+\alpha\pi,$
(12)
$\alpha$ is a coefficient that can be found from the requirement (3.9):
$\left\\{G_{0},\;H_{T}\right\\}=\beta\pi;$ (13)
$\displaystyle\left\\{G_{0},\;H_{T}\right\\}$ $\displaystyle=$
$\displaystyle-\left\\{T,\;H_{T}\right\\}+\alpha\left\\{\pi,\;H_{T}\right\\}=-\left\\{T,\;\pi\dot{\mu}-2\mu
T\right\\}+\alpha T$ (14) $\displaystyle=$
$\displaystyle-\left\\{T,\;\pi\right\\}\dot{\mu}+\alpha
T=\frac{1}{2\mu}\;\dot{\mu}T+\alpha T;$
$\beta=0;\qquad\alpha=-\frac{1}{2\mu}\;\dot{\mu};$ (15)
$G_{0}=-\frac{1}{2\mu}\;\dot{\mu}\pi-T.$ (16)
The full generator $G$ (3.6) can be written as
$G=\left(-\frac{1}{2\mu}\;\dot{\mu}\pi-T\right)\theta+\pi\dot{\theta}.$ (17)
The transformation of the variable $\mu$ is
$\delta\mu=\left\\{\mu,\;G\right\\}=-\frac{1}{2\mu}\;\dot{\mu}\theta+\dot{\theta}.$
(18)
The same expression (up to the multiplier being equal to 2) can be obtained
from general transformations of the metric tensor,
$\delta
g_{\mu\nu}=\theta^{\lambda}\partial_{\lambda}g_{\mu\nu}+g_{\mu\lambda}\partial_{\nu}\theta^{\lambda}+g_{\nu\lambda}\partial_{\mu}\theta^{\lambda};$
(19) $\delta g_{00}=\dot{g}_{00}\theta^{0}+2g_{00}\dot{\theta}^{0},$ (20)
if one keeps in mind that $g_{00}=\mu$ and in the above formulas
$\theta=\theta_{0}=g_{00}\theta^{0}$.
Ir is easy to see that the correct expression (3.18) is entirely due to the
replacement of the canonical Hamiltonian (3.3) by the total Hamiltonian (3.4),
otherwise one would miss the contribution from the Poisson bracket
$\\{T,\;\pi\\}$ to the generator (3.17) (see the second line of (3.14)).
On the other hand, making use of the total Hamiltonian may not lead to a
correct result for another parametrization. Let us return to the Lagrangian
(3.1). Now the total Hamiltonian is
$H_{T}=\pi\dot{N}-\frac{1}{2}\frac{N}{a}\;p^{2}-\frac{1}{2}\;N\,a$ (21)
Again, $\pi$ is the momentum conjugate to the gauge variable $N$, and $\pi=0$
is the only primary constraint. The secondary constraint does not depend on
$N$ in this case:
$\dot{\pi}=\left\\{\pi,\;H_{T}\right\\}=-\frac{\partial H_{T}}{\partial
N}=\frac{1}{2a}\;p^{2}+\frac{1}{2}\;a=T,$ (22)
therefore, the Poisson bracket $\left\\{T,\;\pi\right\\}$ in (3.14) is equal
to zero, and one would obtain an incorrect expression for the generator,
$G=-T\theta+\pi\dot{\theta}.$ (23)
It cannot produce the correct variation of $N$, that reads
$\delta N=-\dot{N}\theta-N\dot{\theta}.$ (24)
As we can see, this algorithm fails to produce correct results for an
arbitrary parametrization. In the next section we shall construct Hamiltonian
dynamics in extended phase space and discuss its features and advantages.
## 4\. Extended phase space: the isotropic model
We shall consider the effective action including gauge and ghost sectors as it
appears in the path integral approach to gauge field theories,
$S=\int dt\left(L_{(grav)}+L_{(gauge)}+L_{(ghost)}\right)$ (1)
As was mentioned above, it is not enough just to extend pase space by
including formally gauge degrees of freedom in it. One should also introduce
missing velocities into the Lagrangian. It can be done by means of special
(differential) gauge conditions that actually extends the phase space and
enables one to avoid the mixed formalism. For our model (3.1) the equation
$N=f(a)$ determines in a general form a relation between the only gauge
variable $N$ and the scale factor $a$. The differential form of this relation
is
$\dot{N}=\frac{df}{da}\;\dot{a}.$ (2)
The ghost sector of the model reads
$L_{(ghost)}=\dot{\bar{\theta}}N\dot{\theta}+\dot{\bar{\theta}}\left(\dot{N}-\frac{df}{da}\;\dot{a}\right)\theta,$
(3)
so that
$\displaystyle L$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{a\dot{a}^{2}}{N}+\frac{1}{2}Na+\lambda\left(\dot{N}-\frac{df}{da}\;\dot{a}\right)+\dot{\bar{\theta}}\left(\dot{N}-\frac{df}{da}\;\dot{a}\right)\theta+\dot{\bar{\theta}}N\dot{\theta}=$
(4) $\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{a\dot{a}^{2}}{N}+\frac{1}{2}Na+\pi\left(\dot{N}-\frac{df}{da}\;\dot{a}\right)+\dot{\bar{\theta}}N\dot{\theta}.$
The conjugate momenta are:
$\pi=\lambda+\dot{\bar{\theta}}\theta;\quad
p=-\frac{a\dot{a}}{N}-\pi\frac{df}{da};\quad\bar{\cal
P}=N\dot{\bar{\theta}};\quad{\cal P}=N\dot{\theta}.$ (5)
Let us now go to a new variable
$N=v(\tilde{N},a).$ (6)
At the same time, the rest variables are unchanged:
$a=\tilde{a};\quad\theta=\tilde{\theta};\quad\bar{\theta}=\tilde{\bar{\theta}}.$
(7)
It is the analog of the transformation from the original gravitational
variables $g_{\mu\nu}$ to the ADM variables. Indeed, in the both cases only
gauge variables are transformed while the rest variables remain unchanged.
After the change (4.6) the Lagrangian is written as (below we shall omit the
tilde over $a$ and ghost variables which remain unchanged)
$L=-\frac{1}{2}\;\frac{a\dot{a}^{2}}{v(\tilde{N},a)}+\frac{1}{2}\;v(\tilde{N},a)\;a+\pi\left(\frac{\partial
v}{\partial\tilde{N}}\;\dot{\tilde{N}}+\frac{\partial v}{\partial
a}\;\dot{a}-\frac{df}{da}\;\dot{a}\right)+v(\tilde{N},a)\;\dot{\bar{\theta}}\dot{\theta}.$
(8)
The new momenta are:
$\tilde{\pi}=\pi\frac{\partial
v}{\partial\tilde{N}};\qquad\tilde{p}=-\frac{a\dot{a}}{v(\tilde{N},a)}+\pi\frac{\partial
v}{\partial a}-\pi\frac{df}{da}=p+\pi\frac{\partial v}{\partial a};$ (9)
$\tilde{\bar{\cal P}}=v(\tilde{N},a)\;\dot{\bar{\theta}}=\bar{\cal
P};\qquad\tilde{\cal P}=v(\tilde{N},a)\;\dot{\theta}={\cal P}.$ (10)
It is easy to demonstrate that the transformations (4.6), (4.7), (4.9), (4.10)
are canonical in extended phase space. The generating function will depend on
new coordinates and old momenta,
$\Phi\left(\tilde{N},\;\tilde{a},\;\tilde{\bar{\theta}},\;\tilde{\theta},\;\pi,\;p,\;\bar{\cal
P},\;{\cal P}\right)=-\pi\,v(\tilde{N},\tilde{a})-p\,\tilde{a}-\bar{\cal
P}\,\tilde{\theta}-\tilde{\bar{\theta}}\,{\cal P}.$ (11)
One can see that the generating function has the same form as in (2.5). The
relations
$N=-\frac{\partial\Phi}{\partial\pi};\qquad a=-\frac{\partial\Phi}{\partial
p};\qquad\tilde{\pi}=-\frac{\partial\Phi}{\partial\tilde{N}};\qquad\tilde{p}=-\frac{\partial\Phi}{\partial\tilde{a}};$
(12) $\theta=-\frac{\partial\Phi}{\partial\bar{\cal
P}\vphantom{\sqrt{N}}};\qquad\bar{\theta}=-\frac{\partial\Phi}{\partial{\cal
P}};\qquad\tilde{\cal
P}=-\frac{\partial\Phi}{\partial\tilde{\bar{\theta}}};\qquad\tilde{\bar{\cal
P}}=-\frac{\partial\Phi}{\partial\tilde{\theta}}$ (13)
give exactly the transformation (4.6), (4.7), (4.9), (4.10). On the other
hand, one can check that Poisson brackets among all phase variables maintain
their canonical form.
Now we are going to write down equations of motion in extended phase space.
Firstly, we rewrite the Lagrangian (4.8) through the momentum $\tilde{\pi}$.
$\displaystyle L$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\;\frac{a\dot{a}^{2}}{v(\tilde{N},a)}+\frac{1}{2}\;v(\tilde{N},a)\;a$
(14) $\displaystyle+$
$\displaystyle\tilde{\pi}\left[\dot{\tilde{N}}+\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\frac{\partial v}{\partial
a}\;\dot{a}-\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\frac{df}{da}\;\dot{a}\right]+v(\tilde{N},a)\;\dot{\bar{\theta}}\dot{\theta}.$
The variation of (4.14) gives, accordingly, the equation of motion (4.15), the
constraint (4.16), the gauge condition (4.17) and the ghost equations (4.18) –
(4.19):
$\displaystyle\frac{a\ddot{a}}{v(\tilde{N},a)}$ $\displaystyle+$
$\displaystyle\frac{1}{2}\;\frac{\dot{a}^{2}}{v(\tilde{N},a)}-\frac{1}{2}\;\frac{a\dot{a}^{2}}{v^{2}(\tilde{N},a)}\;\frac{\partial
v}{\partial a}-\frac{a\dot{a}}{v^{2}(\tilde{N},a)}\;\frac{\partial
v}{\partial\tilde{N}}\dot{\tilde{N}}$ (15) $\displaystyle+$
$\displaystyle\frac{1}{2}\;\frac{\partial v}{\partial
a}\;a+\frac{1}{2}v(\tilde{N},a)-\dot{\tilde{\pi}}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\frac{\partial v}{\partial
a}+\dot{\tilde{\pi}}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\frac{df}{da}$ $\displaystyle+$
$\displaystyle\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-2}\frac{\partial^{2}v}{\partial\tilde{N}^{2}}\;\frac{\partial
v}{\partial a}\;\dot{\tilde{N}}-\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\frac{\partial^{2}v}{\partial\tilde{N}\partial
a}\;\dot{\tilde{N}}$ $\displaystyle-$
$\displaystyle\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-2}\frac{\partial^{2}v}{\partial\tilde{N}^{2}}\;\frac{df}{da}\;\dot{\tilde{N}}+\frac{\partial
v}{\partial a}\;\dot{\bar{\theta}}\dot{\theta}=0;$
$\displaystyle\frac{1}{2}\;\frac{a\dot{a}^{2}}{v^{2}(\tilde{N},a)}\;\frac{\partial
v}{\partial\tilde{N}}$ $\displaystyle+$
$\displaystyle\frac{1}{2}\;\frac{\partial
v}{\partial\tilde{N}}\;a-\dot{\tilde{\pi}}-\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-2}\frac{\partial^{2}v}{\partial\tilde{N}^{2}}\;\frac{\partial
v}{\partial a}\;\dot{a}+\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\frac{\partial^{2}v}{\partial\tilde{N}\partial
a}\;\dot{a}$ (16) $\displaystyle+$
$\displaystyle\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-2}\frac{\partial^{2}v}{\partial\tilde{N}^{2}}\;\frac{df}{da}\;\dot{a}+\frac{\partial
v}{\partial\tilde{N}}\;\dot{\bar{\theta}}\dot{\theta}=0;$ $\frac{\partial
v}{\partial\tilde{N}}\;\dot{\tilde{N}}+\frac{\partial v}{\partial
a}\;\dot{a}-\frac{df}{da}\;\dot{a}=0;$ (17)
$v(\tilde{N},\;a)\;\ddot{\theta}+\frac{\partial
v}{\partial\tilde{N}}\;\dot{\tilde{N}}\dot{\theta}+\frac{\partial v}{\partial
a}\;\dot{a}\dot{\theta}=0;$ (18)
$v(\tilde{N},\;a)\;\ddot{\bar{\theta}}+\frac{\partial
v}{\partial\tilde{N}}\;\dot{\tilde{N}}\dot{\bar{\theta}}+\frac{\partial
v}{\partial a}\;\dot{a}\dot{\bar{\theta}}=0.$ (19)
The Hamiltonian in extended phase space looks like
$\displaystyle H$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\;\frac{v(\tilde{N},a)}{a}\left[\tilde{p}^{2}+2\tilde{p}\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\frac{df}{da}+\tilde{\pi}^{2}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-2}\left(\frac{df}{da}\right)^{2}\right.$ (20)
$\displaystyle-$ $\displaystyle\left.2\tilde{p}\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\frac{\partial v}{\partial
a}-2\tilde{\pi}^{2}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-2}\frac{\partial v}{\partial
a}\;\frac{df}{da}+\tilde{\pi}^{2}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-2}\left(\frac{\partial v}{\partial
a}\right)^{2}\right]$ $\displaystyle-$
$\displaystyle\frac{1}{2}\;v(\tilde{N},a)\;a+\frac{1}{v(\tilde{N},a)}\;\bar{\cal
P}{\cal P}.$
The Hamiltonian equations in extended phase space are:
$\displaystyle\dot{\tilde{p}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\frac{1}{a}\frac{\partial v}{\partial
a}-\frac{v(\tilde{N},a)}{a^{2}}\right]\left[\tilde{p}+\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\frac{df}{da}-\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\frac{\partial v}{\partial a}\right]^{2}$
(21) $\displaystyle-$
$\displaystyle\frac{v(\tilde{N},a)}{a}\left[\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-2}\frac{\partial^{2}v}{\partial\tilde{N}\partial
a}\;\frac{df}{da}-\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\frac{d^{2}f}{da^{2}}\right.$
$\displaystyle-$ $\displaystyle\left.\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-2}\frac{\partial^{2}v}{\partial\tilde{N}\partial
a}\;\frac{\partial v}{\partial a}+\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\frac{\partial^{2}v}{\partial a^{2}}\right]$
$\displaystyle\times$
$\displaystyle\left[\tilde{p}+\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\frac{df}{da}-\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\frac{\partial v}{\partial a}\right]$
$\displaystyle+$ $\displaystyle\frac{1}{2}\;\frac{\partial v}{\partial
a}\;a+\frac{1}{2}\;v(\tilde{N},a)+\frac{1}{v^{2}(\tilde{N},a)}\;\bar{\cal
P}{\cal P};$ $\displaystyle\dot{a}$ $\displaystyle=$
$\displaystyle-\frac{v(\tilde{N},a)}{a}\left[\tilde{p}+\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\frac{df}{da}-\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\frac{\partial v}{\partial a}\right];$ (22)
$\displaystyle\dot{\tilde{\pi}}$ $\displaystyle=$
$\displaystyle\frac{1}{2a}\;\frac{\partial
v}{\partial\tilde{N}}\left[\tilde{p}+\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\frac{df}{da}-\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\frac{\partial v}{\partial a}\right]^{2}$
(23) $\displaystyle-$
$\displaystyle\frac{v(\tilde{N},a)}{a}\left[\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-2}\frac{\partial^{2}v}{\partial\tilde{N}^{2}}\;\frac{df}{da}-\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-2}\frac{\partial^{2}v}{\partial\tilde{N}^{2}}\;\frac{\partial
v}{\partial a}\right.$ $\displaystyle+$
$\displaystyle\left.\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\frac{\partial^{2}v}{\partial\tilde{N}\partial
a}\right]\left[\tilde{p}+\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\frac{df}{da}-\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\frac{\partial v}{\partial a}\right]$
$\displaystyle+$ $\displaystyle\frac{1}{2}\;\frac{\partial
v}{\partial\tilde{N}}\;a+\frac{1}{v^{2}(\tilde{N},a)}\;\frac{\partial
v}{\partial\tilde{N}}\;\bar{\cal P}{\cal P};$ $\displaystyle\dot{\tilde{N}}$
$\displaystyle=$
$\displaystyle-\frac{v(\tilde{N},a)}{a}\left[\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\frac{df}{da}-\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\frac{\partial v}{\partial a}\right]$ (24)
$\displaystyle\times$
$\displaystyle\left[\tilde{p}+\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\frac{df}{da}-\tilde{\pi}\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\frac{\partial v}{\partial a}\right];$
$\displaystyle\dot{\bar{\cal P}}$ $\displaystyle=$ $\displaystyle 0;$ (25)
$\displaystyle\dot{\theta}$ $\displaystyle=$
$\displaystyle\frac{1}{v(\tilde{N},a)}\;{\cal P};$ (26)
$\displaystyle\dot{\cal P}$ $\displaystyle=$ $\displaystyle 0;$ (27)
$\displaystyle\dot{\bar{\theta}}$ $\displaystyle=$
$\displaystyle\frac{1}{v(\tilde{N},a)}\;\bar{\cal P}.$ (28)
One can check that the Hamiltonian equations (4.21) – (4.28) are completely
equivalent to the Lagrangian equations (4.15) – (4.19), the constraint (4.23)
and the gauge condition (4.24) being true Hamiltonian equations.
The Hamiltonian equations (4.21) – (4.28) in extended phase space, as well as
the equations (4.15) – (4.19), include gauge-dependent terms. In this
connection one can object that the equations are not equivalent to the
original Einstein equation, which are known to be gauge-invariant. However, we
remember that any solution to the gauge-invariant Einstein equation is
determined up to arbitrary functions which have to be fix by a choice of a
reference frame (a state of the observer). It is usually done on the final
stage of solving the Einstein equations. It is important that one cannot avoid
fixing a reference frame to obtain a final form of the solution. By varying
the gauged action (4.1), in fact, we deal with a generalized mathematical
problem, its generalization has come from the development of quantum field
theory.
In the case of the extended set of equations (4.21) – (4.28) (or, (4.15) –
(4.19)) one can keep the function $f(a)$ non-fixed up to the final stage of
their resolution. Further, under the conditions $\bar{\pi}=0$, $\theta=0$,
$\bar{\theta}=0$ all gauge-dependent terms are excluded, and the extended set
of equations is reduced to gauge-invariant equations, therefore, any solution
of the Einstein equations can be found among solutions of the extended set.
Solutions with non-trivial $\bar{\pi}$, $\theta$, $\bar{\theta}$ should be
considered and physically interpreted separately.
One can also reveal that there exists a quantity conserved if the Hamiltonian
(or, equivalently, Lagrangian) equations hold. It plays the role of the BRST
generator for our model:
$\Omega=-H\theta-\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}\tilde{\pi}{\cal P}.$ (29)
It generates correct transformations for the variables $a$, $\theta$,
$\bar{\theta}$ and for any gauge variable $\tilde{N}$ given by the relation
(4.6),
$\delta\tilde{N}=-\frac{\partial
H}{\partial\tilde{\pi}}\;\theta-\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}{\cal
P}=-\dot{\tilde{N}}\theta-\left(\frac{\partial
v}{\partial\tilde{N}}\right)^{-1}v(\tilde{N},a)\;\dot{\theta}.$ (30)
In particular, for the original variable $N$ one gets the transformation
(3.24).
## 5\. The canonicity of transformations in extended phase space
for the full gravitational theory
In this section we shall demonstrate for the full gravitational theory that
different parametrizations from a wide enough class (2.9) are related by
canonical transformations. Again, we shall start from the gauged action
$S=\int d^{4}x\left({\cal L}_{(grav)}+{\cal L}_{(gauge)}+{\cal
L}_{(ghost)}\right)$ (1)
We shall use a gauge condition in a general form, $f^{\mu}(g_{\nu\lambda})=0$.
The differential form of this gauge condition introduces the missing
velocities and actually extends phase space,
$\frac{d}{dt}f^{\mu}(g_{\nu\lambda})=0,\qquad\frac{\partial f^{\mu}}{\partial
g_{00}}\dot{g}_{00}+2\frac{\partial f^{\mu}}{\partial
g_{0i}}\dot{g}_{0i}+\frac{\partial f^{\mu}}{\partial g_{ij}}\dot{g}_{ij}=0.$
(2)
Then,
${\cal L}_{(gauge)}=\lambda_{\mu}\left(\frac{\partial f^{\mu}}{\partial
g_{00}}\dot{g}_{00}+2\frac{\partial f^{\mu}}{\partial
g_{0i}}\dot{g}_{0i}+\frac{\partial f^{\mu}}{\partial
g_{ij}}\dot{g}_{ij}\right).$ (3)
Taking into account the gauge transformations,
$\delta
g_{\mu\nu}=\partial_{\lambda}g_{\mu\nu}\theta^{\lambda}+g_{\mu\lambda}\partial_{\nu}\theta^{\lambda}+g_{\nu\lambda}\partial_{\mu}\theta^{\lambda},$
(4)
one can write the ghost sector:
${\cal L}_{(ghost)}=\bar{\theta}_{\mu}\frac{d}{dt}\left[\frac{\partial
f^{\mu}}{\partial
g_{\nu\lambda}}\left(\partial_{\rho}g_{\nu\lambda}\theta^{\rho}+g_{\lambda\rho}\partial_{\nu}\theta^{\rho}+g_{\nu\rho}\partial_{\lambda}\theta^{\rho}\right)\right].$
(5)
It is convenient to write down the action (5.1), (5.3), (5.5) in the form
$\displaystyle S$ $\displaystyle=$ $\displaystyle\int d^{4}x\left[{\cal
L}_{(grav)}+\Lambda_{\mu}\left(\frac{\partial f^{\mu}}{\partial
g_{00}}\dot{g}_{00}+2\frac{\partial f^{\mu}}{\partial
g_{0i}}\dot{g}_{0i}+\frac{\partial f^{\mu}}{\partial
g_{ij}}\dot{g}_{ij}\right)\right.$ (6) $\displaystyle-$
$\displaystyle\dot{\bar{\theta_{\mu}}}\left(\frac{\partial f^{\mu}}{\partial
g_{00}}\left(\partial_{i}g_{00}\theta^{i}+2g_{0\nu}\dot{\theta}^{\nu}\right)+2\frac{\partial
f^{\mu}}{\partial
g_{0i}}\left(\partial_{j}g_{0i}\theta^{j}+g_{0\nu}\partial_{i}\theta^{\nu}+g_{i\nu}\dot{\theta}^{\nu}\right)\right.$
$\displaystyle+$ $\displaystyle\left.\left.\frac{\partial f^{\mu}}{\partial
g_{ij}}\left(\partial_{k}g_{ij}\theta^{k}+g_{i\nu}\partial_{j}\theta^{\nu}+g_{j\nu}\partial_{i}\theta^{\nu}\right)\right)\right].$
Here $\Lambda_{\mu}=\lambda_{\mu}-\dot{\bar{\theta_{\mu}}}\theta^{0}$. One can
see that the generalized velocities enter into the bracket multiplied by
$\Lambda_{\mu}$, in addition to the gravitational part ${\cal L}_{(grav)}$.
This very circumstance will ensure the canonicity of the transformation to new
variables.
Our goal now is to introduce new variables by
$g_{0\mu}=v_{\mu}\left(N_{\nu},g_{ij}\right).\qquad
g_{ij}=\gamma_{ij};\qquad\theta^{\mu}=\tilde{\theta}^{\mu};\qquad\bar{\theta}_{\mu}=\tilde{\bar{\theta}_{\mu}}.$
(7)
This is the inverse transformation for (2.9) and concerns only $g_{0\mu}$
metric components. After the transformation the action will read
$\displaystyle S$ $\displaystyle=$ $\displaystyle\int d^{4}x\left[{\cal
L^{\prime}}_{(grav)}+\Lambda_{\mu}\left(\frac{\partial f^{\mu}}{\partial
g_{00}}\;\frac{\partial v_{0}}{\partial N_{\nu}}\;\dot{N}_{\nu}+\frac{\partial
f^{\mu}}{\partial g_{00}}\;\frac{\partial v_{0}}{\partial
g_{ij}}\;\dot{g}_{ij}\right.\right.$ (8) $\displaystyle+$
$\displaystyle\left.2\;\frac{\partial f^{\mu}}{\partial
g_{0i}}\;\frac{\partial v_{i}}{\partial
N_{\nu}}\;\dot{N}_{\nu}+2\;\frac{\partial f^{\mu}}{\partial
g_{0k}}\;\frac{\partial v_{k}}{\partial g_{ij}}\;\dot{g}_{ij}+\frac{\partial
f^{\mu}}{\partial g_{ij}}\;\dot{g}_{ij}\right)$ $\displaystyle-$
$\displaystyle\dot{\bar{\theta_{\mu}}}\left(\frac{\partial f^{\mu}}{\partial
g_{00}}\;\frac{\partial v_{0}}{\partial
N_{\nu}}\;\partial_{i}N_{\nu}\theta^{i}+\frac{\partial f^{\mu}}{\partial
g_{00}}\;\frac{\partial v_{0}}{\partial
g_{ij}}\;\partial_{k}g_{ij}\theta^{k}+2\;\frac{\partial f^{\mu}}{\partial
g_{00}}\;v_{\nu}(N_{\lambda},g_{ij})\;\dot{\theta}^{\nu}\right.$
$\displaystyle+$ $\displaystyle 2\;\frac{\partial f^{\mu}}{\partial
g_{0i}}\;\frac{\partial v_{i}}{\partial
N_{\nu}}\;\partial_{j}N_{\nu}\theta^{j}+2\;\frac{\partial f^{\mu}}{\partial
g_{0i}}\;\frac{\partial v_{i}}{\partial g_{jk}}\;\partial_{l}g_{jk}\theta^{l}$
$\displaystyle+$ $\displaystyle 2\;\frac{\partial f^{\mu}}{\partial
g_{0i}}\left[v_{\nu}(N_{\lambda},g_{jk})\;\partial_{i}\theta^{\nu}+v_{i}(N_{\lambda},g_{jk})\;\dot{\theta}^{0}+g_{ij}\dot{\theta}^{j}\right]$
$\displaystyle+$ $\displaystyle\frac{\partial f^{\mu}}{\partial
g_{ij}}\left[\partial_{k}g_{ij}\theta^{k}+v_{i}(N_{\lambda},g_{kl})\;\partial_{j}\theta^{0}+g_{ik}\partial_{j}\theta^{k}\right.$
$\displaystyle+$
$\displaystyle\left.\left.\left.v_{j}(N_{\lambda},g_{kl})\;\partial_{i}\theta^{0}+g_{jk}\partial_{i}\theta^{k}\right]\right)\right]$
We can write down the “old” momenta,
$\displaystyle\pi^{ij}$ $\displaystyle=$ $\displaystyle\frac{\partial{\cal
L}_{(grav)}}{\partial\dot{g}_{ij}}+\Lambda_{\mu}\;\frac{\partial
f^{\mu}}{\partial g_{ij}};$ (9) $\displaystyle\pi^{0}$ $\displaystyle=$
$\displaystyle\frac{\partial{\cal
L}_{(grav)}}{\partial\dot{g}_{00}}+\Lambda_{\mu}\;\frac{\partial
f^{\mu}}{\partial g_{00}};$ (10) $\displaystyle\pi^{i}$ $\displaystyle=$
$\displaystyle\frac{\partial{\cal
L}_{(grav)}}{\partial\dot{g}_{0i}}+2\Lambda_{\mu}\;\frac{\partial
f^{\mu}}{\partial g_{0i}},$ (11)
and the “new” momenta are:
$\displaystyle\Pi^{ij}$ $\displaystyle=$ $\displaystyle\frac{\partial{\cal
L^{\prime}}_{(grav)}}{\partial\dot{g}_{ij}}+\Lambda_{\mu}\left(\frac{\partial
f^{\mu}}{\partial g_{00}}\;\frac{\partial v_{0}}{\partial
g_{ij}}+2\;\frac{\partial f^{\mu}}{\partial g_{0k}}\;\frac{\partial
v_{k}}{\partial g_{ij}}+\frac{\partial f^{\mu}}{\partial g_{ij}}\right);$ (12)
$\displaystyle\Pi^{0}$ $\displaystyle=$ $\displaystyle\frac{\partial{\cal
L^{\prime}}_{(grav)}}{\partial\dot{N}_{0}}+\Lambda_{\mu}\left(\frac{\partial
f^{\mu}}{\partial g_{00}}\;\frac{\partial v_{0}}{\partial
N_{0}}+2\;\frac{\partial f^{\mu}}{\partial g_{0i}}\;\frac{\partial
v_{i}}{\partial N_{0}}\right);$ (13) $\displaystyle\Pi^{i}$ $\displaystyle=$
$\displaystyle\frac{\partial{\cal
L^{\prime}}_{(grav)}}{\partial\dot{N}_{i}}+\Lambda_{\mu}\left(\frac{\partial
f^{\mu}}{\partial g_{00}}\;\frac{\partial v_{0}}{\partial
N_{i}}+2\;\frac{\partial f^{\mu}}{\partial g_{0j}}\;\frac{\partial
v_{j}}{\partial N_{i}}\right).$ (14)
The relations between the “old” and “new” momenta:
$\displaystyle\Pi^{ij}$ $\displaystyle=$
$\displaystyle\pi^{ij}+\left(\pi^{\mu}-\frac{\partial{\cal
L}_{(grav)}}{\partial\dot{g}_{0\mu}}\right)\frac{\partial v_{\mu}}{\partial
g_{ij}};$ (15) $\displaystyle\Pi^{\mu}$ $\displaystyle=$
$\displaystyle\frac{\partial{\cal
L^{\prime}}_{(grav)}}{\partial\dot{N}_{\mu}}+\left(\pi^{\nu}-\frac{\partial{\cal
L}_{(grav)}}{\partial\dot{g}_{0\nu}}\right)\frac{\partial v_{\nu}}{\partial
N_{\mu}}.$ (16)
It is easy to check that the momenta conjugate to ghosts remain unchanged,
$\tilde{\cal P}^{\mu}={\cal P}^{\mu}$, $\tilde{\bar{\cal P}}_{\mu}=\bar{\cal
P}_{\mu}$.
As any Lagrangian is determined up to total derivatives, the gravitational
Lagrangian density ${\cal L}_{(grav)}$ can be modified in such a way for the
primary constraints to take the form $\pi^{\mu}=0$, where $\pi^{\mu}$ are the
momenta conjugate to gauge variables $g_{0\mu}$. This change of the Lagrangian
density does not affect the equation of motion. It was made by Dirac [3] to
simplify the calculations. A similar change of the Lagrangian density by
omitting a divergence and a total time derivative was made also in the ADM
paper [9]. Therefore, one can put
$\frac{\partial{\cal
L}_{(grav)}}{\partial\dot{g}_{0\mu}}=0,\qquad\frac{\partial{\cal
L^{\prime}}_{(grav)}}{\partial\dot{N}_{\mu}}=0.$ (17)
Then, the relations (5.15) – (5.16) would become simpler and take the form
$\Pi^{ij}=\pi^{ij}+\pi^{\mu}\frac{\partial v_{\mu}}{\partial
g_{ij}};\qquad\Pi^{\mu}=\pi^{\nu}\frac{\partial v_{\nu}}{\partial N_{\mu}}.$
(18)
It is easy to demonstrate that the transformations (2.9), (5.18) are canonical
in extended phase space. The generating function again depends on new
coordinates and old momenta and has the same form as for a non-constrained
system (see (2.5), compare also with (4.11)),
$\Phi\left(N_{\mu},\;\gamma_{ij},\;\tilde{\theta}^{\mu},\;\tilde{\bar{\theta}}_{\mu},\;\pi^{\mu},\;\pi^{ij},\;\bar{\cal
P}_{\mu},\;{\cal
P}^{\mu}\right)=-\pi^{\mu}v_{\mu}(N_{\nu},\gamma_{ij})-\pi^{ij}\gamma_{ij}-\bar{\cal
P}_{\mu}\tilde{\theta}^{\mu}-\tilde{\bar{\theta}}_{\mu}{\cal P}^{\mu}.$ (19)
Then the following relations take place
$g_{0\mu}=-\frac{\partial\Phi}{\partial\pi^{\mu}};\qquad
g_{ij}=-\frac{\partial\Phi}{\partial\pi^{ij}};\qquad\theta^{\mu}=-\frac{\partial\Phi}{\partial\bar{\cal
P}\vphantom{\sqrt{N}}_{\mu}};\qquad\bar{\theta}_{\mu}=-\frac{\partial\Phi}{\partial{\cal
P}^{\mu}};$ (20) $\Pi^{\mu}=-\frac{\partial\Phi}{\partial
N_{\mu}};\qquad\Pi^{ij}=-\frac{\partial\Phi}{\partial\gamma_{ij}};\qquad\tilde{\bar{\cal
P}}_{\mu}=-\frac{\partial\Phi}{\partial\tilde{\theta}^{\mu}};\qquad\tilde{\cal
P}^{\mu}=-\frac{\partial\Phi}{\partial\tilde{\bar{\theta}}\vphantom{\sqrt{N}}_{\mu}},$
(21)
that give exactly the transformations
$g_{0\mu}=v_{\mu}(N_{\nu},\gamma_{ij});\qquad
g_{ij}=\gamma_{ij};\qquad\qquad\qquad\qquad\theta^{\mu}=\tilde{\theta}^{\mu};\qquad\bar{\theta}_{\mu}=\tilde{\bar{\theta}}_{\mu};$
(22) $\Pi^{\mu}=\pi^{\nu}\frac{\partial v_{\nu}}{\partial
N_{\mu}};\qquad\qquad\Pi^{ij}=\pi^{ij}+\pi^{\mu}\frac{\partial
v_{\mu}}{\partial g_{ij}};\qquad\quad\tilde{\bar{\cal P}}_{\mu}=\bar{\cal
P}_{\mu};\qquad\tilde{\cal P}^{\mu}={\cal P}^{\mu}.$ (23)
We can now check if the Poisson brackets maintain their form. Differentiating
the first relation in (2.9) with respect to $g_{ij}$ one gets
$\frac{\partial V_{\mu}}{\partial g_{ij}}+\frac{\partial V_{\mu}}{\partial
g_{0\lambda}}\frac{\partial v_{\lambda}}{\partial g_{ij}}=0,$ (24)
Similarly, differentiating the same relation with respect to $N_{\nu}$ gives
$\delta_{\mu}^{\nu}-\frac{\partial V_{\mu}}{\partial
g_{0\lambda}}\frac{\partial v_{\lambda}}{\partial N_{\nu}}=0.$ (25)
Making use of (5.24), (5.25), it is not difficult to calculate the Poisson
brackets. So, we can recalculate the bracket (2.10) to see that it will be
zero in our extended phase space formalism,
$\displaystyle\left.\\{N_{\mu},\,\Pi^{ij}\\}\right|_{g_{\nu\lambda},p^{\rho\sigma}}$
$\displaystyle=$ $\displaystyle\frac{\partial N_{\mu}}{\partial
g_{0\rho}}\;\frac{\partial\Pi^{ij}}{\partial\pi^{\rho}}+\frac{\partial
N_{\mu}}{\partial
g_{kl}}\;\frac{\partial\Pi^{ij}}{\partial\pi^{kl}}=\left\\{V_{\mu}(g_{0\nu},g_{kl}),\;\pi^{ij}+\pi^{\lambda}\frac{\partial
v_{\lambda}}{\partial g_{ij}}\right\\}$ (26) $\displaystyle=$
$\displaystyle\frac{\partial V_{\mu}}{\partial g_{0\rho}}\;\frac{\partial
v_{\lambda}}{\partial g_{ij}}\delta_{\rho}^{\lambda}+\frac{\partial
V_{\mu}}{\partial
g_{kl}}\;\frac{1}{2}\left(\delta_{k}^{i}\delta_{l}^{j}+\delta_{l}^{j}\delta_{k}^{i}\right)=\frac{\partial
V_{\mu}}{\partial g_{0\lambda}}\;\frac{\partial v_{\lambda}}{\partial
g_{ij}}+\frac{\partial V_{\mu}}{\partial g_{ij}}=0.$
To give another example, let us check the following bracket:
$\left.\\{N_{\mu},\,\Pi^{\nu}\\}\right|_{g_{\lambda\rho},p^{\sigma\tau}}=\frac{\partial
N_{\mu}}{\partial
g_{0\rho}}\;\frac{\partial\Pi^{\nu}}{\partial\pi^{\rho}}=\left\\{V_{\mu}(g_{0\rho},g_{ij}),\;\pi^{\lambda}\frac{\partial
v_{\lambda}}{\partial N_{\nu}}\right\\}=\frac{\partial V_{\mu}}{\partial
g_{0\rho}}\;\frac{\partial v_{\rho}}{\partial N_{\nu}}=\delta_{\mu}^{\nu}.$
(27)
The rest of the brackets can be checked by analogy. This completes the proof
of canonicity of the transformation (2.9) for the full gravitational theory.
## 6\. Discussion
A starting point for the present investigation was the paper [11] and the
statement made by its authors that components of metric tensor and the ADM
variables are not related by a canonical transformation. However, it is
misunderstanding to pose the question about canonicity of the transformation
(2.8) which involves, from the viewpoint of the Dirac approach, non-canonical
variables. Let us remind that Dirac himself consider these variables,
$g_{0\mu}$ (along with the zero component of vector potential of
electromagnetic field $A_{0}$) as playing the role of Lagrange multipliers
while the phase space in his approach includes pairs of generalized
coordinates and momenta for which corresponding velocities can be expressed
through the momenta.
We should remember also that the Einstein equations were originally formulated
in Lagrangian formalism. Dirac’s Hamiltonian formulation for gravity is
equivalent to Einstein’s formulation at the level of equations. It means that
Hamiltonian equations for canonical variables (in Dirac’s sense) are
equivalent to the ($ij$) Einstein equations, and the gravitational constraints
are equivalent to the ($0\mu$) Einstein equations. On the other hand, it
implies that a group of transformations in Hamiltonian formalism must involve
the full group of gauge transformations of the original theory. However, in
the limits of the Dirac approach we fail to construct a generator that would
produce correct transformations for all variables. We inevitably have to
modify the Dirac scheme, and attempts to do it were presented yet in [7, 8].
Therefore, we cannot consider the Dirac approach as fundamental and undoubted.
The ADM formulation of Hamiltonian dynamics for gravity is, first of all, the
choice of parametrization, which is preferable because of its geometrical
interpretation. There is no any special ”ADM procedure”: Arnowitt, Deser and
Misner constructed the Hamiltonian dynamics following exactly the Dirac
scheme, just making use of another variables. The fact that two Hamiltonian
formulations (both according to the Dirac scheme, but the one for original
variables and the other for the ADM variables) are not related by canonical
transformations, should not lead to any bad-grounded conclusions like the one
made in [11], p. 68, that the gravitational Lagrangian used by Dirac and the
ADM Lagrangian are not equivalent. At the Lagrangian level, the transition to
the ADM variables is nothing more as a change of variables in the Einstein
equations, and there are no mathematical rules that would prohibit such change
of variables. It is the Lagrangian formulation of General Relativity which is
original and fundamental while its Hamiltonian formulation still remains
questionable, in spite of fifty years passed after Dirac’s paper [3]. The
extended phase space approach treating all degrees of freedom on an equal
footing may be a real alternative to the Dirac generalization of Hamiltonian
dynamics.
The example considered in Section 4 shows that the BRST charge can play the
role of a sought generator in extended phase space. Nevertheless, the
algorithm suggested by BFV for constructing the BRST charge again relies upon
the algebra of constraints. Even for the model from Section 4 it would not
lead to the correct result (4.30). Another way is to construct the BRST charge
as a conserved quantity based on BRST-invariance of the action and making use
of the first Noether theorem. This method works satisfactory for simple models
with a given symmetry. Below we mentioned that the gravitational Lagrangian
density can be modified for the primary constraints to take the simplest form
$\pi^{\mu}=0$ without affecting the equation of motion. However, after this
modification the full action may not be BRST-invariant. Some authors (see, for
example, [12, 13]) use some boundary conditions to exclude total derivatives
and ensure BRST-invariance. The boundary conditions (as a rule, these are
trivial boundary conditions for ghosts and $\pi^{\mu}$) correspond to
asymptotic states and are well-grounded in ordinary quantum field theory. This
way does not seem to be general enough, and for gravitational field the
justification of the boundary conditions, as well as the control of BRST-
invariance of the action, requires special study.
In [3] Dirac pointed out that “any dynamical theory must first be put in the
Hamiltonian form before one can quantize it”. Based upon Hamiltonian dynamics
in extended phase space, a new approach to quantum theory of gravity has been
proposed in [14, 15]. Ir was argued that it is impossible to construct a
mathematically consistent quantum theory of gravity without taking into
account the role of gauge degrees of freedom in description of quantum
gravitational phenomena from the point of view of different observers. The
present paper show that even at the classical level gauge degrees of freedom
cannot be excluded from consideration. As we have seen, the extension of phase
space by introducing the missing velocities changes the relations between the
“old” and “new” momenta (see (5.18)). As a consequence, the transformation
(2.9) is canonical. In that way, we consider extended phase space not just as
an auxiliary construction which enables one to compensate residual degrees of
freedom and regularize a path integral, as it was in the Batalin – Fradkin –
Vilkovisky approach [4, 5, 6], but rather as a structure that ensures
equivalence of Hamiltonian dynamics for a constrained system and Lagrangian
formulation of the original theory.
## Acknowledgements
I would like to thank Giovanni Montani and Francesco Cianfrani for attracting
my attention to the paper [11] and discussions.
## References
* [1] P. A. M. Dirac, Can. J. Math. 2 (1950), P. 129–148.
* [2] P. A. M. Dirac, Proc. Roy. Soc. A246 (1958), P. 326–332.
* [3] P. A. M. Dirac, Proc. Roy. Soc. A246 (1958), P. 333–343.
* [4] E. S. Fradkin and G. A. Vilkovisky, Phys. Lett B55 (1975), P. 224–226.
* [5] I. A. Batalin and G. A. Vilkovisky, Phys. Lett B69 (1977), P. 309–312.
* [6] E. S. Fradkin and T. E. Fradkina, Phys. Lett B72 (1978), P. 343–348.
* [7] L. Castellani, Ann. Phys. 143 (1982), P. 357–371.
* [8] R. Banerjee, H. J. Rothe and K. D. Rothe, Phys. Lett. B463 (1999), P. 248–251.
* [9] R. Arnowitt, S. Deser and C. W. Misner, “The Dynamics of General Relativity”, in: Gravitation: an Introduction to Current Research, ed. by L. Witten, John Wiley & Sons, New York (1962), P. 227–265.
* [10] L. D. Faddeev, Usp. Fiz. Nauk 136 (1982), P. 435–457 [Sov. Phys. Usp. 25 (1982). P. 130–142].
* [11] N. Kiriushcheva and S. V. Kuzmin, “The Hamiltonian formulation of General Relativity: myth and reality”, E-print arXiv: gr-qc/0809.0097.
* [12] J. J. Halliwell, Phys. Rev. D38 (1988), P. 2468–2481.
* [13] M. Hennaux, Phys. Rep. 126 (1985), P. 1–66.
* [14] V. A. Savchenko, T. P. Shestakova and G. M. Vereshkov, Gravitation & Cosmology 7 (2001), P. 18–28.
* [15] V. A. Savchenko, T. P. Shestakova and G. M. Vereshkov, Gravitation & Cosmology 7 (2001), P. 102–116.
|
arxiv-papers
| 2011-02-01T08:35:34 |
2024-09-04T02:49:16.763869
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "T. P. Shestakova",
"submitter": "Tatyana P. Shestakova",
"url": "https://arxiv.org/abs/1102.0097"
}
|
1102.0098
|
eurm10 msam10 119–126
# Disentangle plume-induced anisotropy in the velocity field in buoyancy-
driven turbulence
Quan ZHOU1,2 and Ke-Qing XIA2 Email address for correspondence:
qzhou@shu.edu.cnEmail address for correspondence: kxia@phy.cuhk.edu.hk
1Shanghai Key Laboratory of Mechanics in Energy and Environment Engineering,
Shanghai Institute of Applied Mathematics and Mechanics, E-Institutes of
Shanghai Universities, Shanghai University, Shanghai 200072, China
2Department of Physics, The Chinese University of Hong Kong, Shatin, Hong
Kong, China
(1996; ?? and in revised form ??)
###### Abstract
We present a method of disentangling the anisotropies produced by the cliff
structures in turbulent velocity field and test it in the system of turbulent
Rayleigh-Bénard (RB) convection. It is found that in the RB system the cliff
structures in the velocity field are generated by thermal plumes. These cliff
structures induce asymmetry in the velocity increments, which leads us to
consider the plus and minus velocity structure functions (VSF). The plus
velocity increments exclude cliff structures, while the minus ones include
them. Our results show that the scaling exponents of the plus VSFs are in
excellent agreement with those predicted for homogeneous and isotropic
turbulence (HIT), whereas those of the minus VSFs exhibit significant
deviations from HIT expectations in places where thermal plumes abound. These
results demonstrate that plus and minus VSFs can be used to quantitatively
study the effect of cliff structures in the velocity field and to effectively
disentangle the associated anisotropies caused by these structures.
###### keywords:
Cliff structures, structure functions, thermal plumes, turbulent thermal
convection
††volume: 538
## 1 Introduction
A paradigm for studying turbulent flows is the so-called homogeneous and
isotropic turbulence (HIT) Monin & Yaglom (1975). Studying this idealized
model allows one to focus on the essential physics of small-scale turbulence
in the simplest possible case and use it as a first step to understand more
complicated turbulence problems. However, in almost all flow systems existing
in nature, anisotropy is always present and unavoidable. How to disentangle
the effects of anisotropy in the experimentally or numerically measured
physical quantities has been a major focus in turbulence research in recent
years and several methods, such as the SO(3) group decomposition, have been
put forward to separate the isotropic and anisotropic contributions in
turbulent flows (Arad _et al._ 1998, 1999; Grossmann, VON DER Heydt $\&$ Lohse
2001; Biferale _et al._ 2002).
Buoyancy-driven turbulent flows occur widely in geophysical and astrophysical
systems and in numerous engineering applications. In buoyancy-driven thermal
turbulence, buoyant structures, such as thermal plumes, are predominant
coherent structures that transport heat and drive the flow (Shang _et al._
2003; Xia, Sun $\&$ Zhou 2003). Earlier visualization experiments (Moses,
Zocchi $\&$ Libchaber 1993; Xi, Lam $\&$ Xia 2004) have shown that these
structures consist of a cap with sharp temperature gradient and a stem that is
relatively diffusive and hence would generate cliff-ramp-like structures in
temperature time series when passing a thermal probe Belmonte & Libchaber
(1996); Zhou & Xia (2002). It is well known that the so-called cliff-ramp
structures would induce strong anisotropic effects. This has been widely
studied in passive scalars Warhaft (2000), but the study on the effects of
buoyancy-induced anisotropy is very limited. Here, we use turbulent Rayleigh-
Bénard (RB) convection, a fluid layer heated from below and cooled on the top,
as an example to study the anisotropic effects induced by buoyancy. In the
past few decades, turbulent RB convection has become a model system for
studying the phenomena and the generic physics associated with turbulent flows
driven by buoyancy (Ahlers, Grossmann $\&$ Lohse 2009; Lohse $\&$ Xia 2010).
In the field of turbulence, the velocity structure functions (VSF), namely
$S_{p}(r)=\langle|\delta_{r}v|^{p}\rangle,$ (1)
are usually used to characterize the turbulent kinetic energy cascades, and
hence they are of prime importance and have been the central focus in the
study of fluid turbulence (Sreenivasan $\&$ Antonia 1997; Ishihara, Gotoh $\&$
Kaneda 2009). Here, $\delta_{r}v=v(x+r)-v(x)$ is defined as the velocity
increment over a separation $r$ and $\langle\cdots\rangle$ denotes an ensemble
average. In particular, the situation for thermally-driven turbulence is more
complicated. Bolgiano (1959) and Obukhov (1959), denoted hereafter as BO59 for
short, have long argued that within the so-called inertial range and above a
certain buoyant scale, i.e. the Bolgiano length scale $\ell_{B}$, buoyant
forces drive the cascade processes and scale as $S_{p}(r)\sim r^{3p/5}$.
However, experimentally whether the BO59 scaling exists in turbulent RB system
remains unsettled (for a recent review, see Lohse & Xia, 2010). The maturing
of spatial-velocity-field measuring techniques, e.g. the particle image
velocimetry (PIV), has provided a great impetus for our understanding of such
issues (Sun, Zhou $\&$ Xia 2006; Zhou, Sun $\&$ Xia 2008; Kunnen _et al._
2008; Zhou $\&$ Xia 2010). In particular, the work of Sun et al. (2006) have
shown that, at the center of the convection cell, the velocity field exhibits
the same scaling behavior that one would find in HIT, whereas, near the cell
sidewall, a “new scaling” was found. While results at both places imply no
BO59, the observed scaling behavior near sidewall remains unexplained by any
existing theoretical models. We note that in a closed RB cell, the spatial
distribution of the buoyancy-driven thermal plumes is highly inhomogeneous,
they abound near the sidewall but are scarcely found at the center Qiu & Tong
(2001); Shang et al. (2003, 2004); Xi et al. (2004). These give us a clue that
plumes may be responsible for the different scaling behavior observed at
different places of the system and suggest that buoyancy plays an important
role in the cascade process, but how this comes about is still missing. The
objective of the present study is to address such a question, i.e., how
buoyant forces influence the cascade properties in turbulent RB system?
## 2 Experimental setup and parameters
The convection cell has been described in detail elsewhere Zhou & Xia (2010);
Zhou et al. (2011) and here we give only its main features. It is a vertical
cylinder of height $H=50$ cm and inner diameter $D=50$ cm and thus of unity
aspect ratio. The top and bottom plates are made of 1.5 cm thick pure copper
with nickel-plated fluid-contact surface and the sidewall is made of a
plexiglas tube of 5 mm in wall thickness. Deionized and degassed water was
used as the convecting fluid. A square-shaped jacket made of flat plexiglas
plates and filled with water is fitted to the outside of the sidewall, which
greatly reduced the distortion effect to the PIV images caused by the
curvature of the cylindrical sidewall.
Two series of measurements of the spatial velocity field were carried out
using the PIV technique. In the first series, the measuring positions were
fixed near the cell sidewall and the experiments covered the range $5.9\times
10^{9}\lesssim Ra\lesssim 1.1\times 10^{11}$ of the Rayleigh number $Ra=\alpha
g\Delta TH^{3}/\nu\kappa$, with $g$ being the gravitational acceleration,
$\Delta T$ the temperature difference across the fluid layer, and $\alpha$,
$\nu$ and $\kappa$ being, respectively, the thermal expansion coefficient, the
kinematic viscosity, and the thermal diffusivity of water, whereas in the
second series, the measurements were made from near the cell sidewall to the
cell center at fixed Rayleigh number $Ra=4.0\times 10^{10}$. For both series,
the cell was tilted by a small angle of about 0.5∘ so that both series were
made within the vertical plane of the large-scale circulation and at midheight
of the cell. During the experiment the entire cell was wrapped by several
layers of Styrofoam and the mean temperature of the convecting fluid was kept
at $29^{\circ}$, corresponding to a Prandtl number $Pr=\nu/\kappa=5.5$. The
details of the PIV measurement could be found elsewhere (Xia _et al._ 2003;
Sun, Xia $\&$ Tong 2005), here we give only its main features. Hollow glass
spheres of 10 $\mu$m in diameter were chosen as seed particles and the
thickness of the laser lightsheet was $\sim 0.5$ mm. The spatial resolution of
the measured velocity field is 0.59 mm, which is much smaller than the lower
end of the inertial range and hence is sufficient to reveal the scaling
properties in the inertial range. In each measurement, the measuring region
has an area of $4.7\times 3.7$ cm2 (see the left panel of figure 1),
corresponding to $79\times 63$ velocity vectors, and the experiment lasted 3
hours in which a total of 25000 two-dimensional vector maps were acquired with
a sampling rate $\sim 2.3$ Hz. Since buoyant forces are exerted on the fluid
in the vertical direction, we focus our attention mainly on the longitudinal
vertical velocity increments $\delta_{r}w=w(x,z+r)-w(x,z)$, where $w(x,z)$ is
the vertical velocity component obtained at position $(x,z)$. To acquire the
accurate statistics, both temporal average and spatial average of moments of
velocity increments within an area of $1.8\times 3.7$ cm2, i.e. $31\times 63$
velocity vectors, were used when calculating VSFs, as the flow is
approximately locally homogeneous in turbulent RB system Sun et al. (2006);
Zhou et al. (2008).
Figure 1: Left: A snapshot of the instantaneous vertical velocity field
$w(x,z)$ near the cell sidewall for $Ra=4.0\times 10^{10}$. Positive is
defined for the upward motion and color coding is in cm/s. Right: Vertical
slices of the vertical velocity component $w(x,z)$ through the left figure.
The red curves mark the extracted cliff structures, corresponding to the
regions of plume fronts.
## 3 Results and discussions
### 3.1 Cliff structures in the vertical velocity field
The left panel of figure 1 shows a typical snapshot of the vertical velocity
field $w(x,z)$ near the cell sidewall, where $x$ is the horizontal distance
from the wall. The arrow in the figure indicates the region with large
vertical velocity, which is typically caused by a group of hot plumes passing
by. The plumes are generally believed to be detached thermal boundary layers
by buoyant forces. As introduced in $\S$ 1, thermal plumes usually generate
cliff structures in the temperature field. Here, the snapshot of $w(x,z)$ and
the associated slices through $w(x,z)$ in the vertical direction (the right
panel of figure 1) further show that in addition to temperature, plume fronts
can also generate cliff structures in the vertical velocity field. The
formation of cliff structures in the velocity field may be understood as
follows: Under the action of buoyant forces thermal plumes possess a higher
speed in the vertical direction in comparison to the background fluids, which
would deform the plumes such that they are compressed in the vertical
direction and stretched in the horizontal directions. This deformation
shortens the distance between the (relatively) high speed plume front and the
low-speed downstream fluids, therefore resulting a steep velocity gradient,
i.e. a cliff structure.
Figure 2: (_a_) PDFs of the normalized increments of vertical velocity
components at $x/D=0.04$. From the inner to the outer PDF, $r/\eta=16.6$
(dark-green up-triangles), 33.3 (pink squares), 57.1 (blue down-triangles),
and 107 (red circles), all within the inertial range. (_b_) Skewness of
$\delta_{r}w$ as a function of $r/\eta$ for different measuring positions
$x/D$ from near the cell sidewall to the cell center. The data were obtained
at $Ra=4.0\times 10^{10}$. (_c_) Skewness of $\delta_{r}w$ as a function of
$r/\eta$ for five different $Ra$ obtained near the cell sidewall ($x/D=0.04$).
One can expect that the persistence of such cliff structures would violate the
local isotropy of turbulent flows. Isotropy is the central hypothesis for most
theories and models of small-scale turbulence. Although a number of
investigations in the atmosphere and in laboratory flows have shown that the
skewness of velocity derivative is nontrivial Sreenivasan & Antonia (1997),
implying local anisotropy at small scales, the presence of cliffs would
enhance the degree of this small-scale anisotropy. This can be reflected by
the probability density functions (PDF) of velocity increments over different
scales. Figure 2(_a_) shows PDFs of $\delta_{r}w$ normalized by the standard
deviation of $w$, $\sigma_{w}=\sqrt{\langle(w-\langle w\rangle)^{2}\rangle}$,
over several different length scales within the inertial range. The data were
obtained near the cell sidewall ($x/D=0.04$) at $Ra=4.0\times 10^{10}$. The
asymmetry of the distributions with long left tails is clear, especially for
the (relatively) larger scales, which signifies the anisotropy at these
scales. To quantify this anisotropy, we plot in figure 2(_b_) the skewness of
$\delta_{r}w$ as a function of $r/\eta$ 111The Kolmogorov length scale $\eta$
is estimated from $\eta=(\nu^{3}/\varepsilon)^{1/4}$, where
$\varepsilon=\sigma_{w}^{3}/L$ is the energy dissipation rate per unit mass
and $L$ is the largest length scale of the turbulence. It is found that $\eta$
changes from 0.27 mm near the sidewall to 0.26 mm at cell center for
$Ra=4.0\times 10^{10}$. for different measuring positions ranging from
$x/D=0.04$ (near the sidewall) to $x/D=0.50$ (at cell center). It is seen that
all skewness are negative, but their magnitudes decrease continuously when
moving away from the wall and appear to remain invariant for $x/D\gtrsim 0.4$.
If the nonvanishing skewness observed here is indeed induced by cliff
structures of thermal plumes, the behaviors of the skewness may then be
understood from the inhomogeneous spatial distributions of thermal plumes in
the convection cell, i.e., plumes abound near the sidewall but are scarce in
the central region Qiu & Tong (2001); Shang et al. (2003, 2004); Xi et al.
(2004).
Figure 2(_c_) shows the skewness of $\delta_{r}w$ as a function of $r/\eta$
for five different $Ra$ obtained near the cell sidewall ($x/D=0.04$). Again,
all skewnesses are found to be negative. Another noticeable feature is that
the magnitudes of the skewness decrease with increasing $Ra$. This may be
understood as the cliff structures are being smoothed out by the increased
turbulent fluctuations. As the cliff structures of the velocity field are
generated by buoyancy, this result suggests that temperature becomes more
passive when the convective flow becomes more turbulent. Note that previous
experimental study has shown that temperature may become progressively passive
for $Ra>10^{10}$ Belmonte & Libchaber (1996); Zhou & Xia (2002), which is
qualitatively consistent with the picture obtained in the present study.
### 3.2 Plus and minus velocity structure functions
Figure 3: (_a_) Compensated third-order VSFs $S_{3}^{+}(r)/r$, $S_{3}(r)/r$,
and $S_{3}^{-}(r)/r$ measured near the cell sidewall. The arrow marks
$r=1.5\ell_{B}$ for reference. (_b_) Comparison of VSF exponents
$\zeta_{p}^{+}$, $\zeta_{p}$, and $\zeta_{p}^{-}$ with various model
predictions. The data were measured at $x/D=0.04$ for $Ra=4.0\times 10^{10}$.
(_c_) The third-order VSF exponents $\zeta_{3}^{+}$, $\zeta_{3}$, and
$\zeta_{3}^{-}$ as a function of the normalized distance from the cell
sidewall $x/D$ for $Ra=4.0\times 10^{10}$. (_d_) $Ra$-dependence of
$\zeta_{3}^{+}$, $\zeta_{3}$, and $\zeta_{3}^{-}$ obtained near the cell
sidewall ($x/D=0.04$).
The fact that plumes can induce anisotropy suggests that the cascades of
velocity field would possess different dynamics in different directions, i.e.,
the positive and negative velocity increments may have different scaling
behaviors. To study this quantitatively, we examine the statistical properties
of the plus and minus longitudinal VSFs Vainshtein & Sreenivasan (1994);
Sreenivasan et al. (1996), defined as
$S_{p}^{\pm}(r)=\langle[(|\delta_{r}w|\pm\delta_{r}w)/2]^{p}\rangle.$ (2)
From this definition, it is clear that cliff structures such as those shown in
the right panel of figure 1) will be excluded from $S_{p}^{+}(r)$ but included
in $S_{p}^{-}(r)$. This enables one to study separately the contributions of
positive and negative velocity increments to the corresponding structure
functions. A similar analysis has been performed previously in a turbulent
channel flow and it was found that the plus VSFs are less affected by the
presence of the wall Onorato & Iuso (2001). Figure 3(_a_) plots in log-log
scale the compensated third-order VSFs $S_{3}^{-}(r)/r$ (triangles),
$S_{3}(r)/r$ (circles), and $S_{3}^{+}(r)/r$ (squares) vs $r/\eta$ near the
cell sidewall, which exhibit slightly different scaling ranges. The
compensated plot shows a flat range for $S_{3}^{+}(r)/r$, i.e.
$S_{3}^{+}(r)\sim r$, suggesting that $S_{3}^{+}(r)$ possesses the same
scaling behavior as that for HIT, whereas both $S_{3}(r)$ and $S_{3}^{-}(r)$
exhibit much steeper scalings. To reveal this more clearly, we show in figure
3(_b_) the measured scaling exponents $\zeta_{p}^{+}$, $\zeta_{p}$, and
$\zeta_{p}^{-}$ of the VSFs of orders $p=1$ to 8. When comparing
$\zeta_{p}^{+}$ with the predictions of the hierarchy models of She & Leveque
(1994) (SL94) for HIT, we find excellent agreement. These results suggest that
the scaling behaviors of $S_{p}^{+}(r)$ are consistent with what one would
expect for HIT. Moreover, because cliff structures contribute mainly to the
negative velocity increments, $\zeta_{p}^{-}$ should exhibit some deviation
from the K41-type scaling. This is indeed observed. Figure 3(_b_) shows that
$\zeta_{p}^{-}$ and $\zeta_{p}$ are both much larger than the predictions of
SL94. Therefore, the plus and minus structures functions are effective means
to study the effects of cliff structures in the velocity field and can be used
to effectively disentangle the associated anisotropies caused by these
structures. These cliff structures are produced by thermal plumes in the
present case, but in general they can be produced by coherent structures in
other types of flows.
Let’s move on now to the location- and $Ra$-dependencies of the VSF scaling
exponents. We focus mainly on the third-order because $\zeta_{3}=1$ is an
exact result for HIT. Figure 3(_c_) shows the exponents $\zeta_{3}^{+}$,
$\zeta_{3}$, and $\zeta_{3}^{-}$ as a function of the measuring position
$x/D$. It is seen that both $\zeta_{3}$ and $\zeta_{3}^{-}$ are much larger
than the K41-value of 1 near the cell sidewall and drop from the wall. For
$x/D>0.3$ $\zeta_{3}$ and $\zeta_{3}^{-}$ are both essentially 1. On the other
hand, apart from some data scatter, $\zeta_{3}^{+}$ assumes essentially the
K41 value for all $x/D$. The decreases of $\zeta_{3}$ and $\zeta_{3}^{-}$
correspond to the reduced anisotropy associated to the cliff structures, i.e.,
the number of plumes decreases as the measuring position moves away from the
cell sidewall, confirming quantitatively the results shown in figure 2(_b_).
For $x/D>0.3$, i.e. the cell’s central region, thermal plumes are scarce and
hence it is not surprising to obtain approximately the K41 scaling for all
three VSFs.
Figure 3(_d_) shows the measured scaling exponents versus $Ra$. Again,
$\zeta_{3}^{+}$ is seen to remain nearly constant around the value of 1, but
$\zeta_{3}$ and $\zeta_{3}^{-}$, despite certain scatter, show an overall
decreasing trend with increasing $Ra$. Here, the decrease of $\zeta_{3}^{-}$
implies that the influence of buoyancy on the cascade processes becomes weaker
when the flow becomes more turbulent, which could also be reflected by the
behavior of the skewness of $\delta_{r}w$, whose magnitude is found to
increase with decreasing $Ra$ as shown in figure 2(_c_).
Figure 4: Comparison of ESS VSF exponents $\zeta_{p,3}^{+}$,
$\zeta_{p,3}^{-}$, and the SL94 scaling exponents.
It should be noted that near the sidewall the result
$\zeta_{3}^{-}\gtrsim\zeta_{3}^{+}$, shown in figure 3, does not imply that
the positive velocity increments are more intermittent than the negative ones.
To illustrate this, we examine the scaling behavior of VSFs via the extended
self-similarity (ESS) method Benzi et al. (1993), i.e., $S_{p}(r)$ is plotted
against $S_{3}(r)$, instead of $r$, in a log-log scale. Figure 4 shows the
measured ESS (relative) scaling exponents $\zeta_{p,3}^{+}$ and
$\zeta_{p,3}^{-}$ for the plus and minus VSFs, respectively. One sees that
$\zeta_{p,3}^{-}$ are slightly smaller than $\zeta_{p,3}^{+}$. This result
suggests that it is the minus velocity increments that possesses a higher
degree of intermittency, which should be attributed to the persistence of
cliff structures. A detailed analysis of such cliff structures would therefore
be helpful for understanding the present results. To do this, we use a
criterion to identify cliff structures in the vertical slices of $w(x,z)$
which is similar to those used for passive and active scalars Moisy et al.
(2001); Zhou & Xia (2002): a cliff is identified when $-\partial w/\partial
z>\sigma_{w}/\ell_{B}$, where $\ell_{B}$ is based on the global quantities Sun
et al. (2006). When such a cliff is found, we define its position $z_{0}$ as
the point maximizing $|\partial w/\partial z|$, and its width $\lambda_{C}$ as
the separation in space between the two extrema of $w$ surrounding $z_{0}$.
Applying this procedure, over 50 000 cliffs were identified near the sidewall
($x/D=0.04$) for each $Ra$. Three examples of the extracted cliff structures
are shown as red curves in the right panel of figure 1. Figure 5 shows the
mean cliffs’ width $\langle\lambda_{C}\rangle$, normalized by $\ell_{B}$. One
sees that $\langle\lambda_{C}\rangle=(1.5\pm 0.2)\ell_{B}$ is nearly
independent of $Ra$. We further note that the cliff structures mainly occur at
the scales near the upper end of the VSF scaling range [see figure 3(_a_)].
This could also be reflected qualitatively from figure 2(_a_), which shows
that the distributions of $\delta_{r}w$ over large scales seem to be more
asymmetric than those over relatively small scales, and quantitatively from
figure 2(_b_), from which one sees that within the inertial range ($16\lesssim
r/\eta\lesssim 110$) the magnitude of the skewness of $\delta_{r}w$ increases
with the scale $r$ for the near wall data ($x/D=0.04$, dark-green up-
triangles). Cliff structures contain large velocity variations and would
enhance the magnitude of velocity increments over the scales around their mean
width $\langle\lambda_{C}\rangle$, i.e. the scale near the upper end of the
VSF scaling range. This can also be seen from figure 3(_a_) that $S_{p}^{+}$
and $S_{p}^{-}$ differ the most around $1.5\ell_{B}$, signifying that scale is
representative of the typical size of a cliff structure. The cliffs’
influences to the scales near the lower end of the scaling range, however, are
much weaker [see figure 3(_a_)]. Therefore, under the influences of cliff
structures, the value of $S_{p}^{-}(r)$ near the upper end of the VSF scaling
range would increase, while those near the lower end do not change
significant. As a result, the measured scaling exponents of $S_{p}^{-}(r)$
would increase. With decreasing buoyancy effects, the scaling exponents are
also expected to drop, which is indeed observed in figure 3(_d_).
Figure 5: Mean cliffs’ width $\langle\lambda_{C}\rangle$, normalized by the
Bolgiano length scale $\ell_{B}$, as a function of $Ra$ obtained near the cell
sidewall ($x/D=0.04$). The error bars mark the standard deviation of
$\lambda_{C}$.
## 4 Conclusion
To summarize, we have demonstrated that as a manifestation of buoyancy effects
thermal plumes can generate cliff structures in the vertical velocity field,
which would in turn generate asymmetry in the velocity increments. We further
show that such effects can be quantified by examining respectively the plus
and minus velocity increments. For the plus increments, which largely exclude
the cliff structures and hence removing the buoyancy effects, the scaling of
their moments are the same as those expected of homogeneous and isotropic
turbulence (HIT). For the minus increments, the scaling is found to be much
steeper than that for HIT, which is caused by the presence of cliff
structures. Such effects of buoyant forces are found to vanish gradually when
moving away from the cell sidewall, owing to the inhomogeneous distributions
of thermal plumes in a closed convection cell. It is also shown that, as a
result of cliff structures being smoothed out by the increased turbulent
fluctuations, the effect of buoyancy on the velocity field decreases with
increasing $Ra$. Despite its simpleness, the analysis presented here provides
a useful method to quantitatively study the effect of buoyancy and disentangle
its contribution in the velocity field in buoyancy-driven turbulence. More
generally, it may be used to quantify the effect of anisotropy in other
complex phenomena when sharp fronts exist in the related random fields.
###### Acknowledgements.
This work was supported in part by Natural Science Foundation of China (No.
11002085), “Pu Jiang” project of Shanghai (No. 10PJ1404000), “Cheng Guang”
project of Shanghai (No. 09CG41), and E-Institutes of Shanghai Municipal
Education Commission (Q.Z.) and by the Research Grants Council of Hong Kong
SAR (Grant No. CUHK403806 and No. CUHK403807) (K.Q.X.).
## References
* Ahlers et al. (2009) Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection. Rev. Mod. Phys. 81, 503–537.
* Arad et al. (1999) Arad, I., Biferale, L., Mazzitelli, I. & Procaccia, I. 1999 Disentangling scaling properties in anisotropic and inhomogeneous turbulence. Phys. Rev. Lett. 82, 5040–43.
* Arad et al. (1998) Arad, I., Dhruva, B., Kurien, S., L’vov, V. S., Procaccia, I. & Sreenivasan, K. R. 1998 Extraction of anisotropic contributions in turbulent flows. Phys. Rev. Lett. 81, 5330–33.
* Belmonte & Libchaber (1996) Belmonte, A. & Libchaber, A. 1996 Thermal signature of plumes in turbulent convection: The skewness of the derivative. Phys. Rev. E 53, 4893–98.
* Benzi et al. (1993) Benzi, R., Ciliberto, S., Tripiccione, R., Baudet, C., Massaioli, F. & Succi, S. 1993 Extended self-similarity in turbulent flows. Phys. Rev. E 48, R29–32.
* Biferale et al. (2002) Biferale, L., Lohse, D., Mazzitelli, I. & Toschi, F. 2002 Probing structures in channel flow through SO(3) and SO(2) decomposition. J. Fluid Mech. 452, 39–59.
* Bolgiano (1959) Bolgiano, R. 1959 Turbulent spectra in a stably stratified atmosphere. J. Geophys. Res. 64, 2226–29.
* Grossmann et al. (2001) Grossmann, S., von der Heydt, A. & Lohse, D. 2001 Scaling exponents in weakly anisotropic turbulence from the Navier-Stokes equation. J. Fluid Mech. 440, 381–390.
* Ishihara et al. (2009) Ishihara, T., Gotoh, T. & Kaneda, Y. 2009 Study of high-Reynolds number isotropic turbulence by direct numerical simulation. Annu. Rev. Fluid Mech. 41, 165–180.
* Kunnen et al. (2008) Kunnen, R. P. J., Clercx, H. J. H., Geurts, B. J., van Bokhoven, L. J. A., Akkermanns, R. A. D. & Verzicco, R. 2008 Numerical and experimental investigation of structure-function scaling in turbulent Rayleigh-Bénard convection. Phys. Rev. E 77, 016302.
* Lohse & Xia (2010) Lohse, D. & Xia, K.-Q. 2010 Small-scale properties of turbulent Rayleigh-Bénard convection. Annu. Rev. Fluid Mech. 42, 335–64.
* Moisy et al. (2001) Moisy, F., Willaime, H., Andersen, J. S. & Tabeling, P. 2001 Passive scalar intermittency in low temperature helium flows. Phys. Rev. Lett. 86, 4827–30.
* Monin & Yaglom (1975) Monin, A. S. & Yaglom, A. M. 1975 Statistical fluid mechanics. vol. 2. MIT Press.
* Moses et al. (1993) Moses, E., Zocchi, G. & Libchaber, A. 1993 An experimental study of laminar plumes. J. Fluid Mech. 251, 581–601.
* Obukhov (1959) Obukhov, A. M. 1959 On the influence of archimedean forces on the structure of the temperature filed in a turbulent flow. Dokl. Akad. Nauk. SSSR 125, 1246–48.
* Onorato & Iuso (2001) Onorato, M. & Iuso, G. 2001 Probability density function and “plus” and “minus” structure functions in a turbulent channel flow. Phys. Rev. E 63, 025302(R).
* Qiu & Tong (2001) Qiu, X.-L & Tong, P. 2001 Onset of coherent oscillations in turbulent Rayleigh-Bénard convection.
* Shang et al. (2003) Shang, X.-D., Qiu, X.-L., Tong, P. & Xia, K.-Q. 2003 Measured local heat transport in turbulent Rayleigh-Bénard convection. Phys. Rev. Lett. 90, 074501.
* Shang et al. (2004) Shang, X.-D., Qiu, X.-L., Tong, P. & Xia, K.-Q. 2004 Measured local convective heat flux in turbulent Rayleigh-Bénard convection. Phys. Rev. E 70, 026308.
* She & Leveque (1994) She, Z.-S. & Leveque, E. 1994 Universal scaling laws in fully developed turbulence. Phys. Rev. Lett. 72, 336–39.
* Sreenivasan & Antonia (1997) Sreenivasan, K. R. & Antonia, R. A. 1997 The phenomenology of small-scale turbulence. Annu. Rev. Fluid Mech. 29, 435–472.
* Sreenivasan et al. (1996) Sreenivasan, K. R., Vainshtein, S. I., Bhiladvala, R., Gil, I. San, Chen, S. & Cao, N. 1996 Asymmetry of velocity increments in fully developed turbulence and the scaling of low-order moments. Phys. Rev. Lett. 77, 1488–91.
* Sun et al. (2005) Sun, C., Xia, K.-Q. & Tong, P. 2005 Three-dimensional flow structures and dynamics of turbulent thermal convection in a cylindrical cell. Phys. Rev. E 72, 026302.
* Sun et al. (2006) Sun, C., Zhou, Q. & Xia, K.-Q. 2006 Cascades of velocity and temperature fluctuations in buoyancy-driven thermal turbulence. Phys. Rev. Lett. 97, 144504.
* Vainshtein & Sreenivasan (1994) Vainshtein, S. I. & Sreenivasan, K. R. 1994 Kolmogorov’s 4/5th law and intermittency in turbulence. Phys. Rev. Lett. 73, 3085–88.
* Warhaft (2000) Warhaft, Z. 2000 Passive scalars in turbulent flows. Annu. Rev. Fluid Mech. 32, 203–240.
* Xi et al. (2004) Xi, H.-D., Lam, S. & Xia, K.-Q. 2004 From laminar plumes to organized flows: The onset of large-scale circulation in turbulent thermal convection. J. Fluid Mech. 503, 47–56.
* Xia et al. (2003) Xia, K.-Q., Sun, C. & Zhou, S.-Q. 2003 Particle image velocimetry measurement of the velocity field in turbulent thermal convection. Phys. Rev. E 68, 066303.
* Zhou et al. (2011) Zhou, Q., Li, C.-M., Lu, Z.-M. & Liu, Y.-L. 2011 Experimental investigation of longitudinal space-time correlations of the velocity field in turbulent Rayleigh-Bénard convection. J. Fluid Mech. submitted.
* Zhou et al. (2008) Zhou, Q., Sun, C. & Xia, K.-Q. 2008 Experimental investigation of homogeneity, isotropy and circulation of the velocity field in buoyancy-driven turbulence. J. Fluid Mech. 598, 361–372.
* Zhou & Xia (2010) Zhou, Q. & Xia, K.-Q. 2010 Universality of local dissipation scales in buoyancy-driven turbulence. Phys. Rev. Lett. 104, 124301.
* Zhou & Xia (2002) Zhou, S.-Q. & Xia, K.-Q. 2002 Plume statistics in thermal turbulence: Mixing of an active scalar. Phys. Rev. Lett. 89, 184502.
|
arxiv-papers
| 2011-02-01T08:41:52 |
2024-09-04T02:49:16.770152
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Quan Zhou and Ke-Qing Xia",
"submitter": "Quan Zhou",
"url": "https://arxiv.org/abs/1102.0098"
}
|
1102.0099
|
Automatic Network Fingerprinting Through Single-Node Motifs
Christoph Echtermeyer1, Luciano da Fontoura Costa2, Francisco A. Rodrigues3,
Marcus Kaiser1,4,5,∗
1 School of Computing Science, Claremont Tower, Newcastle University,
Newcastle-upon-Tyne NE1 7RU, UK
2 Instituto de Física de São Carlos, Universidade de São Paulo, São Carlos, PO
Box 369, 13560-970 São Carlos, São Paulo, Brazil
3 Departamento de Matemática Aplicada e Estatística, Instituto de Ciências
Matemáticas e de Computação, Universidade de São Paulo, São Carlos, PO Box
668, 13560-970 São Carlos, São Paulo, Brazil
4 Institute of Neuroscience, The Medical School, Framlington Place, Newcastle
University, Newcastle-upon-Tyne NE2 4HH, UK
5 Department of Brain and Cognitive Sciences, Seoul National University, Seoul
151-746, Korea
$\ast$ E-mail: m.kaiser@newcastle.ac.uk
## Abstract
Complex networks have been characterised by their specific connectivity
patterns (network motifs), but their building blocks can also be identified
and described by node-motifs—a combination of local network features. One
technique to identify single node-motifs has been presented by Costa et al.
(L. D. F. Costa, F. A. Rodrigues, C. C. Hilgetag, and M. Kaiser, Europhys.
Lett., 87, 1, 2009). Here, we first suggest improvements to the method
including how its parameters can be determined automatically. Such automatic
routines make high-throughput studies of many networks feasible. Second, the
new routines are validated in different network-series. Third, we provide an
example of how the method can be used to analyse network time-series. In
conclusion, we provide a robust method for systematically discovering and
classifying characteristic nodes of a network. In contrast to classical motif
analysis, our approach can identify individual components (here: nodes) that
are specific to a network. Such special nodes, as hubs before, might be found
to play critical roles in real-world networks.
## Introduction
Networks appear in a variety of real-world systems ranging from biology to
engineering [1, 2]. Examples include neural [3, 4, 5], social [6, 7, 8], and
computer networks [9, 10] to name but a few. Networks have been used to study
the emergence of cooperative behaviour [11, 12, 13]; to address
epidemiological questions [14, 15] especially in scale free networks [16, 17];
and to investigate the causes of cascade effects [18, 19] for a more complete
understanding of why networks differ in robustness against error and attack
[20, 21]. Attempts to classify network-topologies [22] were accompanied by
detailed studies of scale-free [23] and small-world networks [24,
25]—properties that were identified in many real networks. Additional to
investigations of concrete structures, theoretical studies of random networks
collected valuable information about large classes of networks [26, 27, 28].
Mapping complex systems to networks revealed that some nodes are remarkably
different from other nodes of the same network. For instance, hubs,
characterized by a high number of connections (a high node degree), often play
a fundamental role in protein-protein interaction networks and their removal
can be lethal for an organism [29, 30]. Hubs are similarly important for
socio-economic systems, where defective hubs can cause cooperation to decline
[31]. Also, in engineered systems like the Internet, hubs are important to
maintain the communication between autonomous systems [20]. These outlier
nodes have been identified since the introduction of complex network theory,
e.g. in the World Wide Web [9] and the Internet [10], but hubs are outliers
only in terms of their degree; other network properties can also define
special nodes. For instance, Internet topology has been shown decompose onion-
like into different shells around a relatively small core network [32]. The
closer a node’s layer is to the core, the higher is the node’s shell-index
(coreness) [33]. Nodes with high coreness are not necessarily hubs, which one
might suspect to be the most efficient spreaders of information. Instead, the
position of a node close to the network-core has more impact on successful
dissemination than having a high degree [34]. In networks where hubs are not
present, as in most geographical networks, nodes whose neighbours are also
connected to each other are special (high local clustering coefficient).
Further examples of outlier nodes can be found with different measures some of
which examine more than the direct neighbourhood of a node [26, 35], such that
they specify rather global (network specific) than local (node specific)
characteristics. Global measures, such as characteristic path length or
clustering coefficient [24], summarise the whole network in a single value.
Local measurements, on the other hand, analyse each node or edge individually,
yielding a more fine-grained picture of the network. Nodes that express common
features and outliers that are different can be identified with pattern
recognition approaches, which group nodes of similar characteristics.
Corresponding techniques have been proposed recently [36, 37, 38] and revealed
important network properties. For example, in protein-protein interaction
networks the relative number of outliers tends to decrease with the complexity
of organism, i.e. proteins in more complex species show higher homogeneity in
their interplay [38]. This demonstrates that, by considering multiple node-
features jointly, pattern recognition based methods can point out exceptional
network components.
Networks can describe complex systems whose interactivity between dynamical
components changes over time. Altered connections between the elements
(represented by nodes) may in turn feed back on the dynamics, such that the
dynamical process and the network topology evolve in an adaptive fashion [39].
In the context of game theory, corresponding coevolution of behaviour and
connectivity has been studied in socio-economic systems [12, 13]. In complex
scenarios like these, analysing a single network is often insufficient and
several networks must be compared to gain insights. Further examples for the
need of network-comparisons are families of protein-protein interaction
networks, brain connectivity networks in patient- and control-populations, or
time-dependent (developing or declining) networks [40]. Comparing such sets of
networks requires consistent approaches, which are often non-trivial, because
networks differ in size (number of nodes or edges) or they comprise a disjoint
sets of nodes (some nodes occur in one network but not in others). Direct
comparisons between structures may thus be ruled out. Based on outlier-
detection as described above, we previously proposed motif-regions for which
the relative frequencies of outliers falling into one of them yields a network
specific fingerprint [41]. Relating different networks to each other has
thereby become as easy as comparing bar-graphs. Nevertheless, although this
methodology has been demonstrated to be suitable and accurate for outlier
identification as well as for network comparisons, it suffers from several
limitations, which we address in this paper.
Here, we describe a novel workflow for detecting characteristic single-node
motifs and for using fingerprints for network comparison. Improvements
compared to the previous approach include (a) automatic parameter
determination, which facilitates high throughput analysis without user
interaction, and (b) replacing the k-means clustering algorithm with a
deterministic method to simplify the workflow and to improve robustness of
results. In addition, we provide (c) a validation of our method and (d) an
application to networks where the topology changes over time (addition or
deletion of nodes or edges).
### Previous work
The application of single measures to complex networks has revealed important
insights in many cases. However, as Newman and Leicht recognised [36],
detecting exceptions is limited to network features that are quantified by the
measures in use. Otherwise, if the chosen characteristics do not reflect the
properties that are specific for a network or its components, important
features remain unnoticed.
To solve this problem, two complementary approaches have been suggested. The
first approach by Newman and Leicht groups nodes based on their connectivity
without any further prior information [36]. By fitting the parameters of a
mixture model (using an expectation-maximization algorithm), each node is
assigned a probability of belonging to any one group that has been identified.
The probabilistic nature of this approach has the advantage that nodes that
can not be unambiguously categorised are not crudely assigned to one
particular group, but the conflict becomes evident, such that it can be dealt
with. The structure of networks can thereby be investigated without requiring
any other parameter than the number of groups that are to be created. This
elegant method has been examined thoroughly and improvements to it have been
suggested [42, 37].
Analyses with focus on only one particular aspect of a network at a time might
fail to detect irregularities or similarities in structure. The second
approach is to avoid single measures and to use a combination of multiple ones
[41]. Instead of reducing network components down to one dimension, joint
measures map it into a multi-dimensional feature space [43]; each vector-point
in that space corresponds to a combination of node-characteristics and
statistical methods are used to identify motif regions, such that each vertex
falls into one of them: A node is either classified regular—showing features
like the majority of nodes—or singular, i.e. its features deviate by following
a particular single node-motif. The term motif refers to the concept of
network motifs, i.e. patterns incorporating multiple nodes [44].
Each of these two approaches to identify patterns in complex networks has its
drawbacks and advantages. The Newman and Leicht algorithm (NLA) does not
depend on one or few network measures, but it works on network links directly.
Networks are not restricted to undirected ones, but directed links and even
weighted ones can be considered. The NLA requires the number of node-groups to
be specified; this is also true for the approach by Costa et al. [Beyond the
Average (BtA)], where the number of motif regions needs to be chosen a priori
[41]. Unfortunately, for real-world networks this number is often unknown. The
BtA-workflow requires two additional parameters to control which nodes will
constitute individual motif regions. Both methods differ in their output, as
BtA not only provides a grouping of nodes, but also a network-fingerprint,
which can be used to compare networks from different domains. Most
importantly, however, is the conceptional distinction between NLA and BtA, as
they rely on local edge connectivity and local node measures, respectively.
BtA will fail to pinpoint features of the network, if the chosen set of
measures can not formulate a corresponding motif. Similarly NLA can fail, as
it only takes into account direct connections between nodes: NLA does not
consider how the neighbours of a node are connected, for example, but BtA can
deal with such information (by evaluating the local clustering coefficient).
Indeed, the extensibility concerning features to assess is the biggest
advantage of BtA; (un-)directed and weighted links can be processed likewise
and in spatial networks the location of nodes can be taken into account. In
conclusion, NLA is readily applicable to a broad variety of network domains;
however, considering direct connections only is a weakness. BtA can be nicely
adopted to these cases, but care has to be taken at all times to ensure the
set of measures is diverse enough to cover as many patterns that might occur
in networks as possible.
In the next section we suggest several improvements to the BtA-workflow (Fig.
1), which can be sketched as follows: Initially, multiple local network
measures are applied to each node, which yields a multi-dimensional
characterisation in form of a feature vector. Correlation between different
measures is accounted for by principal component analysis (PCA), which is used
to map feature vectors of all nodes to two dimensional space [45, Chapter 8].
Next, nodes are assigned probabilities in order to distinguish nodes with
common and rare features. The required probability density function (PDF) is
gained by smoothing over points in the two dimensional PCA-plane (Parzen
window approach [46, 47, Chapter 4.3]). Now, the least probable nodes, i.e
those with uncommon features, can be identified from the PDF. These singular
nodes are then clustered in order to distinguish different motif-groups. Each
of the two dimensional motif-groups corresponds to a higher dimensional motif-
region into which the feature vectors split up and the distribution of feature
vectors among the different motif-regions is the fingerprint of the network.
Apart from the initial decision on which measures to use, the user needs to
choose the bandwidth of the smoothing kernel, the number of singular nodes
$w$, and the number of motif-groups $k$, respectively (steps 3, 4, and 5 in
Fig. 1). Additionally, when comparing multiple networks, a limit must be
specified below which motif regions are considered too close to each other to
constitute different motifs (join threshold; Step 7). So far, these settings
had to be chosen manually, but here we suggest how to determine all three
parameters (bandwidth, $w$, and $k$) automatically. The last setting (join
threshold), however, is not considered for automation: So far we could not
identify a procedure that yields results as good as manual selection by the
user. We thus concentrated our efforts on the parameters that need to be set
for every network (bandwidth, $w$, and $k$), such that high-throughput
applications become possible. Automating the setting of the main parameters is
thus of higher benefit than for the threshold that determines Voronoi cells to
be joined; this needs to be chosen only once, when all networks are compared
to each other at the end.
## Results
In this paper we propose how to choose all relevant parameters of the BtA-
workflow automatically (see Methods section), which allows for the analysis of
many networks without the need for human interaction. The only remaining
limiting factor for high throughput analyses are the computational costs of
the analysis, which predominantly depend on the measures that are chosen to
characterise each node. Using implementations of common local measures (see
supporting information), the estimated run-time scales linearly to cubic with
network size (Fig. S1). Costs are thus comparatively cheap considering methods
that identify specific connectivity patterns by counting occurrences of
particular sub-graphs (e.g. [48, 49, 44, 50, 51, 52]); such motif-counts also
scale at least linearly in network size, but they show exponentially growing
costs as the size of the motif-pattern increases [50]. In practice this often
means that counts can not be determined for patterns involving 10 nodes or
more [53], which renders some domains computationally intractable for this
approach, but eventually not for BtA. However, before processing huge networks
or many different structures with BtA, we first need to verify that parameters
are indeed chosen adequately, which is confirmed in the next section.
### Method Verification
The first validation is on a network that is small enough to confirm BtA-
results by eye: We use a family-tree from The Simpsons [54] to create a
network with nodes representing characters and directed links pointing to
their offspring (Fig. 2a). Nodes that have a sparsely connected and
homogeneous neighbourhood are suitably highlighted as outliers by BtA.
With these reassuring results from a single network, we proceed by testing BtA
on whole series: We generate structures with both regular components and
exceptional ones, which BtA has to identify. In our first series we compose
networks of two components: a regular ring lattice and a smaller Erdős-Rényi
(ER) [55] random network (Fig. 2b). While the ring lattice remains unchanged,
the size of the random module increases throughout the series, such that its
proportion of the full network grows gradually111 The ring lattice is
comprised of 100 nodes, each of which is connected to its four closest
neighbours (Fig. 2b). ER-random networks ($n=1,\ldots,50$ nodes) have an
average edge-density of 25%. . Composed networks are analysed with BtA: Of all
outlier-nodes less than 2% are missed while over 96% are classified correctly,
if the random component contributes less than 25% of nodes to the network.
Beyond that limit, the number of nodes in the random-part does no longer match
the number of identified outliers $w$. But this does not imply a mis-
classification by BtA: The larger a random network, the more likely it is that
a few nodes are connected regularly (or close to that). Quantifying these
nodes with local network measures yields the same values as (or similar to)
those of the ring lattice, which is why it would be incorrect to consider them
singular. Additional to regular connection patterns in large random networks,
other local motifs can also be frequent enough, such that they constitute a
common rather than an exceptional feature of the network. Thus, network
components that seem clearly separable at first may actually be very similar
or—although intended to form outliers—they may contain common elements, due to
random effects. Together this explains the observed deviations in numbers of
outliers for growing ER-components in this test-series.
Finally, we reverse the nature of the networks: The major component is set to
a random network [ER, Barabási and Albert (BA) [56], or Watts-Strogatz (WS)
[24] model] in which we embed a small, but highly regular structure (Fig. 2c).
The inserted structure was chosen, such that its nodes are highly clustered
(both on level 1 and 2); the six outer nodes further show significant
variability in their neighbours’ degrees. These characteristics are rarely
observed in our random networks, which is why BtA should identify these nodes
(alongside with other outliers that might emerge). We confirm this in a series
of networks with varying sparseness222Random networks ($n=100$ nodes) are
generated according to the ER, BA, and WS model; edge-density is gradually
increased from 1% to 50% (step-size 1%). The regular structure (7 nodes)
illustrated in Fig. 2c is added to each random network before BtA is applied.
: The 6 outer nodes of the regular structure are classified singular in over
97% of all networks. Additionally, the inner node (with less extreme features)
is regarded uncommon in 81% of all cases.
In conclusion, the automatic parameter determination gives very satisfying
results, which yield confidence in BtA’s ability to identify outliers in
complex networks autonomously.
### Network Time-Series: A Small-World Emerging
Large complex networks are challenging to analyse; time-series of such are
even more so. We attempt to approach this challenge by first condensing
networks to a compact representation—mapping a series of changing structures
to a uniform representation benefits the identification of trends and changes
of such. Therefore, all networks have to be characterised, which we do using
single node-motifs. These are identified with BtA using six common local
measures: (1) the normalised average degree $r$, (2) the coefficient of
variation of the degrees of the immediate neighbours of a node $cv$, (3) the
clustering coefficient $cc$ [24, 57], (4) the locality index $loc$, (5) the
hierarchical clustering coefficient of level two $cc_{2}$ [58], and (6) the
normalised node degree $K$. (For definitions of these measures see Methods
section.) Next, we describe the time-series of 600 networks and the results
found with BtA.
Similar to random graphs, small-world networks have a small characteristic
path length, but at the same time they exhibit a high degree of clustering, as
regular ring lattices, for example. It has been discovered early that the
combination of short paths plus grouping is inherent to social networks; a
phenomenon that became known as six degrees of separation [59, 60, 61]. Today
it is known that small-world networks can be found in many other domains (e.g.
[26, 27, 2, 28]). We thus created a network-time series in which structures
gradually change from a completely regular ring lattice to a small-world
network (see Methods section, Fig. S2).
In total we identified 5 single node-motifs, which differ in characteristics,
frequency, and time of emergence (Fig. 3): A node according to motif 1 has
relatively few connections in contrast to its well connected neighbourhood.
Different from that, nodes corresponding to motif 2 are signified by many
connections to a rather sparsely connected neighbourhood. Motif 3-nodes have
relatively few connections and nodes in their neighbourhood are similar in
number of links and corresponding targets. Motif 4 describes rarely connected
nodes whose neighbours have a diverse number of connections; but instead of
being linked between each other, neighbours share other common targets. The
final motif 5 can be best characterised by its relation to the rest of the
network, which shows a higher degree of connectivity than any node involved in
the motif. Neighbours of the motif-node further vary in their number of
connections and do not link to each other. Motifs 2, 3 and 5 appear right from
the beginning of the rewiring process; motifs 2 and 5 gradually become more
common over time, whereas 3 levels out after a transient peak. The remaining
motifs 1 and especially 4 only become apparent at later stages towards which
both become more frequent. Together, BtA reveals the increasing irregularity
in network structure and it also provides details on the characteristic
connectivity patterns at different times. Both would be valuable information
if real networks were analysed; here, with precise knowledge about the
network-changing process, the temporally dependent motif expression levels
yield another validation of the technique (detailed discussion in supporting
information).
Overall, results are very satisfying and we are confident that BtA could be
successfully applied to real networks using the automatic parameter
determination.
## Discussion
In this paper we presented a method to detect single node-motifs
automatically. The main parameters of the previous routine [41]—the smoothing
kernel bandwidth plus the number of singular nodes and motif groups—are now
selected based on the data. We further proposed a deterministic replacement
for the k-means algorithm, which is used to form the different motif-groups.
In contrast to k-means, our alternative approach can determine the number of
motifs itself and due to the lack of random elements, clustering results are
robust over multiple repetitions.
Despite our improvements to BtA certain issues and room for further
advancements remain. For example, reducing feature vectors in dimension
inevitably leads to a loss of information, but which has to be kept withing
reasonable bounds. In other words, although 6-dimensional feature vectors were
suitably represented in the 2-dimensional plane so far [41], different
networks may require the use of more than just the first 2 principal
components in order to ensure that network characteristics are represented
properly. Thus, if the chosen number of principal components does not account
for at least 80% of the variance, their number should be increased (Kaiser’s
rule). The degree to which feature vectors can be reduced thereby depends on
the correlation between measured values, which is specific to the analysed
network.
In cases where feature vectors can not be suitably represented in 2
dimensions, their display becomes more complicated and verifying a good fit of
the estimated probability density function (PDF) is challenging. However, a
good PDF estimate is needed in the BtA workflow to determine outlier nodes.
Problems that might arise in these situations could possibly be circumvented
by a major change to the workflow: The use of PCA to compact information
offers the possibility to replace both the PDF estimation and the subsequent
outlier selection with a more direct and non-parametric standard technique,
which is Hotelling’s $T^{2}$ (a generalisation of Student’s t-statistic). This
modification would allow to identify outliers without the need to estimate a
PDF, but the exploration of the resulting workflow will be addressed in
another publication.
Considering the BtA workflow as presented in this paper, the technique can be
easily adapted by including different local network measures in the analysis.
Measures that take spacial aspects of the network into account, for instance,
or those including link-weights can increase quality of the analysis. Finally,
interest might not only lie on motifs formed by outlier nodes, but on all
single node-motifs occurring in the network. In this case regular and singular
nodes are not distinguished, but all of them have to be included in the
network fingerprint.
BtA-fingerprinting of many networks has so far been prevented by the need to
choose parameters during the analysis manually. With the improvements
presented in this paper, however, it is now possible to process large numbers
of networks fully unsupervised. Identified outliers are characteristic nodes
that can provide a fingerprint of a network; fingerprinting networks from
numerous domains allows easy characterisation and comparisons. As already
demonstrated [41], such studies can reveal important characteristics and
differences between network domains. Additionally, the example on an emerging
small-world network in this paper showed that BtA can also be used to analyse
time-series of networks.
To encourage the use of the BtA methodology by other researchers, we provide
our implementation of the workflow including the automatic parameter
determination for download (http://www.biological-networks.org/). Two versions
of the code exist: The first one requires Matlab (Mathworks Inc, Natick, USA)
and allows the user to apply the workflow using a graphical user interface
(Fig. S3). The other one is a command line utility that either requires Matlab
or the free alternative Octave [62] and it can be easily used to batch process
many networks.
In conclusion, we provide a robust method for systematically discovering and
classifying characteristic nodes of a network. The distribution of node-
classes results in a fingerprint, which in turn can give a classification of
whole networks, as for network motifs of multiple nodes [63]. In contrast to
classical motif analysis, our approach can identify the individual components
that are specific to a network. Such special nodes, as hubs before, might be
found to play critical roles in real-world networks.
## Methods
#### Local Network Measures
Network nodes were characterised with six common local measures whose
definitions are given in the following. Therefore, let $A=(a_{ij})$ denote the
adjacency matrix of the network, i.e. $a_{ij}=1$, if a link from node $i$ to
node $j$ exists, and otherwise $a_{ij}=0$. Row- and column-sums of $A$
correspond to the in- and out-degrees of nodes, respectively. In undirected
networks, in- and out-degree are equal and either of them can be used as a
node’s degree. If links are directed, the degree is the sum of in- and out-
degree. Dividing a node’s degree by the number of all links in the network
yields the normalised node degree $K$. The normalised average degree $r_{i}$
of a node $i$ is the average over all its neighbours’ degrees. (Nodes that are
directly linked to node $i$ are called neighbours.) Likewise, the coefficient
of variation $cv$ of the degrees of the immediate neighbours of a node can be
calculated. The neighbours’ connectivity with each other is quantified by the
clustering coefficient $cc_{i}$, which is the proportion of existing
connections between node $i$’s neighbours to the number of all possible links
between them [24, 57]. The clustering coefficient thus reflects the relative
number of triangle-shaped paths a node has—a concept that is extended to
connections between neighbours’ neighbours (further away node node $i$) by the
hierarchical clustering coefficient of level two $cc_{2}$ [58]. Whereas the
cluster coefficients quantify connectivity within a node’s neighbourhood, the
locality index $loc_{i}$, which is based on the matching index (e.g. [64]), is
the fraction of neighbours’ links that connect to the same node (not
necessarily a neighbour of node $i$). Further details and measures can be
found in the literature [26, 27, 28, 35].
In the following sections we describe how appropriate settings for the
parameters of the BtA-workflow can be found automatically. Kernel-bandwidth,
the number of singular nodes $w$, and the number of motif regions $k$ are
discussed separately below.
#### Kernel-Bandwidth
In step 3 of the workflow (Fig. 1), the Parzen window approach is used to
estimate a probability density function (PDF) over all nodes [46, 47, Chapter
4.3]. This is achieved by smoothing the overall arrangement of reduced feature
vectors, which were obtained using principal component analysis (PCA) [45,
Chapter 8] in the previous step 2. The dimensions of the smoothing kernel,
i.e. the width and breadth of the Gaussian function
$\mathcal{N}_{2}(\mu,\,\Sigma)$ can be controlled through its covariance
matrix $\Sigma=(\sigma_{ij})$. (Mean vectors $\mu$ are fixed to equal the
data-points.) The original publication made use of the fact that the absolute
covariance values ($\forall k:\,\sigma_{kk}=0.05$) do not matter for the
estimated PDF. However, their values relative to each other do matter and we
therefore scale them according to the standard deviation along each principal
component (PC) axis. Variability-based re-shaping of the kernel function
improves the overall fit of the PDF to the points (Fig. S4). A further
refinement would be to tilt the Gaussian in order to account for correlation
between axes (Fig. S5); however, the PCs are expected to show weak correlation
only, which is why we chose un-tilted kernels (for which the covariance matrix
$\Sigma$ is zero except for the variances on the diagonal).
#### Number of Singular Nodes $w$
After assigning probabilities to all nodes (Step 3), nodes with an
exceptionally low probability come into focus: These outliers correspond to
points in the PCA-plane that are spatially separated from larger clusters; and
this separation corresponds to abnormalities of measured features. Due to
their uncommon characteristics, these nodes are considered singular. For
humans it is usually straightforward to identify these non-regular nodes, if
interactive visual aids are provided; we therefore implemented a graphical
user interface for the whole workflow (Fig. S3). In the following, however, we
discuss how the number of singular nodes $w$ can be adjusted without
interaction.
To determine singular nodes, automated methods can query the PDF that has been
estimated earlier (Step 3). For example, for a fixed number $w$ of singular
nodes, the $w$ least probable ones can be selected easily. Alternatively, a
probability cut-off can be set, e.g. at 1% or 5%, to separate nodes into
regular and singular ones. Both these simple methods involve constants, but
which have to be chosen depending on network size to yield sensible
results333Choosing one fixed number of singular nodes $w$ for differently
sized networks can render the majority of nodes non-regular in comparatively
small networks; vice versa, $w$ may be too small compared to the number of
exceptional nodes in large networks. A fixed probability cut-off does not
circumvent this problem, because the nodes’ absolute probability values are
dependent on network size.. In the following, we therefore propose a flexible
probability-threshold: The cut-off does not occur at a fixed pre-defined
level, but where it yields the best separation between singular and regular
nodes.
A necessary condition for a node being considered singular is a sufficiently
low probability compared to other nodes. Additionally, it is desirable that
singular nodes appear somewhat separated from the regular ones, which renders
their classification non-arbitrary. We therefore suggest to set the borderline
between regular and singular nodes where the steepest increase in probability
among the low probability nodes appears. Nodes with a probability below mean
$\bar{p}$ minus one standard deviation $\sigma(p)$ of all nodes’ probabilities
$p=(p_{k})_{k=1,\ldots,n}$ are potentially singular. Given that the
probabilities $p=(p_{k})_{k=1,\ldots,n}$ are sorted increasingly, the number
of singular nodes $w$ is then chosen as
$w=\arg\max_{k\ :\ p_{k}<\ \bar{p}-\sigma(p)}{p_{k+1}-p_{k}}\ ,$ (1)
or $w=0$, if probabilities undershoot the mean only minimally (i.e. $\nexists\
k:p_{k}<\bar{p}-\sigma(p)$).
#### Number of Motif Groups $k$
Once nodes are classified as either regular or singular (Step 4), clusters of
singular nodes (motif-groups) are identified using $k$-means [65, Chapter
20.1]. The $k$-means clustering algorithm requires the number of clusters $k$
to be chosen a priori; the actual procedure then determines $k$ centroids and
assigns each node to the closest one of them. Choosing $k$ too low results in
clustering errors, because multiple motif-groups are falsely considered as
one. Conversely, too many clusters split motif-groups into non-existing sub-
groups. Determining a suitable $k$ is thus crucial for automating the workflow
and we come back to this issue later. Even if $k$ is chosen adequately,
clustering results are not guaranteed to be satisfactory when using k-means:
The algorithm initially chooses the cluster-centroids at random, but their
actual distribution impacts on the quality of clustering results [66].
Attempts to optimise the centroid initialisation have been made (e.g. the
$k$++-algorithm [67]), but random effects still remain; we therefore suggest a
deterministic replacement for $k$-means.
Optimal groupings of singular nodes consider well separated nodes to be in
different clusters, whereas relatively close ones are grouped together. The
standard deviations along each PC-axis can serve as a threshold for closeness
and we consider each of the singular nodes to occupy a certain volume in the
PCA-plane, i.e. an ellipse-shaped area centred on it. All ellipses have the
same dimensions, which equal the standard deviations along the two axes. Nodes
are then assigned to the same motif-group if all their ellipses constitute a
connected area (Fig. 4). Practically, this idea can be implemented in 3 steps:
1. 1.
Similar to an adjacency matrix, create a binary overlap-matrix $O=(o_{ij})$ in
which nodes are connected if their ellipses overlap; otherwise they are not.
For two nodes $i$ and $j$ let $x=(x_{1},x_{2})$ and $y=(y_{1},y_{2})$ denote
their corresponding points on the PCA-plane, i.e. the centres of their
ellipses with dimensions $\sigma_{1}$ and $\sigma_{2}$. Using the rescaled
centres $c_{x}=(x_{1}/\sigma_{1},x_{2}/\sigma_{2})$ and
$c_{y}=(y_{1}/\sigma_{1},y_{2}/\sigma_{2})$ the entry of the overlap-matrix is
defined by
$o_{ij}=\begin{cases}1,&d_{2}(c_{x},c_{y})<1\ ,\\\ 0,&\text{otherwise}\
,\end{cases}$ (2)
where $d_{2}(\cdot,\cdot)$ is the Euclidean distance.
2. 2.
Determine a corresponding clique-matrix $C=(c_{ij})$ that specifies whether a
path—a connected area of ellipses—between any two nodes exists or not. Paths
or cliques can be determined through powers $O^{k}=\left(o_{ij}^{(k)}\right)$
of the overlap-matrix $O$ via
$c_{ij}=\begin{cases}1,&\exists\ k\in\mathbb{N}:o_{ij}^{(k)}>0\ ,\\\
0,&\text{otherwise}\ .\end{cases}$ (3)
3. 3.
Colour all cliques differently, which finally yields the motif-groups.
Note that this procedure has no parameter controlling the number of motif-
groups, but these are identified automatically. Instead of using this method
to actually group nodes it might also serve as a pre-processing step in order
to determine the number of clusters $k$ for k-means. The drawback of this
simple approach is that long elongated clusters can result when nodes are
widely distributed, but connected by a chain of nodes that are just less than
one standard deviation apart from each other. However, we have not observed
such formation in practical applications.
### Generation of Small-World Networks
The prevalence of small-world networks has risen questions about their
generating mechanisms and different explanatory models have been proposed [24,
68]. We use one of them here in order to generate a series of networks: Watts
and Strogatz described a rewiring procedure by which a regular ring-lattice is
randomly rewired by which it becomes a small-world network [24]. This is a
step-wise process, which allows to sample a network at each intermediate
stage. Starting with a completely regular structure, over time, networks
become increasingly perturbed (Fig. S2). In total, we sampled 600 networks (à
200 nodes), which were then analysed with BtA, to determine the single node-
motifs that evolve over time.
## Acknowledgments
Marcus Kaiser and Christoph Echtermeyer were supported by EPSRC (EP/G03950X/1)
and the CARMEN e-science project (http://www.carmen.org.uk) funded by EPSRC
(EP/E002331/1). Marcus Kaiser also acknowledges support by the WCU program
through the National Research Foundation of Korea funded by the Ministry of
Education, Science and Technology (R32-10142). Luciano da F. Costa is grateful
to CNPq (301303/06-1) and FAPESP (05/00587-5) for financial support. Francisco
A. Rodrigues is grateful to FAPESP (2007/50633-9) for sponsorship.
## References
* 1. Bornholdt S, Schuster HG, editors (2003) Handbook of graphs and networks: from the Genome to the Internet. John Wiley and Sons, 1st edition.
* 2. Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang D (2006) Complex networks: Structure and dynamics. Physics Reports 424: 175–308.
* 3. Sporns O, Chialvo DR, Kaiser M, Hilgetag CC (2004) Organization, development and function of complex brain networks. Trends in Cognitive Sciences 8: 418–25.
* 4. Kaiser M (2007) Brain architecture : a design for natural computation. Philosophical Transactions of the Royal Society A 365: 3033–45.
* 5. Bullmore E, Sporns O (2009) Complex brain networks: graph theoretical analysis of structural and functional systems. Nature Reviews Neuroscience 10: 186–98.
* 6. Lazer D, Pentland A, Adamic L, Aral S, Barabási AL, et al. (2009) Social science. Computational social science. Science 323: 721–3.
* 7. Borgatti SP, Mehra A, Brass DJ, Labianca G (2009) Network analysis in the social sciences. Science 323: 892–5.
* 8. Centola D (2010) The Spread of Behavior in an Online Social Network Experiment. Science 329: 1194–7.
* 9. Albert R, Jeong H, Barabási AL (1999) Diameter of the World-Wide Web. Nature 401: 130–1.
* 10. Faloutsos M, Faloutsos P, Faloutsos C (1999) On Power-Law Relationships of the Internet Topology. In: Proceedings of the conference on Applications, technologies, architectures, and protocols for computer communication. ACM, pp. 251–62.
* 11. Nowak MA (2006) Five rules for the evolution of cooperation. Science (New York, NY) 314: 1560–3.
* 12. Szabo G, Fath G (2007) Evolutionary games on graphs. Physics Reports 446: 97–216.
* 13. Perc M, Szolnoki A (2010) Coevolutionary games–a mini review. BioSystems 99: 109–25.
* 14. Barthélemy M, Barrat A, Pastor-Satorras R, Vespignani A (2005) Dynamical patterns of epidemic outbreaks in complex heterogeneous networks. Journal of theoretical biology 235: 275–88.
* 15. Funk S, Salathe M, Jansen VAA (2010) Modelling the influence of human behaviour on the spread of infectious diseases: a review. Journal of The Royal Society Interface .
* 16. Pastor-Satorras R, Vespignani A (2001) Epidemic Spreading in Scale-Free Networks. Physical Review Letters 86: 3200–3.
* 17. Meloni S, Arenas A, Moreno Y (2009) Traffic-driven epidemic spreading in finite-size scale-free networks. Proceedings of the National Academy of Sciences of the United States of America 106: 16897–902.
* 18. Watts DJ (2002) A simple model of global cascades on random networks. Proceedings of the National Academy of Sciences of the United States of America 99: 5766–71.
* 19. Buldyrev SV, Parshani R, Paul G, Stanley HE, Havlin S (2010) Catastrophic cascade of failures in interdependent networks. Nature 464: 1025–8.
* 20. Albert R, Jeong H, Barabási AL (2000) Error and attack tolerance of complex networks. Nature 406: 378–82.
* 21. Kaiser M, Martin R, Andras P, Young MP (2007) Simulation of robustness against lesions of cortical networks. European Journal of Neuroscience 25: 3185–92.
* 22. Estrada E (2007) Topological structural classes of complex networks. Physical Review E 75: 1–12.
* 23. Barabási AL (2009) Scale-free networks: a decade and beyond. Science 325: 412–3.
* 24. Watts DJ, Strogatz SH (1998) Collective dynamics of ’small-world’ networks. Nature 393: 440–2.
* 25. Schnettler S (2009) A structured overview of 50 years of small-world research. Social Networks 31: 165–78.
* 26. Albert R, Barabási AL (2002) Statistical mechanics of complex networks. Reviews of modern physics 74: 47–97.
* 27. Newman MEJ (2003) The Structure and Function of Complex Networks. SIAM Review 45: 167–256.
* 28. Newman MEJ, Barabási AL, Watts DJ (2006) The Structure and Dynamics of Networks. NJ: Princeton University Press.
* 29. Jeong H, Mason SP, Barabási AL, Oltvai ZN (2001) Lethality and centrality in protein networks. Nature 411: 41–2.
* 30. Rodrigues FA, Costa LDF (2009) Protein lethality investigated in terms of long range dynamical interactions. Molecular BioSystems 5: 385–90.
* 31. Perc M (2009) Evolution of cooperation on scale-free networks subject to error and attack. New Journal of Physics 11: 033027.
* 32. Carmi S, Havlin S, Kirkpatrick S, Shavitt Y, Shir E (2007) A model of Internet topology using k-shell decomposition. Proceedings of the National Academy of Sciences of the United States of America 104: 11150–4.
* 33. Seidman SB (1983) Network Structure and Minimum Degree. Social Networks 5.
* 34. Kitsak M, Gallos LK, Havlin S, Liljeros F, Muchnik L, et al. (2010) Identification of influential spreaders in complex networks. Nature Physics 6: 888–93.
* 35. Costa LDF, Rodrigues FA, Travieso G, Boas PRV (2007) Characterization of complex networks: A survey of measurements. Advances in Physics 56: 167–242.
* 36. Newman MEJ, Leicht EA (2007) Mixture models and exploratory analysis in networks. Proceedings of the National Academy of Sciences of the United States of America 104: 9564–9.
* 37. Wang J, Lai CH (2008) Detecting groups of similar components in complex networks. New Journal of Physics 10: 123023.
* 38. Costa LDF, Rodrigues FA (2009) Seeking for simplicity in complex networks. Europhysics Letters 85: 48001.
* 39. Gross T, Blasius B (2008) Adaptive coevolutionary networks: a review. Journal of the Royal Society, Interface 5: 259–71.
* 40. Saavedra S, Reed-Tsochas F, Uzzi B (2008) Asymmetric disassembly and robustness in declining networks. Proceedings of the National Academy of Sciences of the United States of America 105: 16466–71.
* 41. Costa LDF, Rodrigues FA, Hilgetag CC, Kaiser M (2009) Beyond the average: Detecting global singular nodes from local features in complex networks. Europhysics Letters 87: 18008.
* 42. Ramasco JJ, Mungan M (2008) Inversion method for content-based networks. Physical Review E 77: 036122.
* 43. Costa LDF, Kaiser M, Hilgetag CC (2007) Predicting the connectivity of primate cortical networks from topological and spatial node properties. BMC Systems Biology 1: 16.
* 44. Milo R, Shen-Orr S, Itzkovitz S, Kashtan N, Chklovskii D, et al. (2002) Network motifs: simple building blocks of complex networks. Science 298: 824–7.
* 45. Johnson RA, Wichern DW (2007) Applied multivariate statistical analysis. Prentice Hall Englewood Cliffs, NJ, USA, 6th edition, 800 pp.
* 46. Parzen E (1962) On Estimation of a Probability Density Function and Mode. The annals of mathematical statistics 33: 1065–76.
* 47. Duda RO, Hart PE, Stork DG (2001) Pattern classification. Wiley Interscience, 2nd edition, 680 pp.
* 48. Wasserman S, Faust K (1994) Social Network Analysis: Methods and Applications (Structural Analysis in the Social Sciences). Cambridge University Press.
* 49. Kuramochi M, Karypis G (2001) Frequent Subgraph Discovery. In: Proceedings of the 2001 IEEE International Conference on Data Mining. IEEE Computer Society, pp. 313–20.
* 50. Kashtan N, Itzkovitz S, Milo R, Alon U (2004) Efficient sampling algorithm for estimating subgraph concentrations and detecting network motifs. Bioinformatics 20: 1746–58.
* 51. Middendorf M, Ziv E, Wiggins CH (2005) Inferring network mechanisms: the Drosophila melanogaster protein interaction network. Proceedings of the National Academy of Sciences of the United States of America 102: 3192–7.
* 52. Bordino I, Donato D, Gionis A, Leonardi S (2008) Mining Large Networks with Subgraph Counting. In: 8th IEEE International Conference on Data Mining (ICDM). IEEE.
* 53. Ribeiro P, Silva F, Kaiser M (2009) Strategies for Network Motifs Discovery. In: 5th IEEE International Conference on e-Science. IEEE, pp. 80–7.
* 54. Groening M (2005) The Simpsons Uncensored Family Album. HarperCollins Entertainment.
* 55. Erdös P, Rényi A (1959) On Random Graphs I. Publ Math (Debrecen) 6: 290–7.
* 56. Barabási AL, Albert R (1999) Emergence of Scaling in Random Networks. Science 286: 509–12.
* 57. Kaiser M, Goerner M, Hilgetag CC (2007) Criticality of spreading dynamics in hierarchical cluster networks without inhibition. New Journal of Physics 9: 110.
* 58. Costa LDF, Silva FN (2006) Hierarchical Characterization of Complex Networks. Journal of Statistical Physics 125: 841–76.
* 59. Milgram S (1967) The small world problem. Psychology today 2: 60–7.
* 60. Kochen M, editor (1989) The Small World. Ablex Publishing, Norwood, NJ.
* 61. Guare J (1994) Six degrees of separation: a play. Vintage Books, 2 edition.
* 62. Eaton JW (2002) GNU Octave Manual. Limited, Network Theory.
* 63. Milo R, Itzkovitz S, Kashtan N, Levitt R, Shen-Orr S, et al. (2004) Superfamilies of evolved and designed networks. Science 303: 1538–42.
* 64. Kaiser M, Hilgetag CC (2004) Edge vulnerability in neural and metabolic networks. Biological cybernetics 90: 311–7.
* 65. MacKay DJ (2003) Information theory, inference, and learning algorithms. Cambridge University Press, 1st edition.
* 66. Jain AK, Murty MN, Flynn PJ (1999) Data clustering: a review. ACM Computing Surveys 31: 264–323.
* 67. Arthur D, Vassilvitskii S (2007) k-means++: The advantages of careful seeding. In: Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms. Society for Industrial and Applied Mathematics, pp. 1027–35.
* 68. Ozik J, Hunt BR, Ott E (2004) Growing networks with geographical attachment preference: Emergence of small worlds. Physical Review E 69: 026108(5).
* 69. Mahalanobis PC (1936) On the generalized distance in statistics. Proceedings of the National Institute of Science of India 2: 49–55.
## Figures
Figure 1: Analysis work-flow to identify global singular nodes from local
features [41]. Step 1: Choose set of local measures to characterise network
nodes [35]. Calculate local measurements for all nodes in the network (feature
vectors). Step 2: Map each node’s feature vector to lower dimensional space
using principal component analysis (PCA plane) [45, Chapter 8]. Step 3:
Estimate each node’s probability using the Parzen window approach (PDF) [46,
47, Chapter 4.3]. Step 4: Query PDF to identify least probable nodes (singular
nodes). Step 5: Cluster singular nodes in PCA plane using k-means (motif
groups) [65, Chapter 20.1]. Step 6: Determine Voronoi cells for grouped nodes
using a modified Mahalanobis distance (potential motif regions) [69]. Step 7:
Join potential motif regions that are close to each other (motif regions).
Step 8: Calculate relative frequencies of nodes falling into motif-regions
(A–F) or non-motif region (NO) (fingerprint).
Figure 2: Network types used for testing BtA: a Network derived from The
Simpsons family-tree [54]. Nodes in very regular parts of the network were
identified singular (shaded grey) because of two characteristics: Their
neighbours’ degrees are comparatively low and show no variation (values $r$
and $cv$ significantly below average). b Schematic of large regular ring
lattice combined with a minor ER-random component (shaded grey). c A small
regular structure (white nodes) embedded into a large random network (ER, BA,
or WS model).
Figure 3: Single node-motifs in emerging small-world network (Fig. S2).
Vertical axes in subfigures a–c correspond to number of outlier nodes $w$,
number of single node-motifs $k$, and their frequencies, respectively. a
Number of identified outliers $w$ rising from 0 to 54. b Diversity of node-
motifs $k$ quickly rising during the 1st re-wiring round; less increase during
2nd round; and stable during the 3rd. c Proportions of nodes expressing
identified motifs (motif frequencies). Nodes classified regular not shown. d
Schematics of identified single node-motifs and their distinguishing
characteristics.
Figure 4: Example of 2 clusters (left, right) with 3 points each (1–3, 4–6).
Ellipses are centred on each point with dimensions corresponding to standard
deviations $\sigma$ along PC-axes. A set of points is considered a clique, if
the area of all their ellipses is connected (e.g. {1}, {1, 2}, or {1, 2, 3};
but not $\\{2,3\\}$). A maximal clique is called a cluster (i.e. {1,2,3} or
{4,5,6}) and is used to define a distinct motif-group.
|
arxiv-papers
| 2011-02-01T08:46:22 |
2024-09-04T02:49:16.776026
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Christoph Echtermeyer, Luciano da Fontoura Costa, Francisco A.\n Rodrigues, Marcus Kaiser",
"submitter": "Christoph Echtermeyer",
"url": "https://arxiv.org/abs/1102.0099"
}
|
1102.0128
|
# Adiabatic Conditions and the Uncertainty Relation
Qian-Heng Duan, Ping-Xing Chen pxchen@nudt.edu.cn Wei Wu Department of
Physics, National University of Defense Technology, Changsha, 410073, China
pxchen@nudt.edu.cn Department of Physics, National University of Defense
Technology, Changsha 410073, P. R. China
###### Abstract
The condition for adiabatic approximation are of basic importance for the
applications of the adiabatic theorem. The traditional quantitative condition
was found to be necessary but not sufficient, but we do not know its physical
meaning and the reason why it is necessary from the physical point of view. In
this work, we relate the adiabatic theorem to the uncertainty relation, and
present a clear physical picture of the traditional quantitative condition. It
is shown that the quantitative condition is just the amplitude of the
probability of transition between two levels in the time interval which is of
the order of the time uncertainty of the system. We also present a new
sufficient condition with clear physical picture.
###### pacs:
03.65.Ta, 03.65.Ca, 03.67.Lx
The adiabatic theorem s1 ; s2 is one of the basic results in quantum theory
and has applications in many fields, for example, in quantum field theory s3 ,
geometric phase s4 as well as in quantum control and adiabatic quantum
computation s5 . As described in many publications, the traditional adiabatic
theorem s6 ; s7 states that if a quantum system with a time-dependent
Hamiltonian $\hat{H}(\mathbf{t})$ is initially in the $n\\_th$ instantaneous
eigenstate of $\hat{H}(0)$, $\hat{H}(\mathbf{t})$ evolves slowly enough and
the energy levels don’t cross in the evolution process, then the state of the
system will stay at the $n\\_th$ instantaneous eigenstate of
$\hat{H}(\mathbf{t})$ up to a phase factor at a later time. But the
application of the theorem depends on the criterion of the “slowness”.
Usually, the “slowness” is described as follows s8
$\left|{\frac{{\left\langle{E_{n}\left(t\right)}\mathrel{\left|{\vphantom{{E_{n}\left(t\right)}{\dot{E}_{m}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{m}\left(t\right)}}\right\rangle}}{{E_{m}\left(t\right)-E_{n}\left(t\right)}}}\right|\ll
1,m\neq n,t\in\left[{0,T}\right]$ (1)
where $E_{n}\left(t\right)$ and $\left|{E_{n}\left(t\right)}\right\rangle$ are
the instantaneous eigenvalues and eigenstates of $\hat{H}\left(t\right)$, and
$T$ is the total evolution time.
In recent years, many doubts have been raised in the traditional criterion s8
; s9 ; s10 ; s11 ; s12 . It was first shown by Marzlin and Sanders s8 and
then by Tong et al s9 that if two systems which we call system $S^{A}$ and
$S^{B}$ are related though
$\hat{H}^{B}\left(t\right)=-\hat{U}^{A+}\left(t\right)\hat{H}^{A}\left(t\right)\hat{U}^{A}\left(t\right)$
(2)
The two systems can’t have an adiabatic evolution at the same time unless
$\left|{\left\langle{E_{n}^{A}\left(t\right)}\mathrel{\left|{\vphantom{{E_{n}^{A}\left(t\right)}{E_{n}^{A}\left(0\right)}}}\right.\kern-1.2pt}{{E_{n}^{A}\left(0\right)}}\right\rangle}\right|\approx
1$, even if both of the system satisfy condition (1). Many authors
investigated the reasons of the insufficiency s18 ; s19 ; s20 ; s22 .
Recently, Amin pointed out that the violations of the traditional criterion
all arise from resonant transitions between energy levels s11 . At the same
time, some authors proposed some new alternative criterions s13 ; s16 ; s14 ;
s15 ; s23 ; s24 ; s25 ; s26 . In 2008, Du et al experimentally examined the
traditional criterion s10 .
However, the physical pictures of the criterions proposed before are not
clear. Even though Tong proved that the traditional condition (1) is necessary
in guaranteeing the validity of the adiabatic approximation s12 , we still do
not know the reason why it is necessary from the physical point of view. It is
foundmentally important to find a new condition with clear physics picture or
probe the physical meaning of the existed conditions. In this letter, we
relate the adiabatic condition to the uncertainty relation. We first propose a
new sufficient condition for adiabatic process, and then give clear physical
pictures of the new condition and the necessary condition (1) in terms of the
uncertainty relation. It is shown that the state of a system cannot be
appreciably modified by an evolution until a least evolution time has elapsed,
and
$\left|{\frac{{\left\langle{E_{n}\left(t\right)}\mathrel{\left|{\vphantom{{E_{n}\left(t\right)}{\dot{E}_{m}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{m}\left(t\right)}}\right\rangle}}{{E_{n}\left(t\right)-E_{m}\left(t\right)}}}\right|$
in Eq. (1) is just the amplitude of the probability of the transition between
$\left|{E_{n}\left(t\right)}\right\rangle$ and
$\left|{E_{m}\left(t\right)}\right\rangle$ in the least evolution time. The
least evolution time is of the order of the time-uncertainty of the system.
In an adiabatic process, if the system is initially in the $n\\_th$
instantaneous eigenstate ${\left|{E_{n}\left(0\right)}\right\rangle}$, then at
the end of the adiabatic evolution process the state
$\left|{\psi\left(t\right)}\right\rangle$ fulfills
$\left|{\left\langle{E_{n}\left(T\right)}\mathrel{\left|{\vphantom{{E_{n}\left(T\right)}{\psi\left(T\right)}}}\right.\kern-1.2pt}{{\psi\left(T\right)}}\right\rangle}\right|^{2}\approx
1.$ (3)
Let
$\left|{\psi\left(t\right)}\right\rangle=\sum\limits_{m}{a_{m}\left(t\right)\left|{E_{m}\left(t\right)}\right\rangle,}$
(4)
where $a_{m}\left(t\right)$ is a complex number. Substituting the above
equation into the Schrödinger equation, we obtain
$\frac{d}{{dt}}a_{n}\left(t\right)=-\sum\limits_{m}{a_{m}\left(t\right)\left\langle{E_{n}\left(t\right)}\mathrel{\left|{\vphantom{{E_{n}\left(t\right)}{\dot{E}_{m}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{m}\left(t\right)}}\right\rangle}-\frac{i}{\hbar}a_{n}\left(t\right)E_{n}\left(t\right)$
(5)
and
$\frac{d}{{dt}}a_{n}^{\ast}\left(t\right)=-\sum\limits_{m}{a_{m}^{\ast}\left(t\right)\left\langle{\dot{E}_{m}\left(t\right)}\mathrel{\left|{\vphantom{{\dot{E}_{m}\left(t\right)}{E_{n}\left(t\right)}}}\right.\kern-1.2pt}{{E_{n}\left(t\right)}}\right\rangle}+\frac{i}{\hbar}a_{n}^{\ast}\left(t\right)E_{n}\left(t\right)$
(6)
Using equations (5) and (6), and denoting
$P_{n}(t)={a_{n}\left(t\right)a_{n}^{\ast}\left(t\right)}$, we have
$\displaystyle\frac{d}{{dt}}P_{n}\left(t\right)$ $\displaystyle=$
$\displaystyle
a_{n}\left(t\right)\frac{d}{{dt}}a_{n}^{\ast}\left(t\right)+a_{n}^{\ast}\left(t\right)\frac{d}{{dt}}a_{n}\left(t\right)$
(7) $\displaystyle=$
$\displaystyle-\sum\limits_{m}{a_{n}\left(t\right)a_{m}^{\ast}\left(t\right)\left\langle{\dot{E}_{m}\left(t\right)}\mathrel{\left|{\vphantom{{\dot{E}_{m}\left(t\right)}{E_{n}\left(t\right)}}}\right.\kern-1.2pt}{{E_{n}\left(t\right)}}\right\rangle}$
$\displaystyle-\sum\limits_{m}{a_{n}^{\ast}\left(t\right)a_{m}\left(t\right)\left\langle{E_{n}\left(t\right)}\mathrel{\left|{\vphantom{{E_{n}\left(t\right)}{\dot{E}_{m}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{m}\left(t\right)}}\right\rangle}$
$\displaystyle=$
$\displaystyle-2\sum\limits_{m}\mathit{{Re}\left\\{{a_{n}^{\ast}\left(t\right)a_{m}\left(t\right)\chi_{nm}}\right\\}}$
where
$\chi_{nm}=\left\langle{E_{n}\left(t\right)}\mathrel{\left|{\vphantom{{E_{n}\left(t\right)}{\dot{E}_{m}\left(t\right)}}}\right.\kern-1.2pt}{\dot{E}_{m}\left(t\right)}\right\rangle,$
and we use a gauge in which $\chi_{nn}=0$. Integrating equation (7), we get
$\displaystyle P_{n}\left(T\right)$ $\displaystyle=$ $\displaystyle
1-2\sum\limits_{m}{\int_{0}^{T}\mathit{{Re}{\left({a_{n}\left(t\right)a_{m}^{\ast}\left(t\right)\chi_{nm}}\right)dt}}}$
(8) $\displaystyle\geq$ $\displaystyle 1-2\sum\limits_{m\neq
n}{\int_{0}^{T}{\left|{\chi_{nm}}\right|dt}}$ $\displaystyle\geq$
$\displaystyle 1-2\sum\limits_{m\neq
n}{T\max\left\\{{\left|{\chi_{nm}}\right|}\right\\}}$
When the dimension of the system is finite, if we have
$2T\max\left\\{{\left|{\left\langle{E_{m}\left(t\right)}\mathrel{\left|{\vphantom{{E_{m}\left(t\right)}{\dot{E}_{n}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{n}\left(t\right)}}\right\rangle}\right|}\right\\}\ll
1,$ (9)
the sum in the equation (8) can always be a small number so that
$P_{n}\left(T\right)\approx 1,$ which means that condition (9) is a sufficient
condition for adiabatic theorem.
Condition (9) can sufficiently guarantee the validity of the adiabatic
approximation, but we do not understand its physical meaning clearly, just as
we do with the necessary condition (1). Especially, condition (9) means
seemingly that only if $T$ is small enough and
$\max\left\\{{\left|{\left\langle{E_{m}\left(t\right)}\mathrel{\left|{\vphantom{{E_{m}\left(t\right)}{\dot{E}_{n}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{n}\left(t\right)}}\right\rangle}\right|}\right\\}$
is finite, it can always be fulfilled and the adiabatic approximation can be
guaranteed. This conflicts seemingly with condition (1) in which the time $T$
seems be not involved. How to attemper this conflict? Let’s go to the central
purpose of this letter, we will present the clear physical pictures of
conditions (1) and (9). From these pictures conditions (1) and (9) are
consistent. Interestingly, the uncertainty relation plays a key role here.
We first show that the evolution time must be more than the least evolution
time to get an obvious state change, and the least evolution time is in the
order of the time uncertainty of the system.
For simplicity, we consider a two level system. The Hamiltonian
$\hat{H}\left(t\right)$ has two eigenstates
$\left|{E_{k}\left(t\right)}\right\rangle$ and
$\left|{E_{n}\left(t\right)}\right\rangle$ which satisfy the following
equation
$\hat{H}\left(t\right)\left|{E_{n,k}\left(t\right)}\right\rangle=E_{n,k}\left(t\right)\left|{E_{n,k}\left(t\right)}\right\rangle.$
(10)
The state of the system at time $t,$
$\left|{\psi\left(t\right)}\right\rangle,$ can be expanded as
$\left|{\psi\left(t\right)}\right\rangle=\sum\limits_{n}{a_{n}\left(t\right)e^{i\beta_{n}\left(t\right)}\left|{E_{n}\left(t\right)}\right\rangle}$
(11)
where $a_{n}\left(t\right)$ and $\beta_{n}\left(t\right)$ are real, and the
phase $\beta_{n}\left(t\right)$ can be expressed as s17
$\beta_{n}\left(t\right)\mathrm{\
=}-\frac{1}{\hbar}\int_{0}^{t}{E_{n}\left({t^{\prime}}\right)}dt^{\prime}+i\int_{0}^{t}{\left\langle{{E_{n}\left({t^{\prime}}\right)}}\mathrel{\left|{\vphantom{{E_{n}\left({t^{\prime}}\right)}{\dot{E}_{n}\left({t^{\prime}}\right)}}}\right.\kern-1.2pt}{{\dot{E}_{n}\left({t^{\prime}}\right)}}\right\rangle
dt^{\prime}.}$ (12)
Substituting equations (11) and (12) into the Schrödinger equation, we obtain
$\frac{{da_{k}\left(t\right)}}{{dt}}=-{a_{n}\left(t\right)e^{i\beta_{nk}\left(t\right)}\left\langle{E_{k}\left(t\right)}\mathrel{\left|{\vphantom{{E_{k}\left(t\right)}{\dot{E}_{n}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{n}\left(t\right)}}\right\rangle}$
(13)
where
$\beta_{nk}\left(t\right)=\beta_{n}\left(t\right)-\beta_{k}\left(t\right)$.
Let us consider two systems $S^{A}$ and $S^{B}$, the Hamiltonian of which are
related though Eq. (2) as shown in s8 ; s9 . The instantaneous eigenvalues and
eigenstates of the two system satisfy s9
$\begin{array}[]{l}E_{{}_{n}}^{B}\left(t\right)=-E_{{}_{n}}^{A}\left(t\right)\\\
\left|{E_{{}_{n}}^{B}\left(t\right)}\right\rangle=\hat{U}^{A+}\left(t\right)\left|{E_{{}_{n}}^{A}\left(t\right)}\right\rangle\end{array}$
(14)
and their evolution operator
$\hat{U}^{B}\left(t\right)=\hat{U}^{A+}\left(t\right).$ (15)
From Eqs. (14) and (15) we have
$\left\langle{{E_{k}^{B}\left(t\right)}}\mathrel{\left|{\vphantom{{E_{k}^{B}\left(t\right)}{\dot{E}_{n}^{B}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{n}^{B}\left(t\right)}}\right\rangle=\frac{i}{\hbar}E_{n}^{A}\left(t\right)\delta_{nk}+\left\langle{{E_{k}^{A}\left(t\right)}}\mathrel{\left|{\vphantom{{E_{k}^{A}\left(t\right)}{\dot{E}_{n}^{A}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{n}^{A}\left(t\right)}}\right\rangle$
(16)
Since $a_{k}\left(t\right)$ is real, from Eqs. (13) and (16) we can get that
$\beta_{nk}^{A}\left(t\right)+\widetilde{\omega}_{nk}^{A}=q^{A}\pi;$
$\beta_{nk}^{B}\left(t\right)+\widetilde{\omega}_{nk}^{B}=q^{B}\pi,$ and then
$\beta_{{}_{nk}}^{B}\left(t\right)=\beta_{{}_{nk}}^{A}\left(t\right)+q\pi$
(17)
where $q^{A},q^{B},q$ are integer, and
$\widetilde{\omega}_{nk}^{A}=\widetilde{\omega}_{nk}^{B}$ are the phases of
${\left\langle{E_{k}^{A}\left(t\right)}\mathrel{\left|{\vphantom{{E_{k}\left(t\right)}{\dot{E}_{n}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{n}^{A}\left(t\right)}}\right\rangle}$
and
${\left\langle{E_{k}^{B}\left(t\right)}\mathrel{\left|{\vphantom{{E_{k}\left(t\right)}{\dot{E}_{n}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{n}^{B}\left(t\right)}}\right\rangle}$.
From Eqs. (12), (14) and (16), we obtain
$\displaystyle\beta_{{}_{nk}}^{B}\left(t\right)=-\frac{1}{\hbar}\int_{0}^{t}{\left({E_{{}_{n}}^{B}-E_{{}_{k}}^{B}}\right)dt^{\prime}}$
(18)
$\displaystyle+i\int_{0}^{t}{\left\\{{\left\langle{E_{n}^{B}}\mathrel{\left|{\vphantom{{E_{n}^{B}}{\dot{E}_{n}^{B}}}}\right.\kern-1.2pt}{{\dot{E}_{n}^{B}}}\right\rangle-\left\langle{E_{k}^{B}}\mathrel{\left|{\vphantom{{E_{k}^{B}}{\dot{E}_{k}^{B}}}}\right.\kern-1.2pt}{{\dot{E}_{k}^{B}}}\right\rangle}\right\\}dt^{\prime}}$
$\displaystyle=$
$\displaystyle\beta_{{}_{nk}}^{A}\left(t\right)+\frac{1}{\hbar}\int_{0}^{t}{\left({E_{{}_{n}}^{A}-E_{{}_{k}}^{A}}\right)dt^{{}^{\prime}}.}$
By Eqs. (17) and (18), we get
$\frac{1}{\hbar}\int_{0}^{t}{\left({E_{{}_{n}}^{A}-E_{{}_{k}}^{A}}\right)dt^{\prime}}=q\pi.$
(19)
Eq. (19) is very interesting since it shows the relation between the evolution
time and the instantaneous eigenvalues of the Hamiltonian. For any arbitrary
system $S^{A}$ one can always find a corresponding system $S^{B}$ satisfying
Eq. (2), so Eq. (19) is only a result of the Schrödinger equation. If we
denote $\overline{\bigtriangleup
E_{nk}}\equiv\frac{1}{t}\int_{0}^{t}{\left({E_{{}_{n}}^{A}-E_{{}_{k}}^{A}}\right)dt^{{}^{\prime}}}$
as the average of the ${E_{{}_{n}}^{A}-E_{{}_{k}}^{A}}$ in the time interval
$[0,t],$ Eq. (19) can be expressed as
$t=\frac{q\pi\hbar}{\overline{\bigtriangleup E_{nk}}}.$ (20)
Eq. (20) means the least evolution time is
$\frac{\pi\hbar}{\overline{\bigtriangleup E_{nk}}}$(i.e.,$q=1$). Furthermore,
if we regard $\overline{\bigtriangleup E_{nk}}$ as the energy uncertainty of
the system, according to the uncertainty relation $\overline{\bigtriangleup
E_{nk}}t\sim h$, the time uncertainty is $\frac{h}{\overline{\bigtriangleup
E_{nk}}}$ which is in the order of the leat evolution time. In fact, if the
system undergoes a quantum transition between
$\left|{E_{n}\left(t\right)}\right\rangle$ and
$\left|{E_{k}\left(t\right)}\right\rangle$ by the evolution according to the
Schrödinger equation, the energy of the system has uncertainty of
$E_{k}\left(t\right)-E_{n}\left(t\right)$ This can be explained as follows.
Suppose the system is in the state
$\left|{E_{n}\left(t^{\prime}\right)}\right\rangle$ in the time $t^{\prime}$,
after a evolution from $t^{\prime}$ to $t$ the system’s state becomes
$\left|{\psi\left(t\right)}\right\rangle\ $which is a superposition of the
instantaneous eigenstates $\left|{E_{n}\left(t\right)}\right\rangle$ and
$\left|{E_{k}\left(t\right)}\right\rangle$(in this case, there is a quantum
transition between $\left|{E_{n}\left(t\right)}\right\rangle$ and
$\left|{E_{k}\left(t\right)}\right\rangle).$ According to quantum mechanics
theory, when the system is in the superposition state
$\left|{\psi\left(t\right)}\right\rangle$ one cannot distinguish whether the
system is in the state $\left|{E_{n}\left(t\right)}\right\rangle$ or
$\left|{E_{k}\left(t\right)}\right\rangle.$ So we can say the system has
energy uncertainty ${E_{k}\left(t\right)}-{E_{n}\left(t\right)}.$ Owing to the
uncertainty relation the corresponding time-uncertainty is
$\frac{1}{{E_{k}\left(t\right)}-{E_{n}\left(t\right)}}$ (We let $h=1$).
How to understand that the least evolution time is the order of time-
uncertainty? We can say that any evolution in the time much less than the
time-uncertainty $\frac{1}{{E_{k}\left(t\right)}-{E_{n}\left(t\right)}}$ will
be negligible, namely, the evolution time must not be much less than
$\frac{1}{{E_{k}\left(t\right)}-{E_{n}\left(t\right)}}$ to produce an
effective evolution. Otherwise, we can determinate time parameter with
precision more than the time-uncertainty by distinguishing the difference
between the states before and after the effective evolution sc , which
violates the uncertainty relation.
A similar conclusion can also be reached from a different point of view s7 .
Let $\left|{\psi\left(0\right)}\right\rangle$ and
$\left|{\psi\left(t\right)}\right\rangle=u(t)\left|{\psi\left(0\right)}\right\rangle$
denote the initial state and the state at time $t$ of the system, where $u(t)$
is the evolution operator. The expansion of the $u(t)$ is
$\displaystyle u(t)$ $\displaystyle=$ $\displaystyle
1-i\int_{0}^{t}{H(t}_{1}{)dt}_{1}$ (21)
$\displaystyle+\frac{(-i)^{2}}{2}\int_{0}^{t}{{dt}_{1}}\int_{0}^{t_{1}}{dt}_{2}{H(t}_{1}{)H(t}_{2}{)+\cdots.}$
Since $t$ is small, we can keep only the first order approximation. let
$\overline{H}\equiv\frac{1}{t}\int_{0}^{t}{H(t}_{1}{)dt}_{1},$ then at time
$t$ the probability $p$ of finding the system not being in the initial state
$\left|{\psi\left(0\right)}\right\rangle$ is
$\displaystyle p$ $\displaystyle=$
$\displaystyle\left\langle{\psi\left(0\right)}\right|u(t)^{+}[I-\left|{\psi\left(0\right)}\right\rangle\left\langle{\psi\left(0\right)]}\right|u(t)\left|{\psi\left(0\right)}\right\rangle$
(22) $\displaystyle\approx$
$\displaystyle\left\langle{\psi\left(0\right)}\right|(1+it\overline{H})[I-\left|{\psi\left(0\right)}\right\rangle\left\langle{\psi\left(0\right)]}\right|(1-it\overline{H})\left|{\psi\left(0\right)}\right\rangle$
$\displaystyle=$
$\displaystyle\left\langle{\psi\left(0\right)}\right|\overline{H}^{2}\left|{\psi\left(0\right)}\right\rangle
t^{2}-\left\langle{\psi\left(0\right)}\right|\overline{H}\left|{\psi\left(0\right)}\right\rangle^{2}t^{2}$
$\displaystyle\equiv$ $\displaystyle(\bigtriangleup\overline{H})^{2}t^{2},$
where $\bigtriangleup\overline{H}$, the root mean square deviation of the
energy, is the average uncertainty of the energy of the system in the time
interval $[0,t]$, its inversion $\frac{1}{\bigtriangleup\overline{H}}$ is the
uncertainty of the time. If evolution time
$t\ll\frac{1}{\bigtriangleup\overline{H}},$ then $p\ll 1.$ Namely, if the
evolution time is much less than the time-uncertainty, the system will stay in
the initial state.
From the discussion above, we can conclude that any system has the least
effective evolution time (LEET) which is the order of time-uncertainty.
$E_{k}\left(t\right)-E_{n}\left(t\right)$ can be regarded as the energy-
uncertainty when the system undergoes a transition between the two states
$\left|{E_{n}\left(t\right)}\right\rangle$ and
$\left|{E_{k}\left(t\right)}\right\rangle.$ So the time
$\frac{1}{{E_{k}\left(t\right)}-{E_{n}\left(t\right)}}$ can be regarded
roughly as the least effective evolution time which we denote as $T_{LEET}$.
With those in mind, we can discuss the physical pictures of conditions (1) and
(9) easily.
By the basic meaning of the inner product of two vectors in a Hilbert space,
we know that
$\left\langle{{E_{n}\left(t\right)}}\mathrel{\left|{\vphantom{{E_{n}\left(t\right)}{\dot{E}_{m}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{m}\left(t\right)}}\right\rangle$
is proportional to the amplitude of the probability of the transition from
$\left|{E_{m}\left(t\right)}\right\rangle$ to
$\left|{E_{n}\left(t\right)}\right\rangle$ in an unit time interval. By
equation (8) we know
${\int_{0}^{T}\mathit{{Re}{\left({a_{n}\left(t\right)a_{m}^{\ast}\left(t\right)}\left\langle{{E_{n}\left(t\right)}}\mathrel{\left|{\vphantom{{E_{n}\left(t\right)}{\dot{E}_{m}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{m}\left(t\right)}}\right\rangle\right)dt}}}$
is proportional to the probability of the transition from
$\left|{E_{n}\left(t\right)}\right\rangle$ to
$\left|{E_{m}\left(t\right)}\right\rangle)$ in the time interval $[0,T].$ And
then
$2T\max\left\\{{\left|{\left\langle{E_{m}\left(t\right)}\mathrel{\left|{\vphantom{{E_{m}\left(t\right)}{\dot{E}_{n}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{n}\left(t\right)}}\right\rangle}\right|}\right\\}$
is the maximal probability of the transition from
$\left|{E_{n}\left(t\right)}\right\rangle$ to
$\left|{E_{m}\left(t\right)}\right\rangle$ in the time interval $[0,T].$
Condition (9) means just that the transition between
$\left|{E_{n}\left(t\right)}\right\rangle$ and
$\left|{E_{m}\left(t\right)}\right\rangle$ is very small and can be neglected
in the whole time interval $[0,T]$. So it is sufficient to assure adiabatic
process.
In condition (1),
$\frac{\left\langle{{E_{n}\left(t\right)}}\mathrel{\left|{\vphantom{{E_{n}\left(t\right)}{\dot{E}_{m}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{m}\left(t\right)}}\right\rangle}{{E_{m}\left(t\right)}-{E_{n}\left(t\right)}}$
is nothing else but the amplitude of the average probability of the transition
between $\left|{E_{n}\left(t\right)}\right\rangle$ and
$\left|{E_{m}\left(t\right)}\right\rangle$ in one LEET. The condition
$\frac{\left\langle{{E_{n}\left(t\right)}}\mathrel{\left|{\vphantom{{E_{n}\left(t\right)}{\dot{E}_{m}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{m}\left(t\right)}}\right\rangle}{{E_{m}\left(t\right)}-{E_{n}\left(t\right)}}<<1$
for each LEET in the whole time interval $[0,T]$ is necessary for adiabatic
process, otherwise, it is possible for the system has an obvious transition
between $\left|{E_{n}\left(t\right)}\right\rangle$ and
$\left|{E_{m}\left(t\right)}\right\rangle$ in a LEET.
To make the pictures of the necessary condition (1) and the sufficient
condition (9) more clear we discuss when the necessary condition becomes
sufficient, we investigate the effect of the phases of ${a_{n}\left(t\right)\
}$and ${\chi_{nm}(t).}$ Let
${a_{n}\left(t\right)=}\left|{a_{n}\left(t\right)}\right|e^{-i\int_{0}^{t}{E_{n}\left(t^{\prime}\right)dt}^{\prime}};$
(23) ${\chi_{nm}(t)=}\left|{\chi_{nm}(t)}\right|{e}^{i\omega(t)dt},$ (24)
from Eq. (8), the probability of the transition from the
$\left|{E_{n}\left(t\right)}\right\rangle$ to
$\left|{E_{m}\left(t\right)}\right\rangle$ is proportional to $\epsilon_{nm}.$
$\displaystyle\epsilon_{nm}$ $\displaystyle\equiv$
$\displaystyle{\int_{0}^{T}\mathit{{Re}{\left({a_{n}\left(t\right)a_{m}^{\ast}\left(t\right)\chi_{nm}}\right)dt}}}$
$\displaystyle=$
$\displaystyle{\int_{0}^{T}\mathit{{Re}{\left(\left|{a_{n}\left(t\right)}\right|\left|{a_{m}^{\ast}\left(t\right)}\right|{e}^{-i\omega_{nm}(t)t}\left|{\chi_{nm}}\right|{e}^{i\omega(t)t}\right)dt}}}$
$\displaystyle=$
$\displaystyle{\int_{0}^{T}{\left|{a_{n}\left(t\right)}\right|\left|{a_{m}^{\ast}\left(t\right)}\right|\left|{\chi_{nm}}\right|\cos((\omega(t)-\omega_{nm}(t))t)dt}}$
where
${{\omega_{nm}(t)\equiv}}\frac{1}{t}\int_{0}^{t}{E_{n}-E_{m}\left(t^{\prime}\right)dt}^{\prime}.$
As shown in s11 in the presence of resonant oscillation, i.e.,
${{\omega(t)=\omega_{nm}(t),}}$
$\displaystyle\epsilon_{nm}$ $\displaystyle=$
$\displaystyle{\int_{0}^{T}{\left|{a_{n}\left(t\right)}\right|\left|{a_{m}^{\ast}\left(t\right)}\right|\left|{\chi_{nm}}\right|dt}}$
$\displaystyle\leq$ $\displaystyle
T{\max_{t\in[0,T]}{\left|{\chi_{nm}}\right|.}}$
Suppose that $T$ includes $M$ LEET, i.e., $M\approx\frac{T}{T_{LEET}},$ then
$\displaystyle\epsilon_{nm}$ $\displaystyle\leq$ $\displaystyle
T{\max_{t\in[0,T]}{\left|{\chi_{nm}}\right|=}}\sum_{i=1}^{M}{\max{\left|{\chi_{nm}}\right|T}}_{LEET}^{i}$
(27) $\displaystyle=$
$\displaystyle\sum_{i=1}^{M}{\max}\frac{{{\left|{\chi_{nm}}\right|}}^{i}}{{E_{m}\left(t\right)}^{i}-{E_{n}\left(t\right)}^{i}},$
where
${T}_{LEET}^{i}=\frac{{1}}{{E_{m}\left(t\right)}^{i}-{E_{n}\left(t\right)}^{i}}$
is the $i\\_th$ LEET. The conditions (1), which means
${\max}\frac{{{\left|{\chi_{nm}}\right|}}^{i}}{{E_{m}\left(t\right)}^{i}-{E_{n}\left(t\right)}^{i}}\ll
1$ for each LEET, cannot assure the error of the whole process is small since
$M$ may increase as the time $T$ does. But the condition (9) means that
$\sum_{i=1}^{M}{\max{\left|{\chi_{nm}}\right|T}}_{LEET}^{i}\ll 1$, i.e., the
error of the whole process is small, so it is sufficient.
In the absence of resonant oscillation, i.e.,
${{\omega(t)-\omega_{nm}(t)\equiv\omega}}^{\prime}\neq 0,$
$\epsilon_{nm}={\int_{0}^{T}{\left|{a_{n}\left(t\right)}\right|\left|{a_{m}^{\ast}\left(t\right)}\right|\left|{\chi_{nm}}\right|\cos{\omega}^{\prime}tdt}}$.
This means $\epsilon_{nm}$ may not increase as $T$ owing to the different sign
of ${\cos\omega}^{\prime}t$ in the different LEET. In this case, the adiabatic
opproximation holds under condition (1).
It should be noted that if the evolution time $T$ is of the order $T_{LEET}$,
the error of the whole process is small and adiabatic approximation is valid
in many cases. For example, consider a simple two-state system as used by Amin
s11 . The Hamiltonian of the system is
$H\left(t\right)=-\varepsilon\frac{{\sigma_{z}}}{2}-V\sin\left({\omega_{0}t}\right)\sigma_{x}$
(28)
and $V$ is a small positive number. The system’s exact instantaneous
eigenvalues and eigenstates are
$\displaystyle E_{0,1}$ $\displaystyle\mathrm{=}$
$\displaystyle\pm\frac{1}{2}\Omega;$ (29)
$\displaystyle\left|{E_{0,1}}\right\rangle$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{l}\alpha^{\pm}\\\
\pm\alpha^{\mp}\end{array}\right)$ (32)
where
$\Omega=\sqrt{\varepsilon^{2}+4V^{2}\sin^{2}\left({\omega_{0}t}\right)},\alpha^{\pm}=\sqrt{{\raise
3.01385pt\hbox{${\left({\Omega\pm\varepsilon}\right)}$}\\!\mathord{\left/{\vphantom{{\left({\Omega\pm\varepsilon}\right)}{2\Omega}}}\right.\kern-1.2pt}\\!\lower
3.01385pt\hbox{${2\Omega}$}}}.$ If $\varepsilon\approx\omega_{0}$, and the
system starts at its ground state, then at time $T$, the probability of the
system ends at the ground state is
$P_{0}\left(t\right)=\left|{\left\langle{{E_{0}\left(t\right)}}\mathrel{\left|{\vphantom{{E_{0}\left(t\right)}{\psi\left(t\right)}}}\right.\kern-1.2pt}{{\psi\left(t\right)}}\right\rangle}\right|^{2}\approx\frac{{\left({\cos
Vt+1}\right)}}{2}.$ (33)
$E_{0}-E_{1}=\Omega\approx\varepsilon\approx\omega_{0}$, so the
$T_{LEET}\approx\frac{1}{\omega_{0}}.$ If the evolution time $T$ is of order
$\frac{1}{\omega_{0}},$ then $VT\ll 1$ and
$P_{0}\left(t\right)\approx\frac{{\left({\cos Vt+1}\right)}}{2}\approx 1.$
That means adiabatic approximatoin is valid even in the presence of fast
driven oscillations.
In conclusion, we have shown that the evolution time must not be much less
than a lower bound which is in the order of the time uncertainty of the system
to get an obvious change of the state of the system. The quantitative
condition has a clear physical picture: the amplitude of the probability of
transition between two levels in each of the least evolution time is small. We
also present a new sufficient condition with clear physical meaning. Our
results are helpful to clarify the physical images of the some existing
conditions for adiabatic approximation and remove the previous doubts on the
quantitative condition. A possible interesting topic in the further is: what
is the role of the uncertainty relation in the evolution of a quantum system.
We thank Prof. Chengzu Li for helpful discussions. P.-X Chen is very grateful
for friendly help of Prof. Ian Walmsley, Dr. Lijian Zhang and the other
members in the Walmsley’s group when he visited in physics department of
Oxford university. This work was supported by NSFC (no:10774192) and FANEDD in
China (no 200524).
## References
* (1) P. Ehrenfest, Ann. Phys. 51, 327 (1916).
* (2) M. Born, V. Fock, Z. Phys. 51, 165 (1928).
* (3) M. Gell-Mann and F. Low, Phys. Rev. 84, 350 (1951).
* (4) M. V. Berry, Proc. R. Soc. London, Ser. A 392, 45 (1984).
* (5) E. Farhi, et al., Science 292, 472 (2001).
* (6) T. Kato, J. Phys. Soc. Jpn. 5, 435 (1950).
* (7) A. Messiah, Quantum Mechanics $\left(Dover\right)$, New York, (1999).
* (8) K. P. Marzlin, B. C. Sanders, Phys. Rev. Lett. 93, 160408 (2004).
* (9) D. M. Tong, et al., Phys. Rev. Lett. 95, 110407 (2005).
* (10) J. Du, L. Hu, Y. Wang, J. Wu, M. Zhao, and D. Suter, Phys. Rev. Lett. 101, 060403 (2008).
* (11) M.H.S Amin, Phys. Rev. Lett. 102, 220401 (2009).
* (12) D. M. Tong, Phys. Rev. Lett. 104, 120401 (2010).
* (13) J. Ma, Y.P. Zhang, E. G. Wang and B. Wu, Phys. Rev. Lett. 97, 128902 (2006).
* (14) S. Duki, H. Mathur and O. Narayan, Phys. Rev. Lett. 97, 128901 (2006).
* (15) M. Y. Ye, X. F. Zhou, Y. S. Zhang and G. C. Guo, Phys. Lett. A. 368, 18 (2007).
* (16) Y. Zhao, Phys. Rev. A. 77, 032109 (2008).
* (17) R. MacKenzie, E. Marcotte, and H. Paquette, Phys. Rev. A 73, 042104 (2006); R. MacKenzie et al, Phys. Rev. A. 76, 044102 (2007).
* (18) D. M. Tong, K. Singh, L. C. Kwek, and C. H. Oh, Phys. Rev. Lett. 98, 150402 (2007).
* (19) J.-D. Wu, M.-S. Zhao, J.-L. Chen, and Y. D. Zhang, Phys. Rev. A 77, 06214 (2008)
* (20) J.-L. Chen, M.-S. Zhao, J.-D. Wu, and Y. D. Zhang, arXiv:quant-ph/07060299.
* (21) Z. H. Wei and M. S. Ying, Phys. Rev. A. 76, 024304 (2007).
* (22) M. Maamache and Y. Saadi, Phys. Rev. Lett. 101, 150407 (2008).
* (23) V. I. Yukalov, Phys. Rev. A. 79, 052117 (2009).
* (24) X. L. Huang and X. X. Yi, Phys. Rev. A. 80, 032108 (2009).
* (25) J. Samuel, R. Bhandari, Phys. Rev. Lett. 60, 2339 (1988).
* (26) Samuel L. Braunstein and Carlton M. Caves, Phys. Rev. Lett. 72, 3439 (1984).
|
arxiv-papers
| 2011-02-01T11:11:31 |
2024-09-04T02:49:16.783258
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Qian-Heng Duan, Ping-Xing Chen, Wei Wu",
"submitter": "Chen Ping Xing",
"url": "https://arxiv.org/abs/1102.0128"
}
|
1102.0406
|
# Threshold Saturation on Channels with Memory via Spatial Coupling
Shrinivas Kudekar1 and Kenta Kasai2
1 New Mexico Consortium and Center for Non-linear Studies, Los Alamos National
Laboratory, NM, 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. A particular instance are
convolutional LDPC ensembles. It was recently shown that, for transmission
over the memoryless binary erasure channel, this coupling increases the belief
propagation threshold of the ensemble to the maximum a-posteriori threshold of
the underlying component ensemble. This paved the way for a new class of
capacity achieving low-density parity check codes. It was also shown
empirically that the same threshold saturation occurs when we consider
transmission over general binary input memoryless channels.
In this work, we report on empirical evidence which suggests that the same
phenomenon also occurs when transmission takes place over a class of channels
with memory. This is confirmed both by simulations as well as by computing
EXIT curves.
## 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 particular, it was shown in [2] that the BP threshold of the spatially
coupled ensemble (see the last paragraph of this section for a definition) is
essentially equal to the MAP threshold of the underlying component ensemble.
It was also shown that for long chains the MAP performance of the chain cannot
be substantially larger than the MAP threshold of the component ensemble. In
this sense, the BP threshold of the chain is increased to its maximal possible
value. 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. They attribute the
observation of the equality of the two thresholds to G. Liva. The phenomena of
threshold saturation seems not to be restricted to the BEC. It was also shown
recently in [4] that the same phenomena manifests itself when we consider
transmission over more general BMS channels.
The principle which underlies the good performance of spatially coupled
ensembles is very broad. It 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 [5], and a variety of
graphical models in statistical physics and computer science like the so-
called $K$-SAT problem, random graph coloring, or the Curie-Weiss model [6].
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 [7].
It is tempting to conjecture that the same phenomenon occurs for transmission
over general channels with memory. We provide some empirical evidence that
this is indeed the case. In particular, we compute EXIT curves for
transmission over a class of channels with memory known as the Dicode Erasure
Channel (DEC). We show that these curves behave in an identical fashion to the
ones when transmission takes place over the memoryless BEC. We also compute
fixed points (FPs) of the spatial configuration and we demonstrate again
empirically that these FPs have properties identical to the ones in 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 $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$.
This is the ensemble we use throughout the paper as it is the simplest to
analyze.
### I-A $(d_{\mathrm{l}},d_{\mathrm{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{d_{\mathrm{l}}}{d_{\mathrm{r}}}M$ check nodes. It remains to describe
how the connections are chosen. We assume that each of the $d_{\mathrm{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 $d_{\mathrm{r}}$ connections of a
check node at position $i$ is independently chosen from the range
$[i-w+1,\dots,i]$.
A discussion on the above ensemble and a proof of the following lemma can be
found in [2].
###### Lemma 1 (Design Rate)
The design rate of the ensemble $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$, with
$w\leq 2L$, is given by
$\displaystyle R(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$
$\displaystyle=(1-\frac{d_{\mathrm{l}}}{d_{\mathrm{r}}})-\frac{d_{\mathrm{l}}}{d_{\mathrm{r}}}\frac{w+1-2\sum_{i=0}^{w}\bigl{(}\frac{i}{w}\bigr{)}^{d_{\mathrm{r}}}}{2L+1}.$
In the next section we provide the channel model and the joint iterative
decoder. We also present the density evolution analysis of the joint iterative
decoder when we consider $(d_{\mathrm{l}},d_{\mathrm{r}})$-regular LDPC
ensembles. In the section on main results, we demonstrate the threshold
saturation phenomena by using spatially coupled codes.
## II Channels with Memory: The Dicode Erasure Channel
The particular class of channel with memory that we consider is the Dicode
Erasure Channel (DEC). The DEC is a binary-input channel defined as follows.
The output of a binary-input linear filter $(1-D)$ ($D$ is the delay element)
is erased with probability $\epsilon$ and transmitted perfectly with
probability $1-\epsilon$. For this channel we will be interested in the
symmetric information rate (SIR), i.e., the capacity assuming i.i.d Bern(1/2)
signalling. In this case, the Shannon threshold for a given rate $r$ is given
by $\frac{1-r}{4}+\frac{1}{4}\sqrt{(1-r)^{2}+8(1-r)}$. The details on the
definition of the channel and the analytical formula for the SIR can be found
in the thesis of Pfister [8] and in [9].
### II-A Joint Iterative Decoder, Density Evolution and the Extended BP Fixed
Points
We use the joint iterative decoder (JIT) of Pfister and Siegel [9]. More
precisely, we consider a turbo equalization system, which performs one channel
iteration (BCJR step) for each iteration over the LDPC code. As a result, in
every iteration, first the channel detector uses the extrinsic information
provided by the LDPC code to compute its extrinsic erasure fraction. This is
then fed to the LDPC decoder which then again computes the usual variable node
and check node erasure messages.
The simplicity of the DEC gives an analytical formula for the erasure fraction
of the message which is passed from the channel detector to the LDPC code (see
[9] for a derivation). This is given by
$f(x)=\frac{4\epsilon^{2}}{(2-x(1-\epsilon))^{2}},$
where $x$ represents the fraction of erasures entering the channel detector
from the LDPC code. $f(.)$ represents the extrinsic erasure information
provided by the channel detector.
To summarize: the density evolution111See [9] for a rigorous justification of
the density evolution analysis. (DE) equation for the case of
$(d_{\mathrm{l}},d_{\mathrm{r}})$-regular LDPC ensemble is given by
$\displaystyle
x=f((1-(1-x)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}})(1-(1-x)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}-1}.$
Note that the term inside the brackets in $f(.)$ represents the probability
that a variable node is in erasure as given by the LDPC code. Also it is not
hard to see that $f(x)\leq 1$ for any $x$.
###### Example 2
Consider JIT decoding of the DEC with $(5,15)$-regular LDPC ensemble. The
design rate of this code is $2/3$. Using the SIR formula
($=1-2\epsilon^{2}/(1+\epsilon)$) from [9] we get that the Shannon threshold
at rate=2/3 is given by $\epsilon^{\text{Sh}}_{\text{\tiny DEC}}=0.5$. Figure
1 shows the performance of the JIT decoder. We see that the threshold is given
by $\epsilon^{\text{\tiny JIT}}_{\text{\tiny DEC}}(5,15)\approx 0.363471$,
which is far away from the capacity. Throughout the paper we will use
$\epsilon^{\text{\tiny JIT}}_{\text{\tiny
DEC}}(d_{\mathrm{l}},d_{\mathrm{r}})$ to denote the threshold of the JIT
decoder when we use $(d_{\mathrm{l}},d_{\mathrm{r}})$-regular LDPC ensemble
and transmit over the DEC.
(50,-8)(32,0)[cb] ,$0.2$,$0.4$,$0.6$,$0.8$,$1.0$(36,0)(0,32)[l]
,$0.2$,$0.4$,$0.6$,$0.8$,$1.0$$0.0$$\epsilon$
$(1-(1-x)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}}$
Figure 1: The BP curve for the $(d_{\mathrm{l}}=5,d_{\mathrm{r}}=15)$-regular
ensemble and transmission over the DEC. The threshold of the JIT decoder is
given by $\epsilon^{\text{\tiny JIT}}_{\text{\tiny DEC}}(5,15)\approx
0.363471$.
#### The EXIT curve
The EXIT curve222To be very precise, we should call the curves we plot as
EXIT-like curves. The reason being that we do not provide any operation
interpretation of these curves, like the Area theorem [10] in this work. The
curves serve only to illustrate the capacity achieving nature of coupled-
codes. plots all the fixed-points of the DE equation. The curve is given by
the parametric curve
$\\{(1-(1-x)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}},\epsilon(x)\\}$. We obtain
$\epsilon(x)$ by solving for $\epsilon$ in the DE equation.
As an example, we plot the EXIT curve for various
$(d_{\mathrm{l}},d_{\mathrm{r}})$-regular LDPC ensembles as shown in Figure 2.
The JIT threshold is got by dropping a vertical line from the leftmost point
on any given curve. We note that for every $\epsilon>\epsilon^{\text{\tiny
JIT}}_{\text{\tiny DEC}}(d_{\mathrm{l}},d_{\mathrm{r}})$, there are exactly 3
fixed-points. One of them being the trivial 0 fixed-point. This “C” shape of
the EXIT curve is also what we observe when we transmit through a memoryless
BEC using $(d_{\mathrm{l}},d_{\mathrm{r}})$-regular LDPC ensemble. Also we
remark that as the degrees increase, keeping the design rate fixed, the JIT
threshold keeps on decreasing. This is also the case for transmission over
memoryless BEC. In fact, for memoryless BEC case, the BP threshold goes to
zero as we increase the degrees.
(0,-8)(32,0)[cb] ,$0.2$,$0.4$,$0.6$,$0.8$,$1.0$(-14,0)(0,32)[l]
,$0.2$,$0.4$,$0.6$,$0.8$,$1.0$$0.0$$\epsilon$
$(1-(1-x)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}}$
$(3,9)$
$(5,15)$
$(7,21)$
$(10,30)$
$(30,90)$
Figure 2: The EXIT curve for regular LDPC ensembles with
$(d_{\mathrm{l}},d_{\mathrm{r}})$ given by $(3,9)$, $(5,15)$, $(7,21)$,
$(10,30)$, $(30,90)$, and transmission over the DEC. We observe that the JIT
threshold moves to the left and eventually will go to zero as degrees go to
infinity.
We can also show the same result for the DEC. More precisely, we have
###### Lemma 3 (JIT Threshold Goes to Zero)
For any $(d_{\mathrm{l}},d_{\mathrm{r}})$-regular ensemble we have
$\displaystyle\epsilon^{\text{\tiny JIT}}_{\text{\tiny
DEC}}(d_{\mathrm{l}},d_{\mathrm{r}})\leq\sqrt{\frac{1}{\sqrt{d_{\mathrm{r}}-1}(1-(d_{\mathrm{l}}-1)e^{-\sqrt{d_{\mathrm{r}}-1}})}}.$
###### Proof:
We claim that the necessary condition for the JIT decoder to succeed is given
by
$\epsilon^{2}(1-(1-x)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}-1}<x,$
for all $x\in(0,1]$. Indeed, suppose on the contrary that there exists a
$c\in(0,1]$ such that the above inequality is violated. Thus we have
$\epsilon^{2}(1-(1-c)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}-1}\geq c.$ Since
$f(x)\geq\epsilon^{2}$ for all $x\in[0,1]$ we get
$f(c)(1-(1-c)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}-1}\geq c.$
This implies that there exists a FP of DE for the DEC for some value in
$[c,1]$. It is not hard to see that this implies the JIT decoder will get
stuck at this FP, resulting in unsuccessful decoding.
Thus we must have that for all $x\in(0,1]$
$\epsilon^{2}(1-(1-x)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}-1}<x.$
For the choice of $x=\frac{1}{\sqrt{d_{\mathrm{r}}-1}}$ we get the statement
of the lemma. To see this computation first write $(1-x)^{d_{\mathrm{r}}-1}$
as $e^{(d_{\mathrm{r}}-1)\log(1-x)}$. Then use $\log(1-x)\leq-x$ and
$x=\frac{1}{\sqrt{d_{\mathrm{r}}-1}}$ to get $(1-x)^{d_{\mathrm{r}}-1}\leq
e^{-\sqrt{d_{\mathrm{r}}-1}}$. After this use
$\displaystyle(1-e^{-\sqrt{d_{\mathrm{r}}-1}})^{d_{\mathrm{l}}-1}$
$\displaystyle=1-(1-(1-e^{-\sqrt{d_{\mathrm{r}}-1}})^{d_{\mathrm{l}}-1})$
$\displaystyle\geq 1-(d_{\mathrm{l}}-1)e^{-\sqrt{d_{\mathrm{r}}-1}},$
to complete the argument. ∎
As a consequence of Lemma 3 we get that, with the ratio
$d_{\mathrm{l}}/d_{\mathrm{r}}$ kept fixed,
$\lim_{d_{\mathrm{l}}\to\infty}\epsilon^{\text{\tiny JIT}}_{\text{\tiny
DEC}}(d_{\mathrm{l}},d_{\mathrm{r}})=0$.
## III Main Results
In this section we show, empirically, that spatially coupled-codes achieve the
Shannon capacity of the DEC. We recall that we are consider SIR which is give
by the formula SIR$=1-2\epsilon^{2}/(1+\epsilon)$. For the sake of exposition,
we demonstrate our results only for rate equals $2/3$. The Shannon threshold
for this rate is given by $\epsilon^{\text{Sh}}_{\text{\tiny DEC}}=0.5$. For
other rates similar results can be observed. From the preceding section we see
that standard $(d_{\mathrm{l}},d_{\mathrm{r}})$-regular LDPC ensembles do not
saturate the JIT threshold (to the Shannon threshold).
We begin by writing down the DE equation for the coupled-codes.
### III-A Density Evolution
Consider the $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$ ensemble. Recall that there
are $2L+1$ sections of variable nodes. Each section has $M$ variable nodes. We
transmit variable nodes sectionwise over the DEC. More precisely, the variable
nodes in section $-L$ are transmitted first, followed by variable nodes in
section $-L+1$ and so on so forth till we finally transmit all the variable
node in section $L$. As a consequence we have a channel detected factor graph
sitting on top of each section of the coupled-code.
To perform the DE analysis, we already take the limit $M\to\infty$. As a
result of this limit, one can ignore the boundary effects of the channel
detector and treat the channel detectors as disconnected333Another way to
think about this is to imagine that we transmit a known sequence of bits of
length equal to the memory of the channel after we transmit all the variable
nodes in each section. Since the channel memory is finite, this induces a rate
loss going to zero as $M\to\infty$. Now the known sequence is the initial
state for each of the channel detectors and hence we can consider them
disconnected..
Let $x_{i}$, $i\in\mathbb{Z}$, denote the average erasure probability which is
emitted by variable nodes at position $i$. For $i\not\in[-L,L]$ we set
$x_{i}=0$. For $i\in[-L,L]$ the DE is given by
$\displaystyle x_{i}$
$\displaystyle=\epsilon_{i}\Bigl{(}1-\frac{1}{w}\sum_{j=0}^{w-1}\bigl{(}1-\frac{1}{w}\sum_{k=0}^{w-1}x_{i+j-k}\bigr{)}^{d_{\mathrm{r}}-1}\Bigr{)}^{d_{\mathrm{l}}-1},$
(1)
where $\epsilon_{i}$ is given by
$\displaystyle\epsilon_{i}=f\Big{(}\Bigl{(}1-\frac{1}{w}\sum_{j=0}^{w-1}\bigl{(}1-\frac{1}{w}\sum_{k=0}^{w-1}x_{i+j-k}\bigr{)}^{d_{\mathrm{r}}-1}\Bigr{)}^{d_{\mathrm{l}}}\Big{)},$
(2)
where recall that $f(\cdot)$ is the channel extrinsic transfer function. We
will use the notation $\epsilon^{\text{\tiny JIT}}_{\text{\tiny
DEC}}(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$ to denote the threshold of the JIT
decoder when we use the $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$ ensemble for
transmission. As a shorthand we use $g(x_{i-w+1},\dots,x_{i+w-1})$ to denote
$\Bigl{(}1-\frac{1}{w}\sum_{j=0}^{w-1}\bigl{(}1-\frac{1}{w}\sum_{k=0}^{w-1}x_{i+j-k}\bigr{)}^{d_{\mathrm{r}}-1}\Bigr{)}^{d_{\mathrm{l}}-1}.$
###### Definition 4 (FPs of Density Evolution)
Consider DE for the $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$ ensemble. Let
$\underline{x}=(x_{-L},\dots,{x}_{L})$. We call $\underline{x}$ the
constellation. We say that $\underline{x}$ forms a FP of DE with channel
$\epsilon$ if $\underline{x}$ fulfills (1) for $i\in[-L,L]$. As a shorthand we
then say that $(\epsilon,\underline{x})$ is a FP. We say that
$(\epsilon,\underline{x})$ is a non-trivial FP if $\underline{x}$ is not
identically equal to $0\,\forall\,i$. Again, for $i\notin[-L,L]$, $x_{i}=0$. ∎
###### Definition 5 (Forward DE and Admissible Schedules)
Consider forward DE for the $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$ ensemble.
More precisely, pick a channel $\epsilon$. Initialize
$\underline{x}^{(0)}=(1,\dots,1)$. Let $\underline{x}^{(\ell)}$ be the result
of $\ell$ rounds of DE. More precisely, $\underline{x}^{(\ell+1)}$ is
generated from $\underline{x}^{(\ell)}$ by applying the DE equation (1) to
each section $i\in[-L,L]$,
$\displaystyle x_{i}^{(\ell+1)}$
$\displaystyle=\epsilon_{i}g(x_{i-w+1}^{(\ell)},\dots,x_{i+w-1}^{(\ell)}).$
We call this the parallel schedule. The important difference with the
memoryless BEC case is that the channel $\epsilon_{i}$ is not fixed for the
DEC and decreases with increasing iterations according to (2).
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 $\underline{x}^{(\ell)}$ the resulting sequence of
constellations. ∎
One can show that if we perform forward DE under any admissible schedule, then
the constellation $\underline{x}^{(\ell)}$ converges to a FP of DE and this FP
is independent of schedule. This statement can be proved similar to the one in
[2].
### III-B Forward DE – Simulation Results
We consider forward DE for the $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$ ensemble.
More precisely, we fix an $\epsilon$ and initialize all $x_{i}$ for
$i\in[-L,L]$ to 1. Then we run the DE given by (1) till we reach a fixed-
point. We fix $L=250$. For $d_{\mathrm{l}}=3$ and $d_{\mathrm{r}}=9$, we have
that $\epsilon^{\text{\tiny JIT}}_{\text{\tiny DEC}}(3,9,300,3)\approx
0.49815$. If we increase the degrees we get $\epsilon^{\text{\tiny
JIT}}_{\text{\tiny DEC}}(5,15,300,5)\approx 0.49995$, $\epsilon^{\text{\tiny
JIT}}_{\text{\tiny DEC}}(7,21,300,7)\approx 0.499989$ and
$\epsilon^{\text{\tiny JIT}}_{\text{\tiny DEC}}(9,27,300,9)\approx 0.499996$.
We observe that for increasing the degrees the threshold approaches the
Shannon threshold of $0.5$.
### III-C The EXIT Curve for Coupled Ensembles
We now come to the key point of the paper, the computation of the EXIT curve.
Before we do this, we define the entropy of a constellation
$\underline{x}=(x_{-L},\dots,x_{L})$ as
$\displaystyle\chi=\frac{1}{2L+1}\sum_{i=-L}^{L}x_{i}.$
To plot the EXIT curve we first fix $\chi\in[0,1]$ and then run DE such that
the resulting FP constellation has entropy equal to $\chi$. This is the
reverse DE procedure as described in [11]. We remark that $f(x)$ is an
increasing function of $\epsilon$, hence in the reverse DE procedure one can
easily find an appropriate $\epsilon$ by the bisection method.
Figure 3 shows the plot of the EXIT curve for the $(5,15,L,5)$ ensemble with
$L=2,4,8,16,32,64,128,256,512$. We see that the curves look very similar to
the curves when transmitting over a BMS channel. For very small values of $L$,
the curves are far to the right due to significant rate loss that is incurred
at the boundary. As $L$ increases the rate loss diminishes and the JIT
threshold is very close to the Shannon threshold. This picture strongly
suggests that the same threshold saturation effect ($\epsilon^{\text{\tiny
JIT}}_{\text{\tiny
DEC}}(d_{\mathrm{l}},d_{\mathrm{r}},L,w)\approx\epsilon^{\text{\tiny
MAP}}_{\text{\tiny DEC}}(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$) also occurs for
the DEC as it was shown analytically in [2].
$L\\!=\\!2$
$L\\!=\\!4$
$L\\!=\\!8$
$L\\!=\\!16$
(0,-8)(32,0)[cb] ,$0.2$,$0.4$,$0.6$,$0.8$,$1.0$(-16,0)(0,32)[l]
,$0.2$,$0.4$,$0.6$,$0.8$,$1.0$
EXIT
Figure 3: The EXIT curve for the $(d_{\mathrm{l}}=5,d_{\mathrm{r}}=15,L,5)$
ensemble and transmission over the DEC for $L=2,4,8,16,32,64,128,256,512$. The
curves keep moving to the left as $L$ increases similar to the curves when
transmitting over BMS. The “vertical” drop in the EXIT curves occurs at
$\approx 0.5$ for $L\geq 32$. Also shown in light gray is the BP exit curve
for the uncoupled $(5,15)$-regular ensemble.
### III-D Shape of Fixed Point of Density Evolution
We plot the constellation representing the unstable FP of DE. This FP cannot
be reached via forward DE and is obtained via reverse DE procedure. We recall
that this FP played a key role in proving the threshold saturation phenomena
when transmitting over the BEC. 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 $\epsilon\approx 0.49995$.
* (ii)
Let $x_{\text{s}}(\epsilon)$ denote a stable FP of DE. The value in the flat
part in the middle is $\approx 0.4434$ which is very close to the stable FP of
DE for the underlying uncoupled $(5,15)$-regular ensemble.
* (iii)
The transition from close to zero to close to $x_{\text{s}}(\epsilon)$ 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 constellation representing FP of DE for $(5,15,33,5)$ ensemble
and entropy fixed to $\chi=0.2$. This is an unstable FP constellation. The
constellation is very similar to any unstable FP constellation when
transmitting over memoryless BEC. The constellation is unimodal. There is a
long tail of zeros followed by a sharp transition and then a long flat part
with values close to $x_{\text{s}}(\epsilon)$. The constellation has
$\epsilon\approx 0.49995$.
## IV A Possible Proof Approach
Till now we gave empirical evidence of the threshold saturation phenomena when
transmitting over the DEC using coupled-codes. Before we proceed to give the
proof idea for the threshold saturation, we first show that coupling indeed
helps. More precisely we have the following lemma,
###### Lemma 6 (Spatial Coupling Helps)
For $d_{\mathrm{l}},d_{\mathrm{r}}\to\infty$ with the ratio
$d_{\mathrm{l}}/d_{\mathrm{r}}$ kept fixed, we have
$\displaystyle\epsilon^{\text{\tiny JIT}}_{\text{\tiny
DEC}}(d_{\mathrm{l}},d_{\mathrm{r}},L,w)\geq\frac{d_{\mathrm{l}}}{d_{\mathrm{r}}}.$
###### Proof:
Since $\epsilon_{i}$ is an increasing function of $x_{i-w+1},\dots,x_{i+w-1}$,
we have $\epsilon_{i}\leq
f(1)\leq\frac{4\epsilon^{2}}{(1+\epsilon)^{2}}\leq\epsilon.$ Combining this
with the DE equation for the coupled-codes, we get
$\displaystyle x_{i}\leq\epsilon g(x_{i-w+1},\dots,x_{i+w-1}),$
for all $i\in[-L,L]$. But we know from Theorem 10 in [2] that
$\lim_{d_{\mathrm{l}}\to\infty}\epsilon^{\text{\tiny
BP}}_{\text{BEC}}(d_{\mathrm{l}},d_{\mathrm{r}},L,w)\to\frac{d_{\mathrm{l}}}{d_{\mathrm{r}}}$.
Thus for $\epsilon<\frac{d_{\mathrm{l}}}{d_{\mathrm{r}}}$ the right-hand-side
of the above inequality goes to zero. Hence the lemma. ∎
As an example, consider the $(d_{\mathrm{l}},d_{\mathrm{r}})$-regular ensemble
with $d_{\mathrm{l}}/d_{\mathrm{r}}=1/3$ (rate equal to $2/3$) . For
$L\to\infty$, the rate of the $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$ goes to
$2/3$. From Lemma 3 we have that $\epsilon^{\text{\tiny JIT}}_{\text{\tiny
DEC}}(d_{\mathrm{l}},d_{\mathrm{r}})\to 0$ and from Lemma 6 we have that
$\epsilon^{\text{\tiny JIT}}_{\text{\tiny
DEC}}(d_{\mathrm{l}},d_{\mathrm{r}},L,w)\geq\frac{d_{\mathrm{l}}}{d_{\mathrm{r}}}=\frac{1}{3}$.
Thus spatial coupling indeed boosts the JIT threshold. However the empirical
evidence suggests that the boost is all the way up to the Shannon threshold
(which is $0.5$ in this case). Since there is ample similarity between the DEC
and the BEC, the guideline for a proof is similar to when we are transmitting
over the BEC.
(i) Existence of FP: A key ingredient in proving the result for the BEC was to
show the existence of a special FP of DE $(\underline{x},\epsilon^{*})$. In
principle, the BEC proof should extend. The only difference is that instead of
a constant channel $\epsilon$, we have a channel value which depends on the FP
constellation itself. However, since the functions involved are rational, this
should not be a big hurdle.
(ii) Shape of the constellation and the transition length: The next task is to
show that the FP guaranteed by the above theorem has the properties as given
in Section III-D. Proving this would first involve showing that the underlying
regular ensemble has a “C” shaped EXIT curve. Intuitively, this means that the
FP constellation (of the coupled-code) can only hover around the stable FPs of
DE (of the underlying regular ensemble), implying that it has either a large
tail of zeros or a large flat part with values close to
$x_{\text{s}}(\epsilon^{*})$.
(iii) Construction of the EXIT curve and the Area Theorem: Another key part of
the BEC proof was to construct a family of FPs (not necessarily stable FPs)
using the special FP guaranteed by the Existence theorem. The EXIT curve plus
the fast transition would allow us to show that this special FP must have an
associated channel parameter, $\epsilon^{*}$, very close to the Shannon
threshold (for large degrees.)444For finite degrees, $\epsilon^{*}$ should be
very close to the MAP threshold of the
$(d_{\mathrm{l}},d_{\mathrm{r}})$-regular ensemble. One should be able to
prove this by formulating an appropriate Area theorem (see Section 3.20 in
[10]).
Operational interpretation: The proof would be completed by providing an
operation meaning to the EXIT curve. Loosely speaking, the EXIT constructed
above would have a vertical drop at
$\epsilon\approx\epsilon^{\text{Sh}}(d_{\mathrm{l}},d_{\mathrm{r}})$ (cf.
Figure 3). This would help to show that for any
$\epsilon<\epsilon^{\text{Sh}}(d_{\mathrm{l}},d_{\mathrm{r}})$, the JIT
decoder will go to the trivial FP.
## V Conclusions
In this paper we show that empirically coupled-codes saturate the JIT
threshold on the DEC. For the channel extrinsic transfer function we consider
the case when there is no precoding. We list below some comments and open
questions.
* (i)
An obvious future direction is to complete the proof of threshold saturation.
The guidelines provided above serve as a starting point. Following this route,
in principle, it should be possible to prove the capacity achieving nature of
these codes on the DEC.
* (ii)
Another interesting question is that whether the threshold saturation
phenomena can be shown to be true for all channel extrinsic transfer functions
$f(.)$ which are non-decreasing both in $\epsilon$ and $x$ (threshold
saturation holds when $f(.)$ represents precoding).
* (iii)
A proof of the threshold saturation phenomena should also pave the way for the
justification of the Maxwell construction to determine $\epsilon^{\text{\tiny
MAP}}_{\text{\tiny DEC}}(d_{\mathrm{l}},d_{\mathrm{r}})$ for the DEC.
* (iv)
Recently, it was observed that coupled MacKay-Neal (MN) codes with bounded
degree exhibit the BP threshold very close to the Shannon threshold over the
BEC [12]. It is interesting to see if the coupled MN codes have the JIT
threshold close to the SIR over the DEC.
## 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, Misha Chertkov and Henry Pfister for their
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] 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.
* [5] 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.
* [6] 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.
* [7] 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.
* [8] H. D. Pfister, “On the capacity of finite state channels and the analysis of convolutional accumulate-$m$ codes,” Ph.D. dissertation, UCSD, San Diego, CA, USA, 2003.
* [9] H. D. Pfister and P. H. Siegel, “Joint iterative decoding of LDPC codes for channels with memory and erasure noise,” _IEEE J. Sel. Area. Commun._ , vol. 26, no. 2, pp. 320–337, Feb. 2008.
* [10] T. Richardson and R. Urbanke, _Modern Coding Theory_. Cambridge University Press, 2008.
* [11] 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.
* [12] K. Kasai and K. Sakaniwa, “Spatially-coupled bounded-density capacity-achieving codes,” in _Proc. Symp. on Inf. Theory and its Applications_ , Dec. 2010, pp. 1–6, (in Japanese).
|
arxiv-papers
| 2011-02-02T11:26:03 |
2024-09-04T02:49:16.794722
|
{
"license": "Public Domain",
"authors": "Shrinivas Kudekar and Kenta Kasai",
"submitter": "Kenta Kasai",
"url": "https://arxiv.org/abs/1102.0406"
}
|
1102.0432
|
11institutetext: Institute for Condensed Matter Physics, National Academy of
Sciences of Ukraine, Svientsitsky str.1, 79011 Lviv, Ukraine; 11email:
pavlenko@mailaps.org 22institutetext: Institute for Applied Mathematics and
Fundamental Sciences, Lviv Technical University, Ustyianowycha str. 10, 79013
Lviv, Ukraine
# Interstitial Fe-Cr alloys: Tuning of magnetism by nanoscale structural
control and by implantation of nonmagnetic atoms
Interstitial Fe-Cr alloys
N. Pavlenko 11 N. Shcherbovskikh 22 and Z.A. Duriagina 22
###### Abstract
Using the density functional theory, we perform a full atomic relaxation of
the bulk ferrite with $12.5\%$-concentration of monoatomic interstitial Cr
periodically located at the edges of the bcc Feα cell. We show that structural
relaxation in such artificially engineered alloys leads to significant atomic
displacements and results in the formation of novel highly stable
configurations with parallel chains of octahedrically arranged Fe. The
enhanced magnetic polarization in the low-symmetry metallic state of this type
of alloys can be externally controlled by additional inclusion of nonmagnetic
impurities like nitrogen. We discuss possible applications of generated
interstitial alloys in spintronic devices and propose to consider them as a
basis of novel durable types of stainless steels.
## 1 Introduction
Last years demonstrate increased activities in the search for novel materials
exhibiting controlled modification of electronic properties by inclusion or
implantation of different atoms or ionic groups. A prominent example of the
implantation-altered systems is the stainless steel. In the steels, the
implantation of chromium, molibdenium, nitrogen and other chemical elements
substantially changes the microstructure of subsurface layers and modify their
corrosion resistance and hardness steels .
In the development of novel efficient multifunctional materials for
technological applications in the long-term devices, the properties like
hardness, corrosion, heat resistance and other types of mechanical and
chemical durability are of central interest mai ; yokokawa . It frequently
appears in science and technology that well known materials doped by different
chemical elements exhibit unexpected physical properties not revealed
previously.
As an example of such a new unexpected behavior, in the present work we
consider an alloy Fe-Cr. The alloys of Fe and Cr, doped by C, Ni and by other
elements, are widely used as basic components for ferritic and martensitic
steels. Substitutional alloys of Fe and Cr have attracted much attention of
theory and experiment due to their rich magnetic properties characterized by
local antiferromagnetism in the proximity of Cr atoms implanted into
ferromagnetic iron victora ; paxton ; paduani ; davies . Due to small
differences between the atomic radii of iron and chromium, the modification of
the substitutional alloy properties is limited to the local magnetic
transformation due to local changes in the electronic orbital occupancies,
without significant structural modifications. In contrast to the
substitutional structural configurations, the interstitial Fe-Cr alloys
considered in the present work contain Cr impurities which are located in the
interstitial positions of the bcc lattice of Feα. In the recent theoretical
studies of the Cr intersitials in Fe-Cr alloys, different types of
interstitial configurations were analyzed. Among them, a pair configuration
$\langle 111\rangle$ dumbbell is considered as the most energetically
favourable which requires about 4.2eV for its formation under irradiation
klaver ; olsson .
In the present work, we consider a novel monoatomic interstitial configuration
which contains single Cr atoms positioned in the centers of the edges of the
bcc ferrite. In contrast to the substitutional alloys, the significant forces
due to the interstitial atoms induce substantial structural optimization which
enhances the volume due to modified lattice constants and leads to the
relaxation of the atomic positions in the unit cell. We find that the
relaxation of the initial bcc unit cell results in significant atomic
distortions and in the formation of atomic chain-like structures. As appears
in the density-functional-theory (DFT) calculations of the optimized
structures, the energy gain achieved due to the structural relaxation of the
considered interstitial alloy can approach 6.17 eV which makes this type of
systems highly stable and durable. In the present work, we propose to consider
these artificially generated alloys as candidates for novel types of stainless
steels.
The fundamental difference between the industrial alloys and the alloys
studied in the present work is the ordered and periodic character of the
latters. In the industrial steels, the amorphic character of the systems is
related to the random distribution of the impurities. The hardening of the
steels proceeds through the surface treatment and is accompanied by formation
of granular microstructure with the spatially inhomogeneous impurity
concentration and modified subsurface properties afm . In the studies of the
subsurface Cr-doped alloyed ferrite, we consider the supercells containing
periodically located Cr atoms in the cubic lattice of Feα. The interstitial Cr
induces significant atomic reconstruction with consequent break of initial
cubic symmetry and stabilization of a new lower-symmetry state. The appearing
structural transformation has a character of a phase transition which occurs
due to nanoscale tailoring of cubic Fe by interstitial inclusion of Cr atoms,
the effect which can be experimentally verified by the means of modern methods
like AFM spectroscopy .
Using the DFT-based structural optimization, we obtain the optimized atomic
microstructure of a chain-like character where the chains of octahedrically
arranged Fe atoms are formed along the (001)-axis. We find that the competing
ferromagnetic and antiferromagnetic interactions lead to spatially
inhomogeneous spin polarization. The magnetization of the structurally relaxed
system is significantly enhanced as compared to the pure ferrite without Cr
inclusions. The obtained enhancement makes the generated alloys perspective
candidates for spin polarizers in spintronic applications. In the generated
chain-like structures, the relaxation is accompanied by the formation of
spatial channels with extremely low carrier density. We suggest that these
channels can be considered as paths for the low-barrier-migration of light
impurities like H, N, Li or C. As an example of a light atom in the
interstitial alloy, we study of the migration paths of nonmagnetic nitrogen
and calculate the energy barriers along the migration paths. We obtain a
strong influence of the nonmagnetic N on the alloy magnetization. Our findings
show that the structural modifications due to possible nanoscale tuning of Cr
impurities on the edges of bcc cubic cells of iron can play a central role in
the control of their electronic properties.
## 2 Structural relaxation of interstitial alloy Feα-Cr
The present studies of the electronic properties of the considered
interstitial Fe-Cr alloy are based on the DFT calculations of the electronic
structure of the systems generated by periodic translation of specially chosen
supercells. The initial supercell shown in Fig. 1 contains the doubled
$2\times 2$ cubic bcc cell of ferrite (Feα) and a single Cr atom centered in
one of the edges of the Feα cubic unit cell with the lattice constant $a=3.85$
Å. The obtained structure is described by a chemical formula Fe8Cr and
determines an interstitial Fe-Cr alloy with the Cr concentration $n=0.125$
which is typical for stainless steels. The presence of interstitial Cr leads
to significant local forces acting on the neighbouring Fe atoms. To minimize
the forces, the coordinates of all atoms have been relaxed. In the present
studies, the optimization of the supercell has been performed by employing the
DFT approach implemented withing the linearized augmented plane wave (LAPW)
scheme in the full potential Wien2k code wien2k . To study the role of the
spin polarization in the structural relaxation, two different relaxation
procedures have been employed. In the first procedure, the atomic optimal
positions are calculated in the local density approximation (LDA) on a
$2\times 2\times 5$ k-points grid. To explore the role of spin degrees of
freedom in the relaxation, in the second procedure the local spin density
approximation (LSDA) has been used in the optimization of the structure. The
results of both methods of the structural relaxation are presented in Fig. 2.
Figure 1: Schematic view of unrelaxed 2$\times$2 Feα cell which contain
12.5$\%$ of edge-centered interstitial Cr.
A central common feature which characterizes both (LDA- and LSDA-relaxed)
structures is the clusterization of the sublattice of the iron atoms. In the
LDA-optimized structure (Fig. 2(a)), the relaxation results in formation of a
high-symmetry clusterized network. This network consists of the Fe6-octahedra
which form the square plaquettes in the $(x,y)$ ($(a,b)$) plane with Cr atoms
located in the center of each plaquette. The distance from the centered Cr to
each nearest iron octaherda amounts $1.9$ Å. Despite the significant
displacements of the iron atoms from their initial positions, the net electric
polarization of the cell is zero due to high structural symmetry C4/m obtained
after the relaxation.
The formation energy of the relaxed Fe8Cr-configuration can be expressed as
$\displaystyle E_{f}({\rm LDA})=E_{{\rm tot}}({\rm Fe}_{8}{\rm Cr})-8E_{{\rm
tot}}({\rm Fe})-E_{{\rm tot}}({\rm Cr}),$
where the last two terms identify the total energies of the bulk bcc Feα and
Cr, respectively. To determine $E_{{\rm tot}}({\rm Fe})$, we have calculated
the total energy value of the bulk Feα in the ferromagnetic state. As the
LSDA-calculation of the spin-polarized configurations of the bulk Cr are
converged to the paramagnetic state, we consider the total energy $E_{{\rm
tot}}({\rm Cr})$ for the paramagnetic Cr. With these values, we find that
$E_{f}({\rm LDA})=4.82$eV. To analyze the role of the relaxation, we have also
calculated the energy $E_{f}({\rm unrel})$ of the formation of initial
unrelaxed configuration which is equal to 5.02eV. As a consequence, the
significant energy gain due to the structural relaxation
$\displaystyle\Delta E({\rm LDA})=E_{f}({\rm unrel})-E_{f}({\rm
LDA})=0.196eV,$
shows a central importance of the atomic displacements for the stability of
the considered systems.
The optimization procedure based on the LSDA approach accounts for additional
corrections due to spin polarization and produces new ordered structural
patterns presented in Fig. 2(b) and Fig. 2(c) for two different (unrelaxed
$a=b=2.86$Å and relaxed $a=b=3$Å) lattice constants. The volume-optimized
structure (c) is signified by the 13%th increase of the unit cell volume due
to the insertion of the interstitial Cr. The LSDA-optimized structural pattern
is characterized by the chains of atomic Fe-groups along the
$x$($a$)-direction, each group containing six Fe-atoms. The nearest chains are
separated by a distance about $4$Å and are connected to each other by the Fe-
Cr bonds of the length about $2.4$Å for the structure (b) with $a=2.86$Å, and
$2.7$Å for the structure (c) with the optimized $a=3.0$Å. The local
antiferromagnetic ordering in the vicinity of Cr is characterized by the
magnetic moments $\mu_{Cr}=-0.72$ $\mu_{B}$ and $\mu_{5}=2.4$ $\mu_{B}$ and
$\mu_{6}=1.25$ $\mu_{B}$ of the neighbouring atoms Fe5 and Fe6, respectively.
The magnetic moments of more distant iron atoms have the values around 2.5
$\mu_{B}$, which is close to results obtained for substitutional alloys and in
pure Feα klaver .
As compared to the tetragonal structure of the LDA-optimized system, the
chain-like structure of the LSDA-relaxed supercell is characterized by
substantially lower crystal symmetry and by the absence of the inversion
center. In contrast to the LDA-based configurations, the formation energy of
the LSDA-relaxed Fe8Cr configuration $E_{f}({\rm LSDA})=E_{{\rm tot}}({\rm
Fe}_{8}{\rm Cr})-8E_{{\rm tot}}({\rm Fe})-E_{{\rm tot}}(Cr)=-1.15$eV is
negative which implies its high stability. We can also calculate the energy
gain due to the structural relaxation by the LSDA approach
$\displaystyle\Delta E({\rm LSDA})=E_{f}({\rm unrel})-E_{f}({\rm
LSDA})=6.17eV,$
which also demonstrates the high stability of the relaxed spin-polarized
structure and a necessity to account for a spin polarization in the structural
optimization of the systems with strong magnetoelastic effect.
### 2.1 Magnetic properties
In the considered systems, we have also analyzed modification of the local
magnetic properties due to the relaxation of the interatomic distances. To see
how the atomic displacements influence the spin polarization of the
surrounding atoms, in Fig. 3 we present the dependences of the local moments
of Cr and of two nearest neighboring Fe on the Cr displacement along the bond
[Fe5-Cr-Fe6]
$\displaystyle\Delta=[x({\rm Cr})-x({\rm Fe5})]-[x({\rm Cr})-x({\rm
Fe5})]^{0},$
where $[x({\rm Cr})-x({\rm Fe5})]^{0}$ is the optimized [Fe5-Cr]-bond length.
The increase of $\Delta$ leads to the change of $\mu_{\rm Cr}$ from -0.7
$\mu_{B}$ to the value about -0.73 $\mu_{B}$. In addition, the larger $\Delta$
implies the elongation of the [Fe5-Cr] bond and lead to the reduced
$\mu_{5}=2.39$ $\mu_{B}$ due to the tendency for a suppression of
antiferromagnetism in the vicinity of Fe5. The increase of $\Delta$ also
produces an enhancement of $\mu_{6}$ from 1.25 $\mu_{B}$ to the values about
$1.28-1.3$ $\mu_{B}$, an opposite trend which occurs due to the shortening of
the bond between Cr and Fe6.
In Fig. 3, the $\Delta$-dependences of the atomic magnetic moments are highly
asymmetric with respect to $\Delta$. Consequently, the obtained magnetoelastic
coupling produces an anisotropy of the magnetic moments and is accompanied by
the loss of the inversion center due to the atomic displacements, the effect
which can be observed in Fig. 2(b) and (c). In Fig. 2(c), the low-symmetry
structure corresponds to the minimum of the total energy. As a conclusion, the
neglect of the magnetoelastic coupling in the electronic structure
calculations does not allow to achieve a full optimization in this type of
interstitial alloys.
Figure 2: Relaxed structure of Fe with 12.5$\%$ of Cr: (a) LDA calculations,
(b) spin-polarized LSDA calculations in the structure with $a=b=2.86$ Å and
(c) spin-polarized LSDA calculations in the structure with $a=b=3.0$ Å. The
path 1 and path 2 identify possible pathes for diffusion through the channels
formed due to atomic relaxation.
### 2.2 Electronic structure
Fig. 4 shows the $3d$ spin-polarized electronic density contours of the LSDA-
optimized structure in the ($x$, $z$) plane. One can see that the majority
$3d$ spin-up states of Fe are highly occupied by the electrons whereas the
electron concentration of Cr spin up states is substantially lower. In
contrast to this, the spin-down (minority) electrons are characterized by high
electron occupation of Cr and lower electron density on Fe. In Fig. 4, the
chain-like structures Fe-Cr in the $z$-direction are characterized by strong
hybridization between the intra-chain $3d$ spin-down orbitals of Fe and Cr.
The last feature leads to the spatial charge redistribution and to higher
charge densities on the bonds between spin-down Cr and Fe. In the LSDA-
optimized system, the structural optimization produces areas with low charge
density in the $y$ ($b$)-direction, where each area can be identified between
the chains of Fe-octahedra. As can be seen in Fig. 4, these areas are almost
free of the charge and can be considered as channels for the migration of
light atoms like H, Li or N. Similarly to the contours in Fig. 4, the electron
density of the majority Fe and Cr orbitals and on the bonds between Cr and Fe
calculated for the LDA-optimized structure (Fig. 5) is substantially lower
than the charge density on the spin-down contours, although the spatial charge
distribution is more homogeneous as compared to that in Fig. 4.
Figure 3: Local magnetic moments (in $\mu_{B}$) of the atoms in Fe5-Cr-Fe6
triad versus the displacement $\Delta$=[Fe5-Cr]-[Fe5-Cr]0 of Cr along the
(100) axis. Here [Fe5-Cr]0 is the equilibrium distance between Fe6 and Cr.
Figure 4: Contours of electron density maps in the ($x$, $z$)-plane
($y/b=0.25$, $x$ and $z$ given in Å) obtained by integration of electronic
states in the energy window $E$ between $-3$ eV below the Fermi level and the
Fermi level. The results obtained by the structural optimization using the
LSDA approximation. Figure 5: Contours of electron density maps in the ($x$,
$z$)-plane ($y/b=0.25$, $x$ and $z$ given in Å) calculated by integration of
electron states in the energy window $E$ between $-3$ eV below the Fermi level
and the Fermi level. The LSDA results obtained in the initially LDA-relaxed
structure.
For the LDA-relaxed structure, the density of states is characterized by
strong suppression of the majority spin-up DOS at the Fermi level (Fig. 6(a)),
whereas the minority DOS at the Fermi level remains significant. Similar,
although much stronger, suppression of majority DOS is typically observed in
half-metallic systems where the electric current is conducted by the electrons
with the same direction of spin park . In contrast to the half-metallic-like
features of the LDA-relaxed structure, the DOS of the LSDA-optimized system
(Fig. 6(b)) demonstrates substantial values at the Fermi level for both spin
directions which implies an enhancement of the metallic state for the majority
electrons.
In transition metal oxides, the metallic state obtained in the LDA approach is
usually strongly influenced by additional account for the local Coulomb
corrections for the $d$-electronic states anisimov ; czyzyk ; pavlenko3 ;
pavlenko4 . In our work, the Coulomb corrections are incorporated within the
SIC-variant of the LSDA+$U$ approximation introduced in Ref. anisimov . The
results are presented in Fig. 7 for two different values of $U=2$ eV and
$U=4.5$ eV estimated and employed in Ref. bandyopadhyay ; zhang ; korotin to
account for the electron repulsion of 3d electrons of Fe and Cr. Fig. 7 shows
the finite density of states at the Fermi ($E=0$) level, although larger $U$
leads to a significant suppression of the majority DOS at $E_{F}$ which
suggests a prevailing tendency towards a half-metallic behavior.
In the LSDA-optimized structure, we find that the cell magnetic moment $M_{\rm
LSDA}=3.84$ $\mu_{B}$ is larger then the magnetic moment $M_{\rm LDA}=2.88$
$\mu_{B}$ in the LDA-optimized cell. Such an enhancement of the magnetic
polarization is connected with the substantial distortions
$\Delta\boldsymbol{R}_{i}$ in the range $0.2-0.84$ Å and can be considered as
a direct evidence of significant magnetoelastic effect. The local Coulomb
corrections in the LSDA+$U$-calculations result in enhanced spin polarization.
Specifically, we obtain $M_{\rm LSDA}=4.07$ $\mu_{B}$ for $U=2$ eV, and
$M_{\rm LSDA}=4.66$ $\mu_{B}$ for $U=4.5$ eV. It is remarkable that in the
substitutional alloy Fe-Cr with 12.5$\%$ of Cr, the LSDA approach gives the
value 3.8 $\mu_{B}$ for the cell magnetic moment which is slightly lower than
the magnetic moment for the considered LSDA-relaxed substitutional alloy.
Figure 6: Total density of states (in eV-1) for structures optimized using (a)
LDA approach and (b) spin-polarized LSDA approximation. The Fermi level
corresponds to $E=0$.
Figure 7: Total densities of states (in eV-1) for the LSDA-optimized structure
calculated by the LSDA+$U$ method with the local Coulomb corrections for the
3d-orbitals of Fe and Cr $U=2$ eV (black curves) and 4 eV (blue curves). The
Fermi level corresponds to $E=0$.
The obtained high spin polarization of the considered interstitial alloys Fe-
Cr allows us to suggest these materials as possible candidates for spin
polarizers in the spintronic devices. Possible technological applications of
artificially generated interstitial alloys would be related to the thin films
produced for the needs of modern electronic industry. In such artificial
systems, a central question is related to the methods of implantation and
positioning of Cr in centers of the edges of cubic bcc lattice of bulk Fe.
With the current state of the art, such structural nanoscale manipulation can
be confined to the first subsurface layers of ferrite films by the using for
instance the methods of optical trapping by lasers optic or by atomic force
microscopy afm . To estimate the stability of ultra-thin films of iron-
chromium alloys which can be also considered as a basis for stainless steels,
we have also extended our calculations to the nanoscale two-monolayers-thick
iron films containing one interstitial Cr atom per each 20-25 subsurface Fe
atoms pavlenko2 . For such type of films, we have performed the calculations
of the surface formation energy and of the electronic work functions, which
were also compared to the corresponding quantities in the films of standard
substitutional Fe-Cr alloys. In the interstitial Fe-Cr films, we find that the
energy of the surface formation is about 1.69eV and the electronic work
function amounts to 1eV, whereas for the substitutional Fe-Cr films we obtain
2.56eV for the surface energy and 0.57eV for the work function. These results
allow us to expect high stability and durability of films generated on the
basis of nanoscale-manipulated interstitial Fe-Cr alloys, as compared to the
iron films with substitutional Cr impurities.
## 3 Migration paths of N in interstitial alloys Feα-Cr
A central question related to the stability of the considered interstitial
alloys is how various atomic impurities can modify the electronic properties.
In the structurally relaxed alloys, the chains of atomic Fe-groups are
separated by 4Å-wide atomic empty channels, which are expected to contain
pathways for light impurity atoms like H, N or Li. To explore a possibility of
the migration of the impurities, we consider possible migration paths of a
single nitrogen atom in the vicinity of the atomic Fe-chains in the
interstitial Feα-Cr.
In the studies of the migration paths of N impurities, we employed the nudged-
elastic-band (NEB) method implemented in the Quantum-Espresso (QE) Package for
the DFT calculations with the use of plane-wave basis sets and
pseudopotentials qe ; jonsson . In these calculations, for the atomic cores of
Fe and Cr we employ the Perdew-Burke-Ernzerhof (PBE) norm-conserving
pseudopotentials pbe . For each stage of the nitrogen transport, the NEB
method involves a relaxation of the atomic positions and of the distances
between the different atoms in the supercell until the forces acting on the
atoms reach their minima. In these calculations, we use the plane-wave cutoff
680 eV and the energy cutoff for charge and potential given by 1360 eV. In the
NEB-approach, the relaxation of the atomic positions along the nitrogen
migration path is performed by the minimization of the total energy of each
intermediate configuration (image). These images correspond to different
positions of N on the migration path and they are produced by the optimization
of a specially generated object functional (action) with the consequent
minimization of the spring forces perpendicular to the path. In our
calculations, the convergence criteria for the norm of the force orthogonal to
the path is achieved at the values below 0.05eV/Å. As the initial atomic
configuration, the supercell Fe-Cr relaxed by the full-potential LSDA-approach
wien2k has been considered.
Recent studies of the migration paths of single hydrogen atoms by the
pseudopotential NEB method demonstrate a good agreement of the obtained
transport mechanisms and energy barriers with the experimental measurements
pavlenko . Similarly to Ref. pavlenko , in the present studies, each NEB-
generated configuration has been modified by the introduction of the N atom
and the obtained in this way extended supercell has been fully structurally
optimized.
To study the migration of N, we consider two different migration paths across
the atomic Fe-chains in the Fe-Cr supercell. The first path (path (a))
describes the migration of N from the initial position inside the cell
($z/c=0.5$) near the chain (1) across the channel to the chain (2)
schematically presented in Fig. 8(a). In distinction to the path (a), the
second path (b) reflects path2 in Fig. 2(c) and contains additional migration
step of N from the supercell boundary ($z\approx 0$) along the $c$-direction
inside the supercell, with the further relocation through the channel to the
atomic Fe-chain (2) indicated in Fig. 8(b).
Fig. 9 shows the profiles of the total energy calculated along the N migration
paths (a) and (b). The path (a) is characterized by the high energy barriers
about $0.8$ eV in the path coordinate range ($0\leq r/R_{N}\leq 0.4$) and
($0.8\leq r/R_{N}\leq 1$) which corresponds to the migration of N within the
two Fe-chains (1) and (2). Here $R_{N}$ denotes the maximal length of the N
path in the supercell which reaches about 1 nm for the path (a). The
interchain motion inside the channel is signified by a low energy barrier
about $0.2$ eV ($0.4\leq r/R_{N}\leq 0.7$ in Fig. 9(a)).
In contrast to this, the energy profile for the migration path (b) (Fig. 9(b))
contains a plateau-like region at $0.1\leq r/R_{N}\leq 0.4$. This miration
step indicates the intracell replacement of N near the Fe-chain (1) along the
[001]-direction demonstrated in Fig. 4, with a further relocation between the
atomic chains across the atomic-empty channel with a low energy barrier about
$0.2$ eV.
As a conclusion, we can note that the possible migration paths of the light
atoms in the considered interstitial Fe-Cr alloys contain a combination of the
motion (i) within the atomic-empty channels and (ii) along the $c$-direction
along to the Fe-contained atomic chains.
Figure 8: Two different migration paths of nitrogen through the channel of the
optimized crystal cell of Fe-Cr. The top picture (a) represents the interchain
migration of N inside the cell with the coordinate $z/c$ near $0.5$. The
bottom picture (b) corresponds to the migration of N from the cell boundary
($z=0$) along the $z$ direction with the further migration between two
neighbouring Fe-chains. The symbols (1) and (2) denote the different atomic
chains; (i) and (f) correspond to the initial and final positions of nitrogen
in the migration paths. Figure 9: Total energy profiles of the system along
the migration paths of N. Here $E_{0}$ denotes the energy of the system in the
initial position of N and $R_{N}$ is the N coordinate in the final position of
the path. Figure 10: Cell ($M_{\rm tot}$) and atomic magnetic moments (in
$\mu_{B}$) along the migration paths of N. The red arrows identify the maximal
cell magnetic moments approached upon the minimization of the distance [N-Cr]
along the N migration paths.
The question which arise due to the inclusion of N into the magnetic Fe-Cr
alloy is how the N impurities modify the magnetic properties of the system. In
the work of I. Mazinmazin , a comparison of the degrees of spin-polarization
(DSP) calculated for Fe in the static limit through the density of electronic
states, via the current densities and in the ballistic limit is presented. It
is shown that all three definitions of the DSP give very similar behavior for
Fe due to strong hybridization of the $sp$ and $d$ states at the Fermi level.
Thus we expect that in the considered Fe alloy with relatively low
concentration of Cr it is sufficient to study the static spin polarization in
order to capture the main properties of the alloy. Fig. 10 presents the change
of the cell and atomic magnetic moments at the migration of N along the path
(a) and path (b). Although the nitrogen is initially nonmagnetic in the bulk,
it becomes weakly magnetic inside the Cr-Fe alloy with a small magnetic moment
$-0.04$ $\mu_{B}$ induced by the magnetism of the surrounding. It is
noteworthy that the cell magnetic moment is increased to 4 $\mu_{B}$ as N
approaches Cr and the distance [N-Cr] becomes about 1.95 Å. Such an
enhancement of $M_{tot}$ is explained by the strong atomic distortions in the
range between $0.04$ Å(Fe7) up to $0.2$ Å(Fe3) caused by the replacement of N
and by the consequentmagnetoelastic effect. In Fig. 10, the increase of the
distance from N to Cr suppresses the magnetic moment of N and decreases the
cell magnetic polarization to the typical values about $3.5-3.8$ $\mu_{B}$
obtained in LSDA-calculations for the artificial Fe-Cr alloys. The obtained
drastic change of the magnetic polarization clearly demonstrates a crucial
importance of the location of nonmagnetic impurities like $N$ for the
electronic properties of alloy. As follows from our findings, a control of the
location of N, for example by external electric field, can lead to externally
tuned changes of the magnetic polarization, a feature which is of central
importance for possible spintronic devices based on the artificial Fe-Cr
alloys.
## 4 Conclusion
We have performed the DFT studies of the bulk ferrite with
$12.5\%$-concentration of monoatomic interstitial Cr periodically located at
the edges of the bcc Feα cell. We have shown that the full atomic relaxation
of the obtained interstitial Fe-Cr stabilizes a new chain-like low-symmetry
structure. In this structure, the monoatomic Cr at the edges of ferrite bcc
cells leads to the local atomic distortions and results in the formation of
parallel chains of Fe6-ochahedra, which are connected by the interchain Fe-Cr
bonds. The significant energy gain caused by such a structural relaxation
approaches 6.17 eV which makes this type of interstitial alloy highly stable
and energetically favorable with the negative formation energy approaching
$-1.15$ eV. The novel electronic state of the system can be characterized as
metallic, where the metallic properties is the result of strong Fe-Cr
hybridization of the structurally relaxed alloy. In the investigations of the
magnetic state of the generated relaxed structures, we have obtained a local
antiferromagnetic order in the close proximity of Cr atoms, whereas the more
distant Fe atoms are coupled ferromagnetically. We also find that the
nonmagnetic impurities like nitrogen can substantially modify the magnetic
properties of the interstitial alloy which can be considered as an additional
manifestation of the strong magnetoelastic effect in this type of alloys. We
propose to consider the generated interstitial alloys as perspective
candidates for fabrication of novel highly durable stainless steels and for
possible applications in spintronic and multifunctional devices.
## 5 Acknowledgements
This work has beed partially supported through the project ”Models of quantum
statistical description of catalytic processes on metallic substrates” of the
Ministry of Education and Sciences of Ukraine and the grant 0108U002091 of the
National Academy of Science of Ukraine. A grant of computer time from the
Ukrainian Academic Greed is acknowledged.
## References
* (1) D. Peckner and I.M. Bernstein. Handbook on Stainless Steels, McGraw-Hill Book Co., New York, 1977.
* (2) A. Mai, V.A.C. Haanappel, S. Uhlenbruck, F. Tietz, and D. Stöver, Solid State Ionics, 176, 1341 (2005).
* (3) H. Yokokawa, H. Tu, B. Iwanschitz, and A. Mai, J. Power Sources, 182, 400 (2008).
* (4) R.H. Victora and L.M. Falicov, Phys.Rev.B 31, 7335 (1985).
* (5) A.T. Paxton and M.W.Finnis, Phys.Rev.B 77, 024428 (2008).
* (6) C. Paduani and J.C. Krause, Braz. Journ. of Physics, 36, 1262 (2006).
* (7) A. Davies, J.A. Stroscio, D.T. Pierce, and R.J. Celotta, Phys.Rev.Lett, 76 4175 (1996).
* (8) T.P.C. Klaver, P. Olsson, and M.W. Finnis, Phys.Rev.B 76, 214110 (2007).
* (9) P. Olsson, C. Domain, and J. Wallenius, Phys.Rev.B 75, 014110 (2007).
* (10) Z.A.Duriagina and M.I.Pashechko, Metal Science and Treatment of Metals, 4, 34 (2000).
* (11) Y. Sugimoto et al., Science 322, 413 (2008).
* (12) N. Pavlenko, unpublished.
* (13) P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, and J. Luitz, WIEN2K, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties, ISBN 3-9501031-1-2 (TU Wien, Austria, 2001).
* (14) J.-H. Park et al., Nature 392, 794 (1998).
* (15) V.I. Anisimov, I.V. Solovyov, M.A. Korotin, M.T. Czyzyk, and G.A. Sawatzky, Phys.Rev. B 48, 16929 (1993).
* (16) M.T. Czyzyk and G.A. Sawatzky, Phys.Rev. B 49, 14211 (1994).
* (17) N. Pavlenko, Phys. Rev. B 80 075105 (2009).
* (18) N. Pavlenko, I. Elfimov, T. Kopp, and G.A. Sawatzky, Phys. Rev. B 75, 140512(R) (2007).
* (19) T. Bandyopadhyay and D.D. Sarma, Phys.Rev. B 39, 3517 (1989).
* (20) Ze Zhang and S. Satpathy, Phys.Rev. B 44, 13319 (1991).
* (21) M.A. Korotin, V.I. Anisimov, D.I. Khomskii, and G.A. Sawatzky, Phys.Rev.Lett. 80, 4305 (1998).
* (22) A. Ashkin. Optical trapping and manipulation of neutral particles using lasers, World Scientific Pub., Singapore, 2006.
* (23) P. Giannozzi et al., J. Phys. Cond. Matter 21, 395502(2009).
* (24) H. Jonsson, G. Mills, and K.W. Jacobsen, Nudged Elastic Band Method for Finding Minimum Energy Paths of Transitions in Classical and Quantum Dynamics in Condensed Phase Simulations in Classical and Quantum Dynamics in Condensed Phase Simulations, ed. by B.J. Berne, G. Ciccoti, and D.F. Coker (Singapore: World Scientific, 1998).
* (25) J.P. Perdew, S. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865(1996).
* (26) N. Pavlenko, A. Pietraszko, A. Pawlowski, M. Polomska, I.V.Stasyuk, and B. Hilczer, Phys. Rev. B 84, 064303 (2011).
* (27) I. Mazin, Phys. Rev. Lett. 83, 1427(1999).
|
arxiv-papers
| 2011-02-02T14:05:22 |
2024-09-04T02:49:16.800618
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "N.Pavlenko, N.Shcherbovskikh, and Z.A.Duriagina",
"submitter": "Natalia Pavlenko",
"url": "https://arxiv.org/abs/1102.0432"
}
|
1102.0452
|
author subject
XV International Conference on
Gravitational Microlensing
Conference Book
January 20–22, 2011
University of Salerno, Italy
Editors
V. Bozza, S. Calchi Novati, L. Mancini, G. Scarpetta
(Local Organizing Committee)
Organized by the Astrophysics Group of the
Physics Department of the University of Salerno
Sponsored by IIASS (International Institute for Advanced Scientific Studies),
INFN (Istituto Nazionale di Fisica Nucelare), International Ph.D. in
Astrophysics, and University of Salerno
http://smc2011.physics.unisa.it
Foreword
This volume collects the abstracts in extended format of the
$15^{\mathrm{th}}$ Microlensing Conference held in the University of Salerno
on January 20-22, 2011. The Conference has gathered 68 scientists from 17
countries confirming microlensing as a mature and established tool of research
over a broad range of astrophysical issues, from dark matter searches to the
detection of new extrasolar planets of very low mass, down to Earth-size or
below. The 3-days Conference has been preceeded by a 2-days school on
“Modelling Planetary Microlensing Events” focused on this last issue. The
abstracts collected here offer an updated snapshot of the current researches
in this field. The topics include: the status of current surveys, planetary
events, dark matter searches, cosmological microlensing, theoretical
investigations and an outlook towards the future, with in particular a
discussion on the possible role to be played by microlensing searches for
exoplanets in the forecoming space missions, WFIRST and EUCLID. Finally, the
conference has been enriched by a series of topical speeches on related issues
from a non-microlensing point of view: the physics of giant planet accretion
and evolution, “new physics” and dark matter in the LHC era and an update on
the GAIA mission and its potential to characterize planetary systems with
high-precision astrometry.
Scientific Rationale
25 years after the seminal intuition by Bohdan Paczynski, microlensing has
rapidly evolved into a new promise for modern astrophysics. Indeed, the
detection of celestial bodies by their gravitational effects on the light of
background sources has proved to be a very powerful tool for the study of many
aspects of our Galaxy and beyond.
The original proposal of using microlensing to estimate the amount of baryonic
dark matter in the form of compact substellar bodies is still topical. The
observation of microlensing events toward nearby galaxies still represents a
hard challenge for present observational facilities. Very strong efforts are
under way for upgrading the current strategies in particular toward the galaxy
M31.
Hundreds of microlensing events are discovered toward the Galactic Bulge every
year. This considerable amount of statistics can be used to characterize the
stellar populations of the bulge and the disc of our Galaxy. After several
years of observations, the significance of the microlensing sample in the
characterization of our Galaxy is growing more and more.
Yet, the most intriguing perspective offered by microlensing is represented by
its power to discover new extrasolar planets of very low mass, down to Earth-
size or below. Several planetary events have already been detected and
studied, while many more are expected as soon as future dedicated telescopes
become operational. Interestingly, microlensing is already being used to
estimate the abundance of planets around stars in the disc of our Galaxy and
to characterize their distributions in distance and mass. As the planetary
anomalies typically last only a few hours, the cooperation among all observing
groups is mandatory, in order to characterize the events properly and maximize
the scientific achievements. Even amateur astronomers are now giving their
fundamental contribution. In this respect, microlensing stands as a perfect
example of how science can unite the whole mankind in a common path toward
pure knowledge.
Salerno Microlensing Conference 2011 will gather all people active in this
field, providing the state of the art of microlensing searches and the
perspectives opened by new methodologies and new observational and
computational facilities. Colloquia on dark matter searches and planet
formation theories are also foreseen as a central part of the conference. The
three-days conference will be preceded by a school dedicated to the delicate
issue of efficient modelling of planetary microlensing events, which requires
major efforts and new ideas from new talents in order to get access to the
precious physical information hidden in microlensing light curves. For one
week in January 2011, Salerno will thus be the place in which the present and
the future of microlensing will be unveiled.
Valerio Bozza
---
Local Organizing Committee
Welcome address
On behalf of the Rector of Salerno University, professor Raimondo Pasquino,
the Dean of the Science Faculty, professor Mariella Transirico, the Decan of
the Department of Physics, professor Ferdinando Mancini and the Organizing
Committee, I welcome all the participants to this exciting Conference.
This University has the peculiar status to be at the same time one of the
youngest Universities in Italy and the oldest one.
The re-establishment of Salerno University in the Modern Era dates back to
about forty years ago, and as you can see, in a very short time it has grown
in the Irno valley as a university campus, with about forty thousand students,
more than one thousand professors and researchers, distributed in ten
faculties.
Five years ago the Campus was completed with the Medical Science Faculty,
realizing the ideal link with one of the oldest Medical Schools of the West,
the so called “Scuola Medica Salernitana”.
Salerno was not the seat of the first Medieval University, but the Medical
School was the first to organize studies and diffuse culture on international
scale during the medieval period.
According to the legend, the “Scuola Medica Salernitana” traces its origins to
four erudite: the Greek Pontus, the Jew Helinus, the Arab Abdela, and the
Latin Salernus. Thanks to the encounter of these four cultures the Medical
School established its knowledge divulgating the Greek, Jewish, Arab and Latin
medical knowledge. The divulgation in the West of the Islamic and Greek
Medical Science is certainly due to Constantino the African (XI century),
thanks to his translations into Latin of the most important Arabic and Greek
medical treatises.
I like to read some words extracted from the nobel lecture of Abdus Salam
(1979)
“Scientific thought and its creation is the common and shared heritage of
mankind. In this respect, the history of science, like the history of all
civilization, has gone through cycles. Perhaps I can illustrate this with an
actual example.
Seven hundred and sixty years ago, a young Scotsman left his native glens to
travel south to Toledo in Spain. His name was Michael, […] Michael reached
Toledo in 1217 AD. […] From Toledo, Michael travelled to Sicily, to the Court
of Emperor Frederick II. Visiting the medical school at Salerno, chartered by
Frederick in 1231, Michael met the Danish physician, Henrik Harpestraeng […]
Henrik had come to Salerno to compose his treatise on blood-letting and
surgery. Henrik’s sources were the medical canons of the great clinicians of
Islam, Al-Razi and Avicenna, which only Michael the Scot could translate for
him. Toledo’s and Salerno’s schools, representing as they did the finest
synthesis of Arabic, Greek, Latin and Hebrew scholarship, were some of the
most memorable of international assays in scientific collaboration”.
The “Scuola Medica Salernitana” was at the height of its fame in the XIV
century and carried on its teaching for nine centuries, until Giocchino
Murat’s decree, dated the 25th of January 1812, prescribed the closure.
After two centuries, we all, coming here in the Salerno University from east
and west countries, renew in some sense the old tradition, establishing an
ideal link with the cultural heritage of the “Scuola Medica Salernitana” .
I hope you will enjoy your stay in Salerno and I wish you good and useful
work.
Gaetano Scarpetta
---
Chair of the Local Organizing Committee
Local Organizing Committee
V. Bozza |
---|---
S. Calchi Novati |
L. Mancini |
G. Scarpetta (Chair) |
Scientific Organizing Committee
---
J.-P. Beaulieu | France
D. Bennett | USA
I. Bond | New Zealand
S. Calchi Novati (Chair) | Italy
M. Dominik | UK
A. Gould | USA
C. Han | South Korea
Ph. Jetzer | Switzerland
E. Kerins | UK
M. Moniez | France
T. Sumi | Japan
Y. Tsapras | UK
A. Udalski | Poland
L. Wyrzykowski | UK
Scientific Secretary
---
O. De Pasquale
V. Di Marino
T. Nappi
S. Russo
CONTENTS
Foreword iii
Scientific Rationale v
welcome vi
Organization ix
Participants xvi
School Program xix
Conference Program xx
School on Modelling Planetary Microlensing Events 1
Introduction to microlensing 2
Philippe Jetzer
From microlensing observations to science 3
Martin Dominik
The theory and phenomenology of planetary microlensing 4
Scott Gaudi
From raw images to lightcurves: how to make sense of your data 5
Yiannis Tsapras
The efficient modeling of planetary microlensing events 6
David Bennett
Contour integration and downhill fitting 7
Valerio Bozza
Microlensing modelling and high performance computing 8
Ian Bond
Conference topical speeches 9
Giant planet accretion and dynamical evolution: considerations
on systems around small-mass stars 10
Alessandro Morbidelli
The dark matter - LHC endeavour to unveil TeV new physics 14
Antonio Masiero
Characterization of planetary systems with high-precision
astrometry: the Gaia potential 15
Alessandro Sozzetti
Status of current surveys 18
Status of the OGLE-IV survey 19
Andrzej Udalski
MOA-II observation in 2010 season 20
Takahiro Sumi
The RoboNet 2010 season 22
Yiannis Tsapras
Cosmological Microlensing 25
Dark matter determinations from Chandra observations of
quadruply lensed quasars 26
David Pooley
Cosmic equation of state from strong gravitational lensing systems 27
Marek Biesiada & Beata Malec
Galactic Microlensing: the dark matter search 31
PAndromeda - the Pan-STARRS M31 survey for dark matter 32
Arno Riffeser, Stella Seitz, Ralf Bender, C.-H. Lee, Johannes Koppenhoefer
Final OGLE-II and OGLE-III results on microlensing towards
the LMC and SMC 34
Lukasz Wyrzykowski
Analysis of microlensing events towards the LMC 37
Luigi Mancini & Sebastiano Calchi Novati
Simulation of short time scale pixel lensing towards the Virgo cluster 39
Sedighe Sajadian & Sohrab Rahvar
M31 pixel lensing and the PLAN project 40
Sebastiano Calchi Novati
Planetary events 44
MOA-2009-BLG-266LB: the first cold Neptune with a measured
mass 45
David Bennett
Increasing the detection rate of low-mass planets in high-magnification
events and MOA-2006-BLG-130 46
Julie Baudry & Philip Yock
The complete orbital solution for OGLE-2008-BLG-513 49
Jennifer Yee
Planetary microlensing event MOA-2010-BLG-328 51
Kei Furusawa
Binary microlensing event OGLE-2009-BLG-020 gives orbit predictions
verifiable by follow-up observations 53
Jan Skowron
Theoretical investigations 59
Microlensing and planet populations - What do we know,
and how could we learn more 60
Martin Dominik
The Frequency of extrasolar planet detections with microlensing
simulations 65
Rieul Gendron & Shude Mao
A semi-analytical model for gravitational microlensing events 66
Denis Sullivan, Paul Chote, Michael Miller
GPU-assisted contouring for modeling binary microlensing events 70
Markus Hundertmark, Frederic V. Hessman, Stefan Dreizler
Red noise effect in space-based microlensing observations 74
Achille Nucita, Daniele Vetrugno, Francesco De Paolis, Gabriele Ingrosso,
Berlinda M. T. Maiolo, Stefania Carpano
Light curve errors introduced by limb-darkening models 76
David Heyrovsky
Isolated, stellar-mass black holes through microlensing 77
Kailash Sahu, Howard E. Bond, Jay Anderson, Martin Dominik,
Andrzej Udalski, Philip Yock
The observability of isolated compact remnants with microlensing 80
Nicola Sartore & Aldo Treves
Gravitational microlensing by the Ellis wormhole 82
Fumio Abe
The deflection of light ray in strong field: a material medium
approach 100
Asoke Kumar Sen
Rapidly rotating lenses - repeating orbital motion features in
close binary microlensing 104
Matthew Penny, Eamonn Kerins, Shude Mao
Towards the future: new facilities/instrumentation/procedures 107
Microlensing with the SONG global network 108
Uffe G. Jørgensen, Kennet B. W. Harpsøe, Per K. Rasmussen, Michael I.
Andersen, Anton N. Sørensen, Jørgen Christensen-Dalsgaard, Søren Frandsen,
Frank Grundahl, Hans Kjeldsen
Next-generation microlensing pilot planet search and the frequency of
planetary systems 112
Dan Maoz & Yossi Shvartzvald
Kohyama Astronomical Observatory: current status 116
Atsunori Yonehara, Mizuki Isogai, Akira Arai, Hiroki Tohyama
Optimal imaging for gravitational microlensing 118
Kennet B.W. Harpsøe, Uffe G. Jørgensen, Per K. Rasmussen, Michael I. Andersen,
Anton N. Sørensen, Jørgen Christensen-Dalsgaard, Søren Frandsen, Frank
Grundahl, Hans Kjeldsen
IPAC s role as the science center for NASA s WFIRST mission 121
Kaspar von Braun
EUCLID microlensing planet hunt 122
Jean-Philippe Beaulieu & Matthew Penny
Space-based microlensing exoplanet survey: WFIRST and/or Euclid 124
David Bennett
Microlensing with Gaia satellite 125
Lukasz Wyrzykowski
Poster session 127
Critical curve topology in special triple lens configurations 128
Kamil Danek
PAndromeda - a dedicated deep survey of M31 with
Pan-STARRS 1 129
Chien-Hsiu Lee, Arno Riffeser, Stella Seitz, Ralf Bender, Johannes
Koppenhoefer
SMC2011 Participants
Abe, Fumio | Nagoya University, Japan
---|---
Baudry, Julie | University of Orsay, France
Bachelet, Etienne | University of Toulouse, France
Beaulieu, Jean-Philippe | Institut d’Astrophysique de Paris, France
Bennett, David | University of Notre Dame, USA
Bond, Ian | Massey University, New Zealand
Bonino, Donata | INAF - Turin Astronomical Observatory, Italy
Bozza, Valerio | University of Salerno, Italy
Browne, Paul | University of St Andrews, UK
Calchi Novati, Sebastiano | University of Salerno, Italy
Danek, Kamil | Charles University, Czech Republic
De Paolis, Francesco | University of Salento, Italy
Dominik, Martin | University of St Andrews, UK
Dominis Prester, Dijana | University of Rijeka, Croatia
Fouqu$\acute{\mathrm{e}}$, Pascal | University of Toulouse, France
Furusawa, Kei | Nagoya University, Japan
Gardiol, Daniele | INAF - Turin Astronomical Observatory, Italy
Gaudi, Scott | Ohio State University, USA
Gendron, Riel | University of Manchester, UK
Gould, Andrew | Ohio State University, USA
Harpsøe, Kennet | Niels Bohr Institute, Denmark
Harris, Pauline | Victoria University of Wellington, New Zealand
Henderson, Calen | Ohio State University, USA
Heyrovsky, David | Charles University, Czech Republic
Horne, Keith | University of St Andrews, UK
Hundertmark, Markus | University of Göttingen, Germany
Jetzer, Philippe | University of Zurich, Switzerland
Jørgensen, Uffe G. | Niels Bohr Institute, Denmark
Lambiase, Gaetano | University of Salerno, Italy
Lee, Chien-Hsiu | University Observatory Munich
Liebig, Christine | University of St Andrews
Lubini, Mario | University of Zurich, Switzerland
Malec, Beata | Copernicus Center for Interdisciplinary Studies, Poland
Mancini, Luigi | University of Salerno, Italy
Maoz, Dan | Tel-Aviv University, Israel
Masiero, Antonio | University of Padova, Italy
Mirzoyan, Sergey | University of Salerno, Italy
Miyake, Noriyuki | Nagoya University, Japan
Morbidelli, Alessandro | Observatoire de la Cote d’Azur, France
Muraki, Yasushi | Konan University, Japan
Nucita, Achille | University of Salento, Italy
Orio, Marina | INAF - Padova Astronomical Observatory, Italy
Paulin-Henriksson, St$\acute{\mathrm{e}}$phane | CEA - Paris, France
Payandeh, Farrin | Payame Noor University Tabriz, Iran
Penny, Matthew | University of Manchester, UK
Pooley, David | Eureka Scientific, USA
Retana Montenegro, Edwin F. | Universidad de Costa Rica, Costa Rica
Riffeser, Arno | Max Planck Institute for Extraterrestrial Physics, Germany
Sahu, Kailash | Space Telescope Science Institute, USA
Sajadian, Sedighe | Sharif University of Technology, Iran
Sartore, Nicola | INAF - IASF Milano, Italy
Scarpetta, Gaetano | University of Salerno, Italy
Sen, Asoke Kumar | Sen Assam University, India
Shvartzvald, Yossi | Tel-Aviv University, Israel
Skowron, Jan | Ohio State University, USA
Sozzetti, Alessandro | INAF - Turin Astronomical Observatory, Italy
Sullivan, Denis | Victoria University of Wellington, New Zealand
Sumi, Takahiro | Nagoya University, Japan
Tortora, Crescenzo | University of Zurich, Switzerland
Tsapras, Yiannis | Queen Mary University, UK
Udalski, Andrzej | Warsaw University Observatory, Poland
Vetrugno, Daniele | University of Salento, Italy
Vilasi, Gaetano | University of Salerno, Italy
von Braun, Kaspar | California Institute of Technology, USA
Yee, Jennifer | Ohio State University, USA
Yock, Philip | University of Auckland, New Zealand
Yonehara, Atsunori | Kyoto Sangyo University, Japan
Wyrzykowski, Lukasz | University of Cambridge, UK
School on Modelling Planetary Microlensing Events
Program
Tuesday, January 18, 2011
---
09:00 | Introduction to microlensing | Philippe Jetzer
10:50 | Coffee Break
11:15 | From microlensing observations to science | Martin Dominik
13:00 | Lunch Break
14:45 | The theory and phenomenology of planetary | Scott Gaudi
| microlensing
16:45 | Coffee Break
17:15 | From raw images to lightcurves: how to make | Yiannis Tsapras
| sense of your data
Wednesday, January 19, 2011
09:00 | The efficient modeling of planetary microlensing | David Bennett
| events
10:50 | Coffee Break
11:15 | Contour integration and downhill fitting | Valerio Bozza
13:00 | Lunch Break
14:45 | Microlensing modelling and high performance | Ian Bond
| computing
16:45 | Coffee Break
SMC2011 Program
Thursday, January 20, 2011
---
09:00 | Registration
09:30 | Greetings
09:40 | Communications |
| | Chair: Gaetano Scarpetta
Status of current Surveys I | |
09:50 | Status of the OGLE-IV survey | Andrzej Udalski
10:20 | MOA-II observation in 2010 season | Takahiro Sumi
10:50 | Coffee Break
Topical Speech I | |
11:20 | Formation and evolution of our | Alessandro Morbidelli
| solar system
Theoretical investigations I | |
12:10 | Microlensing and planet populations: | Martin Dominik
| what do we know, and how could we
| learn more?
12:40 | The frequency of extrasolar planet | Rieul Gendron
| detections with microlensing simulations
13:00 | Lunch Break
| | Chair: Philippe Jetzer
Dark Matter Search | |
14:40 | PAndromeda - the Pan-STARRS M31 | Arno Riffeser
| survey for dark matter
15:10 | Final OGLE-II and OGLE-III results on | Lukasz Wyrzykowski
| microlensing towards the LMC and SMC
15:30 | Analysis of microlensing events towards the LMC | Luigi Mancini
15:50 | Simulation of short time scale pixel lensing | Sedighe Sajadian
| towards the Virgo cluster
16:10 | M31 pixel lensing and the PLAN project | Sebastiano Calchi Novati
16:30 | Coffee Break
Towards the future I | |
17:00 | Microlensing with the SONG global network | Uffe G. Jørgensen
17:20 | Next-generation microlensing pilot planet search | Dan Maoz
| and the frequency of planetary systems
17:40 | Kohyama Astronomical Observatory: | Atsunori Yonehara
| current status
18:00 | The lucky imaging technique for microlensing | Kennet Harpsøe
| observations
Friday, January 21, 2011
| | Chair: P. Yock
Status of current Surveys II | |
09:00 | The 2010 MicroFUN season | Jennifer Yee
09:30 | The RoboNet 2010 season | Yiannis Tsapras
Theoretical investigations II | |
10:00 | A semi-analytical model for gravitational | Denis Sullivan
| microlensing events
10:20 | GPU-assisted contouring for modeling | Markus Hundertmark
| binary microlensing events
10:40 | Rapidly rotating lenses - repeating orbital motion | Matthew Penny
| features in close binary microlensing
11:00 | Coffee Break
Topical Speech II | |
11:30 | The dark matter - LHC endeavour to unveil | Antonio Masiero
| TeV new physics
Cosmological microlensing | |
12:20 | Dark matter determinations from Chandra | David Pooley
| observations of quadruply lensed quasars
12:40 | Cosmic equation of state from strong | Beata Malec
| gravitational lensing systems
13:00 | Lunch Break
| | Chair: Pascal Fouqu$\acute{\mathrm{{\bf e}}}$
Planetary events | |
14:30 | MOA-2009-BLG-266LB: the first cold Neptune | David Bennett
| with a measured mass
14:50 | Increasing the detection rate of low-mass | Julie Baudry
| planets in high-magnification events and
| MOA-2006-BLG-130
15:10 | The complete orbital solution for | Jennifer Yee
| OGLE-2008-BLG-513
15:30 | Planetary microlensing event MOA-2010-BLG-328 | Kei Furusawa
15:50 | Binary microlensing event OGLE-2009-BLG-020 | Jan Skowron
| gives orbit predictions verifiable by follow-up
| observations
16:10 | Coffee Break
17:30 | Tour of the old town of Salerno
20:00 | Social dinner
Saturday, January 22, 2011
| | Chair: Francesco De Paolis
Towards the future II | |
09:00 | IPAC s role as the science center for | Kaspar von Braun
| NASA s WFIRST mission
09:30 | EUCLID microlensing planet hunt | Jean-Philippe Beaulieu
09:50 | Simulating the planet hunting capability of Euclid | Matthew Penny
10:10 | Space-based microlensing exoplanet survey: | David Bennett
| WFIRST and/or Euclid
10:40 | Coffee Break
Topical Speech III | |
11:10 | Characterization of planetary systems with | Alessandro Sozzetti
| high-precision astrometry: the Gaia Potential
12:00 | Microlensing with Gaia satellite | Lukasz Wyrzykowski
Theoretical investigations III | |
12:20 | Red noise effect in space-based microlensing | Achille Nucita
| observations
12:40 | Light curve errors introduced by limb-darkening | David Heyrovsky
| models
13:00 | Lunch Break
| | Chair: Scott Gaudi
14:40 | Isolated, stellar-mass black holes through | Kailash Sahu
| microlensing
15:00 | The observability of isolated compact | Nicola Sartore
| remnants with microlensing
15:20 | Gravitational microlensing by the Ellis wormhole | Fumio Abe
15:40 | The deflection of light ray in strong field: | Asoke Kumar Sen
| a material medium approach
16:00 | How to stop a runaway (Monte Carlo Markov) | Keith Horne
16:20 | Coffee Break
16:50 | Open session |
Poster Session
| Critical curve topology in special triple lens | Kamil Danek
| configurations
| PAndromeda - a dedicated deep survey of M31 | Chien-Hsiu Lee
| with Pan-STARRS 1
|
arxiv-papers
| 2011-02-02T14:53:01 |
2024-09-04T02:49:16.808138
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Valerio Bozza, Sebastiano Calchi Novati, Luigi Mancini, Gaetano\n Scarpetta (Eds.)",
"submitter": "Luigi Mancini",
"url": "https://arxiv.org/abs/1102.0452"
}
|
1102.0509
|
# Considerations on the subgroup commutativity degree and related notions
Francesco G. Russo Dipartimento di Matematica e Informatica, Universitá di
Palermo, Via Archirafi 34, 90123, Palermo, Italy. francescog.russo@yahoo.com
###### Abstract.
The concept of subgroup commutativity degree of a finite group $G$ is arising
interest in several areas of group theory in the last years, since it gives a
measure of the probability that a randomly picked pair $(H,K)$ of subgroups of
$G$ satisfies the condition $HK=KH$. In this paper, a stronger notion is
studied and relations with the commutativity degree are found.
###### Key words and phrases:
Subgroup commutativity degree, permutable subgroups, centralizers, subgroup
lattices.
###### 2010 Mathematics Subject Classification:
Primary: 20D60, 20P05; Secondary: 20D08.
## 1\. Introduction
In the present paper we deal only with finite group, even if there is a recent
interest to the subject in the context of infinite groups [1, 11, 10, 17, 25].
The commutativity degree of a group $G$, given by
(1.1) $d(G)=\frac{|\\{(x,y)\in G\times G\ |\
[x,y]=1\\}|}{|G|^{2}}=\frac{1}{|G|^{2}}\sum_{x\in G}|\\{y\in G\ |\
y^{-1}xy=x\\}|$ $=\frac{1}{|G|^{2}}\sum_{x\in G}|C_{G}(x)|,$
was studied extensively in [2, 4, 6, 9, 12, 16, 18, 19, 20, 21, 22, 23, 26] an
generalized in various ways. Its importance is testified in the theory of the
groups of prime power orders in [5, Chapter 2], where it is called measure of
commutativity by Y. Berkovich in order to emphasize the fact that it really
gives a measure of how far is the group from being abelian. In [7, 8, 9] it
was introduced the following variation,
(1.2) $d(H,K)=\frac{|\\{(h,k)\in H\times K\ |\
[h,k]=1\\}|}{|H||K|}=\frac{1}{|H||K|}\underset{h\in H}{\sum}|C_{K}(h)|,$
where $H$ and $K$ are two arbitrary subgroups of $G$. Of course,
$d(G,G)=d(G)$, whenever $H=K=G$, and, consequently, the bounds known in
literature for $d(G)$ may be sharpened by examining $d(H,K)$. In recent years,
there is an increasing interest in studying the problem from the point of view
of the lattice theory (see [13, 14, 15, 27, 28]). Tǎrnǎuceanu [30, 31] has
introduced the subgroup commutativity degree of a finite group, that is, the
ratio
(1.3) $sd(G)=\frac{|\\{(H,K)\in\mathcal{L}(G)\times\mathcal{L}(G)\ |\
HK=KH\\}|}{|\mathcal{L}(G)|^{2}},$
where $\mathcal{L}(G)$ denote the subgroup lattice of $G$. It turns out that
(1.4)
$sd(G)=\frac{1}{|\mathcal{L}(G)|^{2}}\underset{H\in\mathcal{L}(G)}{\sum}|\mathcal{C}(H)|,$
where
(1.5) $\mathcal{C}(H)=\\{K\in\mathcal{L}(G)\ |\ HK=KH\\}.$
Variations on this theme have been considered in [3, 13, 14, 15, 24, 27, 28],
involving weaker notions of permutability among subgroups.
Of course, if $[H,K]=1$, then $HK=KH$, where $[H,K]=\langle[h,k]\ |\ h\in
H,k\in K\rangle$. Conversely, $HK=KH$ does not imply that $[H,K]=1$. In fact,
the equality among the sets $\\{hk\ |\ h\in H,k\in K\\}$ and $\\{kh\ |\ k\in
K,h\in H\\}$ does not imply, in general, that all the elements of $H$ permute
with all elements of $K$. Many examples can be given. Therefore it is
meaningful to define the following ratio
(1.6) $ssd(G)=\frac{|\\{(H,K)\in\mathcal{L}(G)^{2}\ |\
[H,K]=1\\}|}{|\mathcal{L}(G)|^{2}},$
which we will call strong subgroup commutativity degree of $G$. It is easy to
see that
(1.7)
$ssd(G)=\frac{1}{|\mathcal{L}(G)|^{2}}\underset{H\in\mathcal{L}(G)}{\sum}|Comm_{G}(H)|,$
where
(1.8) $Comm_{G}(H)=\\{K\in\mathcal{L}(G)\ |\ [H,K]=1\\},$
and that $ssd(G)$ is the probability that the subgroup $[H,K]$ of an
arbitrarily chosen pair of subgroups $H,K$ of a group $G$ is equal to the
trivial subgroup of $G$. Equivalently, $ssd(G)$ expresses the probability
that, randomly picked two subgroups of $G$, the subgroup generated by their
commutators is trivial, and, in particular, the two subgroups are permutable.
The present paper is devoted to study this notion, showing that it is related
to the previous investigations in the area of the measure theory of finite
groups.
## 2\. Some basic properties
There are some considerations which come by default with the strong subgroup
commutativity degree. A group $G$ is quasihamiltonian, if all pairs of its
subgroups are permutable. $G$ is called modular, if $\mathcal{L}(G)$ satisfies
the well–known modular law (see [29]). Quasihamiltonian groups were classified
by Iwasawa (see [5, Chapter 6] or [29, Chapter 2]), who proved that they are
nilpotent and modular. This is equivalent to say that a group $G$ is
quasihamiltonian if and only if all its Sylow $p$-subgroups are modular (see
[29, Exercise 3 at p.87]), being $p$ a prime. Therefore the knowledge of
quasihamiltonian groups may be reduced to that of modular $p$-groups. In
literature, for $m\geq 3$ the groups
(2.1) $M(p^{m})=\langle x,y\ |\
x^{p^{m-1}}=y^{p}=1,y^{-1}xy=x^{p^{m-2}+1}\rangle=\langle
y\rangle\ltimes\langle x\rangle,$
are nonabelian modular $p$-groups and their properties have interested the
researches of many authors in various contexts (see [5, 29, 30]). An immediate
observation is the following. If $G=M(p^{m})$, then $[\langle x\rangle,\langle
y\rangle]\not=1$ and consequently $sd(G)=1$ but $ssd(G)\not=1$. In this sense,
it is important to know when the strong subgroup commutativity degree is
trivial.
###### Proposition 2.1.
A group $G$ has $ssd(G)=1$ if and only if it is abelian.
###### Proof.
We have that $ssd(G)=1$ if, and only if, $[H,K]=1$ for all subgroups $H$ and
$K$ of $G$, if, and only if, $[h,k]=1$ for all $h\in H$, $k\in K$ and for all
$H$ and $K$ in $G$. This implies, in particular, that $[h,k]=1$ for all
$h,k\in G$, that is, $G$ is abelian. Conversely, if $G$ is abelian, then it is
clear that $ssd(G)=1$. ∎
On another hand, the following relation shows that $ssd(G)$ is related to
$d(H,K)$ in a deep way.
###### Theorem 2.2.
Let $H$ and $K$ be two subgroups of a group $G$. Then
$ssd(G)<\frac{|G|^{2}}{|\mathcal{L}(G)|^{2}}\sum_{H,K\in\mathcal{L}(G)}d(H,K).$
###### Proof.
We claim that
(2.2) $\bigcup_{K\in\mathcal{L}(G)}C_{K}(H)=Comm_{G}(H).$
Let $T={\underset{K\in\mathcal{L}(G)}{\bigcup}}C_{K}(H)$ and $t\in T$. Then
there exists a $K_{t}\in\mathcal{L}(G)$ containing $t$ such that $t\in
C_{K_{t}}(H)$, that is, $[t,H]=1$, which means that $t$ permutes with all
elements of $H$. In particular, the powers of $t$ permutes with all elements
of $H$ and so $[\langle t\rangle,H]=1$, which means $\langle t\rangle$ is in
$Comm_{G}(H)$. We conclude that $T\subseteq Comm_{G}(H)$. Conversely, if
$K\in\mathcal{L}(G)$ is in $Comm_{G}(H)$, then $[K,H]=1$ and so $K\subseteq
C_{G}(H)$, then $K\subseteq T$. The claim follows.
Therefore
(2.3) $|\mathcal{L}(G)|^{2}\
ssd(G)=\sum_{H\in\mathcal{L}(G)}|Comm_{G}(H)|=\sum_{H\in\mathcal{L}(G)}\left|\bigcup_{K\in\mathcal{L}(G)}C_{K}(H)\right|$
$<\sum_{K\in\mathcal{L}(G)}\sum_{H\in\mathcal{L}(G)}|C_{K}(H)|$
and we note that the equality cannot occur here as the identity $1\in
C_{K}(H)$ for all $H$ and $K$ in $\mathcal{L}(G)$. Since $C_{K}(H)\subseteq
C_{K}(h)$ whenever $h\in H$, we may continue, finding the following upper
bound
(2.4)
$\leq\sum_{K\in\mathcal{L}(G)}{\underset{H\in\mathcal{L}(G)}{\underset{h\in
H}{\sum}}}|C_{K}(h)|=\sum_{H,K\in\mathcal{L}(G)}\Big{(}\sum_{h\in
H}|C_{K}(h)|\Big{)}$ $=\sum_{H,K\in\mathcal{L}(G)}d(H,K)\ |H|\ |K|\leq|G|^{2}\
\sum_{H,K\in\mathcal{L}(G)}d(H,K).$
∎
###### Remark 2.3.
We want just to illustrate two points of views which allow us to decide
whether a group $G$ is abelian or not. The first deals with the subgroups:
from Proposition 2.1 $G$ is abelian if and only if $ssd(G)$ is trivial. The
second deals with the elements: $G$ is abelian if and only if $d(G)$ is
trivial. Theorem 2.2 is relevant, because it correlates $d(G)$ with $ssd(G)$.
This is very helpful, because we have literature on $d(G)$ but few is known
about $ssd(G)$ and $sd(G)$.
In virtue of the previous remark, the following result is significative and
answers partially some open questions in [31]. We will see, concretely, that
the argument of Theorem 2.2 is very general and can be adapted to the context
of $sd(G)$.
###### Theorem 2.4.
Let $H$ and $K$ be two subgroups of a group $G$. Then
$sd(G)\geq\frac{1}{|\mathcal{L}(G)|^{2}}\sum_{H\in\mathcal{L}(G)}\left|\bigcap_{h\in
H}C_{K}(h)\right|$
with
$\sum_{H,K\in\mathcal{L}(G)}d(H,K)\ |H|\
|K|\geq\sum_{H,K\in\mathcal{L}(G)}\left|\bigcap_{h\in H}C_{K}(h)\right|.$
###### Proof.
From Theorem 2.2 (more precisely from (3.18)), we may restrict to prove only
the first inequality. In order to do this, we claim that
(2.5)
$C_{K}(H)\subseteq\bigcup_{K\in\mathcal{L}(G)}C_{K}(H)\subseteq\mathcal{C}(H).$
The first inclusion is trivial. Let
$S={\underset{K\in\mathcal{L}(G)}{\bigcup}}C_{K}(H)$ and $s\in S$. Then there
exists a $K_{s}\in\mathcal{L}(G)$ containing $s$ such that $s\in
C_{K_{s}}(H)$, that is, $[s,H]=1$, which means that $s$ permutes with all
elements of $H$. In particular, $[\langle s\rangle,H]=1$ then $\langle
s\rangle H=H\langle s\rangle$, which means $\langle
s\rangle\in\mathcal{C}(H)$. We conclude that $S\subseteq\mathcal{C}(H)$.
Therefore
(2.6) $|\mathcal{L}(G)|^{2}\
sd(G)=\sum_{H\in\mathcal{L}(G)}|\mathcal{C}(H)|\geq\sum_{H\in\mathcal{L}(G)}\left|\bigcup_{K\in\mathcal{L}(G)}C_{K}(H)\right|\geq\sum_{H\in\mathcal{L}(G)}|C_{K}(H)|$
but we observe that in general the following is true
(2.7) ${\underset{h\in H}{\bigcap}}C_{K}(h)=C_{K}(H)$
so that
(2.8) $=\sum_{H\in\mathcal{L}(G)}\left|\bigcap_{h\in H}C_{K}(h)\right|.$
On another hand, we note that
(2.9) $\sum_{H,K\in\mathcal{L}(G)}d(H,K)\ |H|\
|K|=\sum_{H,K\in\mathcal{L}(G)}\ \Big{(}\sum_{h\in H}|C_{K}(h)|\Big{)}$
$=\sum_{K\in\mathcal{L}(G)}\
\Big{(}{\underset{H\in\mathcal{L}(G)}{\underset{h\in
H}{\sum}}}|C_{K}(h)|\Big{)}\geq\sum_{K\in\mathcal{L}(G)}\
\Big{(}\sum_{H\in\mathcal{L}(G)}\left|\bigcap_{h\in H}C_{K}(h)\right|\Big{)}.$
∎
In the rest of this section we reformulate $ssd(G)$ in terms of arithmetic
functions. It is possible to rewrite $ssd(G)$ in the following form:
(2.10)
$ssd(G)=\frac{1}{|\mathcal{L}(G)|^{2}}{\underset{X,Y\in\mathcal{L}(G)}{\sum}\varphi(X,Y)},$
where $\varphi:\mathcal{L}(G)^{2}\rightarrow\\{0,1\\}$ is the function defined
by
(2.11) $\varphi(X,Y)=\left\\{\begin{array}[]{lcl}1,&&\mathrm{if}\ [X,Y]=1,\\\
0,&&\mathrm{if}\ [X,Y]\not=1.\end{array}\right.$
The reader may note that $\varphi(X,Y)=\varphi(Y,X)$, that is, $\varphi$ is
symmetric in the variables $X$ and $Y$. There is a corresponding property of
symmetry for the subgroup commutativity degree in [30, Section 2], but, in
general, this property depends on the permutability which we are going to
study. For instance, this does not happen for weaker forms of permutability
with respect to the maximal sugroups, as shown in [24]. However, the
introduction of the function $\varphi$ allows us to simplify the notations. In
fact, if $Z$ is a given subgroup of $G$ and we consider the sets
$\mathcal{B}_{1}=\\{(X\in\mathcal{L}(G):Z\subseteq X\\}$ and
$\mathcal{B}_{2}=\\{X\in\mathcal{L}(G):X\subset Z\\},$ then
$\mathcal{B}_{1}\cup\mathcal{B}_{2}\subseteq\mathcal{L}(G)$ and so
(2.12) $|\mathcal{L}(G)|^{2}\
ssd(G)\geq\sum_{X,Y\in\mathcal{B}_{1}\cup\mathcal{B}_{2}}\varphi(X,Y)$
$=\sum_{X,Y\in\mathcal{B}_{1}}\varphi(X,Y)+\sum_{X,Y\in\mathcal{B}_{2}}\varphi(X,Y)+2\sum_{X\in\mathcal{B}_{1}}\sum_{Y\in\mathcal{B}_{2}}\varphi(X,Y).$
A consequence of this equation is examined below and overlaps a similar
situation for $sd(G)$ in [30].
###### Proposition 2.5.
Let $G$ be a group and $N$ be a normal subgroup of $G$. Then
$ssd(G)\geq\frac{1}{|\mathcal{L}(G)|^{2}}\
\Big{(}\Big{(}|\mathcal{L}(N)|+|\mathcal{L}(G/N)|-1\Big{)}^{2}$
$+(ssd(N)-1)|\mathcal{L}(N)|^{2}+(ssd(G/N)-1)|\mathcal{L}(G/N)|^{2}\Big{)}.$
###### Proof.
We are going to rewrite more properly the terms in the left side of (2.12).
(2.13) $|\mathcal{L}(G/N)|^{2}\
ssd(G/N)={\underset{X,Y\in\mathcal{B}_{1}}{\sum}}\varphi(X,Y);$ (2.14)
$|\mathcal{L}(N)|^{2}\
ssd(G/N)-2|\mathcal{L}(N)|+1={\underset{X,Y\in\mathcal{B}_{2}\cup\\{N\\}}{\sum}}\varphi(X,Y)$
$-2{\underset{X\in\mathcal{B}_{2}\cup\\{N\\}}{\sum}}\varphi(X,N)+1={\underset{X,Y\in\mathcal{B}_{2}}{\sum}}\varphi(X,Y);$
(2.15)
$2|\mathcal{L}(G/N)|(|\mathcal{L}(N)|-1)=2|\mathcal{B}_{1}||\mathcal{B}_{2}|=2{\underset{X\in\mathcal{B}_{1}}{\sum}}{\underset{Y\in\mathcal{B}_{2}}{\sum}}\varphi(X,Y).$
Replacing these expressions in (2.12), the result follows. ∎
We list three consequences of Proposition 2.5, overlapping similar situations
for $sd(G)$ in [30]. Their proof is omitted, since it is enough to note that
for a normal abelian subgroup $N$ of $G$ we have $ssd(G/N)=1$ by Proposition
2.5, and, if it is of prime index in $G$, then $|\mathcal{L}(G/N)|=2$ .
###### Corollary 2.6.
Let $G$ be a group and $N$ be a normal subgroup of $G$ such that $G/N$ and $N$
are abelian. Then
$ssd(G)\geq\frac{1}{|\mathcal{L}(G)|}\Big{(}|\mathcal{L}(N)|+|\mathcal{L}(G/N)|-1\Big{)}^{2}.$
###### Corollary 2.7.
Let $G$ be a group and $N$ be a normal subgroup of $G$ of prime index. Then
$ssd(G)\geq\frac{1}{|\mathcal{L}(G)|^{2}}\Big{(}ssd(N)|\mathcal{L}(N)|^{2}+2|\mathcal{L}(N)|+1\Big{)}.$
###### Corollary 2.8.
A nonabelian solvable group $G$ has
$ssd(G)\geq\frac{1}{|\mathcal{L}(G)|^{2}}\Big{(}ssd(G^{\prime})|\mathcal{L}(G^{\prime})|^{2}+2|\mathcal{L}(G^{\prime})|+1\Big{)}.$
In particular, if $G$ is metabelian, then
$ssd(G)\geq\frac{1}{|\mathcal{L}(G)|^{2}}\Big{(}|\mathcal{L}(G^{\prime})|^{2}+2|\mathcal{L}(G^{\prime})|+1\Big{)}.$
Now we list some general bounds, related to subgroups and quotients. In a
different context, these relations have been found in [24].
###### Theorem 2.9.
Let $H$ be a subgroup of a group $G$. Then
$\frac{|\mathcal{L}(H)|^{2}}{|\mathcal{L}(G)|^{2}}\ ssd(H)\leq ssd(G)$
and for all subgroups $L$ and $M$ of $H$
$\frac{1}{|\mathcal{L}(G)|^{2}}\sum_{L\in\mathcal{L}(H)}\left|\bigcap_{l\in
L}C_{M}(l)\right|\leq sd(H)\leq sd(G).$
###### Proof.
We proceed to prove the first inequality. The result is obviously true for
$H=G$ and then we may assume $H\not=G$. Since
$\mathcal{L}(H)\subseteq\mathcal{L}(G)$,
(2.16) $|\mathcal{L}(H)|^{2}\
ssd(H)={\underset{X,Y\in\mathcal{L}(H)}{\sum}}\varphi(X,Y)\leq{\underset{X,Y\in\mathcal{L}(G)}{\sum}}\varphi(X,Y)=|\mathcal{L}(G)|^{2}\
ssd(G).$
The inequality follows.
Now we proceed to prove the remaining part. When we consider the corresponding
function $\psi$, related to $sd(G)$ (details can be found in [30, 31]),
instead of $\varphi$, we may overlap the previous argument and find that
$\frac{|\mathcal{L}(H)|^{2}}{|\mathcal{L}(G)|^{2}}sd(H)\leq sd(G)$. From
Theorem 2.4, it follows that
(2.17)
$\frac{1}{|\mathcal{L}(H)|^{2}}\sum_{L\in\mathcal{L}(H)}\left|\bigcap_{l\in
L}C_{M}(l)\right|\leq\ sd(H)$
then
(2.18) $\frac{|\mathcal{L}(H)|^{2}}{|\mathcal{L}(G)|^{2}}\
\Big{(}\frac{1}{|\mathcal{L}(H)|^{2}}\sum_{L\in\mathcal{L}(H)}\left|\bigcap_{l\in
L}C_{M}(l)\right|\Big{)}\leq sd(H)$
and the result follows. ∎
In [29, Chapter 1], it is shown that $\mathcal{L}(G_{1}\times
G_{2})\neq\mathcal{L}(G_{1})\times\mathcal{L}(G_{2})$ in general, but if
$G_{1}$ and $G_{2}$ have coprime orders then it is true. This motivates our
assumption in the following proposition.
###### Proposition 2.10.
For two groups $G_{1}$ and $G_{2}$ of coprime orders,
$ssd(G_{1}\times G_{2})=ssd(G_{1})\cdot ssd(G_{2}).$
###### Proof.
We have $\mathcal{L}(G_{1}\times
G_{2})=\mathcal{L}(G_{1})\times\mathcal{L}(G_{2})$, because $G_{1}$ and
$G_{2}$ have coprime orders. Therefore, with obvious meaning of symbols,
(2.19) $ssd(G_{1}\times G_{2})=\frac{1}{|\mathcal{L}(G_{1}\times
G_{2})|^{2}}\underset{A_{1}\times A_{2}\in\mathcal{L}(G_{1}\times
G_{2})}{\sum}|Comm_{G_{1}\times G_{2}}(A_{1}\times A_{2})|$
$=\frac{1}{|\mathcal{L}(G_{1})\times\mathcal{L}(G_{2})|^{2}}\underset{A_{1}\times
A_{2}\in\mathcal{L}(G_{1})\times\mathcal{L}(G_{2})}{\sum}|Comm_{G_{1}}(A_{1})\times
Comm_{G_{2}}(A_{2})|$
$=\left(\frac{1}{|\mathcal{L}(G_{1})|^{2}}\underset{A_{1}\in\mathcal{L}(G_{1})}{\sum}|Comm_{G_{1}}(A_{1})|\right)\left(\frac{1}{|\mathcal{L}(G_{2})|^{2}}\underset{A_{2}\in\mathcal{L}(G_{2})}{\sum}|Comm_{G_{2}}(A_{2})|\right)$
$=ssd(G_{1})\cdot ssd(G_{2}).$
Hence the proposition follows. ∎
###### Corollary 2.11.
Proposition 2.10 is still true for finitely many factors.
###### Proof.
We can mimick the proof of Proposition 2.10. ∎
## 3\. Multiple strong subgroup commutativity degree
In analogy with $d^{(n)}(H,G)$ ($n\geq 1$), introduced in [12], the notion of
strong subgroup commutativity degree, given in Section 1, can be further
generalized in the following way:
(3.1)
$ssd^{(n)}(H,G)=\frac{|\\{(L_{1},\dots,L_{n},K)\in\mathcal{L}(H)^{n}\times\mathcal{L}(G)\
|\ [L_{1},\dots,L_{n},K]=1\\}|}{|\mathcal{L}(H)|^{n}\ |\mathcal{L}(G)|}.$
In particular, if $n=1$ and $H=G$, then $ssd^{(1)}(G,G)=ssd(G)$. Briefly,
$ssd^{(n)}(H)$ denotes
(3.2)
$ssd^{(n)}(H,H)=\frac{|\\{(L_{1},\dots,L_{n},L_{n+1})\in\mathcal{L}(H)^{n+1}\
|\ [L_{1},\dots,L_{n},L_{n+1}]=1\\}|}{|\mathcal{L}(H)|^{n+1}}.$
On another hand, we note that
(3.3)
$[L_{1},\dots,L_{n},K]=[[L_{1},\dots,L_{n}],K]=\ldots=[[\ldots[[L_{1},L_{2}],L_{3}]\ldots
L_{n}],K]=1$
and so
(3.4) $Comm_{G}(L_{1},\dots,L_{n})=\\{K\in\mathcal{L}(G)\ |\
[L_{1},\dots,L_{n},K]=1\\},$ (3.5) $Comm_{H\times
G}(L_{1},\dots,L_{n-1})=\\{(L_{n},K)\in\mathcal{L}(H)\times\mathcal{L}(G)\ |\
[[[L_{1},\dots,L_{n-1}],L_{n}],K]=1\\}$
$\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots$
$Comm_{H^{n-1}\times
G}(L_{1})=\\{(L_{2},L_{3},\ldots,L_{n},K)\in\mathcal{L}(H)^{n-1}\times\mathcal{L}(G)\
|$
$\ [\ldots[L_{1},L_{2}],\ldots,L_{n}],K]=1\\}.$
Of course, all these sets are nonempty, since they contain at least the
trivial subgroup. By construction, $Comm_{H^{n-1}\times G}(L_{1})\subseteq
Comm_{H^{n-2}\times G}(L_{1},L_{2})\subseteq\ldots\subseteq Comm_{H\times
G}(L_{1},\dots,L_{n-1})\subseteq Comm_{G}(L_{1},\dots,L_{n}).$ From the above
inclusions we observe that for $n$ which is growing the $Comm_{H^{n-1}\times
G}(L_{1})$ is getting to the trivial subgroup. Therefore
(3.6) $|\mathcal{L}(H)|^{n}\ |\mathcal{L}(G)|\
ssd^{(n)}(H,G)={\underset{L_{1},\dots,L_{n}\in\mathcal{L}(H)}{\sum}}|Comm_{G}(L_{1},\dots,L_{n})|$
$={\underset{L_{1},\dots,L_{n}\in\mathcal{L}(H)}{\sum}}|Comm_{H^{n-1}\times
G}(L_{1})|$
and to the extreme case we have
(3.7) $\lim_{n\rightarrow\infty}\
ssd^{(n)}(H,G)=\lim_{n\rightarrow\infty}\frac{1}{|\mathcal{L}(H)|^{n}\
|\mathcal{L}(G)|}\ \cdot\
\lim_{n\rightarrow\infty}{\underset{L_{1},\dots,L_{n}\in\mathcal{L}(H)}{\sum}}|Comm_{H^{n-1}\times
G}(H_{1})|$ $=\frac{1}{|\mathcal{L}(G)|}\ \cdot\
\lim_{n\rightarrow\infty}\frac{1}{|\mathcal{L}(H)|^{n}}\ \cdot\ 1=0.$
This is a qualitative argument which shows that it is meaningful to consider
values of probabilities of $ssd^{(n)}(H,G)$ for a small number of commuting
subgroups. At the same time, the above construction shows that
$ssd^{(n)}(H,G)$ is a strictly decreasing sequence of numbers in $[0,1]$ in
the variable $n$. Namely,
(3.8) $ssd^{(1)}(H,G)\geq ssd^{(2)}(H,G)\geq\ldots\geq ssd^{(n)}(H,G)\geq
ssd^{(n+1)}(H,G)\geq\ldots$
We want to point out that a similar treatment can be done for $sd(G)$, as
proposed in a series of opens problems in [31], where the corresponding
version of $ssd^{(n)}(H,G)$ is called relative subgroup commutativity degree.
As done in Section 2, we may rewrite $ssd^{(n)}(H,G)$ in the following form:
(3.9) $ssd^{(n)}(H,G)=\frac{1}{|\mathcal{L}(H)|^{n}\
|\mathcal{L}(G)|}{\underset{\underset{Y\in\mathcal{L}(G)}{X_{1},\ldots,X_{n}\in\mathcal{L}(H)}}{\sum}\varphi_{n}(X_{1},\ldots,X_{n},Y)},$
where $\varphi_{n}:\mathcal{L}(H)^{n}\times\mathcal{L}(G)\rightarrow\\{0,1\\}$
is the function defined by
(3.10)
$\varphi_{n}(X_{1},\ldots,X_{n},Y)=\left\\{\begin{array}[]{lcl}1,&&\mathrm{if}\
[X_{1},\ldots,X_{n},Y]=1,\\\ 0,&&\mathrm{if}\
[X_{1},\ldots,X_{n},Y]\not=1\end{array}\right.$
and continues to be symmetric.
###### Proposition 3.1.
Given subgroup $H$ of a group $G$,
$ssd^{(n)}(H,G)\leq ssd^{(n)}(G,G)\leq ssd(G)\leq sd(G).$
###### Proof.
We begin to prove the first inequality. Since
$\mathcal{L}(H)\subseteq\mathcal{L}(G)$,
(3.11) $ssd^{(n)}(H,G)\leq|\mathcal{L}(H)|^{n}\ |\mathcal{L}(G)|\
ssd^{(n)}(H,G)={\underset{\underset{Y\in\mathcal{L}(G)}{X_{1},\ldots,X_{n}\in\mathcal{L}(H)}}{\sum}\varphi_{n}(X_{1},\ldots,X_{n},Y)}$
(3.12)
$\leq\sum_{X_{1},\ldots,X_{n},Y\in\mathcal{L}(G)}\varphi_{n}(X_{1},\ldots,X_{n},Y)=|\mathcal{L}(G)|^{n}\
|\mathcal{L}(G)|\ ssd^{(n)}(G,G).$
The second inequality follows once we note that $ssd^{(n)}(H,G)$ is a
decreasing sequence. Therefore, if we fix $H=G$, then
$ssd(G)=ssd^{(1)}(G,G)\geq ssd^{(2)}(G,G)\geq\ldots\geq
ssd^{(n)}(G,G)\geq\ldots$. The last inequality follows once we note that
$Comm_{G}(H)\subseteq\mathcal{C}(H)$ and that
(3.13)
$ssd(G)=\frac{1}{|\mathcal{L}(G)|^{2}}\sum_{H\in\mathcal{L}(G)}|Comm_{G}(H)|\leq\frac{1}{|\mathcal{L}(G)|^{2}}\sum_{H\in\mathcal{L}(G)}|\mathcal{C}(H)|=sd(G).$
∎
###### Proposition 3.2.
For two groups $C$ and $D$ of coprime orders and two subgroups $A\leq C$ and
$B\leq D$,
$ssd^{(n)}(A\times B,C\times D)=ssd^{(n)}(A,C)\cdot ssd^{(n)}(B,D).$
###### Proof.
(3.14) $ssd^{(n)}(A\times B,C\times D)$ $=\frac{1}{|\mathcal{L}(A\times
B)|^{n}\ |\mathcal{L}(C\times D)|}\underset{A_{1}\times
B_{1},\ldots,A_{n}\times B_{n}\in\mathcal{L}(A\times B)}{\sum}|Comm_{A\times
B}(A_{1}\times B_{1},\dots,A_{n}\times B_{n})|$
$=\frac{1}{|\mathcal{L}(A)|^{n}\cdot|\mathcal{L}(B)|^{n}\cdot|\mathcal{L}(C)|\cdot|\mathcal{L}(D)|}\Big{(}\underset{A_{1}\times
B_{1},\ldots,A_{n}\times B_{n}\in\mathcal{L}(A\times
B)}{\sum}|Comm_{A}(A_{1},\ldots,A_{n})|$
$\cdot|Comm_{B}(B_{1},\ldots,B_{n})|\Big{)}=\frac{1}{|\mathcal{L}(A)|^{n}\cdot|\mathcal{L}(B)|^{n}\cdot|\mathcal{L}(C)|\cdot\mathcal{L}(D)|}$
$=\Big{(}\underset{A_{1},\ldots,An\in\mathcal{L}(A)}{\sum}|Comm_{A}(A_{1},\dots,A_{n})|\Big{)}\cdot\Big{(}\underset{B_{1},\ldots,B_{n}\in\mathcal{L}(B)}{\sum}|Comm_{B}(B_{1},\ldots,B_{n})|\Big{)}$
$=\frac{1}{|\mathcal{L}(A)|^{n}\
|\mathcal{L}(C)|}\Big{(}\underset{A_{1},\ldots,An\in\mathcal{L}(A)}{\sum}|Comm_{A}(A_{1},\dots,A_{n})|\Big{)}$
$\cdot\frac{1}{|\mathcal{L}(B)|^{n}\
|\mathcal{L}(D)|}\Big{(}\underset{B_{1},\ldots,B_{n}\in\mathcal{L}(B)}{\sum}|Comm_{B}(B_{1},\ldots,B_{n})|\Big{)}$
$=ssd^{(n)}(A,C)\cdot ssd^{(n)}(B,D).$
∎
We note that Proposition 2.10 follows from Proposition 3.2, when $n=1$,
$A=C=G_{1}$, $B=D=G_{2}$.
###### Corollary 3.3.
Proposition 3.2 is still true for finitely many factors.
###### Proof.
We can mimick the proof of Proposition 3.2. ∎
We end with a variation on the theme of Theorems 2.2 and 2.4.
###### Theorem 3.4.
Let $H$ and $K$ be two subgroups of a group $G$. Then for all $n\geq 1$
$ssd^{(n)}(H,H)<\frac{|H|^{n+1}}{|\mathcal{L}(H)|^{n+1}}\
\sum_{K\in\mathcal{L}(H)}d^{(n)}(K,K).$
###### Proof.
Overlapping the argument in the proof of Theorem 2.2,we firstly prove that
(3.15)
$\bigcup_{(L_{2},\ldots,L_{n},L_{n+1})\in\mathcal{L}(H)^{n}}C_{H^{n}}(L_{1})=Comm_{H^{n}}(L_{1}),$
where
(3.16) $Comm_{H^{n}}(L_{1})=Comm_{H^{n-1}\times H}(L_{1})$
$=\\{(L_{2},L_{3},\ldots,L_{n},L_{n+1})\in\mathcal{L}(H)^{n-1}\times\mathcal{L}(H)\
|\ [\ldots[L_{1},L_{2}],\ldots,L_{n}],L_{n+1}]=1\\}$
and then
(3.17) $|\mathcal{L}(H)|^{n+1}\
ssd^{(n)}(H,H)=\sum_{L_{1}\in\mathcal{L}(H)}|Comm_{H^{n}}(L_{1})|$
$=\sum_{L_{1}\in\mathcal{L}(H)}\left|\bigcup_{(L_{2},\ldots,L_{n},L_{n+1})\in\mathcal{L}(H)^{n}}C_{H^{n}}(L_{1})\right|$
$<\sum_{(L_{2},\ldots,L_{n},L_{n+1})\in\mathcal{L}(H)^{n}}\sum_{L_{1}\in\mathcal{L}(H)^{n}}|C_{H^{n}}(L_{1})|$
and we note that the equality must be strict for the same motivation of the
corresponding step in the proof of Theorem 2.2. Since
$C_{H^{n}}(L_{1})\subseteq C_{H^{n}}(l_{1})$ whenever $l_{1}\in L_{1}$, we may
continue, finding that
(3.18)
$\leq\sum_{(L_{2},\ldots,L_{n},L_{n+1})\in\mathcal{L}(H)^{n}}{\underset{L_{1}\in\mathcal{L}(H)}{\underset{l_{1}\in
L_{1}}{\sum}}}|C_{H^{n}}(l_{1})|$
$=\sum_{(L_{1},L_{2},\ldots,L_{n},L_{n+1})\in\mathcal{L}(H)^{n+1}}\Big{(}\sum_{l_{1}\in
L_{1}}|C_{H^{n}}(l_{1})|\Big{)}$ $=\sum_{K\in\mathcal{L}(H)}d^{(n)}(K,K)\
|K|^{n+1}\leq|H|^{n+1}\ \sum_{K\in\mathcal{L}(H)}d^{(n)}(K,K).$
∎
Roughly speaking, in the proof of Theorem 2.9 we may replace the role of
$\varphi=\varphi_{2}$ with that of $\varphi_{n}$ for $n>2$. We will find the
following generalization of Theorem 2.9, whose proof is easy to check and so
it is omitted.
###### Theorem 3.5.
Let $H$ be a subgroup of a group $G$. Then for all $n\geq 1$
$\frac{|\mathcal{L}(H)|^{n+1}}{|\mathcal{L}(G)|^{n+1}}\ ssd^{(n)}(H)\leq
ssd^{(n)}(G).$
We note that a similar treatment can be done for the relative subgroup
commutativity degree in [31], since the arguments involve only combinatorial
properties and set theory. This fact motivates to conjecture that the context
of infinite compact groups, once a suitable Haar measure is replaced with
$ssd(G)$ or with $sd(G)$, may be subject to an analogous treatment.
## 4\. Two applications
Here we illustrate an application to the theory of characters and another to
the dihedral groups. Relations with the theory of characters are due to the
fact that in a group $G$
(4.1) $d(G)=\frac{|\mathrm{Irr}(G)|}{|G|},$
where $\mathrm{Irr}(G)$ denotes the set of all irreducible complex characters
of $G$. For an element $g$ of $G$, let
(4.2) $\xi(g)=|(X,Y)\in\mathcal{L}(\langle g\rangle)\times\mathcal{L}(G)\ |\
[X,Y]=1\\}|.$
Thus,
(4.3) $ssd(\langle g\rangle,G)=\frac{\xi(g)}{|\mathcal{L}(\langle
g\rangle)||\mathcal{L}(G)|}.$
###### Lemma 4.1.
$\xi(g)$ is a class function.
###### Proof.
It is enough to note that, for each $a\in G$, the map
(4.4) $f:(X,Y)\mapsto f(X,Y)=(aXa^{-1},aYa^{-1})$
defines a one to one correspondence between the sets
$\\{(X,Y)\in\mathcal{L}(\langle g\rangle)\times\mathcal{L}(G)\ |\ [X,Y]=1\\}$
and $\\{(X,Y)\in\mathcal{L}(\langle aga^{-1}\rangle)\times\mathcal{L}(G)\ |\
[X,Y]=1\\}$. ∎
Thus, it is meaningful to write
(4.5) $\xi(g)=\underset{\chi\in\mathrm{Irr}(G)}{\sum}[\xi,\chi]\chi(g)$
where $[\,,\,]$ denotes the usual inner product of characters, defined by
(4.6) $[\xi,\chi]=\dfrac{1}{|G|}\sum_{g\in
G}\xi(g)\overline{\chi(g)}=\dfrac{1}{|G|}\sum_{g\in G}\xi(g)\chi(g^{-1}).$
We recall that a class function defined on a finite group $G$ is said to be an
$R$–generalized character of $G$, for any ring $\mathbb{Z}\subseteq
R\subseteq\mathbb{C}$, if it is an $R$–linear combination of irreducible
complex characters of $G$.
###### Theorem 4.2.
$\xi$ is a $\mathbb{Q}$-generalized character of $G$.
###### Proof.
Clearly, if two elements $x$ and $y$ of $G$ generate the same cyclic group
then $\xi(x)=\xi(y)$. Suppose that $o(x)=o(y)=n$. Let $\varepsilon$ be a
primitive $n$th root of unity. We have $y=x^{m}$ for some $m$ with $(m,n)=1$
and thus $\varepsilon^{m}$ is a primitive $n$th root of unity. As usual,
$\mathrm{Gal}(\mathbb{Q}[\varepsilon]/\mathbb{Q})$ denotes the Galois group,
related to the algebraic extension $\mathbb{Q}[\varepsilon]$ over
$\mathbb{Q}$, obtained adding $\varepsilon$. Therefore, for any
$\sigma\in\mathrm{Gal}(\mathbb{Q}[\varepsilon]/\mathbb{Q})$ we have
(4.7)
$\chi(x)^{\sigma}=\sum{\epsilon_{i}}^{\sigma}=\sum{\epsilon_{i}}^{m}=\chi(x^{m}).$
Thus for any $\chi\in\mathrm{Irr}(G)$ and $g\in G$,
(4.8) $\chi(g)^{\sigma}=\chi(g^{m})$
and hence
$\left(\delta(g)\chi(g^{-1})\right)^{\sigma}=\delta(g^{m})\chi(g^{-m})$. Hence
$\sigma$ fixes $\sum_{g\in G}\delta(g)\chi(g^{-1})$ and this completes the
proof. ∎
###### Corollary 4.3.
$|G|[\xi,\chi]$ is an integer for all $\chi\in\mathrm{Irr}(G)$.
###### Proof.
Since $\chi(g)$ is an algebraic integer the result follows from Lemma 4.1 and
Theorem 4.2. ∎
For the second application, the dihedral group
(4.9) $D_{2n}=\langle x,y\ |\ x^{2}=y^{n}=1,x^{-1}yx=y^{-1}\rangle$
of symmetries of a regular polygon with $n\geq 1$ edges has order $2n$ and a
well–known de- scription of $|\mathcal{L}(D_{2n})|$ can be found in [29, 30,
31]. For instance, it is easy to see that $D_{2n}\simeq C_{2}\ltimes C_{n}$ is
the semidirect product of a cyclic group $C_{2}$ of order 2 acting by
inversion on a cyclic group $C_{n}$ of order $n$. For every divisor $r$ of
$n$, $D_{2n}$ has a subgroup isomorphic to $C_{r}$ , namely $\langle
x^{\frac{n}{r}}\rangle$, and $\frac{n}{r}$ subgroups isomorphic to $D_{2r}$,
namely $\langle x^{\frac{n}{r}},x^{i-1},y\rangle$ for
$i=1,2,\ldots,\frac{n}{r}$. Then
(4.10) $|\mathcal{L}(D_{2n})|=\sigma(n)+\tau(n),$
where $\sigma(n)$ and $\tau(n)$ are the sum and the number of all divisors of
$n$, respectively. The next result generalizes the above considerations, when
we have a group with a structure very close to that of $D_{2n}$.
###### Corollary 4.4.
Assume that $G$ is a metabelian group of even order. If
$|\mathcal{L}(G)|=\sigma(\frac{|G|}{2})+\tau(\frac{|G|}{2})$ and $G^{\prime}$
is cyclic, then
$\frac{(\tau(G^{\prime})+1)^{2}}{\Big{(}\sigma\Big{(}\frac{|G|}{2}\Big{)}+\tau\Big{(}\frac{|G|}{2}\Big{)}\Big{)}^{2}}\leq\sum_{H,K\in\mathcal{L}(G)}\varphi(H,K)\leq\frac{|G|^{2}}{\Big{(}\sigma\Big{(}\frac{|G|}{2}\Big{)}+\tau\Big{(}\frac{|G|}{2}\Big{)}\Big{)}^{2}}\sum_{H,K\in\mathcal{L}(G)}d(H,K).$
###### Proof.
Since $G^{\prime}$ is cyclic, $|\mathcal{L}(G^{\prime})|=\tau(G^{\prime})$.
Then the lower bound follows from Corollary 2.8, specifying the numerical
values of the subgroup lattices. From Theorem 2.2, we get the upper bound,
adapted to our case. The result follows. ∎
Corollary 4.4 is a counting formula for the number of permuting subgroups via
$\varphi$, or, equivalently, via the strong subgroup commutativity degree and
the commutativity degree. This observation is important in virtue of the fact
that we know explicitly $d(H,K)$ by results in [2, 7, 8, 9, 12, 18, 19].
## References
* [1] A.M. Alghamdi, D.E. Otera and F.G. Russo, On some recent investigations of probability in group theory, Boll. Mat. Pura Appl. 3 (2010), 87–96.
* [2] A.M. Alghamdi and F.G. Russo, A generalization of the probability that the commutator of two group elements is equal to a given element, preprint,Cornell University Library, 2010, arXiv: 1004.0934.
* [3] R. Barman, Quasinormality degrees of subgroups of a finite group and a class function, preprint, 2011.
* [4] F. Barry, D. MacHale and Á. Ní Shé, Some supersolvability conditions for finite groups, Math. Proc. Royal Irish Acad. 106 A (2) (2006), 163–177.
* [5] Y. Berkovich, Groups of prime power order Vol. I, de Gruyter, Berlin, 2008.
* [6] K. Chiti, M.R.R. Moghaddam and A. Salemkar, $n$-isoclinism classes and $n$-nilpotency degree of finite groups, Algebra Colloq. 12 (2005), 255–261.
* [7] 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.
* [8] A.K. Das and R.K. Nath, On a lower bound of commutativity degree, Rend. Circ. Mat. Palermo 59 (2010), 137–142.
* [9] 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.
* [10] A. Erfanian and F.G. Russo, Probability of mutually commuting $n$-tuples in some classes of compact groups, Bull. Iran. Math. Soc. 34 (2008), 27–37.
* [11] A. Erfanian and R. Rezaei, On the commutativity degree of compact groups, Arch. Math. (Basel) 93 (2009), 345–356.
* [12] A. Erfanian, R. Rezaei and F.G. Russo, Relative $n$-isoclinism classes and relative $n$-th nilpotency degree of finite groups, preprint, Cornell University Library, 2010, arXiv: 1003.2306.
* [13] M. Farrokhi, H. Jafari and F. Saeedi, Subgroup normality degree of finite groups I, Arch. Math. (Basel) 96 (2011), 215–224.
* [14] M. Farrokhi and F. Saeedi, Subgroup normality degree of finite groups II, preprint, 2011.
* [15] M. Farrokhi, Finite groups with two subgroup normality degrees, preprint, 2011.
* [16] R.M. Guralnick and G.R. Robinson, On the commuting probability in finite groups, J. Algebra 300 (2006), 509–528.
* [17] K.H. Hofmann and F.G. Russo, The probability that $x$ and $y$ commute in a compact group, preprint, Cornell University Library, 2010, arXiv:1001.4856.
* [18] P. Lescot, Isoclinism classes and commutativity degrees of finite groups, J. Algebra 177 (1995), 847–869.
* [19] P. Lescot, Central extensions and commutativity degree, Comm. Algebra 29 (2001), 4451–4460.
* [20] H. Mohammadzadeh, A. Salemkar and H. Tavallaee, A remark on the commuting probability in finite groups, Southeast Asian Bull. Math. 34 (2010), 755–763,
* [21] P. Niroomand and R. Rezaei, On the exterior degree of finite groups, Comm. Algebra 39 (2011), 335–343.
* [22] P. Niroomand and R. Rezaei, The exterior degree of a pair of finite groups, preprint, Cornell University Library, arXiv:1101.4312v1, 2011.
* [23] P. Niroomand, R. Rezaei and F.G. Russo, Commuting powers and exterior degree of finite groups, preprint, Cornell University Library, arXiv:1102.2304, 2011.
* [24] D.E. Otera and F.G. Russo, Subgroup $S$-commutativity degree of finite groups, preprint, Cornell University Library, 2010, arXiv:1009.2171.
* [25] R. Rezaei and F.G. Russo, $n$-th relative nilpotency degree and relative $n$-isoclinism classes, Carpathian J. Math. 27 (2011), 123–130.
* [26] D.J. Rusin, What is the probability that two elements of a finite group commute?, Pacific J. Math. 82 (1979), 237–247.
* [27] F.G. Russo, A probabilistic meaning of certain quasinormal subgroups, Int. J. Algebra 1 (2007), 385–392.
* [28] F.G. Russo, The generalized commutativity degree in a finite group, Acta Univ. Apulensis Math. Inform. 18 (2009), 161–167.
* [29] R. Schmidt, Subgroup Lattices of Groups, de Gruyter, Berlin, 1994.
* [30] M. Tǎrnǎuceanu, Subgroup commutativity degrees of finite groups, J. Algebra 321 (2009), 2508–2520.
* [31] M. Tǎrnǎuceanu, Addendum [Subgroup commutativity degrees of finite groups], J. Algebra (2011), in press.
|
arxiv-papers
| 2011-02-02T17:32:29 |
2024-09-04T02:49:16.813414
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Francesco G. Russo (Universita' degli Studi di Palermo, Palermo,\n Italy)",
"submitter": "Francesco G. Russo",
"url": "https://arxiv.org/abs/1102.0509"
}
|
1102.0739
|
# The Renormalization Group and the Effective Action
D.G.C. McKeon Department of Applied Mathematics, The University of Western
Ontario, London, ON N6A 5B7, Canada
###### Abstract
The renormalization group is used to sum the leading-log (LL) contributions to
the effective action for a large constant external gauge field in terms of the
one-loop renormalization group (RG) function $\beta$, the next-to-leading-log
(NLL) contributions in terms of the two-loop RG function etc. The log
independent pieces are not determined by the RG equation, but can be fixed by
considering the anomaly in the trace of the energy-momentum tensor. Similar
considerations can be applied to the effective potential $V$ for a scalar
field $\phi$; here the log independent pieces are fixed by the condition
$V^{\prime}(\phi=v)=0$.
11footnotetext: Email: dgmckeo2@uwo.ca
The effective Lagrangian for a constant external gauge field has been
considered in a number of papers [1-7]. In the limit of a strong external
field strength, these lead to logarithmic corrections to the classical
Lagrangian $-\frac{1}{4}F_{\mu\nu}^{2}\equiv-\frac{1}{4}\Phi$. (We can regard
$F$ as being either the electromagnetic field strength in QED or a non-Abelian
field strength which may be coupled to matter.) A systematic summation of
these effects by using the RG equation has been discussed in ref. [8]; the
summation of logarithmic effects arising due to radiative processes in other
contents has been considered in [9]. In this note we show that the RG
equation, when applied to the effective Lagrangian $L$ for a strong external
gauge field, can be rewritten as a sequence of coupled ordinary differential
equations for functions $S_{n}$ with $S_{0}$ giving the LL contribution to
$L$, $S_{1}$ the NLL contribution to $L$, etc. The boundary conditions for
these equations are the log-independent contributions to $L$. These can be
fixed by examining the anomaly in the energy momentum tensor [10] as this
anomaly can be used to find a formal expression for $L$ [11]. The approach
used is similar to one employed with the effective potential when there is a
fundamental scalar field in the model [12].
If $\mu$ is the renormalization scale in a model, $F_{\mu\nu}$ is the constant
external field strength and $\lambda$ the gauge coupling, then the effective
Lagrangian $L(F_{\mu\nu},\lambda,\mu)$ must be independent of $\mu$ and hence
the RG equation follows,
$\frac{dL}{d\mu}=\left(\mu\frac{\partial}{\partial\mu}+\beta(\lambda)\frac{\partial}{\partial\lambda}+\gamma(\lambda)F_{\mu\nu}\frac{\partial}{\partial
F_{\mu\nu}}\right)L=0.$ (1)
Since $\lambda F_{\mu\nu}$ is not renormalized,
$\beta(\lambda)=-\lambda\gamma(\lambda)$ [13] and so if
$\Phi=F^{\mu\nu}F_{\mu\nu}$, eq. (1) becomes
$\left[\mu\frac{\partial}{\partial\mu}+\beta(\lambda)\left(\frac{\partial}{\partial\lambda}-\frac{2}{\lambda}\Phi\frac{\partial}{\partial\Phi}\right)\right]L=0.$
(2)
With $t=\frac{1}{4}\ln\left(\frac{\lambda^{2}\Phi}{\mu^{4}}\right)$, the form
of $L$ when $\lambda\Phi>>\mu^{2}$ is [8]
$L=\sum_{n=0}^{\infty}\sum_{m=0}^{n}\,T_{n,m}\lambda^{2n}t^{m}\Phi=\sum_{n=0}^{\infty}S_{n}(\lambda^{2}t)\lambda^{2n}\Phi$
(3)
where
$S_{n}(\lambda^{2}t)=\displaystyle{\sum_{m=0}^{\infty}}T_{n+m,m}(\lambda^{2}t)^{m}$.
If $\beta(\lambda)=\displaystyle{\sum_{n=1}^{\infty}}b_{2n+1}\lambda^{2n+1}$,
eq. (2) is satisfied at progressively higher orders in $\lambda$ provided
these functions satisfy a set of coupled ordinary differential equations,
$\displaystyle wS_{0}^{\prime}(w)\\!\\!$ $\displaystyle-$
$\displaystyle\\!\\!S_{0}=0$ (4a) $\displaystyle
b_{3}wS_{1}^{\prime}(w)\\!\\!$ $\displaystyle-$
$\displaystyle\\!\\!b_{5}S_{0}(w)+(1+w)b_{5}S_{0}^{\prime}(w)=0$ (4b)
$\displaystyle-b_{3}S_{2}^{\prime}+b_{3}S_{2}\\!\\!$ $\displaystyle-$
$\displaystyle\\!\\!b_{7}S_{0}+(1+w)(b_{7}S_{0}^{\prime}+b_{5}S_{1}^{\prime}+b_{3}S_{2}^{\prime})=0$
(4c)
etc. with $w=-1+2b_{3}(\lambda^{2}t)$. In general $S_{n}(\xi)$ can be found
once $S_{0}\ldots S_{n-1}$ have been determined provided $b_{3}\ldots
b_{2n+3}$ are known and the boundary conditions $S_{n}(\lambda^{2}t=0)=T_{n0}$
have been specified. In particular, $S_{0}=T_{00}w$,
$S_{1}=-\frac{T_{00}b_{5}}{b_{3}}\ln|w|+T_{10}$ and
$S_{2}=\left[\left(\frac{b_{5}}{b_{3}}\right)^{2}-\frac{b_{7}}{b_{3}}\right]T_{00}\ln|w|-\left(\frac{b_{3}}{b_{5}}\right)^{2}T_{00}\left(\frac{1}{w}-1\right)+T_{20}$.
To find these boundary conditions, an extra condition must be found. To do
this, we reexpress $L$ in eq. (3) as
$L=\displaystyle{\sum_{n=0}^{\infty}}A_{n}(\lambda)t^{n}\Phi$
where$A_{n}=\displaystyle{\sum_{m=n}^{\infty}}T_{m,n}\lambda^{2m}$. Eq. (2) is
now satisfied at each order in $t$ provided
$\frac{1}{\lambda^{2}}A_{n+1}(\lambda)=\frac{1}{n+1}\beta(\lambda)\frac{d}{d\lambda}\left(\frac{1}{\lambda^{2}}A_{n}(\lambda)\right).$
(5)
If now $A_{n}(\lambda)=\lambda^{2}\overline{A}_{n}(\lambda)$ and
$\eta=\displaystyle{\int_{\lambda_{0}}^{\lambda(\eta)}}\frac{dx}{\beta(x)}$
then
$\overline{A}_{n+1}(\lambda(\eta))=\frac{1}{(n+1)!}\,\frac{d^{n+1}}{d\eta^{n+1}}\overline{A}_{0}(\lambda(\eta))$
(6)
so that
$L=\lambda^{2}(\eta)\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\,\frac{d^{n}}{d\eta^{n}}\overline{A}_{0}(\lambda(\eta))\Phi=\lambda^{2}(\eta)\overline{A}_{0}(\lambda(\eta+t))\Phi=\frac{\lambda^{2}(\eta)}{\lambda^{2}(\eta+t)}A_{0}(\lambda(\eta+t))\Phi\,.$
(7)
Since $A_{0}$ is determined by the $T_{n0}$, we see from eq. (7) that again
the log independent contributions to $L$ fix the log dependent contributions
once $\beta$ is known. When $\eta=0$, we take the value of the function
$\lambda(\eta)$ to be $\lambda_{0}$.
We now recall that the trace anomaly of the energy momentum tensor [10]
$\left\langle\theta^{\mu}_{\,\,\mu}\right\rangle=\frac{\beta(\overline{\lambda})}{2\overline{\lambda}(t)}\,\frac{\lambda^{2}_{0}}{\overline{\lambda}^{2}(t)}\Phi$
(8)
where
$\frac{d\overline{\lambda}(t)}{dt}=\beta(\overline{\lambda}(t))\;\;\left(t=\int_{\lambda_{0}}^{\overline{\lambda}(t)}\,\frac{dx}{\beta(x)}\right)$
(9)
leads to [11]
$L=-\frac{1}{4}\frac{\lambda^{2}_{0}}{\overline{\lambda}^{2}(t)}\Phi$ (10)
since
$\left\langle\theta^{\mu\nu}\right\rangle=-\eta^{\mu\nu}L+2\frac{\partial
L}{\partial\eta_{\mu\nu}}$. (It can be verified that eq. (10) satisfies eq.
(2).) The usual “running coupling function” $\overline{\lambda}(t)$ has the
boundary condition $\overline{\lambda}(0)=\lambda_{0}$ with $\lambda_{0}$ also
being equal to $\lambda(\eta=0)$. It is now apparent that eqs. (7) and (10)
are identical provided $\eta=0$, and so we now have the boundary conditions
for eq. (4)
$T_{n0}=-\frac{1}{4}\,\delta_{n0}.$ (11)
Upon equating $L$ in eqs. (10) and (3) we see that
$\frac{1}{\overline{\lambda}^{2}(t)}=\frac{-4}{\lambda_{0}^{2}}\left[\displaystyle{\sum_{n=0}^{\infty}}S_{n}(\lambda_{0}^{2}t)\lambda_{0}^{2n}\right]$,
which is a novel expression for the running coupling in terms of
$\beta(\lambda)$. Consequently the log-independent contributions to $L$ are
fixed by the trace anomaly.
The effective potential $V$ for a massless scalar field with the classical
potential $V_{C1}=\lambda\phi^{4}$ can be treated in an analogous fashion. We
will now review how [12] the RG equation can be used to express the log-
dependent part of $V$ in terms of the log-independent parts, and how these
log-independent parts can be determined by considering an extra condition
(which in this case is $V^{\prime}(\phi=v)=0)$. The expansion
$V=\sum_{n=0}^{\infty}\,\sum_{m=0}^{n}\lambda^{n+1}T_{n,m}L^{m}\phi^{4}\quad\left(L=\log\frac{\phi}{\mu}\right)$
(12)
when expressed as
$V=\sum_{n=0}^{\infty}A_{n}(\lambda)L^{n}\phi^{4}$ (13)
(where $A_{n}=\displaystyle{\sum_{m=n}^{\infty}}T_{m,n}\lambda^{m+1}$)
satisfies the RG equation
$\left(\mu\frac{\partial}{\partial\mu}+\beta(\lambda)\frac{\partial}{\partial\lambda}+\gamma(\lambda)\phi\frac{\partial}{\partial\phi}\right)V=0$
(14)
provided
$A_{n+1}(\lambda)=\frac{1}{n+1}\left(\hat{\beta}\frac{\partial}{\partial\lambda}+4\hat{\gamma}\right)A_{n}(\lambda)$
(15)
where $\hat{\beta}=\beta/(1-\gamma)$ and $\hat{\gamma}=\gamma/(1-\gamma)$. If
now
$\eta=\int_{\lambda_{0}}^{\lambda(\eta)}\,\frac{dx}{\hat{\beta}(x)}$ (16)
and
$\hat{A}_{n}(\lambda)=A_{n}(\lambda)\exp\left(4\int_{\lambda_{0}}^{\lambda}\frac{\hat{\gamma}(x)}{\hat{\beta}(x)}dx\right)$
(17)
then by eq. (15)
$\hat{A}_{n+1}(\lambda(\eta))=\frac{1}{n+1}\frac{d}{d\eta}\hat{A}_{n}(\lambda(\eta))=\frac{1}{(n+1)!}\frac{d^{n+1}}{d\eta^{n+1}}\hat{A}_{0}(\lambda(\eta)).$
(18)
The sum of eq. (13) now leads to
$V=A_{0}(\lambda(\eta+L))\exp\left(4\int_{\lambda(\eta)}^{\lambda(\eta+L)}\frac{\gamma(x)}{\beta(x)}dx\right)\phi^{4}.$
(19)
As with eq. (7), eq. (19) shows that effective potential is determined by its
log-independent contributions and the RG functions.
To fix these log-independent contributions to $V$, we need a second condition.
The trace of the energy-momentum tensor does not help us to do this. However,
we can invoke the condition
$\frac{dV(\phi)}{d\phi}\left|{}_{\phi=v}=0\right.$ (20)
where $v$ is the vacuum expectation value of $V$. If the renormalization scale
parameter $\mu$ is chosen to be equal to $v$, then by eqs. (13) and (20)
$[A_{1}(\lambda)+4A_{0}(\lambda)]v^{3}=0.$ (21)
This equation has been derived for a particular value of $\mu$, but as
$\lambda$ at this value of $\mu$ is not fixed, eq. (21) implies the functional
relation
$A_{1}(\lambda)=-4A_{0}(\lambda)$ (22)
provided $v\neq 0$. Eq. (22) and eq. (15) with $n=0$ together lead to
$\left[\hat{\beta}\frac{d}{d\lambda}+4(1+\hat{\gamma})\right]A_{0}=0$ (23)
so that
$A_{0}(\lambda)=A_{0}(\lambda_{0})\exp\left(-4\int_{\lambda{{}_{0}}}^{\lambda}\frac{dx}{\beta(x)}\right),$
(24)
and hence eq. (19) becomes
$\displaystyle V$
$\displaystyle=A_{0}(\lambda_{0})\exp\left(-4\int_{\lambda_{0}}^{\lambda(\eta+L)}\,\frac{dx}{\beta(x)}\right)\exp\left(4\int_{\lambda_{(\eta)}}^{\lambda(\eta+L)}\,\frac{\gamma(x)}{\beta(x)}dx\right)\phi^{4}$
$\displaystyle=A_{0}(\lambda_{0})\exp\left(-4\int_{\lambda{{}_{0}}}^{\lambda}\frac{dx}{\beta(x)}\right)\mu^{4},$
(25)
upon using eq. (16). Consequently, $V$ is independent of $\phi$ provided
$v\neq 0$; either there is no spontaneous symmetry breakdown or the potential
is “flat”. (Of course, this flatness does not preclude spontaneous symmetry
breaking.)
We thus see that the effective Lagrangian for a constant gauge field and the
effective potential for a massless scalar field are completely determined by
the RG functions when the RG equation is supplemented by a suitable extra
condition. In the case of the effective action for an external electromagnetic
field this extra condition is provided by the anomalous trace of the energy
momentum tensor. For the effective potential $V(\phi)$ in a massless
$\phi^{4}$ model, it is the fact that $V^{\prime}(0)$ disappears when $\phi=v$
which determines $V(\phi)$ completely in terms of the RG functions.
## 1 Acknowledgments
The author would like to thank F.T. Brandt, F. Chishtie, T. Hanif, J. Jia,
T.N. Sherry and C. Schubert for discussions. R. Macleod had a helpful
suggestion.
## References
* [1] W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936).
* [2] J. Schwinger, Phys. Rev. 82, 664 (1951).
* [3] V. Weisskopf, K. Danske Vidensk 14, no. 6 (1936).
* [4] V.I. Ritus, ZhETF 69, 1517 (1975)(JETP 42, 774 (1976)).
* [5] S. Vanyashin and M.V. Terentev, ZhETF 48, 565 (1965) (JETP 21, 375 (1965)).
* [6] I.A. Batalin, S.G. Matinyan and G.K. Savvidi, Yad. Fiz. 26, 407 (1977)(Sov. J. Nucl. Phys. 26, 214 (1977)).
* [7] M.R. Brown and M. Duff, Phys. Rev., D11, 2124 (1975).
M. Duff and M. Ram$\acute{\rm{o}}$n-Medrano, Phys. Rev., D12, 3357 (1975).
* [8] W. Dittrich and M. Reuter, Effective Lagrangian in Quantum Electrodynamics, Springer-Verlag (Berlin 1984).
* [9] M.R. Ahmady, F.A. Chishtie, V. Elias, A. Fariborz, N. Fattahi, D.G.C. McKeon, T.N. Sherry and T.G. Steele, Phys. Rev. D66, 014010 (2002).
F.A. Chishtie, T. Hanif, D.G.C. McKeon and T.G. Steele, Phys. Rev. D77, 065007
(2008).
B.M. Kastening, Phys. Lett. B283, 287 (1982).
* [10] R.J. Crewther, Phys. Rev. Lett. 28, 1421 (1972).
M.S. Chanowitz and J.R. Ellis, Phys. Lett. B40, 397 (1972).
S.L. Adler, J.C. Collins and A. Duncan, Phys. Rev. D15, 1712 (1977).
J.C. Collins, A. Duncan and S.D. Joglekar, Phys. Rev. D16, 438 (1977).
N.K. Nielsen, Nucl. Phys. B 120, 212 (1977).
* [11] H. Pagels and E. Tomboulis, Nucl. Phys. B 143, 485 (1978).
H. Leutwyler, Nucl. Phys. B 179, 129 (1981).
G.V. Dunne, H. Gies and C. Schubert, JHEP 0211, 032 (2006).
* [12] V. Elias, R.B. Mann, D.G.C. McKeon and T.G. Steele, Nucl. Phys. B 678, 147 (2004) (E B703, 413 (2004); ibid. Phys. Rev. D72, 037902 (2005).
F.T. Brandt, F.A. Chishtie and D.G.C. McKeon, Mod. Phys. Lett. A20, 2215
(2005); ibid. Int. J. Mod. Phys A22, 1 (2007).
F.A. Chishtie, T. Hanif, J. Jia, D.G.C. McKeon and T.N. Sherry, Int. J. Mod.
Phys. (in press).
* [13] S.G. Matinyan and G.V. Savvidy, Nucl. Phys. B134, 539 (1978).
L. Abbott, Nucl. Phys. B185, 189 (1981).
|
arxiv-papers
| 2011-02-03T17:31:31 |
2024-09-04T02:49:16.824464
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D.G.C. McKeon",
"submitter": "Gerry McKeon",
"url": "https://arxiv.org/abs/1102.0739"
}
|
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